\documentclass[11pt,letterpaper]{article} % Armstrong-Vicol, "Anomalous regularization of passive scalars up to the Obukhov-Corrsin exponent". % The whole paper in one file (Section 1: introduction; Part I: Sections 2-6; Part II: Sections 7-15; index of notation), with main.bib and % the bibliography style alpha-initials.bst (alpha with initials and an optional sortdate field). \usepackage[nottoc,notlof,notlot]{tocbibind}% References in the table of contents % --- preamble --- \usepackage[T1]{fontenc} \usepackage{amsmath,amsthm,amssymb,mathtools,nicefrac,mathrsfs} \usepackage[margin=1.05in]{geometry} \usepackage{microtype} \setlength{\emergencystretch}{1em} \usepackage{xcolor} \definecolor{paperlink}{RGB}{0,0,114} \usepackage{tikz} \usetikzlibrary{arrows.meta,calc,positioning,decorations.pathreplacing,shapes.geometric} \usepackage{xr-hyper} \usepackage{hyperref} \hypersetup{colorlinks=true,linktoc=all,linkcolor=paperlink, citecolor=paperlink,urlcolor=paperlink,pdfstartview=Fit, pdfauthor={Scott Armstrong and Vlad Vicol}} \usepackage{enumitem} \setlist[itemize]{leftmargin=2em,itemsep=0.25em,topsep=0.35em} \setlist[enumerate]{leftmargin=2.25em,itemsep=0.25em,topsep=0.35em} \setlength{\parindent}{1.25em} \setlength{\parskip}{0pt} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \setlength{\headheight}{14pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \date{} % Theorem environments. Only the principal results are lettered. \newtheorem{maintheorem}{Theorem} \renewcommand{\themaintheorem}{\Alph{maintheorem}} \newtheorem{maincorollary}[maintheorem]{Corollary} \newtheorem*{corollaryAprime}{Corollary A$'$} \newtheorem{proposition}{Proposition}[section] \newtheorem{lemma}[proposition]{Lemma} \newtheorem{theorem}[proposition]{Theorem} \newtheorem{corollary}[proposition]{Corollary} \theoremstyle{definition} \newtheorem{definition}[proposition]{Definition} \theoremstyle{remark} \newtheorem{remark}[proposition]{Remark} \theoremstyle{plain} % Figures. Defining \NOFIGURES before the preamble is read leaves them out. \newif\iffigures \figurestrue \ifdefined\NOFIGURES\figuresfalse\fi \newcommand{\thetain}{\theta_{\mathsf{in}}} \newcommand{\tin}{t_{\mathsf{in}}} \newcommand{\paperfigure}[1]{\iffigures#1\fi} % --- macros --- % Shared macros. The notation follows Armstrong and Vicol, "Anomalous diffusion by fractal % homogenization". \newcommand{\R}{\mathbb R} \newcommand{\T}{\mathbb T} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\dd}{\,\mathrm d} \newcommand{\nf}[2]{\nicefrac{#1}{#2}} \DeclareMathOperator{\tr}{tr} \newcommand{\eps}{\varepsilon} \renewcommand{\b}{\mathbf b} \newcommand{\K}{\mathbf K} \newcommand{\Khom}{\overline{\mathbf K}} % the slashed (average) integral \makeatletter \@ifundefined{fint}{% \def\Xint#1{\mathchoice{\XXint\displaystyle\textstyle{#1}}{\XXint\textstyle\scriptstyle{#1}}{\XXint\scriptstyle\scriptscriptstyle{#1}}{\XXint\scriptscriptstyle\scriptscriptstyle{#1}}\!\int} \def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$}\vcenter{\hbox{$#2#3$}}\kern-.5\wd0}} \def\fint{\Xint-}}{} \makeatother % --- end of macros --- % --- end of preamble --- \usetikzlibrary{fit} \makeatletter \def\thm@space@setup{\thm@preskip=.7\topsep \thm@postskip=\thm@preskip} \makeatother % Layout and headings of Armstrong-Vicol, "Anomalous diffusion by fractal homogenization" (the % author's house style): 1in margins, no running head, centred section titles, depth-2 contents. \geometry{margin=1in} \pagestyle{plain} \setcounter{tocdepth}{2} \makeatletter\renewcommand*\l@subsection{\@dottedtocline{2}{1.5em}{2.7em}}\makeatother% room for 12.10 \hypersetup{linkcolor=blue,citecolor=blue,urlcolor=blue,pdfstartview=FitV, pdftitle={Anomalous regularization of passive scalars up to the Obukhov-Corrsin exponent}} \usepackage{titlesec} \newcommand{\addperiod}[1]{#1.} \titleformat{\section} {\centering\normalfont\Large}{\thesection.}{0.5em}{} \titleclass{\part}{straight} \titleformat{\part}[block] {\centering\normalfont\LARGE}{\partname\ \thepart.}{0.5em}{} \titlespacing*{\part}{0pt}{3ex plus 1ex minus .5ex}{2ex plus .5ex} \titleformat*{\subsection}{\bfseries} \titleformat{\subsubsection}[runin] {\normalfont\bfseries} {\thesubsubsection.} {0.5em} {\addperiod} \titleformat*{\subsubsection}{\normalfont\itshape} \titleformat*{\paragraph}{\bfseries} \titleformat*{\subparagraph}{\large\bfseries} \title{Anomalous regularization of passive scalars up to the Obukhov--Corrsin exponent} \author{Scott Armstrong \thanks{CNRS \& Laboratoire Jacques-Louis Lions, Sorbonne Universit\'e; Courant Institute School of Mathematics, Computing, and Data Science, New York University. {\footnotesize \href{mailto:scottnarmstrong@gmail.com}{scottnarmstrong@gmail.com}.} } \and Vlad Vicol \thanks{Courant Institute School of Mathematics, Computing, and Data Science, New York University. {\footnotesize \href{mailto:vicol@cims.nyu.edu}{vicol@cims.nyu.edu}.} } } \date{November 2, 2026} % Numbering: equations, figures and tables within the section; lemmas, propositions and % corollaries on one counter within the section; only the main results are lettered. \numberwithin{figure}{section} \numberwithin{table}{section} % Macros of Part I. \newcommand{\Kset}{\mathcal K} \newcommand{\bstar}{\mathbf b_*} \newcommand{\J}{\mathbf J} \newcommand{\Nrep}{N_{\mathrm{rep}}} \newcommand{\Nout}{N_{\mathrm{out}}} \newcommand{\Ncor}{N_{\mathrm{cor}}} \newcommand{\Nsrc}{N_{\mathrm{src}}} \newcommand{\tpr}{t_{\mathrm{pr}}} \newcommand{\size}{\operatorname{size}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\expp}{\exp_p} \newcommand{\deps}{\alpha,\allowbreak s,\allowbreak q,\allowbreak\zeta,\allowbreak\Lambda,\allowbreak n,\allowbreak t_0,\allowbreak T} % Macros of Section 14. \newcommand{\Nret}{N_{\mathrm{ret}}} \newcommand{\Nmom}{N_{\mathrm{mom}}} \newcommand{\rhomol}{\rho_{\mathrm{mol}}} \newcommand{\cmol}{c_{\mathrm{mol}}} \newcommand{\tA}{t_{\mathrm A}} \newcommand{\tc}{t_{\mathrm c}} \newcommand{\tf}{t_{\mathrm f}} \newcommand{\tS}{t_{\mathrm S}} % Macros of Section 10. \newcommand{\Hc}{\mathsf H} \newcommand{\Vc}{\mathcal V} \newcommand{\Mc}{\mathcal M} \newcommand{\Lc}{\mathcal L} \newcommand{\Lsrc}{\mathcal L^{\mathrm{src}}} \newcommand{\eperp}{e_{(k)}^\perp} \newcommand{\fac}{\operatorname{fac}} \newcommand{\maj}{\operatorname{maj}} \newcommand{\cellwin}{[(k-61/48)\tau_m,(k+61/48)\tau_m]} % Macros of Section 11. \newcommand{\gen}{\operatorname{gen}} \newcommand{\res}{\operatorname{res}} \newcommand{\Ngen}{N_{\mathrm{gen}}} \newcommand{\Nlen}{N_{\mathrm{len}}} \newcommand{\Nword}{N_{\mathrm{word}}} \newcommand{\Npath}{N_{\mathrm{path}}} \newcommand{\Ncoef}{N_{\mathrm{coef}}} \newcommand{\Nan}{N_{\mathrm{an}}} \newcommand{\Noff}{N_{\mathrm{off}}} \newcommand{\Cadm}{C_{\mathrm{adm}}} \newcommand{\Cseed}{C_{\mathrm{seed}}} \newcommand{\Cmaj}{C_{\mathrm{maj}}} \newcommand{\Cwid}{C_{\mathrm{wid}}} \newcommand{\Cder}{C_{\mathrm{der}}} \newcommand{\Cjet}{C_{\mathrm{jet}}} \newcommand{\mS}{m_{\mathrm S}} \newcommand{\mc}{m_{\mathrm c}} \newcommand{\can}{c_{\mathrm{an}}} \newcommand{\Ch}{\mathcal C} \newcommand{\Xc}{\mathcal X} \newcommand{\xb}{\bar x} \newcommand{\tb}{\bar t} \newcommand{\precise}[2]{\par\smallskip\noindent\refstepcounter{equation}\label{#1}(\theequation)~\emph{#2}\ } % Macros of Section 12. \newcommand{\Bl}{\mathcal B} \newcommand{\Rb}{\mathcal R} \newcommand{\Zs}{\mathcal Z} \newcommand{\Pt}{\widetilde\Pi} \newcommand{\Ud}[1]{U^{[#1]}} \newcommand{\Ndep}{N_{\mathrm{dep}}} \newcommand{\Nbuf}{N_{\mathrm{buf}}} \newcommand{\cbr}{c_{\mathrm{br}}} \newcommand{\Cbr}{C_{\mathrm{br}}} \newcommand{\clo}{c_{\mathrm{lo}}} \newcommand{\cdis}{c_{\mathrm{dis}}} \newcommand{\chigh}{c_{\mathrm{hi}}} \newcommand{\Chigh}{C_{\mathrm{hi}}} \newcommand{\ebar}{\bar\eps_m} % Macros of Section 13. \newcommand{\Ncl}{N_{\mathrm{cl}}} \newcommand{\Nrec}{N_{\mathrm{rec}}} \newcommand{\Fsm}{F^{\mathrm{sm}}} \newcommand{\Ffr}{F^{\mathrm{fr}}} \newcommand{\Fcom}{F^{\mathrm{com}}} % Macros of Section 15. \newcommand{\Ncw}{N_{\mathrm{cor,w}}} \newcommand{\Nlw}{N_{\mathrm{len,w}}} \newcommand{\Nmw}{N_{\mathrm{mom,w}}} \newcommand{\Ntw}{N_{\mathrm{tr,w}}} \newcommand{\Nce}{N_{\mathrm{cor,e}}} \newcommand{\pe}{p_{\mathrm e}} \newcommand{\Nle}{N_{\mathrm{len,e}}} \newcommand{\Ncle}{N_{\mathrm{cl,e}}} \newcommand{\Nme}{N_{\mathrm{mom,e}}} \newcommand{\Nrde}{N_{\mathrm{rd,e}}} \newcommand{\Nse}{N_{\mathrm{src,e}}} \newcommand{\Noe}{N_{\mathrm{out,e}}} \newcommand{\Qe}{Q_{\mathrm e}} \newcommand{\The}{\Theta_{\mathrm e}} \newcommand{\ce}{c_{\mathrm e}} \newcommand{\Sw}{S_{\mathrm w}} \newcommand{\rw}{r_{\mathrm w}} \newcommand{\Cw}{\Corr^{\mathrm w}} \newcommand{\Ce}{\Corr^{\mathrm e}} \newcommand{\Uw}{\Upsilon^{\mathrm w}} \newcommand{\Ue}{\Upsilon^{\mathrm e}} \newcommand{\W}{\mathcal W} \newcommand{\te}{t_{\mathrm e}} \newcommand{\tpM}{\tpr^{(M)}} \newcommand{\GM}{\widetilde G_M} \newcommand{\UM}{\widetilde U_M} \begin{document} \maketitle \begin{abstract} We consider the advection--diffusion equation $\partial_t\theta+\mathbf{b}\cdot\nabla\theta-\kappa\Delta\theta=0$ on the two-dimensional torus~$\T^2$, where~$\mathbf{b}$ is a divergence-free velocity field constructed as in~\cite{AV}, H\"older continuous with an exponent $\alpha<\nf13$ and built from shear flows alternating in direction at infinitely many scales. We prove anomalous regularization: for every~$s$ below the Obukhov--Corrsin exponent $\frac12(1-\alpha)$, with~$\mathbf{b}$ depending on~$s$, the solutions are bounded in~$C^{0,s}(\T^2)$ at positive times by a constant times the $L^2(\T^2)$ norm of the initial data, uniformly in the diffusivity~$\kappa$ in a union of bands accumulating at~$0$. As a consequence, we obtain anomalous dissipation at a uniform rate: solutions dissipate a fixed fraction of their variance in unit time, so that the variance decays exponentially, with constants independent of~$\kappa$, of the initial datum and of the initial time. The proof combines iterated quantitative homogenization with large-scale regularity: at each scale the solution is compared with a solution of the homogenized equation of that scale. Since the derivatives consumed by each comparison are regenerated at every scale, the H\"older norm is carried from the coarsest scale to the finest by induction. Each of our main results, together with its proof, is fully formalized in the Lean~4 proof assistant. \end{abstract} \tableofcontents % ===== 00-introduction ===== % The introduction: Section 1, with numbered subsections. \section{Introduction} \label{s.intro} We study the advection--diffusion equation \begin{equation} \left\{ \begin{aligned} & \partial_t\theta+\b\cdot\nabla\theta-\kappa\Delta\theta=0 & \mbox{in} & \ (\tin,\infty)\times\T^2\,,\\ & \theta(\tin,\cdot)=\thetain & \mbox{on} & \ \T^2 \end{aligned} \right. \label{e.intro.pde} \end{equation} on the torus $\T^2\coloneqq\R^2/\Z^2$, for a divergence-free velocity field~$\b$, a diffusivity $\kappa>0$, an initial time $\tin\in\R$ and an initial datum $\thetain\in L^2(\T^2)$. The solution~$\theta$ is a passive scalar, carried by~$\b$ and diffused with the molecular diffusivity~$\kappa$. Since $\nabla\cdot\b=0$, the transport term conserves the $L^2(\T^2)$ norm, and diffusion alone decreases it: for every $t>\tin$, \begin{equation*} \|\theta(t,\cdot)\|_{L^2(\T^2)}^2+2\kappa\|\nabla\theta\|_{L^2((\tin,t)\times\T^2)}^2=\|\thetain\|_{L^2(\T^2)}^2\,. \end{equation*} We are interested in small~$\kappa$ and in velocity fields which, like the velocity of a turbulent fluid, are continuous but rough:~$\mathbf{b}$ is H\"older continuous with an exponent $\alpha<1$. The question is how regular the solutions are, uniformly in~$\kappa$. For the velocity of a turbulent fluid, whose increments over a distance scale like the power~$\nf13$ of the distance in Kolmogorov's theory~\cite{Kolmogorov41}, Obukhov~\cite{Obukhov} and Corrsin~\cite{Corrsin} predicted by a scaling argument that the increments of the scalar scale like the same power~$\nf13$, over a range of distances that depends on~$\kappa$. The same argument predicts the exponent $\frac12(1-\alpha)$ when the increments of the velocity scale like the power~$\alpha$~\cite[Section~1.3]{AV}. We call $\frac12(1-\alpha)$ the \emph{Obukhov--Corrsin exponent} of a velocity field which is H\"older continuous with exponent~$\alpha$. In~\cite{AV} we constructed, for every~$\alpha < \nf13$, a multiscale ``fractal'' velocity field~$\mathbf{b}$ which is continuous in time with values in~$C^{0,\alpha}(\T^2)$. For this field, the solutions~$\theta^\kappa$ of~\eqref{e.intro.pde} with diffusivity~$\kappa$, initial time $\tin=0$ and mean-zero initial data~$\thetain \in H^1(\T^2)$ exhibit anomalous dissipation, \begin{equation} \inf_{\kappa \in \Kset_{\mathrm{AV}}} \kappa\|\nabla\theta^\kappa\|_{L^2((0,1)\times\T^2)}^2\ge c\,\|\thetain\|_{L^2(\T^2)}^2\,. \label{e.intro.anomalous} \end{equation} Here~$\Kset_{\mathrm{AV}}$ is a countable union of bands accumulating at~$0$, the permissible diffusivities of~\cite[(3.43)]{AV}, which is the set~$\Kset$ of~\eqref{e.bands} for the value of~$q$ fixed in~\cite{AV}, and the constant $c>0$ depends on the ratio~$\|\thetain\|_{L^2(\T^2)}/\|\nabla \thetain\|_{L^2(\T^2)}$~\cite[Theorem~1.1 and the discussion after it]{AV}. In this paper we prove that, for every exponent~$s$ below~$\frac12(1-\alpha)$ and a velocity field~$\mathbf{b}$ of the same construction depending on~$s$, the solutions are uniformly H\"older continuous with exponent~$s$ at positive times, for every datum in~$L^2(\T^2)$, uniformly in~$\kappa$ in the corresponding union of bands. Consequently, for this velocity field, an estimate of the form~\eqref{e.intro.anomalous} holds for every mean-zero datum in~$L^2(\T^2)$ and from every initial time, with a constant~$c$ independent of the datum (Theorem~\ref{t.B} below). \subsection{Main results} \paragraph{Construction of the velocity field.} The velocity field of~\cite{AV} is built level by level. Level $m\ge1$ has a wavelength~$\eps_m$, and the wavelengths decrease supergeometrically: $\eps_{m-1}/\eps_m\ge\Lambda$ for a large integer~$\Lambda$, and $\eps_m$ is of the order of~$\eps_{m-1}^q$ for $m\ge2$, for a parameter $q>1$. Level~$m$ adds to the velocity field~$\b_{m-1}$ of the coarser levels short pulses of a shear flow of wavelength~$\eps_m$ and speed of order~$\eps_m^\alpha$, alternating between the two coordinate directions and written in the Lagrangian coordinates of~$\b_{m-1}$. A fixed cutoff function~$\zeta$ of time switches the pulses on and off. The limit~$\b$ of the velocity fields~$\b_m$ is periodic in space and time, divergence free, and H\"older continuous with exponent~$\alpha$ in space and time. The velocity field of this paper is that of~\cite{AV} with one change: the parameter~$q$, which~\cite{AV} fixes in terms of~$\alpha$, is here a free parameter, taken close to~$1$ depending on~$\alpha$ and on the exponent~$s$ of the main result (Section~\ref{s.construction}). With $\beta\coloneqq1+\alpha$ and $\gamma\coloneqq(q-1)\beta/(q+1)$, the diffusivities that we allow are the \emph{permissible diffusivities} of~\cite[(3.43)]{AV} for the present~$q$: the union~$\Kset$ of the bands \begin{equation*} \Kset_M=\bigl[\tfrac12\eps_M^{\beta-\gamma},\,2\eps_M^{\beta-\gamma}\bigr]\,,\qquad M\ge1\,, \end{equation*} defined in~\eqref{e.bands}, which are disjoint and accumulate at~$0$. The exponent~$\gamma$ corrects~$\beta$ in the exponent of the effective diffusivities, described in the strategy below; since $q(\beta-\gamma)=\beta+\gamma$, a diffusivity in~$\Kset_M$, $M\ge2$, is of the order of~$\eps_{M-1}^{\beta+\gamma}$. \paragraph{Interior regularity.} Our main result is Theorem~\ref{t.A}, stated precisely in Section~\ref{s.result}. Let $\alpha\in(0,\nf13)$ and $s\in(0,\frac12(1-\alpha))$, let~$q>1$, close to~$1$ depending on~$\alpha$ and~$s$, and the cutoff function~$\zeta$ be as in Section~\ref{s.construction}, and let~$\Lambda$ be an integer at least a threshold $\Lambda_0(\alpha,s,q,\zeta)\ge128$. Then the velocity field~$\b$ of Section~\ref{s.construction}, which depends on~$s$ through~$q$, has the following property: for every $00$ such that, for every $\tin\in\R$, $\kappa\in\Kset$ and $\thetain\in L^2(\T^2)$, with $(\thetain)_{\T^2}$ the mean of~$\thetain$, every weak solution of~\eqref{e.intro.pde} satisfies \begin{equation*} \kappa\|\nabla\theta\|_{L^2((\tin,\tin+1)\times\T^2)}^2\ge c\,\bigl\|\thetain-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}^2\,, \end{equation*} and, for every $t\ge \tin$, \begin{equation*} \bigl\|\theta(t,\cdot)-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}\le2\exp(-c(t-\tin))\bigl\|\thetain-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}\,. \end{equation*} Over every unit of time, every solution thus dissipates a fixed fraction of its variance. The proof is short: by Theorem~\ref{t.A}, the solution operator over a unit of time maps~$L^2(\T^2)$ boundedly into~$C^{0,s}(\T^2)$, and a positive operator with this property which preserves constants and integrals contracts the functions of mean zero, with a fixed contraction constant (Lemma~\ref{l.mr.operator}). \paragraph{Local regularity.} Theorem~\ref{t.C}, stated and proved in Section~\ref{s.mainresults}, localizes Theorem~\ref{t.A}. Consider a solution of the advection--diffusion equation with $\kappa\in\Kset$, in the sense of distributions and with finite energy, in a space-time tube that follows the trajectories of the velocity field~$\b_m$ of the first~$m$ levels over the time~$8h_{m+1}$ (with~$h_{m+1}$ the renewal time of level~$m+1$, Subsection~\ref{ss.scales}), with no lateral data prescribed. Theorem~\ref{t.C} bounds its norm in~$C^{0,s}$ on the trajectories of a smaller concentric disc, from a waiting time on, by a constant times its root mean square over the tube, and the constant depends neither on~$\kappa$ nor on the final time of the tube. The proof compares the solution with the solution on the whole torus from a cut-off datum, to which Theorem~\ref{t.A} applies. \paragraph{Vanishing diffusivity.} Corollary~\ref{c.D}, also in Section~\ref{s.mainresults}, passes to the limit $\kappa\to0$. Let~$\kappa$ tend to~$0$ along a sequence~$\Kset'$ in~$\Kset$, and let~$\theta^\kappa$ be the solutions, from a common initial time~$\tin$, with data~$\thetain^\kappa$ bounded in~$L^2(\T^2)$. A subsequence converges locally uniformly on $(\tin,\infty)\times\T^2$ to a continuous solution~$\theta_*$ of the transport equation $\partial_t\theta_*+\nabla\cdot(\b\theta_*)=0$, in the sense of distributions. The limit takes the weak limit of the data along the subsequence as its initial value, weakly in~$L^2(\T^2)$, satisfies the bound of Theorem~\ref{t.A} with $\sup_{\kappa\in\Kset'}\|\thetain^\kappa\|_{L^2(\T^2)}$ in place of $\|\thetain\|_{L^2(\T^2)}$, and is H\"older continuous in time with exponent $s/(1+s)$ on every set $[\tin+t_0,\tin+T]\times\T^2$. Corollary~\ref{c.D}(b) is the local form, in the tubes of Theorem~\ref{t.C}. The conclusion is subsequential: for the velocity field of~\cite{AV} with $\alpha\ge\nf1{67}$, vanishing diffusivity does not select a unique limit~\cite[Proposition~5.5]{AV}; see also~\cite{CCS22,HuysmansTiti25}. \subsection{Prior results on anomalous dissipation and the Obukhov--Corrsin exponent} If~$\b$ is Lipschitz continuous in space, the transport equation $\partial_t\theta+\b\cdot\nabla\theta=0$ is well posed, its solutions conserve the $L^2(\T^2)$ norm, and, by the energy identity, the dissipation $\kappa\|\nabla\theta^\kappa\|_{L^2((\tin,\tin+1)\times\T^2)}^2$ tends to zero with~$\kappa$~\cite[Section~1.1]{AV}. \emph{Anomalous dissipation}, the failure of this limit for rougher velocity fields, was first discussed by Obukhov~\cite{Obukhov}; it is a basic assumption of the phenomenological theories of scalar turbulence, and it is supported by strong experimental and numerical evidence~\cite{shraiman,Warhaft,DSY}. Rigorous examples include the white-in-time stochastic model of~\cite{K1,Bernard}, for which a quantitative proof of anomalous dissipation is given in~\cite{Rowan24}, and deterministic constructions. Those of~\cite{DEIJ,CCS22,ElgindiLiss24} are built on alternating shear flows or on mixing, up to a singular time at which all the dissipation takes place~\cite[p.~4]{CCS22}, \cite[Section~1.2.2]{ElgindiLiss24}, \cite[Remark~2.1]{HessChildsRowan25a}; those of~\cite{HessChildsRowan25a,HessChildsRowan25b} iterate a construction of this type and dissipate at countably many times. Related constructions exist for forced fluid equations~\cite{BrueDeLellis22,BCCDLS22,Cheskidov24,HofmanovaPappaletteraZhuZhu24}. Autonomous examples were obtained in~\cite{JohanssonSorella26a,JohanssonSorella26b} (see also~\cite{BagnaraBoutrosDeLellisMayboroda26}); the second of these uses the relation between spontaneous stochasticity and scalar dissipation established in~\cite{DrivasEyink17}. The velocity field of~\cite{AV} has infinitely many active scales at almost every time, and the proof of~\cite[Theorem~1.1]{AV} homogenizes these scales one at a time, from the scale at which molecular diffusion acts up to the unit scale, where the effective diffusivity is of order one. The Obukhov--Corrsin exponent is a threshold for anomalous dissipation. As observed in~\cite[Section~1.3]{AV}, the commutator estimate of~\cite{CET} shows that, for a velocity field which is continuous in time with values in~$C^{0,\alpha}(\T^2)$, solutions which are bounded uniformly in~$\kappa$, continuously in time, in a H\"older space of exponent larger than $\frac12(1-\alpha)$ dissipate no variance in the limit $\kappa\to0$; see~\cite[Theorem~4]{DEIJ} for the scalar statement. Below the exponent, Colombo, Crippa and Sorella~\cite{CCS22} obtained anomalous dissipation together with uniform H\"older bounds in the full range of exponents, for a particular datum and in~$L^2$ in time~\cite[Sections~1.1 and~1.3]{AV}. A construction with anomalous dissipation for every nonzero smooth mean-zero datum and sharp time-integrated regularity bounds was subsequently given in~\cite[Theorem~2]{ElgindiLiss24}. In contrast, the estimate of Theorem~\ref{t.A} is in~$L^\infty$ in time at positive times and holds for every datum in~$L^2(\T^2)$, uniformly in~$\kappa\in\Kset$. Regularization from rough data has also been proved for other velocity fields. In~\cite[Theorem~1.5]{HessChildsRowan25b}, the deterministic construction mentioned above gives a positive Sobolev gain from data in~$L^2(\T^2)$, on the time interval~$[0,1]$ from the initial time~$0$, measured in~$L^2$ in time and uniformly in the diffusivity. For rough white-in-time Kraichnan velocities, anomalous Sobolev regularization was proved in~\cite{GaleatiGrottoMaurelli26}, and expected Sobolev bounds at each positive time for~$L^2$ data were obtained in~\cite[Theorem~1.3]{DrivasGaleatiPappalettera25}. On the torus, uniform-in-diffusivity estimates for the expected regularity yield bounds on the stationary scalar spectrum in~\cite{Rowan25}. The Sobolev threshold in these white-in-time models is~$1-\alpha$, rather than the exponent~$\frac12(1-\alpha)$ for time-regular velocities~\cite[Remark~1.4]{Rowan25}. For the elliptic, time-independent problem in self-similar random incompressible flows, in a perturbative regime, iterated homogenization gives H\"older estimates uniform in the molecular diffusivity, with a random constant determined by a minimal scale~\cite[Theorem~C]{ArmstrongBouRabeeKuusi26}. For regularization in nonlinear models, see~\cite{CheskidovZaya16,Drivas22}. Burczak, Sz\'ekelyhidi and Wu~\cite{BSW23} combined the iterated homogenization of~\cite{AV} with the convex integration methods used in the proof of Onsager's conjecture~\cite{Isett18,BDLSV19}, and obtained anomalous dissipation for every nonzero mean-zero datum in~$H^1(\T^3)$ by H\"older continuous weak solutions of the unforced incompressible Euler equations on~$\T^3$, with any exponent below~$\nf13$~\cite[Theorem~1.1]{BSW23}. In~\cite{BSW26} they proved anomalous dissipation for mean-zero data with more than~$\frac12(1+\alpha)$ derivatives in~$L^2(\T^2)$, a class larger than~$H^1(\T^2)$, together with uniform H\"older bounds, along a sequence of diffusivities tending to~$0$. Their velocity fields are divergence free on~$\T^2$, with space-time H\"older exponent~$\alpha<\nf13$, and built, as they explain in~\cite[Section~1.3]{BSW26}, essentially as in~\cite{AV,BSW23}. Given~$s<\frac12(1-\alpha)$, their H\"older bound reads, in our notation, for mean-zero data~$\thetain\in H^{1+s'}(\T^2)$ with~$s'>s$, \begin{equation*} \|\theta\|_{L^\infty([0,1];C^{0,s}(\T^2))}\le C\|\thetain\|_{H^{1+s'}(\T^2)}\,, \end{equation*} with~$C$ independent of the datum and of the diffusivity along the sequence~\cite[Theorem~1.1]{BSW26}. In~\cite[Section~1.2.1]{BSW26}, the authors write that they ``do not follow the `large scale regularity' approach suggested'' in~\cite{AV}. Of a ``naive approach to regularity'', they write that its second difficulty, an order-one background transport against a vanishing diffusivity at the larger scales, ``seems to be a fundamental feature of our setup, prohibiting the use of `large-scale regularity' techniques''~\cite[Section~1.3]{BSW26}. Their proof transfers regularity from coarser homogenized equations to finer ones, the mechanism described in~\cite[Section~4.1]{AVecm}, by high-order corrector expansions and estimates for systems of forced equations~\cite[Theorems~6.10 and~7.2]{BSW26}, started separately for the Littlewood--Paley pieces of the datum~\cite[Section~8.3]{BSW26}. For smooth data, their bound gives, for a velocity field depending on the exponent, the uniform H\"older regularity below the Obukhov--Corrsin exponent sought in~\cite[Section~1.3]{AV}. The interior regularization announced at the end of~\cite[Section~1.3]{AV} is a bound of a different kind: a bound at positive times of the H\"older norm of the solution by the norm of the datum in~$L^2(\T^2)$, for every datum in~$L^2(\T^2)$, with a constant that tends to infinity as the waiting time tends to~$0$. The bound of~\cite{BSW26} propagates regularity: it holds up to the initial time, and its right side requires a datum which is already H\"older continuous. Theorem~\ref{t.A} proves this interior regularization for every $s<\frac12(1-\alpha)$, with the velocity field constructed here depending on~$s$. Its proof carries out the large-scale regularity program announced in~\cite[Section~1.3]{AV}, by the induction described in~\cite[Section~4.1]{AVecm}. The comparison of two levels, in~$L^2(\T^2)$ at a high power of~$\eps_{m-1}$, is read in~$C^{0,s}(\T^2)$ through representations, which also measure derivatives at each wavelength, and the derivatives that the correctors consume are regenerated at every level (Proposition~\ref{p.cycle}); both are explained in the strategy below. The constants in the lower bounds of~\cite[Theorem~1.1]{AV}, \cite[Theorem~1.1]{BSW23} and~\cite[Theorem~1.1]{BSW26} depend on the datum through a ratio of its $L^2$ norm to a Sobolev seminorm of positive order, while the constant of Theorem~\ref{t.B} is independent of the datum, of the initial time and of~$\kappa\in\Kset$. Independence from the datum was announced in~\cite[Section~1.3]{AV} and~\cite[Section~4.1]{AVecm} as a consequence of uniform regularity. Among deterministic velocity fields, dissipation uniform over the initial datum was obtained by a different construction in~\cite[Corollary~1.3]{HessChildsRowan25a}, which gives total dissipation in the limit $\kappa\to0$ for every mean-zero datum in~$L^2(\T^2)$; for white-in-time velocity fields, see~\cite{Rowan24,Rowan25}. A related construction gives accelerating exponential decay for its time-periodic extension~\cite[Corollary~1.2]{HessChildsRowan25b}. These estimates are stated from the initial time~$0$, for velocity fields constructed on a unit time interval, whose dissipation takes place at singular times~\cite[Section~1.1]{HessChildsRowan25b}. Theorem~\ref{t.B} holds from every initial time~$\tin\in\R$, for a velocity field which is periodic and H\"older continuous in space and time, with infinitely many active scales at almost every time. \subsection{Strategy of the proof} \label{ss.strategy} The proof of Theorem~\ref{t.A} rests on nine propositions and one lemma, all stated in Part~I; their logical dependence is shown in Figure~\ref{f.map}. We next outline the main ideas of the proof, and postpone the technical estimates to Part~II. \paragraph{Reduction to finitely many scales.} Let $\kappa\in\Kset_M$. The levels of~$\b$ much finer than level~$M$ do not matter: their stream functions are much smaller than~$\kappa$, a perturbation of the stream function of size~$\kappa\varpi$ moves the solution by at most $\varpi\|\thetain\|_{L^2(\T^2)}$ in~$L^2(\T^2)$, and a H\"older estimate for small perturbations of the heat equation turns this into a bound in~$C^{0,s}(\T^2)$. It therefore suffices to prove the bound of Theorem~\ref{t.A} for the velocity field~$\b_{M+n}$, for a fixed number~$n$ of extra levels and for~$M$ large; the finitely many remaining bands follow from the same H\"older estimate directly (Section~\ref{s.reduction}). \paragraph{Effective diffusivities and three kinds of level.} For $\kappa\in\Kset_M$, the \emph{effective diffusivities} $\kappa_M\coloneqq\kappa,\kappa_{M-1},\dots,\kappa_0$ of~\cite{AV} are defined by a recursion: $\kappa_{m-1}$ is the diffusivity that the solution feels once the pulses of level~$m$, at diffusivity~$\kappa_m$, have been averaged out (Subsection~\ref{ss.diffusivities}). By Taylor dispersion~\cite{Taylor53} (see also~\cite{FannjiangPapanicolaou94,MajdaKramer99}), the pulses of level~$m$, whose stream function has amplitude~$\eps_m^\beta$, add about $\frac9{80}\eps_m^{2\beta}/\kappa_m$ to the diffusivity (Proposition~\ref{p.scales}(a)). When the enhancement dominates, $\kappa_{m-1}\kappa_m\asymp\eps_m^{2\beta}$, and the scaling $\kappa_m\asymp\eps_m^{\beta+\gamma}$ is consistent with this exactly when $q(\beta-\gamma)=\beta+\gamma$, which defines~$\gamma$. For $\kappa\in\Kset_M$ this scaling holds at every level $1\le m0$ is the small exponent of~\eqref{e.exponents}. The correctors and the whole analysis of level~$m$ are written in these coordinates. The other difficulty is that the correctors consume derivatives, while the datum is only in~$L^2(\T^2)$ and the induction cannot lose derivatives at each of infinitely many levels: the derivatives are regenerated at every level, within one period of its pulses. \paragraph{Two dual cascades.} Fix $M$ and $\kappa\in\Kset_M$. By the definition of the effective diffusivities, homogenizing the pulses of level~$m\le M$ turns the operator $\partial_t+\b_m\cdot\nabla-\kappa_m\Delta$ of level~$m$ into the operator of level $m-1$; for $m>M$, the operator of level~$m$ has~$\kappa$ in place of~$\kappa_m$. The proof is an induction on the levels made of two dual cascades (Subsection~\ref{ss.overview}), both run by Proposition~\ref{p.cascade} between the levels $M-1$ and~$m_0$. The \emph{homogenization cascade} goes from fine to coarse, \emph{up the levels} and forward in time: the \emph{entry} (Proposition~\ref{p.entry}) crosses the levels $M+n,\dots,M$, and then one \emph{cycle} at a time (Proposition~\ref{p.cycle}), each in a short interval of time, crosses the levels $M-1,\dots,m_0+1$, up to a fixed top level~$m_0$, independent of~$M$ and~$\kappa$, at which the solution is smooth at a fixed scale. Each cycle compares the solution of a level with one of the homogenized equation, the equation of the next coarser level; this cascade carries the \emph{marked amplitude}, a weighted $L^2$ norm taken in~$L^\infty$ in time (Subsection~\ref{ss.overview}). The \emph{regularity cascade} goes from coarse to fine, back down the levels at one fixed time: as in large-scale regularity, it transfers a H\"older bound from the homogenized equation to the heterogeneous one, one finer level at a time. Each cascade needs the other: the first produces the homogenized solutions with which the second compares, and the second turns the $L^2$ control of the first into a H\"older bound. \paragraph{Families of coupled equations.} The comparison of a level with the next coarser one leaves error terms that are not small and that carry derivatives of the coarse solution. They become new unknowns, each solving the equation of its level with a source made of smoothed derivatives of earlier unknowns, so that the object passed from one level to the next is a coupled system of equations with the operator of its level, a \emph{family} (Subsection~\ref{ss.families}). The marked amplitude of a solution of a family, which weights the $L^2$ norms of its \emph{components}, the unknowns of the system, is multiplied per cycle by at most one plus a constant times a fixed positive power of~$\eps_{m-1}$; since $\eps_m\le\Lambda^{-m}$, these factors have a bounded product, uniformly in the number of levels. \paragraph{The entry.} At the levels finer than~$M$, the enhancement of a level is small compared with~$\kappa$. A \emph{weak passage} (Subsection~\ref{ss.entry.weak}) therefore keeps the diffusivity~$\kappa$, carries the enhancement as a source of the coarser family (Subsection~\ref{ss.families}), and regains derivatives on the quiet margins, the pulse-free intervals of length~$\tau'_m$ at both ends of each block of time of length~$h_m$ on which the pulses of level~$m$ are renewed (Section~\ref{s.construction}). Over a margin, molecular diffusion reaches the wavelength~$\eps_{m-1}$. Level~$M$ also has a small P\'eclet number, but its margins are too short for this, $\kappa\tau'_M\ll\eps_{M-1}^2$, so it is crossed by a \emph{terminal cycle} (Subsection~\ref{ss.entry.terminal}), which passes to~$\kappa_{M-1}$ and produces the first family of the homogenization cascade (Proposition~\ref{p.cascade}), the \emph{seed} (Subsection~\ref{ss.families}). The weak passages and the terminal cycle form the entry (Section~\ref{s.entry}). \paragraph{One homogenization cycle.} A cycle has two parts. The first, the \emph{preparation} (Section~\ref{s.preparation}), regenerates derivatives: within one period~$4h_m$ of the pulses, the solution of level~$m$ becomes, up to a small error, smooth at the \emph{effective diffusion length} $\rho(m)=(\kappa_{m-1}h_m)^{\nf12}$, which lies between the two wavelengths, $\eps_m\ll\rho(m)\ll\eps_{m-1}$ (Proposition~\ref{p.scales}(c) and Lemma~\ref{l.fl.scales}(c)). The proof uses the Bloch transform, the Fourier transform adapted to periodic coefficients. For the periodic elliptic theory of corrector expansions and Bloch approximation, see~\cite{BLP78,ConcaVanninathan97,ConcaOriveVanninathan02}. In the Lagrangian coordinates of a renewal block, with the conductivity frozen at a point, the operator of level~$m$ is periodic with period~$\eps_m$, and Bloch waves split its evolution into evolutions on the period cell, one for each frequency~$\xi$. The principal branch of these evolutions, the one that continues the constants, has, to leading order, the symbol $-\xi\cdot\J_m^{\kappa_m}(t)\xi$, where the \emph{averaged flux} $\J_m^{\kappa_m}(t)$ is the diffusivity matrix produced by the pulses through Taylor dispersion, enhanced along the direction of the active pulses~\eqref{e.flux}. Since every renewal block contains pulses of both directions, the real part of this symbol, integrated over a block, is at most a negative multiple of $\kappa_{m-1}h_m|\xi|^2$, in every direction and uniformly in the level: the principal branch is damped like a heat flow at the effective diffusion length. At the small frequencies, the part of each of these evolutions off the branch decays at the rate $\kappa_m/\eps_m^2$ of the cell problem, and at the remaining frequencies the whole evolution does. The second part, the \emph{comparison} (Section~\ref{s.comparison}), builds from the solution of the next family, the \emph{successor} (Subsection~\ref{ss.families}), a \emph{lift} to level~$m$, in three steps (Subsection~\ref{ss.lifts}). It adds each new component of the successor to its parent component (Subsection~\ref{ss.families}), with a small weight. It undoes the oscillation in time of the averaged flux, which alternates between the two directions of the pulses and has the average~$\kappa_{m-1}I$ over a period: it applies the truncated exponential of the second-order operator $\Gamma_m(t):\nabla^2$, where the \emph{covariance}~$\Gamma_m(t)$ is the accumulated difference between~$\J_m^{\kappa_m}$ and~$\kappa_{m-1}I$. Since $|\Gamma_m(t)|\le C\eps_{m-1}^\delta\rho(m)^2$, each power of the covariance gains~$\eps_{m-1}^\delta$ on functions that are smooth at the scale~$\rho(m)$, so the exponential can be truncated. Finally, it adds the correctors of the pulses, which oscillate at the wavelength~$\eps_m$. The errors that are not small become the sources of the next family; the rest is a residual of the size of a fixed power of~$\eps_{m-1}$, which the energy estimate for families converts into a bound on the difference between the solution and the lift. A cycle that starts at a time~$t_{\mathrm c}$ is prepared by a time~$\tpr$ within~$4h_m$ of~$t_{\mathrm c}$, and the comparison holds from~$\tpr$ on (Proposition~\ref{p.cycle}). \paragraph{Representations and the H\"older norm.} The regularity cascade carries the H\"older norm by \emph{representations}: decompositions of a solution of a family into a smooth base and oscillating profiles, each measured at its wavelength (Subsection~\ref{ss.representations}); the size of a representation plays the role of the excess. In terms of representations, a cycle is a \emph{passage} (Proposition~\ref{p.passage}): from a representation of the solution of level $m-1$ at the wavelength~$\eps_{m-1}$, it builds one of the solution of level~$m$ at the wavelength~$\eps_m$, at the same time. The old profiles are copied and never differentiated again; the new ones are the parts of the lift and the error of the comparison. A passage multiplies the size of a representation by at most one plus a constant times a fixed positive power of~$\eps_{m-1}$, and adds at most a constant times~$\eps_{m-1}$ times the marked amplitude of the solution; the balance~$\eps_{m-1}^\mu$ above pays for the correctors. Running the two cascades bounds the size of the representation at the finest level, and with it the $C^{0,s}(\T^2)$ norm, by the $L^2$ norm of the datum, uniformly in the number of levels (Proposition~\ref{p.cascade}). At the finest level, the lifts through the entry recover~$\theta$ from the solution of the seed (Section~\ref{s.entry}). \paragraph{Energy estimates and paired smoothing.} The families are coupled systems of parabolic equations, and their energy does not grow when the coupling is small (Proposition~\ref{p.energy}). The sources that a homogenization cycle creates are smoothed by products of averages over two thin ellipses, whose long axes come from the two orientations of the pulses; such pairs smooth in every direction, uniformly in their eccentricity (Proposition~\ref{p.paired}). For the sources of the seed the two ellipses are round discs. \subsection{Lean formalization} A formalization of this paper in the Lean~4 proof assistant is available at \begin{center} \url{https://github.com/scottnarmstrong/AVAnomalousRegularization}\,. \end{center} It contains Theorems~\ref{t.A}, \ref{t.B} and~\ref{t.C}, Corollary~\ref{c.D} and every statement of Part~I; the propositions and lemmas of Part~II follow from its formal statements, directly or by elementary steps, or are proved, where they are used, inside its formal proofs. A correspondence table, published with the repository, maps every numbered statement and display of the paper to the formal statements that prove it. Some results are stated here in a weaker form than their formal versions: for example, where the paper bounds a norm in~$L^\infty$ in time, the formal statement gives the bound at every time. Theorems~\ref{t.A}, \ref{t.B} and~\ref{t.C} and Corollary~\ref{c.D} are also stated in a self-contained form, whose definitions use only Mathlib and the statement definitions of the formalization of~\cite{AV}, with formal proofs connecting these statements to the development. The formal proofs follow the arguments given here; in some places the paper combines formally proved statements by elementary steps, such as an induction or a sequence of inequalities, which the formal proof combines differently. The formalization uses Lean~4 and Mathlib at version v4.35.0-rc2 and builds on the formalization of~\cite{AV}, available at \begin{center} \url{https://github.com/scottnarmstrong/AVAnomalousDiffusion}\,. \end{center} It contains no unproved placeholders and no custom axioms: the main theorems depend only on the three standard axioms of Lean~4 (\texttt{propext}, \texttt{Classical.choice} and \texttt{Quot.sound}). \subsection{Use of artificial intelligence} The main ideas of this work were developed by the authors several years ago; the program was announced in~\cite[Section~1.3]{AV} and described in~\cite[Section~4.1]{AVecm}. Starting from a detailed outline of the proof written by the authors, a very long and unpolished manuscript (612 pages of slop), in which every detail of the proof was filled in, was first produced by the authors together with AI agents (Claude Opus~5.5 and GPT-6.1 Sol). This manuscript was then formalized in Lean by AI agents directed by the authors: Claude Opus~5.5 orchestrated the development throughout, and the Lean code was written by Claude Opus~5.5 and Claude Sonnet~5.5, with parts written by GPT-6.1 Sol and GPT-6 Luna. Only then was the present paper written. Its first draft was written by Claude Opus~5.5 following the formalization, and Claude Sonnet~5.5 and GPT-6-Astra checked its statements against the formal ones and reviewed it. The authors checked this draft and rewrote parts of it themselves; the rest was rewritten by Claude Opus~5.5 agents under the direction of the authors, over several rounds of revision and audit. \subsection{Organization of the paper} Part~I (Sections~\ref{s.construction}--\ref{s.tools}) is a detailed outline of the proof, which can be read by itself. Section~\ref{s.construction} constructs the velocity field and states the properties of its effective diffusivities, Section~\ref{s.result} states Theorems~\ref{t.A} and~\ref{t.B}, and Section~\ref{s.reduction} reduces Theorem~\ref{t.A} to a velocity field with finitely many scales. Section~\ref{s.cascade} explains the two dual cascades, defines families, lifts and representations, and states the entry, the induction on the levels, one cycle and one passage. Section~\ref{s.tools} states the two tools. Part~II (Sections~\ref{s.flows}--\ref{s.entry}) states the notions used in Part~I in full and proves the statements of Part~I that are not proved there. Its sections treat the properties of the construction and of the effective diffusivities, relying on~\cite{AV}, and the flows and renewal charts (Section~\ref{s.flows}); the two tools (Section~\ref{s.familytools}); the reduction and the main results, including the statements of Theorem~\ref{t.C} and Corollary~\ref{c.D} (Section~\ref{s.mainresults}); the cell problems of the pulses, which give the correctors and the averaged flux (Section~\ref{s.cell}); admissible families (Section~\ref{s.admissible}); the two parts of one cycle, the preparation and the comparison (Sections~\ref{s.preparation} and~\ref{s.comparison}); the passage and the induction on the levels (Section~\ref{s.passage}); and the entry (Section~\ref{s.entry}). % ===== 01-construction ===== \part{A map of the proof} \label{part.map} \section{Construction of the velocity field} \label{s.construction} In this section we fix the parameters, define the scales, the velocity field~$\b$ and the effective diffusivities, and state their properties. The scales, shear streams, cutoff function and recursion are those of~\cite{AV}, except for the parameter~$q$, which~\cite[(2.2)]{AV} fixes as $q=\frac12\bigl(1+\frac{2-\beta}{2(\beta-1)}\bigr)$. Equation and statement numbers of~\cite{AV} refer to its arXiv version, arXiv:2305.05048v3. \subsection{Parameters} Throughout we fix \begin{itemize} \item $\alpha\in(0,\nf13)$, the H\"older exponent of the velocity field~$\b$; we also set $\beta\coloneqq1+\alpha$; \item $s\in(0,\frac12(1-\alpha))$, the H\"older exponent of the solutions in Theorem~\ref{t.A}; \item $q$ satisfying \begin{equation}\label{e.q} 10$ separates the time scales of Subsection~\ref{ss.scales}. The exponents $\mu>0$ and $\eta>0$ are used from Section~\ref{s.cascade} on, $\mu$ as the exponent of the cost~\eqref{e.mu} of one level. We write $x\asymp y$ if $C^{-1}y\le x\le Cy$ for a constant~$C$ depending only on $\alpha,q,\zeta$, and, for $m\ge2$, we write $x\ll y$, or $y\gg x$, if $x\le C\eps_{m-1}^cy$ for constants $c>0$ and~$C$ depending only on $\alpha,q,\zeta$. Throughout, $|\cdot|$ is the absolute value of a number, the Euclidean norm of a vector, and the operator norm, for the Euclidean norms, of a matrix, or of a multilinear map regarded as a linear map on arrays. We denote by~$(g)_E\coloneqq\fint_Eg$ the average of a function~$g$ over a set~$E$ of finite positive measure. On~$\R^2$ and~$\mathbb C^2$ we also write $|\cdot|_\infty$ for the largest modulus of a coordinate and, for a $2\times2$ matrix~$A$, $|A|_\infty$ for its operator norm for~$|\cdot|_\infty$, the largest sum of the moduli of the entries of a row. For a map or a vector field~$X$, $\nabla X$ is its Jacobian matrix, $(\nabla X)_{ii'}=\partial_{i'}X_i$. \subsection{Scales}\label{ss.scales} For $m\ge1$, level~$m$ has the wavelength~$\eps_m$, the renewal time~$h_m$, the margin time~$\tau'_m$ and the pulse time~$\tau_m$, defined by $\eps_0\coloneqq1$ and \cite[(2.8), (2.12)--(2.14)]{AV} \begin{equation}\label{e.scales} \begin{gathered} \eps_m\coloneqq\bigl\lceil\Lambda^{q^m/(q-1)}\bigr\rceil^{-1},\qquad h_m\coloneqq\frac{2^{-25}}{\lceil\eps_{m-1}^{-(2-\beta+2\delta)}\rceil}\,,\\ \tau'_m\coloneqq\frac{h_m}{4\lceil\eps_{m-1}^{-\delta}\rceil+1}\,,\qquad \tau_m\coloneqq\frac{h_m}{(4\lceil\eps_{m-1}^{-\delta}\rceil+1)^2}\,. \end{gathered} \end{equation} Then, for every $m\ge2$, $\eps_m\asymp\eps_{m-1}^q$, and, for every $m\ge1$, $\eps_{m-1}/\eps_m\ge\Lambda$, $4\lceil\eps_{m-1}^{-\delta}\rceil+1\asymp\eps_{m-1}^{-\delta}$ and \begin{equation}\label{e.times} h_m\asymp\eps_{m-1}^{2-\beta+2\delta}\,,\qquad \tau'_m\asymp\eps_{m-1}^{2-\beta+3\delta}\,,\qquad \tau_m\asymp\eps_{m-1}^{2-\beta+4\delta}\,, \end{equation} so that $\tau_m\ll\tau'_m\ll h_m$ for $m\ge2$; the ratios $\tau'_m/\tau_m$ and~$h_m/\tau'_m$ both equal $4\lceil\eps_{m-1}^{-\delta}\rceil+1$. \subsection{Pulses and the slow cutoff} The slots of level~$m$ are indexed by $k\in\Z$, and slot~$k$ carries the shear stream \cite[(2.19)--(2.20)]{AV} \begin{equation}\label{e.shear} \psi_{m,k}(y)\coloneqq\begin{cases}\eps_m^\beta\sin(2\pi y_1/\eps_m)&\text{if }k\in4\Z+1,\\ \eps_m^\beta\sin(2\pi y_2/\eps_m)&\text{if }k\in4\Z+3,\\ 0&\text{if }k\in2\Z,\end{cases} \end{equation} so even slots are empty and the odd ones, the \emph{pulses}, alternate between the two directions. Slot~$k$ is active for a time of about~$\tau_m$ around~$k\tau_m$, through the fast cutoff \begin{equation}\label{e.fast} \zeta_{m,k}(t)\coloneqq\zeta\Bigl(\frac{t-k\tau_m}{\tau_m}\Bigr). \end{equation} The renewal blocks are the intervals $[(l-\frac12)h_m,(l+\frac12)h_m]$, $l\in\Z$; slot~$k$ lies in the block of \begin{equation}\label{e.label} l_{m,k}\coloneqq\Bigl\lfloor\frac{k\tau_m+\frac12(h_m-\tau_m)}{h_m}\Bigr\rfloor\,. \end{equation} This is the label for which~\cite[(2.18)]{AV} holds: the slot $[(k-\frac12)\tau_m,(k+\frac12)\tau_m]$ lies in the block of~$l_{m,k}$ (Lemma~\ref{l.fl.scales}(d)). The slow cutoff of the block of~$l$ is \begin{equation}\label{e.slow} \widehat\zeta_{m,l}(t)\coloneqq\mathsf H\Bigl(\frac{t-(l-\frac12)h_m}{\tau'_m}-1\Bigr)\,\mathsf H\Bigl(\frac{(l+\frac12)h_m-t}{\tau'_m}-1\Bigr), \end{equation} where~$\mathsf H$ is the smooth step with $\mathsf H=0$ on $(-\infty,0]$, $\mathsf H=1$ on $[1,\infty)$ and \begin{equation*} \mathsf H(r)\coloneqq\frac{\exp(-1/r)}{\exp(-1/r)+\exp(-1/(1-r))}\qquad\text{for }00$, the \emph{averaged flux of the level-$m$ pulses} at diffusivity~$\nu$ is the matrix \begin{equation}\label{e.flux} \J_m^\nu(t)\coloneqq\nu I+2\pi^2\eps_m^{2\beta-2}\sum_{k\in2\Z+1}\widetilde\zeta_{m,k}(t)\Bigl(\int_{-\infty}^t\exp\Bigl(-\frac{4\pi^2\nu(t-t')}{\eps_m^2}\Bigr)\widetilde\zeta_{m,k}(t')\dd t'\Bigr)\,e_{(k)}\otimes e_{(k)}\,, \end{equation} where $e_{(k)}\coloneqq e_2$ for $k\in4\Z+1$ and $e_{(k)}\coloneqq e_1$ for $k\in4\Z+3$. This is the spatial average of the flux of the correctors of the level-$m$ pulses \cite[(3.15)]{AV}, written explicitly as in~\cite[(3.17)--(3.18)]{AV}: the shear dispersion of the pulses. Pulse~$k$ enhances diffusion in the direction~$e_{(k)}$ of its flow, and the time integral averages its envelope over the past at the rate $4\pi^2\nu/\eps_m^2$ at which the diffusivity~$\nu$ damps the modes $\sin(2\pi y_i/\eps_m)$. For a pulse held on forever, the time integral equals $\eps_m^2/(4\pi^2\nu)$ and the enhancement is $\eps_m^{2\beta}/(2\nu)$, Taylor's dispersion formula for the steady shear~$\psi_{m,k}$. The matrix $\J_m^\nu(t)$ is diagonal, satisfies \begin{equation*} \nu I\le\J_m^\nu(t)\le\Bigl(\nu+\frac{\eps_m^{2\beta}}{2\nu}\Bigr)I\,, \end{equation*} and equals~$\nu I$ when no pulse is active. Since the number $h_m/\tau_m$ of slots in a block is odd, a shift of time by~$2h_m$ maps slot~$k$ to slot $k+2h_m/\tau_m\equiv k+2\pmod4$ and exchanges the two diagonal entries; a shift by~$4h_m$ leaves the matrix unchanged. Its average over every interval of length~$4h_m$ is the same scalar matrix \cite[(3.16)]{AV}: for every $t\in\R$, \begin{equation}\label{e.mean} \fint_t^{t+4h_m}\J_m^\nu(t')\dd t'=\overline K_m(\nu)\,I\,,\qquad \overline K_m(\nu)\coloneqq\frac12\fint_0^{4h_m}\tr\J_m^\nu(t')\dd t'\,. \end{equation} The \emph{permissible diffusivities} are the union $\Kset\coloneqq\bigcup_{M\ge1}\Kset_M$ of the bands \cite[(3.43)]{AV} \begin{equation}\label{e.bands} \Kset_M\coloneqq\bigl[\tfrac12\eps_M^{\beta-\gamma},\,2\eps_M^{\beta-\gamma}\bigr]\,, \end{equation} which are disjoint and accumulate at~$0$. For $\kappa\in\Kset_M$, we call~$M$ the \emph{terminal level}, and the \emph{effective diffusivities} $\kappa_M,\kappa_{M-1},\dots,\kappa_0$ are defined by the recursion \cite[(3.42)]{AV} \begin{equation}\label{e.recursion} \kappa_M\coloneqq\kappa\,,\qquad \kappa_{m-1}\coloneqq\overline K_m(\kappa_m)\,,\quad m=M,M-1,\dots,1\,. \end{equation} Thus~$\kappa_{m-1}$ is the diffusivity the solution feels once the pulses of level~$m$, at diffusivity~$\kappa_m$, have been averaged out. \begin{proposition}[Effective diffusivities and scales]\label{p.scales} There exist $\Lambda_0(\alpha,q,\zeta)\ge128$ and $C(\alpha,q,\zeta)<\infty$ such that, for every $\Lambda\ge\Lambda_0$, $M\ge1$, $\kappa\in\Kset_M$ and $1\le m\le M$, the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item At each level the recursion adds about $\frac9{80}\eps_m^{2\beta}/\kappa_m$: \begin{equation*} \Bigl|\kappa_{m-1}-\kappa_m-\frac9{80}\,\frac{\eps_m^{2\beta}}{\kappa_m}\Bigr|\le C\,\frac{\eps_m^{2\beta}}{\kappa_m}\Bigl(\frac{\eps_m^2}{\kappa_m\tau_m}+\eps_{m-1}^\delta\Bigr). \end{equation*} \item If $m0$, is a weak solution on $(\tin,\tin+T)$. \begin{maintheorem}[Interior regularity]\label{t.A} Let $\alpha\in(0,\nf13)$, $s\in(0,\frac12(1-\alpha))$, $q$ satisfy~\eqref{e.q}, and let $\zeta$ be an even function in $C_c^\infty((-\frac23,\frac23))$ satisfying~\eqref{e.zeta}. There exists $\Lambda_0(\alpha,s,q,\zeta)\ge128$ such that, for every integer $\Lambda\ge\Lambda_0$, the velocity field~$\b$ of Section~\ref{s.construction} has the following property. For every $00$ such that, for every $\tin\in\R$, $\kappa\in\Kset$ and $\thetain\in L^2(\T^2)$, every weak solution~$\theta$ of the initial-value problem \begin{equation}\label{e.ivp.D} \left\{ \begin{aligned} & \partial_t\theta+\b\cdot\nabla\theta-\kappa\Delta\theta=0 & \mbox{in} & \ (\tin,\infty)\times\T^2\,,\\ & \theta(\tin,\cdot)=\thetain & \mbox{on} & \ \T^2 \end{aligned} \right. \end{equation} satisfies \begin{equation}\label{e.dissipation} \kappa\|\nabla\theta\|_{L^2((\tin,\tin+1)\times\T^2)}^2\ge c\,\bigl\|\thetain-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}^2\,, \end{equation} and, for every $t\ge \tin$, \begin{equation}\label{e.decay} \bigl\|\theta(t,\cdot)-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}\le2\exp(-c(t-\tin))\bigl\|\thetain-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}\,. \end{equation} \end{maintheorem} \paperfigure{% \begin{figure}[!htb] \centering \begin{tikzpicture}[x=1cm,y=1cm,font=\small, st/.style={draw,rounded corners=2pt,align=center,inner sep=3pt,minimum height=0.9cm,fill=white}, use/.style={-{Stealth[length=2.2mm]},semithick}, meth/.style={-{Stealth[length=2.2mm]},semithick,densely dashed}, lab/.style={fill=white,inner sep=1pt,font=\footnotesize}] \node[st] (A) at (0,0) {Theorem~\ref{t.A}\\ interior regularity}; \node[st] (D) at (-4.2,0) {Theorem~\ref{t.B}\\ uniform dissipation}; \node[st] (T1) at (-2.7,-1.9) {Proposition~\ref{p.finite}\\ truncated velocity field}; \node[st] (P2) at (1.2,-1.9) {Proposition~\ref{p.transfer}\\ transfer}; \node[st] (L3) at (5.1,-1.9) {Lemma~\ref{l.micro}\\ perturbed heat equations}; \node[st] (P4) at (-5.0,-3.8) {Proposition~\ref{p.entry}\\ initialization}; \node[st] (T5) at (-0.6,-3.8) {Proposition~\ref{p.cascade}\\ induction on the levels}; \node[st,very thick] (T7) at (-0.6,-5.7) {Proposition~\ref{p.passage}\\ passage between\\ consecutive levels}; \node[st,very thick] (T6) at (-0.6,-7.6) {Proposition~\ref{p.cycle}\\ one cycle}; \node[st] (P8) at (-4.4,-9.6) {Proposition~\ref{p.scales}\\ scales}; \node[st] (P9) at (-0.6,-9.6) {Proposition~\ref{p.paired}\\ paired smoothing}; \node[st] (P10) at (3.2,-9.6) {Proposition~\ref{p.energy}\\ energy}; \node[draw,dashed,rounded corners=3pt,fit=(P8)(P9)(P10),inner sep=6pt,label={[font=\small]below:tools}] (tools) {}; \draw[use] (T1) -- (A); \draw[use] (A) -- (D); \draw[use] (P2) -- (A); \draw[use] (L3) -- (P2); \draw[use] (L3.north west) -- (A.east); \draw[use] (P4) -- (T1); \draw[use] (T5) -- (T1); \draw[use] (T7) -- (T5); \draw[use] (T6) -- (T7); \draw[use] (T6.west) to[out=160,in=200] (T5.west); \draw[meth] (T6.south west) -| node[lab,pos=0.25,below] {method} (P4.south); \end{tikzpicture} \caption{The statements and their uses. A solid arrow from one statement to another means that the proof of the second uses the first. The dashed arrow means that the entry repeats the method of the cycle at the terminal level. The two main steps are framed in bold. The dashed frame collects Proposition~\ref{p.scales} and the two tools of Section~\ref{s.tools}, whose uses are not drawn.} \label{f.map} \end{figure}} We compare Theorem~\ref{t.B} with~\cite[Theorem~1.1]{AV}, recalled in Subsection~\ref{ss.idea}, which concerns the velocity field of~\cite{AV} and gives a lower bound for $\limsup_{\kappa\to0}\kappa\|\nabla\theta^\kappa\|_{L^2((0,1)\times\T^2)}^2$ for mean-zero data in~$H^1$. The restriction to the bands and a lower bound for every~$\kappa$ in them are already in~\cite{AV}: the discussion after~\cite[Theorem~1.1]{AV} states the lower bound for every~$\kappa$ in an explicit union of bands, the permissible diffusivities~\cite[(3.43)]{AV}, with a constant depending on the datum. For the velocity field of this paper, Theorem~\ref{t.B} gains three things: the datum is only in~$L^2(\T^2)$; the constant~$c$ is independent of the datum and of the initial time~$\tin$, whereas the constant in~\cite{AV} depends on the ratio $\|\thetain\|_{L^2(\T^2)}/\|\nabla \thetain\|_{L^2(\T^2)}$; and the variance decays exponentially, at a rate independent of~$\kappa$. As a consequence, since the bands accumulate at~$0$, for every mean-zero $\thetain\in L^2(\T^2)$ the weak solutions~$\theta^\kappa$ of~\eqref{e.ivp.D} with $\tin=0$, one for each $\kappa\in\Kset$, which are unique (Section~\ref{s.mainresults}), satisfy \begin{equation*} \liminf_{\kappa\to0,\ \kappa\in\Kset}\kappa\|\nabla\theta^\kappa\|_{L^2((0,1)\times\T^2)}^2\ge\inf_{\kappa\in\Kset}\kappa\|\nabla\theta^\kappa\|_{L^2((0,1)\times\T^2)}^2\ge c\,\|\thetain\|_{L^2(\T^2)}^2\,, \end{equation*} with the constant~$c$ of Theorem~\ref{t.B}: a lower bound on the limit inferior, where~\cite[Theorem~1.1]{AV} bounds the limit superior. \emph{Idea of the proof of Theorem~\ref{t.B}.} Fix~$\tin$ and~$\kappa$. The solution operator~$\mathcal S$, which maps~$\thetain$ to $\theta(\tin+1,\cdot)$, is linear, positive by the maximum principle, and preserves constants and integrals. Theorem~\ref{t.A} on $(\tin,\tin+1)$, with $t_0=\frac12$, bounds it from $L^2(\T^2)$ into $C^{0,s}(\T^2)$ by a constant $C\ge1$ independent of~$\tin$ and $\kappa\in\Kset$. With~$N$ the least integer such that $N^s\ge C$, Lemma~\ref{l.mr.operator} gives $\|\mathcal S\thetain\|_{L^2(\T^2)}^2\le(1-1/(2N^4))\|\thetain\|_{L^2(\T^2)}^2$ for every mean-zero $\thetain\in L^2(\T^2)$, and the energy identity turns this drop into~\eqref{e.dissipation}, with $c=1/(4N^4)$, and into a contraction by $1-2c$ of $\|\theta(t,\cdot)-(\thetain)_{\T^2}\|_{L^2(\T^2)}^2$ over each unit of time, which gives~\eqref{e.decay}. The proof is in Subsection~\ref{ss.mr.D}. % ===== 03-reduction ===== \section{Reduction to a velocity field with finitely many scales} \label{s.reduction} For $\kappa\in\Kset_M$, the levels of~$\b$ much finer than level~$M$ are irrelevant to the solution: their stream functions are much smaller than~$\kappa$, and molecular diffusion averages them out. In divergence form, a perturbation of the stream function of size~$\kappa\varpi$ is a skew-symmetric perturbation of relative size~$\varpi$ of the diffusion matrix~$\kappa I$, and it moves the solution by at most $\varpi\|\thetain\|_{L^2(\T^2)}$ in~$L^2(\T^2)$ (Lemma~\ref{l.mr.stream}, as in the comparison of~$\theta$ with~$\theta_M$ in~\cite[Section~5.4]{AV}). Proposition~\ref{p.transfer} turns this into a bound in~$C^{0,s}(\T^2)$, at a cost which is bounded once the levels finer than $M+n$ are removed, for a fixed number~$n$ of extra levels. We therefore prove the estimate for the smooth velocity field~$\b_{M+n}$, which has finitely many scales (Proposition~\ref{p.finite}, proved in Section~\ref{s.cascade}), and transfer it to~$\b$. \begin{proposition}[Regularity for the truncated velocity field]\label{p.finite} There exists $\Lambda_0(\alpha,s,q,\zeta)\ge128$ such that, for every integer $\Lambda\ge\Lambda_0$, integer $n\ge1$ and $00$ and $C(s)<\infty$ with the following property. Let $\Phi,\widetilde\Phi\colon\R\times\R^2\to\R$ be $\Z^2$-periodic in~$x$, with~$\Phi$ continuous and~$C^1$ in~$x$, $\nabla^\perp\Phi$ bounded and continuous, and~$\widetilde\Phi$ smooth. Let $\kappa>0$, $0s$, the two bounds give the bound in~$C^{0,s}(\T^2)$ at all distances (Step~4 of the proof in Subsection~\ref{ss.mr.transfer}). \begin{lemma}[H\"older estimate for perturbed heat equations]\label{l.micro} There exist $\varpi_1(s)>0$ and $C(s)<\infty$ with the following property. Let $D_1\coloneqq\{x\in\R^2:|x|<1\}$ and $D_{1/2}\coloneqq\{x\in\R^2:|x|<\frac12\}$, and let $[\cdot]_{C^{0,(1+s)/2}_{\mathrm{par}}((-\frac14,0]\times D_{1/2})}$ be the H\"older seminorm on $(-\frac14,0]\times D_{1/2}$ for the parabolic distance $|x-y|+|t-t'|^{\nf12}$. Let~$\mathbf a$ and~$\bstar$ be measurable functions on $(-1,0)\times D_1$ with $\mathbf a-I$ skew-symmetric, $\bstar(t,\cdot)$ divergence free in the sense of distributions for almost every~$t$, and $|\mathbf a-I|+|\bstar|\le\varpi_1$ almost everywhere, and let $U\in L^\infty((-1,0);L^2(D_1))$ with $\nabla U\in L^2((-1,0)\times D_1)$ be a weak solution of \begin{equation*} \partial_tU-\nabla\cdot(\mathbf a\nabla U)+\bstar\cdot\nabla U=0\quad\mbox{in }(-1,0)\times D_1\,. \end{equation*} Then $U(t,\cdot)$ converges in~$L^2(D_{1/2})$ as $t\uparrow0$, and, with this limit as $U(0,\cdot)$, \begin{align*} \lefteqn{[U]_{C^{0,(1+s)/2}_{\mathrm{par}}((-\frac14,0]\times D_{1/2})}+\bigl\|U-(U)_{(-1,0)\times D_1}\bigr\|_{L^\infty((-\frac14,0]\times D_{1/2})}} \qquad & \notag \\ & \le C\Bigl(\fint_{(-1,0)\times D_1}\bigl|U(t,x)-(U)_{(-1,0)\times D_1}\bigr|^2\dd x\dd t\Bigr)^{\nf12}. \end{align*} \end{lemma} \emph{Idea of the proof.} Since $\mathbf a-I$ and~$\bstar$ are small, $U$ is close in~$L^2$ to a caloric function, a solution of the heat equation, on every smaller cylinder, and Campanato's characterization of H\"older continuity by the decay of these $L^2$ distances gives the bound. The proof is in Subsection~\ref{ss.mr.cylinders}. \emph{Idea of the proof of Theorem~\ref{t.A}.} For $\kappa\in\Kset_M$ with~$M$ large, we continue~$\theta$ to $(\tin,\tin+T+1)$, as Proposition~\ref{p.transfer} uses the solutions after~$t_2$, and compare it with the solution~$\widetilde\theta$ of~\eqref{e.ivp.finite}, with $\Phi=\phi$ and $\widetilde\Phi=\phi_{M+n}$; Proposition~\ref{p.finite} bounds~$\widetilde\theta$ in $L^\infty([\tin+\frac12t_0,\tin+T+1];C^{0,s}(\T^2))$. By~\eqref{e.tails} and Proposition~\ref{p.fl.velocity}(ii), we may take~$\varpi$ a multiple of $\eps_{M+n+1}^\beta/\kappa$ and~$r^2$ a small multiple of $\kappa\eps_{M+n}^{2-\beta}$, and a choice of~$n$ depending only on $\alpha,s,q$ makes $\varpi r^{-1-s}$ bounded (Step~2 of the proof in Subsection~\ref{ss.mr.A}). The remaining finitely many bands lie in a compact subinterval of $(0,2]$; there the H\"older continuity of~$\b$ makes the drift small at the scale $\kappa^{1/\beta}$ along its trajectories, and Lemma~\ref{l.micro} applies in a frame moving along a trajectory of~$\b$ (Lemma~\ref{l.mr.cylinders}). % ===== 04-cascade ===== \section{Iterated homogenization} \label{s.cascade} This section reduces Proposition~\ref{p.finite} to four propositions, proved in Part~II, by an induction on the levels of~$\b_{M+n}$, which are homogenized one at a time, from the finest to a fixed coarse level. \subsection{From \texorpdfstring{$L^2$}{L2} to \texorpdfstring{$L^\infty$}{L-infinity}, scale by scale} \label{ss.idea} This subsection is heuristic, except for the definition of the corrector amplitude and the inequality~\eqref{e.mu}, which is proved here and used in the proof; the statements of the proof begin in Subsection~\ref{ss.overview}. \paragraph{Anomalous dissipation in~\cite{AV}.} For the velocity field of~\cite{AV}, that is, for the choice of~$q$ made there, \cite[Theorem~1.1]{AV} states that the scalar variance is dissipated anomalously: for every mean-zero $\thetain\in H^1(\T^2)$ with $\thetain\ne0$, the solutions~$\theta^\kappa$ of \begin{equation*} \left\{ \begin{aligned} & \partial_t\theta^\kappa+\b\cdot\nabla\theta^\kappa-\kappa\Delta\theta^\kappa=0 & \mbox{in} & \ (0,1)\times\T^2\,,\\ & \theta^\kappa(0,\cdot)=\thetain & \mbox{on} & \ \T^2 \end{aligned} \right. \end{equation*} satisfy \begin{equation*} \limsup_{\kappa\to0}\,\kappa\|\nabla\theta^\kappa\|_{L^2((0,1)\times\T^2)}^2>0\,, \end{equation*} with a lower bound depending only on $\|\thetain\|_{L^2(\T^2)}$ and the ratio $\|\thetain\|_{L^2(\T^2)}/\|\nabla \thetain\|_{L^2(\T^2)}$. The proof compares, for $\kappa\in\Kset_M$, the solutions~$\theta_m$ of the equations of level~$m$, with velocity field~$\b_m$, diffusivity~$\kappa_m$ and the same initial datum, and shows that the dissipation $\kappa_m\|\nabla\theta_m\|_{L^2((0,1)\times\T^2)}^2$ is multiplied only by $1+O(\eps_{m-1}^\delta)$ from one level to the next, at all levels finer than a threshold depending on the initial datum \cite[Proposition~5.2]{AV}. \paragraph{The expected regularity.} By the energy identity, the dissipation of each level satisfies $\kappa_m\|\nabla\theta_m\|_{L^2((0,\infty)\times\T^2)}^2\le\frac12\|\thetain\|_{L^2(\T^2)}^2$. Heuristically, $\theta_m$ is the solution seen at the scale~$\eps_m$, with the oscillations of smaller wavelength averaged out, so at the scale~$\eps_m$ the gradient of the solution is of size at most $\kappa_m^{-\nf12}\asymp\eps_m^{-(\beta+\gamma)/2}$ in $L^2((0,\infty)\times\T^2)$, for $1\le m0$ is the condition that~$s$ lies below the exponent in~\eqref{e.heuristic}. \paragraph{Large-scale regularity.} In the quantitative argument of large-scale regularity recalled in Subsection~\ref{ss.strategy}, the homogenization errors are summed over the scales in an iteration of Campanato type, in~$L^2$ in~\cite{AKMbook} and in~$L^\infty$ in~\cite{AS}. Here the comparison is made once per level, on the whole torus. The seminorm $\lambda^{-s}[\,\cdot\,]_{C^{\Nrep}_\lambda(\T^2)}$ (see~\eqref{e.seminorms}) of a solution, bounded by the size of a representation (Subsection~\ref{ss.representations}) at the wavelength~$\lambda$, plays the role of the excess at the scale~$\lambda$, and~\eqref{e.mu} is the cost of one level. \subsection{Overview: two dual cascades} \label{ss.overview} Fix $M$, $\kappa\in\Kset_M$ and a weak solution~$\theta$ of~\eqref{e.ivp.finite}. For each level $m\le M$ the effective diffusivity~$\kappa_m$ was defined so that, once the pulses of level~$m$ are homogenized, the operator $\partial_t+\b_m\cdot\nabla-\kappa_m\Delta$ of level~$m$ becomes the operator of level~$m-1$, with~$\b_{m-1}$ and~$\kappa_{m-1}$. The proof runs two dual cascades on the levels (Figure~\ref{f.induction}). The \emph{homogenization cascade} goes \emph{up the levels} one at a time, from~$M$ to a fixed top level~$m_0$, toward coarser levels and forward in time; the \emph{regularity cascade} then goes back down, toward finer levels, at one fixed time. The proof has four stages, and Proposition~\ref{p.cascade} runs both cascades between the seed and the top level. \paperfigure{% \begin{figure}[!htb] \centering \begin{tikzpicture}[x=1cm,y=0.72cm,font=\small, use/.style={-{Stealth[length=2.2mm]},semithick}, lab/.style={fill=white,inner sep=1pt,font=\footnotesize}] % levels (y), coarser levels higher, and the solutions U_m on [t_(m), t_f] (grey) \foreach \y/\t/\l in {0.2/0/M+n,1.4/0.75/M,2.2/1.5/M-1,3.8/2.4/m,4.6/3.1/m-1,6.2/4.1/m_0+1,7.0/5.0/m_0} { \draw[black!30] (\t,\y) -- (12,\y); \node[anchor=east] at (-0.25,\y) {$\l$};} \foreach \y in {0.75,2.95,5.35} \node at (-0.75,\y) {$\vdots$}; \draw[use] (-1.75,0.3) -- node[midway,above,rotate=90,font=\footnotesize] {coarser levels} (-1.75,6.9); % time axis \draw[-{Stealth[length=2.2mm]}] (-0.2,-0.5) -- (12.7,-0.5) node[right] {$t'$}; \foreach \x/\n in {0/\tin,1.5/\te,7.0/\tin+t_0,9.0/t,12.0/\tf=\tin+T} { \draw (\x,-0.62) -- (\x,-0.38); \node[below] at (\x,-0.62) {$\n$};} \draw[decorate,decoration={brace,amplitude=3pt,mirror}] (1.5,-1.4) -- (5.0,-1.4) node[midway,below=3pt] {$<\frac12t_0$}; % the entry \fill[black!7,rounded corners=2pt] (-0.1,-0.1) rectangle (1.75,2.4); \node[font=\footnotesize] at (0.82,2.7) {entry}; \draw[semithick] (0,0.2) -- (0.3,0.2); \draw[use] (0.3,0.2) -- (0.3,0.58); \draw[densely dotted,semithick] (0.3,0.58) -- (0.75,1.4); \draw[semithick] (0.75,1.4) -- (1.5,1.4); \draw[use] (1.5,1.4) -- (1.5,2.2); \node[lab,anchor=west] at (0.9,0.75) {weak passages}; \node[lab,anchor=west] at (1.6,1.8) {terminal cycle}; \fill (1.5,2.2) circle (1.6pt); \node[lab,anchor=east] at (1.4,2.2) {seed}; % the cascade: cycles step up and forward in time \draw[semithick] (1.5,2.2) -- (1.85,2.2); \draw[use] (1.85,2.2) -- (1.85,2.58); \draw[densely dotted,semithick] (1.85,2.58) -- (2.4,3.8); \draw[semithick] (2.4,3.8) -- (3.1,3.8); \draw[use] (3.1,3.8) -- (3.1,4.6); \node[lab,anchor=west,align=left] at (3.2,4.2) {marked amplitude\\$\times\bigl(1+C\eps_{m-1}^Q+C\eps_{m-1}^\delta\bigr)$}; \draw[semithick] (3.1,4.6) -- (3.5,4.6); \draw[use] (3.5,4.6) -- (3.5,4.98); \draw[densely dotted,semithick] (3.5,4.98) -- (4.1,6.2); \draw[semithick] (4.1,6.2) -- (5.0,6.2); \draw[use] (5.0,6.2) -- (5.0,7.0); % the way back down, at the fixed time t \draw[use,very thick] (9.0,7.0) -- (9.0,6.2); \draw[dotted,semithick] (9.0,6.2) -- (9.0,4.6); \draw[use,very thick] (9.0,4.6) -- (9.0,3.8); \draw[dotted,semithick] (9.0,3.8) -- (9.0,2.2); \draw[-{Stealth[length=2.2mm]},very thick,dashed] (9.0,2.2) -- (9.0,0.2); \node[lab,anchor=west,align=left] at (9.12,4.2) {size $\times\bigl(1+C\eps_{m-1}^\eta\bigr)$\\$+\,C\eps_{m-1}\cdot{}$marked amplitude}; \node[lab,anchor=west] at (9.12,7.0) {smooth at the scale $\rho(m_0+1)$}; \node[lab,anchor=west] at (9.12,1.2) {lift through the entry}; \node[lab,anchor=west] at (9.12,0.2) {$\theta(t)$ in $C^{0,s}(\T^2)$}; \end{tikzpicture} \caption{The induction of Subsection~\ref{ss.overview}, levels against time~$t'$ (not to scale), coarser levels higher. The homogenization cascade, up the levels and forward in time: the entry (Proposition~\ref{p.entry}) takes~$\theta$, at level $M+n$ and time~$\tin$, to the seed, at level $M-1$ and time~$\te$; each up arrow is then a cycle (Proposition~\ref{p.cycle}), with the factor of~\eqref{e.down}, and the cycles end less than~$\frac12t_0$ after~$\te$. Grey: the solution~$U_m$ of each family, on $[t_{(m)},\tf]$. The regularity cascade, back down the levels at one time $t\in[\tin+t_0,\tin+T]$: each down arrow is a passage (Proposition~\ref{p.passage}), with the bound~\eqref{e.up}, and the dashed arrow is the lift through the entry. Dotted: levels not drawn.} \label{f.induction} \end{figure}} \begin{enumerate} \item \emph{The entry} (Proposition~\ref{p.entry}) disposes of the levels $M+n,\dots,M$. At the levels finer than~$M$, the P\'eclet number $\eps_m^\beta/\kappa$ is small and molecular diffusion over the quiet margins smooths at the wavelength of the next coarser level, so these levels are crossed at the same diffusivity~$\kappa$. At level~$M$ a full homogenization cycle passes to~$\kappa_{M-1}$. The output is a system at level~$M-1$, the \emph{seed}, whose solution, lifted back to level $M+n$, reproduces~$\theta$ up to a small error in~$C^{0,s}(\T^2)$. \item \emph{The homogenization cascade} (Proposition~\ref{p.cascade}) then homogenizes the levels $M-1,\dots,m_0+1$ one at a time. Each step up is a homogenization cycle (Proposition~\ref{p.cycle}). \item \emph{At the top level}~$m_0$, a fixed level independent of~$M$ and~$\kappa$, the solution is smooth at a fixed scale. \item \emph{The regularity cascade} (Proposition~\ref{p.cascade}) then goes back down the levels $m_0+1,\dots,M-1$ at one fixed time. Each step down is a passage (Proposition~\ref{p.passage}), and the lifts through the entry (Proposition~\ref{p.entry}) carry the H\"older bound from the seed to~$\theta$. \end{enumerate} \paragraph{What the homogenization cascade carries up the levels.} A cycle compares the solution at level~$m$ with a solution at level~$m-1$, regenerating derivatives and keeping the marked amplitude, defined below, up to the factor in~\eqref{e.down}. The object passed from level~$m$ to level~$m-1$ is a coupled system of equations with the operator of its level, a \emph{family} (Subsection~\ref{ss.families}). Starting from the seed, the cycles produce families~$G_m$, with solutions~$U_m$ on time windows $[t_{(m)},t_{\mathrm{f}}]$ that end at a common final time~$t_{\mathrm{f}}$, where $t_{(M-1)}=\te$ and, for $m0$; \item the bounds on generation and word length of the successors (Subsection~\ref{ss.families}); \item the \emph{top level}~$m_0\ge2$, chosen after the others: the largest of the thresholds on the level in Sections~\ref{s.admissible}--\ref{s.passage} (item~(4) of Subsection~\ref{ss.admissible.parameters}). \end{itemize} Table~\ref{t.admissible.schedule} lists them, with the parameters of Part~II, in the order in which they are chosen. The preparations of level~$m$ are smooth at the effective diffusion length over one renewal time (Section~\ref{s.construction}), \begin{equation}\label{e.rho} \rho(m)=(\kappa_{m-1}h_m)^{\nf12}\,. \end{equation} The notions of family, admissible family, lift and representation below use these choices, families use the effective diffusivities of $\kappa\in\Kset_M$, and time windows have length at most~$T$. \subsection{Families} \label{ss.families} A comparison between levels~$m$ and~$m-1$ leaves error terms carrying derivatives of the coarse solution. Estimating them would cost derivatives at every level, which the induction cannot afford, so they become new unknowns, each solving the equation of its level forced by smoothed derivatives of earlier unknowns. These new sources are the higher-order terms of the averaged equation that the lift of Subsection~\ref{ss.lifts} produces: the defect of the averaged flux, the means of the lift along the paths of sources of the family, and their conjugates by the truncated exponential. They, and the old sources carried over through their cell averages, are listed in Definition~\ref{d.admissible.successor}. \paragraph{Families and their solutions.} A family is the data of a system of linear equations of level~$m$, one for each of its components, coupled through smoothed derivatives; a solution is a vector of functions on $[t_1,t_2]\times\T^2$, one for each component. Precisely, a \emph{family}~$G$ of level~$m$ consists of \begin{itemize} \item a finite index set~$V$, whose elements are the \emph{components} (labels, which are words in the families of the proof, as described below), with \emph{masses} $\omega_v>0$ and \emph{markers} $\Psi_v\ge1$, $v\in V$; \item finitely many \emph{sources}~$\sigma$, each from a component~$v'$ to a component~$v$ (written $\sigma\colon v'\to v$; there may be several sources, or none, from~$v'$ to~$v$), with a linear smoothing operator~$S_\sigma(t)$ of width $r_\sigma>0$, and coefficients~$A_{\sigma,d}$, $2\le d\le\Nsrc$; \item constants $c_j\ge0$ such that the operators~$S_\sigma(t)$, which commute with derivatives, satisfy, for every source~$\sigma$, time~$t$, $f\in L^2(\T^2)$ and $j\le\Nsrc-2$, \begin{equation*} \|\nabla^jS_\sigma(t)f\|_{L^2(\T^2)}\le c_jr_\sigma^{-j}\|f\|_{L^2(\T^2)}\,. \end{equation*} \end{itemize} A \emph{datum} of~$G$ is $g=(g_v)_{v\in V}\in L^2(\T^2)^V$, and the \emph{solution} of~$G$ from~$g$ at time~$t_1$, on a time window $[t_1,t_2]$ on which the hypotheses of Proposition~\ref{p.energy} hold, is the solution $U=(U_v)_{v\in V}$ of \begin{equation}\label{e.family.system} \left\{ \begin{aligned} & \partial_tU_v+\b_m\cdot\nabla U_v-\kappa_m\Delta U_v\\ & \qquad=\nabla\cdot\sum_{v'\in V}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}A_{\sigma,d}\nabla^{d-1}(S_\sigma U_{v'}) & \mbox{in} & \ (t_1,t_2)\times\T^2\,,\\ & U_v(t_1,\cdot)=g_v & \mbox{on} & \ \T^2\,, \end{aligned} \right. \qquad v\in V\,, \end{equation} in the class of Proposition~\ref{p.energy}, where the inner sum runs over the sources from~$v'$ to~$v$, so that the right side collects all the sources into~$v$; this is~\eqref{e.energy.system} with $\phi_m$ and~$\kappa_m$ in place of~$\Phi$ and~$\nu$. The smoothing operators are products of two smoothings along ellipses (Proposition~\ref{p.paired}). For the sources that a cycle creates at level~$m$ the ellipses are thin. The averaged flux of a pulse is enhanced only along its direction of flow, and the two ellipses follow the averaged flux of a horizontal and of a vertical pulse, so that their axes are in a ratio of order $(\kappa_m/\kappa_{m-1})^{\nf12}$, which is small below the terminal level~$M$ (Proposition~\ref{p.scales}(c)); together they smooth in every direction. \paragraph{Norms.} Fields of~$G$ are measured in the norms \begin{equation}\label{e.norms} \begin{aligned} \|g\|_{L^2_\omega(\T^2)}^2&\coloneqq\sum_{v\in V}\frac{\|g_v\|_{L^2(\T^2)}^2}{\omega_v}\,,\\ \|g\|_{L^2_G(\T^2)}^2&\coloneqq\sum_{v\in V}\frac{\Psi_v^2\,\|g_v\|_{L^2(\T^2)}^2}{\omega_v}\,,\\ \|g\|_{H^N_{\rho,G}(\T^2)}^2&\coloneqq\sum_{v\in V}\frac{\Psi_v^2\,\|g_v\|_{H^N_\rho(\T^2)}^2}{\omega_v}\,, \end{aligned} \end{equation} where, for $\rho>0$ and an integer $N\ge0$, \begin{equation}\label{e.HNrho} \|f\|_{H^N_\rho(\T^2)}^2\coloneqq\sum_{j\le N}\frac{\rho^{2j}}{(j!)^2}\,\|\nabla^jf\|_{L^2(\T^2)}^2 \end{equation} measures~$N$ derivatives at the length scale~$\rho$. The norm $\|\cdot\|_{L^2_G(\T^2)}$ is the \emph{marked norm}. \paragraph{The coupling matrix.} The coupling of the components is measured by the \emph{coupling matrix} of~$G$ on a time window $[t_1,t_2]$, \begin{equation}\label{e.coupling} B_{vv'}\coloneqq\Bigl(\frac{\omega_{v'}}{\omega_v}\Bigr)^{\nf12}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}c_{d-2}\,\frac{\sup_{[t_1,t_2]\times\T^2}|A_{\sigma,d}|}{\kappa_m\,r_\sigma^{d-2}}\,,\qquad v,v'\in V\,, \end{equation} with the constants~$c_j$ of~$G$. By Schur's test, the operator norm of $(B_{vv'})$ on~$\R^V$ with the Euclidean norm is at most the geometric mean of its largest row sum and its largest column sum. By Proposition~\ref{p.energy}, if all the row and column sums of the coupling matrix are at most~$\frac18$, then the energy $\|U(t)\|_{L^2_\omega(\T^2)}$ does not grow. Indeed, testing the equation for~$U_v$ with~$U_v/\omega_v$ and writing $\nabla^{d-1}S_\sigma U_{v'}=\nabla^{d-2}S_\sigma\nabla U_{v'}$, we bound the contribution of the sources from~$v'$ to~$v$ by $\kappa_mB_{vv'}x_vx_{v'}$, where $x_v\coloneqq\omega_v^{-\nf12}\|\nabla U_v\|_{L^2(\T^2)}$. By Schur's test the coupling is then at most one eighth of the dissipation $\kappa_m\sum_{v\in V}x_v^2$. The coupling matrix \emph{for the masses} $\omega_v/\Psi_v^2$ is~\eqref{e.coupling} with~$\omega_v/\Psi_v^2$ in place of~$\omega_v$; with these masses the norm $\|\cdot\|_{L^2_\omega(\T^2)}$ is the marked norm. \paragraph{Successors.} A cycle turns a family~$G$ of level~$m$ into a \emph{successor}~$G'$ of level~$m-1$. Each component~$v$ of~$G$ is kept and becomes the \emph{parent} of new components, its \emph{children}, labelled by words whose letters name the source terms the comparison creates. A new component has mass~$\omega_v$ times a positive power of~$\eps_{m-1}$ and marker~$\Psi_v$ times a negative power of~$\eps_{m-1}$, so the markers grow with the generation (the number of creations). A successor is a family of Definition~\ref{d.admissible.successor}, which gives its letters, the words of its components and its sources; it keeps the components of bounded generation and bounded word length. The solution is controlled in the marked norm, so the components of high generation are small in $\|\cdot\|_{L^2_\omega(\T^2)}$ and the flux into the discarded ones is negligible (Lemma~\ref{l.cmp.frontier}); components with longer words are small because each letter carries a positive power of~$\eps_{m-1}$. \paragraph{Seeds and admissible families.} A \emph{seed} is a family whose components have generation~$0$ and marker~$1$. A family~$G$ of level~$m$ is \emph{admissible} on a time window if it descends by cycles from an \emph{admissible seed}, a seed whose coefficients obey the bounds of Part~II (Definition~\ref{d.admissible.seed}), each cycle, at a level~$m'$, moving the start of the window forward by~$3\tau_{m'}$ (Definition~\ref{d.admissible.admissible}). Its coefficients then obey the bounds of Part~II (Proposition~\ref{p.admissible.bounds}), and the row and column sums of its coupling matrices, for the masses~$\omega_v$ and~$\omega_v/\Psi_v^2$, are at most a constant times a fixed positive power of~$\eps_{m-1}$. \subsection{Lifts} \label{ss.lifts} Let~$G'$ be a successor of a family~$G$ of level~$m$. The \emph{lift}~$Y^\sharp$ of a field~$Y$ of~$G'$ to the components of~$G$ is built in three steps: \begin{equation}\label{e.lift} (\Upsilon_mY)_v\coloneqq Y_v+\sum_{v'\text{ child of }v}\eps_{m-1}^{\eta\,\#v'}\,Y_{v'}\,,\qquad Y^\sharp(t)\coloneqq\Corr_m\bigl[\expp\bigl(\Gamma_m(t):\nabla^2\bigr)\,\Upsilon_mY(t)\bigr], \end{equation} where the sum runs over the children of~$v$ (Subsection~\ref{ss.families}), $\#v'$ is the number of letters of the word of~$v'$, $\eta$ is the exponent of~\eqref{e.exponents}, and~$\Corr_m$ is the corrector operator of Definition~\ref{d.cell.corr}, described below. First, the readout~$\Upsilon_m$ adds each new component to its parent with a weight that is a positive power of~$\eps_{m-1}$. Second, the truncated exponential of $\Gamma_m(t):\nabla^2\coloneqq\sum_{i,i'}(\Gamma_m)_{ii'}(t)\partial_i\partial_{i'}$ is applied, where \begin{equation}\label{e.covariance} \expp\bigl(\pm\Gamma_m(t):\nabla^2\bigr)\coloneqq\sum_{l=0}^p\frac1{l!}\bigl(\pm\Gamma_m(t):\nabla^2\bigr)^l\,,\qquad \Gamma_m(t)\coloneqq\int_0^t\bigl(\J_m^{\kappa_m}(t')-\kappa_{m-1}I\bigr)\dd t'\,. \end{equation} The \emph{covariance}~$\Gamma_m(t)$ of the level-$m$ pulses is the accumulated surplus of their averaged flux~$\J_m^{\kappa_m}$ over its average~$\kappa_{m-1}I$. Since $\partial_t\Gamma_m(t)=\J_m^{\kappa_m}(t)-\kappa_{m-1}I$ and operators with constant coefficients in space commute, the full exponential $\exp(\Gamma_m(t):\nabla^2)$ would turn solutions of $\partial_t-\kappa_{m-1}\Delta$ into solutions of $\partial_t-\J_m^{\kappa_m}(t):\nabla^2$; as $\Gamma_m(t)$ is symmetric and in general indefinite, this operator is unbounded on~$L^2(\T^2)$, so the lift truncates it. The covariance is small at the scale~$\rho(m)$: by Lemmas~\ref{l.cell.covariance}(b) and~\ref{l.cmp.conjugation}(iii), for every~$t$, \begin{align*} |\Gamma_m(t)| & \le C\kappa_{m-1}\tau'_m\le C\eps_{m-1}^\delta\rho(m)^2\,,\\ \bigl\|\bigl(\expp(\pm\Gamma_m(t):\nabla^2)-I\bigr)F\bigr\|_{H^N_{\rho(m)}(\T^2)} & \le C\eps_{m-1}^\delta\|F\|_{H^{N+2p}_{\rho(m)}(\T^2)}\,; \end{align*} this is the origin of the term~$\eps_{m-1}^\delta$ in the first bound of Proposition~\ref{p.cycle}(b). Third, $\Corr_m$ (Definition~\ref{d.cell.corr}) adds on each pulse the correctors of degrees up to~$\Ncor$, which oscillate at the wavelength~$\eps_m$ in the Lagrangian coordinates of the pulse; they change a field by the corrector amplitude $\ebar$ times its derivatives. The \emph{retained trace} at time~$\tpr$ of a datum~$g$ of~$G$ is the datum of~$G'$ given by the truncated inverse of the second step, exact up to terms of degree at least~$p+1$ in~$\Gamma_m(\tpr):\nabla^2$: it is $\expp(-\Gamma_m(\tpr):\nabla^2)\,g_v$ on each component~$v$ of~$G$ and~$0$ on the new components. \subsection{Representations} \label{ss.representations} Fields~$U$ of a family take values in~$\R^V$, normed pointwise by $|U(x)|_\omega\coloneqq(\sum_v\omega_v^{-1}U_v(x)^2)^{\nf12}$, and~$\|\cdot\|_{L^\infty(\T^2)}$ denotes the supremum of this pointwise norm; the H\"older norm and seminorm~\eqref{e.holdernorm} of such fields use it in place of~$|\cdot|$. For an integer $N\ge1$, $0<\lambda\le1$ and $F\in C^N(\T^2)^V$ we use the norm and seminorm at the scale~$\lambda$ \begin{equation}\label{e.seminorms} [F]_{C^N_\lambda(\T^2)}\coloneqq\sum_{1\le|j|\le N}\lambda^{|j|}\,\frac{\|\partial^jF\|_{L^\infty(\T^2)}}{j!}\,,\qquad \|F\|_{C^N_\lambda(\T^2)}\coloneqq\|F\|_{L^\infty(\T^2)}+[F]_{C^N_\lambda(\T^2)}\,, \end{equation} where $j=(j_1,j_2)$ runs over multi-indices, $|j|\coloneqq j_1+j_2$ and $j!\coloneqq j_1!\,j_2!$. A \emph{representation}~$R$ of~$U$ at wavelength~$\lambda_*$ is a decomposition into functions in $C^{\Nrep}(\T^2)^V$, a \emph{base}~$E$ and a finite collection~$\mathcal P$ of \emph{profiles}~$P$ (repetitions allowed), each with a wavelength $\lambda_P\in[\lambda_*,1]$, such that $U=E+\sum_{P\in\mathcal P}P$. Its \emph{size} is \begin{equation}\label{e.representation} \size_{\lambda_*}(R)\coloneqq\|E\|_{L^\infty(\T^2)}+[E]_{C^{0,s}(\T^2)}+\lambda_*^{-s}[E]_{C^{\Nrep}_{\lambda_*}(\T^2)}+\sum_{P\in\mathcal P}\lambda_P^{-s}\|P\|_{C^{\Nrep}_{\lambda_P}(\T^2)}\,. \end{equation} Then $\|U\|_{C^{0,s}(\T^2)}\le3\size_{\lambda_*}(R)$, by Lemma~\ref{l.II8.rep}(a). The readout~$\Upsilon_m$, like every constant linear map, maps a representation~$R$ of a field~$Y$ of~$G'$ to a representation of~$\Upsilon_mY$ of size at most $|\Upsilon_m|\size_{\lambda_*}(R)$, where $|\Upsilon_m|$ is its operator norm for the norms~$|\cdot|_\omega$. Each profile keeps its wavelength: measured at $\lambda_*<\lambda_P$, its term $\lambda_P^{-s}\|P\|_{L^\infty(\T^2)}$ would grow by the factor $(\lambda_P/\lambda_*)^s$. \subsection{Initialization and induction} \label{ss.entry} \begin{proposition}[Initialization from rough data]\label{p.entry} There exist $M_*(\deps)\in\N$ and $C(\deps)<\infty$ with the following property. For every $M\ge M_*$, $\kappa\in\Kset_M$ and $\tin\in\R$, with $t_{\mathrm{f}}\coloneqq \tin+T$, there exist $t_{\mathrm{e}}\in[\tin,\tin+\frac14t_0]$ and an admissible seed~$G$ of level~$M-1$ on $[t_{\mathrm{e}},t_{\mathrm{f}}]$ such that the following holds. For every $\thetain\in L^2(\T^2)$ and every weak solution~$\theta$ of~\eqref{e.ivp.finite}, there exists a datum~$g$ of~$G$ with $\|g\|_{L^2_\omega(\T^2)}\le C\|\thetain\|_{L^2(\T^2)}$ such that the solution~$U$ of~$G$ from~$g$ at time~$t_{\mathrm{e}}$ has the following property: for every interval $[t_1,t_2]\subset[\tin+\frac14t_0,t_{\mathrm{f}}]$ and every $\Xi\ge0$, if, for almost every $t\in[t_1,t_2]$, there exists a representation~$R(t)$ of~$U(t)$ at wavelength~$\eps_{M-1}$ with $\size_{\eps_{M-1}}(R(t))\le\Xi$, then \begin{equation*} \|\theta\|_{L^\infty([t_1,t_2];C^{0,s}(\T^2))}\le C\bigl(\Xi+\|\thetain\|_{L^2(\T^2)}\bigr). \end{equation*} \end{proposition} \emph{Idea of the proof.} At the levels $m\in(M,M+n]$ the P\'eclet number $\eps_m^\beta/\kappa$ is at most $C\eps_{m-1}^{(q+2)\gamma}$, and molecular diffusion over a quiet margin reaches the wavelength~$\eps_{m-1}$ of the next coarser level: $\eps_{m-1}^2/(\kappa\tau'_m)\le C\eps_{m-1}^{\gamma-3\delta}$, with $\gamma>3\delta$. So a \emph{weak passage} compares the solution with a family of level~$m-1$ at the same diffusivity, which keeps the enhancement of level~$m$ as a source instead of adding it to the diffusivity, and regains derivatives on the margins. After~$n$ of them the sources are smoothed once, by isotropic paired convolutions of width slightly below~$\eps_M$. At level~$M$ a margin is too short, $\kappa\tau'_M\asymp\eps_{M-1}^{2+\gamma+3\delta}\ll\eps_{M-1}^2$, and a \emph{terminal cycle}, prepared over a time of order~$h_M$ by the method of Proposition~\ref{p.cycle}, passes to $\kappa_{M-1}\in[\kappa,C\kappa]$ and yields the seed. For Proposition~\ref{p.finite}, $\theta$ is, up to $C\eps_{M-1}^2\|\thetain\|_{L^2(\T^2)}$ in $L^\infty([\tin+\frac14t_0,\tin+T];C^{0,s}(\T^2))$, the lift of~$U$ through the terminal cycle and the~$n$ weak passages (variants of~\eqref{e.lift}, defined in Section~\ref{s.entry}), and these lifts carry a representation at~$\eps_{M-1}$ to one at~$\eps_{M+n}$ with fewer than~$\Nrep$ derivatives, at a constant cost. \begin{proposition}[Induction on the levels]\label{p.cascade} There exists $C(\deps)<\infty$ with the following property. Let $m_00$, the operator \begin{equation}\label{e.Srh} S_{\ell,r,\Omega}g(y)\coloneqq\int_{\R^2}W_\ell(z)\,g(y-r\Omega z)\dd z \end{equation} takes the mean of~$g$ over the ellipse $\{y-r\Omega z:|z|<1\}$ (a segment or a point if~$\Omega$ is singular) with the signed weight~$W_\ell$. \begin{proposition}[Paired smoothing in eccentric geometry]\label{p.paired} Let $\ell\ge1$ and $1\le N_1\le N_2$ be integers with $N_1<2\ell$. There exists $C(u,\ell,N_1,N_2)<\infty$ with the following property. Let $C_\Omega>0$, and let~$\Omega_1,\Omega_2$ be invertible linear maps of~$\R^2$ with $|\Omega_i|_\infty\le C_\Omega$ whose vectors $\Omega_1e_1$, $\Omega_2e_2$ are uniformly transverse, in the sense that the matrix with these columns has an inverse of $|\cdot|_\infty$-norm at most~$2$. Then, for every $\rho>0$, $00$. The constant of the third bound depends only on~$u$ and~$N_1$. \end{proposition} Two ellipses whose vectors $\Omega_1e_1$, $\Omega_2e_2$ are transverse together smooth in every direction, whatever their eccentricity; in the applications these vectors are the long axes. For the sources of a seed the ellipses are round discs. \emph{Idea of the proof.} Each~$\partial_i$ is a combination of the derivatives along~$\Omega_1e_1$ and~$\Omega_2e_2$, and along each of these vectors the corresponding smoothing gains one power of~$\rho/r$ per derivative, while its vanishing moments give the approximation. \begin{proposition}[Energy of coupled systems]\label{p.energy} Let $\nu>0$ and $t_10$. Let finitely many sources $\sigma\colon v'\to v$, $v,v'\in V$, be given, with coefficients~$A_{\sigma,d}$, $2\le d\le\Nsrc$, that are Borel measurable in~$t$ and bounded, continuous and periodic in~$x$, and linear operators~$S_\sigma(t)$ on $L^2(\T^2)$ that commute with derivatives and such that $t\mapsto S_\sigma(t)g\in L^2(\T^2)$ is Borel measurable for every $g\in L^2(\T^2)$. Let constants $c_j\ge0$ and widths $r_\sigma>0$ be such that, for every~$\sigma$, $t\in[t_1,t_2]$, $g\in L^2(\T^2)$ and $j\le\Nsrc-2$, $\|\nabla^jS_\sigma(t)g\|_{L^2(\T^2)}\le c_jr_\sigma^{-j}\|g\|_{L^2(\T^2)}$. Consider the problem \begin{equation}\label{e.energy.system} \left\{ \begin{aligned} & \partial_tU_v+\nabla^\perp\Phi\cdot\nabla U_v-\nu\Delta U_v\\ & \qquad=\nabla\cdot\sum_{v'\in V}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}A_{\sigma,d}\nabla^{d-1}(S_\sigma U_{v'}) & \mbox{in} & \ (t_1,t_2)\times\T^2\,,\\ & U_v(t_1,\cdot)=g_v & \mbox{on} & \ \T^2\,, \end{aligned} \right. \qquad v\in V\,. \end{equation} If all row and column sums of the coupling matrix \begin{equation*} B_{vv'}\coloneqq\Bigl(\frac{\omega_{v'}}{\omega_v}\Bigr)^{\nf12}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}c_{d-2}\,\frac{\sup_{[t_1,t_2]\times\T^2}|A_{\sigma,d}|}{\nu\,r_\sigma^{d-2}} \end{equation*} are at most~$\frac18$, then, for every datum $g=(g_v)_{v\in V}\in L^2(\T^2)^V$, there exists a unique solution $U\in C([t_1,t_2];L^2(\T^2)^V)\cap L^2((t_1,t_2);H^1(\T^2)^V)$ of~\eqref{e.energy.system} (in the integrated sense of~\eqref{e.ft.weak}, without forcing), and, for every $t',t\in[t_1,t_2]$ with $t'\le t$, \begin{equation*} \|U(t)\|_{L^2_\omega(\T^2)}^2+\nu\,\|\nabla U\|_{L^2((t',t);L^2_\omega(\T^2))}^2\le\|U(t')\|_{L^2_\omega(\T^2)}^2\,. \end{equation*} \end{proposition} \emph{Idea of the proof.} We test the equation for~$U_v$ with $U_v/\omega_v$ and bound the coupling by Schur's test, as after~\eqref{e.coupling}. Its analogue at higher order, Proposition~\ref{p.ft.energy}, gives the derivatives of~$Y$ used after Proposition~\ref{p.cycle} and the $C^{\Nrep}$ regularity in Proposition~\ref{p.passage}. % ===== 06-flows ===== \part{The proofs} \label{part.proofs} \section{Properties of the velocity field: flows and effective diffusivities} \label{s.flows} This section proves the properties of the construction that Section~\ref{s.construction} states without proof, and Proposition~\ref{p.scales}. Other sections use the flows of Subsection~\ref{ss.fl.flows}: the short flows carry the tubes of Theorem~\ref{t.C}, and the renewal charts are the coordinates of Sections~\ref{s.cell}--\ref{s.comparison}. Throughout, $\Lambda\ge128$ is an integer, and~$C$ and~$c$ denote positive constants depending only on $\alpha,q,\zeta$, unless another dependence is stated; they may change from line to line. Matrices and arrays are normed as in Section~\ref{s.construction}. \subsection{Scales and cutoffs} \label{ss.fl.scales} \begin{lemma}[Scales and cutoffs] \label{l.fl.scales} For every integer $m\ge1$ the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item \emph{Wavelengths.} $\eps_{m-1}/\eps_m\ge\Lambda$ and $\eps_m\le\Lambda^{-m}$. If $m\ge2$, then $\frac12\eps_{m-1}^q\le\eps_m\le2\eps_{m-1}^q$. \item \emph{Times.} $4\eps_{m-1}^{-\delta}\le4\lceil\eps_{m-1}^{-\delta}\rceil+1\le9\eps_{m-1}^{-\delta}$; \eqref{e.times} and~\eqref{e.renewal} hold, and $\eps_m\le2^{35}\tau_m$. \item \emph{Exponents.} $q(\beta-\gamma)=\beta+\gamma$, $q(2-\beta-\gamma)=2-\beta+8\delta$, $(q-1)(\beta+\gamma)=2q\gamma$, $3\delta<\gamma$, and $q-(2-\beta+4\delta)=\alpha+(q+\frac12)\gamma>0$. \item \emph{Cutoffs.} Let $k,l\in\Z$ and let $j\ge0$ be an integer. There exist $C(\zeta,j)$ and~$C(j)$ such that $\|\partial_t^j\zeta_{m,k}\|_{L^\infty(\R)}\le C(\zeta,j)\tau_m^{-j}$ and $\|\partial_t^j\widehat\zeta_{m,l}\|_{L^\infty(\R)}\le C(j)(\tau'_m)^{-j}$. The fast cutoff~$\zeta_{m,k}$ vanishes outside $[(k-\frac23)\tau_m,(k+\frac23)\tau_m]$. At each time at most two fast cutoffs are nonzero, and at most one with~$k$ odd. The slow cutoff satisfies $0\le\widehat\zeta_{m,l}\le1$ and \begin{equation}\label{e.fl.slow} \mathbf 1_{[(l-\frac12)h_m+2\tau'_m,\,(l+\frac12)h_m-2\tau'_m]}\le\widehat\zeta_{m,l}\le\mathbf 1_{((l-\frac12)h_m+\tau'_m,\,(l+\frac12)h_m-\tau'_m)}\,, \end{equation} where~$\mathbf 1_E$ is the indicator of a set~$E$. The slot $[(k-\frac12)\tau_m,(k+\frac12)\tau_m]$ lies in the block $[(l_{m,k}-\frac12)h_m,(l_{m,k}+\frac12)h_m]$. \end{enumerate} Finally, for $\alpha,s>0$ a number~$q$ satisfying~\eqref{e.q} exists if and only if $\alpha<\frac13$ and $s<\frac12(1-\alpha)$; and for $q>1$, $\delta>0$ if and only if $q<\frac{1-\alpha}{2\alpha}$, while $\mu>0$ if and only if $q<\frac{1-s}{\alpha+s}$. \end{lemma} \begin{proof} (a) Let $m\ge1$. Since $\Lambda^{q^m/(q-1)}\ge\Lambda^{1/(q-1)}\ge\log\Lambda/(q-1)$ and $\Lambda^{q^m-1}-1\ge(q-1)\log\Lambda$, \begin{equation*} \Lambda^{q^{m+1}/(q-1)}-\Lambda\cdot\Lambda^{q^m/(q-1)}=\Lambda\cdot\Lambda^{q^m/(q-1)}\bigl(\Lambda^{q^m-1}-1\bigr)\ge\Lambda(\log\Lambda)^2\ge\Lambda\,. \end{equation*} So $\Lambda\eps_m^{-1}\le\Lambda(\Lambda^{q^m/(q-1)}+1)\le\eps_{m+1}^{-1}$ by~\eqref{e.scales}; with $\eps_0=1$ and $\eps_1^{-1}\ge\Lambda^{q/(q-1)}\ge\Lambda$, the first two bounds follow. For $m\ge2$, $\Lambda^{q^{m-1}/(q-1)}\ge128$, so its ceiling is at most $\frac{129}{128}$ times it, and $(\frac{129}{128})^q<2$ gives $\eps_{m-1}^{-q}\le2\Lambda^{q^m/(q-1)}$. Hence $\eps_m\le\Lambda^{-q^m/(q-1)}\le2\eps_{m-1}^q$, and $\eps_m\ge\frac12\Lambda^{-q^m/(q-1)}\ge\frac12\eps_{m-1}^q$, since the ceiling of a number at least~$1$ is at most twice the number. (b) By~\eqref{e.scales}, $h_m=2^{-25}/\lceil\eps_{m-1}^{-(2-\beta+2\delta)}\rceil$, and~$\tau'_m$ and~$\tau_m$ are $h_m$ divided once and twice by $4\lceil\eps_{m-1}^{-\delta}\rceil+1$, so the bounds $y\le\lceil y\rceil\le2y$ for $y\ge1$ give the first bounds, \eqref{e.times}, $h_m\le2^{-25}\eps_{m-1}^{2-\beta+2\delta}$, which is~\eqref{e.renewal}, and $\tau_m\ge2^{-26}\eps_{m-1}^{2-\beta+4\delta}/81$. For $m\ge2$, (a) and~(c) then give $\eps_m\le2\eps_{m-1}^q\le2\eps_{m-1}^{2-\beta+4\delta}\le2^{35}\tau_m$; for $m=1$, $\eps_1\le\Lambda^{-1}\le2^{35}\tau_1$. (c) and the equivalences are algebra on~\eqref{e.q} and~\eqref{e.exponents}. (d) follows from~\eqref{e.zeta}, \eqref{e.fast}, \eqref{e.slow} and the properties of~$\mathsf H$; the centres of odd slots are $2\tau_m$ apart, and~$\zeta_{m,k}$ has support of length $\frac43\tau_m$. Finally, $h_m/\tau_m$ is the odd integer $(4\lceil\eps_{m-1}^{-\delta}\rceil+1)^2$, so $k+\frac12(h_m/\tau_m-1)$ is an integer, and~\eqref{e.label} gives $l_{m,k}h_m-\frac12(h_m-\tau_m)\le k\tau_m\le l_{m,k}h_m+\frac12(h_m-\tau_m)$. \end{proof} In particular, the envelope $\widetilde\zeta_{m,k}=\widehat\zeta_{m,l_{m,k}}\zeta_{m,k}$ of slot~$k$ vanishes outside the block of~$l_{m,k}$ and within~$\tau'_m$ of its ends. Cutoffs as in~\eqref{e.zeta} exist: for $0\frac23$ gives $\Lambda^{2-\beta}\ge128^{2/3}>16$. So every entry of $\nabla^2\phi_m$ is at most $\frac{16}{15}\cdot36500\,\eps_m^{\beta-2}\le39000\,\eps_m^{\beta-2}$ in absolute value. The norm of a $2\times2$ matrix is at most twice its largest entry, and~$\nabla\b_m$ is~$\nabla^2\phi_m$ composed with a rotation, which gives~\eqref{e.fl.hessian}. (iii) By Lemma~\ref{l.fl.increments} and $\eps_{m'+1}\le\Lambda^{-1}\eps_{m'}$, the sums $\sum_{m'>m}|\phi_{m'}-\phi_{m'-1}|$ and $\sum_{m'>m}|\b_{m'}-\b_{m'-1}|$ are at most $10\eps_{m+1}^\beta/(1-\Lambda^{-\beta})$ and $2\cdot320\,\eps_{m+1}^\alpha/(1-\Lambda^{-\alpha})$, uniformly in~$(t,x)$. Hence~$\phi_m$ and~$\b_m=\nabla^\perp\phi_m$ converge uniformly, and~\eqref{e.tails} holds. The limit~$\phi$ is continuous; since~$\phi_m(t,\cdot)$ and its gradient converge uniformly, $\phi(t,\cdot)$ is~$C^1$ with $\nabla^\perp\phi=\b$; and the periodicities persist. (iv) By~(iii), $\b$ is continuous and periodic, and $|\b|\le\sum_{m'\ge1}640\,\eps_{m'}^\alpha\le C$. Each $\b_m(t,\cdot)$ is divergence free, and this passes to the uniform limit in the sense of distributions. For~\eqref{e.holder}, let $t,t'\in\R$ and $x,y\in\R^2$. If $(t,x)=(t',y)$ there is nothing to prove, and if $|t-t'|+|x-y|\ge1$, \eqref{e.holder} follows from the bound on~$\b$. Otherwise, let~$\overline m\ge1$ be the least level with $\eps_{\overline m}\le|t-t'|+|x-y|$. We write $\b=\sum_{m'\ge1}(\b_{m'}-\b_{m'-1})$, bound the terms with $m'<\overline m$ by~\eqref{e.fl.lipschitz} and those with $m'\ge\overline m$ by~\eqref{e.fl.increments} at each of the two points: \begin{align*} |\b(t,x)-\b(t',y)|&\le C\bigl(|t-t'|+|x-y|\bigr)\sum_{1\le m'<\overline m}\eps_{m'}^{\beta-2}+1280\sum_{m'\ge\overline m}\eps_{m'}^\alpha\\ &\le C\bigl(|t-t'|+|x-y|\bigr)\eps_{\overline m-1}^{\alpha-1}+C\eps_{\overline m}^\alpha\le C\bigl(|t-t'|+|x-y|\bigr)^\alpha\,, \end{align*} since $\beta-2=\alpha-1$, the two sums are geometric (Lemma~\ref{l.fl.scales}(a)) and $\eps_{\overline m}\le|t-t'|+|x-y|<\eps_{\overline m-1}$; for $\overline m=1$ the first sum is empty. This is~\eqref{e.holder}. \end{proof} \subsection{Flows and renewal charts} \label{ss.fl.flows} \begin{lemma}[Short flows] \label{l.fl.short} Let $m\ge0$ and let $t,t_1\in\R$ satisfy $|t-t_1|\,\eps_m^{\beta-2}\le2^{-22}$. Then $y\mapsto X_m(t,y,t_1)$ is a smooth, volume-preserving diffeomorphism of~$\R^2$ commuting with the translations by~$\Z^2$, with inverse $x\mapsto X_m(t_1,x,t)$, and, for every $x,y\in\R^2$, \begin{equation}\label{e.fl.short} |\nabla_yX_m(t,y,t_1)-I|\le\tfrac12\,,\qquad |\nabla_xX_m(t_1,x,t)-I|\le\tfrac12\,; \end{equation} in particular both maps are $2$-Lipschitz. This applies when $|t-t_1|\le8h_{m+1}$, by~\eqref{e.renewal}. \end{lemma} \begin{proof} By Proposition~\ref{p.fl.velocity}(i), $\b_m$ is smooth, bounded, $\Z^2$-periodic and divergence free, so the two maps are smooth, mutually inverse by the group law of the flow, commute with the translations by~$\Z^2$ by uniqueness, and preserve Lebesgue measure by Liouville's theorem. The derivative solves $\partial_t\nabla_yX_m(t,y,t_1)=\nabla\b_m(t,X_m(t,y,t_1))\,\nabla_yX_m(t,y,t_1)$, so Gr\"onwall's inequality and~\eqref{e.fl.hessian} give \begin{equation*} |\nabla_yX_m(t,y,t_1)-I|\le\exp\bigl(78000\,\eps_m^{\beta-2}|t-t_1|\bigr)-1\le\exp\bigl(\tfrac1{50}\bigr)-1<\tfrac12\,, \end{equation*} and likewise for $\nabla_xX_m(t_1,x,t)$. \end{proof} \paragraph{Complex extensions.} Section~\ref{s.admissible} bounds the derivatives of its families in their parameters by Cauchy's estimates, which needs holomorphic extensions of the flows to complex labels; they come from those of the velocity fields. For a vector of~$\mathbb C^2$, $\operatorname{Im}$ is its imaginary part, taken coordinatewise, $|\cdot|_1$ is the sum of the moduli of its coordinates, and $|\cdot|_\infty$ is as in Section~\ref{s.construction}, on vectors and on matrices. On the strips $\{|\operatorname{Im}x|_10$. Let $F\in L^2((t_1,t_2)\times\T^2;\R^2)$, $F^0\in L^2((t_1,t_2)\times\T^2)$ and $g\in L^2(\T^2)$. A function~$\theta$ solves \begin{equation*} \left\{ \begin{aligned} & \partial_t\theta+\b_m\cdot\nabla\theta-\nu\Delta\theta=\nabla\cdot F+F^0 & \mbox{in} & \ (t_1,t_2)\times\T^2\,,\\ & \theta(t_1,\cdot)=g & \mbox{on} & \ \T^2 \end{aligned} \right. \end{equation*} if and only if $U(t,y)\coloneqq\theta(t,X_{m-1}(t,y,lh_m))$ solves \begin{equation}\label{e.fl.chart} \left\{ \begin{aligned} & \partial_tU+\nabla^\perp\psi_m\cdot\nabla U-\nu\nabla\cdot(\K_{m,l}\nabla U)\\ & \qquad=\nabla\cdot\bigl((\nabla_yX_{m-1})^{-1}F(t,X_{m-1})\bigr)+F^0(t,X_{m-1}) & \mbox{in} & \ (t_1,t_2)\times\T^2\,,\\ & U(t_1,\cdot)=g\bigl(X_{m-1}(t_1,\cdot,lh_m)\bigr) & \mbox{on} & \ \T^2\,, \end{aligned} \right. \end{equation} where~$X_{m-1}$ and~$\nabla_yX_{m-1}$ are evaluated at $(t,y,lh_m)$, and $\nabla$ acts in~$y$. Solutions are energy solutions, as in~\eqref{e.ft.weak}. \end{lemma} In the chart the drift~$\b_{m-1}$ disappears, the identity in the diffusion term becomes the conductivity~$\K_{m,l}$, which is close to~$I$ by Lemma~\ref{l.fl.charts}, and the pulses become the explicit shears~\eqref{e.fl.psim}. Since, for $U\in H^1(\T^2)$, $-\nabla\cdot(\psi_m\nabla^\perp U)=\nabla^\perp\psi_m\cdot\nabla U$, the equation in~\eqref{e.fl.chart} can equally be written in divergence form, with the matrix $\nu\K_{m,l}+\psi_m\bigl(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr)$. \begin{proof} By Proposition~\ref{p.fl.velocity}(i) and as in the proof of Lemma~\ref{l.fl.short}, at each time the chart is a smooth, volume-preserving diffeomorphism commuting with the translations by~$\Z^2$, so the change of variables preserves the class of solutions and the weak formulation: for $\varphi\in C^\infty(\T^2)$ and $\varphi_t\coloneqq\varphi(X_{m-1}(lh_m,\cdot,t))$, $\langle U(t),\varphi\rangle_{L^2(\T^2)}=\langle\theta(t),\varphi_t\rangle_{L^2(\T^2)}$, so the equation for~$U$ tested against~$\varphi$ is the equation for~$\theta$ tested against~$\varphi_t$. As~$\varphi_t$ is transported by~$\b_{m-1}$, $\partial_t\varphi_t+\b_{m-1}\cdot\nabla\varphi_t=0$, so the drift~$\b_{m-1}$ is absorbed into the time derivative of~$U$. Since $\nabla_yU=(\nabla_yX_{m-1})^{\mathsf T}\nabla\theta(t,X_{m-1})$, the second identity of~\eqref{e.fl.pulled} turns $(\b_m-\b_{m-1})\cdot\nabla\theta$ into $\nabla^\perp\psi_m\cdot\nabla_yU$. Finally, since $\det\nabla_yX_{m-1}=1$, the Piola identity (the rows of the cofactor matrix of a Jacobian matrix are divergence free) gives, for $C^1$ fields and, in the weak formulation, for fields in~$L^2$, $(\nabla\cdot F)(t,X_{m-1})=\nabla_y\cdot((\nabla_yX_{m-1})^{-1}F(t,X_{m-1}))$. Applied to~$F$ and to $\nu\nabla\theta=\nu(\nabla_yX_{m-1})^{-\mathsf T}\nabla_yU$, it gives the forcing and the term $\nu\nabla\cdot(\K_{m,l}\nabla U)$. \end{proof} \subsection{The averaged flux} \label{ss.fl.flux} \begin{lemma}[The averaged flux] \label{l.fl.flux} Let $m\ge1$ and $\nu>0$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $t\in\R$, $\J_m^\nu(t)$ is diagonal, $\nu I\le\J_m^\nu(t)\le(\nu+\eps_m^{2\beta}/(2\nu))I$, and, if $\widetilde\zeta_{m,k}(t)=0$ for every odd~$k$, then $\J_m^\nu(t)=\nu I$. \item For every $t\in\R$, $\J_m^\nu(t+2h_m)$ is~$\J_m^\nu(t)$ with its two diagonal entries exchanged, and $\J_m^\nu(t+4h_m)=\J_m^\nu(t)$. \item \eqref{e.mean} holds, and $\overline K_m(\nu)\ge\nu$. \end{enumerate} \end{lemma} \begin{proof} (i) The matrices $e_{(k)}\otimes e_{(k)}$ are diagonal, and $0\le\widetilde\zeta_{m,k}\le1$, so the time integral in~\eqref{e.flux} lies between~$0$ and $\int_{-\infty}^t\exp(-4\pi^2\nu(t-t')/\eps_m^2)\dd t'=\eps_m^2/(4\pi^2\nu)$. At most one pulse is active at each time (Lemma~\ref{l.fl.scales}(d)), so $0\le\J_m^\nu(t)-\nu I\le2\pi^2\eps_m^{2\beta-2}\cdot\eps_m^2/(4\pi^2\nu)\,I=\eps_m^{2\beta}/(2\nu)\,I$. (ii) Let $j\in2\Z$. Since $h_m/\tau_m$ is an integer, \eqref{e.label}, \eqref{e.fast} and~\eqref{e.slow} give $l_{m,k+jh_m/\tau_m}=l_{m,k}+j$ and $\widetilde\zeta_{m,k+jh_m/\tau_m}(t+jh_m)=\widetilde\zeta_{m,k}(t)$, so the terms of $\J_m^\nu(t+jh_m)$ are those of $\J_m^\nu(t)$ with~$k$ replaced by $k+jh_m/\tau_m$. The integer $h_m/\tau_m$ is odd, so for $j=2$ the shift keeps~$k$ odd and exchanges $4\Z+1$ and $4\Z+3$, hence $e_1\otimes e_1$ and $e_2\otimes e_2$, and for $j=4$ it fixes the classes. (iii) By~(ii), $\J_m^\nu$ is $4h_m$-periodic and a shift by~$2h_m$ exchanges its diagonal entries. So its averages over $[t,t+4h_m]$ and $[t+2h_m,t+6h_m]$ are equal and have exchanged entries; by~(i) they are diagonal, hence half their trace times~$I$, which is~\eqref{e.mean}; and $\overline K_m(\nu)\ge\nu$. \end{proof} Section~\ref{s.cell} identifies~$\J_m^\nu$ with the spatial average of the flux of the correctors of the level-$m$ pulses. The bands~\eqref{e.bands} are disjoint and accumulate at~$0$: by Lemma~\ref{l.fl.scales}(a), $2\eps_{M+1}^{\beta-\gamma}\le2\Lambda^{-(\beta-\gamma)}\eps_M^{\beta-\gamma}<\frac12\eps_M^{\beta-\gamma}$, since $\beta-\gamma=2\beta/(q+1)>\frac{32}{33}$ and $\Lambda\ge128$. \subsection{Effective diffusivities} \label{ss.fl.diffusivities} This subsection proves Proposition~\ref{p.scales}; its part~(a) holds for every $\Lambda\ge128$, with~$C$ depending only on~$\zeta$. \begin{proof}[Proof of Proposition~\ref{p.scales}(a)] Fix $m\ge1$ and $\nu>0$. By the recursion~\eqref{e.recursion}, we must show that $\overline K_m(\nu)$ is within $C(\eps_m^{2\beta}/\nu)(\eps_m^2/(\nu\tau_m)+\eps_{m-1}^\delta)$ of $\nu+\frac9{80}\eps_m^{2\beta}/\nu$. \emph{Step 1: The time integral.} Each pulse acts on the averaged flux as if it were held at its current value. Let~$k$ be odd. By Lemma~\ref{l.fl.scales}(d) and $\tau_m\le\tau'_m$, $|\partial_t\widetilde\zeta_{m,k}|\le C/\tau_m$, so \begin{equation*} \Bigl|\int_{-\infty}^t\exp\Bigl(-\frac{4\pi^2\nu(t-t')}{\eps_m^2}\Bigr)\widetilde\zeta_{m,k}(t')\dd t'-\frac{\eps_m^2}{4\pi^2\nu}\,\widetilde\zeta_{m,k}(t)\Bigr|\le\frac C{\tau_m}\int_0^\infty t'\exp\Bigl(-\frac{4\pi^2\nu t'}{\eps_m^2}\Bigr)\dd t'\le\frac{C\eps_m^4}{\nu^2\tau_m}\,. \end{equation*} As $\tr(e_{(k)}\otimes e_{(k)})=1$, $0\le\widetilde\zeta_{m,k}\le1$ and at most one pulse is active at a time, \eqref{e.flux} gives, for every $t\in\R$, \begin{equation}\label{e.fl.trace} \Bigl|\tr\J_m^\nu(t)-2\nu-\frac{\eps_m^{2\beta}}{2\nu}\sum_{k\in2\Z+1}\widetilde\zeta_{m,k}(t)^2\Bigr|\le C\,\frac{\eps_m^{2\beta}}\nu\,\frac{\eps_m^2}{\nu\tau_m}\,. \end{equation} \emph{Step 2: The average of the squared envelopes.} The function $\sum_{k\in2\Z+1}\zeta_{m,k}^2$ is $2\tau_m$-periodic, and its integral over a period is $\tau_m\int_\R\zeta^2=\frac9{10}\tau_m$ by~\eqref{e.zeta}. Since $h_m/\tau_m$ is an integer, its average over $[0,4h_m]$ is exactly~$\frac9{20}$: half of the slots are odd, and each carries~$\frac9{10}$. The slow cutoffs change this sum only where $\widehat\zeta_{m,l_{m,k}}(t)<1$ for an active pulse~$k$. By Lemma~\ref{l.fl.scales}(d), the slot of pulse~$k$ lies in its block and~$\zeta_{m,k}$ vanishes outside the slot enlarged by $\frac16\tau_m<2\tau'_m$; so this happens only within $2\tau'_m$ of the ends of blocks, a fraction at most $4\tau'_m/h_m$ of $[0,4h_m]$. As both sums lie in $[0,1]$, Lemma~\ref{l.fl.scales}(b) gives \begin{equation}\label{e.fl.squares} \Bigl|\fint_0^{4h_m}\sum_{k\in2\Z+1}\widetilde\zeta_{m,k}(t)^2\dd t-\frac9{20}\Bigr|\le\frac{4\tau'_m}{h_m}=\frac4{4\lceil\eps_{m-1}^{-\delta}\rceil+1}\le\eps_{m-1}^\delta\,. \end{equation} \emph{Step 3: Conclusion.} By~\eqref{e.mean}, $\overline K_m(\nu)$ is half the average of $\tr\J_m^\nu$ over $[0,4h_m]$, so~\eqref{e.fl.trace} and~\eqref{e.fl.squares} give \begin{equation*} \Bigl|\overline K_m(\nu)-\nu-\frac{\eps_m^{2\beta}}{4\nu}\cdot\frac9{20}\Bigr|\le C\,\frac{\eps_m^{2\beta}}\nu\Bigl(\frac{\eps_m^2}{\nu\tau_m}+\eps_{m-1}^\delta\Bigr)\,, \end{equation*} and $\frac{\eps_m^{2\beta}}{4\nu}\cdot\frac9{20}=\frac9{80}\frac{\eps_m^{2\beta}}\nu$. This is~(a). \end{proof} \begin{proof}[Proof of Proposition~\ref{p.scales}(b)] We may assume $M\ge2$. Going up from~$M$, we bound the logarithm of the ratio $\mathsf s_m\coloneqq\kappa_m/((9/80)^{\nf12}\eps_m^{\beta+\gamma})$, uniformly in~$m$. This normalization is the one a level reproduces: by~\eqref{e.fl.onelevel} and~\eqref{e.fl.balance} below, $\mathsf s_m=1$ gives $\mathsf s_{m-1}\approx(\eps_m/\eps_{m-1}^q)^{\beta-\gamma}(1+\eps_m^{2\gamma})$, a number close to~$1$. \emph{Step 1: One level.} Let $2\le m\le M$. By~(a), since $\kappa_m+\frac9{80}\eps_m^{2\beta}/\kappa_m\ge\frac9{80}\eps_m^{2\beta}/\kappa_m$, \begin{equation}\label{e.fl.onelevel} \Bigl|\frac{\kappa_{m-1}}{\kappa_m+\frac9{80}\eps_m^{2\beta}/\kappa_m}-1\Bigr|\le C\Bigl(\frac{\eps_m^2}{\kappa_m\tau_m}+\eps_{m-1}^\delta\Bigr)\,. \end{equation} Since $q(\beta-\gamma)=\beta+\gamma$ (Lemma~\ref{l.fl.scales}(c)), the comparison value satisfies the exact identity \begin{equation}\label{e.fl.balance} \frac{\kappa_m+\frac9{80}\eps_m^{2\beta}/\kappa_m}{(9/80)^{\nf12}\eps_{m-1}^{\beta+\gamma}}=\Bigl(\frac{\eps_m}{\eps_{m-1}^q}\Bigr)^{\beta-\gamma}\Bigl(\eps_m^{2\gamma}\mathsf s_m+\frac1{\mathsf s_m}\Bigr)\,. \end{equation} So, up to the factor $(\eps_m/\eps_{m-1}^q)^{\beta-\gamma}$ and the error in~\eqref{e.fl.onelevel}, a level maps~$\mathsf s_m$ to $\eps_m^{2\gamma}\mathsf s_m+1/\mathsf s_m$, essentially~$1/\mathsf s_m$, and inversion preserves~$|\log|$: precisely, $|\log(\eps_m^{2\gamma}\mathsf s_m+1/\mathsf s_m)|\le|\log\mathsf s_m|+\eps_m^{2\gamma}$, since $\eps_m^{2\gamma}\mathsf s_m+1/\mathsf s_m$ lies between~$1/\mathsf s_m$ and $(1+\eps_m^{2\gamma})\max\{\mathsf s_m,1/\mathsf s_m\}$ and $\log(1+\eps_m^{2\gamma})\le\eps_m^{2\gamma}$. Moreover: \begin{itemize} \item \emph{The error of~(a).} By Lemma~\ref{l.fl.scales}(a)--(c), $\eps_m^{2-\beta-\gamma}\le2\eps_{m-1}^{2-\beta+8\delta}$ and $\tau_m\ge c\eps_{m-1}^{2-\beta+4\delta}$, so $\eps_m^2/(\kappa_m\tau_m)=(\eps_m^{2-\beta-\gamma}/\tau_m)(\eps_m^{\beta+\gamma}/\kappa_m)\le C\eps_{m-1}^{4\delta}/\mathsf s_m$. \item \emph{The factor.} $|(\eps_m/\eps_{m-1}^q)^{\beta-\gamma}-1|\le C\eps_{m-1}$: with $\lambda\coloneqq\Lambda^{-q^{m-1}/(q-1)}$, $\eps_{m-1}^{-1}$ and~$\eps_m^{-1}$ are the ceilings of~$\lambda^{-1}$ and of~$\lambda^{-q}$, so $\eps_m/\eps_{m-1}^q=\lceil\lambda^{-1}\rceil^q/\lceil\lambda^{-q}\rceil$ lies between $(1+\lambda^q)^{-1}$ and $(1+\lambda)^q$, and $\lambda\le2\eps_{m-1}$. \end{itemize} If the right side of~\eqref{e.fl.onelevel} is at most~$\frac12$, then, since the logarithm of a number in $[\frac12,\frac32]$ is at most twice its distance to~$1$ in absolute value, \eqref{e.fl.balance} gives \begin{equation}\label{e.fl.logstep} |\log\mathsf s_{m-1}|\le|\log\mathsf s_m|+C\eps_{m-1}+\eps_m^{2\gamma}+C\Bigl(1+\frac1{\mathsf s_m}\Bigr)\eps_{m-1}^\delta\,. \end{equation} \emph{Step 2: The start.} Let $m=M$. Since $\frac12\eps_M^{\beta-\gamma}\le\kappa_M\le2\eps_M^{\beta-\gamma}$, the two terms of the sum on the right side of~\eqref{e.fl.balance} lie in $[\frac12(80/9)^{\nf12},2(80/9)^{\nf12}]\subset[1.49,5.97]$ and in $[0,2(9/80)^{\nf12}\eps_M^{2\gamma}]\subset[0,0.68]$, and the factor $(\eps_M/\eps_{M-1}^q)^{\beta-\gamma}$ lies in $[2^{-4/3},2^{4/3}]$, by Lemma~\ref{l.fl.scales}(a) and $0<\beta-\gamma<\frac43$. By the first bullet, if $\Lambda\ge C$, the right side of~\eqref{e.fl.onelevel} is at most $C\eps_{M-1}^\delta\le\frac12$. So $\mathsf s_{M-1}$ lies in $[\frac12\cdot2^{-4/3}\cdot1.49,\frac32\cdot2^{4/3}\cdot(5.97+0.68)]\subset[\exp(-4),\exp(4)]$. \emph{Step 3: Up the levels.} The constants~$C$ of Step~1 come from~(a) and Lemma~\ref{l.fl.scales} and do not depend on~$\Lambda$, and the start bound~$4$ is absolute; so we fix the bound $5=4+1$ first and~$\Lambda_0$ last. We choose $\Lambda_0\ge128$ so large that, for $\Lambda\ge\Lambda_0$: the requirement $\Lambda\ge C$ of Step~2 holds; for every $m\ge2$, $C(1+\exp(5))(\eps_{m-1}^{4\delta}+\eps_{m-1}^\delta)\le\frac12$; and \begin{equation*} \sum_{m\ge2}\Bigl(C\eps_{m-1}+\eps_m^{2\gamma}+C(1+\exp(5))\eps_{m-1}^\delta\Bigr)\le1\,, \end{equation*} which is possible since $\eps_{m'}\le\Lambda^{-m'}$ by Lemma~\ref{l.fl.scales}(a). We prove by induction on~$m$, for $m=M-1,M-2,\dots,1$ in turn, that \begin{equation*} |\log\mathsf s_m|\le4+\sum_{m'=m+1}^{M-1}\Bigl(C\eps_{m'-1}+\eps_{m'}^{2\gamma}+C(1+\exp(5))\eps_{m'-1}^\delta\Bigr)\,. \end{equation*} For $m=M-1$ this is Step~2. If it holds at some $m\ge2$, then $|\log\mathsf s_m|\le5$ by the display above, so the right side of~\eqref{e.fl.onelevel} is at most~$\frac12$, and~\eqref{e.fl.logstep} with $1/\mathsf s_m\le\exp(5)$ gives it at~$m-1$. Since the sum is at most~$1$, $|\log\mathsf s_m|\le5$ for $1\le m\le M-1$: this is~(b). \end{proof} \begin{proof}[Proof of Proposition~\ref{p.scales}(c)] Let $2\le m\le M$. First, $\kappa_m\le\overline K_m(\kappa_m)=\kappa_{m-1}$ by Lemma~\ref{l.fl.flux}(iii). Since $1\le m-10$ and $g=(g_v)_{v\in V}$, we write \begin{equation}\label{e.ft.norm} \|g\|_{H^N_{\rho,\omega}(\T^2)}^2\coloneqq\sum_{v\in V}\frac{\|g_v\|_{H^N_\rho(\T^2)}^2}{\omega_v}\,,\qquad \|g\|_{H^{(N_v)}_{\rho,\omega}(\T^2)}^2\coloneqq\sum_{v\in V}\frac{\|g_v\|_{H^{N_v}_\rho(\T^2)}^2}{\omega_v}\,, \end{equation} the second for integers $N_v\ge0$; for $N=0$ the first is the norm $\|g\|_{L^2_\omega(\T^2)}$ of~\eqref{e.norms}. \subsection{Products and derivatives in~\texorpdfstring{$H^N_\rho$}{H\^{}N\_rho}} \label{ss.ft.factorial} \begin{lemma}[Products and derivatives at the scale~$\rho$]\label{l.ft.product} Let $N\ge0$ be an integer and $\rho>0$. \begin{enumerate}[label=\textup{(\alph*)}] \item For every coefficient~$A$ of class~$C^N$ on~$\T^2$ and every $g\in H^N(\T^2)$ with values in the domain of~$A$, \begin{equation*} \|Ag\|_{H^N_\rho(\T^2)}\le\Bigl(\sum_{j=0}^N\frac{\rho^j}{j!}\sup_{\T^2}|\nabla^jA|\Bigr)\|g\|_{H^N_\rho(\T^2)}\,. \end{equation*} \item For every integer $i\ge0$ and every $g\in H^{N+i}(\T^2)$, $\|\nabla^ig\|_{H^N_\rho(\T^2)}\le\rho^{-i}((N+i)!/N!)\|g\|_{H^{N+i}_\rho(\T^2)}$. \end{enumerate} \end{lemma} \begin{proof} (a) The weights turn the Leibniz rule into a convolution: there are~$\binom ji$ ways to let~$i$ of the~$j$ derivatives of~$\nabla^j(Ag)$ fall on~$A$, each giving an array of norm at most $|\nabla^iA|\,|\nabla^{j-i}g|$ at every point, and $\binom ji\rho^j/j!=(\rho^i/i!)(\rho^{j-i}/(j-i)!)$. So the sequence $(\rho^j/j!)\|\nabla^j(Ag)\|_{L^2(\T^2)}$ is bounded by the convolution of $(\rho^i/i!)\sup_{\T^2}|\nabla^iA|$ and $(\rho^i/i!)\|\nabla^ig\|_{L^2(\T^2)}$, and Young's inequality for a summable sequence convolved with a square-summable one gives~(a). (b) For $j\le N$, $\rho^j/j!=\rho^{-i}((j+i)!/j!)\,\rho^{j+i}/(j+i)!$ and $(j+i)!/j!\le(N+i)!/N!$. \end{proof} \subsection{Paired smoothing} \label{ss.ft.paired} \paragraph{The kernel.} We prove the properties of~$W_\ell$ stated before Proposition~\ref{p.paired}. With $u_\ell(z)\coloneqq(2\ell)^2u(2\ell z)$, the autocorrelation $u_\ell*u_\ell(-\,\cdot\,)$ has Fourier transform $|\widehat u(\xi/(2\ell))|^2$, so expanding the power in~\eqref{e.kernel} gives $W_\ell=\sum_{j=1}^\ell(-1)^{j+1}\binom\ell j(u_\ell*u_\ell(-\,\cdot\,))^{*j}$, where~$(\cdot)^{*j}$ is the $j$-fold convolution. The autocorrelation is smooth with support in the disc of radius~$1/\ell$, so~$W_\ell$ is smooth with support in the unit disc, and $\int_{\R^2}W_\ell=\widehat W_\ell(0)=1$. Since $u\ge0$ has integral one, $0\le|\widehat u|^2\le1$, so~$\widehat W_\ell$ takes values in~$[0,1]$. The smooth function $1-|\widehat u|^2$ vanishes at its minimum, the origin, so $1-\widehat W_\ell=(1-|\widehat u(\xi/(2\ell))|^2)^\ell=O(|\xi|^{2\ell})$, and the derivatives of~$\widehat W_\ell$ of orders~$1$ to $2\ell-1$, which are constant multiples of the moments of~$W_\ell$, vanish at the origin. Consequently, each average~$S_{\ell,r,\Omega}$ multiplies the Fourier coefficient of a function on~$\T^2$ at the frequency~$\xi$ by $\widehat W_\ell(r\Omega^{\mathsf T}\xi)\in[0,1]$: it commutes with derivatives and, by Plancherel's theorem, contracts every~$H^N_\rho(\T^2)$. \paperfigure{% \begin{figure}[!htbp] \centering \begin{tikzpicture}[scale=1.1] \draw[semithick,fill=black!10,cm={2.1,0.55,0.12,0.12,(0,0)}] (0,0) circle (1); \draw[semithick,fill=black!10,fill opacity=0.8,cm={-0.1,0.12,-0.45,1.75,(0,0)}] (0,0) circle (1); \draw[-{Stealth[length=2.2mm]},thick] (0,0) -- (2.1,0.55) node[right] {$r\Omega_1e_1$}; \draw[-{Stealth[length=2.2mm]},thick] (0,0) -- (-0.45,1.75) node[above] {$r\Omega_2e_2$}; \fill (0,0) circle (1.2pt) node[below right] {$y$}; \end{tikzpicture} \caption{The two thin ellipses $\{y-r\Omega_1z:|z|<1\}$ and $\{y-r\Omega_2z:|z|<1\}$, centred at~$y$, over which the two averages of $S_{\ell,r,\Omega_1}S_{\ell,r,\Omega_2}$ are taken, and the vectors $r\Omega_1e_1$ and $r\Omega_2e_2$.} \label{f.ft.ellipses} \end{figure}} \begin{proof}[Proof of Proposition~\ref{p.paired}] \emph{Step 1: Derivatives along~$\Omega_1e_1$ and~$\Omega_2e_2$.} Each of these derivatives falls on one average and costs~$r^{-1}$, whatever the short axes of the ellipses (Figure~\ref{f.ft.ellipses}). Let $\partial_{(1)}\coloneqq\Omega_1e_1\cdot\nabla$ and $\partial_{(2)}\coloneqq\Omega_2e_2\cdot\nabla$. On $y\mapsto g(y-r\Omega_1z)$, $\partial_{(1)}$ is~$-r^{-1}\partial_{z_1}$, and integration by parts in~$z_1$ in~\eqref{e.Srh} gives \begin{equation*} \partial_{(1)}S_{\ell,r,\Omega_1}g(y)=r^{-1}\int_{\R^2}\partial_1W_\ell(z)\,g(y-r\Omega_1z)\dd z\,, \end{equation*} and, by Minkowski's inequality, for every $j\ge0$, $\|\partial_{(1)}^jS_{\ell,r,\Omega_1}g\|_{L^2(\T^2)}\le r^{-j}\|\partial_1^jW_\ell\|_{L^1(\R^2)}\|g\|_{L^2(\T^2)}$; similarly for~$\partial_{(2)}$, $S_{\ell,r,\Omega_2}$ and~$\partial_2$. The pair $(\partial_{(1)},\partial_{(2)})$ is the image of $(\partial_1,\partial_2)$ under the transpose of the matrix with columns $\Omega_1e_1$, $\Omega_2e_2$, so each~$\partial_i$ is a combination of~$\partial_{(1)}$ and~$\partial_{(2)}$ with coefficients that are entries of the inverse of that matrix, of absolute value at most its $|\cdot|_\infty$-norm, hence at most~$2$. An ordered derivative of order~$j$ is thus a sum of~$2^j$ products of~$j$ derivatives~$\partial_{(1)}$ or~$\partial_{(2)}$, with coefficients of absolute value at most~$2^j$. The two averages commute with each other and with derivatives; letting~$\partial_{(1)}$ act on~$S_{\ell,r,\Omega_1}$ and~$\partial_{(2)}$ on~$S_{\ell,r,\Omega_2}$ gives, for every $j\ge0$, \begin{equation*} \|\nabla^jSg\|_{L^2(\T^2)}\le c_j\,r^{-j}\|g\|_{L^2(\T^2)}\,, \end{equation*} where~$c_j$ depends only on~$u$, $\ell$ and~$j$. Proved for smooth~$g$, this bound extends to every $g\in L^2(\T^2)$ by density, as the averages are bounded on~$L^2(\T^2)$. \emph{Step 2: The first two bounds.} The averages contract~$H^{N_1}_\rho(\T^2)$, which gives the second bound and the terms of order $j\le N_1$ of the first. For $N_1N_1$ and $\sup_{\ell>0}(C_u/(4\ell^2))^\ell=\exp(\sqrt{C_u}/\mathrm e)$, for every $\xi\in\R^2$, \begin{equation*} 0\le1-\widehat W_\ell(\xi)=\Bigl(1-\Bigl|\widehat u\Bigl(\frac\xi{2\ell}\Bigr)\Bigr|^2\Bigr)^\ell\le\min\Bigl\{1,\Bigl(\frac{C_u|\xi|^2}{4\ell^2}\Bigr)^\ell\Bigr\}\le\exp\Bigl(\frac{\sqrt{C_u}}{\mathrm e}\Bigr)|\xi|^{N_1}\,. \end{equation*} Let $i\in\{1,2\}$. As $S_{\ell,r,\Omega_i}$ multiplies the Fourier coefficient at~$\xi$ by $\widehat W_\ell(r\Omega_i^{\mathsf T}\xi)$, and $|\Omega_i^{\mathsf T}\xi|\le\sqrt2\,|\Omega_i|_\infty|\xi|$, Plancherel's theorem gives \begin{equation*} \|g-S_{\ell,r,\Omega_i}g\|_{L^2(\T^2)}\le\exp\Bigl(\frac{\sqrt{C_u}}{\mathrm e}\Bigr)\bigl(\sqrt2\,rC_\Omega\bigr)^{N_1}\|\nabla^{N_1}g\|_{L^2(\T^2)}\le C\,C_\Omega^{N_1}\Bigl(\frac r\rho\Bigr)^{N_1}\|g\|_{H^{N_1}_\rho(\T^2)}\,, \end{equation*} with~$C$ depending only on~$u$ and~$N_1$. Since $g-Sg=(g-S_{\ell,r,\Omega_1}g)+S_{\ell,r,\Omega_1}(g-S_{\ell,r,\Omega_2}g)$ and $0\le\widehat W_\ell\le1$, this gives the third bound with the constant~$2C$. This step uses neither the invertibility of~$\Omega_i$, nor the transversality of $\Omega_1e_1$ and~$\Omega_2e_2$, nor $r\le\rho$, which proves the last sentence of the proposition. \end{proof} \begin{lemma}[Time-dependent paired smoothing]\label{l.ft.moving} Let $\ell\ge1$ and $r>0$, and let $\Omega_1(t)$, $\Omega_2(t)$, $t\in\R$, be invertible linear maps of~$\R^2$, Borel measurable in~$t$, such that, for every $t\in\R$, the matrix with columns $\Omega_1(t)e_1$, $\Omega_2(t)e_2$ has an inverse of $|\cdot|_\infty$-norm at most~$2$. Then the operators $S(t)\coloneqq S_{\ell,r,\Omega_1(t)}S_{\ell,r,\Omega_2(t)}$ commute with derivatives; for every $g\in L^2(\T^2)$, $t\mapsto S(t)g\in L^2(\T^2)$ is Borel measurable; and there exist constants $c_j(u,\ell)$ such that, for every $t\in\R$, $j\ge0$ and $g\in L^2(\T^2)$, $\|\nabla^jS(t)g\|_{L^2(\T^2)}\le c_jr^{-j}\|g\|_{L^2(\T^2)}$. Thus~$S(t)$ is a smoothing operator of width~$r$ (Subsection~\ref{ss.families} and the hypotheses of Proposition~\ref{p.energy}), with no bound on~$|\Omega_i(t)|$. \end{lemma} \begin{proof} Step~1 of the proof of Proposition~\ref{p.paired} at the time~$t$ gives the bound, and the multiplier form of~$S(t)$ gives the commutation. The multiplier at each frequency depends continuously on the matrices, so $t\mapsto S(t)g$ is Borel measurable, by Plancherel's theorem and dominated convergence. \end{proof} \subsection{Energy of coupled systems} \label{ss.ft.energy} We keep the setting of Proposition~\ref{p.energy}, without its condition on~$B$, except that~$U_v$, $g_v$ and~$F^0_v$ may take values in a finite-dimensional Euclidean space~$E_v$ ($E_v=\R$ for families), and~$F_v$ in~$E_v^2$; $A_{\sigma,d}$ maps arrays of order~$d-1$ over~$E_{v'}$ to~$E_v^2$, $S_\sigma(t)$ acts on each coordinate, and, in the component~$v$, $L^2(\T^2)$ and $H^N(\T^2)$ stand for $L^2(\T^2;E_v)$ and $H^N(\T^2;E_v)$. We consider, for $v\in V$, \begin{equation}\label{e.ft.system} \left\{ \begin{aligned} & \partial_tU_v+\nabla^\perp\Phi\cdot\nabla U_v-\nu\Delta U_v\\ & \qquad=\nabla\cdot\Bigl(\sum_{v'\in V}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}A_{\sigma,d}\nabla^{d-1}(S_\sigma U_{v'})+F_v\Bigr)+F^0_v & \mbox{in} & \ (t_1,t_2)\times\T^2\,,\\ & U_v(t_1,\cdot)=g_v & \mbox{on} & \ \T^2\,, \end{aligned} \right. \end{equation} with a flux forcing~$F_v$ and a scalar forcing~$F^0_v$. A \emph{solution} of~\eqref{e.ft.system} is a function $U\in C([t_1,t_2];\allowbreak L^2(\T^2)^V)\cap\allowbreak L^2((t_1,t_2);\allowbreak H^1(\T^2)^V)$ such that, for every $v\in V$ and $t\in[t_1,t_2]$, \begin{equation}\label{e.ft.weak} U_v(t)-g_v =\int_{t_1}^t\Bigl(-\nabla^\perp\Phi\cdot\nabla U_v+\nu\Delta U_v +\nabla\cdot\Bigl(\sum_{v',\sigma,d}A_{\sigma,d}\nabla^{d-1}(S_\sigma U_{v'})+F_v\Bigr)+F^0_v\Bigr)(t'')\dd t'' \,. \end{equation} Here $\nabla^{d-1}(S_\sigma U_{v'})=\nabla^{d-2}S_\sigma\nabla U_{v'}$ is measurable in time (by the linearity of~$S_\sigma(t)$, its bound and the measurability of $t\mapsto S_\sigma(t)h$), with norm at most $c_{d-2}r_\sigma^{2-d}\|\nabla U_{v'}\|_{L^2((t_1,t_2);L^2(\T^2))}$ in $L^2((t_1,t_2);L^2(\T^2))$; \eqref{e.ft.weak} is an identity in~$H^{-1}(\T^2)$. For an integer $N\ge0$ and $\rho>0$, the \emph{coupling matrix of order~$N$ at scale~$\rho$} is \begin{equation}\label{e.ft.coupling} B^{N,\rho}_{vv'}\coloneqq\Bigl(\frac{\omega_{v'}}{\omega_v}\Bigr)^{\nf12}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}\frac{c_{d-2}}{\nu\,r_\sigma^{d-2}}\sum_{j=0}^N\frac{\rho^j}{j!}\sup_{[t_1,t_2]\times\T^2}|\nabla^jA_{\sigma,d}|\,,\qquad v,v'\in V\,. \end{equation} For $N=0$ and every~$\rho$, it is the coupling matrix~$B$ of Proposition~\ref{p.energy}. \begin{proposition}[Energy of coupled systems at higher order]\label{p.ft.energy} In the setting of~\eqref{e.ft.system}, let $N\ge0$ be an integer and $\rho>0$, and assume the following. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $(t,x)\in[t_1,t_2]\times\T^2$ and every integer $1\le i\le N$, \begin{equation*} \rho^{i+1}\,|\nabla^i\nabla^\perp\Phi(t,x)|\le2^{-i-2}\,i!\,\nu\,. \end{equation*} \item For every $t\in[t_1,t_2]$, the coefficients $A_{\sigma,d}(t,\cdot)$ are of class~$C^N$ on~$\T^2$, and, for every $j\le N$, $\nabla^jA_{\sigma,d}$ is bounded on $[t_1,t_2]\times\T^2$. \item All row and column sums of~$B^{N,\rho}$ are at most~$\frac18$. \end{enumerate} Then, for every datum~$g$ and forcing~$F$, $F^0$ such that, for every $v\in V$, $g_v\in H^N(\T^2)$, $F_v\in L^2((t_1,t_2);H^N(\T^2)^2)$ and $F^0_v\in L^1((t_1,t_2);H^N(\T^2))$, there exists a unique solution~$U$ of~\eqref{e.ft.system}. It belongs to $C([t_1,t_2];H^N(\T^2)^V)\cap L^2((t_1,t_2);H^{N+1}(\T^2)^V)$, and the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item The solution satisfies \begin{align*} \|U\|_{L^\infty([t_1,t_2];H^N_{\rho,\omega}(\T^2))}^2+\frac\nu2\,\|\nabla U\|_{L^2((t_1,t_2);H^N_{\rho,\omega}(\T^2))}^2 & \le2\|g\|_{H^N_{\rho,\omega}(\T^2)}^2+\frac8\nu\,\|F\|_{L^2((t_1,t_2);H^N_{\rho,\omega}(\T^2))}^2 \notag \\ & \qquad +10\,\|F^0\|_{L^1((t_1,t_2);H^N_{\rho,\omega}(\T^2))}^2\,. \end{align*} \item Without forcing, for every $t_1\le t'\le t\le t_2$, \begin{equation*} \|U(t)\|_{H^N_{\rho,\omega}(\T^2)}^2+\nu\,\|\nabla U\|_{L^2((t',t);H^N_{\rho,\omega}(\T^2))}^2\le\|U(t')\|_{H^N_{\rho,\omega}(\T^2)}^2\,. \end{equation*} \item Without forcing, the components satisfy, each with a separate supremum in time, \begin{equation*} \sum_{v\in V}\frac1{\omega_v}\Bigl(\|U_v\|_{L^\infty([t_1,t_2];H^N_\rho(\T^2))}^2+\nu\,\|\nabla U_v\|_{L^2((t_1,t_2);H^N_\rho(\T^2))}^2\Bigr)\le2\|g\|_{H^N_{\rho,\omega}(\T^2)}^2\,. \end{equation*} \end{enumerate} \end{proposition} By~(i), the velocity gradient times the diffusion time~$\rho^2/\nu$ of the scale~$\rho$ is at most~$\frac18$. The sources cost at most an eighth of the dissipation $2\nu\|\nabla U\|_{H^N_{\rho,\omega}(\T^2)}^2$ (Lemma~\ref{l.ft.source}), and the drift a quarter (Lemma~\ref{l.ft.commutator}). \begin{lemma}[The source term]\label{l.ft.source} Let $N\ge0$, $\rho>0$ and $\vartheta\ge0$, assume~(ii) of Proposition~\ref{p.ft.energy}, and assume that all row and column sums of~$B^{N,\rho}$ are at most~$\vartheta$. Then, for every $t\in[t_1,t_2]$ and every $U=(U_v)_{v\in V}$ with $U_v\in H^{N+1}(\T^2)$, \begin{equation*} \Bigl\|\Bigl(\sum_{v',\sigma,d}A_{\sigma,d}(t)\nabla^{d-1}(S_\sigma(t)U_{v'})\Bigr)_{v\in V}\Bigr\|_{H^N_{\rho,\omega}(\T^2)}\le\vartheta\nu\,\|\nabla U\|_{H^N_{\rho,\omega}(\T^2)}\,. \end{equation*} Under~(iii) of Proposition~\ref{p.ft.energy}, this holds with $\vartheta=\frac18$. \end{lemma} \begin{proof} Fix $\sigma\colon v'\to v$ and~$d$; we omit~$t$. Since~$S_\sigma$ commutes with derivatives, its bound in~$L^2(\T^2)$ holds in~$H^N_\rho(\T^2)$, and with Lemma~\ref{l.ft.product}(a), \begin{equation*} \bigl\|A_{\sigma,d}\nabla^{d-1}(S_\sigma U_{v'})\bigr\|_{H^N_\rho(\T^2)}\le\frac{c_{d-2}}{r_\sigma^{d-2}}\Bigl(\sum_{j=0}^N\frac{\rho^j}{j!}\sup_{[t_1,t_2]\times\T^2}|\nabla^jA_{\sigma,d}|\Bigr)\|\nabla U_{v'}\|_{H^N_\rho(\T^2)}\,. \end{equation*} So the $v$-th component of the source term, divided by~$\omega_v^{\nf12}$, has norm in~$H^N_\rho(\T^2)$ at most $\nu\sum_{v'}B^{N,\rho}_{vv'}\,\omega_{v'}^{-\nf12}\|\nabla U_{v'}\|_{H^N_\rho(\T^2)}$, and Schur's test (Subsection~\ref{ss.families}) gives the lemma. \end{proof} \begin{lemma}[The drift commutator]\label{l.ft.commutator} Let $N\ge1$, $\rho>0$ and $t\in[t_1,t_2]$, and assume~(i) of Proposition~\ref{p.ft.energy} at~$t$. Then, for every smooth~$g$ on~$\T^2$, with~$\Phi$ taken at the time~$t$, \begin{equation*} \sum_{j=1}^N\frac{\rho^{2j}}{(j!)^2}\,\Bigl|\Bigl\langle\nabla^j(\nabla^\perp\Phi\cdot\nabla g)-\nabla^\perp\Phi\cdot\nabla(\nabla^jg),\nabla^jg\Bigr\rangle_{L^2(\T^2)}\Bigr|\le\frac\nu4\,\|\nabla g\|_{H^N_\rho(\T^2)}^2\,. \end{equation*} \end{lemma} \begin{proof} By the Leibniz rule, as in the proof of Lemma~\ref{l.ft.product}(a), the commutator is the sum over $1\le i\le j$ of~$\binom ji$ arrays in which~$i$ derivatives fall on~$\nabla^\perp\Phi$ and $j-i$ on~$\nabla g$, each of norm at most $|\nabla^i\nabla^\perp\Phi|\,|\nabla^{j-i+1}g|$ at every point. Since $\binom ji\rho^{2j}/(j!)^2$ is the product of $\rho^{i+1}/(i!\,j)$, $\rho^{j-i}/(j-i)!$ and $\rho^{j-1}/(j-1)!$, the hypothesis and $1/j\le1$ bound the contribution of~$i$ to the term of~$j$ by \begin{equation*} 2^{-i-2}\nu\cdot\frac{\rho^{j-i}}{(j-i)!}\|\nabla^{j-i+1}g\|_{L^2(\T^2)}\cdot\frac{\rho^{j-1}}{(j-1)!}\|\nabla^jg\|_{L^2(\T^2)}\,. \end{equation*} For each~$i$, the sum over~$j$ of these products is at most $2^{-i-2}\nu\|\nabla g\|_{H^N_\rho(\T^2)}^2$, by the Cauchy--Schwarz inequality, and $\sum_{i\ge1}2^{-i-2}=\frac14$. \end{proof} \begin{proof}[Proof of Proposition~\ref{p.ft.energy}] \emph{Step 1: The energy inequalities.} Let~$U$ be a solution without forcing that is smooth in~$x$, or one of the Galerkin system of Step~3 (the system projected onto trigonometric polynomials). We apply~$\nabla^j$, $j\le N$, to the equation of~$U_v$ and take the inner product in~$L^2(\T^2)$ with $(\rho^{2j}/(j!)^2)\nabla^jU_v$: the transport term $\langle\nabla^\perp\Phi\cdot\nabla(\nabla^jU_v),\nabla^jU_v\rangle_{L^2(\T^2)}$ vanishes because~$\nabla^\perp\Phi$ is divergence free, the commutator is bounded by Lemma~\ref{l.ft.commutator} applied to~$U_v(t)$, and the source term pairs with~$\nabla U_v$ after an integration by parts. Summing over~$j$, we obtain \begin{equation}\label{e.ft.component} \frac{\mathrm d}{\mathrm dt}\|U_v(t)\|_{H^N_\rho(\T^2)}^2+\frac32\,\nu\,\|\nabla U_v(t)\|_{H^N_\rho(\T^2)}^2\le2\Bigl|\Bigl\langle\sum_{v',\sigma,d}A_{\sigma,d}\nabla^{d-1}(S_\sigma U_{v'}),\nabla U_v\Bigr\rangle_{H^N_\rho(\T^2)}\Bigr|\,, \end{equation} with the inner product of the norm~\eqref{e.HNrho}. Divided by~$\omega_v$ and summed over~$v$, the right sides are at most $\frac\nu4\|\nabla U(t)\|_{H^N_{\rho,\omega}(\T^2)}^2$, by the Cauchy--Schwarz inequality and Lemma~\ref{l.ft.source}, so \begin{equation}\label{e.ft.monotone} \frac{\mathrm d}{\mathrm dt}\|U(t)\|_{H^N_{\rho,\omega}(\T^2)}^2+\frac54\,\nu\,\|\nabla U(t)\|_{H^N_{\rho,\omega}(\T^2)}^2\le0\,. \end{equation} Integrating~\eqref{e.ft.monotone} over~$[t',t]$ gives~(b). For~(c), we integrate~\eqref{e.ft.component} over~$[t_1,t]$, drop its dissipation, take the supremum in~$t$, divide by~$\omega_v$ and sum. By the bound of the right sides above, and then by~\eqref{e.ft.monotone} integrated over~$[t_1,t_2]$, the left side of~(c) is at most \begin{equation*} \|g\|_{H^N_{\rho,\omega}(\T^2)}^2+\frac54\,\nu\,\|\nabla U\|_{L^2((t_1,t_2);H^N_{\rho,\omega}(\T^2))}^2\le2\|g\|_{H^N_{\rho,\omega}(\T^2)}^2\,. \end{equation*} \emph{Step 2: The forced estimate.} With forcing, the computation of Step~1, summed over~$v$, has the further terms $-2\langle F,\nabla U\rangle_{H^N_{\rho,\omega}(\T^2)}$ and $2\langle F^0,U\rangle_{H^N_{\rho,\omega}(\T^2)}$, with the inner product of the norm~\eqref{e.ft.norm}. Bounding the first by $\frac\nu4\|\nabla U\|_{H^N_{\rho,\omega}(\T^2)}^2+\frac4\nu\|F\|_{H^N_{\rho,\omega}(\T^2)}^2$ and integrating gives, for every $t\in[t_1,t_2]$, \begin{align}\label{e.ft.integrated} \|U(t)\|_{H^N_{\rho,\omega}(\T^2)}^2+\nu\,\|\nabla U\|_{L^2((t_1,t);H^N_{\rho,\omega}(\T^2))}^2 & \le\|g\|_{H^N_{\rho,\omega}(\T^2)}^2+\frac4\nu\,\|F\|_{L^2((t_1,t_2);H^N_{\rho,\omega}(\T^2))}^2 \notag \\ & \qquad +2\,\|U\|_{L^\infty([t_1,t_2];H^N_{\rho,\omega}(\T^2))}\|F^0\|_{L^1((t_1,t_2);H^N_{\rho,\omega}(\T^2))}\,. \end{align} Young's inequality absorbs the last term, in two stages. On the right side of~\eqref{e.ft.integrated}, let~$Y$ be the sum of the first two terms and~$X$ and~$Z$ the two norms in the last, and let $D\coloneqq\nu\|\nabla U\|_{L^2((t_1,t_2);H^N_{\rho,\omega}(\T^2))}^2$. The supremum in~$t$ of~\eqref{e.ft.integrated} gives the first line below, and this supremum plus half of~\eqref{e.ft.integrated} at $t=t_2$ gives the second, which is~(a), with the first line in its last step: \begin{align*} X^2&\le Y+2XZ\le Y+\frac12X^2+2Z^2\,,\qquad\text{so}\qquad X^2\le2Y+4Z^2\,,\\ X^2+\frac12D&\le\frac32Y+3XZ\le\frac32Y+\frac14X^2+9Z^2\le2Y+10Z^2\,. \end{align*} \emph{Step 3: Existence, uniqueness and regularity.} By the standard Galerkin argument, with the Fourier truncations~$P_n$, which are self-adjoint and commute with derivatives, Steps~1 and~2 hold for the Galerkin approximations uniformly in~$n$, and a subsequence converges weakly-$*$ in $L^\infty([t_1,t_2];H^N(\T^2)^V)\cap L^2((t_1,t_2);H^{N+1}(\T^2)^V)$ to a solution~$U$. For $j\le N$, $\partial_t\nabla^jU_v$ lies in $L^2((t_1,t_2);H^{-1}(\T^2))+L^1((t_1,t_2);L^2(\T^2))$, by~(i), Lemma~\ref{l.ft.source} and the classes of~$F$ and~$F^0$; testing with Fourier truncations then gives the energy identity for~$\nabla^jU_v$, hence $U\in C([t_1,t_2];H^N(\T^2)^V)$ and Steps~1 and~2 at order~$N$ (Lemma~\ref{l.ft.commutator} extends to $H^{N+1}(\T^2)$ by density), and so~(a)--(c). At order~$0$ the same lines apply to every solution, with only the boundedness of~$\nabla^\perp\Phi$, so the difference of two solutions vanishes by Step~1 with $N=0$, since~(i) is then void and $B^{0,\rho}\le B^{N,\rho}$. \end{proof} By the uniqueness at order~$0$ (Step~3), the restriction of a solution to~$[t',t_2]$ is the solution from~$U(t')$ with the restricted forcing, as it satisfies~\eqref{e.ft.weak} there and Proposition~\ref{p.ft.energy} applies on~$[t',t_2]$. \begin{proof}[Proof of Proposition~\ref{p.energy}] Take $N=0$ in Proposition~\ref{p.ft.energy}, without forcing and with real components: (i) is void, (ii) is the boundedness of the coefficients, $B^{0,\rho}=B$, and~(b) is the energy inequality. \end{proof} \begin{proposition}[Systems without smoothing]\label{p.ft.plain} In the setting of~\eqref{e.ft.system}, let $\rho>0$, let $N_v\ge0$, $v\in V$, be integers, and assume~(i) of Proposition~\ref{p.ft.energy} at the order $\max_vN_v$. Let~$A_{vv',d}$, for $v,v'\in V$ and finitely many $d\ge2$, be coefficients as in Proposition~\ref{p.energy} and in~(ii) of Proposition~\ref{p.ft.energy} at the order~$N_v$, vanishing unless $N_{v'}\ge N_v+d-2$, such that all row and column sums of the matrix with entries \begin{equation*} \Bigl(\frac{\omega_{v'}}{\omega_v}\Bigr)^{\nf12}\sum_d\frac{(N_v+d-2)!}{N_v!\,\nu\,\rho^{d-2}}\sum_{j=0}^{N_v}\frac{\rho^j}{j!}\sup_{[t_1,t_2]\times\T^2}|\nabla^jA_{vv',d}|\,,\qquad v,v'\in V\,, \end{equation*} are at most~$\frac18$. Then, for every~$g$, $F$, $F^0$ such that, for every $v\in V$, $g_v\in H^{N_v}(\T^2)$, $F_v\in L^2((t_1,t_2);H^{N_v}(\T^2)^2)$ and $F^0_v\in L^1((t_1,t_2);H^{N_v}(\T^2))$, the problem, for $v\in V$, \begin{equation*} \left\{ \begin{aligned} & \partial_tU_v+\nabla^\perp\Phi\cdot\nabla U_v-\nu\Delta U_v=\nabla\cdot\Bigl(\sum_{v'\in V}\sum_dA_{vv',d}\nabla^{d-1}U_{v'}+F_v\Bigr)+F^0_v & \mbox{in} & \ (t_1,t_2)\times\T^2\,,\\ & U_v(t_1,\cdot)=g_v & \mbox{on} & \ \T^2\,, \end{aligned} \right. \end{equation*} has a unique solution~$U$ (as in~\eqref{e.ft.weak}) with $U_v\in C([t_1,t_2];H^{N_v}(\T^2))\cap L^2((t_1,t_2);H^{N_v+1}(\T^2))$, and \begin{align*} \|U\|_{L^\infty([t_1,t_2];H^{(N_v)}_{\rho,\omega}(\T^2))}^2+\frac\nu2\,\|\nabla U\|_{L^2((t_1,t_2);H^{(N_v)}_{\rho,\omega}(\T^2))}^2 & \le2\|g\|_{H^{(N_v)}_{\rho,\omega}(\T^2)}^2+\frac8\nu\,\|F\|_{L^2((t_1,t_2);H^{(N_v)}_{\rho,\omega}(\T^2))}^2 \notag \\ & \qquad +10\,\|F^0\|_{L^1((t_1,t_2);H^{(N_v)}_{\rho,\omega}(\T^2))}^2\,. \end{align*} \end{proposition} The system~\eqref{e.ft.system} with every~$S_\sigma$ replaced by the identity is of this form, with $A_{vv',d}=\sum_{\sigma\colon v'\to v}A_{\sigma,d}$. \begin{proof} The proof is that of Proposition~\ref{p.ft.energy}, for each~$U_v$ at the order~$N_v$ and with uniqueness by~(a) for differences, except for the source term: by Lemma~\ref{l.ft.product}, with $i=d-2$, \begin{equation*} \|A_{vv',d}\nabla^{d-1}U_{v'}\|_{H^{N_v}_\rho(\T^2)}\le\Bigl(\sum_{j=0}^{N_v}\frac{\rho^j}{j!}\sup_{[t_1,t_2]\times\T^2}|\nabla^jA_{vv',d}|\Bigr)\frac{(N_v+d-2)!}{N_v!\,\rho^{d-2}}\,\|\nabla U_{v'}\|_{H^{N_v+d-2}_\rho(\T^2)}\,, \end{equation*} and, when $A_{vv',d}\ne0$, $\|\nabla U_{v'}\|_{H^{N_v+d-2}_\rho(\T^2)}\le\|\nabla U_{v'}\|_{H^{N_{v'}}_\rho(\T^2)}$; Schur's test concludes. \end{proof} % ===== 08-main-results ===== \section{Reduction and proofs of the main results} \label{s.mainresults} In this section we prove Lemma~\ref{l.micro} and Proposition~\ref{p.transfer}, deduce Theorem~\ref{t.A} from them and Proposition~\ref{p.finite}, state and prove Theorem~\ref{t.C} and Corollary~\ref{c.D}, and prove Theorem~\ref{t.B}. \paragraph{Conventions.} Throughout, $C$ and~$c$ denote positive constants depending only on $\alpha,s,q,\zeta,\Lambda$ and on the quantities named in each statement. We write $D_r(x)\coloneqq\{y\in\R^2:|y-x|0$, the initial-value problem~\eqref{e.ivp} has exactly one weak solution on every $(\tin,\tin+T)$; its restriction to $(\tin',\tin+T)$ is the weak solution with datum $\theta(\tin',\cdot)$ at time~$\tin'$, and, for every $t\in(\tin,\tin+T)$, \begin{equation}\label{e.mr.energy} \|\theta(t,\cdot)\|_{L^2(\T^2)}^2+2\kappa\|\nabla\theta\|_{L^2((\tin,t)\times\T^2)}^2=\|\thetain\|_{L^2(\T^2)}^2\,. \end{equation} These facts are standard (Galerkin approximation, Lions--Magenes), and so is the first assertion below; the second follows from it, since the balls of $C^{0,s}(\T^2)$ are weakly closed in~$L^2(\T^2)$: for every $[t_1,t_2]\subset[\tin,\tin+T]$ and $t\in[t_1,t_2]$, \begin{equation}\tag{\ref*{e.mr.energy}$'$}\label{e.mr.continuity} \theta\in C([\tin,\tin+T];L^2(\T^2))\,,\qquad \|\theta(t,\cdot)\|_{C^{0,s}(\T^2)}\le\|\theta\|_{L^\infty([t_1,t_2];C^{0,s}(\T^2))}\,. \end{equation} So the limit in $L^2_{\mathrm{loc}}$ of $\theta(t',\cdot)$ as $t'\uparrow t$, when it exists, is $\theta(t,\cdot)$. For an open set $\mathcal O\subset\R\times\R^2$ with time sections $\mathcal O_t\coloneqq\{x:(t,x)\in\mathcal O\}$, a \emph{local weak solution} of $\partial_t\theta+\b\cdot\nabla\theta-\kappa\Delta\theta=0$ in~$\mathcal O$ is a function $\theta\in L^2_{\mathrm{loc}}(\mathcal O)$ such that, for almost every~$t$, $\theta(t,\cdot)$ has a weak gradient on~$\mathcal O_t$, with $\nabla\theta\in L^2_{\mathrm{loc}}(\mathcal O)$, and, for every $g\in C^\infty_c(\mathcal O)$, \begin{equation}\label{e.mr.local} \int_{\mathcal O}\bigl(-\theta\,\partial_tg+(\b\cdot\nabla\theta)g+\kappa\nabla\theta\cdot\nabla g\bigr)\dd t\dd x=0\,. \end{equation} A weak solution on $(\tin,\tin+T)$ is a local weak solution in $(\tin,\tin+T)\times\R^2$. Finally, by Lemma~\ref{l.fl.short} and~\eqref{e.renewal}, for every $m\ge0$ and $|t-\tf|\le8h_{m+1}$, the map $y\mapsto X_m(t,y,\tf)$ of~\eqref{e.flow} is a smooth, volume-preserving diffeomorphism of~$\R^2$ commuting with the translations by~$\Z^2$, with inverse $x\mapsto X_m(\tf,x,t)$, and both maps are $2$-Lipschitz, since \begin{equation}\label{e.mr.short} |\nabla_yX_m(t,y,\tf)-I|\le\tfrac12\,,\qquad |\nabla_xX_m(\tf,x,t)-I|\le\tfrac12\,. \end{equation} \subsection{Perturbed heat equations and moving cylinders} \label{ss.mr.cylinders} \begin{proof}[Proof of Lemma~\ref{l.micro}] We adapt Campanato's method of~\cite[Chapter~III, Section~2]{Giaquinta1983} to parabolic equations, with caloric comparison functions and the energy estimates of~\cite{LadyzhenskayaSolonnikovUraltseva1968}. In this proof, $C$ depends only on~$s$. We write $\mathcal Q_r(t,x)\coloneqq(t-r^2,t)\times D_r(x)$ and $\mathcal Q_r\coloneqq\mathcal Q_r(0,0)$, and take $\varpi_1\le1$. \emph{Step 1: Energy and a caloric comparison function.} Testing the equation with $(U-(U)_{\mathcal Q_1})g^2$, for a cutoff~$g$ equal to~$1$ on~$\mathcal Q_{3/4}$ and vanishing near the parabolic boundary of~$\mathcal Q_1$, gives \begin{equation}\label{e.mr.cacc} \int_{\mathcal Q_{3/4}}|\nabla U|^2\dd t\dd x\le C\int_{\mathcal Q_1}\bigl|U-(U)_{\mathcal Q_1}\bigr|^2\dd t\dd x\,: \end{equation} the skew part has no principal term, since $\nabla U\cdot(\mathbf a-I)\nabla U=0$, and the divergence-free drift contributes only $\frac12\int(U-(U)_{\mathcal Q_1})^2\bstar\cdot\nabla g^2\dd t\dd x$. Since $\bstar\cdot\nabla U=\nabla\cdot((U-(U)_{\mathcal Q_1})\bstar)$, $U$ solves $\partial_tU-\Delta U=\nabla\cdot F$ with $F\coloneqq(\mathbf a-I)\nabla U-(U-(U)_{\mathcal Q_1})\bstar$, and $|F|\le\varpi_1(|\nabla U|+|U-(U)_{\mathcal Q_1}|)$. Let~$W$ solve this forced heat equation in~$\mathcal Q_{3/4}$ with zero data, and set $U_{\mathrm{cal}}\coloneqq U-W$. Then $U_{\mathrm{cal}}-(U)_{\mathcal Q_1}$ is caloric in~$\mathcal Q_{3/4}$, and by the energy estimate and~\eqref{e.mr.cacc} \begin{equation*} \int_{\mathcal Q_{3/4}}|W|^2\dd t\dd x\le C\int_{\mathcal Q_{3/4}}|F|^2\dd t\dd x\le C\varpi_1^2\int_{\mathcal Q_1}\bigl|U-(U)_{\mathcal Q_1}\bigr|^2\dd t\dd x\,. \end{equation*} \emph{Step 2: One step of decay.} By the heat kernel representation of a cut-off caloric function, for every $(t,x),(t',x')\in(-\frac14,0]\times D_{1/2}$, \begin{equation*} |U_{\mathrm{cal}}(t,x)-U_{\mathrm{cal}}(t',x')|\le C(|t-t'|+|x-x'|)\int_{\mathcal Q_{3/4}}\bigl|U_{\mathrm{cal}}(\tau,z)-(U)_{\mathcal Q_1}\bigr|\dd\tau\dd z\,. \end{equation*} With Step~1 and $|\mathcal Q_1|/|\mathcal Q_r|=r^{-4}$, this gives, for every $0\frac14$ the seminorm follows from the $L^\infty$ term, since $|U(t,x)-U(t',x')|\le2\|U-(U)_{\mathcal Q_1}\|_{L^\infty((-\frac14,0]\times D_{1/2})}$ and $1<(4r)^{(1+s)/2}$. The uniform continuity of~$U$ on $(-\frac14,0]\times D_{1/2}$ gives the convergence of $U(t,\cdot)$ as $t\uparrow0$. \end{proof} A trajectory of~$\b$ on a time interval is a $C^1$ curve~$X$ with $\partial_{t'}X(t')=\b(t',X(t'))$; since~$\b$ is bounded and continuous, a trajectory ends at every point at every time. \begin{lemma}[H\"older bounds along moving cylinders] \label{l.mr.cylinders} There exist $c_{\mathrm{cyl}}>0$ and $C<\infty$, depending only on $\alpha,s,q,\zeta,\Lambda$, with the following property. Let $\kappa>0$, let $\mathcal O\subset\R\times\R^2$ be open, let~$\theta$ be a local weak solution of $\partial_t\theta+\b\cdot\nabla\theta-\kappa\Delta\theta=0$ in~$\mathcal O$, and let $\Xi\ge0$, $t\in\R$, $E\subset\R^2$ and $00$ and $\varpi\ge0$ with $\|\Phi-\widetilde\Phi\|_{L^\infty(\R\times\T^2)}\le\kappa\varpi$. Let $\tin\in\R$, $T>0$, $\thetain\in L^2(\T^2)$, and let~$\theta$ and~$\widetilde\theta$ be weak solutions on $(\tin,\tin+T)$ of the two equations of Proposition~\ref{p.transfer}, with the datum~$\thetain$ at time~$\tin$. Then \begin{equation*} \|\theta-\widetilde\theta\|_{L^\infty([\tin,\tin+T];L^2(\T^2))}\le\varpi\,\|\thetain\|_{L^2(\T^2)}\,. \end{equation*} \end{lemma} \begin{proof} Since $\nabla^\perp(\Phi-\widetilde\Phi)\cdot\nabla\widetilde\theta=-\nabla\cdot((\Phi-\widetilde\Phi)\nabla^\perp\widetilde\theta)$, the difference $\theta-\widetilde\theta$ solves \begin{equation*} \partial_t(\theta-\widetilde\theta)+\nabla^\perp\Phi\cdot\nabla(\theta-\widetilde\theta)-\kappa\Delta(\theta-\widetilde\theta)=\nabla\cdot\bigl((\Phi-\widetilde\Phi)\nabla^\perp\widetilde\theta\bigr) \end{equation*} with zero datum. Its energy identity, Young's inequality and~\eqref{e.mr.energy} for~$\widetilde\theta$ give, for $\tin\beta$ and, as $\beta=1+\alpha$ and $s<\frac12(1-\alpha)$, $(1+s)(2-\beta)<\frac12(3-\alpha)(1-\alpha)<2\beta$. Fix an integer $n\ge1$ with \begin{equation}\label{e.mr.levels} q^n\Bigl(\beta q-\frac{(1+s)(2-\beta)}2\Bigr)-\frac{(\beta-\gamma)(3+s)}2\ge0\,; \end{equation} it depends only on $\alpha,s,q$. \emph{Step 2: The bands far out.} Let $\kappa\in\Kset_M$, so that $\frac12\eps_M^{\beta-\gamma}\le\kappa\le2\eps_M^{\beta-\gamma}$. We apply Proposition~\ref{p.transfer}, whose regularity hypotheses hold by Proposition~\ref{p.fl.velocity}(i), (iii) and~(iv), with $\Phi=\phi$, $\widetilde\Phi=\phi_{M+n}$ and \begin{equation*} r\coloneqq\Bigl(\frac{\varpi_0\kappa}{2C_0}\Bigr)^{\nf12}\eps_{M+n}^{(2-\beta)/2}\,,\qquad \varpi\coloneqq\frac{C_0\eps_{M+n+1}^\beta}\kappa\,, \end{equation*} so that, by~\eqref{e.mr.C0}, $\|\phi-\phi_{M+n}\|_{L^\infty(\R\times\T^2)}\le\kappa\varpi$ and $\frac{r^2}\kappa\|\nabla^2\phi_{M+n}\|_{L^\infty(\R\times\T^2)}\le\frac12\varpi_0$. Since $\eps_m\asymp\eps_{m-1}^q$, we have $\eps_{M+n+1}\le C\eps_M^{q^{n+1}}$ and $\eps_{M+n}\ge c\eps_M^{q^n}$, with constants depending on~$n$, and, as $\kappa\ge\frac12\eps_M^{\beta-\gamma}$, \begin{equation*} \varpi\,r^{-1-s}\le C\eps_{M+n+1}^\beta\kappa^{-(3+s)/2}\eps_{M+n}^{-(2-\beta)(1+s)/2}\le C\eps_M^{q^n(\beta q-(1+s)(2-\beta)/2)-(\beta-\gamma)(3+s)/2}\le C \end{equation*} by~\eqref{e.mr.levels}. Moreover $r\le C\eps_M^{(\beta-\gamma)/2}$, $r^2/\kappa\le C\eps_M^{2-\beta}$ and $\varpi\le C\eps_M^{\beta q^{n+1}-(\beta-\gamma)}$ tend to zero as $M\to\infty$, uniformly in $\kappa\in\Kset_M$; we fix~$M_{\mathrm A}$, depending only on $\alpha,s,q,\zeta,\Lambda,t_0,T$, such that, for $M\ge M_{\mathrm A}$, $r\le\frac18$, $r^2/\kappa\le\min\{\frac12t_0,1\}$, $\varpi\le\frac12\varpi_0$, and~$M$ is at least the threshold~$M_1$ of Proposition~\ref{p.finite} for the times $\frac12t_00$ and a final time $\tf\in\R$, the \emph{tube} is \begin{equation}\label{e.mr.tube} \mathcal T\coloneqq\bigl\{(t,X_m(t,y,\tf)):\tf-8h_{m+1}0$, a \emph{local energy solution} of $\partial_t\theta+\b\cdot\nabla\theta-\kappa\Delta\theta=0$ in~$\mathcal T$ is a function $\theta\in L^2(\mathcal T)$ such that, for every $g\in C^\infty_c(\mathcal T)$, \begin{equation}\label{e.mr.distributional} \int_{\mathcal T}\theta\,\bigl(\partial_tg+\b\cdot\nabla g+\kappa\Delta g\bigr)\dd t\dd x=0\,, \end{equation} which is the equation, written as $\partial_t\theta+\nabla\cdot(\b\theta)-\kappa\Delta\theta=0$ since $\nabla\cdot\b=0$, and such that, in the coordinates~$y$, $U(t,y)\coloneqq\theta(t,X_m(t,y,\tf))$ satisfies $U\in L^\infty((\tf-8h_{m+1},\tf);L^2(D_{r_{\mathrm{tube}}}))$ and $\nabla_yU\in L^2((\tf-8h_{m+1},\tf)\times D_{r_{\mathrm{tube}}})$. No lateral data are prescribed. \setcounter{maintheorem}{2} \begin{maintheorem}[Local interior regularity]\label{t.C} In the setting of Theorem~\ref{t.A}, with~$\Lambda_0$ as there, let $\Lambda\ge\Lambda_0$ be an integer. There exists $C_1(\alpha,s,q,\zeta,\Lambda)\ge1$ with the following property. Let $m\ge0$ be an integer and let~$r_{\mathrm{tube}}$ satisfy \begin{equation}\label{e.mr.radius} C_1\max\bigl\{\eps_{m+1}^\alpha h_{m+1},\ (h_{m+1}\sup\Kset)^{\nf12}\bigr\}\le r_{\mathrm{tube}}\le\frac18\,. \end{equation} For every $00$, depending only on $\alpha,s,q,\zeta,\Lambda$, with the following property. Let $m\ge0$, let $r_{\mathrm{tube}}\le\frac18$ satisfy $C_1\eps_{m+1}^\alpha h_{m+1}\le r_{\mathrm{tube}}$, let $\tf\in\R$, let $\kappa>0$ satisfy $C_1(\kappa h_{m+1})^{\nf12}\le r_{\mathrm{tube}}$, let~$\theta$ be a local energy solution in~$\mathcal T$, and let $\tin\coloneqq \tf-\frac92h_{m+1}$. There exists $\thetain\in L^2(\T^2)$ with $\|\thetain\|_{L^2(\T^2)}\le Cr_{\mathrm{tube}}(\fint_{\mathcal T}\theta(t,x)^2\dd t\dd x)^{\nf12}$ such that the weak solution~$\theta^{\mathrm{per}}$ of~\eqref{e.ivp} on $(\tin,\tin+5h_{m+1})$ with datum~$\thetain$ satisfies, with $U^{\mathrm{per}}(t,y)\coloneqq\theta^{\mathrm{per}}(t,X_m(t,y,\tf))$, \begin{equation}\label{e.mr.localization} \|U-U^{\mathrm{per}}\|_{L^\infty((\tin,\tf);L^2(D_{r_{\mathrm{tube}}/8}))}\le Cr_{\mathrm{tube}}\Bigl(\fint_{\mathcal T}\theta(t,x)^2\dd t\dd x\Bigr)^{\nf12}\exp\Bigl(-\frac{c\,r_{\mathrm{tube}}^2}{\kappa h_{m+1}}\Bigr)\,. \end{equation} \end{lemma} \begin{proof} \emph{Step 1: The equation in the coordinates~$y$.} For $\tf-8h_{m+1}\le t\le \tf$, changing variables by the volume-preserving map $y\mapsto X_m(t,y,\tf)$, whose time derivative is~$\b_m$, turns~\eqref{e.mr.distributional} into \begin{equation}\label{e.mr.label} \begin{gathered} \partial_tU=\nabla_y\cdot\bigl(\kappa\mathbf M\nabla_yU-U\bstar\bigr)\,,\\ \mathbf M\coloneqq(\nabla_yX_m)^{-1}(\nabla_yX_m)^{-\mathsf T},\qquad \bstar\coloneqq(\nabla_yX_m)^{-1}(\b-\b_m)(t,X_m(t,y,\tf))\,, \end{gathered} \end{equation} in $(\tf-8h_{m+1},\tf)\times D_{r_{\mathrm{tube}}}$, where $\nabla_yX_m$ is evaluated at $(t,y,\tf)$, and~$U^{\mathrm{per}}$ solves it in $(\tin,\tf)\times\R^2$. Since $\det\nabla_yX_m=1$, the Piola identity (the cofactor matrix of a smooth map is divergence free) gives $\nabla_y\cdot\bstar=(\nabla\cdot(\b-\b_m))\circ X_m=0$, and, applied to~$\nabla\theta$ in place of $\b-\b_m$, it puts the equation in this divergence form. By~\eqref{e.mr.C0} and~\eqref{e.mr.short}, $|\bstar|\le2C_0\eps_{m+1}^\alpha$, and, as $\det\nabla_yX_m=1$, $\mathbf M$ has the eigenvalues of $(\nabla_yX_m)^{\mathsf T}\nabla_yX_m$, which lie in $[\frac14,\frac94]$. \emph{Step 2: Energy after the start of the tube.} Let $g\in C^\infty_c(D_{r_{\mathrm{tube}}})$ with $0\le g\le1$ and $|\nabla_yg|\le C/r_{\mathrm{tube}}$. Testing~\eqref{e.mr.label} with~$g^2U$ times a time cutoff that rises on $(\tf-8h_{m+1},\tin)$ shows that $gU$ is continuous in time with values in $L^2(D_{r_{\mathrm{tube}}})$ on $[\tin,\tf]$ (Lions--Magenes) and, since $\int_{(\tf-8h_{m+1},\tf)\times D_{r_{\mathrm{tube}}}}U^2=8\pi h_{m+1}r_{\mathrm{tube}}^2\fint_{\mathcal T}\theta^2$ by volume preservation, \begin{equation}\label{e.mr.localenergy} \|gU\|_{L^\infty([\tin,\tf];L^2(D_{r_{\mathrm{tube}}}))}^2\le C\Bigl(\frac1{h_{m+1}}+\frac\kappa{r_{\mathrm{tube}}^2}+\frac{\eps_{m+1}^\alpha}{r_{\mathrm{tube}}}\Bigr)\int_{\mathcal T}\theta^2\dd t\dd x\le Cr_{\mathrm{tube}}^2\fint_{\mathcal T}\theta^2\dd t\dd x\,, \end{equation} by the hypotheses on~$r_{\mathrm{tube}}$ and~$\kappa$, with $C_1\ge1$. We apply~\eqref{e.mr.localenergy} with $g=g_1$ and with $g=g_2$, where $g_1(y)\coloneqq\mathsf H(3-4|y|/r_{\mathrm{tube}})$ equals~$1$ on $D_{r_{\mathrm{tube}}/2}$ and~$0$ outside $D_{3r_{\mathrm{tube}}/4}$, and $g_2(y)\coloneqq\mathsf H(7-8|y|/r_{\mathrm{tube}})$ equals~$1$ on $D_{3r_{\mathrm{tube}}/4}$ and~$0$ outside $D_{7r_{\mathrm{tube}}/8}$. \emph{Step 3: The datum.} The function $g_1U(\tin,\cdot)$ is supported in $D_{3r_{\mathrm{tube}}/4}\subset D_{1/8}$, so its translates by~$\Z^2$ have disjoint supports. Let~$\thetain$ be their sum, composed with $x\mapsto X_m(\tf,x,\tin)$. It is $\Z^2$-periodic, $U^{\mathrm{per}}(\tin,\cdot)$ equals $g_1U(\tin,\cdot)$ on~$D_{r_{\mathrm{tube}}}$, and $\|\thetain\|_{L^2(\T^2)}\le Cr_{\mathrm{tube}}(\fint_{\mathcal T}\theta^2)^{\nf12}$ by~\eqref{e.mr.localenergy} with $g=g_1$. By~\eqref{e.mr.energy} and the four translates of the unit cell, since $X_m(t,D_{r_{\mathrm{tube}}},\tf)$ lies in a disc of radius $2r_{\mathrm{tube}}<\frac12$ by~\eqref{e.mr.short}, $\|U^{\mathrm{per}}\|_{L^\infty((\tin,\tf);L^2(D_{r_{\mathrm{tube}}}))}^2\le4\|\thetain\|_{L^2(\T^2)}^2\le Cr_{\mathrm{tube}}^2\fint_{\mathcal T}\theta^2$. \emph{Step 4: Exponential separation.} We use energy estimates with an exponential weight of the form used in~\cite[Section~2]{Davies87}. For a length $r'>0$, the weight \begin{equation*} \exp\Bigl(-\frac{r_{\mathrm{tube}}}{8r'}\,\mathsf H\Bigl(\frac{4|y|}{r_{\mathrm{tube}}}-\frac12\Bigr)\Bigr) \end{equation*} equals~$1$ on $D_{r_{\mathrm{tube}}/8}$, equals $\exp(-r_{\mathrm{tube}}/(8r'))$ outside $D_{3r_{\mathrm{tube}}/8}$, and the gradient of its logarithm is at most~$C/r'$. Testing the equation of $U-U^{\mathrm{per}}$ shows that the integral over $D_{r_{\mathrm{tube}}}$ of the square of this weight times $g_2(y)^2(U-U^{\mathrm{per}})(t,y)^2$ grows at most at the rate $C(\kappa/r'^2+\eps_{m+1}^\alpha/r')$: the derivatives of the weight cost~$1/r'$ against the diffusion and the divergence-free drift, and Young's inequality absorbs the rest into the dissipation. The other terms contain derivatives of~$g_2$, which vanish on~$D_{3r_{\mathrm{tube}}/4}$, so they carry the factor $\exp(-r_{\mathrm{tube}}/(4r'))$, the square of the weight outside $D_{3r_{\mathrm{tube}}/8}$. After Young's inequality, which keeps~$g_2$ in front of the gradient of $U-U^{\mathrm{per}}$, their integral in time is at most $C(\kappa/r_{\mathrm{tube}}^2+\eps_{m+1}^\alpha/r_{\mathrm{tube}})$ times the integral of $U^2+(U^{\mathrm{per}})^2$ over $(\tin,\tf)\times D_{r_{\mathrm{tube}}}$, which is at most $Ch_{m+1}r_{\mathrm{tube}}^2\fint_{\mathcal T}\theta^2$ by Steps~2 and~3; and $(\kappa/r_{\mathrm{tube}}^2+\eps_{m+1}^\alpha/r_{\mathrm{tube}})h_{m+1}\le2$ by the hypotheses. The initial value is bounded by~\eqref{e.mr.localenergy} with $g=g_2$, since at time~$\tin$ the difference $U-U^{\mathrm{per}}=(1-g_1)U(\tin,\cdot)$ vanishes on~$D_{r_{\mathrm{tube}}/2}$. Since $\tf-\tin=\frac92h_{m+1}$, Gr\"onwall's inequality gives, for $\tin\tin$. In the coordinates~$y$, the point $X(t')$ moves with the velocity~$\bstar$ of~\eqref{e.mr.label}, which equals $\nabla_xX_m(\tf,x,t')(\b-\b_m)(t',x)$ at $x=X(t')$ and has size at most $\frac32C_0\eps_{m+1}^\alpha$ by~\eqref{e.mr.C0} and~\eqref{e.mr.short}, so it stays within $\frac32C_0\eps_{m+1}^\alpha r^2/\kappa$ of~$D_{r_{\mathrm{obs}}}$, a quarter of the term $6C_0\eps_{m+1}^\alpha r^2/\kappa$ allowed by~\eqref{e.mr.contain}; and $x\mapsto X_m(\tf,x,t')$ is $2$-Lipschitz. By the first bound in~\eqref{e.mr.contain}, the moving cylinder of radius~$r$ around~$X$ lies in~$\mathcal O$. \emph{Step 4: The bound before the end of the tube.} Lemma~\ref{l.mr.cylinders} applies to $\theta-\theta^{\mathrm{per}}$ in~$\mathcal O$ at every $t\in[\tf-8h_{m+1}+t_0,\tf)$, with $E=E_t$ and~$\Xi$ of Step~2, and gives $\|(\theta-\theta^{\mathrm{per}})(t,\cdot)\|_{C^{0,s}(E_t)}\le Cr^{-1-s}\Xi$. The power of~$\kappa$ in~$r^{-1-s}$ is absorbed by the exponential in~$\Xi$, since $\kappa\mapsto\kappa^{-(1+s)/\beta}\exp(-cr_{\mathrm{tube}}^2/(\kappa h_{m+1}))$ and $\kappa\mapsto\kappa^{-(1+s)/2}\exp(-cr_{\mathrm{tube}}^2/(\kappa h_{m+1}))$ are bounded on $(0,\infty)$: $r^{-1-s}\Xi\le C(\fint_{\mathcal T}\theta^2)^{\nf12}$, with~$C$ depending on $m$, $r_{\mathrm{tube}}$, $r_{\mathrm{obs}}$ and~$t_0$. With Step~1, $\theta(t,\cdot)$ is continuous on~$E_t$ and $\|\theta(t,\cdot)\|_{C^{0,s}(E_t)}\le C(\fint_{\mathcal T}\theta^2)^{\nf12}$. Since $y\mapsto X_m(t,y,\tf)$ is $2$-Lipschitz by~\eqref{e.mr.short}, $U(t,\cdot)$ is continuous and $\|U(t,\cdot)\|_{C^{0,s}(D_{r_{\mathrm{obs}}})}\le2^s\|\theta(t,\cdot)\|_{C^{0,s}(E_t)}$; the convergence of $U(t',\cdot)$ to $U(t,\cdot)$ in~$L^2(D_{r_{\mathrm{obs}}})$ as $t'\uparrow t$ follows from that of Lemma~\ref{l.mr.cylinders} for $\theta-\theta^{\mathrm{per}}$, the continuity in time of~$\theta^{\mathrm{per}}$ and the continuity of~$X_m$. \emph{Step 5: The end of the tube.} By Step~2 of the proof of Lemma~\ref{l.mr.localization}, $U(t',\cdot)$ converges in~$L^2(D_{r_{\mathrm{obs}}})$ as $t'\uparrow \tf$; by the uniform bound of Step~4, the limit is continuous and satisfies the same bound. \end{proof} \subsection{Vanishing diffusivity: Corollary~\ref{c.D}} \label{ss.mr.C} As $\nabla\cdot\b=0$, the transport equation reads $\partial_t\theta+\nabla\cdot(\b\theta)=0$, meaningful for continuous~$\theta$. \begin{maincorollary}[Vanishing-diffusivity limits]\label{c.D} In the setting of Theorem~\ref{t.A}, with~$\Lambda_0$ as there, let $\Lambda\ge\Lambda_0$ be an integer. \begin{enumerate}[label=\textup{(\alph*)},leftmargin=*] \item There exist constants $C(\alpha,s,q,\zeta,\Lambda,t_0,T)$, for $01$ the bound $CA_0$ on the slices gives it. So every~$\theta^\kappa$ is continuous, and the~$\theta^\kappa$ are bounded and equicontinuous on every $[\tin+t_0,\tin+T]\times\T^2$; the theorem of Arzel\`a and Ascoli with a diagonal argument, and the weak compactness of the balls of~$L^2(\T^2)$, give~(i). \emph{(a), the limit.} For $g\in C^\infty_c((\tin,\infty)\times\R^2)$, $\int\theta^\kappa(\partial_tg+\b\cdot\nabla g+\kappa\Delta g)=0$; since $\b\theta^\kappa\to\b\theta_*$ uniformly on the support of~$g$ as $\kappa\to0$ in~$\Kset''$, the limit is~(ii). The bound of~(iii) passes to the limit pointwise, and the computation above with $\kappa=0$, for~$\theta_*$, and $r=|t-t'|^{1/(1+s)}$ for $|t-t'|\le1$ give the exponent $s/(1+s)$. For the initial trace, let~$g$ be smooth and $\Z^2$-periodic. By the weak form of the equation, the Cauchy--Schwarz inequality and~\eqref{e.mr.energy}, for $t>\tin$, \begin{equation*} \Bigl|\int_{\T^2}\bigl(\theta^\kappa(t,x)-\thetain^\kappa(x)\bigr)g(x)\dd x\Bigr|\le(t-\tin)A_0\bigl\|\b\cdot\nabla g+\kappa\Delta g\bigr\|_{L^\infty((\tin,t)\times\T^2)}\,. \end{equation*} Letting first $\kappa\to0$ in~$\Kset''$ and then $t\downarrow \tin$ gives $\int\theta_*(t,\cdot)g\to\int \theta_{\mathsf{in},*}g$; since $\|\theta_*(t,\cdot)\|_{L^2(\T^2)}\le A_0$, this extends to every $g\in L^2(\T^2)$. \emph{(a), part~(iv).} Each finite union of bands is bounded away from zero, so $M\to\infty$ as $\kappa\to0$ in~$\Kset'$, where $\kappa\in\Kset_M$. Lemma~\ref{l.mr.stream}, with $\Phi=\phi$, $\widetilde\Phi=\phi_M$ and, by~\eqref{e.mr.C0} and $\kappa\ge\frac12\eps_M^{\beta-\gamma}$, \begin{equation*} \varpi\coloneqq\frac{C_0\eps_{M+1}^\beta}{\kappa}\le C\eps_M^{q\beta-\beta+\gamma}=C\eps_M^{(q+2)\gamma}\to0\,, \end{equation*} where $\beta(q-1)=(q+1)\gamma$, gives, for every $T>0$, $\|\theta^\kappa-\widetilde\theta^\kappa\|_{L^\infty([\tin,\tin+T];L^2(\T^2))}\le\varpi A_0$, hence~(iv). \emph{(b).} This is the same argument in the tube, with Theorem~\ref{t.C} in place of Theorem~\ref{t.A}. The sets $\{(t,X_m(t,y,\tf)):\tf-8h_{m+1}+t_0\le t\le \tf,\ |y|0$ small, depending on~$m$, $\frac18r_{\mathrm{tube}}-r_{\mathrm{obs}}$ and $t_0-4h_{m+1}$, are supported in a slightly larger such set at every time $\tau\in[t',t]$, since, for $z\in D_r(x)$, $|X_m(\tf,z,\tau)-y|\le2(r+\|\b_m\|_{L^\infty(\R\times\T^2)}(t-\tau))$ by~\eqref{e.mr.short}. They give the same modulus in time for $t<\tf$, and, by the limit of Theorem~\ref{t.C} at the end of the tube, for $t=\tf$. \end{proof} \subsection{Uniform anomalous dissipation: Theorem~\ref{t.B}} \label{ss.mr.D} \begin{lemma}[Positive H\"older-smoothing operators contract] \label{l.mr.operator} Let $N\ge1$ be an integer, and let $\mathcal S$ be a linear map from $L^2(\T^2)$ to the continuous functions on~$\T^2$ such that $\mathcal S1=1$ and, for every $f\in L^2(\T^2)$, $\int_{\T^2}\mathcal Sf(x)\dd x=\int_{\T^2}f(x)\dd x$, $\|\mathcal Sf\|_{C^{0,s}(\T^2)}\le N^s\|f\|_{L^2(\T^2)}$ and, if $f\ge0$, $\mathcal Sf\ge0$. Then, for every $f\in L^2(\T^2)$ with $\int_{\T^2}f(x)\dd x=0$, \begin{equation*} \|\mathcal Sf\|_{L^2(\T^2)}^2\le\Bigl(1-\frac1{2N^4}\Bigr)\|f\|_{L^2(\T^2)}^2\,. \end{equation*} \end{lemma} \begin{proof} \emph{Step 1: Rows.} For every $x\in\T^2$, $f\mapsto\mathcal Sf(x)$ is bounded on~$L^2(\T^2)$, so, by the Riesz representation theorem, $\mathcal Sf(x)=\langle\mathcal S_x,f\rangle_{L^2(\T^2)}$ for some $\mathcal S_x\in L^2(\T^2)$, the row of~$\mathcal S$ at~$x$. The H\"older bound gives $\|\mathcal S_x-\mathcal S_y\|_{L^2(\T^2)}\le N^s|x-y|_{\T^2}^s$. Positivity gives $\mathcal S_x\ge0$, since $0\le\mathcal S(\mathcal S_x^-)(x)=-\|\mathcal S_x^-\|_{L^2(\T^2)}^2$ for $\mathcal S_x^-\coloneqq\max\{-\mathcal S_x,0\}$; $\mathcal S1=1$ gives $\int_{\T^2}\mathcal S_x(z)\dd z=1$, hence $\|\mathcal S_x\|_{L^2(\T^2)}\ge1$; and the preservation of integrals gives $\int_{\T^2}\mathcal S_x\dd x=1$. \emph{Step 2: The kernel of $\mathcal S\mathcal S^*$.} The function $\langle\mathcal S_x,\mathcal S_y\rangle_{L^2(\T^2)}$ is continuous, symmetric and nonnegative, for every~$x$ its integral in~$y$ is~$1$, and, by polarization, it equals $\frac12(\|\mathcal S_x\|_{L^2(\T^2)}^2+\|\mathcal S_y\|_{L^2(\T^2)}^2-\|\mathcal S_x-\mathcal S_y\|_{L^2(\T^2)}^2)\ge1-\frac12N^{2s}|x-y|_{\T^2}^{2s}$, which, if $|x-y|_{\T^2}\le1/N$, is at least~$\frac12$. The adjoint is $\mathcal S^*g=\int_{\T^2}g(x)\mathcal S_x\dd x$, and expanding $\|\mathcal S^*g\|_{L^2(\T^2)}^2$ gives, for continuous~$g$, \begin{equation}\label{e.mr.adjoint} \|g\|_{L^2(\T^2)}^2-\|\mathcal S^*g\|_{L^2(\T^2)}^2=\frac12\int_{\T^2\times\T^2}\langle\mathcal S_x,\mathcal S_y\rangle_{L^2(\T^2)}\,|g(x)-g(y)|^2\dd x\dd y\,. \end{equation} \emph{Step 3: A Poincar\'e inequality by translations.} Let~$g$ be continuous, periodic and of mean zero. Then the average over $z\in[-\frac12,\frac12)^2$ of $\int_{\T^2}|g(x+z)-g(x)|^2\dd x$ is $2\|g\|_{L^2(\T^2)}^2$. Writing $g(\cdot+Nz)-g$ as a sum of~$N$ increments by~$z$, Cauchy--Schwarz bounds the integral at~$Nz$ by~$N^2$ times the integral at~$z$, and the change of variables from~$Nz$ to~$z$ gives \begin{equation*} 2\|g\|_{L^2(\T^2)}^2\le N^4\int_{[-\frac1{2N},\frac1{2N})^2\times\T^2}|g(x+z)-g(x)|^2\dd z\dd x\,. \end{equation*} For $z\in[-\frac1{2N},\frac1{2N})^2$, $|z|\le1/N$, so, by Step~2 and the change of variables $y=x+z$ in~\eqref{e.mr.adjoint}, the right side of~\eqref{e.mr.adjoint} is at least $\frac14\int_{[-\frac1{2N},\frac1{2N})^2\times\T^2}|g(x+z)-g(x)|^2\dd z\dd x\ge\frac1{2N^4}\|g\|_{L^2(\T^2)}^2$. \emph{Step 4: Conclusion.} Let $\int_{\T^2}f=0$ and $g\coloneqq\mathcal Sf$, which is continuous and of mean zero. Then $\|g\|_{L^2(\T^2)}^2=\langle f,\mathcal S^*g\rangle_{L^2(\T^2)}\le\|f\|_{L^2(\T^2)}\|\mathcal S^*g\|_{L^2(\T^2)}\le(1-\frac1{2N^4})^{\nf12}\|f\|_{L^2(\T^2)}\|g\|_{L^2(\T^2)}$ by Step~3. \end{proof} \begin{proof}[Proof of Theorem~\ref{t.B}] Fix $\tin\in\R$ and $\kappa\in\Kset$, and let $\mathcal S\thetain\coloneqq\theta(\tin+1,\cdot)$, with~$\theta$ the weak solution with datum~$\thetain$. By uniqueness, $\mathcal S$ is linear; constants are solutions, so $\mathcal S1=1$; testing the equation with a function of time only gives $\int_{\T^2}\mathcal S\thetain=\int_{\T^2}\thetain$, since $\nabla\cdot\b=0$. The operator~$\mathcal S$ is positive by the maximum principle: if $\thetain\ge0$, testing the equation with $\min\{\theta,0\}$ (Stampacchia's truncation) gives $\int_{\T^2}\min\{\theta(t,\cdot),0\}^2\dd x\le\int_{\T^2}\min\{\thetain,0\}^2\dd x=0$ for every $t>\tin$: the diffusion term has a sign, and the transport term vanishes, as $\nabla\cdot\b=0$. By Theorem~\ref{t.A} on $(\tin,\tin+1)$ with $t_0=\frac12$ and~\eqref{e.mr.continuity}, $\mathcal S$ is bounded from $L^2(\T^2)$ into $C^{0,s}(\T^2)$, and, by Lemma~\ref{l.mr.operator} with~$N$ the least integer such that~$N^s$ is at least the constant of Theorem~\ref{t.A}, we obtain, with $c\coloneqq1/(4N^4)$, \begin{equation*} \|\mathcal Sg\|_{L^2(\T^2)}^2\le(1-2c)\|g\|_{L^2(\T^2)}^2\qquad\mbox{for every }g\in L^2(\T^2)\mbox{ with }\int_{\T^2}g(x)\dd x=0\,. \end{equation*} Since the constant of Theorem~\ref{t.A} is independent of~$\tin$ and of $\kappa\in\Kset$, so are~$N$ and~$c$. Since constants are solutions, $\theta-(\thetain)_{\T^2}$ is the weak solution with datum $\thetain-(\thetain)_{\T^2}$, and $\theta(\tin+1,\cdot)-(\thetain)_{\T^2}=\mathcal S(\thetain-(\thetain)_{\T^2})$. The energy identity~\eqref{e.mr.energy} for $\theta-(\thetain)_{\T^2}$ gives \begin{align*} 2\kappa\|\nabla\theta\|_{L^2((\tin,\tin+1)\times\T^2)}^2&=\bigl\|\thetain-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}^2-\bigl\|\mathcal S(\thetain-(\thetain)_{\T^2})\bigr\|_{L^2(\T^2)}^2\\ &\ge2c\,\bigl\|\thetain-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}^2\,, \end{align*} which is~\eqref{e.dissipation}. For $\tin'>\tin$, the restriction of~$\theta$ to $(\tin',\infty)$ is the weak solution with datum $\theta(\tin',\cdot)$, whose mean is $(\thetain)_{\T^2}$. So the contraction, with $\tin+k$ in place of~$\tin$, gives $\|\theta(\tin+k+1,\cdot)-(\thetain)_{\T^2}\|_{L^2(\T^2)}^2\le(1-2c)\|\theta(\tin+k,\cdot)-(\thetain)_{\T^2}\|_{L^2(\T^2)}^2$ for every integer $k\ge0$, and, by~\eqref{e.mr.energy}, $\|\theta(t,\cdot)-(\thetain)_{\T^2}\|_{L^2(\T^2)}$ does not increase in~$t$. Hence, for every $t\ge \tin$, as $1-2c\le\exp(-2c)$ and $c\le\frac12$, we obtain~\eqref{e.decay}: \begin{equation*} \bigl\|\theta(t,\cdot)-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}^2\le(1-2c)^{\lfloor t-\tin\rfloor}\bigl\|\thetain-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}^2\le4\exp(-2c(t-\tin))\bigl\|\thetain-(\thetain)_{\T^2}\bigr\|_{L^2(\T^2)}^2\,.\qedhere \end{equation*} \end{proof} % ===== 09-cell ===== \section{Cell problems and the averaged flux} \label{s.cell} In this section we construct, on each pulse, the finite lift of a system with sources: correctors of every degree up to~$\Ncor$, which oscillate at the wavelength~$\eps_m$, and the means they leave behind, the sources of the successor. In the chart of its block (Section~\ref{s.flows}), a pulse of level~$m$ is a single shear, whose cell problems are one-dimensional and explicitly solvable. The first correctors are those of~\cite[Section~3.2]{AV}. Throughout, $\Lambda\ge128$ is an integer, and~$C$, $c$ denote positive constants depending only on $\alpha,s,q,\zeta$, unless another dependence is stated. For $M\ge1$, $\kappa\in\Kset_M$ and $1\le m\le M$, recall that $\ebar=\eps_m(\kappa_{m-1}/\kappa_m)^{\nf12}$ is the corrector amplitude of Subsection~\ref{ss.idea}, and~$\rho(m)=(\kappa_{m-1}h_m)^{\nf12}$ is the effective diffusion length~\eqref{e.rho}. \paragraph{The window of a pulse.} Let $m\ge1$ and~$k$ be odd, and let $\eperp\coloneqq e_1$ for $k\in4\Z+1$ and $\eperp\coloneqq e_2$ for $k\in4\Z+3$, the coordinate vector other than~$e_{(k)}$. By~\eqref{e.shear}, $\psi_{m,k}(y)=\eps_m^\beta\sin(2\pi y\cdot\eperp/\eps_m)$ and \begin{equation}\label{e.cell.shear} \nabla^\perp\psi_{m,k}(y)=\pm2\pi\eps_m^{\beta-1}\cos\bigl(2\pi y\cdot\eperp/\eps_m\bigr)\,e_{(k)}\,, \end{equation} with~$+$ for $k\in4\Z+1$ and~$-$ for $k\in4\Z+3$. The \emph{window of pulse~$k$} and the \emph{time cutoff} of~$k$ are \begin{equation}\label{e.cell.window} |t-k\tau_m|\le\frac{61}{48}\tau_m\,,\qquad \Hc_{m,k}(t)\coloneqq\Hc\Bigl(\frac52-\frac{2|t-k\tau_m|}{\tau_m}\Bigr), \end{equation} with~$\Hc$ of~\eqref{e.slow}, which is~$0$ on $(-\infty,0]$ and~$1$ on $[1,\infty)$. As the envelope $\widetilde\zeta_{m,k}$ satisfies $0\le\widetilde\zeta_{m,k}\le\zeta_{m,k}$ (by~\eqref{e.zeta}, Lemma~\ref{l.fl.scales}(d)), it vanishes for $|t-k\tau_m|\ge\frac23\tau_m$; as the centres of odd slots are~$2\tau_m$ apart, no other pulse is active on the window of pulse~$k$, where therefore $\psi_m(t,y)=\widetilde\zeta_{m,k}(t)\psi_{m,k}(y)$ (see~\eqref{e.fl.psim}). The cutoff~$\Hc_{m,k}$ is~$1$ for $|t-k\tau_m|\le\frac34\tau_m$, hence where $\widetilde\zeta_{m,k}\ne0$; it vanishes for $|t-k\tau_m|\ge\frac54\tau_m$; and $\sum_{k\in2\Z+1}\Hc_{m,k}=1$, since $\Hc(x)+\Hc(1-x)=1$. By Lemma~\ref{l.fl.stream}(i), in the chart $y\mapsto X_{m-1}(t,y,l_{m,k}h_m)$ of~$k$ the velocity added at level~$m$ is $\nabla^\perp\psi_m(t,y)$ in the block of~$k$, and it vanishes on the rest of the window; the chart and~$\K_{m,l_{m,k}}$ are data on the whole window. A \emph{two-scale form} on the window of pulse~$k$ is a function $\mathcal A(t,y,z)$ of~$t$ in the window and $y,z\in\R^2$, $1$-periodic in the \emph{fast variable}~$z$, read at $z=y/\eps_m$; it is \emph{transverse} if it depends on~$z$ only through $z\cdot\eperp$. We write $(\mathcal A)_{\T^2}(t,y)\coloneqq\fint_{[0,1]^2}\mathcal A(t,y,z)\dd z$ for its average in~$z$, and, for a transverse form, $\widehat{\mathcal A}(t,y;\xi)$ for its $\xi$-th Fourier coefficient in $z\cdot\eperp$, $\xi\in\Z$, as in Section~\ref{s.admissible}. \subsection{The corrector of a pulse and the averaged flux} \label{ss.cell.corrector} \begin{lemma}[The corrector of a pulse]\label{l.cell.corrector} Let $m\ge1$, $\nu>0$ and~$k$ odd, and let \begin{equation}\label{e.cell.corrector} \chi_{m,k}^\nu(t,y)\coloneqq-\Bigl(\int_0^\infty\exp\Bigl(-\frac{4\pi^2\nu t'}{\eps_m^2}\Bigr)\widetilde\zeta_{m,k}(t-t')\dd t'\Bigr)\nabla^\perp\psi_{m,k}(y)\,. \end{equation} \begin{enumerate}[label=\textup{(\alph*)}] \item $\chi_{m,k}^\nu$ is $\Z^2$-periodic and smooth in~$y$, continuously differentiable in~$t$, of mean zero over $[0,1]^2$, parallel to~$e_{(k)}$, a function of $y\cdot\eperp$, and it solves, componentwise, \begin{equation}\label{e.cell.problem} \left\{ \begin{aligned} & \partial_t\chi_{m,k}^\nu+\widetilde\zeta_{m,k}\nabla^\perp\psi_{m,k}\cdot\nabla\chi_{m,k}^\nu-\nu\Delta\chi_{m,k}^\nu=-\widetilde\zeta_{m,k}\nabla^\perp\psi_{m,k} & \mbox{in} & \ \R\times\R^2\,,\\ & \chi_{m,k}^\nu=0 & \mbox{on} & \ (-\infty,(k-\tfrac23)\tau_m]\times\R^2\,, \end{aligned} \right. \end{equation} whose transport term vanishes identically. \item $\|\chi_{m,k}^\nu\|_{L^\infty(\R\times\T^2)}\le\eps_m^{\beta+1}/(2\pi\nu)$. \item If $m\ge2$, there exist $\Lambda_0(\alpha,s,q,\zeta)\ge128$ and $C(\alpha,s,q,\zeta)<\infty$ such that, for every $\Lambda\ge\Lambda_0$, $M\ge m$ and $\kappa\in\Kset_M$, \begin{equation}\label{e.cell.amplitude} \frac{\eps_m^{2\beta}}{\kappa_m}\le20\kappa_{m-1}\,,\qquad \|\chi_{m,k}^{\kappa_m}\|_{L^\infty(\R\times\T^2)}\le C\ebar\le C\eps_{m-1}^{3\delta}\rho(m)\,. \end{equation} \end{enumerate} \end{lemma} This is the corrector of~\cite[(3.6)]{AV}; (b) sharpens~\cite[(3.10)]{AV}, and~(c) bounds~$\chi_{m,k}^{\kappa_m}$ by a constant times the corrector amplitude~$\ebar$ of Part~I. \begin{proof} (a) The integral in~\eqref{e.cell.corrector} is $\int_{-\infty}^t\exp(-4\pi^2\nu(t-t')/\eps_m^2)\widetilde\zeta_{m,k}(t')\dd t'$; it vanishes before the pulse, and its derivative is $\widetilde\zeta_{m,k}(t)$ minus $4\pi^2\nu/\eps_m^2$ times itself. By~\eqref{e.cell.shear}, $-\nu\Delta\nabla^\perp\psi_{m,k}=(4\pi^2\nu/\eps_m^2)\nabla^\perp\psi_{m,k}$, which gives the equation; the transport term vanishes because the gradient of a function of $y\cdot\eperp$ is orthogonal to~$e_{(k)}$. (b) The integral is at most $\eps_m^2/(4\pi^2\nu)$, and $|\nabla^\perp\psi_{m,k}|\le2\pi\eps_m^{\beta-1}$. (c) By Proposition~\ref{p.scales}(a) and~(c), and as $\eps_{m-1}\le\Lambda^{-1}$ (Lemma~\ref{l.fl.scales}(a)), for~$\Lambda_0$ large, \begin{align*} \kappa_{m-1}&\ge\Bigl(\frac9{80}-C\eps_{m-1}^{4\delta}-C\eps_{m-1}^\delta\Bigr)\frac{\eps_m^{2\beta}}{\kappa_m}\ge\frac1{20}\,\frac{\eps_m^{2\beta}}{\kappa_m}\,,\\ \frac{\eps_m^{\beta+1}}{\kappa_m}&=\eps_m\Bigl(\frac{\eps_m^{2\beta}}{\kappa_m}\Bigr)^{\nf12}\kappa_m^{-\nf12}\le C\eps_m\Bigl(\frac{\kappa_{m-1}}{\kappa_m}\Bigr)^{\nf12}=C\ebar\,: \end{align*} the first line is the first bound of~\eqref{e.cell.amplitude}, and the second, with~(b), gives the second. The third follows from $(\kappa_{m-1}/\kappa_m)^{\nf12}\le C\eps_{m-1}^{-q\gamma}$, $\rho(m)\ge c\eps_{m-1}^{1+\gamma/2+\delta}$ (Proposition~\ref{p.scales}(c)), $\eps_m\le2\eps_{m-1}^q$ (Lemma~\ref{l.fl.scales}(a)) and $q-q\gamma-1-\frac\gamma2-\delta=3\delta$, by~\eqref{e.exponents}. \end{proof} \begin{lemma}[The averaged flux is the flux of the correctors]\label{l.cell.flux} Let $m\ge1$ and $\nu>0$. \begin{enumerate}[label=\textup{(\alph*)}] \item For every odd~$k$, $t\in\R$ and $e\in\R^2$, with the term~$k$ of~\eqref{e.flux} on the right, \begin{align*} \lefteqn{\fint_{[0,1]^2}\Bigl(\nu I+\widetilde\zeta_{m,k}(t)\psi_{m,k}(y)\begin{pmatrix}0&-1\\1&0\end{pmatrix}\Bigr)\bigl(e+\nabla(e\cdot\chi_{m,k}^\nu(t,y))\bigr)\dd y} \qquad & \notag \\ & =\nu e+2\pi^2\eps_m^{2\beta-2}\,\widetilde\zeta_{m,k}(t)\Bigl(\int_0^\infty\exp\Bigl(-\frac{4\pi^2\nu t'}{\eps_m^2}\Bigr)\widetilde\zeta_{m,k}(t-t')\dd t'\Bigr)(e_{(k)}\otimes e_{(k)})\,e\,. \end{align*} \item The glued corrector $\chi_m^\nu\coloneqq\sum_{k\in2\Z+1}\Hc_{m,k}\chi_{m,k}^\nu$ satisfies, for every $t\in\R$ and $e\in\R^2$, \begin{equation}\label{e.cell.flux} \fint_{[0,1]^2}\Bigl(\nu I+\psi_m(t,y)\begin{pmatrix}0&-1\\1&0\end{pmatrix}\Bigr)\bigl(e+\nabla(e\cdot\chi_m^\nu(t,y))\bigr)\dd y=\J_m^\nu(t)e\,, \end{equation} and there exists $C(\alpha,q,\zeta)<\infty$ such that \begin{equation}\label{e.cell.glued} \begin{gathered} \partial_t\chi_m^\nu+\nabla^\perp\psi_m\cdot\nabla\chi_m^\nu-\nu\Delta\chi_m^\nu+\nabla^\perp\psi_m=\sum\nolimits_{k\in2\Z+1}\Hc'_{m,k}\chi_{m,k}^\nu\,,\\ \Bigl\|\sum\nolimits_{k\in2\Z+1}\Hc'_{m,k}\chi_{m,k}^\nu\Bigr\|_{L^\infty(\R\times\T^2)}\le\frac{C\eps_m^{\beta+1}}{\nu\tau_m}\exp\Bigl(-\frac{\pi^2\nu\tau_m}{3\eps_m^2}\Bigr). \end{gathered} \end{equation} \end{enumerate} \end{lemma} The integrand of~\eqref{e.cell.flux} is the flux of the corrected coordinate $y\cdot e+e\cdot\chi_m^\nu$ (the divergence form after Lemma~\ref{l.fl.equation}, with~$I$ in place of~$\K_{m,l}$; \cite[(3.15), (3.18)]{AV}), and the glued correctors solve the cell problem up to~\eqref{e.cell.glued} (\cite[(3.11)--(3.13)]{AV}). \begin{proof} (a) Let $k\in4\Z+1$ (the other case is symmetric). Then $\psi_{m,k}=\eps_m^\beta\sin(2\pi y_1/\eps_m)$, and $\chi_{m,k}^\nu$, $\nabla(e\cdot\chi_{m,k}^\nu)$ are $-2\pi\eps_m^{\beta-1}\cos(2\pi y_1/\eps_m)e_2$ and $4\pi^2\eps_m^{\beta-2}\sin(2\pi y_1/\eps_m)(e\cdot e_2)e_1$ times the integral in~\eqref{e.cell.corrector}. Only the product of~$\psi_{m,k}$ and~$\nabla(e\cdot\chi_{m,k}^\nu)$ has nonzero mean; since $\bigl(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr)e_1=e_2$ and $\fint\sin^2=\frac12$, its mean is $2\pi^2\eps_m^{2\beta-2}\widetilde\zeta_{m,k}(t)$ times the integral times $(e\cdot e_2)e_2$, and $e_{(k)}=e_2$. (b) If $\psi_m(t,\cdot)\ne0$, exactly one pulse~$k$ is active, $\Hc_{m,k}(t)=1$ and the other cutoffs vanish at~$t$, as $|t-k'\tau_m|\ge\frac43\tau_m$ for odd $k'\ne k$; so $\chi_m^\nu(t)=\chi_{m,k}^\nu(t)$, and~(a) gives~\eqref{e.cell.flux}. Otherwise both sides are~$\nu e$ (Lemma~\ref{l.fl.flux}(i)). The product rule and~\eqref{e.cell.problem} give the equation, since $\Hc_{m,k}=1$ where $\widetilde\zeta_{m,k}\ne0$ and the transport term vanishes. Finally, $\Hc'_{m,k}\ne0$ only for $\frac34\tau_m\le|t-k\tau_m|\le\frac54\tau_m$, where $|\Hc'_{m,k}|\le C/\tau_m$ and either $\chi_{m,k}^\nu=0$ or the pulse ended at least $\frac1{12}\tau_m$ earlier; after the pulse the integral in~\eqref{e.cell.corrector} decays at the rate $4\pi^2\nu/\eps_m^2$, so $|\chi_{m,k}^\nu|\le(\eps_m^{\beta+1}/2\pi\nu)\exp(-\pi^2\nu\tau_m/(3\eps_m^2))$ there. \end{proof} \subsection{The covariance} \label{ss.cell.covariance} \begin{lemma}[The covariance]\phantomsection\label{l.cell.covariance} \begin{enumerate}[label=\textup{(\alph*)}] \item There exists $C(\zeta)<\infty$ such that, for every $m\ge1$, $\nu>0$ and $t,t'\in\R$, \begin{equation}\label{e.cell.covgeneral} \Bigl|\int_{t'}^t\bigl(\J_m^\nu(t'')-\overline K_m(\nu)I\bigr)\dd t''\Bigr|\le C\,\frac{\eps_m^{2\beta}}{\nu}\Bigl(\tau'_m+\frac{\eps_m^2h_m}{\nu\tau_m}\Bigr). \end{equation} \item There exist $\Lambda_0(\alpha,s,q,\zeta)\ge128$ and $C(\alpha,s,q,\zeta)<\infty$ such that, for every $\Lambda\ge\Lambda_0$, $M\ge2$, $\kappa\in\Kset_M$, $2\le m\le M$ and $t,t'\in\R$, the covariance~\eqref{e.covariance} satisfies \begin{equation}\label{e.cell.covariance} |\Gamma_m(t)|+|\Gamma_m(t)-\Gamma_m(t')|\le C\kappa_{m-1}\tau'_m\le C\eps_{m-1}^\delta\rho(m)^2\,. \end{equation} \end{enumerate} \end{lemma} The matrices are diagonal (Lemma~\ref{l.fl.flux}(i)); Section~\ref{s.entry} uses the two-time form~(b). \begin{proof} (a) We compare~$\J_m^\nu$ with the flux $\J_m^{\nu,\mathrm{fast}}\coloneqq\nu I+(\eps_m^{2\beta}/2\nu)\sum_{k\in2\Z+1}\zeta_{m,k}^2e_{(k)}\otimes e_{(k)}$ of pulses held at their current value and without slow cutoffs. By Step~1 of the proof of Proposition~\ref{p.scales}(a), holding the pulses changes~$\J_m^\nu$ by at most $C(\eps_m^{2\beta}/\nu)\eps_m^2/(\nu\tau_m)$ at each time; removing the slow cutoffs changes it only within~$2\tau'_m$ of the ends of the blocks, where the slow cutoffs differ from~$1$ by~\eqref{e.fl.slow}, as in~\eqref{e.fl.squares}. So $\int|\J_m^\nu-\J_m^{\nu,\mathrm{fast}}|\le C(\eps_m^{2\beta}/\nu)(\tau'_m+\eps_m^2h_m/(\nu\tau_m))$ over every interval of length~$4h_m$. The fast flux is $4\tau_m$-periodic with mean $(\nu+\frac9{80}\eps_m^{2\beta}/\nu)I$, since each period holds two odd slots, one for each direction~$e_{(k)}$, and $\int_\R\zeta_{m,k}^2=\frac9{10}\tau_m$, so its centred primitive is at most $C(\eps_m^{2\beta}/\nu)\tau_m$. The flux~$\J_m^\nu$ is $4h_m$-periodic with mean $\overline K_m(\nu)I$ (Lemma~\ref{l.fl.flux}), and $4h_m$ is a multiple of~$4\tau_m$, so the two means differ by at most $(4h_m)^{-1}$ times the integral above. The left side of~\eqref{e.cell.covgeneral} is unchanged when~$t$ or~$t'$ moves by~$4h_m$, since the integral of $\J_m^\nu-\overline K_m(\nu)I$ over a period vanishes; so it suffices to bound the centred primitive over one period. Adding the errors, with $\tau_m\le\tau'_m$, gives~\eqref{e.cell.covgeneral}. (b) By~\eqref{e.covariance}, $\Gamma_m(0)=0$ and $\Gamma_m(t)-\Gamma_m(t')=\int_{t'}^t(\J_m^{\kappa_m}(t'')-\kappa_{m-1}I)\dd t''$. We take $\nu=\kappa_m$ in~(a), so that $\overline K_m(\kappa_m)=\kappa_{m-1}$ by~\eqref{e.recursion}, and it suffices to bound the right side of~\eqref{e.cell.covgeneral} by $C\kappa_{m-1}\tau'_m$. By~\eqref{e.cell.amplitude}, Proposition~\ref{p.scales}(c) and Lemma~\ref{l.fl.scales}(b), $\eps_m^{2\beta}/\kappa_m\le C\kappa_{m-1}$ and $\eps_m^2h_m/(\kappa_m\tau_m)\le C\eps_{m-1}^{4\delta}h_m\le C\tau'_m$, because $h_m/\tau'_m=4\lceil\eps_{m-1}^{-\delta}\rceil+1\le9\eps_{m-1}^{-\delta}$; and $\kappa_{m-1}\tau'_m\le\frac14\eps_{m-1}^\delta\kappa_{m-1}h_m=\frac14\eps_{m-1}^\delta\rho(m)^2$, as $\rho(m)=(\kappa_{m-1}h_m)^{\nf12}$. \end{proof} \subsection{The cell inverse, the finite lift and the means} \label{ss.cell.lift} In this subsection and the next one, $M\ge2$, $\kappa\in\Kset_M$, $2\le m\le M$, $k$ is odd, $\Ncor\ge1$, and~$\K(t,y)$ is a \emph{conductivity} on the window of pulse~$k$: a symmetric matrix, measurable in~$t$, smooth in~$y$, with $\frac34I\le\K\le\frac54I$, for instance~$\K_{m,l_{m,k}}$ (Lemma~\ref{l.fl.charts}(a), $\Lambda$ large). The \emph{cell operator} is the part of $\partial_t+\nabla^\perp\psi_m\cdot\nabla-\kappa_m\nabla\cdot(\K\nabla)$, applied to a form~$\mathcal A$ read at $z=y/\eps_m$, without derivative in~$y$: \begin{equation}\label{e.cell.cellop} \Lc_0\mathcal A\coloneqq\partial_t\mathcal A+\eps_m^{-1}(\nabla^\perp\psi_m)(t,\eps_mz)\cdot\nabla_z\mathcal A-\kappa_m\eps_m^{-2}\nabla_z\cdot(\K\nabla_z\mathcal A)\,. \end{equation} On transverse forms its shear term vanishes by~\eqref{e.cell.shear}, and $\Lc_0=\partial_t-\kappa_m\eps_m^{-2}(\eperp\cdot\K\eperp)(\eperp\cdot\nabla_z)^2$: the cell problems across the pulse are one-dimensional heat equations. \begin{lemma}[The cell inverse]\label{l.cell.inverse} Let~$F^0$ and~$F$ be transverse forms of zero average in~$z$, measurable in~$t$, of class~$C^2$ and~$C^3$ in~$z$ respectively, whose derivatives in~$z$ up to these orders are bounded on $\cellwin\times\R^2\times\R^2$. There exists exactly one \emph{energy solution}~$\mathcal A$, a transverse form of zero average in~$z$ with, for every $y\in\R^2$, $\nabla_z\mathcal A(\cdot,y,\cdot)\in L^2(\cellwin\times[0,1]^2)$ and $\mathcal A(\cdot,y,\cdot)\in C(\cellwin;L^2([0,1]^2))$, of the following problem, in the integrated form in~$t$ tested against smooth periodic functions of~$z$, for every~$y$: \begin{equation}\label{e.cell.inverse} \left\{ \begin{aligned} & \Lc_0\mathcal A=F^0+\eperp\cdot\nabla_zF & \mbox{in} & \ \bigl((k-\tfrac{61}{48})\tau_m,(k+\tfrac{61}{48})\tau_m\bigr)\times\R^2\times\R^2\,,\\ & \mathcal A=0 & \mbox{on} & \ \{(k-\tfrac{61}{48})\tau_m\}\times\R^2\times\R^2\,; \end{aligned} \right. \end{equation} it is the \emph{cell inverse} of $F^0+\eperp\cdot\nabla_zF$. For every $\xi\in\Z\setminus\{0\}$, $y\in\R^2$ and $t\in\cellwin$, \begin{gather} \widehat{\mathcal A}(t,y;\xi)=\int_{(k-61/48)\tau_m}^t\exp\Bigl(-\frac{4\pi^2\xi^2\kappa_m}{\eps_m^2}\int_{t'}^t\eperp\cdot\K(t'',y)\eperp\dd t''\Bigr)\bigl(\widehat{F^0}+2\pi\mathrm i\xi\widehat F\bigr)(t',y;\xi)\dd t'\,,\label{e.cell.mode}\\ \begin{aligned} \|\widehat{\mathcal A}(\cdot,y;\xi)\|_{L^\infty(\cellwin)}\le{}&\frac{\eps_m^2}{3\pi^2\xi^2\kappa_m}\|\widehat{F^0}(\cdot,y;\xi)\|_{L^\infty(\cellwin)}\\ &+\frac{2\eps_m^2}{3\pi|\xi|\kappa_m}\|\widehat F(\cdot,y;\xi)\|_{L^\infty(\cellwin)}\,. \end{aligned}\notag \end{gather} If~$\K$ is instead a complex symmetric matrix, measurable in~$t$, whose entry $\eperp\cdot\K\eperp$ is bounded with real part at least~$\frac34$ (as in Section~\ref{s.admissible}), the right side of~\eqref{e.cell.mode} defines a transverse form~$\mathcal A$ of zero average in~$z$, the \emph{cell inverse} at~$\K$, with the same bound. The cell inverse of $-(\nabla^\perp\psi_m)(t,\eps_mz)$ is the \emph{corrector in the conductivity}~$\K$, written $\chi_\K(t,y,z)$; $\chi_I(t,y,z)=\chi_{m,k}^{\kappa_m}(t,\eps_mz)$. \end{lemma} \begin{proof} On the mode~$\xi$, $\Lc_0$ is $\partial_t+(4\pi^2\xi^2\kappa_m/\eps_m^2)\eperp\cdot\K\eperp$: Duhamel's formula gives~\eqref{e.cell.mode}; for a conductivity, the energy identity in~$z$ gives uniqueness. As $\operatorname{Re}\eperp\cdot\K\eperp\ge\frac34$, the exponential is at most $\exp(-3\pi^2\xi^2\kappa_m(t-t')/\eps_m^2)$, of integral in~$t'$ at most $\eps_m^2/(3\pi^2\xi^2\kappa_m)$, which gives the bound. The source $F^0+\eperp\cdot\nabla_zF$ is of class~$C^2$ in~$z$ with bounded derivatives, so its modes are $O(\xi^{-2})$ and, by~\eqref{e.cell.mode}, those of~$\mathcal A$ are $O(\xi^{-4})$; the series of~$\mathcal A$ converges with two derivatives in~$z$, continuously in~$t$ with values in $L^2([0,1]^2)$, and is an energy solution. For $\K=I$, \eqref{e.cell.mode} on the modes $\xi=\pm1$ is~\eqref{e.cell.corrector}, since the pulse starts at $(k-\frac23)\tau_m$, inside the window. \end{proof} A \emph{system on the window} of pulse~$k$ has finitely many components~$v$, with masses $\omega_v>0$, and finitely many sources $\sigma\colon v'\to v$. A source has a birth order $2\le d_\sigma\le\Nsrc$, its order when created (Subsection~\ref{ss.admissible.parameters}), a factor $\fac(\sigma)>0$, a majorant $\maj(\sigma)\ge0$, and transverse two-scale forms~$\mathcal A_{\sigma,d}$, $2\le d\le d_\sigma$, whose values are coefficients from arrays of order $d-1$ to~$\R^2$ (Section~\ref{s.familytools}); we set $\mathcal A_{\sigma,d}\coloneqq0$ for $d>d_\sigma$. The system acts on fields $Y=(Y_v)_v$ by \begin{align}\label{e.cell.system} & \partial_tY_v+\nabla^\perp\psi_m\cdot\nabla Y_v-\kappa_m\nabla\cdot(\K\nabla Y_v) \notag \\ & \qquad -\sum\nolimits_{\sigma\colon v'\to v}\sum\nolimits_{d=2}^{d_\sigma}\nabla\cdot\bigl(\kappa_m\eps_m^{d-2}\fac(\sigma)\mathcal A_{\sigma,d}(t,y,y/\eps_m)\nabla^{d-1}Y_{v'}\bigr); \end{align} for the families of Section~\ref{s.admissible} these are the coefficients~\eqref{e.admissible.twoscale} in the chart of~$k$. A \emph{path} $\sigma_1\cdots\sigma_i$ is a sequence of sources, each starting where the previous one ends; the empty path~$\emptyset$ has $i=0$. We assume that every path has at most~$\Npath$ sources, as for the families of Section~\ref{s.admissible}. On a form~$\mathcal A$ read at $z=y/\eps_m$, the chain rule $\nabla=\nabla_y+\eps_m^{-1}\nabla_z$ splits~\eqref{e.cell.system} by the number of derivatives in~$y$ into~$\Lc_0$, \begin{align}\label{e.cell.split} \Lc_1\mathcal A & \coloneqq(\nabla^\perp\psi_m)(t,\eps_mz)\cdot\nabla_y\mathcal A-\kappa_m\eps_m^{-1}\bigl(\nabla_z\cdot(\K\nabla_y\mathcal A)+\nabla_y\cdot(\K\nabla_z\mathcal A)\bigr), \notag \\ \Lc_2\mathcal A & \coloneqq-\kappa_m\nabla_y\cdot(\K\nabla_y\mathcal A)\,, \end{align} and the terms~$-\Lsrc_j[\sigma]$, where the \emph{source parts}~$\Lsrc_j[\sigma]$ are the parts with exactly~$j$ derivatives in~$y$ of the term of~$\sigma$ in~\eqref{e.cell.system} without its minus sign, those that fall on~$\mathcal A_{\sigma,d}$ included. The finite lift below collects by degree the terms of the operator~\eqref{e.cell.system} applied to the sum of $\Vc_j[\sigma_1\cdots\sigma_i]Y$ over the degrees and the paths, for a field~$Y$ that solves the averaged equation stated after Proposition~\ref{p.cell.means}. The time derivative falls on the coefficients of the correctors, which is the term~$\partial_t$ of~$\Lc_0$, and on~$Y$, where the averaged equation replaces it; this gives the last sum of~\eqref{e.cell.recursion}, in which a path splits at the component where the mean ends and the corrector begins. The proof of Lemma~\ref{l.cmp.cancel} carries out this computation. In~$\Mc_{j,d}$, as in~$\mathcal A_{\sigma,d}$ and in the coefficients of~\eqref{e.family.system}, the order~$d$ counts the derivatives on the field together with the outer divergence, while in~$\Vc_{j,d}$ it counts the derivatives on the field. The means start at $d=2$: in~\eqref{e.cell.means} the field~$Y$ appears only differentiated, since $\Vc_0[\emptyset]Y=Y$ appears only under a derivative and the correctors of positive degree annihilate constants (Proposition~\ref{p.cell.means}(a)). {\allowdisplaybreaks % let (9.15)--(9.16) break across a page \begin{definition}[The finite lift on a pulse]\label{d.cell.lift} The \emph{finite lift} of a system on the window of pulse~$k$ assigns to every path $\sigma_1\cdots\sigma_i$ and every \emph{degree} $0\le j\le\Ncor$, the number of derivatives in~$y$ it carries (on the field or the coefficients), a \emph{corrector} $\Vc_j[\sigma_1\cdots\sigma_i]$, a linear map from smooth fields~$Y(y)$ to transverse two-scale forms, $\Vc_j[\sigma_1\cdots\sigma_i]Y=\sum_{d=0}^j\Vc_{j,d}[\sigma_1\cdots\sigma_i]\nabla^dY$, and to every degree $2\le j\le\Ncor+\Nsrc$ the \emph{means} $\Mc_{j,d}[\sigma_1\cdots\sigma_i](t,y)$, $2\le d\le j$, arrays symmetric in the order of the derivatives. With $\Vc_0[\emptyset]\coloneqq I$, $\Vc_0[\sigma_1\cdots\sigma_i]\coloneqq0$ for $i\ge1$, and $\Vc_j[\sigma_1\cdots\sigma_i]\coloneqq0$ for $j<0$ and $j>\Ncor$, they are defined by induction on~$i$, then on~$j$. The means are the coefficients in \begin{align}\label{e.cell.means} & \sum_{d=2}^j\Mc_{j,d}[\sigma_1\cdots\sigma_i]\nabla^{d-1}Y\coloneqq\mathbf 1_{\{j=2,\,i=0\}}\kappa_m\K\nabla Y-\bigl((\nabla^\perp\psi_m)(t,\eps_mz)\,\Vc_{j-1}[\sigma_1\cdots\sigma_i]Y\bigr)_{\T^2} \notag \\ & \qquad +\Bigl(\text{the part with $j-1$ derivatives in $y$ of}\ \sum_{d'=2}^{d_{\sigma_i}}\kappa_m\eps_m^{d'-2}\fac(\sigma_i)\mathcal A_{\sigma_i,d'}(\nabla_y+\eps_m^{-1}\nabla_z)^{d'-1} \notag \\ & \qquad\qquad \sum_{j''=0}^{\Ncor}\Vc_{j''}[\sigma_1\cdots\sigma_{i-1}]Y\Bigr)_{\T^2}, \end{align} where, for $i=0$, the last term is absent, and $\Vc_{j''}$ counts as~$j''$ derivatives in~$y$. For every $1\le j\le\Ncor$, $\Vc_j[\sigma_1\cdots\sigma_i]$ is the cell inverse, coefficient by coefficient, of the part of zero average in~$z$ of \begin{align}\label{e.cell.recursion} -& \Bigl(\Lc_1\Vc_{j-1}[\sigma_1\cdots\sigma_i]Y+\Lc_2\Vc_{j-2}[\sigma_1\cdots\sigma_i]Y -\sum_{j'=0}^{\min\{j,\Nsrc\}}\Lsrc_{j'}[\sigma_i]\Vc_{j-j'}[\sigma_1\cdots\sigma_{i-1}]Y\Bigr) \notag \\ & \qquad -\sum_{j''=0}^{j-2}\sum_{i'=0}^i\Vc_{j''}[\sigma_{i'+1}\cdots\sigma_i]\nabla\cdot\Bigl(\sum_{d=2}^{j-j''}\Mc_{j-j'',d}[\sigma_1\cdots\sigma_{i'}]\nabla^{d-1}Y\Bigr), \end{align} with, for $i=0$, no source sum; the parenthesis, for any $j\ge1$, is the \emph{bracket of degree~$j$}. The finite lift of the system~\eqref{e.cell.system} itself is~\eqref{e.cell.means}--\eqref{e.cell.recursion} read for matrices over the components, with the source sum over the sources into each component and matrix products in the last sum. \end{definition} } As~$\Lsrc_0$ acts along a shorter path and the means of degree~$j$ use only lower degrees or shorter paths, the induction is well founded. The definition extends to any cell operator with an inverse as in Lemma~\ref{l.cell.inverse}; the statements below are for transverse forms. \begin{proposition}[The finite lift and its means]\label{p.cell.means} There exists $\Lambda_0(\alpha,s,q,\zeta)\ge128$ with the following property. Let $\Lambda\ge\Lambda_0$ and $N\ge0$, and consider a system on the window of pulse~$k$ whose \emph{majorant matrix}, with the entries $(\omega_{v'}/\omega_v)^{\nf12}\sum_{\sigma\colon v'\to v}\fac(\sigma)\maj(\sigma)$ as in~\eqref{e.admissible.majorant}, has row and column sums at most~$1$, and whose conductivity and forms are measurable in~$t$ and satisfy, with constants $C(j_1)$, $C(j_1,j_2)$, for every~$t$ in the window of pulse~$k$, $y,z\in\R^2$, $2\le d\le\Nsrc$, $j_1\le N+4(\Npath+1)(\Ncor+\Nsrc)^2$ and $j_2\le N+2+\Npath\Nsrc+2\Ncor+2\Nsrc$, \begin{equation}\label{e.cell.hyp} \eps_{m-1}^{j_1}|\nabla_y^{j_1}\K(t,y)|\le C(j_1)\,,\qquad \eps_{m-1}^{j_1}|\nabla_y^{j_1}\nabla_z^{j_2}\mathcal A_{\sigma,d}(t,y,z)|\le C(j_1,j_2)\maj(\sigma)\,. \end{equation} Then: \begin{enumerate}[label=\textup{(\alph*)}] \item \emph{Existence and means.} The correctors of positive degree exist, are unique, transverse, of zero average in~$z$, zero at the start of the window of pulse~$k$, and annihilate constants, and the means are $N+1$ times differentiable in~$y$. For every path $\sigma_1\cdots\sigma_i$ and $1\le j\le\Ncor+\Nsrc$, the average in~$z$ of the bracket of degree~$j$ along it is $-\nabla\cdot\bigl(\sum_{d=2}^j\Mc_{j,d}[\sigma_1\cdots\sigma_i]\nabla^{d-1}Y\bigr)$, and, for $j=1$, it vanishes. \item \emph{The cell equations.} For every $1\le j\le\Ncor$, \eqref{e.cell.recursion} has zero average in~$z$, and $\Vc_j[\sigma_1\cdots\sigma_i]Y$ solves~\eqref{e.cell.inverse} with right side~\eqref{e.cell.recursion}. The means of degree $j\ge\Ncor+2$ along~$\emptyset$ vanish. \item \emph{The averaged flux.} If $\K=I$, then $\Mc_{2,2}[\emptyset]=\J_m^{\kappa_m}(t)$. If, for every~$t$ in the window of pulse~$k$, $y\in\R^2$ and~$j_1$ as in~\eqref{e.cell.hyp}, $\eps_{m-1}^{j_1}|\nabla_y^{j_1}(\K-I)|\le C(j_1)\eps_{m-1}^{2\delta}$, with~$C(j_1)$ of~\eqref{e.cell.hyp}, then there exists~$C$, depending only on $\alpha,s,q,\zeta,\Ncor,\Nsrc,N$ and the constants of~\eqref{e.cell.hyp}, such that, for every such~$t$ and~$y$ and every $j_1\le N$, $\eps_{m-1}^{j_1}|\nabla_y^{j_1}(\Mc_{2,2}[\emptyset]-\J_m^{\kappa_m}(t))|\le C\kappa_{m-1}\eps_{m-1}^{2\delta}$. \item \emph{Quiet times.} Let $N\ge\Nsrc-2$, and let~$t$ be a time of the window of pulse~$k$ at which $\widetilde\zeta_{m,k}=0$ and the forms do not depend on~$z$. Then, at~$t$: $\Mc_{2,2}[\emptyset]=\kappa_m\K$; for every path~$\sigma$ of one source, every degree $2\le j\le\Ncor+\Nsrc$ and every smooth field~$Y$, $\Mc_{j,j}[\sigma]\nabla^{j-1}Y=\kappa_m\eps_m^{j-2}\fac(\sigma)\mathcal A_{\sigma,j}\nabla^{j-1}Y$; and all other coefficients of the means vanish. \item \emph{Quiet vanishing.} Let~$t$ be in the window of pulse~$k$. If, at all times of the window up to~$t$, $\widetilde\zeta_{m,k}=0$ and the forms do not depend on~$z$, then, for every $1\le j\le\Ncor$ and every path $\sigma_1\cdots\sigma_i$, $\Vc_j[\sigma_1\cdots\sigma_i]=0$ at~$t$. \item \emph{The system.} The finite lift of the system~\eqref{e.cell.system} has from~$v'$ to~$v$ the correctors and means $\sum\Vc_j[\sigma_1\cdots\sigma_i]$ and $\sum\Mc_{j,d}[\sigma_1\cdots\sigma_i]$, summed over the paths $\sigma_1\cdots\sigma_i$ from~$v'$ to~$v$. \end{enumerate} \end{proposition} So, up to the cell tail (Section~\ref{s.comparison}), the averaged equation of the finite lift is $\partial_tY_v=\sum\nabla\cdot(\Mc_{j,d}[\sigma_1\cdots\sigma_i]\nabla^{d-1}Y_{v'})$, over~$j$, $d$ and the paths, with the sign of~\eqref{e.family.system}. \begin{proof} \emph{Centring.} The average in~$z$ of a derivative in~$z$ vanishes, and $(\nabla^\perp\psi_m)(t,\eps_mz)$ does not depend on~$y$. So, for a transverse form~$\mathcal A$ of zero average in~$z$, $(\Lc_1\mathcal A)_{\T^2}=\nabla\cdot((\nabla^\perp\psi_m)(t,\eps_mz)\mathcal A)_{\T^2}$ and $(\Lc_2\mathcal A)_{\T^2}=0$, while $(\Lc_1Y)_{\T^2}=0$ and $(\Lc_2Y)_{\T^2}=-\kappa_m\nabla\cdot(\K\nabla Y)$. A source part is $\nabla_y+\eps_m^{-1}\nabla_z$ applied to a flux, with average $\nabla\cdot$ of the averaged flux. \emph{(a), (b).} By induction on~$i$ and then on~$j$. The centring and~\eqref{e.cell.means} give the average of the bracket of degree~$j$. At $j=1$ every term vanishes: along~$\emptyset$ since the shear has mean zero; along a path of positive length since~$\Lsrc_0$ is a divergence in~$z$ and, on~$\Vc_0$, which does not depend on~$z$, the part of a source flux with no derivative in~$y$ vanishes, as $d'\ge2$. So in~\eqref{e.cell.recursion} the term $j''=0$, to which only $i'=i$ contributes, cancels the average of the bracket, and the terms $j''\ge1$ have zero average: \eqref{e.cell.recursion} has zero average in~$z$. It is transverse, since~$\K$ does not depend on~$z$, and Lemma~\ref{l.cell.inverse} gives~$\Vc_j[\sigma_1\cdots\sigma_i]$ and~(b). On constants, $\Lc_1$, $\Lc_2$ and every~$\Lsrc_{j'}[\sigma]$ vanish, hence so do the correctors of positive degree. The regularity of the means is that of Proposition~\ref{p.cell.sizes}, which also gives the derivatives in~$y$ of the lower correctors and means used to form~\eqref{e.cell.recursion}. Existence and these bounds are proved together, by the induction of its proof. \emph{(c).} Along~$\emptyset$, at $j=1$ the bracket is $(\nabla^\perp\psi_m)(t,\eps_mz)\cdot\nabla Y$, so $\Vc_1[\emptyset]Y=\chi_\K\cdot\nabla Y$ and, by~\eqref{e.cell.means}, $\Mc_{2,2}[\emptyset]=\kappa_m\K-((\nabla^\perp\psi_m)(t,\eps_mz)\otimes\chi_\K)_{\T^2}$. For $\K=I$, $\chi_I=\chi_{m,k}^{\kappa_m}(t,\eps_mz)$, and the mean of $\cos^2$ is~$\frac12$, so $\Mc_{2,2}[\emptyset]$ is~$\kappa_mI$ plus the term~$k$ of~\eqref{e.flux}, that is,~$\J_m^{\kappa_m}(t)$. For general~$\K$, by~\eqref{e.cell.mode} the modes $\xi=\pm1$ of~$\chi_\K$ and~$\chi_I$ differ by the integral of~$\widetilde\zeta_{m,k}$ against the difference of two exponentials, at most $(4\pi^2\kappa_m/\eps_m^2)(t-t')\sup|\K-I|\exp(-3\pi^2\kappa_m(t-t')/\eps_m^2)$, of integral in~$t'$ at most $(4\eps_m^2/9\pi^2\kappa_m)\sup|\K-I|$. With $|\nabla^\perp\psi_{m,k}|\le2\pi\eps_m^{\beta-1}$ and~\eqref{e.cell.amplitude}, $|\Mc_{2,2}[\emptyset]-\J_m^{\kappa_m}(t)|\le C(\kappa_m+\eps_m^{2\beta}/\kappa_m)\sup|\K-I|\le C\kappa_{m-1}\eps_{m-1}^{2\delta}$; derivatives in~$y$ fall on~$\K$ and cost $C\eps_{m-1}^{-1}$ each. \emph{(d), (e).} At a time as in~(d), the shear term of~\eqref{e.cell.means} vanishes, and so does the average of the source flux of a corrector of positive degree; what remains is $\kappa_m\K\nabla Y$ and the sources applied to $\Vc_0[\emptyset]Y=Y$, of degree~$d$ for the order~$d$. For~(e), by induction: if the lower correctors of positive degree vanish up to~$t$, then, up to~$t$, the bracket is $\Lc_2Y$ (for $i=0$ and $j=2$), a source part applied to~$Y$ (for $i=1$), or zero, and it does not depend on~$z$. The terms $j''\ge1$ of the last sum of~\eqref{e.cell.recursion} vanish, and its term $j''=0$ does not depend on~$z$. So~\eqref{e.cell.recursion}, of zero average by~(b), vanishes up to~$t$, and~\eqref{e.cell.mode} gives $\Vc_j[\sigma_1\cdots\sigma_i]=0$ at~$t$. \emph{(f).} The definition is linear and the cell inverse unique. In the finite lift of the system, a term carrying~$i$ sources is a product of the forms along one path, in order; its source parts extend a shorter path by its last source, and its products split a path at one point. So the sums over paths satisfy the definition for the system. Its equations are triangular in the length of the longest path into~$v$, which is at most~$\Npath$. At degree~$j$, the source part~$\Lsrc_0[\sigma]$ of a source $\sigma\colon w\to v$ couples the corrector from~$v'$ to~$v$ to the corrector from~$v'$ to~$w$ at the same degree, and the longest path into~$w$ is shorter than that into~$v$. So the definition for the system has exactly one solution, and uniqueness gives~(f). \emph{Paths and the threshold.} The system along a path (its components, one source per step) satisfies the same hypotheses, and~(a) along the path is~(a) for it. The threshold~$\Lambda_0$ is the larger of that of Lemma~\ref{l.cell.corrector}(c) and the one beyond which $\ebar\le\rho(m)\le\eps_{m-1}$, by~\eqref{e.cell.amplitude} and Proposition~\ref{p.scales}(c). \end{proof} \subsection{Bounds on the correctors and the means} \label{ss.cell.sizes} For a transverse form~$\mathcal A$ and integers $N,j_1\ge0$, the \emph{Wiener norm} \begin{equation}\label{e.cell.wiener} \|\mathcal A\|_{N,j_1}\coloneqq\sum_{\xi\in\Z}(1+|\xi|)^N\bigl\|\nabla_y^{j_1}\widehat{\mathcal A}(\cdot,\cdot\,;\xi)\bigr\|_{L^\infty(\cellwin\times\R^2)} \end{equation} is summed over the entries for arrays. For every $j_2\le N$, it bounds $(2\pi)^{-j_2}$ times the norm of $\nabla_y^{j_1}\nabla_z^{j_2}\mathcal A$ in $L^\infty(\cellwin\times\R^2\times\R^2)$, and obeys the Leibniz rule, as $1+|\xi+\xi'|\le(1+|\xi|)(1+|\xi'|)$. A matrix over the components is bounded \emph{in row and column sums} if the sum of its entries over each row and over each column is. \emph{Normalized variables.} In the time $\kappa_mt/\eps_m^2$ and the slow variable $y/\rho(m)$, $(\eps_m^2/\kappa_m)\Lc_0$ has the diffusivity $\eperp\cdot\K\eperp\in[\frac34,\frac54]$ in the fast variable. Relative to $(\eps_m^2/\kappa_m)\Lc_0$ and to the derivatives in the slow variable, $(\eps_m^2/\kappa_m)\Lc_1$, $(\eps_m^2/\kappa_m)\Lc_2$, $(\eps_m^2/\kappa_m)\Lsrc_{j'}[\sigma]$ have the sizes of the powers $1,2,j'$ of $\ebar/\rho(m)\le C\eps_{m-1}^{3\delta}$: the drift $\eps_m(\kappa_m\kappa_{m-1})^{-\nf12}\nabla^\perp\psi_m$ of~$\Lc_1$ is at most~$C$ by~\eqref{e.cell.amplitude}, and the sources carry $(\eps_m/\rho(m))^{j'}$, smaller as $\kappa_m\le\kappa_{m-1}$. \begin{proposition}[Sizes]\label{p.cell.sizes} In the setting of Proposition~\ref{p.cell.means}, there exists~$C$, depending only on $\alpha,s,q,\zeta,\Ncor,\Nsrc,\Npath,N$ and the constants of~\eqref{e.cell.hyp}, such that, with sums over the paths $\sigma_1\cdots\sigma_i$ from~$v'$ to~$v$: \begin{enumerate}[label=\textup{(\alph*)}] \item for every $1\le d\le j\le\Ncor$ and $j_1\le N$, in row and column sums over the components~$v$, $v'$, \begin{equation}\label{e.cell.sizecorr} \Bigl(\frac{\omega_{v'}}{\omega_v}\Bigr)^{\nf12}\eps_{m-1}^{j_1}\Bigl\|\sum\Vc_{j,d}[\sigma_1\cdots\sigma_i]\Bigr\|_{N,j_1}\le C\ebar^{\,j}\eps_{m-1}^{d-j}\,; \end{equation} \item for every $2\le d\le j\le\Ncor+\Nsrc$ and $j_1\le N$, in row and column sums as in~(a), \begin{align}\label{e.cell.sizemean} & \Bigl(\frac{\omega_{v'}}{\omega_v}\Bigr)^{\nf12}\eps_{m-1}^{j_1}\Bigl\|\nabla_y^{j_1}\sum\Mc_{j,d}[\sigma_1\cdots\sigma_i]\Bigr\|_{L^\infty(\cellwin\times\R^2)} \notag \\ & \qquad \le C\kappa_{m-1}\ebar^{\,j-2}\eps_{m-1}^{d-j}\,; \end{align} \item for every path $\sigma_1\cdots\sigma_i$, $2\le d\le j\le\Ncor+\Nsrc$ and $j_1\le N+1$, \begin{align}\label{e.cell.sizepath} & \eps_{m-1}^{j_1}\|\nabla_y^{j_1}\Mc_{j,d}[\sigma_1\cdots\sigma_i]\|_{L^\infty(\cellwin\times\R^2)} \notag \\ & \qquad \le C\fac(\sigma_1)\maj(\sigma_1)\cdots\fac(\sigma_i)\maj(\sigma_i)\,\kappa_{m-1}\ebar^{\,j-2}\eps_{m-1}^{d-j}\,. \end{align} \end{enumerate} \end{proposition} \begin{proof} By induction on the length of the paths and then on the degree; for~(c), the majorant matrix is replaced by the entries $\fac(\sigma)\maj(\sigma)$ of the sources of the path. \emph{The inverse and the four products.} By Lemma~\ref{l.cell.inverse}, in the norms~\eqref{e.cell.wiener} with the weights~$\eps_{m-1}^{j_1}$, the cell inverse costs $C\eps_m^2/\kappa_m$ and returns two derivatives in~$z$ on~$F^0$ and one on~$F$ of~\eqref{e.cell.inverse}. The cross terms of~$\Lc_1$ on a transverse form~$\mathcal B$ have the form $\eperp\cdot\nabla_zF$, as $\nabla_z\cdot(\K\nabla_y\mathcal B)=\eperp\cdot\nabla_z(\eperp\cdot\K\nabla_y\mathcal B)$ and $\nabla_y\cdot(\K\nabla_z\mathcal B)=\eperp\cdot\nabla_z\nabla_y\cdot(\K\eperp\mathcal B)$. A derivative in~$y$ of the exponential of~\eqref{e.cell.mode} brings down $-(4\pi^2\xi^2\kappa_m/\eps_m^2)\int_{t'}^t\eperp\cdot\nabla_y\K(t'',y)\eperp\dd t''$, of size at most $(4\pi^2\xi^2\kappa_m/\eps_m^2)(t-t')\sup|\nabla_y\K|$. So $j_1$ derivatives bring down products of at most~$j_1$ such terms, with $j_1$ derivatives on~$\K$ in all. After the integration in~$t'$ each derivative costs $C\eps_{m-1}^{-1}$, as $\int_0^\infty(a\tau)^l\mathrm e^{-3a\tau/4}\dd\tau=l!(\frac43)^{l+1}/a$ for $a>0$. Each term of~\eqref{e.cell.recursion} is a lower corrector or mean times one of four coefficients, which the inverse turns into powers of the corrector amplitude; by~\eqref{e.cell.amplitude}, the definition of~$\ebar$ and $\eps_m\le\ebar$ (Proposition~\ref{p.scales}(c)), the four products satisfy \begin{equation}\label{e.cell.factors} \frac{\eps_m^2}{\kappa_m}\eps_m^{\beta-1}\le C\ebar,\quad \frac{\eps_m^2}{\kappa_m}\cdot\frac{\kappa_m}{\eps_m}=\eps_m\,,\quad \frac{\eps_m^2}{\kappa_m}\kappa_m\eps_m^{j'-2}=\eps_m^{j'},\quad \frac{\eps_m^2}{\kappa_m}\kappa_{m-1}=\eps_m^2\,\frac{\kappa_{m-1}}{\kappa_m}=\ebar^2\,. \end{equation} \emph{The step.} Assume the bounds along shorter paths, and along the same path at lower degrees. In each term, a derivative in~$y$ raises the order of the derivatives of~$Y$ or falls on a coefficient, at the cost~$\eps_{m-1}^{-1}$, which keeps the power~$\eps_{m-1}^{d-j}$. The shear term of~$\Lc_1\Vc_{j-1}$, of size $2\pi\eps_m^{\beta-1}$ with two modes, gets the first product; the cross terms of~$\Lc_1\Vc_{j-1}$, with one derivative in~$z$ that the inverse returns, the second; $\Lc_2\Vc_{j-2}$ the third, with $j'=2$. A term $\Lsrc_{j'}[\sigma_i]\Vc_{j-j'}[\sigma_1\cdots\sigma_{i-1}]$ has the coefficient $\kappa_m\eps_m^{d'-2}\fac(\sigma_i)\mathcal A_{\sigma_i,d'}$ and $d'-j'$ derivatives in~$z$, each costing~$\eps_m^{-1}$, so it gets the third product times $C\fac(\sigma_i)\maj(\sigma_i)$. A product term of~\eqref{e.cell.recursion} with $j''\ge1$ carries~$\kappa_{m-1}$ and $j-2$ powers of the corrector amplitude, and the fourth product turns~$\kappa_{m-1}$ into two more. This gives~(a) at~$j$. \emph{The means.} By~\eqref{e.cell.means}, the means are $\kappa_m\K$, at most $\frac54\kappa_{m-1}$, at the degree~$2$, plus two averages. For the shear term, $\eps_m^{\beta-1}\ebar=(\kappa_{m-1}\eps_m^{2\beta}/\kappa_m)^{\nf12}\le C\kappa_{m-1}$ by~\eqref{e.cell.amplitude}, so the bounds of the induction on~$\Vc_{j-1}$, those of~(a) with the $N+1$ derivatives in~$y$ of its reserve and, for~(c), along the path, give~(b) and~(c). For the source term, the part with~$j'$ derivatives in~$y$ of a source applied to~$\Vc_{j''}$, $j''+j'=j-1$, is $\kappa_m\eps_m^{j'-1}\fac(\sigma_i)\maj(\sigma_i)$ times the bound of the induction for~$\Vc_{j''}$, and this is at most the bound~(b): by $\eps_m\le\ebar$ if $j'\ge1$, and by $(\kappa_m/\eps_m)\ebar=(\kappa_m\kappa_{m-1})^{\nf12}\le\kappa_{m-1}$ if $j'=0$. \emph{Components and orders.} Over the components, each term is a product of the majorant matrix and of lower correctors and means; row and column sums are submultiplicative, and at most~$1$ for the majorant matrix. The derivatives are counted by reserves: the induction at path length~$i$ and degree~$j$ carries bounds on the correctors with \begin{equation*} N+1+(\Ncor+\Nsrc-j)\ \text{derivatives in~$y$}\quad\text{and}\quad N+2+(\Npath-i)\Nsrc+2(\Ncor+\Nsrc-j)\ \text{in~$z$}\,, \end{equation*} its \emph{reserves} at $(i,j)$, and on the means, which do not depend on~$z$, with $N+2+(\Ncor+\Nsrc-j)$ derivatives in~$y$, one more than a corrector of the same degree. A reserve~$r$ in~$z$ is carried in the Wiener norm~\eqref{e.cell.wiener} of order $r-2$, as the cell inverse acts mode by mode. A form with $r$ bounded derivatives in~$z$ has this norm, since $(1+|\xi|)^r|\widehat{\mathcal A}(\xi)|\le2^r\bigl(\|\mathcal A\|_{L^\infty}+(2\pi)^{-r}\|(\eperp\cdot\nabla_z)^r\mathcal A\|_{L^\infty}\bigr)$ and $\sum_{\xi\in\Z}(1+|\xi|)^{-2}<\infty$. So the bases $N+1$ and $N+2$ give the $N+1$ derivatives in~$y$ of~(c) and the Wiener norms~\eqref{e.cell.wiener} of order~$N$. Lowering the degree by~$j'$ raises the reserve in~$y$ by~$j'$ and that in~$z$ by~$2j'$, and shortening the path by one raises the reserve in~$z$ by~$\Nsrc$. A source part $\Lsrc_{j'}[\sigma_i]$, $i\ge1$, $0\le j'\le\min\{j,\Nsrc\}$, puts at most~$j'$ derivatives in~$y$ and $\Nsrc-j'$ in~$z$ on its predecessor, of path length $i-1$ and degree $j-j'$, whose reserves exceed those at $(i,j)$ by~$j'$ in~$y$ and by $\Nsrc+2j'\ge\Nsrc-j'$ in~$z$. The terms with~$\Lc_1$ and~$\Lc_2$ put at most~$j'$ derivatives in~$y$ and~$2j'$ in~$z$ on~$\Vc_{j-j'}$, $j'=1,2$, whose reserves exceed those at $(i,j)$ by exactly these numbers. By~\eqref{e.cell.means}, a mean of degree~$j$ asks of~$\Vc_{j-1}$ the reserve in~$y$ of the means at~$j$, which is that of~$\Vc_{j-1}$, and of a source part with~$j'$ derivatives in~$y$ on~$\Vc_{j''}$, $j''+j'=j-1$, that reserve plus~$j'$, which is that of~$\Vc_{j''}$. The term $\kappa_m\K$ of~\eqref{e.cell.means} asks at most $N+\Ncor+\Nsrc$ derivatives of~$\K$. In a product term with $j''\ge1$ (the term $j''=0$ does not depend on~$z$, and the part of zero average removes it), the path splits into two paths, of total length~$i$, with the degrees $j''$ and $j-j''$, both below~$j$. By Leibniz's rule, the mean, of degree $j-j''$, receives at most $j''+1$ derivatives in~$y$ beyond those of the target, and its reserve in~$y$ exceeds that of the correctors at $(i,j)$ by exactly $j''+1$. The corrector, of degree~$j''$, receives at most those of the target, within its reserve, and both have reserves in~$z$ at least those at $(i,j)$. So the induction closes. This induction runs together with that of Proposition~\ref{p.cell.means}(a): at~$(i,j)$, the bounds at the lower stages bound~\eqref{e.cell.recursion} in the Wiener norm of order $r-4$, with $r\ge N+2+2\Nsrc\ge6$ the reserve in~$z$ at~$(i,j)$, so that~\eqref{e.cell.recursion}, as a scalar source, has the two bounded derivatives in~$z$ that Lemma~\ref{l.cell.inverse} assumes. The largest reserves, at $(0,0)$, are $N+2+\Npath\Nsrc+2\Ncor+2\Nsrc$ in~$z$, which is the order in~$z$ of~\eqref{e.cell.hyp}, and $N+1+\Ncor+\Nsrc\le N+4(\Npath+1)(\Ncor+\Nsrc)^2$ in~$y$, since $\Nsrc\ge2$, which is within its order in~$y$. \end{proof} \subsection{The corrector operator and the mean flux} \label{ss.cell.corr} Let~$G$ be a family of level~$m$, $2\le m\le M$ (Subsection~\ref{ss.families}), whose sources have coefficients~\eqref{e.admissible.twoscale} satisfying the hypotheses of Proposition~\ref{p.cell.means} on the window of every pulse~$k$, in its chart, with $\K=\K_{m,l_{m,k}}$. Let~$\Vc_j[\sigma_1\cdots\sigma_i]$ and~$\Mc_{j,d}[\sigma_1\cdots\sigma_i]$ be the correctors and means of its finite lift on the window of pulse~$k$, and, for a field~$Y$ on the components of~$G$, let $\widetilde Y_k(t,y)\coloneqq Y(t,X_{m-1}(t,y,l_{m,k}h_m))$. \begin{definition}[The corrector operator and the mean flux]\label{d.cell.corr} For a field~$Y$ as above, a component~$v$, $t\in\R$ and $x\in\R^2$, with $y\coloneqq X_{m-1}(l_{m,k}h_m,x,t)$, $\nabla_yX_{m-1}$ taken at $(t,y,l_{m,k}h_m)$ and $\Vc_j[\sigma_1\cdots\sigma_i]$, $\Mc_{j,d}[\sigma_1\cdots\sigma_i]$ those of the window of pulse~$k$, in the term of~$k$, with the sums over the paths $\sigma_1\cdots\sigma_i$ ending at~$v$, from the component~$v'$, \begin{gather} (\Corr_{m,k}Y)_v(t,x)\coloneqq\sum_{j=0}^{\Ncor}\sum\bigl(\Vc_j[\sigma_1\cdots\sigma_i]\widetilde Y_{k,v'}\bigr)(t,y,y/\eps_m)\,,\quad \Corr_mY\coloneqq\sum_{k\in2\Z+1}\Hc_{m,k}\Corr_{m,k}Y\,, \notag \\[2pt] (\Mc_mY)_v(t,x)\coloneqq\sum_{k\in2\Z+1}\Hc_{m,k}(t)\,\nabla_yX_{m-1}\sum_{j=2}^{\Ncor+\Nsrc}\sum\sum_{d=2}^j\bigl(\Mc_{j,d}[\sigma_1\cdots\sigma_i]\nabla^{d-1}\widetilde Y_{k,v'}\bigr)(t,y)\,.\label{e.cell.corr} \end{gather} \end{definition} The operator~$\Corr_m$ is that of~\eqref{e.lift}. In~\eqref{e.cell.corr}, the term of~$k$ vanishes outside the window of~$k$, where $\Hc_{m,k}=0$, as $\frac54<\frac{61}{48}$, and $\nabla^{d-1}\widetilde Y_{k,v'}$ is taken in the label~$y$. The \emph{mean flux}~$\Mc_m$ glues the means, carried to Eulerian coordinates by the Piola identity of Lemma~\ref{l.fl.equation}, under which a flux in the chart is multiplied by $\nabla_yX_{m-1}$. By Propositions~\ref{p.cell.means}(c) and~\ref{p.cell.sizes}(b) and Lemma~\ref{l.fl.charts}(a), the part along the empty paths is close to $\J_m^{\kappa_m}(t)\nabla$; the difference, of all orders, is the defect of the averaged flux, a source of the successor (Section~\ref{s.comparison}). % ===== 10-admissible ===== \section{Admissible families} \label{s.admissible} In this section we define the admissible families, those that descend from an admissible seed by cycles (Subsection~\ref{ss.families}), and prove by induction along the cycles that they satisfy the invariant of Definition~\ref{d.admissible.invariant} (Proposition~\ref{p.admissible.bounds}), whose clauses (a2)--(a4), (a6), (b)--(d) are used in Sections~\ref{s.preparation}--\ref{s.passage}. \subsection{Parameters and sources} \label{ss.admissible.parameters} \paragraph{Parameters.} The following parameters depend only on~$\deps$. With those of Subsection~\ref{ss.parameters} and Section~\ref{s.passage}, they are fixed in the order of Table~\ref{t.admissible.schedule}, which omits only the absolute constant~$\can$ of Subsection~\ref{ss.admissible.precise} and computed orders such as~$\Nrec$ and~$\Nword$. The orders of the entry, among them~$\Nse$ and~$\Noe$, and the bound~$B$ on the paths of the seed are defined in Subsection~\ref{ss.entry.parameters}, in~\eqref{e.entry.termdefs}, item~(3) there and~\eqref{e.entry.outorders}. \begin{enumerate}[label=(\arabic*)] \item The orders of the entry, which fix Part~I's~$\Nrep$, and the seed's bounds (Section~\ref{s.entry}): orders at most~$\Nse$, and at most~$B$ sources on a path. Then, with the other parameters of Subsection~\ref{ss.parameters} and the orders of Section~\ref{s.passage}: $\Ngen$, the number of generations kept; $\Nsrc$; the width exponent~$\Theta$ of~\eqref{e.II8.width}; $\Nlen$, the length of the longest word kept, and~$\Npath$; $\Nmom$, the moment order of the smoothings, with $2\ell-1\ge\Nmom$ and the conditions of~\eqref{e.cmp.schedule}; $\Ncoef$, the order of the derivative bounds, at least~$\Nret$ and the orders at which the preparation uses these bounds (Section~\ref{s.preparation}). A generation adds at most $2\Ncor+4p+4$ to the orders and multiplies the number of components on a path by at most $\Nlen+2$; this gives the orders~$\Nsrc$ at generation $\Ngen+1$ and the \emph{path bound}~$\Npath$ at generation~$\Ngen$ (the paragraph after Definition~\ref{d.admissible.admissible}), and~$\Theta$ is small: \begin{equation*} \Nsrc\coloneqq\Nse+(\Ngen+1)(2\Ncor+4p+4),\quad \Npath\coloneqq(B+1)(\Nlen+2)^{\Ngen}-1,\quad (\Nsrc-2)\Theta\le\frac\delta8\,. \end{equation*} \item The constant $\cmol\in(0,1]$ of the \emph{molecular radius} \begin{equation}\label{e.admissible.rhomol} \rhomol(m)\coloneqq\cmol\min\bigl\{\eps_m,\,(\kappa_m\eps_m^{2-\beta})^{\nf12}\bigr\}\,, \end{equation} the scale at which a solution of a family of level~$m$ has derivatives after the time $\rhomol(m)^2/\kappa_m$; $\cmol$ is fixed in Section~\ref{s.preparation}, before the orders of item~(1). Then the seed's orders and constants (Section~\ref{s.entry}): its derivative order, at least $\max\{\Ncoef,\Noe\}$ and large enough in terms of the orders of item~(1); its weight order (the exponent of $1+|\xi|$ in~\eqref{p.admissible.norm}), at least its derivative order; the offset $\Noff\ge n$ of the analytic norm (Subsection~\ref{ss.admissible.precise}; used in Definition~\ref{d.entry.weakfamily}); $\Cseed\ge1$ (its mass, coefficients and majorant sums); and its constants of the kinds in item~(3). \item The invariant's constants, weaker than the seed's (constants larger; orders smaller): $C_\Omega\ge3$ ($|\cdot|_\infty$-norms of the matrices of the smoothings and of the inverses of their long axes); $c_j$ (the smoothings); $\Cwid$ (the widths); $\Cder$, $\Cjet(j)$ (the derivative bounds); the weight order $\Nan\ge\Ncoef$ of the analytic norm. They are so large that Steps~6 and~7 of the proof of Lemma~\ref{l.admissible.cycle} close; in particular, the~$c_j$ are at least the $c_j(u,\ell)$ of Lemma~\ref{l.ft.moving} for the kernel orders in use. \item $\Cadm\ge\Cseed$, the constant of the majorant and coupling sums, and~$\Cmaj$, the factor that a cycle applies to majorants, both large enough given item~(3) (Lemma~\ref{l.admissible.seeds}; Step~5 of the proof of Lemma~\ref{l.admissible.cycle}); the \emph{carrier level}~$\mc$, the coarsest level of the chart parameters~\eqref{e.admissible.carrier}; the top level~$m_0\ge\max\{\mc,2\}$ of Subsection~\ref{ss.parameters}, the largest of the thresholds on the level in Sections~\ref{s.admissible}--\ref{s.passage}, in particular so large that $\Cadm\eps_{m-1}^{q\delta/4}\le\frac18$ for every $m\ge m_0$, that $\sum_{i>m_0}(3\tau_i+4h_i)<\frac12t_0$ (Lemma~\ref{l.admissible.successors}(iv)), and that the \emph{fresh smoothings} (those of the sources a cycle creates, Definition~\ref{d.admissible.successor}(iv)) created at the levels finer than~$m_0$ have matrices of long axes with inverses of $|\cdot|_\infty$-norm $\le2$. Also, $\Cmaj$ is large enough given the constants of Lemma~\ref{l.entry.coefficients}(b) and of Steps~4 and~5 of the proof of Lemma~\ref{l.admissible.cycle} for the terminal cycle (Proposition~\ref{p.entry.terminal}). \end{enumerate} \begin{table}[!htbp] \centering \footnotesize \renewcommand{\arraystretch}{1.08} \begin{tabular}{@{}p{0.34\textwidth}p{0.2\textwidth}p{0.4\textwidth}@{}} \hline \raggedright chosen, in this order & \raggedright uses & \raggedright conditions, where stated\tabularnewline \hline \raggedright $\Lambda_0(\alpha,s,q,\zeta)\ge128$ & \raggedright $\alpha,s,q,\zeta$ & \raggedright at least the threshold $\Lambda_0(\alpha,s,q,\zeta)$ of every statement that declares one (Theorem~\ref{t.A})\tabularnewline \raggedright $\Lambda\ge\Lambda_0$, $n\ge1$, $00$, the constant in front of its two-scale forms in~\eqref{e.admissible.twoscale}; and a \emph{majorant} $\maj(\sigma)\ge0$, which bounds its two-scale forms, so that $\fac(\sigma)\maj(\sigma)$ bounds the source. The \emph{generation} $\gen(v)$ counts the cycles that created~$v$. \paragraph{Two-scale forms.} On the window $|t-k\tau_m|\le\frac{61}{48}\tau_m$ of a pulse~$k$ of level~$m$ (Section~\ref{s.cell}), $\sigma$ acts in the chart of~$k$ (Section~\ref{s.flows}), at the label~$y$, through the coefficients \begin{equation}\label{e.admissible.twoscale} \kappa_m\eps_m^{d-2}\fac(\sigma)\,\mathcal A_{\sigma,d}\Bigl(t,y,\frac y{\eps_m}\Bigr)\,,\qquad 2\le d\le d_\sigma\,, \end{equation} with \emph{two-scale forms} $\mathcal A_{\sigma,d}(t,y,z)$ in the sense of Section~\ref{s.cell}, periodic in the fast variable~$z$, with values indexed by the two coordinates of the flux and the entries of~$\nabla^{d-1}$. They also depend on \emph{chart parameters}, the shears of the coarser levels (precise form~\eqref{p.admissible.chains}), and~\eqref{e.admissible.twoscale} uses those of the chart of~$k$. The~$A_{\sigma,d}$ of~\eqref{e.family.system} are the sums over~$k$, with time cutoffs, of the \emph{transports} of the coefficients~\eqref{e.admissible.twoscale} from~$y$ to~$x$, order~$d$ giving orders $2,\dots,d$ (precise form~\eqref{p.admissible.transport}). A two-scale form~$\mathcal A$ has an \emph{analytic norm} $\|\mathcal A\|_{\mathrm{an},v}$, a weighted sum over its Fourier modes in~$z$ of its suprema over a complex neighbourhood of the chart parameters that shrinks by a fixed step per generation of~$v$ (precise form~\eqref{p.admissible.norm}). Cauchy's estimate bounds the derivatives of a cycle at the cost of this step, for at most $\Ngen$ generations. The \emph{majorant matrix} and its \emph{priced} form are \begin{align}\label{e.admissible.majorant} & (\omega_{v'}/\omega_v)^{\nf12}\sum\nolimits_{\sigma\colon v'\to v}\fac(\sigma)\maj(\sigma), \notag \\ & (\omega_{v'}/\omega_v)^{\nf12}\sum\nolimits_{\sigma\colon v'\to v}\fac(\sigma)\max\{1,(\rho_\sigma/r_\sigma)^{d_\sigma-2}\}\maj(\sigma); \end{align} the price compensates for the $d-2$ derivatives that~\eqref{e.coupling} puts on the kernel of width~$r_\sigma$; for the masses $\omega_v/\Psi_v^2$, with the markers~$\Psi_v$ of Subsection~\ref{ss.families}, the entries are multiplied by $\Psi_v/\Psi_{v'}$. \subsection{Definitions and the main proposition} \label{ss.admissible.definitions} Each clause is stated here in words; its precise form, given in Subsection~\ref{ss.admissible.precise}, is the definition. \begin{definition}[The invariant]\label{d.admissible.invariant} A family of level~$m$ satisfies the \emph{invariant} on $[\tA,\tf]$ if, with the constants of items (1), (3) and~(4) of Subsection~\ref{ss.admissible.parameters} (for a seed: its constants, its derivative order (item~(2)) in place of~$\Ncoef$ in (b2), (b2$'$), (c3) and~(d), and $\Cseed$ for~$\Cadm$): \begin{enumerate}[label=(\alph*),leftmargin=1.6em] \item \emph{Cell structure} (cycles solve one-dimensional cell problems; charts exact; precise forms~\eqref{p.admissible.a1}--\eqref{p.admissible.a7}): \begin{enumerate}[label=(a\arabic*)] \item the two-scale forms are built from the construction's functions and the family's \emph{cell solutions} (built from the solutions of its two-dimensional cell problems) by sums, products, derivatives, averages and primitives in~$z$, the cell inverse across the pulse and changes of chart; \item on the window of pulse~$k$, at all chart parameters, the two-scale forms of~$k$ are transverse (Section~\ref{s.cell}), that is, independent of the coordinate of~$z$ along~$e_{(k)}$; \item at every $t\in[\tA,\tf]$ with $|t-k\tau_m|\le\frac34\tau_m$, the forms of all pulses whose windows contain~$t$ are invariant along~$e_{(k)}$; \item at every $t\in[\tA,\tf]$ such that the velocity added at level~$m$ vanishes on $[t-\frac{\tau_m}{24},t+\frac{\tau_m}{24}]$, the forms of every pulse whose window contains~$t$ do not depend on~$z$; \item an integer shift of the base label does not change the forms; \item where the velocity added at level~$m$ vanishes, consecutive charts describe one coefficient: the change of chart maps~\eqref{e.admissible.twoscale} of one to the next's, with~$z$ fixed; \item near each chart parameter, a pulse's forms depend only on the flow over its \emph{history}. \end{enumerate} \item \emph{Majorants} (the derivative bounds of Sections~\ref{s.preparation}--\ref{s.comparison}). Here $\|\cdot\|_{L^\infty_m([\tA,\tf])}$ is the norm of~\eqref{p.admissible.b2}, and (b2)--(b3) hold for both masses. \begin{enumerate}[label=(b\arabic*)] \item for every source~$\sigma\colon v'\to v$ and $2\le d\le d_\sigma$, the entries of~$\mathcal A_{\sigma,d}$ extend as in~\eqref{p.admissible.norm} and $(\eps_m/\rho_\sigma)^{d-2}\|\mathcal A_{\sigma,d}\|_{\mathrm{an},v}\le\maj(\sigma)$; \item[(b2)] for every~$v$, $v'$, in each pulse's chart, the sum \begin{equation}\label{e.admissible.derivative} (\omega_{v'}/\omega_v)^{\nf12}\sum\nolimits_{\sigma\colon v'\to v}\fac(\sigma)\sum\nolimits_{d=2}^{d_\sigma}\sum\nolimits_{j,j'\le\Ncoef}\eps_{m-1}^{j}\bigl\|\nabla_y^{j}\nabla_z^{j'}\mathcal A_{\sigma,d}\bigr\|_{L^\infty_m([\tA,\tf])} \end{equation} is at most~$\Cder$ times the entry $(v,v')$ of the majorant matrix; \item[(b2$'$)] for every $N,j\le\Ncoef$, the sums of~\eqref{e.ft.coupling} of order~$N$ at scale $\rhomol(m)$ for~\eqref{e.admissible.twoscale}, after~$j$ more slow derivatives (in~$y$), are at most $\Cjet(j)\,\eps_{m-1}^{-j}$ times the majorant matrix (precise form~\eqref{p.admissible.b2mol}); \item[(b3)] the priced majorant matrix has row and column sums at most $\Cadm\eps_m^{\delta/4}$. \end{enumerate} \item \emph{Smoothing} (the~$c_j$ of Subsection~\ref{ss.families}; moments; small widths), for every source~$\sigma$: \begin{enumerate}[label=(c\arabic*)] \item $S_\sigma(t)=S_{\ell,r_\sigma,\Omega_1(t)}S_{\ell,r_\sigma,\Omega_2(t)}$ as in~\eqref{e.Srh} ($\ell$, $\Omega_i$ depending on~$\sigma$), $2\ell-1\ge\Nmom$, $\Omega_i$ Borel measurable in~$t$, $|\Omega_i(t)|_\infty\le C_\Omega$, and the matrix of long axes $\Omega_1(t)e_1$, $\Omega_2(t)e_2$ is invertible, with inverse of $|\cdot|_\infty$-norm $\le C_\Omega$; \item $r_\sigma\le\Cwid\rhomol(m)\,\eps_m^{\delta}$ and $r_\sigma\le\Cwid\rho(m)\,\eps_{m-1}^{4\delta}$; \item for every $t\in[\tA,\tf]$, $\rho>0$, $N\le\Ncoef$ and $j\le\Nsrc-2$, $\|\nabla^jS_\sigma(t)g\|_{H^N_\rho(\T^2)}\le c_jr_\sigma^{-j}\|g\|_{H^N_\rho(\T^2)}$ (precise form~\eqref{p.admissible.c3}). \end{enumerate} \item \emph{Coupling} (for Proposition~\ref{p.ft.energy}): for every $N\le\Ncoef$ and both masses, the coupling matrices~\eqref{e.ft.coupling} on $[\tA,\tf]$ of order~$N$ at scale $\rhomol(m)$ have row and column sums at most $\Cadm\eps_{m-1}^{q\delta/4}$. \end{enumerate} \end{definition} \begin{definition}[Admissible seed]\label{d.admissible.seed} An \emph{admissible seed} of level $\mSm_0$ with finite lifts along its paths of sources (Section~\ref{s.cell}). A \emph{successor}~$G'$ of~$G$, of level $m-1$, is a family that the comparison of Section~\ref{s.comparison} constructs with the lift~\eqref{e.lift}, solving no new two-dimensional cell problem; its data are: \begin{enumerate}[label=(\roman*)] \item \emph{Letters:} the \emph{defect letter} (the defect of the averaged flux at level~$m$; birth order $\Ncor+1+4p$); for every $3\le d\le\Nsrc-\Ncor-4p-4$ and every nonzero path length up to the path bound~$\Npath$, a \emph{mean letter} (birth order~$d$) for the means of degree~$d$ of the lift along paths of that length (for a path of one source, less the cell average of that source in the order~$d$, which its transport carries); for every $2\le d\le\Nsrc-\Ncor-4p-4$, every nonzero path length up to~$\Npath$ and every $j,j'\le p$ with $j+j'\ge1$, a \emph{covariance letter} (birth order $d+2(j+j')$) for the means of degree~$d$ along paths of that length, with no subtraction, conjugated by $(-\Gamma_m:\nabla^2)^j/j!$ on the left and $(\Gamma_m:\nabla^2)^{j'}/j'!$ on the right (summed over $j,j'\le p$, the term $j=j'=0$ included, these give the mean times the truncated exponentials~\eqref{e.covariance} on both sides, as in the flux~\eqref{e.cmp.successorflux}). \item \emph{Components:} those of~$G$, with their masses and markers, and their children~$v'$, labelled by nonempty words of letters, with $\gen(v')=\gen(v)+1\le\Ngen$, $\#v'\le\Nlen$ and \begin{equation}\label{e.admissible.weights} \omega_{v'}\coloneqq\frac{1-128^{-\eta/2}}{2\Nword}\,\eps_{m-1}^{\eta/2}\,\omega_v\,,\qquad \Psi_{v'}\coloneqq\eps_{m-1}^{-\eta/8}\,\Psi_v\,, \end{equation} where~$\Nword$ counts the nonempty words of length $\le\Nlen+1$. \item \emph{Old sources,} in each word: the transport of the cell average (the average in~$z$) of every source, in its orders $d\ge3$ (the same factor, width, smoothing, birth radius, birth order and current order; majorant times~$\Cmaj$); the mean of degree~$2$ along each nonempty path of sources (birth order~$2$; width, smoothing and birth radius of its widest source; product of factors; $\Cmaj\times$ product of majorants). \item \emph{New sources:} for each letter and path it labels (each component, for the defect letter), a source from the first component's word to the last one's word plus the letter: coefficients multiplied by~$\eps_{m-1}^{-\eta}$, the fresh smoothing~\eqref{e.II8.S} with the ellipses of~\eqref{e.admissible.ellipses}, birth radius $\rho(m)$, the letter's birth order, factor the path's product of factors ($1$ for the defect letter), majorant $\Cmaj\eps_{m-1}^{\delta-\eta}$ times the product of the majorants along the path. \end{enumerate} The components of~$G'$ are thus the pairs $(v,w)$ of a component~$v$ of~$G$ and a word~$w$ of length at most~$\Nlen$, with $\gen(v)<\Ngen$ if~$w$ is nonempty (the empty word gives~$v$, a nonempty one a child of~$v$). In each word~$w$, a source of~(iii) along a path from~$v'$ to~$v$ (one source, for a transport) goes from $(v',w)$ to $(v,w)$, and a source of~(iv) of a letter~$\ell$ along such a path (with $v'=v$ for the defect letter) from $(v',w)$ to $(v,w\ell)$. The children of word length $\Nlen+1$ or generation $\Ngen+1$ are created with the masses and markers of~\eqref{e.admissible.weights}, then dropped with the sources at them (Subsection~\ref{ss.cmp.successor}). \end{definition} \begin{definition}[Admissible family]\label{d.admissible.admissible} A family~$G$ of level $m\ge m_0$ is \emph{admissible} on $[\tA,\tf]$ if there exist an admissible seed~$G_{\mS}$ of level $\mS\ge m$, $\mS>m_0$, on $[\tS,\tf]$ with $\tS+\frac12t_0<\tf$, and families $G_{\mS-1},\dots,G_m=G$, each~$G_{i-1}$ a successor of~$G_i$, such that $\tA=\tA^{(m)}$, where $\tA^{(\mS)}\coloneqq\tS$ and, for every $m0$ and $z\in\mathbb C^2$ in the strip of the phases of~\eqref{p.admissible.norm}, the chain \emph{extended at~$(t,z)$} has the data of~$\Ch$, $k_{m+1}=k'$, the phase $z^{(m)}=z$ and the centre $y^{(m)}=(\int_t^{\tb_{m+1}}\widetilde\zeta_{m,k}(t'')\dd t'')\nabla^\perp\psi_{m,k}(\eps_mz)$. Its centre point at the time~$t$ is that of~$\Ch$, as the shear over $[\tb_{m+1},t]$ moves $\xb_{m-1}+y^{(m)}$ to~$\xb_{m-1}$; the centre is along~$e_{(k)}$, depends on~$z$ only through $z\cdot e_{(k)}^\perp$, and has $|y^{(m)}|_1\le4\cdot10^{-6}\can\eps_m$ by~\eqref{e.admissible.carrier}, as $|t-\tb_{m+1}|\le\frac12h_{m+1}+\frac34\tau_{m+1}$. At $\eps_mz=\xb_{m-1}$ its map on the history of~$k'$ is the shear displacement of~\eqref{e.fl.shear}. \precise{p.admissible.transport}{Transport.} Let $x\mapsto y$ be an area-preserving change of variables. The Piola identity writes the divergence in~$y$ of a vector field~$F$ as $\nabla_x\cdot\bigl((\nabla_xy)^{-1}F\bigr)$, and the chain rule, applied one derivative at a time in the order of an entry of~$\nabla_y^{d-1}$, writes that entry as a combination of the entries of $\nabla_x,\dots,\nabla_x^{d-1}$. Substituting both into $\nabla_y\cdot(\text{coefficient}\cdot\nabla_y^{d-1}g)$ for the orders $2\le d\le d_\sigma$ and collecting the terms in each entry of each $\nabla_x^{d'-1}g$ gives the transported coefficients, of orders $2\le d'\le d_\sigma$, indexed by the entries of~$\nabla_x^{d'-1}$ (not symmetrized). \precise{p.admissible.norm}{Analytic norm.} For a component~$v$, the \emph{reserve} $\res(v)\coloneqq(\Ngen+2-\gen(v))/(\Ngen+2+\Noff)$, with~$\Noff$ the offset of item~(2). Let~$\mathcal D_v$ be the set of complex chart parameters, with the discrete data of an allowed chain, such that $|\operatorname{Im}y^{(m'-1)}|_1<(1+\res(v)/4)\,\eps_{m'-1}/100$ and, for every $m'\le im_0$ on $[\tS,\tf]$ with $\tS+\frac12t_0<\tf$ satisfies the invariant on $[\tS,\tf]$ with the constants of items (1), (3) and~(4) of Subsection~\ref{ss.admissible.parameters}. \end{lemma} \begin{proof} By Definition~\ref{d.admissible.seed}(iv), the seed satisfies the invariant with its constants. Each clause of Definition~\ref{d.admissible.invariant} other than~(d) weakens when a constant grows or an order decreases (in~(b1), a smaller weight order decreases the analytic norm, and the reserves are the same, since the offset is), and~(d) is increasing in the~$c_j$. As the constants of families are weaker than the seed's (items (1)--(3)), this gives the invariant except for the sums of~(d); for the sums of~(b3), $\Cseed\eps_{\mS}^{\delta/4}\le\Cadm\eps_{\mS}^{\delta/4}$. By~(b1) for the seed, the coefficient~\eqref{e.admissible.twoscale} of order~$d$ of a source~$\sigma$ is at most $\kappa_{\mS}\rho_\sigma^{d-2}\fac(\sigma)\maj(\sigma)$. Its transport to~$x$ gives the orders $d'\le d$ at the cost $\eps_{\mS-1}^{d'-d}\le\rho_\sigma^{d'-d}$: for $d'm_0$ and $\tA+3\tau_m+4h_m\le\tf$. Let~$G$ be a family of level~$m$ obtained by successors from an admissible seed on $[\tS,\tf]$, with $\tS\le\tA$, that satisfies the invariant on $[\tA,\tf]$ with the constants of items (1), (3) and~(4) of Subsection~\ref{ss.admissible.parameters}, and let~$G'$ be a successor of~$G$. Then~$G'$ satisfies the invariant on $[\tA+3\tau_m,\tf]$ with the same constants. \end{lemma} \begin{proof}[Proof of Proposition~\ref{p.admissible.bounds}] Let~$G$ be admissible of level~$m$ on $[\tA,\tf]$, with the seed~$G_{\mS}$ and the families~$G_i$ of Definition~\ref{d.admissible.admissible}. By Lemma~\ref{l.admissible.seeds}, $G_{\mS}$ satisfies the invariant on $[\tA^{(\mS)},\tf]=[\tS,\tf]$, and for $i=\mS,\dots,m+1$ in turn Lemma~\ref{l.admissible.cycle} passes it from~$G_i$ on $[\tA^{(i)},\tf]$ to~$G_{i-1}$ on $[\tA^{(i-1)},\tf]$, since $i>m_0$, $\tS\le\tA^{(i)}$ and $\tA^{(i)}+3\tau_i+4h_i\le\tf$ by~\eqref{e.admissible.windows}. The fresh smoothings of the sources of~$G$ were created above $m\ge m_0$, so Step~7 of Subsection~\ref{ss.admissible.cycle} gives the inverse bound~$2$ and~(c3) with the constants of Lemma~\ref{l.ft.moving}; the bound~$\frac18$ holds by item~(4). \end{proof} \begin{lemma}[Successors]\label{l.admissible.successors} Let~$G$ be a family of level $m>m_0$ obtained from an admissible seed by successors, and let~$G'$ be a successor of~$G$. \begin{enumerate}[label=\textup{(\roman*)}] \item Each component of~$G$ is one of~$G'$, with the same masses and markers. \item The total mass of~$G'$ is at most twice that of the seed. \item The readout~\eqref{e.lift} satisfies $|\Upsilon_m|\le1+\Nword^{\nf12}\eps_{m-1}^{5\eta/4}$. \item For every admissible seed~$G_{\mS}$ of level $\mS>m_0$ on $[\tS,\tf]$ with $\tS+\frac12t_0<\tf$, there exist families $G_{\mS-1},\dots,G_{m_0}$, each~$G_{i-1}$ a successor of~$G_i$, for which~\eqref{e.admissible.windows} holds with $m=m_0$. In particular, each~$G_i$ is an admissible family of level~$i$ on $[\tA^{(i)},\tf]$. \end{enumerate} \end{lemma} \begin{proof} (i) A component of~$G$ is labelled in~$G'$ by the empty word and has generation at most~$\Ngen$, so Definition~\ref{d.admissible.successor}(ii) keeps it with its mass and marker. (ii) By~\eqref{e.admissible.weights}, a cycle at level~$i$ gives each component~$v$ at most~$\Nword$ children, of total mass at most $\frac12(1-128^{-\eta/2})\eps_{i-1}^{\eta/2}\omega_v$, and $\eps_{i-1}\le128^{-(i-1)}$ by Lemma~\ref{l.fl.scales}(a). So the cycles multiply the total mass by at most $\exp(\frac12(1-128^{-\eta/2})\sum_{i\ge2}128^{-(i-1)\eta/2})\le \exp(1/2)<2$. (iii) In the coordinates scaled by the masses to the power~$-\frac12$, the readout~\eqref{e.lift} is the identity on the components of~$G$ plus a matrix with one entry $\eps_{m-1}^{\eta\#v'}(\omega_{v'}/\omega_v)^{\nf12}\le\eps_{m-1}^{5\eta/4}$, by~\eqref{e.admissible.weights}, for each component~$v$ and child~$v'$ of~$v$. A row has at most~$\Nword$ entries and a column one, so Schur's test (Subsection~\ref{ss.families}) bounds this matrix by $\Nword^{\nf12}\eps_{m-1}^{5\eta/4}$. (iv) By Proposition~\ref{p.admissible.bounds} and the setting of the lift (Section~\ref{s.comparison}), where~(b1), (b3), Cauchy's estimate and~\eqref{e.fl.charts} give the hypotheses of Proposition~\ref{p.cell.means}, an admissible family of a level finer than~$m_0$ has finite lifts up to~$\tf$ on the windows that start at or after~$\tA$, as the cell inverse is solved forward from the start of the window~\eqref{p.admissible.a1}; these include every window that meets $[\tA+3\tau_m,\tf]$, of length $\frac{61}{24}\tau_m<3\tau_m$ by~\eqref{e.cell.window}. So Section~\ref{s.comparison} constructs its successor. As Proposition~\ref{p.admissible.bounds} for~$G_i$ uses only $G_{\mS},\dots,G_i$, the families are constructed one level at a time, up to~$G_{m_0}$ as long as~\eqref{e.admissible.windows} holds, and it holds: as $\tA^{(i)}=\tS+\sum_{i'=i+1}^{\mS}3\tau_{i'}$, item~(4) gives, for every $m_0m_0}(3\tau_{i'}+4h_{i'})<\tS+\tfrac12t_0<\tf\,.\qedhere \end{equation*} \end{proof} \subsection{Holomorphic extension of the flows} \label{ss.admissible.chains} The analytic norm takes suprema over complex chart parameters, so the flows~$\Xc_i$ of a chain~\eqref{p.admissible.chains} must extend holomorphically to the outer domain. They are compositions of a coarse flow, extended by Lemma~\ref{l.fl.complex}, with entire shear displacements. The domains close because the renewal times of a chain are short, which the carrier level provides: $\mc$ satisfies, for every level $i\ge\mc-1$, \begin{equation}\label{e.admissible.carrier} \eps_i^{\beta-2}h_{i+1}\le10^{-6}\can\,,\qquad 48\,h_{i+2}<\tau_{i+1}\,, \end{equation} and such a level exists, by~\eqref{e.renewal} and, for the second bound, by~\eqref{e.times} and $\eps_{i+1}\le2\eps_i^q$, as $q(2-\beta+2\delta)-(2-\beta+4\delta)=(q-1)(1-\frac\beta{2q+2})+2q\delta>0$ by~\eqref{e.exponents}. For a chain of a pulse~$k$ of level~$m$ and $m'-1\le i\le m-1$, the flow~$\Xc_i$ is anchored at the centre~$\tb_{i+1}$ of the block of~$k_{i+1}$ ($k_m=k$); we call $|t-\tb_{i+1}|\le\frac35h_{i+1}$ the \emph{interval} of~$\Xc_i$, and write $\xb_i(t)\coloneqq\Xc_i(t,\xb_i,\tb_{i+1})$ for the centre point at time~$t$. The intervals of the flows and the histories of the pulses are nested. Let $i\ge\mc$, and let a pulse~$k_{i+1}$ of level $i+1$ have its chart used at a time~$s$ of the window of a pulse~$k_i$ of level~$i$, so that $|s-k_{i+1}\tau_{i+1}|<\frac54\tau_{i+1}$ and $|s-k_i\tau_i|\le\frac{61}{48}\tau_i$ by~\eqref{e.cell.window}. The time of a pulse of level~$j$ is within~$\frac12h_j$ of the centre of its block (Lemma~\ref{l.fl.scales}(d)), and $\tau_j\le h_j/25$ by~\eqref{e.scales}. Hence, by the second bound of~\eqref{e.admissible.carrier} at the level $i-1\ge\mc-1$ and induction on the level, starting from the finest level (where a history is the window and~$\tb_{M+n}$), the interval $|t-\tb_{i+1,k_{i+1}}|\le\frac35h_{i+1}$ and the history of~$k_{i+1}$ lie within \begin{equation*} \Bigl(\frac35+\frac12\Bigr)h_{i+1}+\frac54\tau_{i+1}\le\frac65h_{i+1}<\frac{\tau_i}{40}\ \ \text{of~$s$, hence within}\ \ \frac12h_i+\frac{61}{48}\tau_i+\frac{\tau_i}{40}<\frac35h_i\ \ \text{of~$\tb_{i,k_i}$}, \end{equation*} as $\frac12+\frac1{25}(\frac{61}{48}+\frac1{40})<\frac35$; so does the history of~$k_i$. \begin{lemma}[Complex chains]\label{l.admissible.chains} Fix the levels and pulses of a chain of a pulse~$k$ of level~$m$, with $m'\ge\mc$. For every $m'-1\le i\le m-1$, the maps $\Xc_i(t',y,t)$, for~$t,t'$ in the interval of~$\Xc_i$, and the centre points~$\xb_i(t)$ defined above extend from the allowed chains to the parameters of the outer domain of~\eqref{p.admissible.norm} and to the points~$y$ with $|y-\xb_i(t)|_\infty<8\can\eps_i$, holomorphically in the parameters and in~$y$, such that, for every $t,t',t''$ in the interval of~$\Xc_i$ and every~$y$ with $|y-\xb_i(t)|_\infty<8\can\eps_i$: \begin{enumerate}[label=\textup{(\roman*)}] \item $\Xc_i(t,y,t)=y$ and $|\nabla_y\Xc_i(t',y,t)-I|_\infty\le2^{25}\eps_i^{\beta-2}|t'-t|$; \item $\Xc_i(t',\xb_i(t),t)=\xb_i(t')$, and, if $|y-\xb_i(t)|_\infty<2\can\eps_i$, then $\Xc_i(t'',\Xc_i(t',y,t),t')=\Xc_i(t'',y,t)$. \end{enumerate} \end{lemma} By~(i), (ii) and $2^{25}\eps_i^{\beta-2}\cdot\frac65h_{i+1}\le\frac1{100}$, from~\eqref{e.admissible.carrier}, the map $\Xc_i(t',\cdot,t)$ sends every ball about~$\xb_i(t)$ of radius at most $8\can\eps_i$ into the ball about~$\xb_i(t')$ of $\frac{101}{100}$ times that radius, and, by Cauchy's estimate, its derivative is Lipschitz on the ball of radius $4\can\eps_i$, with constant at most $1/(4\can\eps_i)$. \begin{proof} By induction on~$i$. \emph{The first level.} We apply Lemma~\ref{l.fl.complex} at the level $m'-1$ on the interval of $\Xc_{m'-1}=X_{m'-1}$, of length $t_2-t_1=\frac65h_{m'}$, where $\xb_{m'-1}(t)=X_{m'-1}(t,y^{(m'-1)},\tb_{m'})$. By~\eqref{e.admissible.carrier}, the number $1-2^{21}\eps_{m'-1}^{\beta-2}(t_2-t_1)$ of that lemma satisfies \begin{equation*} 1-2^{21}\eps_{m'-1}^{\beta-2}\cdot\tfrac65h_{m'}\ge1-\tfrac65\cdot2^{21}\cdot10^{-6}\can\ge0.9998\,. \end{equation*} In units of $\eps_{m'-1}/1000$, in which $\can\eps_{m'-1}=\frac1{20}$, Lemma~\ref{l.fl.complex} with this bound gives holomorphy and~\eqref{e.fl.complex} where $|\operatorname{Im}y|_1<0.9998\cdot19$, and the group law where $|\operatorname{Im}y|_1\le0.9998^2\cdot19$. In the outer domain $|\operatorname{Im}y^{(m'-1)}|_1<15$, so $|\operatorname{Im}\xb_{m'-1}(t)|_1<15/0.9998<15.2$ by~\eqref{e.fl.complex}. As $|\cdot|_1\le2|\cdot|_\infty$, the balls about~$\xb_{m'-1}(t)$ of radii $8\can\eps_{m'-1}$ and $2\can\eps_{m'-1}$, that is, $\frac25$ and~$\frac1{10}$, lie in $|\operatorname{Im}y|_1<16$ and $|\operatorname{Im}y|_1<15.4$. This gives~(i), as $10\cdot2^{21}\le2^{25}$, and, with the group law at the base label,~(ii). \emph{From $i-1$ to~$i$.} As the chart of~$k_{i+1}$ is used in the window of~$k_i$, the interval of~$\Xc_i$ lies within $\frac35h_i$ of~$\tb_i$, inside that of~$\Xc_{i-1}$, by the nesting above ($i\ge m'\ge\mc$). For~$t,t'$ in it, \eqref{p.admissible.chains} composes $\Xc_{i-1}(\tb_i,\cdot,t)$, the \emph{shear map} (the bracket in~\eqref{p.admissible.chains}) and $\Xc_{i-1}(t',\cdot,\tb_i)$. The shear $\nabla^\perp\psi_{i,k_i}$ is entire and depends on its argument only through the coordinate along~$e_{(k_i)}^\perp$, which the shear map therefore does not change. Where the imaginary part of this coordinate is less than $15\can\eps_i$, the shear is at most $2\pi\eps_i^{\beta-1}\exp(30\pi\can)$ and its derivative at most $4\pi^2\eps_i^{\beta-2}\exp(30\pi\can)$, by~\eqref{e.shear}. As $0\le\widetilde\zeta_{i,k_i}\le1$ by~\eqref{e.zeta} and~\eqref{e.pulse}, \eqref{e.admissible.carrier} shows that the shear map over an interval of length at most~$\frac65h_{i+1}$ moves each point of the strip by at most $\frac65h_{i+1}\cdot2\pi\eps_i^{\beta-1}\exp(30\pi\can)\le10^{-5}\can\eps_i$. The point $\xb_{i-1}+y^{(i)}$, and its image under the shear map over $[\tb_{i+1},t]$, which $\Xc_{i-1}(t,\cdot,\tb_i)$ maps to~$\xb_i(t)$ by~\eqref{p.admissible.chains} and~(ii) for~$\Xc_{i-1}$, lie within $(\frac34+10^{-5})\can\eps_i<2\can\eps_{i-1}$ of $\xb_{i-1}=\xb_{i-1}(\tb_i)$, as $|y^{(i)}|_1<\frac34\can\eps_i$. So $|\xb_i(t)-\xb_{i-1}(t)|_\infty\le\frac{101}{100}(\frac34+10^{-5})\can\eps_i\le\can\eps_i$, by~(i) for~$\Xc_{i-1}$, and all points below lie within $10\can\eps_i\le\frac{10}{128}\can\eps_{i-1}$ of~$\xb_{i-1}(t)$ or~$\xb_{i-1}$, in the balls of~(i), (ii) and of the Lipschitz bound for~$\Xc_{i-1}$. Hence $\Xc_{i-1}(\tb_i,\cdot,t)$ sends the ball $|y-\xb_i(t)|_\infty<8\can\eps_i$ into the ball of radius $\frac{101}{100}\cdot8\can\eps_i$ about the image of~$\xb_i(t)$, which is $\xb_{i-1}+y^{(i)}$ moved along~$e_{(k_i)}$ by a shear map. For~$y'$ in the image ball, the bound on $\operatorname{Im}z^{(i)}\cdot e_{(k_i)}^\perp$ of the outer domain gives \begin{equation*} \bigl|\operatorname{Im}\bigl((y'+\eps_iz^{(i)}-\xb_{i-1}-y^{(i)})\cdot e_{(k_i)}^\perp\bigr)\bigr|<\frac{101}{100}\cdot8\can\eps_i+\Bigl(3+\frac12+\frac1{64}\Bigr)\can\eps_i<12\can\eps_i\,: \end{equation*} the argument of the shear stays in the strip, and~$\Xc_i$ and~$\xb_i(t)$ are holomorphic. At $t'=t$ the shear map is the identity, and the shear maps over $[t,t']$ and $[t',t'']$ compose to the one over $[t,t'']$, since the integrals of~$\widetilde\zeta_{i,k_i}$ add and the shear map does not change the coordinate along~$e_{(k_i)}^\perp$; with~(i), (ii) for~$\Xc_{i-1}$, this gives the identities of~(i), (ii). By the chain rule and the group law~(ii) for~$\Xc_{i-1}$, $\nabla_y\Xc_i(t',y,t)-I$ is the sum of three terms: $\nabla_y\Xc_{i-1}(t',y,t)-I$, bounded by~(i) for~$\Xc_{i-1}$; the change of $\nabla_y\Xc_{i-1}(t',\cdot,\tb_i)$ over the shear displacement (at most $|t'-t|\,2\pi\eps_i^{\beta-1}\exp(30\pi\can)$), bounded by the Lipschitz bound above, times $\nabla_y\Xc_{i-1}(\tb_i,y,t)$, of norm at most~$\frac{101}{100}$; and $\int_t^{t'}\widetilde\zeta_{i,k_i}(t'')\dd t''$, at most $|t'-t|$, times the derivative of the shear between two matrices of norm at most~$\frac{101}{100}$. As $\eps_{i-1}^{\beta-2}\le128^{\beta-2}\eps_i^{\beta-2}\le\frac12\eps_i^{\beta-2}$, since $2-\beta=1-\alpha>\frac23$, \begin{align*} \frac{|\nabla_y\Xc_i(t',y,t)-I|_\infty}{|t'-t|} & \le2^{25}\eps_{i-1}^{\beta-2}+\frac{101}{100}\cdot\frac{2\pi\eps_i^{\beta-1}\exp(30\pi\can)}{4\can\eps_{i-1}}+\Bigl(\frac{101}{100}\Bigr)^2\,4\pi^2\eps_i^{\beta-2}\exp(30\pi\can) \notag \\ & \le(2^{24}+10^3)\eps_i^{\beta-2}\,, \end{align*} which is at most $2^{25}\eps_i^{\beta-2}$. At real parameters the extensions are given by the formulas~\eqref{p.admissible.chains}. \end{proof} \subsection{Proof of Lemma~\ref{l.admissible.cycle}} \label{ss.admissible.cycle} \begin{proof}[Proof of Lemma~\ref{l.admissible.cycle}] Each clause for~$G'$, of level $m-1$, is read off its explicit data (Definition~\ref{d.admissible.successor}), those of~$G$ and the invariant of~$G$, on $[\tA+3\tau_m,\tf]$, the time window of Proposition~\ref{p.cycle}(b). At a time~$t$, the forms of~$G'$ read those of~$G$ at the pulses~$k$ of level~$m$ with $\Hc_{m,k}(t)>0$ (Step~1) at times of the window of~$k$ up to~$t$, the lift being solved forward on it~\eqref{p.admissible.a1}, and only the flows of the chains, functions of the construction, reach back along the histories; for $t\ge\tA+3\tau_m$ these times lie in~$[\tA,\tf]$, where~$G$ satisfies the invariant, as in the proof of Lemma~\ref{l.admissible.successors}(iv). Step~3, the recursion for~(b3), is the main estimate. It comes before~(b1) because the majorants of~$G'$ are numbers fixed by Definition~\ref{d.admissible.successor}, so~(b3) is a recursion on numbers, while~(b1) says that the forms obey them, which needs the analytic calculus of Step~4. Step~5 proves~(b1), and~(b2), (b2$'$), (d) follow. Components of~$G$ are written $v,v'$; a component~$\tilde v$ of~$G'$ \emph{over}~$v$ is~$v$ or a child of~$v$. \emph{Step 1: (a1)--(a4).} By Definition~\ref{d.admissible.successor} and Section~\ref{s.comparison}, the forms of a source of~$G'$ at a pulse~$k'$ of level $m-1$ are of four kinds: transports~\eqref{p.admissible.transport} of cell averages of forms of~$G$; means of the lift along paths of sources of~$G$; these means conjugated by powers of $\Gamma_m:\nabla^2$; and the defect of the averaged flux, with the commutator of the covariance with~$\b_{m-1}$, of flux~\eqref{e.cmp.commflux}. At a time~$t$ of the window of~$k'$, they read the construction's functions and the forms of~$G$ at the pulses~$k$ of level~$m$ with $\Hc_{m,k}(t)>0$, at the chains of~$k$ extended at~$(t,z)$ from those of~$k'$ (end of~\eqref{p.admissible.chains}, one level lower), whose phase~$z$ is the fast variable of~$k'$; as the forms of~$G$ may contain evaluations toward the level $m+1$, these evaluations are nested, which~\eqref{p.admissible.a1} allows. The lift of Section~\ref{s.cell} is built from these forms by the cell inverse across the pulse, derivatives and averages in the fast variable of level~$m$, products and slow derivatives. So the forms of~$G'$ are built as in~\eqref{p.admissible.a1} from the construction's functions and the cell solutions of the seed, and no two-dimensional cell problem is solved: this is~(a1). They depend on~$z$ only through the coordinate along~$e_{(k')}^\perp$ of the phase of the shear of~$k'$: so do the shear map and the centre of the extended chain, and so do the forms of~$G$, since forms of finite analytic norm do not depend on the coordinates of the phases along the pulses~\eqref{p.admissible.norm}. So the forms of~$G'$ are transverse: this is~(a2). The same argument, with~(b1) for~$G$, shows that they are periodic in the phases and independent of their coordinates along the pulses, as~\eqref{p.admissible.norm} requires. For~(a4), let~$t$, in the window of~$k'$, be as in~(a4) at the level $m-1$. The forms of~$k'$ at~$t$ read those of~$G$ at the pulses~$k$ of level~$m$ with $\Hc_{m,k}(t)>0$ on their histories, within $\frac65h_m<\frac{\tau_{m-1}}{40}$ of~$t$ by the nesting after~\eqref{e.admissible.carrier} at the level $m-1\ge\mc$. There the shear of~$k'$ vanishes, so the extended chains have centre~$0$ and their maps on the history of~$k$ and centre points do not depend on~$z$; by~(a7) for~$G$, neither do the forms of~$G$ and the correctors at these chains, nor the forms of~$k'$ at~$t$. For~(a3), let $|t-k'\tau_{m-1}|\le\frac34\tau_{m-1}$: the forms of~$k'$ are transverse by~(a2), and the window of another pulse meets this interval only where $|t-k'\tau_{m-1}|\ge\frac{35}{48}\tau_{m-1}$, and there no pulse is active within~$\frac{\tau_{m-1}}{24}$ of~$t$, as $\zeta_{m-1,k''}$ vanishes outside $|t-k''\tau_{m-1}|\le\frac23\tau_{m-1}$ (Lemma~\ref{l.fl.scales}(d)), so that~(a4) applies. The commutator term of the defect letter reads no form of~$G$: in its flux~\eqref{e.cmp.commflux}, $\nabla\b_{m-1}$ read in the chart of~$k'$ is the strain of~$\Xc_{m-2}$ plus $\nabla_y(\nabla_y\Xc_{m-2}\,w)\,(\nabla_y\Xc_{m-2})^{-1}$, with~$w$ the drift of~$k'$, which depends only on $z\cdot e^\perp_{(k')}$ and vanishes at the times of~(a4) and~(a6). So this term is built as in~\eqref{p.admissible.a1} and is transverse; at those times it is the strain, the transport of the one Eulerian coefficient~$\nabla\b_{m-1}$, which gives~(a4) and~(a6); and, a function of the flow, it satisfies~(a5) and~(a7). \emph{Step 2: (a5)--(a7).} Moving the base label by an integer vector commutes with every operation of~\eqref{p.admissible.a1}: the flows commute with the translations by~$\Z^2$ (Lemma~\ref{l.fl.short}), the arguments of the shear streams and the centres of the extended chains do not change, the cutoffs do not depend on the label, and the forms of~$G$ are invariant by~(a5) for~$G$: this gives~(a5). Clauses~(a6) and~(a7) follow from~(a7) for~$G$, as a form of~$G'$ at~$k'$ is built from forms of~$G$ over the histories of the pulses of level~$m$ whose charts are used in the window of~$k'$, which lie in the history of~$k'$. For~(a7), two allowed chains of~$k'$ with the same map over this history, extended at the same~$(t,z)$, have the same centre, which depends only on~$k'$, $k$, $t$, $z$, and the same maps over these histories; so the forms of~$G$ agree there near the chain, by~(a7) for~$G$, and so do the correctors, solved from zero, and the forms of~$G'$. For~(a6), the windows of two pulses~$k'$, $k''$ of level $m-1$ meet only if their indices differ by~$2$, and their overlap lies at distance at least $(\frac{35}{48}-\frac23)\tau_{m-1}=\frac{\tau_{m-1}}{16}$ from the supports of both fast cutoffs, so both shears vanish within~$\frac{\tau_{m-1}}{16}$ of a time~$t$ of the overlap, and, as in Step~1 and by~(a7) for~$G$, the extended chains reduce to the charts of~$k'$ and~$k''$, related by Lemma~\ref{l.fl.charts}(b); the forms of~$G'$ at~$k'$, $k''$ are then transports of the same forms of~$G$ through the two charts, and transport composes exactly, as the chain rule taken one derivative at a time is associative: this is~\eqref{p.admissible.a6}. \emph{Step 3: the recursion for (b3).} Fix one of the two masses. By~(b3) for~$G$, every row and column sum of the priced majorant matrix~\eqref{e.admissible.majorant} of~$G$ is at most $\Cadm\eps_m^{\delta/4}$, at most~$\frac12$ for $m>m_0$ by item~(4) and Lemma~\ref{l.fl.scales}(a). For a path of sources of~$G$ from~$v'$ to~$v$, the product of the corresponding entries of this matrix is at least $(\omega_{v'}/\omega_v)^{\nf12}\prod_{\sigma\in\text{path}}\fac(\sigma)\maj(\sigma)$: the mass ratios telescope, and every price is at least~$1$. Their sums over the paths of a given length into or out of a component are at most the row or column sums of a power of the matrix, so over all nonzero lengths at most $2\Cadm\eps_m^{\delta/4}$. \emph{Old sources.} Each word contains a copy of~$G$ (Definition~\ref{d.admissible.successor}(iii)), its sources joining the components of the word over~$v'$ and~$v$, whose masses and markers are those of~$v'$ and~$v$ times numbers depending only on the word (Lemma~\ref{l.admissible.successors}(i), \eqref{e.admissible.weights}); so the masses in the entries are those of~$G$. A transported source keeps its factor, birth radius, birth order and width, and its majorant is multiplied by~$\Cmaj$; a second-order mean has birth order~$2$, hence price~$1$, factor the product of the factors and majorant~$\Cmaj$ times the product of the majorants. Counting transports as paths of one source, the copies into or out of a component of a word contribute at most $2\Cmaj\Cadm\eps_m^{\delta/4}$ per kind. \emph{New sources.} A letter labelling a path from~$v'$ to~$v$ ($v'=v$ for the defect letter) gives a source from the component of a word over~$v'$ to the child over~$v$ of the extended word. By~\eqref{e.admissible.weights}, the square root of its mass ratio is at most $C\eps_{m-1}^{-\eta/4}(\omega_{v'}/\omega_v)^{\nf12}$, and its marker ratio at most $\eps_{m-1}^{-\eta/8}\Psi_v/\Psi_{v'}$. Its birth radius is $\rho(m)$, its width $r_m=\rho(m)\eps_{m-1}^{\Theta}$ by~\eqref{e.II8.width}, and its birth order at most~$\Nsrc$, so its price is at most $\eps_{m-1}^{-(\Nsrc-2)\Theta}$. Its entry is thus at most \begin{equation*} C\Cmaj\,\eps_{m-1}^{\delta-\eta-\frac\eta4-\frac\eta8-(\Nsrc-2)\Theta} \end{equation*} times the product along its path. Into or out of a component, the entries of one letter add up to at most twice this, by the bound on path sums. \emph{The recursion.} Adding over the at most $\Nword+1$ words and over the letters, every row and column sum of the priced majorant matrix of~$G'$ is at most \begin{equation}\label{e.admissible.recursion} C\eps_{m-1}^{\delta-11\eta/8-(\Nsrc-2)\Theta}+4(\Nword+1)(\Cmaj+1)\Cadm\eps_m^{\delta/4}\,, \end{equation} with~$C$ depending on~$\eta$, $\Nword$, the number of letters and~$\Cmaj$, and \begin{equation*} \delta-\tfrac{11}8\eta-(\Nsrc-2)\Theta\ge\tfrac{45}{64}\delta>\tfrac\delta4\,, \end{equation*} as $\eta,(\Nsrc-2)\Theta\le\delta/8$ by~\eqref{e.exponents} and item~(1). In the second term, $\eps_m^{\delta/4}\le2\eps_{m-1}^{(q-1)\delta/4}\eps_{m-1}^{\delta/4}$ by Lemma~\ref{l.fl.scales}(a): the old sums gain one step of the scales. So, for $m>m_0$, both terms of~\eqref{e.admissible.recursion} are at most $\frac12\Cadm\eps_{m-1}^{\delta/4}$, $m_0$ being chosen after~$\Cadm$, $\Cmaj$ and the exponents: this is~(b3). For the masses $\omega_v/\Psi_v^2$, the markers appear only through the ratio $\eps_{m-1}^{-\eta/8}$ of a first letter, counted above. \emph{Step 4: analyticity.} The construction's functions extend to the outer domain: the flows of the chains and their centre points by Lemma~\ref{l.admissible.chains}; the shear streams are entire; the cutoffs, $\J_m^{\kappa_m}$ and~$\Gamma_m$ depend on time only; and the Jacobians, their inverses, the conductivity and the strain are built from the flows. The strain is defined on the complex domain, as $\partial_t\nabla_yX_i=\nabla\b_i(t,X_i)\nabla_yX_i$ is continuous in~$t$ and holomorphic in~$y$ (Proposition~\ref{p.fl.analytic}, Lemma~\ref{l.fl.complex}), and so are the compositions of~\eqref{p.admissible.chains}, with $C^1$ time weights $\int\widetilde\zeta_{i,k_i}$; near~$\xb_i(t)$, the strain of~$\Xc_i$ is the limit of $h^{-1}(\nabla_y\Xc_i(t+h,\cdot,t)-I)$ as $h\to0$, so its $|\cdot|_\infty$-norm is at most $2^{25}\eps_i^{\beta-2}$, by Lemma~\ref{l.admissible.chains}. As $2^{25}\eps_{m-1}^{\beta-2}\cdot\frac65h_m\le\frac1{100}$ by~\eqref{e.admissible.carrier}, on the outer domain and in the window of~$k$ the Jacobian of $\Xc_{m-1}(t,\cdot,\tb_m)$ is within~$\frac1{100}$ of~$I$, by Lemma~\ref{l.admissible.chains}(i), and its inverse within~$\frac1{99}$. The conductivity of the chain is $(\nabla_y\Xc_{m-1})^{-1}(\nabla_y\Xc_{m-1})^{-\mathsf T}$, as in~\eqref{e.fl.metric}, with the transpose and no complex conjugation, so that it is holomorphic; its entry across the pulse, the only one the cell inverse uses, is the sum of the squares of the entries of the row of the inverse along~$e_{(k)}^\perp$, within~$\frac1{99}$ of~$e_{(k)}^\perp$ in $|\cdot|_1$, so it is within $\frac2{99}+\frac1{99^2}\le\frac1{10}$ of~$1$ and has real part at least~$\frac9{10}$. Let~$v$ be a component of~$G$ and~$\tilde v$ one of~$G'$ over~$v$, so that $\res(\tilde v)\ge\res(v)-1/(\Ngen+2+\Noff)$. As sources never go to a smaller generation (Subsection~\ref{ss.admissible.parameters}), the forms of the sources on a path into~$v$ are bounded on~$\mathcal D_v$ by their analytic norms. We split the step $1/(\Ngen+2+\Noff)$ of the reserve into~$\Nsrc$ equal parts, one for each slow derivative a cycle takes; $\mathcal D$ and $\|\cdot\|_{\mathrm{an}}$ are those of~\eqref{p.admissible.norm} at the level of the form, with a number~$r$ in place of the reserve where written. The extension at~$(t,z)$ of a chain of~$\mathcal D_{\tilde v}$ at the level $m-1$, with $|\operatorname{Im}z\cdot e_{(k')}^\perp|<(3+\frac1{64})\can$, keeps the lower parameters and has a centre of $|\cdot|_1$-norm at most $4\cdot10^{-6}\can\eps_{m-1}$, below all radii of centres, so it lies in~$\mathcal D_{\tilde v}$ at the level~$m$, hence in~$\mathcal D_v$. \emph{A slow derivative costs one part.} Let~$\mathcal A$ have finite analytic norm at the reserve~$\res(v)$, and let $\res(\tilde v)\le\res(v)-1/(\Nsrc(\Ngen+2+\Noff))$. For a chain~$\Ch$ in~$\mathcal D_{\tilde v}$, the recentred chains~$\Ch^{[y]}$ with $|y|_\infty<\can\eps_{m-1}/(9\Nsrc(\Ngen+2+\Noff))$ lie in~$\mathcal D_v$. Indeed, recentring moves the base label by~$y$ if $m'=m$; otherwise, by the group law of Lemma~\ref{l.admissible.chains}(ii), it moves the last centre by $w\coloneqq\Xc_{m-2}(\tb_{m-1},\xb_{m-1}+y,\tb_m)-\Xc_{m-2}(\tb_{m-1},\xb_{m-1},\tb_m)$ and the last phase by $w/\eps_{m-1}$, with $|w|_\infty\le(1+a)|y|_\infty$, $a\coloneqq2^{25}\cdot\frac35\cdot10^{-6}\can<2\cdot10^{-3}$, by Lemma~\ref{l.admissible.chains}(i) and~\eqref{e.admissible.carrier} at the level $m-2$, as $|\tb_m-\tb_{m-1}|\le\frac35h_{m-1}$ by the nesting after~\eqref{e.admissible.carrier} and the segment from~$\xb_{m-1}$ to $\xb_{m-1}+y$ lies within $\can\eps_{m-1}$ of~$\xb_{m-2}(\tb_m)$ (proof of Lemma~\ref{l.admissible.chains}). By~\eqref{p.admissible.norm}, the radii of~$\mathcal D_v$ exceed those of~$\mathcal D_{\tilde v}$ by $\can\eps_{m-1}$, $\can$ and $\eps_{m-1}/100$ times $1/(4\Nsrc(\Ngen+2+\Noff))$, and $2(1+a)/9<\frac14$. Cauchy's estimate in~$y_1$, mode by mode, gives \begin{equation*} \eps_{m-1}\bigl|\partial_{y_1}\widehat{\mathcal A}(t,\Ch^{[y]};\xi)\bigr|_{y=0}\le\frac{9\Nsrc(\Ngen+2+\Noff)}{\can}\sup_{\mathcal D_v}\bigl|\widehat{\mathcal A}(t,\cdot\,;\xi)\bigr|\,, \end{equation*} and likewise in~$y_2$, so that $\eps_{m-1}\|\nabla_y\mathcal A\|_{\mathrm{an},\tilde v}\le18\Nsrc(\Ngen+2+\Noff)\can^{-1}\|\mathcal A\|_{\mathrm{an},v}$. \emph{A transverse primitive costs nothing.} On a transverse form with zero average in~$z$, it multiplies the mode $\xi\ne0$ of $z\cdot e_{(k)}^\perp$ by $(2\pi\mathrm{i}\xi)^{-1}$, of modulus at most $\frac1{2\pi}$, and keeps the mode~$0$ at zero. \emph{The other operations} of~\eqref{p.admissible.a1} cost nothing, a constant or one part, by the same two mechanisms. Products cost a constant, the Fourier weights $\exp(3\pi\can|\xi|)(1+|\xi|)^{\Nan}$ of the analytic norm being submultiplicative in~$\xi$; the average in~$z$ keeps the mode~$0$; the cell inverse of Section~\ref{s.cell} maps the mode~$\xi$ of a flux written $(\kappa_m/\eps_m^2)\eperp\cdot\nabla_zF$ to that of the gradient of the solution by a time integral of $4\pi^2\xi^2\kappa_m/\eps_m^2$ times exponentials whose exponents have real parts at most $-3\pi^2\xi^2\kappa_m/\eps_m^2$ times the elapsed time, as the real part of the entry across the pulse is at least $\frac9{10}\ge\frac34$, so with a constant at most~$\frac43$; on a scalar source it costs the power $\eps_m^2/(3\pi^2\xi^2\kappa_m)$ of Lemma~\ref{l.cell.inverse}, which the proof of Proposition~\ref{p.cell.sizes} pairs with the coefficients of~\eqref{e.cell.recursion}. A derivative in the fast variable of level~$m$ multiplies its mode by $2\pi\mathrm{i}\xi$, which the polynomial weight absorbs: a cycle takes at most $2+\Npath\Nsrc+2\Ncor+2\Nsrc$ of them, the order in~$z$ of~\eqref{e.cell.hyp} at $N=0$, and $\Nan\ge\Ncoef\ge\Nret$ (items~(1) and~(3)) exceeds this number by~\eqref{e.cmp.floor}. A derivative in a centre costs one part. After the averages in the fast variable of level~$m$, the Fourier weights of the analytic norm of~$G'$ are restored: its fast variable is the phase of level $m-1$ of the chains of~$G$, in which the forms of~$G$ are holomorphic on $|\operatorname{Im}z^{(m-1)}\cdot e^\perp_{(k')}|<(3+\frac1{64})\can$, so their modes in it decay faster than $\exp(-6\pi\can|\xi|)$. Transports and second-order means use only operations that cost nothing or a constant. A letter takes at most $\Nsrc-2$ slow derivatives and goes to a child, so its forms have analytic norms at~$\res(\tilde v)$ bounded by constants times those of forms of~$G$ at~$\res(v)$. With $P\coloneqq\Nsrc(\Ngen+2+\Noff)$ and~$\mathcal D(r)$ the set~$\mathcal D$ of the level~$m$ with~$r$ in place of the reserve, these costs give, for a path $\sigma_1\cdots\sigma_i$ of sources of~$G$ into~$v$, $r=\res(v)$ and $2\le d\le j\le\Ncor+\Nsrc$ with $j-d\le\Nsrc-2$, \begin{equation*} \sup_{\mathcal D(r-(j-d)/P)}\bigl|\Mc_{j,d}[\sigma_1\cdots\sigma_i]\bigr|\le C\fac(\sigma_1)\maj(\sigma_1)\cdots\fac(\sigma_i)\maj(\sigma_i)\,\kappa_{m-1}\ebar^{\,j-2}\eps_{m-1}^{d-j}\,. \end{equation*} This is the proof of Proposition~\ref{p.cell.sizes}(c), with Lemma~\ref{l.cell.inverse} in its complex form (it uses only $\operatorname{Re}e_{(k)}^\perp\cdot\K e_{(k)}^\perp\ge\frac34$), the costs above for derivatives, and the monotone generations along the path; Proposition~\ref{p.cell.means}(c) passes in the same way to complex conductivities, as $|e^{-a}-e^{-b}|\le|a-b|\max\{|e^{-a}|,|e^{-b}|\}$ for complex~$a$, $b$. Their hypotheses~\eqref{e.cell.hyp} hold on every window by~(b1) for~$G$, as in Step~6. \emph{Step 5: (b1).} For a source~$\sigma'$ of~$G'$ into~$\tilde v$ we check, kind by kind, that the forms satisfy $(\eps_{m-1}/\rho_{\sigma'})^{d-2}\|\mathcal A_{\sigma',d}\|_{\mathrm{an},\tilde v}\le\maj(\sigma')$; the constants produced by Step~4 are at most~$\Cmaj$, which is large given item~(3). Each form of~$G'$ at~$k'$ is, at a time~$t$, a sum over the at most two pulses~$k$ of level~$m$ with $\Hc_{m,k}(t)>0$ of the transport of a form at~$k$ by the maps of Lemma~\ref{l.admissible.chains}, with the Piola factor of~\eqref{e.cell.corr}; this costs a constant (Lemma~\ref{l.admissible.chains}, Cauchy's estimate on its balls), the coefficient of order~$d'$ from the order~$d$ carrying~$\eps_{m-1}^{d'-d}$, so we bound the forms at~$k$. The birth radius of a source of~$G$ of birth order at least~$3$ is at most~$\eps_m$: for one of the seed by Definition~\ref{d.admissible.seed}(i), and for one created at a level $i>m$ since $\rho(i)\le C\eps_{i-1}^{1+\gamma/2+\delta}\le\eps_{i-1}$ by~\eqref{e.II8.scales}, as $i>m_0$. \emph{Transports.} Let~$\sigma$ be a source of~$G$. Its coefficients~\eqref{e.admissible.twoscale} of orders $d\ge3$, averaged in~$z$, are transported to the chart of~$k'$; the order~$d$ gives the orders $d'\le d$, with the $d-d'$ derivatives of the change of variables at the scale~$\eps_{m-1}$ (Lemma~\ref{l.admissible.chains}). Writing the result of order~$d'$ as $\kappa_{m-1}\eps_{m-1}^{d'-2}\fac(\sigma)\mathcal A_{\sigma',d'}$, we obtain, by~(b1) for~$G$, $\kappa_m\le\kappa_{m-1}$ (Proposition~\ref{p.scales}(c)) and $\rho_\sigma\le\eps_m$, \begin{equation*} \Bigl(\frac{\eps_{m-1}}{\rho_\sigma}\Bigr)^{d'-2}\|\mathcal A_{\sigma',d'}\|_{\mathrm{an},\tilde v}\le C\frac{\kappa_m}{\kappa_{m-1}}\sum_{d=\max\{3,d'\}}^{d_\sigma}\Bigl(\frac{\rho_\sigma}{\eps_{m-1}}\Bigr)^{d-d'}\Bigl(\frac{\eps_m}{\rho_\sigma}\Bigr)^{d-2}\|\mathcal A_{\sigma,d}\|_{\mathrm{an},v}\le C\maj(\sigma)\,. \end{equation*} \emph{Second-order means} are at most~$C$ times the product of the $\fac(\sigma)\maj(\sigma)$ along their path (Section~\ref{s.cell}), in the analytic norm with no loss of reserve, as averages of products of forms and correctors (Step~4). \emph{Mean letters.} Let~$\sigma'$ be the source of a mean letter of birth order~$d_{\sigma'}\ge3$ along a path of~$G$ into~$v$. By Proposition~\ref{p.cell.sizes}(c), the coefficient of order $d\le d_{\sigma'}$ of the mean of degree~$d_{\sigma'}$ along the path is $\kappa_{m-1}\ebar^{\,d_{\sigma'}-2}\eps_{m-1}^{d-d_{\sigma'}}$ times a form bounded by~$C\prod_{\sigma\in\text{path}}\fac(\sigma)\maj(\sigma)$. Its recursion takes $d_{\sigma'}-d$ slow derivatives and otherwise the operations of Step~4, so the bound holds in the analytic norm at the reserve of~$\tilde v$. By Definition~\ref{d.admissible.successor}(iv), $\mathcal A_{\sigma',d}$ is~$\eps_{m-1}^{-\eta}$ times this coefficient divided by $\kappa_{m-1}\eps_{m-1}^{d-2}\prod_{\sigma\in\text{path}}\fac(\sigma)$, and $\rho_{\sigma'}=\rho(m)\le\eps_{m-1}$, by~\eqref{e.II8.scales} as $m>m_0$, so \begin{equation*} \Bigl(\frac{\eps_{m-1}}{\rho_{\sigma'}}\Bigr)^{d-2}\|\mathcal A_{\sigma',d}\|_{\mathrm{an},\tilde v}\le C\eps_{m-1}^{-\eta}\Bigl(\frac{\ebar}{\rho(m)}\Bigr)^{d_{\sigma'}-2}\prod_{\sigma\in\text{path}}\maj(\sigma)\le C\eps_{m-1}^{\delta-\eta}\prod_{\sigma\in\text{path}}\maj(\sigma)\,. \end{equation*} The last step is the smallness of the correctors: by~\eqref{e.cell.amplitude}, $\ebar\le C\eps_{m-1}^{3\delta}\rho(m)$, so the ratio in parentheses is at most $C\eps_{m-1}^{3\delta}$, and $3\delta(d_{\sigma'}-2)\ge\delta$. For a path of one source~$\sigma$, the letter subtracts its cell average, at most $C\eps_{m-1}^{-\eta}(\kappa_m/\kappa_{m-1})\maj(\sigma)$ by the transport bound and $\rho_\sigma\le\eps_m\le\rho_{\sigma'}$; and $\kappa_m/\kappa_{m-1}\le C\eps_{m-1}^{2q\gamma}\le C\eps_{m-1}^\delta$, by Proposition~\ref{p.scales}(c) ($m\le\mSm_0$, both birth radii are at most~$\rhomol(m-1)$. Hence~(b2$'$); for the masses $\omega_v/\Psi_v^2$, both sides gain the factor~$\Psi_v/\Psi_{v'}$. \emph{Step 7: (c).} A transported source keeps its smoothing, and a second-order mean takes that of its widest source, so~(c1) and~(c3) are inherited, and~(c2) follows from~(c2) for~$G$, since $\rhomol(m)\eps_m^{\delta}\le\rhomol(m-1)\eps_{m-1}^{\delta}$ and $\rho(m)\eps_{m-1}^{4\delta}\le\rho(m-1)\eps_{m-2}^{4\delta}$ ($\eps_m\le\eps_{m-1}$, $\kappa_m\le\kappa_{m-1}\le\kappa_{m-2}$, $h_m\le h_{m-1}$). The new sources have the fresh smoothing~\eqref{e.II8.S}, of width~$r_m$, with the ellipses of~\eqref{e.admissible.ellipses}. Then~$\Omega_1$, $\Omega_2$ are piecewise continuous in~$t$, and $|\Omega_i(t)|_\infty\le C\le C_\Omega$, by Lemma~\ref{l.fl.short} ($|t-l_{\mathrm{hor}}h_m|,|t-l_{\mathrm{ver}}h_m|\le\frac92h_m$), Lemma~\ref{l.fl.flux}(i) and Proposition~\ref{p.scales}(a), (c). Both are invertible, as $\det\nabla_yX_{m-1}=1$ (Lemma~\ref{l.fl.short}) and $\J_m^{\kappa_m}\ge\kappa_mI$ (Lemma~\ref{l.fl.flux}(i)). The long axes $\Omega_1(t)e_1$, $\Omega_2(t)e_2$ are the columns $e_1$, $e_2$ of Jacobians within $C\eps_{m-1}^{2\delta}$ of~$I$ (proof of Lemma~\ref{l.fl.short}, \eqref{e.renewal}), times square roots of maxima of the diagonal entries of $\kappa_{m-1}^{-1}\J_m^{\kappa_m}$, at least~$1$ as these entries have average~$1$ over a period by~\eqref{e.mean} and~\eqref{e.recursion}. So the matrix of long axes is within $C\eps_{m-1}^{2\delta}$ of~$I$ times a diagonal matrix with entries $\ge1$, and, for $m>m_0$, its inverse has $|\cdot|_\infty$-norm at most $(1-C\eps_{m-1}^{2\delta})^{-1}\le2\le C_\Omega$. This gives~(c1), with $2\ell-1\ge\Nmom$ by item~(1). Lemma~\ref{l.ft.moving} gives the bound of~(c3) with its constants, which the~$c_j$ of item~(3) dominate; the bound passes to $H^N_\rho(\T^2)$ since~$S(t)$ commutes with derivatives. The widths follow from~\eqref{e.II8.width}, \eqref{e.II8.scales}, $\eps_{m-1}\le2\eps_{m-2}^q$ and $4\deltam_0$, $m_0$ being chosen after~$C$, $\Cadm$ and~$\Cmaj$, both terms are at most $\frac12\Cadm\eps_{m-2}^{q\delta/4}$: this is~(d). \end{proof} % ===== 11-preparation ===== \section{Smoothing within one period: Bloch wave analysis} \label{s.preparation} \newcommand{\Hlo}{{\Hc^{\mathrm{lo}}_m}}% This section proves part~(a) of Proposition~\ref{p.cycle}, the \emph{preparation}, by Bloch wave analysis; part~(b) is proved in Section~\ref{s.comparison}, for the time~$\tpr$ and the map~$\Pi$ constructed here. Part~(a) says that within one period~$4h_m$ of the pulses the solution of an admissible family of level~$m$ becomes, up to a small error, smooth at the scale~$\rho(m)$, with bounds by the $L^2$ norm of the datum. This scale is the distance over which the coarse diffusivity~$\kappa_{m-1}$ spreads in one renewal time; below the terminal level it is far larger than the distance that molecular diffusion reaches in that time. So the smoothing must come from the homogenization of the pulses of level~$m$, and it must hold to the accuracy~$\eps_{m-1}^Q$ of the cycle, uniformly in the contrast, and without any derivative of the datum. We obtain it from an exact decomposition of the evolution. In the chart of a renewal block, with the conductivity frozen at a point, Bloch waves split the evolution into evolutions on the period cell, one for each frequency (Subsection~\ref{ss.prep.bloch}). At small frequencies, the cell mean of the principal branch of these evolutions evolves at a scalar rate, the symbol (Subsection~\ref{ss.prep.branch}). To leading order the symbol is the homogenized operator of the pulses; as a block carries pulses of both orientations, over one block the principal branch is damped by a Gaussian in the frequency at the scale~$1/\rho(m)$, which is the smoothing we need (Subsection~\ref{ss.prep.dissipation}). Everything else decays at the \emph{cell rate}, the inverse of the time that molecular diffusion takes to cross a period cell, and over a block its norm is multiplied by a number smaller than any power of~$\eps_{m-1}$ (Subsection~\ref{ss.prep.high}). Subsections~\ref{ss.prep.tests}--\ref{ss.prep.joint} add the sources of the family, which take derivatives of the solution. We expand the solution in its sources by Duhamel's formula, freeze the coefficients on patches of size~$\rho(m)$, and estimate each term by duality, against solutions of the adjoint cell evolutions, which carry the derivatives that the sources take, except near the ends of a block (Lemma~\ref{l.prep.words}(iii)). On the principal branch a source then costs, over a block, at most a power of $1+\rho(m)|\xi|$, which the Gaussian absorbs; everything else is damped. Subsection~\ref{ss.prep.proof} assembles the proof. \paragraph{Conventions.} In Subsections~\ref{ss.prep.bloch}--\ref{ss.prep.high} every constant of a statement is named in it, and~$C$ is absolute in the proofs of Subsections~\ref{ss.prep.bloch} and~\ref{ss.prep.branch}; Subsection~\ref{ss.prep.tests} states the conventions of Subsections~\ref{ss.prep.tests}--\ref{ss.prep.joint}. Elsewhere, $C$ and~$c$ denote positive constants depending only on~$\deps$ and on the orders and constants fixed in Subsection~\ref{ss.admissible.parameters} and below, which may change from line to line; $C(j)$ depends also on~$j$, and $C(J)$ and~$c(J)$ on~$J$. The corrector amplitude $\ebar=\eps_m(\kappa_{m-1}/\kappa_m)^{\nf12}$ of~\eqref{e.mu}, with the \emph{contrast} $\kappa_{m-1}/\kappa_m$, is at most $C\eps_{m-1}^{3\delta}\rho(m)$, by~\eqref{e.cell.amplitude}. The cell rate is $\kappa_m/\eps_m^2$. Two identities, from the definition of~$\ebar$ and~\eqref{e.rho}, translate between cell units and the coarse scale: \begin{equation*} \frac{\kappa_m}{\eps_m^2}\,\ebar^2=\kappa_{m-1}\,,\qquad \frac{\rho(m)^2}{\ebar^2}=\frac{\kappa_mh_m}{\eps_m^2}\,: \end{equation*} in cell units the homogenized equation damps the frequency~$\xi$ at the rate $(\ebar|\xi|)^2$, and a renewal block lasts $(\rho(m)/\ebar)^2$ cell diffusion times. \paragraph{The constant~$\cmol$.} The constant~$\cmol\in(0,1]$ of the molecular radius~\eqref{e.admissible.rhomol} is fixed here, after~$\Lambda$: it is at most a threshold depending only on $\alpha,s,q,\zeta,\Lambda$, below which, for every $M$, $\kappa\in\Kset_M$ and $2\le m\le M$, \begin{equation}\label{e.prep.cmol} \begin{gathered} \rhomol(m)^{j+1}\bigl\|\nabla^j\b_m\bigr\|_{L^\infty(\R\times\T^2)}\le2^{-j-2}\,j!\,\kappa_m\quad(j\ge1)\,,\\ \frac{\rhomol(m)}{\eps_m}\le2^{-11}\,,\qquad \eps_m^{\beta-2}\,\frac{\rhomol(m)^2}{\kappa_m}\le2^{-20}\,. \end{gathered} \end{equation} The first condition is~(i) of Proposition~\ref{p.ft.energy} for the velocity~$\b_m$ at the scale $\rhomol(m)$ and every order. It holds for small~$\cmol$: Proposition~\ref{p.fl.analytic} and Cauchy's estimate on polydiscs of radius $\eps_m/100$ give $\|\nabla^j\b_m\|_{L^\infty(\R\times\T^2)}\le C^j(j-1)!\,\eps_m^{\beta-1-j}$ with an absolute~$C$, and~\eqref{e.admissible.rhomol} bounds the left side by $\cmol^{j+1}C^j(j-1)!\,\kappa_m$. The last condition makes the molecular time $\rhomol(m)^2/\kappa_m$ short compared with the strain time~$\eps_m^{2-\beta}$ of~$\b_m$ \eqref{e.fl.hessian}. \paragraph{The order of choices.} The \emph{depth}~$\Ndep$ of the Duhamel expansion (Subsection~\ref{ss.prep.tests}) is the least integer with the second condition below, $\Nbuf$ is the order of the molecular smoothing, and the moment order~$\Nmom$ of the smoothings satisfies the first condition (item~(1) of Subsection~\ref{ss.admissible.parameters}, Table~\ref{t.admissible.schedule}): \begin{equation}\label{e.prep.Npr} q\delta\Nmom>Q+3\,,\qquad \frac{q\delta}4(\Ndep+1)>Q+3\,,\qquad \Nbuf\coloneqq(\Ndep+1)(\Nsrc-2)+\Nmom\,. \end{equation} The integers introduced in Subsections~\ref{ss.prep.tests}--\ref{ss.prep.joint} depend only on these orders, on~$\Nsrc$, $Q$ and~$\Nout$, and~$\Ncoef$ is at least each of them and~$\Nbuf$. The top level~$m_0$ is fixed after~$\cmol$. \subsection{Statements} \label{ss.prep.statements} The preparation runs over the interval of Figure~\ref{f.prep.timeline}. From~$\tc$ the solution runs for the molecular time, which gives it derivatives at the scale~$\rhomol(m)$ (Lemma~\ref{l.prep.buffer}), and from the end~$t_1$ of the molecular interval it is expanded in its sources. The rest of the block of~$t_1$ may carry pulses of one orientation only; the next, complete, block smooths, and~$\tpr$ is its end, inside a quiet margin. Proposition~\ref{p.prep.further} gives four further properties of~$\tpr$ and~$\Pi$. By~(i) the comparison of Section~\ref{s.comparison} starts at~$\tpr$, where the lift adds no corrector (Lemma~\ref{l.cmp.quiet}); (iii) gives the derivatives that the passage of Section~\ref{s.passage} spends; and (iv) bounds the datum of the successor, the retained trace of~$\Pi g$. \begin{proposition}[Further properties of the preparation]\label{p.prep.further} There exists $C(\deps)<\infty$ such that, for every $M$, $\kappa$, $m$, $\tA$, $\tc$, $\tf$, $G$ and~$G'$ as in Proposition~\ref{p.cycle}, the time~$\tpr$ and the map~$\Pi$ of Proposition~\ref{p.cycle} can be chosen, satisfying \textup{(a)} and~\textup{(b)} with this~$C$, such that also the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item The time~$\tpr$ comes after the molecular time, and the velocity added at level~$m$ vanishes near it: \begin{equation*} \tc+\frac{\rhomol(m)^2}{\kappa_m}\le\tpr\,,\qquad \b_m=\b_{m-1}\quad\text{on }[\tpr-\tau'_m,\tpr+\tau'_m]\times\T^2\,. \end{equation*} \item $\Pi$ maps every field of~$G$ whose components are constant functions to itself. \item For every datum~$g$, the solution~$U$ of~$G$ from~$g$ at time~$\tc$ satisfies \begin{equation*} \|U\|_{L^\infty([\tpr,\tf];H^{2\Nrep+4}_{\rhomol(m),\omega}(\T^2))}\le C\|g\|_{L^2_G(\T^2)}\,. \end{equation*} \item For every datum~$g$, \begin{equation*} \|\expp(-\Gamma_m(\tpr):\nabla^2)\,\Pi g\|_{H^{\Nret}_{\rho(m),G}(\T^2)}\le C\|g\|_{L^2_G(\T^2)}\,. \end{equation*} \end{enumerate} \end{proposition} \paperfigure{% \begin{figure}[!htbp] \centering \begin{tikzpicture}[x=1.15cm,y=1cm,font=\small] \fill[black!12] (2.55,-0.18) rectangle (3.45,0.18); \fill[black!12] (10.05,-0.18) rectangle (10.95,0.18); \draw[-{Stealth[length=2.2mm]}] (-0.3,0) -- (11.8,0); \foreach \x in {3.0,10.5} \draw (\x,-0.25) -- (\x,0.25); \draw[thick] (0,-0.12) -- (0,0.12) node[above=2pt] {$\tc$}; \draw[thick] (1.0,-0.12) -- (1.0,0.12) node[above=2pt] {$t_1$}; \node[below=3pt] at (3.0,-0.2) {$t_2$}; \node[below=3pt] at (10.5,-0.2) {$\tpr$}; \foreach \k in {0,...,8} { \pgfmathsetmacro{\xx}{3.75+0.75*\k} \ifodd\k \draw[very thick] (\xx,0) -- (\xx,0.45); \else \draw[very thick,densely dotted] (\xx,0) -- (\xx,0.45); \fi} \draw[very thick] (1.9,0) -- (1.9,0.45); \draw[decorate,decoration={brace,amplitude=4pt,mirror}] (0,-0.45) -- (1.0,-0.45) node[midway,below=4pt] {$\rhomol(m)^2/\kappa_m$}; \draw[decorate,decoration={brace,amplitude=4pt}] (1.0,0.75) -- (3.0,0.75) node[midway,above=4pt] {old partial block}; \draw[decorate,decoration={brace,amplitude=4pt}] (3.0,0.75) -- (10.5,0.75) node[midway,above=4pt] {one complete renewal block, of length $h_m$}; \draw[decorate,decoration={brace,amplitude=4pt,mirror}] (0,-1.25) -- (10.5,-1.25) node[midway,below=4pt] {at most $4h_m$}; \end{tikzpicture} \caption{The preparation of a cycle, in time (not to scale): the molecular interval from~$\tc$ to~$t_1$, the rest of the block containing~$t_1$ (the old partial block), and one complete renewal block, from the block boundary~$t_2$ to~$\tpr$. Vertical strokes: pulses, solid and dotted for the two orientations. Shaded: the quiet margins around $t_2$ and~$\tpr$, of half-width~$\tau'_m$.} \label{f.prep.timeline} \end{figure}} \subsection{The frozen operator and its Bloch waves} \label{ss.prep.bloch} In the chart of a renewal block (Lemma~\ref{l.fl.equation}) the coarse velocity is gone: what remains is the periodic shear of the pulses and a conductivity close to the identity that varies only on the scale~$\eps_{m-1}$, much larger than~$\rho(m)$. As a point of the datum influences, over one block, a neighbourhood of size about~$\rho(m)$, we freeze the conductivity at a point~$y_0$; Subsection~\ref{ss.prep.joint} bounds the error. The frozen operator has periodic coefficients, and Bloch waves reduce it to one evolution on the period cell for each frequency, the fibre equation~\eqref{e.prep.fibre}. \paragraph{Standing hypotheses.} Throughout Subsections~\ref{ss.prep.bloch}--\ref{ss.prep.high}, let $\Lambda\ge\Lambda_0$, where $\Lambda_0(\alpha,s,q,\zeta)\ge128$ is at least the thresholds of Proposition~\ref{p.scales} and Lemma~\ref{l.cell.corrector} and is so large that \begin{equation}\label{e.prep.relax} \frac{4\pi^2\kappa_m\tau_m}{\eps_m^2}\ge6\,,\qquad \eps_{m-1}^\delta\le\frac1{16}\qquad\text{for every }M\ge2,\ \kappa\in\Kset_M,\ 2\le m\le M \end{equation} (by Proposition~\ref{p.scales}(c)). Fix $M\ge2$, $\kappa\in\Kset_M$, $2\le m\le M$, $l\in\Z$, a point $y_0\in\R^2$ and a measurable map~$\K$ from~$\R$ to the symmetric $2\times2$ matrices with $\|\K-I\|_{L^\infty(\R)}\le\eps_{m-1}^\delta$. In the applications $\K(t)=\K_{m,l}(t,y_0)$ for $|t-lh_m|\le h_m$, the conductivity of the chart of the block of~$l$ frozen at the label~$y_0$, and $\K(t)=I$ otherwise; by~\eqref{e.fl.charts}, $\|\K-I\|_{L^\infty(\R)}\le C\eps_{m-1}^{2\delta}\le\eps_{m-1}^\delta$ for~$\Lambda_0$ large. By Lemma~\ref{l.fl.equation}, the level-$m$ operator in the chart of the block is $\partial_t+\nabla^\perp\psi_m\cdot\nabla-\kappa_m\nabla\cdot(\K_{m,l}\nabla\,\cdot)$, with $\psi_m$ of~\eqref{e.fl.psim}. With its conductivity frozen at~$y_0$, it becomes the \emph{frozen operator} \begin{equation}\label{e.prep.frozen} \partial_t+\nabla^\perp\psi_m(t,y)\cdot\nabla-\kappa_m\nabla\cdot\bigl(\K(t)\nabla\,\cdot\,\bigr)\qquad\text{on }\R\times\R^2\,, \end{equation} whose coefficients are $\eps_m$-periodic in~$y$ (Section~\ref{s.construction}). \paragraph{Fibres.} We write $z\coloneqq y/\eps_m$ for the fast variable, as in Section~\ref{s.cell}. The velocity $(\nabla^\perp\psi_m)(t,\eps_mz)$ is $1$-periodic in~$z$, divergence free and of mean zero over the cell. On~$\mathbb C^2$ the dot is bilinear, $\xi\cdot\eta\coloneqq\xi_1\eta_1+\xi_2\eta_2$, so that $|\xi|^2=\xi\cdot\bar\xi$. For $\xi\in\mathbb C^2$, the function $\exp(\mathrm i\xi\cdot(y-y_0))U(t,y/\eps_m)$, with~$U$ periodic in~$z$, is annihilated by~\eqref{e.prep.frozen} if and only if~$U$ solves the \emph{fibre equation at~$\xi$}, \begin{equation}\label{e.prep.fibre} \partial_tU=\frac{\kappa_m}{\eps_m^2}(\nabla+\mathrm i\eps_m\xi)\cdot\K(t)(\nabla+\mathrm i\eps_m\xi)U-\frac1{\eps_m}(\nabla^\perp\psi_m)(t,\eps_mz)\cdot(\nabla+\mathrm i\eps_m\xi)U \end{equation} on~$\T^2$, with~$\nabla$ in~$z$ and~$\nabla^\perp\psi_m$ the velocity of~\eqref{e.prep.frozen}, in~$y$. Solutions are energy solutions, one for each datum. The real frequencies with $\eps_m|\xi|_\infty\le\pi$ form the (Brillouin) \emph{zone}. For real~$\xi$ the transport term of~\eqref{e.prep.fibre} is skew-adjoint, as the velocity is divergence free, so \begin{equation}\label{e.prep.contract} \frac{\mathrm d}{\mathrm dt}\|U(t)\|_{L^2(\T^2)}^2=-\frac{2\kappa_m}{\eps_m^2}\bigl\langle\K(t)(\nabla+\mathrm i\eps_m\xi)U,(\nabla+\mathrm i\eps_m\xi)U\bigr\rangle_{L^2(\T^2)}\le0\,, \end{equation} and, by Fourier series, for real~$\xi$ in the zone and every $U\in H^1(\T^2)$, \begin{equation}\label{e.prep.zonegap} \|(\nabla+\mathrm i\eps_m\xi)U\|_{L^2(\T^2)}\ge\max\bigl\{\eps_m|\xi|\,\|U\|_{L^2(\T^2)},\ \pi\|U-(U)_{\T^2}\|_{L^2(\T^2)}\bigr\}\,, \end{equation} since $|2\pi j+\eps_m\xi|\ge\eps_m|\xi|$ for every $j\in\Z^2$, and $\ge\pi$ for $j\ne0$. \paragraph{Bloch sums.} Fields $g=(g_v)_{v\in V}$ on~$\R^2$ carry the masses~$\omega_v$ of~\eqref{e.norms}, and their moments about~$y_0$ at the scale~$\rho(m)$ are measured by \begin{align}\label{e.prep.moments} \|g\|_{L^2_{\omega,N'}(\R^2)} & \coloneqq\Bigl\|\Bigl(1+\frac{|y-y_0|^2}{\rho(m)^2}\Bigr)^{N'/2}g\Bigr\|_{L^2_\omega(\R^2)}\,, \notag \\ \|g\|_{H^N_{\rho,\omega,N'}(\R^2)}^2 & \coloneqq\sum_{j\le N}\frac{\rho^{2j}}{(j!)^2}\|\nabla^jg\|_{L^2_{\omega,N'}(\R^2)}^2\,; \end{align} the centre~$y_0$ is always the point of the frozen operator. For a measurable field~$g$ on~$\R^2$ with $\|g\|_{L^2_{\omega,2}(\R^2)}<\infty$, its \emph{Bloch sums} about~$y_0$ are the sums \begin{equation}\label{e.prep.bloch} \Bl g(\xi,z)\coloneqq\sum_{j\in\Z^2}\exp(-\mathrm i\xi\cdot(\eps_m(z+j)-y_0))g\bigl(\eps_m(z+j)\bigr)\,, \end{equation} periodic in~$z$. At every real~$\xi$ they converge absolutely for almost every~$z$, and in~$L^2(\T^2)$, by the Cauchy--Schwarz inequality against the weight $(1+|y-y_0|^2/\rho(m)^2)^{-2}$, which is summable over the points $y=\eps_m(z+j)$, $j\in\Z^2$. Their mean over~$\T^2$ is $\eps_m^{-2}\widehat g(\xi)$, where $\widehat g(\xi)\coloneqq\int_{\R^2}\exp(-\mathrm i\xi\cdot(y-y_0))g(y)\dd y$ is the Fourier transform about~$y_0$. They satisfy $\Bl(\nabla g)=\eps_m^{-1}(\nabla+\mathrm i\eps_m\xi)\Bl g$, with~$\nabla$ in~$y$ on the left and in~$z$ on the right, and $\Bl(a(\cdot/\eps_m)g)=a\Bl g$ for every $1$-periodic function~$a$. The following lemma, Plancherel's identity for Bloch sums, returns bounds proved at each frequency to the plane. \begin{lemma}[Bloch sums]\label{l.prep.bloch} Let~$g$ be a measurable field on~$\R^2$ with $\|g\|_{L^2_{\omega,2}(\R^2)}<\infty$. Then the Bloch sums of~$g$ satisfy \begin{equation*} \int_{\text{zone}}\|\Bl g(\xi,\cdot)\|_{L^2_\omega(\T^2)}^2\dd\xi=(2\pi)^2\eps_m^{-4}\|g\|_{L^2_\omega(\R^2)}^2\,. \end{equation*} \end{lemma} \begin{proof} For almost every~$z$, $\Bl g(\xi,z)$ is a complex number of modulus one times the trigonometric series $\sum_{j\in\Z^2}\exp(-\mathrm i\eps_m\xi\cdot j)g(\eps_m(z+j))$ of the cell frequency $\eps_m\xi\in[-\pi,\pi]^2$, and Parseval's identity gives $\int_{\text{zone}}|\Bl g(\xi,z)|^2\dd\xi=(2\pi)^2\eps_m^{-2}\sum_j|g(\eps_m(z+j))|^2$; integrating in~$z$ gives the identity. \end{proof} We write~$\Lc_0$ for the cell operator~\eqref{e.cell.cellop} with the conductivity $\K(t)$ of the standing hypotheses, at every $t\in\R$. The next subsection constructs the principal branch as a fixed point of the cell problem forced at all times, which Lemma~\ref{l.prep.inverse} solves with a constant independent of the velocity: the transport is skew and drops out of the energy, and the diffusion damps functions of mean zero at the cell rate. For a Banach space~$X$, forcings and solutions with values in~$X$ are measured over intervals of one cell diffusion time $\eps_m^2/\kappa_m$, \begin{equation*} \|F\|_{L^2_{\mathrm{ul}}(\R;X)}\coloneqq\sup_{t\in\R}\Bigl(\frac{\kappa_m}{\eps_m^2}\int_t^{t+\eps_m^2/\kappa_m}\|F(t')\|_X^2\dd t'\Bigr)^{\nf12}, \end{equation*} and the space $L^\infty(\R;L^2(\T^2))\cap L^2_{\mathrm{ul}}(\R;H^1(\T^2))$ carries the norm $\|w\|_{L^\infty(\R;L^2(\T^2))}+\|\nabla w\|_{L^2_{\mathrm{ul}}(\R;L^2(\T^2))}$. \begin{lemma}[Bounded solutions at every time]\label{l.prep.inverse} Let~$F^0$ and~$F$ be measurable on $\R\times\T^2$, with values in~$\mathbb C$ and~$\mathbb C^2$, with finite $L^2_{\mathrm{ul}}(\R;L^2(\T^2))$ norms. Then the problem \begin{equation*} \Lc_0U=\frac{\kappa_m}{\eps_m^2}\bigl(F^0-(F^0)_{\T^2}+\nabla\cdot F\bigr)\qquad\text{in }\R\times\T^2 \end{equation*} has exactly one energy solution $U\in L^\infty(\R;L^2(\T^2))$ with $(U(t))_{\T^2}=0$ for every~$t$, and \begin{equation*} \|U\|_{L^\infty(\R;L^2(\T^2))}+\|\nabla U\|_{L^2_{\mathrm{ul}}(\R;L^2(\T^2))}\le100\bigl(\|F^0\|_{L^2_{\mathrm{ul}}(\R;L^2(\T^2))}+\|F\|_{L^2_{\mathrm{ul}}(\R;L^2(\T^2))}\bigr). \end{equation*} \end{lemma} \begin{proof} We measure time in units of $\eps_m^2/\kappa_m$, so that the coefficients $\kappa_m/\eps_m^2$ disappear. Testing with~$\bar U$ and using $\K\ge\frac12I$ and Poincar\'e's inequality $\|\nabla U\|_{L^2(\T^2)}\ge2\pi\|U\|_{L^2(\T^2)}$ for mean-zero~$U$, we obtain \begin{equation*} \frac{\mathrm d}{\mathrm dt}\|U\|_{L^2(\T^2)}^2+\pi^2\|U\|_{L^2(\T^2)}^2+\tfrac14\|\nabla U\|_{L^2(\T^2)}^2\le4\bigl(\|F^0\|_{L^2(\T^2)}^2+\|F\|_{L^2(\T^2)}^2\bigr). \end{equation*} Summing over intervals of unit length, we find that every energy solution on $[t',\infty)$ satisfies \begin{equation*} \|U(t)\|_{L^2(\T^2)}\le e^{-\pi^2(t-t')/2}\|U(t')\|_{L^2(\T^2)}+2.01\bigl(\|F^0\|_{L^2_{\mathrm{ul}}(\R;L^2(\T^2))}^2+\|F\|_{L^2_{\mathrm{ul}}(\R;L^2(\T^2))}^2\bigr)^{\nf12}, \end{equation*} and the first display, integrated over an interval of unit length, then bounds~$\nabla U$. The solutions from zero at the times $t'=-1,-2,\dots$ converge to one with the bound, and two bounded solutions coincide (let $t'\to-\infty$). \end{proof} \subsection{The principal branch at small frequencies} \label{ss.prep.branch} At $\xi=0$ the constants solve the fibre equation~\eqref{e.prep.fibre}, and every other solution approaches its cell mean at the cell rate. We show that this splitting persists at small frequencies. As the coefficients depend on time, there is no eigenvalue to continue. We seek instead a cell function~$\Rb_\xi(t)$ of mean one and a scalar rate~$\Zs_\xi(t)$ such that~$\Rb_\xi$, multiplied by a scalar function of time that grows at the rate~$\Zs_\xi$, solves~\eqref{e.prep.fibre}; in the plane this is a plane wave corrected at the cell scale. Averaging the fibre equation over~$\T^2$, with the velocity of mean zero, gives the \emph{symbol} \begin{equation}\label{e.prep.symbol} \Zs_\xi=-\kappa_m\,\xi\cdot\K\xi-\mathrm i\,\xi\cdot\bigl((\nabla^\perp\psi_m)(t,\eps_m\cdot)\,(\Rb_\xi-1)\bigr)_{\T^2}\,, \end{equation} and its part of mean zero is the \emph{graph equation} \begin{align}\label{e.prep.graph} & \Lc_0(\Rb_\xi-1) =\frac{2\mathrm i\kappa_m}{\eps_m}\nabla\cdot\bigl(\K\xi\,(\Rb_\xi-1)\bigr) \notag \\ & \qquad -\mathrm i\bigl(\xi\cdot(\nabla^\perp\psi_m)(t,\eps_m\cdot)\,\Rb_\xi-(\xi\cdot(\nabla^\perp\psi_m)(t,\eps_m\cdot)\,(\Rb_\xi-1))_{\T^2}\bigr) \notag \\ & \qquad +\mathrm i(\Rb_\xi-1)\bigl(\xi\cdot(\nabla^\perp\psi_m)(t,\eps_m\cdot)\,(\Rb_\xi-1)\bigr)_{\T^2}\,. \end{align} This is the cell problem of Lemma~\ref{l.prep.inverse} for $\Rb_\xi-1$, with a forcing that is small in cell units when $\ebar|\xi|$ is small: whatever the P\'eclet number (the velocity in cell units), the transport term on a mode of frequency~$\xi$ has relative size \begin{equation}\label{e.prep.drift} \frac{\eps_m^2}{\kappa_m}\,\|\nabla^\perp\psi_m\|_{L^\infty(\R\times\R^2)}\,|\xi|\le\frac{2\pi\eps_m^{\beta+1}|\xi|}{\kappa_m}\le C\ebar|\xi| \end{equation} with an absolute~$C$ (Step~1 of the proof of Proposition~\ref{p.prep.branch}). So the graph equation has exactly one small solution of mean zero, and~$\Rb_\xi$ is the \emph{principal branch}, the branch that continues the constants. By~(a) below it differs from~$1$ by the relative size of the first corrector of a plane wave, and by~(b) the symbol is at most of the size of the rate of the homogenized equation at the frequency~$\xi$. \begin{proposition}[The principal branch]\label{p.prep.branch} There exist absolute constants $\cbr\in(0,1]$ and $\Cbr$ such that, under the standing hypotheses, there exist $\Rb_\xi(t)\in L^2(\T^2)$ and $\Zs_\xi(t)\in\mathbb C$, for $t\in\R$ and $\xi\in\mathbb C^2$ with $\ebar|\xi|<\cbr$, power series in~$\xi$, such that for every such~$\xi$ the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item For every $t\in\R$, $(\Rb_\xi(t))_{\T^2}=1$, and \begin{equation*} \|\Rb_\xi-1\|_{L^\infty(\R;L^2(\T^2))}\le\Cbr\ebar|\xi|\,. \end{equation*} \item $\Zs_\xi=\sum_{j\ge2}\Zs_{\xi,j}$, where~$\Zs_{\xi,j}$ is homogeneous of degree~$j$ in~$\xi$, and for every $j\ge2$ \begin{equation*} \|\Zs_{\xi,j}\|_{L^\infty(\R)}\le\Cbr\frac{\kappa_m}{\eps_m^2}\Bigl(\frac{2\ebar|\xi|}{\cbr}\Bigr)^j\,. \end{equation*} \item For every $t'\in\R$, the function $t\mapsto\Rb_\xi(t)\exp(\int_{t'}^t\Zs_\xi(t'')\dd t'')$ solves the fibre equation~\eqref{e.prep.fibre} on $[t',\infty)$. \item For every $\xi\in\R^2$ with $\ebar|\xi|<\cbr$ and every $t'\le t$, \begin{equation*} \Bigl|\exp\Bigl(\int_{t'}^t\Zs_\xi(t'')\dd t''\Bigr)\Bigr|\le\Cbr\,. \end{equation*} \item $\Rb_\xi-1$ is the only solution of the graph equation~\eqref{e.prep.graph} that has mean zero at every time and norm at most one in $L^\infty(\R;L^2(\T^2))\cap L^2_{\mathrm{ul}}(\R;H^1(\T^2))$, and~$\Zs_\xi$ is given by~\eqref{e.prep.symbol}. \end{enumerate} \end{proposition} \begin{proof} \emph{Step 1: The branch and the symbol.} As $\|\nabla^\perp\psi_m\|_{L^\infty(\R\times\R^2)}\le2\pi\eps_m^{\beta-1}$ and $\eps_m^{2\beta}/\kappa_m\le20\kappa_{m-1}$ (Lemma~\ref{l.cell.corrector}(c)), the constant of~\eqref{e.prep.drift} is absolute. The right side of the graph equation~\eqref{e.prep.graph} is the cell rate times $F^0-(F^0)_{\T^2}+\nabla\cdot F$, where $F=2\mathrm i\eps_m\K\xi\,(\Rb_\xi-1)$ and~$F^0$ is $\eps_m^2/\kappa_m$ times the transport terms, with $\|F\|_{L^2(\T^2)}\le4\ebar|\xi|\,\|\Rb_\xi-1\|_{L^2(\T^2)}$ and $\|F^0\|_{L^2(\T^2)}\le C\ebar|\xi|(1+\|\Rb_\xi-1\|_{L^2(\T^2)})^2$, by~\eqref{e.prep.drift} and $\eps_m|\xi|\le\ebar|\xi|$. So Lemma~\ref{l.prep.inverse} makes the map that sends a mean-zero function, put in place of $\Rb_\xi-1$ on the right side, to the bounded mean-zero solution of~\eqref{e.prep.graph} a contraction of the unit ball of $L^\infty(\R;L^2(\T^2))\cap L^2_{\mathrm{ul}}(\R;H^1(\T^2))$ when $\ebar|\xi|<\cbr$, with an absolute~$\cbr\le\frac12$; its fixed point has norm at most $C\ebar|\xi|$. The map is a polynomial in~$\xi$ and in the function, so its Picard iterates are polynomials in~$\xi$ and converge uniformly on the complex ball $\ebar|\xi|<\cbr$: $\Rb_\xi$ and~$\Zs_\xi$ are power series in~$\xi$. For $\ebar|\xi|\le\cbr/2$, $|\Zs_\xi|$ is at most an absolute constant times $\kappa_m/\eps_m^2$, by~\eqref{e.prep.symbol} and~\eqref{e.prep.drift}; we take $\Cbr\ge2$, at least this constant and the constant of the bound on the fixed point, which gives the bound of~(a). Cauchy's estimate on the ball of radius $\cbr/2$ bounds the term of degree~$j$ by $(2\ebar|\xi|/\cbr)^j$ times the supremum, which gives~(b); the terms of degree $0$ and~$1$ vanish, as $\Rb_\xi-1$ is of degree at least one in~\eqref{e.prep.symbol}. A mean-zero solution of~\eqref{e.prep.graph} in that unit ball is a fixed point of the map, by the uniqueness in Lemma~\ref{l.prep.inverse}; this is~(e), and~(c) and the mean in~(a) hold by construction. \emph{Step 2: Real frequencies.} By~\eqref{e.prep.contract}, and by~(a) with $\cbr\le\frac12$ and $\Cbr\ge2$, respectively, \begin{equation*} \|\Rb_\xi(t)\|_{L^2(\T^2)}\Bigl|\exp\Bigl(\int_{t'}^t\Zs_\xi(t'')\dd t''\Bigr)\Bigr|\le\|\Rb_\xi(t')\|_{L^2(\T^2)}\,,\qquad1\le\|\Rb_\xi\|_{L^2(\T^2)}\le1+\Cbr\ebar|\xi|\le\Cbr\,, \end{equation*} which gives~(d). \end{proof} \subsection{Dissipation of the branch: the averaged flux} \label{ss.prep.dissipation} The quadratic part of the symbol is given by the averaged flux of Part~I, up to an error small relative to $\kappa_{m-1}|\xi|^2$, from the conductivity and from switching the pulses on and off, because the first-order part of the branch is the corrector of Section~\ref{s.cell}. At each time the averaged flux enhances diffusion only along the flow of the active pulse; a block, however, carries pulses of both orientations, and over a block the averaged flux integrates to at least a constant times~$\rho(m)^2$ in every direction. So over one block the branch is damped by the Gaussian $\exp(-\cdis\rho(m)^2|\xi|^2)$, by~(b) below: this is the homogenized equation of level~$m-1$ seen over the time~$h_m$. Item~(c) also covers intervals that are not complete blocks: the symbol is at most~$\pi^2/16$ times the cell rate, so the branch changes more slowly than diffusion damps functions of mean zero on the cell, and frequency derivatives of its integral, at the scale~$1/\rho(m)$, cost only powers of $1+\rho(m)|\xi|$. Below, the \emph{core} of a pulse~$k$ is $[(k-\frac13)\tau_m,(k+\frac13)\tau_m]$; it is \emph{complete} if the slow cutoff of its block is~$1$ on it. \begin{proposition}[Dissipation of the branch]\label{p.prep.dissipation} There exist $\Lambda_0(\alpha,s,q,\zeta)\ge128$, $C(\alpha,s,q,\zeta)$ and absolute constants $\clo\in(0,\cbr/8]$ and $\cdis>0$ such that, under the standing hypotheses with this~$\Lambda_0$, the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item For every $\xi\in\mathbb C^2$, \begin{equation*} \bigl\|\Zs_{\xi,2}+\xi\cdot\J_m^{\kappa_m}\xi\bigr\|_{L^\infty(\R)}\le C\eps_{m-1}^\delta\kappa_{m-1}|\xi|^2\,. \end{equation*} \item For every $l'\in\Z$, every $\xi\in\R^2$ with $\ebar|\xi|\le2\clo$ and every integer $N\ge2$, on the renewal block $[t_-,t_+]\coloneqq[(l'-\frac12)h_m,(l'+\frac12)h_m]$, \begin{equation*} \operatorname{Re}\int_{t_-}^{t_+}\Zs_\xi(t)\dd t\le-\cdis\,\rho(m)^2|\xi|^2\,,\qquad\operatorname{Re}\int_{t_-}^{t_+}\sum_{j=2}^N\Zs_{\xi,j}(t)\dd t\le-\cdis\,\rho(m)^2|\xi|^2\,. \end{equation*} \item For every $\xi\in\mathbb C^2$ with $\ebar|\xi|\le2\clo$, \begin{equation*} \Cbr\ebar|\xi|\le\frac13\,,\qquad\|\Zs_\xi\|_{L^\infty(\R)}\le\frac{\pi^2\kappa_m}{16\eps_m^2}\,. \end{equation*} For every interval $[t',t'']$ of length at most~$2h_m$ and every $\xi\in\R^2$ with $\ebar|\xi|\le2\clo$, with~$\nabla_\xi^j$ the array of the derivatives of order~$j$ and~$C(j)$ depending only on~$j$, the following hold for every integer $j\ge1$: \begin{equation*} \Bigl|\rho(m)^{-j}\nabla_\xi^j\int_{t'}^{t''}\Zs_\xi(t)\dd t\Bigr|\le\begin{cases}C(1)\rho(m)|\xi|\,,&j=1\,,\\ C(j)\,,&j\ge2\,,\end{cases} \end{equation*} and \begin{equation*} \Bigl|\rho(m)^{-j}\nabla_\xi^j\exp\Bigl(\pm\int_{t'}^{t''}\Zs_\xi(t)\dd t\Bigr)\Bigr|\le C(j)(1+\rho(m)|\xi|)^j\Bigl|\exp\Bigl(\pm\int_{t'}^{t''}\Zs_\xi(t)\dd t\Bigr)\Bigr|\,. \end{equation*} \end{enumerate} \end{proposition} \begin{proof} \emph{Step 1: The quadratic part.} The solution operator of Lemma~\ref{l.prep.inverse} is linear and independent of~$\xi$. So the term~$\Rb_{\xi,1}$ of degree one in~$\xi$ of the branch is the bounded mean-zero solution of~\eqref{e.prep.graph} with $\Rb_{\xi,1}$ in place of $\Rb_\xi-1$ on the left side and with the part of degree one of the right side, $-\mathrm i\xi\cdot(\nabla^\perp\psi_m)(t,\eps_mz)$, on the right. By~\eqref{e.prep.symbol}, \begin{equation}\label{e.prep.D} \Zs_{\xi,2}=-\kappa_m\,\xi\cdot\K(t)\xi-\mathrm i\bigl(\xi\cdot(\nabla^\perp\psi_m)(t,\eps_m\cdot)\,\Rb_{\xi,1}\bigr)_{\T^2}\,. \end{equation} \emph{Step 2: Comparison with the averaged flux.} In this step~$C$ depends only on $\alpha,s,q,\zeta$, as in~(a). We use the glued corrector~$\chi_m^{\kappa_m}$ of Lemma~\ref{l.cell.flux}(b). For $e\in\R^2$, since $(\psi_m)_{\T^2}=0$, $(\nabla(e\cdot\chi_m^{\kappa_m}))_{\T^2}=0$ and $\nabla^\perp\psi_m=\bigl(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr)\nabla\psi_m$, an integration by parts in~\eqref{e.cell.flux} gives the exact identity \begin{equation}\label{e.prep.fluxid} \J_m^{\kappa_m}(t)e=\kappa_me-\fint_{\T^2}\nabla^\perp\psi_m(t,y)\,\bigl(e\cdot\chi_m^{\kappa_m}(t,y)\bigr)\dd y\,, \end{equation} so that $-\xi\cdot\J_m^{\kappa_m}\xi=-\kappa_m\,\xi\cdot\xi+\fint_{\T^2}(\xi\cdot\nabla^\perp\psi_m)(\xi\cdot\chi_m^{\kappa_m})\dd y$ for every $\xi\in\mathbb C^2$, by bilinearity. By~\eqref{e.cell.glued}, read in the fast variable, $\mathrm i\xi\cdot\chi_m^{\kappa_m}(t,\eps_mz)$ solves the equation of~$\Rb_{\xi,1}$ at $\K=I$ up to the switching residual. Hence $\Rb_{\xi,1}-\mathrm i\xi\cdot\chi_m^{\kappa_m}(t,\eps_mz)$ is the bounded mean-zero solution of Lemma~\ref{l.prep.inverse} with $F^0=-(\eps_m^2/\kappa_m)\,\mathrm i\xi\cdot\sum_k\Hc'_{m,k}\chi_{m,k}^{\kappa_m}(t,\eps_mz)$, with $\Hc'_{m,k}$ the derivative of~$\Hc_{m,k}$ in~$t$, and $F=(\K-I)\nabla_z\bigl(\mathrm i\xi\cdot\chi_m^{\kappa_m}(t,\eps_mz)\bigr)$. The gradient in~$z$ of $\chi_m^{\kappa_m}(t,\eps_mz)$ is at most $\eps_m^{\beta+1}/\kappa_m$, by~\eqref{e.cell.corrector}, \eqref{e.cell.shear} and $\sum_k\Hc_{m,k}=1$; with~\eqref{e.cell.glued} and Lemma~\ref{l.prep.inverse}, \begin{equation*} \bigl\|\Rb_{\xi,1}-\mathrm i\xi\cdot\chi_m^{\kappa_m}(\cdot,\eps_m\cdot)\bigr\|_{L^\infty(\R;L^2(\T^2))}\le C|\xi|\,\frac{\eps_m^{\beta+1}}{\kappa_m}\Bigl(\frac{\eps_m^2}{\kappa_m\tau_m}+\eps_{m-1}^\delta\Bigr). \end{equation*} The averages over~$\T^2$ in~$z$ are those over~$\T^2$ in~$y$, as the functions are $\eps_m$-periodic; so~\eqref{e.prep.D} and~\eqref{e.prep.fluxid} give \begin{align*} \Zs_{\xi,2}+\xi\cdot\J_m^{\kappa_m}\xi & =-\kappa_m\,\xi\cdot(\K(t)-I)\xi-\mathrm i\Bigl(\xi\cdot(\nabla^\perp\psi_m)(t,\eps_m\cdot)\,\bigl(\Rb_{\xi,1}-\mathrm i\xi\cdot\chi_m^{\kappa_m}(t,\eps_m\cdot)\bigr)\Bigr)_{\T^2}\,,\\ \bigl|\Zs_{\xi,2}+\xi\cdot\J_m^{\kappa_m}\xi\bigr| & \le\kappa_m\eps_{m-1}^\delta|\xi|^2+C\frac{\eps_m^{2\beta}}{\kappa_m}\Bigl(\frac{\eps_m^2}{\kappa_m\tau_m}+\eps_{m-1}^\delta\Bigr)|\xi|^2\le C\eps_{m-1}^\delta\kappa_{m-1}|\xi|^2\,, \end{align*} the last by $\kappa_m\le\kappa_{m-1}$, $\eps_m^{2\beta}/\kappa_m\le20\kappa_{m-1}$ and $\eps_m^2/(\kappa_m\tau_m)\le C\eps_{m-1}^{4\delta}$ (Proposition~\ref{p.scales}(c) and~\eqref{e.cell.amplitude}). This is~(a). \emph{Step 3: Coercivity over a block.} We show, with $\cdis\coloneqq(1-\exp(-2))/192$, \begin{equation}\label{e.prep.coercive} \int_{t_-}^{t_+}\xi\cdot\J_m^{\kappa_m}(t)\xi\dd t\ge2\cdis\,\kappa_{m-1}h_m|\xi|^2\qquad(\xi\in\R^2)\,. \end{equation} On a complete core $\widetilde\zeta_{m,k}=1$, since $\zeta(\cdot-k)$ is the only term of the partition~\eqref{e.zeta} that is nonzero there. So for $t\in[k\tau_m,(k+\frac13)\tau_m]$ the time integral of~\eqref{e.flux} is at least $\int_0^{\tau_m/3}\exp(-4\pi^2\kappa_m\tilde t/\eps_m^2)\dd\tilde t\ge(1-\exp(-2))\eps_m^2/(4\pi^2\kappa_m)$, by~\eqref{e.prep.relax}, and~\eqref{e.flux} gives \begin{equation*} e_{(k)}\cdot\J_m^{\kappa_m}(t)e_{(k)}\ge\kappa_m+\bigl(1-\exp(-2)\bigr)\frac{\eps_m^{2\beta}}{2\kappa_m}\,. \end{equation*} A block has $h_m/\tau_m=(4\lceil\eps_{m-1}^{-\delta}\rceil+1)^2$ slots, and only those within $2\tau'_m+\tau_m$ of its ends fail to have complete cores (Lemma~\ref{l.fl.scales}(d)); so the complete cores are those of $h_m/\tau_m-4\tau'_m/\tau_m$ consecutive integers, with $\tau'_m/\tau_m=4\lceil\eps_{m-1}^{-\delta}\rceil+1\ge65$ by~\eqref{e.prep.relax}, and it has at least $h_m/(32\tau_m)$ complete cores of each orientation. Since~$\J_m^{\kappa_m}$ is diagonal and at least~$\kappa_mI$ (Lemma~\ref{l.fl.flux}), for $e\in\{e_1,e_2\}$ \begin{equation*} \int_{t_-}^{t_+}e\cdot\J_m^{\kappa_m}e\dd t\ge\kappa_mh_m+\frac{h_m}{32\tau_m}\cdot\frac{\tau_m}3\cdot\bigl(1-\exp(-2)\bigr)\frac{\eps_m^{2\beta}}{2\kappa_m}\ge2\cdis h_m\Bigl(\kappa_m+\frac{\eps_m^{2\beta}}{2\kappa_m}\Bigr)\ge2\cdis h_m\kappa_{m-1}\,, \end{equation*} because $\kappa_{m-1}=\overline K_m(\kappa_m)\le\kappa_m+\eps_m^{2\beta}/(2\kappa_m)$ (Lemma~\ref{l.fl.flux}(i) and~\eqref{e.mean}). This is~\eqref{e.prep.coercive}. \emph{Step 4: The block integral.} For real~$\xi$ with $\ebar|\xi|\le\cbr/4$, by~(a), \eqref{e.prep.coercive}, Proposition~\ref{p.prep.branch}(b) and $\frac{\kappa_m}{\eps_m^2}(\ebar|\xi|)^2=\kappa_{m-1}|\xi|^2$, \begin{align*} \operatorname{Re}\int_{t_-}^{t_+}\Zs_\xi(t)\dd t&\le-\bigl(2\cdis-C\eps_{m-1}^\delta\bigr)\kappa_{m-1}h_m|\xi|^2+\Cbr h_m\frac{\kappa_m}{\eps_m^2}\sum_{j\ge3}\Bigl(\frac{2\ebar|\xi|}{\cbr}\Bigr)^j\\ &\le-\Bigl(2\cdis-C\eps_{m-1}^\delta-\frac{16\Cbr}{\cbr^3}\ebar|\xi|\Bigr)\rho(m)^2|\xi|^2\,. \end{align*} We choose $\clo\coloneqq\min\{\cbr/8,\allowbreak\cbr^3\cdis/(64\Cbr),\allowbreak\pi\cbr/(32\Cbr^{\nf12}),\allowbreak1/(6\Cbr)\}$, and~$\Lambda_0$ so large that $C\eps_{m-1}^\delta\le\cdis/2$. Then the right side, and that of every truncation, is at most $-\cdis\rho(m)^2|\xi|^2$ for $\ebar|\xi|\le2\clo$. \emph{Step 5: Frequency derivatives.} By Proposition~\ref{p.prep.branch}(b), summing a geometric series of ratio at most~$\frac34$, $\|\Zs_{\xi'}\|_{L^\infty(\R)}\le16\Cbr\cbr^{-2}\frac{\kappa_m}{\eps_m^2}(\ebar|\xi'|)^2$ for complex~$\xi'$ with $\ebar|\xi'|\le\frac38\cbr$; as $2\clo\le\frac14\cbr$, the last two entries of the minimum defining~$\clo$ give the first sentence of~(c). There the integral over~$[t',t'']$ is holomorphic and at most $32\Cbr\cbr^{-2}\rho(m)^2|\xi'|^2$, as $t''-t'\le2h_m$ and $\ebar^2\kappa_mh_m/\eps_m^2=\rho(m)^2$. Cauchy's estimates on the polydiscs about a real~$\xi$ of radii $\cbr/(16\ebar)$ and $|\xi|/4$ give the bounds of~(c) on this integral for $j\ge2$ (as $\rho(m)^2/\ebar^2=\kappa_mh_m/\eps_m^2\ge1$) and $j=1$ (at $\xi=0$ the gradient vanishes, as $\Zs_\xi$ has no term of degree one). Fa\`a di Bruno's formula gives the last bound. \end{proof} \subsection{The other frequencies: two orientations} \label{ss.prep.high} At the frequencies with $\ebar|\xi|\ge\clo$ the whole fibre must decay at the cell rate. If the contrast is bounded, as only at the terminal level, the inequality~\eqref{e.prep.zonegap} suffices. If it is large, then at the frequencies with $\eps_m|\xi|\ll1$ molecular diffusion alone damps the cell mean, the plane wave of frequency~$\xi$, far more slowly than at the cell rate. The damping comes instead from the shear: a pulse flowing along~$e_2$ moves the cell mean into the first cell harmonic, at a rate at least comparable to the cell rate once $\ebar|\xi_2|\ge\clo/2$, and diffusion damps the harmonic. Such a pulse does not see the frequencies along~$e_1$, as it moves functions of~$y_1$ along their level lines; so both orientations are needed, and a block has many pulses of each. \begin{proposition}[Damping of the other frequencies]\label{p.prep.high} There exist absolute constants $\chigh,\Chigh>0$ such that the following holds under the standing hypotheses, with~$m$ above a level depending only on $\alpha,s,q,\zeta,\Lambda$. For every $l'\in\Z$, every $\xi\in\R^2$ in the zone with $\ebar|\xi|\ge\clo$ and every solution~$U$ of the fibre equation~\eqref{e.prep.fibre} on the renewal block $[t_-,t_+]\coloneqq[(l'-\frac12)h_m,(l'+\frac12)h_m]$, \begin{equation*} \|U(t_+)\|_{L^2(\T^2)}\le\exp\Bigl(-\chigh\frac{\kappa_mh_m}{\eps_m^2}\Bigr)\|U(t_-)\|_{L^2(\T^2)}\,. \end{equation*} If moreover $\kappa_m\le\frac12\kappa_{m-1}$, then, for every interval $[t',t'']\subset[t_-,t_+]$ of length at least $4\Chigh\eps_m^2/\kappa_m$ that lies in a complete core of a pulse~$k$ with $\ebar|\xi\cdot e_{(k)}|\ge\clo/2$, \begin{equation*} \|U(t'')\|_{L^2(\T^2)}\le\tfrac12\|U(t')\|_{L^2(\T^2)}\,. \end{equation*} \end{proposition} \begin{proof} By~\eqref{e.prep.contract} the norm never grows. We distinguish two cases. \emph{Bounded contrast, $\kappa_m\ge\frac12\kappa_{m-1}$.} Then $\eps_m|\xi|\ge2^{-\nf12}\ebar|\xi|\ge\clo/2$, and~\eqref{e.prep.contract}, \eqref{e.prep.zonegap} and $\K\ge\frac12I$ give $\frac{\mathrm d}{\mathrm dt}\|U\|_{L^2(\T^2)}^2\le-\frac{\kappa_m}{\eps_m^2}\cdot\frac{\clo^2}4\|U\|_{L^2(\T^2)}^2$, whatever the velocity. For $m$ large, this case is $m=M$ (Proposition~\ref{p.scales}). \emph{Large contrast, $\kappa_m\le\frac12\kappa_{m-1}$.} Then $\kappa_{m-1}=\overline K_m(\kappa_m)\le\kappa_m+\eps_m^{2\beta}/(2\kappa_m)$ (Lemma~\ref{l.fl.flux}(i) and~\eqref{e.mean}) forces $\eps_m^{2\beta}/\kappa_m\ge\kappa_{m-1}$. One of the coordinates of~$\xi$, say~$\xi_2$, has $\ebar|\xi_2|\ge\clo/2$. Let~$k\in4\Z+1$ be a pulse, whose flow direction is~$e_2$, with a complete core. On the core only pulse~$k$ is active and $\widetilde\zeta_{m,k}=1$, so by~\eqref{e.cell.shear} the velocity of~\eqref{e.prep.fibre} is the stationary shear $2\pi\eps_m^{\beta-1}\cos(2\pi z_1)e_2$. The shear couples the cell mean $(U)_{\T^2}$ to the first harmonic $(\sqrt2\cos(2\pi z_1)U)_{\T^2}$: testing~\eqref{e.prep.fibre} with~$1$ and with $\sqrt2\cos(2\pi z_1)$, and integrating by parts in~$z_2$ and, where derivatives fall on the test function, in~$z_1$, we obtain \begin{equation}\label{e.prep.meanharm} \begin{aligned} \frac{\mathrm d}{\mathrm dt}(U)_{\T^2}&=-\kappa_m\,\xi\cdot\K(t)\xi\,(U)_{\T^2}-\mathrm i\sqrt2\pi\eps_m^{\beta-1}\xi_2\bigl(\sqrt2\cos(2\pi z_1)U\bigr)_{\T^2}\,,\\ \frac{\mathrm d}{\mathrm dt}\bigl(\sqrt2\cos(2\pi z_1)U\bigr)_{\T^2}&=-\mathrm i\sqrt2\pi\eps_m^{\beta-1}\xi_2\,(U)_{\T^2}+\text{a remainder}\,, \end{aligned} \end{equation} with the remainder at most $C(\eps_m^{\beta-1}|\xi_2|+\kappa_m/\eps_m^2)\|U-(U)_{\T^2}\|_{L^2(\T^2)}$ for an absolute~$C$, as~$\K$ does not depend on~$z$ and $\eps_m|\xi|\le\sqrt2\pi$; here $|(\sqrt2\cos(2\pi z_1)U)_{\T^2}|\le\|U-(U)_{\T^2}\|_{L^2(\T^2)}$, since the harmonic has mean zero. Hence the imaginary part of $\overline{(U)_{\T^2}}\,(\sqrt2\cos(2\pi z_1)U)_{\T^2}$ has the derivative $\sqrt2\pi\eps_m^{\beta-1}\xi_2\allowbreak(|(\sqrt2\cos(2\pi z_1)U)_{\T^2}|^2-\allowbreak|(U)_{\T^2}|^2)$ up to $C(\eps_m^{\beta-1}|\xi_2|+\kappa_m/\eps_m^2)|(U)_{\T^2}|\,\|U-(U)_{\T^2}\|_{L^2(\T^2)}$, the damping of the first line included, as $\eps_m|\xi|\le\sqrt2\pi$. We integrate over an interval $[t',t'']$ of the core; at its ends the imaginary part is at most $\frac12\|U(t')\|_{L^2(\T^2)}^2$, as the norm does not increase, and Young's inequality on the error gives \begin{equation*} \eps_m^{\beta-1}|\xi_2|\int_{t'}^{t''}|(U)_{\T^2}|^2\dd t\le C\|U(t')\|_{L^2(\T^2)}^2+C\eps_m^{\beta-1}|\xi_2|\Bigl(1+\frac{\kappa_m}{\eps_m^{\beta+1}|\xi_2|}\Bigr)^2\int_{t'}^{t''}\|U-(U)_{\T^2}\|_{L^2(\T^2)}^2\dd t\,. \end{equation*} By~\eqref{e.prep.contract} and~\eqref{e.prep.zonegap}, the mean-free part satisfies $\pi^2(\kappa_m/\eps_m^2)\int_{t'}^{t''}\|U-(U)_{\T^2}\|_{L^2(\T^2)}^2\dd t\le\|U(t')\|_{L^2(\T^2)}^2$. Now $\eps_m^{\beta+1}|\xi_2|/\kappa_m\ge\ebar|\xi_2|\ge\clo/2$, by $\eps_m^{2\beta}/\kappa_m\ge\kappa_{m-1}$, so that \begin{equation*} (\eps_m^{\beta-1}|\xi_2|)^{-1}=(\kappa_m/(\eps_m^{\beta+1}|\xi_2|))\,\eps_m^2/\kappa_m\le(2/\clo)\,\eps_m^2/\kappa_m\,. \end{equation*} Dividing the first bound by $\eps_m^{\beta-1}|\xi_2|$ and adding the two bounds, we obtain $\int_{t'}^{t''}\|U\|_{L^2(\T^2)}^2\dd t\le\Chigh(\eps_m^2/\kappa_m)\|U(t')\|_{L^2(\T^2)}^2$ with an absolute~$\Chigh$, as~$\clo$ is absolute; and since the norm does not increase, $\|U(t'')\|_{L^2(\T^2)}\le\frac12\|U(t')\|_{L^2(\T^2)}$ as soon as $t''-t'\ge4\Chigh\eps_m^2/\kappa_m$. This is the last assertion of the proposition for $k\in4\Z+1$; for $k\in4\Z+3$ we exchange the coordinates. A core has length $\frac23\tau_m$, and $\kappa_m\tau_m/\eps_m^2\to\infty$ with~$m$ (Proposition~\ref{p.scales}(c)); for $m$ large, each complete core of the right orientation halves the norm at least $\kappa_m\tau_m/(12\Chigh\eps_m^2)$ times. The block has at least $h_m/(32\tau_m)$ such cores (Step~3 of the proof of Proposition~\ref{p.prep.dissipation}), so the norm decreases by the factor $\exp(-\chigh\kappa_mh_m/\eps_m^2)$, with $\chigh\coloneqq\min\{\log2/(384\Chigh),\clo^2/8\}$. \end{proof} \subsection{The Duhamel expansion and the adjoint equation} \label{ss.prep.tests} \newcommand{\cfac}{{\bigl(1+\ebar/\eps_m\bigr)}}\newcommand{\Nneg}{N_{\mathrm{neg}}}\newcommand{\Nact}{N_{\mathrm{act}}}\newcommand{\Pblk}{\Pi_{\mathrm{blk}}}\newcommand{\Pfut}{\Pi_+}\newcommand{\Pren}{\Pi_{\mathrm{ren}}}\newcommand{\Uexp}[1]{U^{\{#1\}}}% In a family, sources couple the components, and a source of order~$d$ takes~$d$ derivatives. The preparation expands the solution in its sources by Duhamel's formula, and as the sources are small, finitely many terms suffice (Subsection~\ref{ss.prep.proof}). Proposition~\ref{p.prep.joint}, the main result of Subsections~\ref{ss.prep.tests}--\ref{ss.prep.joint}, prepares their sum. A direct estimate of each term would lose derivatives of the datum and, through the transport, powers of the contrast; the map~$\Pt$ of that proposition loses neither, and only its error, of size $\eps_{m-1}^{Q+3}$, uses derivatives of the datum, at the molecular scale. The \emph{Duhamel expansion} of a family~$G$ of level~$m$ from a datum~$g$ at a time~$t_1$ is the sequence $\Uexp0,\Uexp1,\dots$ of solutions of \begin{equation*} \bigl(\partial_t+\b_m\cdot\nabla-\kappa_m\Delta\bigr)\Uexp j_v=\begin{cases}0\,,&j=0\,,\\ \nabla\cdot\sum_{v'}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}A_{\sigma,d}\nabla^{d-1}\Uexp{j-1}_{v'}\,,&j\ge1\,,\end{cases} \end{equation*} with $\Uexp0(t_1)=g$ and $\Uexp j(t_1)=0$ for $j\ge1$: the system~\eqref{e.family.system} with its sources applied one at a time and \emph{without the smoothing operators}~$S_\sigma$. The terms exist in $C([t_1,\tf];H^{\Nbuf-j(\Nsrc-2)}(\T^2)^V)$ for $j\le\Ndep+1$ when $g\in H^{\Nbuf}(\T^2)^V$, by Proposition~\ref{p.ft.plain}, whose condition at the scale $\rhomol(m)$ is~(b2$'$) of Definition~\ref{d.admissible.invariant} with majorant sums at most $C_*^2\eps_m^{\delta/4}$. \begin{proposition}[Preparation of the Duhamel expansion]\label{p.prep.joint} Let $C_*\ge1$. There exist~$C$ and a level, both depending only on~$\deps$ and~$C_*$, such that the following holds for every $M$, $\kappa\in\Kset_M$ and every $m\le M$ above this level, the terminal level included. Let~$G$ be a family of level~$m$ that satisfies the invariant (Definition~\ref{d.admissible.invariant}) on $[\tA,\tf]$, with~$C_*$ in place of the constants $C_\Omega$, $c_j$ ($j\le\Nsrc-2$), $\Cwid$, $\Cder$, $\Cjet(j)$ ($j\le\Ncoef$) and~$\Cadm$ of items~(3)--(4) of Subsection~\ref{ss.admissible.parameters}, and whose majorant matrix has nonzero entries that, as arrows between components, form no path of more than~$\Npath$ arrows, let $t_1\in[\tA,\tf-2h_m]$, and let~$\tpr$ be the end of the block that follows the block containing~$t_1$: with $(l-\frac12)h_m\le t_1<(l+\frac12)h_m$ and $t_2\coloneqq(l+\frac12)h_m$, \begin{equation}\label{e.prep.tpr} \tpr\coloneqq t_2+h_m\,,\qquad h_m<\tpr-t_1\le2h_m\,. \end{equation} Then the velocity added at level~$m$ vanishes near~$\tpr$, \begin{equation*} \b_m=\b_{m-1}\quad\text{on }[\tpr-\tau'_m,\tpr+\tau'_m]\times\T^2\,, \end{equation*} and there exists a linear map $\Pt\colon L^2(\T^2)^V\to H^{\Nout}(\T^2)^V$ that maps fields with constant components to themselves, such that, for both choices~$\omega_v$ and~$\omega_v/\Psi_v^2$ of the masses and for every $g\in H^{\Nbuf}(\T^2)^V$, with~$\Uexp j$ the Duhamel expansion of~$G$ from~$g$ at~$t_1$, \begin{align*} \|\Pt g\|_{H^{\Nout}_{\rho(m),\omega}(\T^2)} & \le C\|g\|_{L^2_\omega(\T^2)}\,,\\ \Bigl\|\sum_{j=0}^{\Ndep}\Uexp j(\tpr)-\Pt g\Bigr\|_{L^2_\omega(\T^2)} & \le C\eps_{m-1}^{Q+3}\|g\|_{H^{\Nbuf}_{\rhomol(m),\omega}(\T^2)}\,. \end{align*} \end{proposition} With the masses~$\omega_v/\Psi_v^2$, these norms are those of~$L^2_G$ and~$H^N_{\rho,G}$ in~\eqref{e.norms}. \emph{The plan.} The proof (end of Subsection~\ref{ss.prep.joint}) cuts the datum into \emph{pieces} within $\rho(m)/4$ of their centres~$y_0$ and freezes the conductivity and the forms of the sources at~$y_0$ as Taylor polynomials, which become \emph{insertions}, operators inserted between free evolutions. Each term of the expansion becomes a finite sum of \emph{Duhamel terms} of the frozen operator~\eqref{e.prep.frozen}, across the old partial block, the change of chart at~$t_2$, and one complete block (Figure~\ref{f.prep.timeline}); Lemma~\ref{l.prep.seam} gives the model of each and its error. \emph{Sources moved onto tests.} In the bounds that determine the model, the derivatives that the sources take are never taken of a term of the expansion: we pair the term with a \emph{test}, a solution of the adjoint fibre equation run backward from the final time, and move each source onto the test, which carries all of them. Against the principal branch~$\Rb_\xi$, a source in divergence form acts at most at a constant times the sum of the rate $\kappa_{m-1}|\xi|^2$ of the homogenized equation and one over the length~$h_m$ of a block (Lemma~\ref{l.prep.insert}), and over a block it costs at most a power of $1+\rho(m)|\xi|$, which the Gaussian of Proposition~\ref{p.prep.dissipation}(b) absorbs. These bounds lose no power of the contrast. Everything else decays over a block by a factor smaller than any power of~$\eps_{m-1}$; it is estimated with losses polynomial in the contrast factor and in the number of cell times $\eps_m^2/\kappa_m$ in a block, and it goes into the error. \paragraph{Conventions for Subsections~\ref{ss.prep.tests}--\ref{ss.prep.joint}.} The standing hypotheses hold with $m\le M$, the terminal level included, and~$m$ above a level that is at least that of Proposition~\ref{p.prep.high} and that each lemma may enlarge; $l\in\Z$ is fixed, and \emph{the block} is $[(l-\frac12)h_m,(l+\frac12)h_m]$. Unless a statement says otherwise, $t'0$, the integers $N_0$, $N_2$, $N_3$ and the level of a statement depend only on~$\deps$, on~$\Npath$ and on the integers that the statement fixes: the lengths of its Duhamel terms, the orders and Taylor degrees of their insertions, the moment order~$N_1$, and its Sobolev orders and weight exponents. Proofs measure time in cell units $\kappa_mt/\eps_m^2$. Quiet times are those of Section~\ref{s.admissible}, at which no pulse of level~$m$ is active; no pulse is active within $\tau'_m\ge5\tau_m$ of the ends of the block, its quiet margins. Windows of pulses are those of~\eqref{e.cell.window}, and complete cores those of Subsection~\ref{ss.prep.dissipation}. We call $\cfac=1+(\kappa_{m-1}/\kappa_m)^{\nf12}$ the \emph{contrast factor}, and we write $\|f\|_{H^J(\T^2)}\coloneqq\|f\|_{H^J_1(\T^2)}$ for the norm~\eqref{e.HNrho} at $\rho=1$. \paragraph{Insertions.} After freezing, the sources become insertions; by~\eqref{e.prep.insertion}, in cell units an insertion of order~$d$ is the cell rate times an operator of order~$d$ in~$z$, with coefficients bounded in~$z$ and polynomial in $y-y_0$. The frozen operator~\eqref{e.prep.frozen} acts componentwise on fields of a finite set~$V$ of components with masses~$\omega_v$; we write $\langle f,f'\rangle_{L^2(\T^2)}\coloneqq(f\,\overline{f'})_{\T^2}$ and $\langle U,U'\rangle_{L^2_\omega(\T^2)}\coloneqq\sum_v\omega_v^{-1}\langle U_v,U'_v\rangle_{L^2(\T^2)}$. An \emph{insertion} is an operator $\mathcal IU\coloneqq\nabla\cdot\sum_{d=2}^{\Nsrc}A_d\nabla^{d-1}U$ between such fields with coefficients \begin{equation}\label{e.prep.insertion} A_d(t,y)=\kappa_m\eps_m^{d-2}\,\mathcal A_d\Bigl(t,y,\frac y{\eps_m}\Bigr)\,, \end{equation} where $\mathcal A_d(t,y,z)=\sum_a((y-y_0)/\rho(m))^a\mathcal A_{d,a}(t,z)$, the sum over the multi-indices $a\in\N^2$ with~$|a|$ at most the \emph{Taylor degree}, has coefficients~$\mathcal A_{d,a}$ that are two-scale forms (Section~\ref{s.cell}) independent of~$y$, measurable in~$t$. On the block they satisfy: \begin{enumerate}[label=\textup{(\roman*)}] \item the structure clauses (a2)--(a4) of Definition~\ref{d.admissible.invariant}: transverse (independent of $z\cdot e_{(k)}$) on the window of each pulse~$k$, invariant along~$e_{(k)}$ where $|t-k\tau_m|\le\frac34\tau_m$, and independent of~$z$ at times with no pulse active within~$\tau_m/24$; \item the derivative bound: with $M_{vv'}\coloneqq\sum_d\sum_a\sum_{j'}\|\nabla_z^{j'}(\mathcal A_{d,a})_{vv'}\|_{L^\infty}$, the sum over~$j'$ up to the derivative counts below, the matrix of entries $(\omega_{v'}/\omega_v)^{\nf12}M_{vv'}$ has row and column sums at most the \emph{strength}; \item either this matrix is diagonal and $\mathcal A_d$ has no term of degree~$0$ in $y-y_0$, or its nonzero entries, as arrows between components, form no path of more than~$\Npath$ arrows. The sources of Section~\ref{s.admissible} satisfy the second alternative, and the conductivity insertions the first. \end{enumerate} The \emph{order} is the largest~$d$ with $A_d\ne0$. We sum the forms of a source over the pulses, with their time cutoffs, each replaced by its Taylor polynomial at~$y_0$ and written in the chart of the block by clause~(a6) of Definition~\ref{d.admissible.invariant}. On the block, extended by zero outside the interval of the Duhamel term, this sum is an insertion of strength~$C$ times its majorant sums. The Taylor polynomial at~$y_0$ of the conductivity minus its value gives diagonal insertions of order~$2$ and strength $C\eps_{m-1}^{2\delta}\rho(m)/\eps_{m-1}$ on fields supported near~$y_0$, each power of $(y-y_0)/\eps_{m-1}$ costing one moment at the scale~$\rho(m)$. \paragraph{Duhamel terms.} A \emph{Duhamel term} of length~$N$ on an interval $[t_1,t_2]$ of the block from a datum~$g$ is the sequence $\Ud0,\dots,\Ud N$ of energy solutions of~\eqref{e.prep.frozen} on $[t_1,t_2]$ with $\Ud0(t_1)=g$ and, for $1\le j\le N$, the forcing $\mathcal I_j\Ud{j-1}$ and $\Ud j(t_1)=0$, where~$\mathcal I_1,\dots,\mathcal I_N$ are insertions of orders $d_1,\dots,d_N$. It is \emph{centred at}~$y_0$ when the conductivity of~\eqref{e.prep.frozen} is frozen at~$y_0$ and the insertions are polynomials in $y-y_0$. For an integer $K\ge0$, let~$\mathcal D_K$ be the class of fields~$g$ on~$\R^2$ with $\nabla^bg\in L^2_{\omega,K}(\R^2)^V$ for every $b\le d_1+\dots+d_N-2N$, with the moments of~\eqref{e.prep.moments}. For data in~$\mathcal D_K$, with~$K$ the sum of the Taylor degrees of the insertions, the Duhamel term exists and is unique, by the energy of Proposition~\ref{p.ft.plain} with the moment weights (the estimate of Lemma~\ref{l.ft.commutator} is pointwise). Along the sequence the orders decrease by~$d_j-2$ and the weights by the Taylor degree of~$\mathcal I_j$ at step~$j$. \paragraph{Tests.} For real~$\xi$, a \emph{test} on $[t',t'']$ is an energy solution~$U^\dagger$ of the \emph{adjoint fibre equation} \begin{equation}\label{e.prep.adjoint} -\partial_tU^\dagger=\frac{\kappa_m}{\eps_m^2}(\nabla+\mathrm i\eps_m\xi)\cdot\K(t)(\nabla+\mathrm i\eps_m\xi)U^\dagger+\frac1{\eps_m}\nabla^\perp\psi_m(t,\eps_mz)\cdot(\nabla+\mathrm i\eps_m\xi)U^\dagger \end{equation} on $[t',t'']\times\T^2$ with a datum at~$t''$ (componentwise for fields): the fibre equation~\eqref{e.prep.fibre} run backward in time with the velocity reversed. The transport term is skew-adjoint and the diffusion self-adjoint, so for every solution~$U$ of~\eqref{e.prep.fibre} with a forcing~$F$, \begin{equation}\label{e.prep.duality} \frac{\mathrm d}{\mathrm dt}\bigl\langle U(t),U^\dagger(t)\bigr\rangle_{L^2_\omega(\T^2)}=\bigl\langle F(t),U^\dagger(t)\bigr\rangle_{L^2_\omega(\T^2)}\,. \end{equation} As $\Rb_\xi(t)\exp(\int_{t'}^t\Zs_\xi(\tilde t)\dd\tilde t)$ solves~\eqref{e.prep.fibre} (Proposition~\ref{p.prep.branch}(c)), $\exp(\int_{t'}^t\Zs_\xi(\tilde t)\dd\tilde t)\langle\Rb_\xi(t),U^\dagger_v(t)\rangle_{L^2(\T^2)}$ is constant. A test is \emph{off-branch} at~$t$ if $\langle\Rb_\xi(t),U^\dagger_v(t)\rangle_{L^2(\T^2)}=0$ for every~$v$; it is then off-branch at every earlier time, and it reads only the part of a solution off the branch. For a time~$t''$ of the block with no pulse of level~$m$ active within~$\tau_m/24$ of it, the \emph{branch test}~$U^\dagger_{\mathrm{br}}$ is the test with $U^\dagger_{\mathrm{br}}(t'')=1$; paired with a Bloch sum at~$t''$, it reads the cell mean, that is, by~\eqref{e.prep.momentfourier} below, the Fourier transform. \paragraph{Moment terms.} The coefficients of insertions are polynomials in $y-y_0$, which do not act fibre by fibre; so we carry the moments of the Duhamel terms as further unknowns. A power of $y-y_0$ then maps moment terms to higher ones, every coefficient becomes periodic in the fast variable, and Bloch sums decouple the fibres. The moments are also the frequency derivatives of the Fourier transform~\eqref{e.prep.momentfourier}, which localize the model near~$y_0$. Let $\Ud0,\dots,\Ud N$ be a Duhamel term on $[t_1,t_2]$, let $N_1\ge0$, let~$N'$ be~$N_1$ plus the sum of the Taylor degrees of its insertions, and let its datum lie in~$\mathcal D_{N'+2}$. For a multi-index $i\in\N^2$ with $|i|$ at most~$N_1$ plus the sum of the Taylor degrees of $\mathcal I_{j+1},\dots,\mathcal I_N$, the \emph{moment term} $(j,i)$ is $U^{[j,i]}\coloneqq((y-y_0)/\rho(m))^{i}\Ud j$. Its \emph{degree} is~$|i|$ plus the sum of the Taylor degrees of $\mathcal I_1,\dots,\mathcal I_j$, at most~$N'$; a moment term is \emph{lower} than $(j,i)$ if its degree is smaller, or equal with a smaller~$j$, and the moment terms with given~$j$ and~$|i|$ form a \emph{group}. Commuting the weight with the frozen operator, we obtain \begin{align}\label{e.prep.moment} \lefteqn{\bigl(\partial_t+\nabla^\perp\psi_m\cdot\nabla-\kappa_m\nabla\cdot(\K(t)\nabla\,\cdot)\bigr)U^{[j,i]}} \qquad & \notag \\ & =\Bigl(\frac{y-y_0}{\rho(m)}\Bigr)^{i}\mathcal I_j\Ud{j-1} +\sum_{e}\frac{i\cdot e}{\rho(m)}\bigl(\nabla^\perp\psi_m\cdot e-2\kappa_m\K(t)e\cdot\nabla\bigr)U^{[j,i-e]} \notag \\ & \qquad +\kappa_m\sum_{e,e'}\frac{(i\cdot e)(i\cdot e'-e\cdot e')}{\rho(m)^2}\,e\cdot\K(t)e'\,U^{[j,i-e-e']} \end{align} (the sums over $e,e'\in\{e_1,e_2\}$; without the first term for $j=0$). Moving the weight through~$\mathcal I_j$, whose coefficients are polynomials in $(y-y_0)/\eps_{m-1}=(\rho(m)/\eps_{m-1})(y-y_0)/\rho(m)$, makes the first term a sum of operators of order at most~$d_j$ on the moment terms $(j-1,i')$, $|i'|$ at most~$|i|$ plus the Taylor degree of~$\mathcal I_j$, with coefficients bounded as those of~$\mathcal I_j$ and $\rho(m)^{-1}$ per derivative on the weight. All moment terms on the right are lower than $(j,i)$. By the rules for Bloch sums after~\eqref{e.prep.bloch}, $\Bl U^{[j,i]}(t)$, short for $\Bl(U^{[j,i]}(t))(\xi,\cdot)$, solves~\eqref{e.prep.fibre} forced by the sum over the lower $(j',i')$ of $\mathcal I_{(j,i),(j',i')}(t)\Bl U^{[j',i']}(t)$, the fibre forms of the terms of~\eqref{e.prep.moment}; $\mathcal I^*_{(j,i),(j',i')}(t)$ is the adjoint for $\langle\cdot,\cdot\rangle_{L^2_\omega(\T^2)}$. As the mean of a Bloch sum is $\eps_m^{-2}$ times the Fourier transform (Subsection~\ref{ss.prep.bloch}), with $\nabla_\xi^i\coloneqq\partial_{\xi_1}^{i_1}\partial_{\xi_2}^{i_2}$, \begin{equation}\label{e.prep.momentfourier} \rho(m)^{-|i|}\nabla_\xi^i\widehat{\Ud j(t)}(\xi)=(-\mathrm i)^{|i|}\widehat{U^{[j,i]}(t)}(\xi)=(-\mathrm i)^{|i|}\eps_m^2\bigl(\Bl U^{[j,i]}(t)\bigr)_{\T^2}\,. \end{equation} The moment terms have two further moments: at step~$j$ the weight available, $N'+2$ minus the Taylor degrees of $\mathcal I_1,\dots,\mathcal I_j$, exceeds the largest~$|i|$ by two. So, by the moment-weighted energy of the Duhamel term, each~$U^{[j,i]}$, its gradient (in~$L^2$ in time) and the flux of its forcing lie in~$L^2_{\omega,2}(\R^2)$, their Bloch sums converge (paragraph of~\eqref{e.prep.bloch}), and Lemma~\ref{l.prep.bloch} applies. For an array of tests $U^\dagger=(U^\dagger_{(j,i)})$ indexed by a group, and for a lower group, $\mathcal I^*U^\dagger$ is the array indexed by the lower group with entries $\sum_{(j,i)}\mathcal I^*_{(j,i),(j',i')}U^\dagger_{(j,i)}$. Arrays are normed by the square sums of the norms of their entries; for a function~$f$ on~$\T^2$ and a field or array~$U$, $|\langle U,f\rangle|=|\langle f,U\rangle|$ is the square root of the sum over the entries and components of $\omega_v^{-1}|\langle U_v,f\rangle_{L^2(\T^2)}|^2$. \paragraph{Derivative counts.} The proofs below differentiate the coefficients of the insertions of a Duhamel term at most $2(N_1+N'+d_1+\dots+d_N+2)+\max\{d_1,\dots,d_N\}+2$ times in the fast variable, and, for a Duhamel term across the renewal (Lemma~\ref{l.prep.seam}), at most $2(3N_1+d_1+\dots+d_{N_-+N_+}+4)+\max\{d_1,\dots,d_{N_-+N_+}\}+2$ plus twice the sum of the Taylor degrees; $\Ncoef$ is at least these counts. \subsection{Estimates for the adjoint equation} \label{ss.prep.estimates} Tests carry the derivatives that the sources take, and in cell units the velocity of the fibre equation has the size of the P\'eclet number, at most a constant times the contrast factor, by~\eqref{e.prep.drift} with $|\xi|=1/\eps_m$; so a derivative of a test may cost a power of the contrast factor. Lemma~\ref{l.prep.fibre} gives the bounds that accept this loss, for the parts that decay at the cell rate. The bounds without loss, Lemma~\ref{l.prep.tests}, rest on a feature of shears: on the window of a pulse, a function independent of the coordinate along its flow is not transported, and on such functions the adjoint fibre equation is a heat equation with a potential of the size of~$\ebar|\xi|$, whatever the contrast. Lemma~\ref{l.prep.insert} says how insertions act on tests. \paragraph{Halving intervals.} For real~$\xi$ in the zone with $\ebar|\xi|\ge\clo$, a \emph{halving interval} is an interval of the block of length $\max\{4\Chigh,16\clo^{-2}\}\eps_m^2/\kappa_m$, contained, if $\kappa_m\le\frac12\kappa_{m-1}$, in a complete core of a pulse~$k$ with $\ebar|\xi\cdot e_{(k)}|\ge\clo/2$. Solutions of~\eqref{e.prep.fibre} lose at least half of their $L^2(\T^2)$ norm across each (Proposition~\ref{p.prep.high}, and for $\kappa_m\ge\frac12\kappa_{m-1}$ the first case of its proof), and so do tests, as by~\eqref{e.prep.duality} their map from the right end to the left end is the adjoint of the solution map. The block contains $c\kappa_mh_m/\eps_m^2$ disjoint ones (the count at the end of that proof). If $\#[t',t'']$ is the largest number of disjoint halving intervals in $[t',t'']$ (and~$0$ if $\ebar|\xi|<\clo$), then $\#[t',t]+\#[t,t'']\ge\#[t',t'']-1$, and shortening $[t',t'']$ by a cell time $\eps_m^2/\kappa_m$ at either end lowers it by at most~$2$. In Lemma~\ref{l.prep.fibre}, (a) and~(b) say that the fibre evolution propagates and creates derivatives at the cell scale ((b) over one cell time, from a datum of negative order and with a forcing), at the cost of a power of the contrast factor, which the decay of Proposition~\ref{p.prep.high} over a block absorbs; by~(c), without pulses each cell mode evolves separately, and the part of mean zero decays with constant one. The negative norm of~(b) is $\|f\|_{H^{-J'}(\T^2)}\coloneqq\sup\{|\langle f,\vartheta\rangle_{L^2(\T^2)}|:\|\vartheta\|_{H^{J'}(\T^2)}\le1\}$. \begin{lemma}[The fibre evolution smooths]\label{l.prep.fibre} For every choice of integers $J,J'\ge0$ there exists~$C$ such that, for every real~$\xi$ in the zone, every solution~$U$ of~\eqref{e.prep.fibre} on $[t',t'']$ satisfies the following. Tests satisfy the same bounds with~$U^\dagger(t')$ on the left side and~$U^\dagger(t'')$ on the right side. \begin{enumerate}[label=\textup{(\alph*)}] \item \emph{Long times.} \begin{align*} \|U(t'')\|_{H^J(\T^2)} & \le C\cfac^{C}2^{-\#[t',t'']} \notag \\ & \qquad \times\min\bigl\{\|U(t')\|_{H^J(\T^2)},(\min\{1,\kappa_m(t''-t')/\eps_m^2\})^{-J/2}\|U(t')\|_{L^2(\T^2)}\bigr\}\,. \end{align*} \item \emph{One cell time.} Let $\kappa_m(t''-t')\le\eps_m^2$, and let~$U$ have a forcing~$F^0$, possibly zero. Then \begin{align*} \|U(t'')\|_{H^J(\T^2)} & \le C\cfac^C\Bigl(\min\Bigl\{\|U(t')\|_{H^J(\T^2)},\Bigl(\frac{\kappa_m(t''-t')}{\eps_m^2}\Bigr)^{-\frac{J+J'}2}\|U(t')\|_{H^{-J'}(\T^2)}\Bigr\} \notag \\ & \qquad +\int_{t'}^{t''}\|F^0(t)\|_{H^J(\T^2)}\dd t\Bigr)\,. \end{align*} \item \emph{Quiet stretches.} If no pulse of level~$m$ is active on $[t',t'']$, then, with constant~$1$, \begin{gather*} (U(t''))_{\T^2}=\exp\Bigl(-\kappa_m\int_{t'}^{t''}\xi\cdot\K(t)\xi\dd t\Bigr)(U(t'))_{\T^2}\,,\\ \bigl\|U(t'')-(U(t''))_{\T^2}\bigr\|_{H^J(\T^2)}\le e^{-\pi^2\kappa_m(t''-t')/(2\eps_m^2)}\bigl\|U(t')-(U(t'))_{\T^2}\bigr\|_{H^J(\T^2)}\,. \end{gather*} \end{enumerate} \end{lemma} \begin{proof} \emph{(a)} The $L^2(\T^2)$ norm does not increase, by~\eqref{e.prep.contract}, and halves across halving intervals. When we differentiate~\eqref{e.prep.fibre}, only the terms in which derivatives fall on the velocity, of size $C(J)\cfac$, are neither skew nor dissipative. In cell time and in the zone, as $|2\pi\mathbf n+\eps_m\xi|\ge\frac12|2\pi\mathbf n|$ for $\mathbf n\in\Z^2$ and $\K\ge\frac{15}{16}I$, \begin{align*} \tfrac14\|\nabla^{j+1}U\|_{L^2(\T^2)}^2 & \le\|(\nabla+\mathrm i\eps_m\xi)\nabla^jU\|_{L^2(\T^2)}^2\le2\|\nabla^{j+1}U\|_{L^2(\T^2)}^2+4\pi^2\|\nabla^jU\|_{L^2(\T^2)}^2\,,\\ \frac{\mathrm d}{\mathrm dt}\|\nabla^jU\|_{L^2(\T^2)}^2 & \le-\tfrac{15}8\|(\nabla+\mathrm i\eps_m\xi)\nabla^jU\|_{L^2(\T^2)}^2 \notag \\ & \qquad +C(j)\cfac\sum_{i\hat t_i$), over which the scalar factor changes in modulus by at most $e^{\pi^2/16}$ (Proposition~\ref{p.prep.dissipation}(c)), with~$C_{\mathrm{tr}}$ independent of the contrast, \begin{equation*} \Bigl\|U^\dagger_{\mathrm{br},i}(t)-\overline{\exp\Bigl(\int_t^{\hat t_i}\Zs_\xi\Bigr)}\Bigr\|_{H^J(\T^2)}\le C_{\mathrm{tr}}\ebar|\xi|\Bigl|\exp\Bigl(\int_t^{\hat t_i}\Zs_\xi\Bigr)\Bigr|\qquad(\hat t_{i+1}\le t\le\hat t_i)\,. \end{equation*} The remainder~$U^\dagger_{\mathrm{off},i}$ below is off-branch at~$\hat t_i$, by the identity, and $U^\dagger_{\mathrm{off},0}=0$: \begin{equation*} \begin{aligned} U^\dagger_{\mathrm{off},i}&\coloneqq U^\dagger_{\mathrm{br}}-\overline{\exp\Bigl(\int_{\hat t_i}^{t''}\Zs_\xi\Bigr)}\,U^\dagger_{\mathrm{br},i}\,,\\ U^\dagger_{\mathrm{br}}(t)-\overline{\exp\Bigl(\int_t^{t''}\Zs_\xi\Bigr)}&=U^\dagger_{\mathrm{off},i}(t)+\overline{\exp\Bigl(\int_{\hat t_i}^{t''}\Zs_\xi\Bigr)}\Bigl(U^\dagger_{\mathrm{br},i}(t)-\overline{\exp\Bigl(\int_t^{\hat t_i}\Zs_\xi\Bigr)}\Bigr)\,. \end{aligned} \end{equation*} By~(b), with its constant~$C_{\mathrm{off}}$ (exponents $N'_{\mathrm{act}}+N_0$ at~$\hat t_i$, where the weight is~$1$, and $N'_{\mathrm{act}}$ at~$t$; at most $\Nact+N_0$, for which the level is taken), and Proposition~\ref{p.prep.dissipation}(c), \begin{equation*} \|U^\dagger_{\mathrm{off},i}(t)\|_{H^J(\T^2);N'_{\mathrm{act}},t}\le2C_{\mathrm{off}}e^{-\pi^2(\hat t_i-t)/16}\,\|U^\dagger_{\mathrm{off},i}(\hat t_i)\|_{H^J(\T^2)}\,\frac{|\exp\int_t^{t''}\Zs_\xi|}{|\exp\int_{\hat t_i}^{t''}\Zs_\xi|}\,. \end{equation*} \emph{Step 3: the induction.} As $\hat t_i-\hat t_{i+1}\ge\frac5{12}\tau_m$, for $m$ large: \begin{equation*} \begin{aligned} &\|U^\dagger_{\mathrm{off},i}(\hat t_i)\|_{H^J(\T^2)}\le4C_{\mathrm{tr}}\ebar|\xi|\Bigl|\exp\Bigl(\int_{\hat t_i}^{t''}\Zs_\xi\Bigr)\Bigr|&&\text{(true for $i=0$)}\\ \Longrightarrow\ &\text{(c) at }t\text{ holds with }\bigl(8C_{\mathrm{off}}C_{\mathrm{tr}}e^{-\pi^2(\hat t_i-t)/16}+2C_{\mathrm{tr}}\bigr)\text{ for }C&&\text{on }[\hat t_{i+1},\hat t_i]\text{, by Step 2}\\ \Longrightarrow\ &\|U^\dagger_{\mathrm{off},i+1}(\hat t_{i+1})\|_{H^J(\T^2)}\le4C_{\mathrm{tr}}\ebar|\xi|\Bigl|\exp\Bigl(\int_{\hat t_{i+1}}^{t''}\Zs_\xi\Bigr)\Bigr|&&\text{at }t=\hat t_{i+1}\text{, as }4C_{\mathrm{off}}e^{-5\pi^2\tau_m/192}\le1. \end{aligned}\qedhere \end{equation*} \end{proof} \paragraph{Insertions against tests.} In a Duhamel term an insertion acts at each time on the previous term, and the result is evolved by the frozen operator; by~\eqref{e.prep.duality}, the pairing of the next term with a test is therefore an integral in time of the pairing of the previous term with the adjoint insertion~$\mathcal I^*$ applied to the test. So what matters is what~$\mathcal I^*$ does to a test at one time: how many derivatives it takes, and how large a multiple of the branch test its output contains. The pairing with~$\Rb_\xi$ measures that multiple; the rest of the output is orthogonal to~$\Rb_\xi$, and the test it starts decays at the cell rate (Lemma~\ref{l.prep.tests}(b)). The insertion acts at the cell rate, and its pairing with~$\Rb_\xi$ is at most $C\kappa_m/\eps_m^2$ times the factor $\ebar(|\xi|+\rho(m)^{-1})$, the corrector amplitude divided by the length on which the field varies. Indeed, paired with~$1$, the outer divergence of the insertion leaves only~$\eps_m\xi$ times a cell mean, $\Rb_\xi$ differs from~$1$ by the size of the corrector, and the commutators of~\eqref{e.prep.moment} carry the corrector amplitude, or~$\eps_m$, divided by~$\rho(m)$. If the test is itself nearly constant, as the branch test is, the innermost derivative of the insertion gives a second factor, at most the same one. The factor, and the rate at which an insertion acts when it carries the factor twice, satisfy \begin{equation}\label{e.prep.zeros} \ebar\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)\le1\,,\qquad\kappa_{m-1}|\xi|^2+\frac1{h_m}\le\frac{\kappa_m}{\eps_m^2}\,\ebar^2\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)^2\le2\Bigl(\kappa_{m-1}|\xi|^2+\frac1{h_m}\Bigr) \end{equation} by~\eqref{e.cell.amplitude} for $m$ large, which gives $\ebar/\rho(m)\le\frac34$, by $\ebar|\xi|\le2\clo\le\frac14$, and, for the bounds on the rate, by the identities of the Conventions. So over a block the rate~\eqref{e.prep.zeros} integrates to at most $2(1+\rho(m)|\xi|)^2$. A single factor is the cost of passing between the branch test and the tests orthogonal to~$\Rb_\xi$. These decay faster than the scalar factor of the branch changes (Proposition~\ref{p.prep.dissipation}(c)), so leaving the branch and returning cost the factor twice over about one cell time, at most a constant times the cost of one cell time on the branch. Lemma~\ref{l.prep.insert}(a),(b) makes these bounds precise, and~(c) is the forward bound used in Lemma~\ref{l.prep.words}(iii). The map~$\mathcal I^*$ keeps the norm~\eqref{e.prep.wnorm}, as, near a pulse, insertions do not depend on the coordinate along its flow. \begin{lemma}[Insertions against tests]\label{l.prep.insert} Consider the moment terms of a Duhamel term on $[t_1,t_2]$, with moment order~$N_1$. Let a group and a lower group be given, let~$d$ be the order of the terms of~\eqref{e.prep.moment} that connect them (the order of the insertion that connects them, or~$1$ for the commutator terms), and let $J,\Nact\ge0$ be integers, with $J+d-1$ at most the derivative counts of Subsection~\ref{ss.prep.tests}. There exists~$C$ such that, for almost every~$t\in[t_1,t_2]$, every real~$\xi$ in the zone, every array~$U^\dagger$ of fields in $H^{J+d}(\T^2)^V$ indexed by the group and every $0\le N'_{\mathrm{act}}\le\Nact$, the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item \emph{Mapping, and the pairing with~$\Rb_\xi$.} The first bound below holds; if moreover $\ebar|\xi|\le2\clo$ and $J+d\ge\max\{d_1,\dots,d_N\}+3$, so does the second: \begin{align*} \|\mathcal I^*U^\dagger\|_{H^J(\T^2);N'_{\mathrm{act}},t} & \le C\frac{\kappa_m}{\eps_m^2}\|U^\dagger\|_{H^{J+d}(\T^2);N'_{\mathrm{act}},t}\,,\\ \bigl|\bigl\langle\Rb_\xi(t),\mathcal I^*U^\dagger\bigr\rangle\bigr| & \le C\frac{\kappa_m}{\eps_m^2}\ebar\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)\|U^\dagger\|_{H^{J+d}(\T^2)}\,. \end{align*} \item \emph{The branch test.} Let $\ebar|\xi|\le2\clo$, let $t''\ge t$ be a time as in Lemma~\ref{l.prep.tests}(c), and let every entry of~$U^\dagger$ be the branch test of that lemma at~$t$ times a constant vector of~$\mathbb C^V$, these vectors having square sum one in~$|\cdot|_\omega$. Then \begin{equation}\label{e.prep.twozeros} \begin{aligned} \bigl|\bigl\langle\Rb_\xi(t),\mathcal I^*U^\dagger\bigr\rangle\bigr| & \le C\Bigl(\kappa_{m-1}|\xi|^2+\frac1{h_m}\Bigr)\Bigl|\exp\Bigl(\int_t^{t''}\Zs_\xi(\tilde t)\dd\tilde t\Bigr)\Bigr|\,,\\ \|\mathcal I^*U^\dagger\|_{H^J(\T^2);N'_{\mathrm{act}},t} & \le C\frac{\kappa_m}{\eps_m^2}\ebar\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)\Bigl|\exp\Bigl(\int_t^{t''}\Zs_\xi(\tilde t)\dd\tilde t\Bigr)\Bigr|\,. \end{aligned} \end{equation} \item \emph{Forward.} For every array~$U'$ in $H^{J+d}(\T^2)^V$ indexed by the lower group and every $(j,i)$ in the group, \begin{equation*} \Bigl\|\sum_{(j',i')}\mathcal I_{(j,i),(j',i')}(t)U'_{(j',i')}\Bigr\|_{H^J(\T^2)}\le C\frac{\kappa_m}{\eps_m^2}\|U'\|_{H^{J+d}(\T^2)}\,. \end{equation*} \end{enumerate} \end{lemma} \begin{proof} We bound one entry and sum by Schur's test in the components with the weighted row and column sums of the matrices of the strengths, so that~$C$ does not depend on the number of components. Recall $\eps_m\le\ebar$, by Proposition~\ref{p.scales}(c), and $\ebar/\rho(m)\le\frac34$ (paragraph of~\eqref{e.prep.zeros}). \emph{(a), the mapping, and~(c).} In cell time the insertion part of an entry of~$\mathcal I^*$ is, up to sign and terms with~$\eps_m/\rho(m)$ in place of a derivative, \begin{equation*} \sum_{d'\le d}(\nabla+\mathrm i\eps_m\xi)^{d'-1}\cdot\bigl(\mathcal A_{d'}^*(\nabla+\mathrm i\eps_m\xi)\,\cdot\,\bigr)\,, \end{equation*} with coefficients bounded with their $z$-derivatives by clause~(ii) of the insertions and $\rho(m)\le C\eps_{m-1}$; the commutator part has coefficients $\eps_m/\rho(m)$, $(\eps_m/\rho(m))^2$ and at most $C\ebar/\rho(m)$, with their $z$-derivatives, by~\eqref{e.prep.drift} and the display in the proof of Lemma~\ref{l.prep.tests}(a). The forward operator of~(c) is the adjoint of this one, with the same conjugated matrix and at most $J+1\le J+d-1$ derivatives on the coefficients of an insertion. For the weights, let~$k$ be the pulse of~\eqref{e.prep.wnorm} at~$t$, with $|t_{\mathrm{act}}(t)-k\tau_m|\le\frac23\tau_m$. Where the activity weight exceeds~$1$, $t$ lies within $C\log\cfac$ cell times after the activity of~$k$, so $|t-k\tau_m|\le\frac34\tau_m$ for $m$ large, and there the coefficients do not depend on $z\cdot e_{(k)}$, by item~(i) of the insertions, the velocity being parallel to~$e_{(k)}$; hence \begin{equation*} \bigl\|\mathcal I^*U^\dagger-(\mathcal I^*U^\dagger)_{e_{(k)}}\bigr\|_{H^J(\T^2)}\le C\frac{\kappa_m}{\eps_m^2}\bigl\|U^\dagger-(U^\dagger)_{e_{(k)}}\bigr\|_{H^{J+d}(\T^2)}\,. \end{equation*} Where the weight is~$1$, the bound is the plain one, by the last sentence after~\eqref{e.prep.wnorm}; elsewhere~$\mathcal I^*$ commutes with the average along~$e_{(k)}$. \emph{(a), the pairing with~$\Rb_\xi$.} For one entry, the mean of a divergence is that of its zeroth-order part: \begin{equation*} \bigl\langle\Rb_\xi,\mathcal I^*U^\dagger\bigr\rangle=\overline{(\mathcal I^*U^\dagger)_{\T^2}}+\bigl\langle\Rb_\xi-1,\mathcal I^*U^\dagger\bigr\rangle\,,\qquad\bigl((\nabla+\mathrm i\eps_m\xi)\cdot F\bigr)_{\T^2}=\mathrm i\eps_m\xi\cdot(F)_{\T^2}\,, \end{equation*} for a vector field~$F$ on~$\T^2$, and the other terms carry $\eps_m/\rho(m)$, $\ebar/\rho(m)$ or $\Cbr\ebar|\xi|$. The first bound with $J=0$ and $N'_{\mathrm{act}}=0$ gives $\|\mathcal I^*U^\dagger\|_{L^2(\T^2)}\le C\frac{\kappa_m}{\eps_m^2}\|U^\dagger\|_{H^d(\T^2)}$, so only $\|U^\dagger\|_{H^d(\T^2)}$ appears. \emph{(b)} Let~$a$ be the array of the constant vectors, so that \begin{equation*} U^\dagger=a\Bigl(U^\dagger_{\mathrm{br}}(t)-\overline{\exp\Bigl(\int_t^{t''}\Zs_\xi\Bigr)}\Bigr)+a\,\overline{\exp\Bigl(\int_t^{t''}\Zs_\xi\Bigr)}\,. \end{equation*} By the first bound of~(a) and Lemma~\ref{l.prep.tests}(c) at the order $J+d$ with the exponent~$N'_{\mathrm{act}}$, the image under~$\mathcal I^*$ of the first part is at most $C\frac{\kappa_m}{\eps_m^2}\ebar|\xi||\exp\int_t^{t''}\Zs_\xi|$ in the norm $\|\cdot\|_{H^J(\T^2);N'_{\mathrm{act}},t}$. On the second part, the innermost derivative of an insertion term gives $\mathrm i\eps_m\xi$, or $\eps_m/\rho(m)$ stands in its place; the commutator terms carry $(\eps_m/\rho(m))\eps_m|\xi|$, $C\ebar/\rho(m)$ and $(\eps_m/\rho(m))^2$. So the image of the second part is at most $C\frac{\kappa_m}{\eps_m^2}\ebar(|\xi|+\frac1{\rho(m)})|\exp\int_t^{t''}\Zs_\xi|$ in $H^J(\T^2)$; where the weight exceeds~$1$ this field is invariant along~$e_{(k)}$, so its weighted norm is at most $\sqrt2$ times this in every case. This is the second line of~\eqref{e.prep.twozeros}. For the first, in the mean a term of order~$d'$ carries $d'\ge2$ quantities among $\eps_m|\xi|$, $\eps_m/\rho(m)$ and the bound $C\ebar|\xi||\exp\int_t^{t''}\Zs_\xi|$ of $(\nabla+\mathrm i\eps_m\xi)U^\dagger_{\mathrm{br}}(t)$ given by Lemma~\ref{l.prep.tests}(c). The commutators carry $(\eps_m/\rho(m))\eps_m|\xi|$ and $(\eps_m/\rho(m))^2$, and the drift, of mean zero against a constant, carries $(\ebar/\rho(m))\ebar|\xi|$; against $\Rb_\xi-1$ the second line gains $\Cbr\ebar|\xi|$. We conclude with~\eqref{e.prep.zeros}. \emph{The derivative count.} In $H^J(\T^2)$, the product rule puts at most $J+d-1$ derivatives on the coefficients of an insertion of order~$d$, as $d-1$ of the derivatives stand outside them. \end{proof} \subsection{Duhamel terms over a renewal block} \label{ss.prep.words} We follow one Duhamel term of the frozen operator, centred at~$y_0$, over a segment that ends at a block boundary, and look at the Fourier transform of its last term at the end of the segment, at the low frequencies. There, by~\eqref{e.prep.momentfourier}, it is~$\eps_m^2$ times the pairing with the branch test, which is~$1$ at the end, and Lemma~\ref{l.prep.piece} follows this pairing across the segment. Over a complete block, Lemma~\ref{l.prep.words} splits the term into a \emph{model}~$\Pblk g$, smooth at the scale~$\rho(m)$, and an error smaller than any power of~$\eps_{m-1}$; the model is the last term at the end of the block, cut off smoothly to low frequencies, which is its part in the principal branch. The low-frequency cutoff is $\Hlo(\xi)\coloneqq\Hc(2-\ebar|\xi|/\clo)$, where~$\Hc$ is the smooth step of~\eqref{e.slow} and~$\clo$ the absolute constant of Proposition~\ref{p.prep.dissipation}: the cutoff is~$1$ where $\ebar|\xi|\le\clo$ and~$0$ where $\ebar|\xi|\ge2\clo$. For a field~$F$ on~$\R^2$ (or any function of~$\xi$ with values in~$\mathbb C^V$ in place of~$\widehat F$) and a datum~$g$, we write \begin{equation*} |\widehat F|_{\rho(m),\omega,N_1;\xi}\coloneqq\sum_{j\le N_1}\frac{|\nabla_\xi^j\widehat F(\xi)|_\omega}{\rho(m)^j\,j!}\,,\qquad\|g\|_{\rho(m),\omega,N';\xi}\coloneqq\sum_{|i|\le N'}\Bigl\|\Bl\Bigl(\Bigl(\frac{y-y_0}{\rho(m)}\Bigr)^{i}g\Bigr)(\xi,\cdot)\Bigr\|_{L^2_\omega(\T^2)}\,, \end{equation*} with~$\nabla_\xi^j$ the array of the derivatives of order~$j$ in~$\xi$, and $|c|_\omega^2\coloneqq\sum_v\omega_v^{-1}|c_v|^2$ for $c\in\mathbb C^V$; the second is the \emph{moment data} of~$g$ at~$\xi$. Without insertions, the left side of~\eqref{e.prep.piece} is, up to the corrector, the Fourier transform of the datum, whose frequency derivatives are its moments. By Lemma~\ref{l.prep.piece}, insertions of any orders change this only by a polynomial factor in $\rho(m)|\xi|$, with no power of the contrast, on a segment that may be a whole block or arbitrarily short. In the proof (Figure~\ref{f.prep.piece}) we pair the moment terms with the branch test and with off-branch tests: insertions act on the branch with the square of the factor $\ebar(|\xi|+\rho(m)^{-1})$ of~\eqref{e.prep.zeros}, a passage between the branch and the off-branch part costs the factor once, and off-branch tests decay at the cell rate, so a triangular Gr\"onwall inequality closes with a polynomial loss in $\rho(m)|\xi|$. The part of the datum off the branch is read by tests of norm at most one, which is why no derivative of the datum enters. \begin{lemma}[The Fourier transform on a segment]\label{l.prep.piece} Let $t_2\coloneqq(l+\frac12)h_m$, $t_1\in[t_2-h_m,t_2]$ and $N_1\ge0$ an integer, and let $\Ud0,\dots,\Ud N$ be a Duhamel term on $[t_1,t_2]$, centred at~$y_0$, from a datum $g\in H^{d_1+\dots+d_N-2N}(\R^2)^V$ with compact support, with~$N'$ as for the moment terms. There exist~$C$ and an integer~$N_2$ such that the function of~$\xi$ inside the norm on the left of~\eqref{e.prep.piece} is $N_1$ times differentiable and \begin{equation}\label{e.prep.piece} \Bigl|\exp\Bigl(-\int_{t_1}^{t_2}\Zs_\xi(t)\dd t\Bigr)\Hlo\,\widehat{\Ud N(t_2)}\Bigr|_{\rho(m),\omega,N_1;\xi}\le C(1+\rho(m)|\xi|)^{N_2}\eps_m^2\|g\|_{\rho(m),\omega,N';\xi}\,. \end{equation} \end{lemma} \begin{figure}[t] \centering \begin{tikzpicture}[x=1.15cm,y=1cm,font=\small] \draw[-{Stealth[length=2.2mm]}] (-0.3,0) -- (10.8,0); \draw[thick] (0,-0.12) -- (0,0.12); \draw[thick] (10,-0.12) -- (10,0.12); \node[below=3pt] at (0,-0.12) {$t_1$}; \node[below=3pt] at (10,-0.12) {$t_2$}; \foreach \x in {2.2,4.4,5.5,6.6,8.4} {\draw (\x,-0.1) -- (\x,0.1); \draw[black!35,densely dotted] (\x,0.1) -- (\x,2.2);} \draw[very thick] (0,2.2) -- (10,2.2) node[right] {branch}; \draw[densely dotted] (0,1.0) -- (10,1.0) node[right] {off-branch}; \node[left] at (-0.55,1.6) {$g$}; \draw[-{Stealth[length=2mm]}] (-0.55,1.6) -- (-0.05,2.15); \draw[-{Stealth[length=2mm]}] (-0.55,1.6) -- (-0.05,1.05); \draw[very thick] (0,1.0) -- (1.5,1.0); \node[below] at (0.75,0.95) {decay}; \fill (2.2,2.2) circle (2.2pt); \node[above] at (2.2,2.3) {power~$2$}; \draw[-{Stealth[length=2mm]}] (4.4,2.15) -- (4.4,1.05); \node[left] at (4.4,1.6) {power~$1$}; \draw[very thick] (4.4,1.0) -- (6.6,1.0); \fill (5.5,1.0) circle (2.2pt); \node[below] at (5.5,0.95) {power~$0$}; \draw[-{Stealth[length=2mm]}] (6.6,1.05) -- (6.6,2.15); \node[right] at (6.6,1.6) {power~$1$}; \fill (8.4,2.2) circle (2.2pt); \node[above] at (8.4,2.3) {power~$2$}; \node[above] at (10,2.3) {$U^\dagger_{\mathrm{br}}=1$}; \end{tikzpicture} \caption{The proof of Lemma~\ref{l.prep.piece} in cell time. Tracks: pairings with the branch test ($1$ at~$t_2$) and with off-branch tests; arrows from~$g$: the split of the datum; ticks: insertion times; dots: insertions within a track; vertical arrows: passages between the tracks; labels: the power of $\ebar(|\xi|+\rho(m)^{-1})$ each costs; thick lower segments: decay at the cell rate.} \label{f.prep.piece} \end{figure} \begin{proof} Fix~$\xi$, and let $\Bl U^{[j,i]}(t)$ be the Bloch sums at~$\xi$ of the moment terms with~$N_1$. \emph{Step 1: Coordinates.} Let~$U^\dagger_{\mathrm{br}}$ be the branch test of Lemma~\ref{l.prep.tests}(c) with $t''=t_2$, which lies in the quiet margin at the end of the block. Each moment term $(j,i)$ gets a test order $N_{j,i}$ and an exponent of the weight, fixed in Step~7; the norm~\eqref{e.prep.wnorm} of this order and exponent is the \emph{weighted norm} of $(j,i)$. The \emph{branch coordinate} and the \emph{off-branch seminorm} of $(j,i)$ at~$t$ are, with the supremum over the~$U^\dagger$ off-branch at~$t$ of weighted norm at most one, \begin{equation*} |\Bl U^{[j,i]}(t)|_{\mathrm{br},t}\coloneqq\frac{|\langle\Bl U^{[j,i]}(t),U^\dagger_{\mathrm{br}}(t)\rangle|}{|\exp\int_{t_1}^{t_2}\Zs_\xi|}\,,\quad\|\Bl U^{[j,i]}(t)\|_{\mathrm{off},t}\coloneqq\frac{\sup_{U^\dagger}|\langle\Bl U^{[j,i]}(t),U^\dagger\rangle_{L^2_\omega(\T^2)}|}{|\exp\int_{t_1}^t\Zs_\xi|}\,; \end{equation*} the scalar factors normalize the branch: on the solution $\Rb_\xi(t)\exp(\int_{t_1}^t\Zs_\xi)$ the branch coordinate is~$1$ and the off-branch seminorm~$0$. \emph{Step 2: The branch inequality.} Fix a constant vector of~$\mathbb C^V$ of norm one in $|\cdot|_\omega$. By~\eqref{e.prep.duality}, the derivative of the pairing of $\Bl U^{[j,i]}$ with $U^\dagger_{\mathrm{br}}$ times this vector is the sum over the lower $(j',i')$ of the pairings of $\Bl U^{[j',i']}$ with $\mathcal I^*_{(j,i),(j',i')}$ applied to it. We split the latter, component by component, into a multiple of $U^\dagger_{\mathrm{br}}(t)$ and a rest off-branch at~$t$: by Lemmas~\ref{l.prep.tests}(c) and~\ref{l.prep.insert}(b) and the chain in~\eqref{e.prep.zeros}, the multiples are at most~$C$ times the square of the factor $\ebar(|\xi|+\rho(m)^{-1})$, and the rest at most~$C$ times the factor times $|\exp\int_t^{t_2}\Zs_\xi|$ in the weighted norm of $(j',i')$. As the scalar factors multiply, integrating gives, with the sum over the terms on the right of~\eqref{e.prep.moment}, \begin{align*} & |\Bl U^{[j,i]}(t)|_{\mathrm{br},t}\le|\Bl U^{[j,i]}(t_1)|_{\mathrm{br},t_1} \notag \\ & \qquad +C\int_{t_1}^t\sum_{(j',i')}\Bigl(\ebar^{\,2}\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)^2|\Bl U^{[j',i']}(t')|_{\mathrm{br},t'}+\ebar\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)\|\Bl U^{[j',i']}(t')\|_{\mathrm{off},t'}\Bigr)\dd t'\,. \end{align*} \emph{Step 3: The off-branch inequality.} Let~$U^\dagger$ be the test on $[t_1,t]$ from a field off-branch at~$t$ of weighted norm at most one for $(j,i)$. By Lemma~\ref{l.prep.tests}(b) it stays off-branch, and, for every term $(j',i')$ on the right of~\eqref{e.prep.moment}, its norm of order $N_{j,i}$ and exponent $N_{j',i'}N_0$ at~$t'$ is at most $Ce^{-\pi^2(t-t')/8}$. By~\eqref{e.prep.duality} its pairing with $\Bl U^{[j,i]}(t)$ is that at~$t_1$ plus the integral of the pairings of the lower terms with $\mathcal I^*U^\dagger$. We split $\mathcal I^*U^\dagger(t')$ as in Step~2: by Lemma~\ref{l.prep.insert}(a), the multiples of $U^\dagger_{\mathrm{br}}(t')$ are at most~$C$ times the factor $\ebar(|\xi|+\rho(m)^{-1})$ times $e^{-\pi^2(t-t')/8}/|\exp\int_{t'}^{t_2}\Zs_\xi|$, and the rest is at most $Ce^{-\pi^2(t-t')/8}$ in the weighted norm. With $|\exp(-\int_{t'}^t\Zs_\xi)|\le e^{\pi^2(t-t')/16}$ (Proposition~\ref{p.prep.dissipation}(c)), \begin{align*} & \|\Bl U^{[j,i]}(t)\|_{\mathrm{off},t}\le Ce^{-\pi^2(t-t_1)/16}\|\Bl U^{[j,i]}(t_1)\|_{L^2_\omega(\T^2)} \notag \\ & \qquad +C\int_{t_1}^te^{-\pi^2(t-t')/16}\sum_{(j',i')}\Bigl(\ebar\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)|\Bl U^{[j',i']}(t')|_{\mathrm{br},t'}+\|\Bl U^{[j',i']}(t')\|_{\mathrm{off},t'}\Bigr)\dd t'\,: \end{align*} the coupling between off-branch parts has no small coefficient. \emph{Step 4: Data.} For $j\ge1$ the moment terms vanish at~$t_1$. For $j=0$, $\|U^\dagger_{\mathrm{br}}(t_1)\|_{L^2(\T^2)}\le C|\exp\int_{t_1}^{t_2}\Zs_\xi|$ by Lemma~\ref{l.prep.tests}(c), and the norm~\eqref{e.prep.wnorm} is at least the $L^2$ norm; so \begin{equation*} |\Bl U^{[0,i]}(t_1)|_{\mathrm{br},t_1}+\|\Bl U^{[0,i]}(t_1)\|_{\mathrm{off},t_1}\le C\|g\|_{\rho(m),\omega,N';\xi}\,. \end{equation*} \emph{Step 5: The triangular Gr\"onwall inequality.} We claim that, at~$t$, the branch coordinate and the off-branch seminorm of $(j,i)$ are at most $\|g\|_{\rho(m),\omega,N';\xi}$ times, respectively, \begin{equation*} C_{j,i}\Bigl(1+\ebar^{\,2}\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)^2(t-t_1)\Bigr)^{N_{j,i}}\,,\qquad\text{the same times }\Bigl(\ebar\Bigl(|\xi|+\frac1{\rho(m)}\Bigr)+e^{-\pi^2(t-t_1)/32}\Bigr)\,, \end{equation*} $C_{j,i}$ depending only on~$N_{j,i}$ and the fixed integers. The off-branch seminorm, a supremum over tests that change with~$t$, is not integrated in~$t$: Steps~2--3 hold with any continuous functions above the lower quantities. We prove the claim by induction on~$N_{j,i}$, inserting the majorants of the lower terms, which are the terms on the right of~\eqref{e.prep.moment}, of test orders lower by at least one (Step~7). In Step~2 the square of the factor times the first majorant and the factor times the second integrate to at most the first majorant, as $\int_0^\infty e^{-\pi^2t'/32}\dd t'\le4$ and the factor is at most~$1$. In Step~3 the integrand is the same power times twice the factor plus $e^{-\pi^2(t'-t_1)/32}$, which against $e^{-\pi^2(t-t')/16}$ gives at most the second majorant. Over the segment $t-t_1\le\kappa_mh_m/\eps_m^2$, and $t-t_1$ times the square of the factor is at most $2(1+\rho(m)|\xi|)^2$ by~\eqref{e.prep.zeros}. \emph{Step 6: Output.} The function of~$\xi$ in~\eqref{e.prep.piece} is $N_1$ times differentiable, as $\Ud N(t_2)$ has $N_1+2$ moments and $\int_{t_1}^{t_2}\Zs_\xi$ is analytic on the support of~$\Hlo$ (Proposition~\ref{p.prep.branch}). At~$t_2$ the branch test is~$1$, so, by~\eqref{e.prep.momentfourier} and Step~5, for $|i|\le N_1$, \begin{equation*} \rho(m)^{-|i|}\bigl|\nabla_\xi^i\widehat{\Ud N(t_2)}(\xi)\bigr|_\omega=\eps_m^2\bigl|\langle\Bl U^{[N,i]}(t_2),1\rangle\bigr|\le C\eps_m^2\Bigl|\exp\Bigl(\int_{t_1}^{t_2}\Zs_\xi\Bigr)\Bigr|(1+\rho(m)|\xi|)^{C}\|g\|_{\rho(m),\omega,N';\xi}\,. \end{equation*} By Leibniz's rule, the derivatives in~\eqref{e.prep.piece} are sums of products of these, of derivatives of $\Hlo$, bounded at the scale~$\rho(m)$ since $\ebar\le\rho(m)$, and of derivatives of $\exp(-\int_{t_1}^{t_2}\Zs_\xi)$, normalized as in~\eqref{e.prep.piece}, at most $C(1+\rho(m)|\xi|)^{N_1}|\exp(-\int_{t_1}^{t_2}\Zs_\xi)|$ by Proposition~\ref{p.prep.dissipation}(c); the factors $|\exp(\pm\int_{t_1}^{t_2}\Zs_\xi)|$ cancel, and $N_2$ is the power above plus~$N_1$. \emph{Step 7: Test orders.} The moment term $(j,i)$ gets the \emph{test order} $N_{j,i}\coloneqq\max\{d_1,\dots,d_N\}+3+d_1+\dots+d_j$ plus its degree, and the exponent of the weight $N_{j,i}N_0$, with~$N_0$ the largest of the integers of Lemma~\ref{l.prep.tests}(b) for these orders (the statement of~(b) holds with any larger integer); Lemmas~\ref{l.prep.tests} and~\ref{l.prep.insert} are applied with~$\Nact$ the largest of these exponents. Along~\eqref{e.prep.moment} the test order drops by at least the order~$d$ of the connecting terms: by at least~$d_j$ from $(j,i)$ to the terms $(j-1,i')$, as $|i'|$ is at most~$|i|$ plus the Taylor degree of~$\mathcal I_j$, and by at least~$1$ to the commutator terms. So Lemma~\ref{l.prep.insert} applies to tests of order~$N_{j,i}$ with $J=N_{j',i'}$, the test order of the lower term, and the exponents drop by at least~$N_0$, as Step~3 requires. Lemma~\ref{l.prep.insert} differentiates the coefficients at most $J+d-1\le N_{j,i}-1\le\max\{d_1,\dots,d_N\}+2+d_1+\dots+d_N+N'$ times, within the derivative counts of Subsection~\ref{ss.prep.tests}; the constants $C_{j,i}$ are fixed in the order of the test orders. \end{proof} Over a complete block the scalar factor $|\exp\int_{t_1}^{t_2}\Zs_\xi|$ is at most the Gaussian of Proposition~\ref{p.prep.dissipation}(b), and Lemma~\ref{l.prep.piece} gives~(ii): the model~$\Pblk g$ is smooth at the scale~$\rho(m)$ and localized near~$y_0$ by its moments. Item~(iii) bounds the rest, which the block damps at the cell rate, with losses in powers of~$\cfac$ and of the number of cell times, and with derivatives taken forward over the last cell time; the damping, smaller than any power of~$\eps_{m-1}$, absorbs these losses. \begin{lemma}[Duhamel terms over a renewal block]\label{l.prep.words} Let $N_1\ge0$ be an integer, and let $\Ud0,\dots,\Ud N$ be a Duhamel term over the block $[t_1,t_2]=[(l-\frac12)h_m,(l+\frac12)h_m]$, centred at~$y_0$, with~$N'$ and the class~$\mathcal D_{N'+2}$ of its data as for the moment terms. There exist~$C$, $c>0$, an integer~$N_2$ and a linear map~$\Pblk$, independent of the datum~$g$, such that: \begin{enumerate}[label=\textup{(\roman*)}] \item for every $g\in\mathcal D_{N'+2}$, $\Pblk g$ is smooth, and $\widehat{\Pblk g}(\xi)=0$ for every~$\xi$ with $\ebar|\xi|>2\clo$; \item for every $g\in\mathcal D_{N'+2}$ and every real~$\xi$, \begin{equation*} |\widehat{\Pblk g}|_{\rho(m),\omega,N_1;\xi}\le C(1+\rho(m)|\xi|)^{N_2}e^{-c\rho(m)^2|\xi|^2}\eps_m^2\|g\|_{\rho(m),\omega,N';\xi}\,; \end{equation*} \item for every $g\in\mathcal D_{N'+2}\cap H^{N_1+d_1+\dots+d_N-2N}(\R^2)^V$, \begin{align*} \lefteqn{\|\Ud N(t_2)-\Pblk g\|_{H^{N_1}_{\rhomol(m),\omega,N_1}(\R^2)}} \qquad & \notag \\ & \le C\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^{N_2}\cfac^{N_2}\exp\Bigl(-\frac{c\kappa_m\tau_m}{\eps_m^2}\Bigr)\|g\|_{L^2_{\omega,N'}(\R^2)}\le C\eps_{m-1}^{Q+3}\|g\|_{L^2_{\omega,N'}(\R^2)}\,. \end{align*} \end{enumerate} \end{lemma} \begin{proof} \emph{The map.} For $g\in\mathcal D_{N'+2}$ let $\widehat{\Pblk g}(\xi)$ be $\Hlo(\xi)^{N'+1}$ times $\widehat{\Ud N(t_2)}(\xi)$. It is linear, with the Fourier support of~(i), and smooth since its Fourier transform has compact support. \emph{(ii).} Lemma~\ref{l.prep.piece} over the block, times $\Hlo(\xi)^{N'}$, bounds the derivatives of $\exp(-\int_{t_1}^{t_2}\Zs_\xi)\widehat{\Pblk g}(\xi)$ for compactly supported $g\in H^{d_1+\dots+d_N-2N}(\R^2)^V$; those of $\exp(\int_{t_1}^{t_2}\Zs_\xi)$ of order~$j$, times $\rho(m)^{-j}$, are at most $C(j)(1+\rho(m)|\xi|)^je^{-\cdis\rho(m)^2|\xi|^2}$ by Proposition~\ref{p.prep.dissipation}(b),(c), and Leibniz's rule gives the bound with $c=\cdis$, the power of $1+\rho(m)|\xi|$ rising by~$N_1$. It extends to $g\in\mathcal D_{N'+2}$ by the limit $g^{(r)}\coloneqq\chi((y-y_0)/r)g\to g$ in~$\mathcal D_{N'+2}$ as $r\to\infty$, $\chi$ a smooth cutoff equal to~$1$ near~$0$: by the moment-weighted energy, $\Ud N_{g^{(r)}}(t_2)\to\Ud N_g(t_2)$ in $L^2_{\omega,N_1+2}(\R^2)$, so the Fourier transforms converge in~$C^{N_1}(\R^2)$; and the moment data of~$g^{(r)}$ converge to those of~$g$ at every~$\xi$ (paragraph of~\eqref{e.prep.bloch}, with two extra moments). \emph{(iii), Step 1: Against tests.} Let~$\xi$ be in the zone and $N_{j,i}$ the test orders of the proof of Lemma~\ref{l.prep.piece}. For every moment term, $t\in[t_1,t_2]$ and $U^\dagger\in H^{N_{j,i}}(\T^2)^V$, no orthogonality required, \begin{equation}\label{e.prep.plain} \bigl|\bigl\langle\Bl U^{[j,i]}(t),U^\dagger\bigr\rangle_{L^2_\omega(\T^2)}\bigr|\le C\Bigl(1+\frac{\kappa_m(t-t_1)}{\eps_m^2}\Bigr)^{C}\cfac^{C}2^{-\#[t_1,t]}\|U^\dagger\|_{H^{N_{j,i}}(\T^2)}\|g\|_{\rho(m),\omega,N';\xi}\,. \end{equation} Indeed, by~\eqref{e.prep.duality} the pairing is that at~$t_1$ with the test from~$U^\dagger$, plus the integral of the pairings of the lower moment terms with~$\mathcal I^*$ of the test, which Lemmas~\ref{l.prep.fibre}(a) and~\ref{l.prep.insert}(a) bound by $C\cfac^C2^{-\#[t',t]}$ in $H^{N_{j',i'}}(\T^2)$. By induction, with $2^{-\#[t_1,t']}2^{-\#[t',t]}\le2\cdot2^{-\#[t_1,t]}$, each pairing is a nested integral over a simplex of insertion times, of volume polynomial in the cell duration, with every derivative on a smooth test: there is no short-time singularity. \emph{Step 2: The full field.} On $N+N'+1$ equal parts of the last cell time of the block, Lemma~\ref{l.prep.fibre}(b) smooths the moment terms one after the other, from the negative order of~\eqref{e.prep.plain} for the part without forcing and with the forcing for the Duhamel part, to the \emph{forward order} of $(j,i)$: $N_1+d_{j+1}+\dots+d_N$ plus $N_1$ plus the Taylor degrees of $\mathcal I_{j+1},\dots,\mathcal I_N$ minus~$|i|$. The rank of $(j,i)$, $j$ plus its degree, lies in $\{0,\dots,N+N'\}$ and drops by at least one along every arrow of~\eqref{e.prep.moment}, as the degree does not increase from $(j,i)$ to the terms $(j-1,i')$ and drops along the commutator terms; so the terms of rank~$r$ are smoothed on the part~$r+1$, their forcings being smooth from its start. The forward order drops along~\eqref{e.prep.moment} by at least the order of the connecting terms, so the forcing lies in the next forward order by Lemma~\ref{l.prep.insert}(c), which differentiates the coefficients at most $N_1+N'+d_1+\dots+d_N+\max\{d_1,\dots,d_N\}-1$ times here, within the derivative counts. So every $(N,i)$ is at most $C(1+\kappa_mh_m/\eps_m^2)^C\cfac^C\|g\|_{\rho(m),\omega,N';\xi}$ in $H^{N_1}(\T^2)$ at~$t_2$. \emph{Step 3: The discarded frequencies.} The Fourier transform of $\Ud N(t_2)-\Pblk g$ is $(1-\Hlo(\xi)^{N'+1})\widehat{\Ud N(t_2)}(\xi)$, supported where $\ebar|\xi|\ge\clo$; we split it by the smooth step $\Hc(2-4\eps_m|\xi|/\pi)$, which is~$1$ for $\eps_m|\xi|\le\pi/4$ and~$0$ for $\eps_m|\xi|\ge\pi/2$. At \emph{moderate} frequencies, $\eps_m|\xi|\le\pi/2$, \eqref{e.prep.momentfourier} makes $\rho(m)^{-|i|}|\nabla_\xi^i\widehat{\Ud N(t_2)}(\xi)|_\omega$ equal to $\eps_m^2|\langle\Bl U^{[N,i]}(t_2),1\rangle|$, which~\eqref{e.prep.plain}, with the test~$1$ and the $c\kappa_mh_m/\eps_m^2$ halving intervals of the block, bounds by \begin{equation*} C\eps_m^2\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^C\cfac^Ce^{-c\kappa_mh_m/\eps_m^2}\|g\|_{\rho(m),\omega,N';\xi}\,. \end{equation*} As $\rhomol(m)|\xi|\le\pi/2$ on this band, Plancherel's identity, Leibniz's rule on the two cutoffs and Lemma~\ref{l.prep.bloch} give the bound of~(iii) for this part. At \emph{high} frequencies, $\eps_m|\xi|\ge\pi/4$, we use the modes of Bloch sums: for~$h$ with $\|h\|_{L^2_{\omega,2}(\R^2)}<\infty$, $\xi'$ in the zone and $\mathbf n\in\Z^2$, by Poisson's summation formula, \begin{equation}\label{e.prep.modes} \int_{\T^2}\Bl h(\xi',z)\,e^{-2\pi\mathrm i\mathbf n\cdot z}\dd z=\frac{e^{-2\pi\mathrm i\mathbf n\cdot y_0/\eps_m}}{\eps_m^2}\,\widehat h\Bigl(\xi'+\frac{2\pi\mathbf n}{\eps_m}\Bigr)\,, \end{equation} and the translates of the zone by $2\pi\Z^2/\eps_m$ tile~$\R^2$. So, with~$\xi'$ the frequency of the zone with $\eps_m(\xi-\xi')\in2\pi\Z^2$, $\widehat{U^{[N,i]}(t_2)}(\xi)$ is~$\eps_m^2$ times a unimodular number times the mode $e^{\mathrm i\eps_m(\xi-\xi')\cdot z}$ of the Bloch sum at~$\xi'$. Let~$t$ be the start of the quiet stretch, of length at least $\tau'_m-\tau_m/24$, that ends the block. By~\eqref{e.prep.plain} with this test, \begin{equation*} \bigl|\bigl\langle\Bl U^{[j,i]}(t)(\xi',\cdot),e^{\mathrm i\eps_m(\xi-\xi')\cdot z}\bigr\rangle\bigr|\le C\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^C\cfac^C(1+\eps_m|\xi|)^{N_{j,i}}\|g\|_{\rho(m),\omega,N';\xi'}\,. \end{equation*} Across the stretch the velocity vanishes and the insertions do not depend on~$z$, by clause~(i) of the definition of insertions, so each mode evolves separately and is damped at the rate $\frac12\eps_m^2|\xi|^2\ge\pi^2/32$ (the proof of Lemma~\ref{l.prep.fibre}(c)). The triangular forcing costs a polynomial in the length of the stretch, and, summed over the modes, the Gaussian damping in $\eps_m|\xi|$ absorbs the growth $(1+\eps_m|\xi|)^{N_{j,i}}$ and leaves the factor $e^{-c\kappa_m\tau'_m/\eps_m^2}$ in $H^{N_1}(\T^2)$ at~$t_2$. We return to the plane: by~\eqref{e.prep.modes}, $\nabla+\mathrm i\eps_m\xi'$ acts on the mode~$\mathbf n$ of $\Bl h(\xi',\cdot)$ as $\mathrm i\eps_m\xi$, with $\xi=\xi'+2\pi\mathbf n/\eps_m$, so, by Parseval's identity, \begin{equation*} \int_{\text{zone}}\bigl\|(\nabla+\mathrm i\eps_m\xi')^j\Bl h(\xi',\cdot)\bigr\|_{L^2_\omega(\T^2)}^2\dd\xi'=(2\pi)^2\eps_m^{2j-4}\|\nabla^jh\|_{L^2_\omega(\R^2)}^2 \end{equation*} whenever the left side is finite, and $\nabla^jh$ then lies in $L^2_\omega(\R^2)$. We apply this with $h=((y-y_0)/\rho(m))^a\Ud N(t_2)$, $|a|\le N_1$, and commute the weight with $\rhomol(m)\nabla$ at the cost $\rhomol(m)/\rho(m)\le1$, the derivatives at the scale $\rhomol(m)\le\eps_m$ costing a constant. As $h_m\ge\tau'_m\ge\tau_m$, this is the first bound of~(iii) for $g\in\mathcal D_{N'+2}$. \emph{Step 4: The power of~$\eps_{m-1}$.} By Proposition~\ref{p.scales}(c) and~\eqref{e.times}, $\kappa_m\tau_m/\eps_m^2\ge c\eps_{m-1}^{-4\delta}$, $\ebar/\eps_m\le C\eps_{m-1}^{-q\gamma}$, and $\kappa_mh_m/\eps_m^2\le\rho(m)^2/\eps_m^2\le C\eps_{m-1}^{-2q}$ as $\kappa_m\le\kappa_{m-1}$, $\rho(m)\le1$ and $\eps_m\ge\frac12\eps_{m-1}^q$ (Lemma~\ref{l.fl.scales}(a)). So the first bound of~(iii) is at most $C\eps_{m-1}^{-C}\exp(-c\eps_{m-1}^{-4\delta})\le C\eps_{m-1}^{Q+3}$ for $m$ large. \end{proof} \subsection{The change of Lagrangian coordinates, and the proof of Proposition~\ref{p.prep.joint}} \label{ss.prep.joint} The old partial block, from~$t_1$ to the block boundary~$t_2$, may be short and carry pulses of one orientation only (Figure~\ref{f.prep.timeline}): it need not damp the frequencies away from the branch, and its scalar factor has no Gaussian. At~$t_2$ the chart changes, by an area-preserving map that does not keep the support of the Fourier transform. So the old term is carried across~$t_2$ at low frequency, where it is bounded only in negative norms, and the next block, the complete \emph{future block}, smooths it (Figure~\ref{f.prep.renewal}): its model sees its datum only through the zone part below, and its Gaussian absorbs any power of $1+\rho(m)|\xi|$. Let $t_2\coloneqq(l+\frac12)h_m$ and $\tpr\coloneqq t_2+h_m$, as in~\eqref{e.prep.tpr}, let~$y_0'$ be a label of the chart of the block of $l+1$, and let $y_0\coloneqq X_{m-1}(lh_m,y_0',(l+1)h_m)$ be the label of the same point in the chart of the block of~$l$ (Lemma~\ref{l.fl.charts}(b)). A \emph{Duhamel term across the renewal} of lengths $N_-,N_+$, from a datum~$g$ at a time $t_1\in[t_2-h_m,t_2]$, is the sequence $\Ud0,\dots,\Ud{N_-+N_+}$ in which $\Ud0,\dots,\Ud{N_-}$ is a Duhamel term on $[t_1,t_2]$ for the block of~$l$, centred at~$y_0$, from~$g$, with the \emph{old} insertions $\mathcal I_1,\dots,\mathcal I_{N_-}$, of orders $d_1,\dots,d_{N_-}$; $\Ud{N_-}$ continues on $[t_2,\tpr]$ by the frozen operator of the block of $l+1$, centred at~$y_0'$, from $\Ud{N_-}(t_2)\circ X_{m-1}(lh_m,\cdot,(l+1)h_m)$; and $\Ud{N_-+1},\dots,\Ud{N_-+N_+}$ start from zero at~$t_2$, forced by~$N_+$ \emph{future} insertions $\mathcal I_{N_-+1},\dots,\mathcal I_{N_-+N_+}$ of the block of $l+1$, of orders $d_{N_-+1},\dots,d_{N_-+N_+}$. Here and below, $\Ud{N_-}(t_2)$ always denotes the value at~$t_2$ in the chart of the block of~$l$. The \emph{low-frequency part} and the \emph{zone part} of a field~$F$ on~$\R^2$ are given, with~$F$ a function of the labels of a chart and the Fourier transform about its centre, by \begin{equation*} \widehat{F^{\mathrm{lo}}}(\xi)\coloneqq\Hlo(\xi)\widehat F(\xi)\,,\qquad\widehat{F^{\mathrm{zone}}}(\xi)\coloneqq\Hc\Bigl(4-\frac{8\eps_m|\xi|}\pi\Bigr)\widehat F(\xi)\,; \end{equation*} the second multiplier, the \emph{zone multiplier}, is~$1$ for $\eps_m|\xi|\le3\pi/8$ and~$0$ for $\eps_m|\xi|\ge\pi/2$, so the Fourier transform of~$F^{\mathrm{zone}}$ is supported in the zone $\eps_m|\xi|_\infty\le\pi$. The zone part is taken after the change of chart. \paragraph{Negative norms.} Fix an integer $N_1\ge0$, the moment order of the output, and let~$N'$ be that of the future block: $N_1$ plus the sum of the Taylor degrees of the future insertions. For an integer $N\ge0$ and $\rho\in\{\eps_m,\rho(m)\}$, over Schwartz fields~$\vartheta$, \begin{equation*} \|F\|_{H^{-N}_{\rho,\omega,N'+2}(\R^2)}\coloneqq\sup\Bigl\{\bigl|\langle F,\vartheta\rangle_{L^2_\omega(\R^2)}\bigr|:\|\vartheta\|_{H^N_{\rho,\omega,-(N'+2)}(\R^2)}\le1\Bigr\}\,, \end{equation*} with $\langle F,\vartheta\rangle_{L^2_\omega(\R^2)}\coloneqq\sum_v\omega_v^{-1}\int_{\R^2}F_v\overline{\vartheta_v}\dd y$, and with the weight $(1+|y-y_*|^2/\rho(m)^2)^{-(N'+2)/2}$ in~\eqref{e.prep.moments}, $y_*$ the centre of the chart of~$F$, about which the moments are taken. A larger~$N$ or the scale~$\rho(m)$ in place of~$\eps_m$ gives a smaller norm. The \emph{negative order} $\Nneg\coloneqq N_1+d_1+\dots+d_{N_-}+2$ exceeds the number of derivatives that the old insertions put on a test field. \begin{figure}[t] \centering \begin{tikzpicture}[x=1.15cm,y=1cm,font=\small,box/.style={draw,rounded corners=2pt,inner sep=3pt,fill=white}] \fill[black!12] (3.05,-0.18) rectangle (3.95,0.18); \draw[-{Stealth[length=2.2mm]}] (-0.3,0) -- (12.4,0); \foreach \x in {0,3.5,11.6} \draw[thick] (\x,-0.12) -- (\x,0.12); \node[below=3pt] at (0,-0.12) {$t_1$}; \node[below=3pt] at (3.5,-0.12) {$t_2$}; \node[below=3pt] at (11.6,-0.12) {$\tpr$}; \draw[decorate,decoration={brace,amplitude=4pt,mirror}] (0,-0.75) -- (3.5,-0.75) node[midway,below=4pt] {old partial block, chart of~$y_0$}; \draw[decorate,decoration={brace,amplitude=4pt,mirror}] (3.5,-0.75) -- (11.6,-0.75) node[midway,below=4pt] {future block, chart of~$y_0'$}; \draw[black!35,densely dotted] (3.5,0.15) -- (3.5,1.1); \node[box] (old) at (3.5,1.5) {$\Ud{N_-}(t_2)$}; \node[left=1pt] at (old.west) {(a)}; \node[box] (lo) at (3.5,2.75) {low-frequency part}; \node[box] (zone) at (3.5,4.0) {zone part}; \node[box] (rest) at (7.0,2.75) {the rest}; \node[box] (model) at (11.6,4.0) {$\Pren g$}; \node[box] (err) at (11.6,1.5) {error}; \draw[-{Stealth[length=2mm]}] (old) -- (lo) node[midway,left] {(b)}; \draw[-{Stealth[length=2mm]}] (lo) -- (zone) node[midway,left] {transport, (c)}; \draw[-{Stealth[length=2mm]}] (old.east) -- (rest.south west); \draw[-{Stealth[length=2mm]}] (zone) -- (model) node[midway,above] {$\Pfut$}; \draw[-{Stealth[length=2mm]},dashed] (rest) -- (err) node[midway,above right=-2pt] {cutoff (c), then $\Pfut$}; \draw[-{Stealth[length=2mm]},dashed] (old.south) .. controls (5,0.55) and (10,0.55) .. (err.south) node[pos=0.5,below] {rest of the future block}; \end{tikzpicture} \caption{The first renewal (Lemmas~\ref{l.prep.old}--\ref{l.prep.seam}). At~$t_2$, in the quiet margin (shaded), $\Ud{N_-}(t_2)$ splits into its low-frequency part, transported to the chart of the future block and cut to its zone part, and the rest. Solid: the model~$\Pren g$; dashed: its two errors; (a)--(c): the items of Lemma~\ref{l.prep.old}; $\Pfut$: the model of the future block.} \label{f.prep.renewal} \end{figure} In Lemma~\ref{l.prep.old}, (a) bounds the old term with losses, and (b) its low-frequency part without loss but, with no Gaussian to absorb the polynomial factor of Lemma~\ref{l.prep.piece}, only in a negative norm at the scale~$\rho(m)$. By~(c) the change of chart preserves these norms, and the high frequencies, far above~$1/\rho(m)$, gain a power of $\ebar/\rho(m)$ at that scale, which absorbs the losses of~(a). \begin{lemma}[The old block, carried across]\label{l.prep.old} In the setting above (a Duhamel term across the renewal from~$g$ at~$t_1$), there exist~$C$, an integer~$N_3$ and, for every integer~$N\ge0$, a constant~$C(N)$ such that: \begin{enumerate}[label=\textup{(\alph*)}] \item \emph{In a negative norm.} For every $g\in H^{d_1+\dots+d_{N_-}-2N_-}(\R^2)^V$ with compact support, \begin{equation*} \|\Ud{N_-}(t_2)\|_{H^{-\Nneg}_{\eps_m,\omega,N'+2}(\R^2)}\le C\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^{N_3}\cfac^{N_3}\|g\|_{L^2_{\omega,N_3}(\R^2)}\,. \end{equation*} \item \emph{At low frequency,} for the same~$g$, with no loss in the contrast or the number of cell times: \begin{equation*} \bigl\|(\Ud{N_-}(t_2))^{\mathrm{lo}}\bigr\|_{H^{-N_3}_{\rho(m),\omega,N'+2}(\R^2)}\le C\|g\|_{L^2_{\omega,N_3}(\R^2)}\,. \end{equation*} \item \emph{Transport and the cutoff.} For every $F\in L^2(\R^2)^V$, every~$N$ and $\rho\in\{\eps_m,\rho(m)\}$, with~$F$ in the chart of the block of~$l$, \begin{gather*} \bigl\|F\circ X_{m-1}(lh_m,\cdot,(l+1)h_m)\bigr\|_{H^{-N}_{\rho,\omega,N'+2}(\R^2)}\le C(N)\|F\|_{H^{-N}_{\rho,\omega,N'+2}(\R^2)}\,,\\ \bigl\|F-F^{\mathrm{lo}}\bigr\|_{H^{-(N+\Nneg)}_{\rho(m),\omega,N'+2}(\R^2)}\le C(N)\Bigl(\frac{\ebar}{\rho(m)}\Bigr)^N\|F\|_{H^{-\Nneg}_{\eps_m,\omega,N'+2}(\R^2)}\,. \end{gather*} \end{enumerate} \end{lemma} \begin{proof} \emph{(a)} By~\eqref{e.prep.duality}, the pairing of $\Ud{N_-}(t_2)$ with a Schwartz field of norm one in $H^{\Nneg}_{\eps_m,\omega,-(N'+2)}(\R^2)$ is a sum of nested integrals, over a simplex of insertion times, of the pairing of~$g$ with this field run backward by the adjoint frozen operator on~$\R^2$ and acted on by the adjoint old insertions, each costing its order in derivatives and its Taylor degree in the weight. The simplex volume, the cost of the adjoint evolution in these weighted norms, and the rate from the transport of the moment weight~$w$, by $\nabla^\perp\psi_m\cdot\nabla w=-\nabla\cdot(\psi_m\nabla^\perp w)$ and the Cauchy--Schwarz inequality against the dissipation (a constant over $t_2-t_1\le h_m$, by Gr\"onwall's inequality), are \begin{equation*} \frac1{N_-!}\Bigl(\frac{\kappa_m(t_2-t_1)}{\eps_m^2}\Bigr)^{N_-},\qquad C\cfac^{C}\Bigl(1+\frac{\kappa_m(t_2-t_1)}{\eps_m^2}\Bigr)^C,\qquad\frac{C|\psi_m|^2}{\kappa_m\rho(m)^2}\le\frac C{h_m}\,, \end{equation*} the last as $|\psi_m|^2\le C\eps_m^{2\beta}$ and $\eps_m^{2\beta}/\kappa_m\le20\kappa_{m-1}$; there is no short-time singularity, as at the scale $\eps_m/(C\cfac)$ the field $\nabla^\perp\psi_m$ satisfies~(i) of Proposition~\ref{p.ft.energy}, whose energy holds with the weight on~$\R^2$, its commutator estimate being pointwise. \emph{(b)} Let $N_3$ be even, at least $N_2+N'+2$, at least the exponents of $1+\kappa_mh_m/\eps_m^2$ and of~$\cfac$ in the proof of~(a), and at least $N'+2$ plus the sum of the Taylor degrees of the old insertions, the weight used in~(a), with~$N_2$ the integer of Lemma~\ref{l.prep.piece} with $N'+2$ in place of~$N_1$. That lemma on $[t_1,t_2]$, and Propositions~\ref{p.prep.dissipation}(c) and~\ref{p.prep.branch}(d) for the scalar factor (no Gaussian, but no growth either), give \begin{equation*} \bigl|\widehat{(\Ud{N_-}(t_2))^{\mathrm{lo}}}\bigr|_{\rho(m),\omega,N'+2;\xi}\le C(1+\rho(m)|\xi|)^{N_2+N'+2}\eps_m^2\|g\|_{\rho(m),\omega,N_3;\xi}\,. \end{equation*} As~$N_3$ is even, $(1-\rho(m)^2\Delta)^{N_3/2}$ moves onto the test fields of the negative norm at the cost~$C$, and the factor $(1+\rho(m)^2|\xi|^2)^{-N_3/2}$ it leaves absorbs the power of $1+\rho(m)|\xi|$; by Plancherel, moments becoming $\xi$-derivatives, and Lemma~\ref{l.prep.bloch} summing over~$\xi$, \begin{equation*} \bigl\|(\Ud{N_-}(t_2))^{\mathrm{lo}}\bigr\|_{H^{-N_3}_{\rho(m),\omega,N'+2}(\R^2)}\le C\eps_m^2\Bigl(\int_{\text{zone}}\|g\|_{\rho(m),\omega,N_3;\xi}^2\dd\xi\Bigr)^{\nf12}\le C\|g\|_{L^2_{\omega,N_3}(\R^2)}\,. \end{equation*} \emph{(c)} The map $X_{m-1}(lh_m,\cdot,(l+1)h_m)$ preserves area, maps~$y_0'$ to~$y_0$, and has a gradient within $C\eps_{m-1}^{2\delta}$ of the identity, with derivatives at the scale~$\eps_{m-1}\ge\rho(m)$ (\eqref{e.fl.charts}, Lemma~\ref{l.fl.charts}(b)); by duality, the first bound follows from the chain rule for the inverse map, as the moment weights $1+|X_{m-1}(lh_m,y,(l+1)h_m)-y_0|^2/\rho(m)^2$ and $1+|y-y_0'|^2/\rho(m)^2$ are comparable. The Fourier transform of $F-F^{\mathrm{lo}}$ is supported in $\ebar|\xi|\ge\clo$; as $\rho(m)^2=\kappa_{m-1}h_m$, \begin{equation}\label{e.prep.cutoff} \frac{\rho(m)}{\ebar}=\Bigl(\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^{\nf12}. \end{equation} By duality it suffices that $1-\Hlo(-\mathrm i\nabla)$ maps $H^{N+\Nneg}_{\rho(m),\omega,-(N'+2)}(\R^2)$ to $H^{\Nneg}_{\rho(m),\omega,-(N'+2)}(\R^2)$, whose norm exceeds that at the scale~$\eps_m$, with norm $C(N)(\ebar/\rho(m))^N$. The kernel of $\Hlo(-\mathrm i\nabla)$ is a Schwartz function at the scale $\ebar/\clo$ with integral~$1$ and vanishing moments, as $\Hlo\equiv1$ near~$0$; so $(1-\Hlo(-\mathrm i\nabla))\vartheta$ is its integral against the Taylor remainder of order~$N$ of~$\vartheta$, of size $|\tilde y|^N|\nabla^N\vartheta|$. Peetre's inequality moves the moment weight through the shift by~$\tilde y$ at the cost $(1+|\tilde y|/\rho(m))^{N'+2}$, and Minkowski's inequality gives the coefficient $C(N)(\ebar/\rho(m))^N$. \end{proof} By Lemma~\ref{l.prep.blocks}(a), the future model maps the zone part from negative norms at the scale~$\rho(m)$ to positive ones, the Gaussian of Lemma~\ref{l.prep.words}(ii) absorbing every power of $1+\rho(m)|\xi|$; by~(b), the future block differs from this model by losses times a coefficient smaller than any power of~$\eps_{m-1}$. \begin{lemma}[The future block]\label{l.prep.blocks} In the setting above, let~$\Pfut$ be the map~$\Pblk$ of Lemma~\ref{l.prep.words} for the future block, centred at~$y_0'$, with the future insertions and~$N_1$. There exist~$C$, $c>0$, an integer~$N_3$ and, for every integer~$N\ge0$, a constant~$C(N)$ such that: \begin{enumerate}[label=\textup{(\alph*)}] \item \emph{The future model on negative norms.} For every $F\in L^2(\R^2)^V$ with finite right side, \begin{equation*} \bigl\|\Pfut(F^{\mathrm{zone}})\bigr\|_{H^N_{\rho(m),\omega,N_1}(\R^2)}\le C(N)\|F\|_{H^{-N}_{\rho(m),\omega,N'+2}(\R^2)}\,. \end{equation*} \item \emph{The rest of the future block.} Let~$F$ satisfy $\nabla^bF\in L^2_{\omega,N'}(\R^2)^V$ for $b\le N_1+d_{N_-+1}+\dots+d_{N_-+N_+}-2N_+$ and $\|F\|_{L^2_{\omega,N'+2}(\R^2)}<\infty$, and let $U_F^{[N_+]}$ be the last term of its Duhamel term over the block of $l+1$ from~$t_2$, with the future insertions. Then \begin{align*} \lefteqn{\bigl\|U_F^{[N_+]}(\tpr)-\Pfut(F^{\mathrm{zone}})\bigr\|_{H^{N_1}_{\rhomol(m),\omega,N_1}(\R^2)}} \qquad & \notag \\ & \le C\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^{N_3}\cfac^{N_3}e^{-c\kappa_m\tau_m/\eps_m^2}\|F\|_{H^{-\Nneg}_{\eps_m,\omega,N'+2}(\R^2)}\,. \end{align*} \end{enumerate} \end{lemma} \begin{proof} \emph{(a)} As the zone multiplier has a Schwartz kernel at the scale~$\eps_m$ and the right side is finite, $F^{\mathrm{zone}}$ and its derivatives have finite moments of order $N'+2$, by Peetre's inequality; so $F^{\mathrm{zone}}$ lies in the class~$\mathcal D_{N'+2}$ of the future block, and Lemma~\ref{l.prep.words}(ii) applies. At~$\xi$ in the zone its Bloch sums are constants, by~\eqref{e.prep.modes}, as $|\eps_m\xi+2\pi\mathbf n|\ge\pi$ for $\mathbf n\in\Z^2\setminus\{0\}$: \begin{equation*} \Bl\Bigl(\Bigl(\frac{y-y_0'}{\rho(m)}\Bigr)^iF^{\mathrm{zone}}\Bigr)(\xi,\cdot)=\frac{\mathrm i^{|i|}}{\eps_m^2\rho(m)^{|i|}}\nabla_\xi^i\widehat{F^{\mathrm{zone}}}(\xi)\,. \end{equation*} So Plancherel's identity and Lemma~\ref{l.prep.words}(ii), with its integer~$N_2$, the Gaussian absorbing $(1+\rho(m)|\xi|)^{4N+2N_2}$, give \begin{equation*} \bigl\|\Pfut(F^{\mathrm{zone}})\bigr\|_{H^N_{\rho(m),\omega,N_1}(\R^2)}^2\le C(N)\sum_{|i|\le N'}\int_{\R^2}\bigl(1+\rho(m)^2|\xi|^2\bigr)^{-N}\rho(m)^{-2|i|}\bigl|\nabla_\xi^i\widehat{F^{\mathrm{zone}}}(\xi)\bigr|_\omega^2\dd\xi\,. \end{equation*} The zone multiplier has derivatives bounded at the scale $\eps_m\le\rho(m)$, so~$F$ may replace~$F^{\mathrm{zone}}$ on the right; by Plancherel this is at most~$C(N)$ times the sum over $|i|\le N'$ of the squared $H^{-N}_{\rho(m),\omega,0}(\R^2)$ norms of $((y-y_0')/\rho(m))^iF$, and, as multiplication by $((y-y_0')/\rho(m))^i$ maps $H^N_{\rho(m),\omega,0}(\R^2)$ into $H^N_{\rho(m),\omega,-(N'+2)}(\R^2)$, duality gives~(a). \emph{(b)} The Duhamel terms from~$F^{\mathrm{zone}}$ and $F^{\mathrm{hi}}\coloneqq F-F^{\mathrm{zone}}$ exist, as both lie in the class of~(b) ($F^{\mathrm{zone}}$ by Peetre's inequality, as in~(a)), and by linearity $U_F^{[N_+]}-\Pfut(F^{\mathrm{zone}})$ is the sum of $U_{F^{\mathrm{zone}}}^{[N_+]}-\Pfut(F^{\mathrm{zone}})$ and~$U_{F^{\mathrm{hi}}}^{[N_+]}$. As in~(a), at the scale~$\eps_m$, $\|F^{\mathrm{zone}}\|_{L^2_{\omega,N'}(\R^2)}\le C\|F\|_{H^{-\Nneg}_{\eps_m,\omega,N'+2}(\R^2)}$, so Lemma~\ref{l.prep.words}(iii) bounds the first term at~$\tpr$. For the second we use the \emph{quiet stretch} $[t_2,t_{\mathrm q}]$, $t_{\mathrm q}\coloneqq t_2+\tau'_m-\tau_m/24$, the first part of the quiet margin of the block of $l+1$. As $\tau'_m\ge5\tau_m$, its length in cell units lies between $\kappa_m\tau'_m/(2\eps_m^2)$, which is at least~$1$ for $m$ large, and $\kappa_mh_m/\eps_m^2$. Let $U_{F^{\mathrm{hi}}}^{[j,i]}$ be the moment terms of the Duhamel term of~$F^{\mathrm{hi}}$ over the block of $l+1$, centred at~$y_0'$. \emph{The quiet stretch.} On $[t_2,t_{\mathrm q}]$ no pulse is active within $\tau_m/24$, so $\psi_m=0$ and the coefficients of the insertions do not depend on~$z$, by clause~(i) of the definition of insertions; once the weight is moved through as in the sentence after~\eqref{e.prep.moment}, they do not depend on~$y$ either. So at each frequency~$\eta$ the system~\eqref{e.prep.moment} for the Fourier transforms of the $U_{F^{\mathrm{hi}}}^{[j,i]}$ is triangular, with the common diagonal $-\kappa_m\eta\cdot\K(t)\eta$ and off-diagonal entries $\kappa_m/\eps_m^2$ times polynomials in $\eps_m\eta$ with bounded coefficients (the commutator terms carry $\eps_m/\rho(m)$ and $(\eps_m/\rho(m))^2$, and $\eps_m\le\rho(m)\le\eps_{m-1}$). As the terms with $j\ge1$ vanish at~$t_2$, \begin{equation*} \widehat{U_{F^{\mathrm{hi}}}^{[j,i]}}(t_{\mathrm q},\eta)=\exp\Bigl(-\kappa_m\int_{t_2}^{t_{\mathrm q}}\eta\cdot\K(t)\eta\dd t\Bigr)\sum_{|i'|\le N'}P_{j,i,i'}(\eps_m\eta)\widehat{U_{F^{\mathrm{hi}}}^{[0,i']}}(t_2,\eta)\,, \end{equation*} with $P_{j,i,i'}(x)$ a linear map on~$\mathbb C^V$, polynomial in the length of the stretch, whose norm on $(\mathbb C^V,|\cdot|_\omega)$ is at most $C(1+\kappa_mh_m/\eps_m^2)^C(1+|x|)^C$. By~\eqref{e.prep.momentfourier}, $\widehat{U_{F^{\mathrm{hi}}}^{[0,i']}}(t_2,\eta)=\mathrm i^{|i'|}\rho(m)^{-|i'|}\nabla_\eta^{i'}\widehat{F^{\mathrm{hi}}}(\eta)$, which vanishes for $\eps_m|\eta|\le3\pi/8$. As $\K\ge\frac{15}{16}I$, the exponential is at most $\exp(-\frac{15}{32}\kappa_m\tau'_m|\eta|^2)$: half of the exponent absorbs $(1+\eps_m|\eta|)^C$, as $\kappa_m\tau'_m/\eps_m^2\ge2$, and the other half is at most $\exp(-\kappa_m\tau'_m/(4\eps_m^2))$ where $\eps_m|\eta|\ge3\pi/8$. Each $\nabla_\eta$ that falls on the zone multiplier costs $\eps_m\le\rho(m)$, and multiplication by $((y-y_0')/\rho(m))^{i'}$, $|i'|\le N'$, maps $H^{\Nneg}_{\eps_m,\omega,0}(\R^2)$ into $H^{\Nneg}_{\eps_m,\omega,-(N'+2)}(\R^2)$, as in~(a) at the scale $\eps_m\le\rho(m)$. So \begin{equation*} \sum_{(j,i)}\int_{\R^2}\bigl(1+\eps_m^2|\eta|^2\bigr)^{-\Nneg}\bigl|\widehat{U_{F^{\mathrm{hi}}}^{[j,i]}}(t_{\mathrm q},\eta)\bigr|_\omega^2\dd\eta\le C\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^C\exp\Bigl(-\frac{\kappa_m\tau'_m}{2\eps_m^2}\Bigr)\|F\|_{H^{-\Nneg}_{\eps_m,\omega,N'+2}(\R^2)}^2\,. \end{equation*} The same formula, with up to two further $\eta$-derivatives, shows that the moment terms have at~$t_{\mathrm q}$ every derivative with two moments, so their Bloch sums are defined; the moments persist for $t\ge t_{\mathrm q}$ by the moment-weighted energy (paragraph \emph{Duhamel terms}), and the Bloch sums solve the fibre system, by the paragraph of~\eqref{e.prep.bloch}. \emph{From the plane to the fibres.} Let~$h$ be a field on~$\R^2$ with two moments and~$\xi$ in the zone. The modes of $\Bl h(\xi,\cdot)$ are given by~\eqref{e.prep.modes}, and, as $|\eps_m\xi|^2\le2\pi^2$, the weights $1+|2\pi\mathbf n|^2$ and $1+|2\pi\mathbf n+\eps_m\xi|^2$ are comparable with the constant $2+4\pi^2$. So, componentwise and then with the masses, for every integer $K\ge0$, with $\|\cdot\|_{H^{-K}(\T^2)}$ the norm dual to $H^K(\T^2)$ for $\langle\cdot,\cdot\rangle_{L^2_\omega(\T^2)}$, \begin{equation*} \int_{\text{zone}}\|\Bl h(\xi,\cdot)\|_{H^{-K}(\T^2)}^2\dd\xi\le C(K)\eps_m^{-4}\int_{\R^2}\bigl(1+\eps_m^2|\eta|^2\bigr)^{-K}\bigl|\widehat h(\eta)\bigr|_\omega^2\dd\eta\,. \end{equation*} \emph{The rest of the block.} Let $t\in[t_{\mathrm q},\tpr]$, let~$\xi$ be in the zone, and let $U^\dagger\in H^{N_{j,i}+\Nneg}(\T^2)^V$. The induction of Step~1 of the proof of Lemma~\ref{l.prep.words}(iii), started at~$t_{\mathrm q}$ with a data term for each moment term and at the test orders raised by~$\Nneg$, gives \begin{equation*} \begin{aligned} \bigl|\bigl\langle\Bl U_{F^{\mathrm{hi}}}^{[j,i]}(t),U^\dagger\bigr\rangle_{L^2_\omega(\T^2)}\bigr|&\le C\Bigl(1+\frac{\kappa_m(t-t_{\mathrm q})}{\eps_m^2}\Bigr)^C\cfac^C\|U^\dagger\|_{H^{N_{j,i}+\Nneg}(\T^2)}\\ &\qquad\times\sum_{(j',i')}\bigl\|\Bl U_{F^{\mathrm{hi}}}^{[j',i']}(t_{\mathrm q})\bigr\|_{H^{-\Nneg}(\T^2)}\,, \end{aligned} \end{equation*} the sum over the moment terms $(j',i')$ equal to or lower than $(j,i)$, with the coefficients differentiated within the derivative counts for a Duhamel term across the renewal. Step~2 of that proof, on $N_++N'+1$ equal parts of the last cell time of the block, which lies in $[t_{\mathrm q},\tpr]$, from the negative order raised by~$\Nneg$, bounds every $(N_+,i)$, $|i|\le N_1$, in $H^{N_1}(\T^2)$ at~$\tpr$ by $C(1+\kappa_mh_m/\eps_m^2)^C\cfac^C$ times the same sum. We square and integrate over the zone: the Parseval identity of Step~3 of that proof on the left side, and the bound from the plane to the fibres with $K=\Nneg$ and the bound at~$t_{\mathrm q}$ on the right side, give, the powers $\eps_m^4$ and $\eps_m^{-4}$ cancelling and as $\tau'_m\ge\tau_m$, \begin{equation*} \bigl\|U_{F^{\mathrm{hi}}}^{[N_+]}(\tpr)\bigr\|_{H^{N_1}_{\rhomol(m),\omega,N_1}(\R^2)}\le C\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^C\cfac^C\exp\Bigl(-\frac{\kappa_m\tau_m}{4\eps_m^2}\Bigr)\|F\|_{H^{-\Nneg}_{\eps_m,\omega,N'+2}(\R^2)}\,. \end{equation*} This is~(b) for the second term, and Lemma~\ref{l.prep.words}(iii) for the future block gives~(b) for the first, for~$c$ at most~$\frac14$ and the~$c$ of that lemma, and~$N_3$ at least the exponents above and its~$N_2$. \end{proof} Lemma~\ref{l.prep.seam} assembles the two lemmas along Figure~\ref{f.prep.renewal}: its model is bounded without loss, and the losses in its two errors, the future block less its model and the high frequencies of the old term, are offset by the damping of the future block and the gain of Lemma~\ref{l.prep.old}(c). \begin{lemma}[The first renewal]\label{l.prep.seam} Let $J\ge0$ be an integer, and consider a Duhamel term across the renewal from a datum $g\in H^{N_1+d_1+\dots+d_{N_-+N_+}-2(N_-+N_+)}(\R^2)^V$ supported within distance~$\rho(m)$ of~$y_0$. There exist~$C$ and a linear map~$\Pren$, independent of~$g$, such that, with the moments about~$y_0'$ on the left sides, \begin{align*} \|\Pren g\|_{H^J_{\rho(m),\omega,N_1}(\R^2)} & \le C\|g\|_{L^2_\omega(\R^2)}\,,\\ \bigl\|\Ud{N_-+N_+}(\tpr)-\Pren g\bigr\|_{H^{N_1}_{\rhomol(m),\omega,N_1}(\R^2)} & \le C\eps_{m-1}^{Q+3}\|g\|_{L^2_\omega(\R^2)}\,. \end{align*} \end{lemma} \begin{proof} Let $N_3$ be the larger of the integers of Lemmas~\ref{l.prep.old} and~\ref{l.prep.blocks}; both hold with it. The model is the solid path of Figure~\ref{f.prep.renewal}, \begin{equation*} \Pren g\coloneqq\Pfut\Bigl(\bigl((\Ud{N_-}(t_2))^{\mathrm{lo}}\circ X_{m-1}(lh_m,\cdot,(l+1)h_m)\bigr)^{\mathrm{zone}}\Bigr)\,. \end{equation*} It is linear, and $\|g\|_{L^2_{\omega,N_3}(\R^2)}\le2^{N_3/2}\|g\|_{L^2_\omega(\R^2)}$ by the support of~$g$. By Lemma~\ref{l.prep.blocks}(a) at the order $\max\{J,N_3\}$, and Lemma~\ref{l.prep.old}(c) at the scale~$\rho(m)$ and Lemma~\ref{l.prep.old}(b), $\|\Pren g\|_{H^J_{\rho(m),\omega,N_1}(\R^2)}\le C\|g\|_{L^2_\omega(\R^2)}$. \emph{The error.} $\Ud{N_-+N_+}(\tpr)$ is the last term at~$\tpr$ of the Duhamel term over the block of $l+1$ from $F\coloneqq\Ud{N_-}(t_2)\circ X_{m-1}(lh_m,\cdot,(l+1)h_m)$, which lies in the class of Lemma~\ref{l.prep.blocks}(b): as~$g$ has compact support, the weighted energy of Proposition~\ref{p.ft.plain} with the moment weights gives finite moments to $\Ud{N_-}(t_2)$ and to its derivatives of order at most $N_1+d_{N_-+1}+\dots+d_{N_-+N_+}-2N_+$, the $j$-th old insertion lowering the order by $d_j-2$, and the change of variables, area preserving with derivatives bounded at the scale $\eps_{m-1}\ge\rho(m)$, keeps the class. So $\Ud{N_-+N_+}(\tpr)-\Pren g$ is the sum of $\Ud{N_-+N_+}(\tpr)-\Pfut(F^{\mathrm{zone}})$ and of~$\Pfut$ applied to the zone part of $(\Ud{N_-}(t_2)-(\Ud{N_-}(t_2))^{\mathrm{lo}})\circ X_{m-1}(lh_m,\cdot,(l+1)h_m)$. By Lemmas~\ref{l.prep.blocks}(b), \ref{l.prep.old}(c) at the scale~$\eps_m$ and~\ref{l.prep.old}(a) for the first, and by Lemmas~\ref{l.prep.blocks}(a) with $N_{\mathrm{err}}+\Nneg$ in place of~$N$ ($N_{\mathrm{err}}\ge N_1$ is chosen below; $\rhomol(m)\le\rho(m)$), \ref{l.prep.old}(c) with~\eqref{e.prep.cutoff}, and~\ref{l.prep.old}(a) for the second, they are at most, times $\|g\|_{L^2_\omega(\R^2)}$, \begin{equation*} C\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^{2N_3}\cfac^{2N_3}e^{-\frac{c\kappa_m\tau_m}{\eps_m^2}}\,,\quad C(N_{\mathrm{err}})\Bigl(\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^{-\frac{N_{\mathrm{err}}}2}\Bigl(1+\frac{\kappa_mh_m}{\eps_m^2}\Bigr)^{N_3}\cfac^{N_3}\,. \end{equation*} \emph{The powers of~$\eps_{m-1}$.} With the bounds of Step~4 of the proof of Lemma~\ref{l.prep.words}, which with $h_m/\tau_m\ge\eps_{m-1}^{-2\delta}$ (by~\eqref{e.scales}) give $\kappa_mh_m/\eps_m^2\ge c\eps_{m-1}^{-6\delta}$, the first term is at most $C\eps_{m-1}^{Q+3}\|g\|_{L^2_\omega(\R^2)}$ for $m$ large, and the second, as $N_{\mathrm{err}}\ge2N_3$ by the choice below, at most $C(N_{\mathrm{err}})\eps_{m-1}^{6\delta(N_{\mathrm{err}}/2-N_3)-q\gamma N_3}\|g\|_{L^2_\omega(\R^2)}$; we choose~$N_{\mathrm{err}}$, after~$N_3$, with $6\delta(N_{\mathrm{err}}/2-N_3)-q\gamma N_3\ge Q+3$. \end{proof} In the proof of Proposition~\ref{p.prep.joint}, moments control the freezing: the coefficients vary on the scale~$\eps_{m-1}$, much larger than~$\rho(m)$, and the moments of a piece are multiplied by at most a constant over each block (Step~3); the models of the pieces, each localized near its centre, are almost orthogonal (Step~5). \begin{proof}[Proof of Proposition~\ref{p.prep.joint}] \emph{Step 1: The time interval.} With~$t_2$ and~$\tpr$ as in~\eqref{e.prep.tpr}, $t_1m_0$ (Proposition~\ref{p.admissible.bounds}); by its part~(b), $\|U(t)\|_{H^j_{\rhomol(m),\omega}(\T^2)}$ does not increase in~$t$. The $(t-\tc)^j$-weighted energy estimate, by induction on~$j$ with $\|U\|_{H^j_{\rhomol(m),\omega}(\T^2)}^2\le\|U\|_{L^2_\omega(\T^2)}^2+\rhomol(m)^2\|\nabla U\|_{H^{j-1}_{\rhomol(m),\omega}(\T^2)}^2$ (by~\eqref{e.HNrho}), gives for $00$ and $|\Omega_i|_\infty\le C_\Omega$, with a constant independent of~$\ell$ as $\Nmom\le2\ell-1$, at the scale~$\rhomol(m)$ with $N_1=\Nmom$, bounds $(I-S_\sigma)\nabla^{d-1}\Uexp j$ by $C(r_\sigma/\rhomol(m))^{\Nmom}\|\nabla^{d-1}\Uexp j\|_{H^{\Nmom}_{\rhomol(m)}(\T^2)}$, and the coefficients~$A_{\sigma,d}$ are summed by the matrix of Step~3 at $N=0$. By Steps~2 and~3, $\eps_m^{\delta/4}\le2\eps_{m-1}^{q\delta/4}$, the choice of~$\Ndep$, \eqref{e.prep.Npr} and Step~1, \begin{equation*} \Bigl\|U-\sum_{j=0}^{\Ndep}\Uexp j\Bigr\|_{L^\infty([t_1,\tf];L^2_\omega(\T^2))}\le C\bigl(\eps_{m-1}^{q\delta(\Ndep+1)/4}+\eps_{m-1}^{q\delta\Nmom}\bigr)\|g\|_{L^2_\omega(\T^2)}\le C\eps_{m-1}^{Q+3}\|g\|_{L^2_\omega(\T^2)}\,, \end{equation*} and the same holds with the masses $\omega_v/\Psi_v^2$, for which~(b2$'$), (b3) and~(d) also hold. \emph{Step 5: The preparation.} Proposition~\ref{p.prep.joint}, with the constants of Definition~\ref{d.admissible.invariant} (Proposition~\ref{p.admissible.bounds}) and with its path hypothesis (an admissible family has paths of at most $\Npath$ sources, item~(1) of Subsection~\ref{ss.admissible.parameters}), gives~$\tpr\in(t_1,t_1+2h_m]\subset[\tc,\tc+4h_m]$ and~$\Pt$; we put $\Pi g\coloneqq\Pt\,U(t_1)$, which is linear in~$g$. Step~4, Proposition~\ref{p.prep.joint} and Step~1, for the masses $\omega_v/\Psi_v^2$, give the first bound of~(a), \begin{equation*} \|\Pi g-U(\tpr)\|_{L^2_G(\T^2)}\le C\eps_{m-1}^{Q+3}\|g\|_{L^2_G(\T^2)}\,. \end{equation*} \emph{Step 6: The other two bounds.} The solution~$U$ does not grow in $L^2_G(\T^2)$, by Proposition~\ref{p.energy} for the masses $\omega_v/\Psi_v^2$, whose coupling sums are at most~$\frac18$ by~(d); the near-unit norm of~$\Pi$ comes from this energy, not from the model. With Step~5 and the first bound of Proposition~\ref{p.prep.joint} for these weights, \begin{align*} \|\Pi g\|_{L^2_G(\T^2)} & \le\|U(\tpr)\|_{L^2_G(\T^2)}+C\eps_{m-1}^Q\|g\|_{L^2_G(\T^2)}\le\bigl(1+C\eps_{m-1}^Q\bigr)\|g\|_{L^2_G(\T^2)}\,,\\ \|\Pi g\|_{H^{\Nout}_{\rho(m),G}(\T^2)} & \le C\|U(t_1)\|_{L^2_G(\T^2)}\le C\|g\|_{L^2_G(\T^2)}\,.\qedhere \end{align*} \end{proof} \begin{proof}[Proof of Proposition~\ref{p.prep.further}] We take~$\tpr$ and~$\Pi$ as just constructed. (i) $\tpr>t_1=\tc+\rhomol(m)^2/\kappa_m$, and the margin is that of Proposition~\ref{p.prep.joint}. (ii) A field with constant components is a solution of~$G$ at every time, so $U(t_1)=g$, and~$\Pt$ fixes it. (iii) Lemma~\ref{l.prep.buffer} with $N=2\Nrep+4\le\Nret\le\Ncoef$, as $\tpr\ge\tc+\rhomol(m)^2/\kappa_m$, and $\|g\|_{L^2_\omega(\T^2)}\le\|g\|_{L^2_G(\T^2)}$, as $\Psi_v\ge1$. (iv) We expand $\expp(-\Gamma_m(\tpr):\nabla^2)$: by Lemma~\ref{l.ft.product}(b) each application of $\Gamma_m(\tpr):\nabla^2$ costs two orders and at most $C(\Nret,p)|\Gamma_m(\tpr)|/\rho(m)^2\le C\eps_{m-1}^\delta$ in $H^{\Nret}_{\rho(m),G}(\T^2)$ (Lemma~\ref{l.cell.covariance}(b)), and $\Nret+2p\le\Nout$; the third bound of Proposition~\ref{p.cycle}(a) concludes. \end{proof} % ===== 12-comparison ===== \section{Two-scale expansion and the homogenization error} \label{s.comparison} In this section we prove Proposition~\ref{p.cycle}(b), the comparison of the solution~$U$ of~$G$ with its two-scale expansion, the lift~$Y^\sharp$ of~\eqref{e.lift}: the successor~$G'$ of Definition~\ref{d.admissible.successor} is admissible, the solution~$Y$ of~$G'$ keeps the amplitude of the datum, and the homogenization error $U-Y^\sharp$ is small. We also give the formulas for the sources of the successor, whose data Definition~\ref{d.admissible.successor} lists. Section~\ref{s.preparation} proves part~(a) and constructs the preparation time~$\tpr$ and the map~$\Pi$ used below. \paragraph{The route.} Each of the three steps of the lift~\eqref{e.lift} turns one equation into another, exactly, up to explicit error terms (Figure~\ref{f.cmp.route}). \begin{itemize} \item \emph{The correctors} (Subsection~\ref{ss.cmp.lift}). The correctors of Section~\ref{s.cell} cancel every term of degree at most~$\Ncor$ of the operator of~$G$ applied to a lifted field. There remain the averaged operator $\partial_t+\b_{m-1}\cdot\nabla-\nabla\cdot\mathcal M_m$, with the mean flux~$\mathcal M_m$ of the lift, whose part of order~$2$ without sources is~$\J_m^{\kappa_m}\nabla$ up to a defect, and a tail of mean zero in the fast variable. Each degree of the tail costs the square root $(\eps_m^2/(\kappa_mh_m))^{\nf12}$ of the ratio of the cell time to the renewal time. The correctors are switched from pulse to pulse in the quiet intervals, where they decay exponentially. \item \emph{The truncated exponential} (Subsection~\ref{ss.cmp.conjugation}). Conjugation by $\expp(\Gamma_m:\nabla^2)$ turns $\kappa_{m-1}\Delta$ into $\J_m^{\kappa_m}(t):\nabla^2$, up to a term of degree~$p$ in~$\Gamma_m$ times $\partial_t\Gamma_m=\J_m^{\kappa_m}-\kappa_{m-1}I$, and commutes with $\b_{m-1}\cdot\nabla$ up to the divergence of an explicit flux. \item \emph{The readout} (Subsection~\ref{ss.cmp.successor}). The part of~$\mathcal M_m$ beyond $\J_m^{\kappa_m}\nabla$ and the commutator, conjugated by the truncated exponential, are not small. They are kept as the sources of~$G'$: this is the construction of the successor. The readout weight~$\eps_{m-1}^{\eta}$ that each letter (Definition~\ref{d.admissible.successor}(i)) adds cancels the power~$\eps_{m-1}^{-\eta}$ of the coefficients of the new source that carries it (Lemma~\ref{l.cmp.readout}), so the letters telescope. \end{itemize} The residual of~$Y^\sharp$ in the system of~$G$ is then a forcing of size $C\eps_{m-1}^Q\|g\|_{L^2_G(\T^2)}$ (Subsection~\ref{ss.cmp.residual}), and Proposition~\ref{p.ft.energy}(a) bounds $U-Y^\sharp$ by it (Subsection~\ref{ss.cmp.proof}). \paperfigure{% \begin{figure}[!htbp] \centering \begin{tikzpicture}[x=1cm,y=1cm,font=\small, st/.style={draw,rounded corners=2pt,align=center,inner sep=3pt,minimum width=5.6cm,minimum height=0.8cm,fill=white}, ar/.style={-{Stealth[length=2.2mm]},semithick}] \node[st] (Y) at (0,0) {$Y$: the successor~$G'$, level $m-1$}; \node[st] (Z) at (0,-1.2) {$\Upsilon_mY$: level $m-1$, drift $\b_{m-1}$, $\kappa_{m-1}\Delta$}; \node[st] (W) at (0,-2.4) {$Y^\flat$: averaged, drift $\b_{m-1}$, $\J_m^{\kappa_m}(t):\nabla^2$}; \node[st] (L) at (0,-3.6) {$Y^\sharp$: the family~$G$, level~$m$}; \draw[ar] (Y) -- node[left] {$\Upsilon_m$\ } node[right] {\ Lemma~\ref{l.cmp.readout}} (Z); \draw[ar] (Z) -- node[left] {$\expp(\Gamma_m:\nabla^2)$\ } node[right] {\ Lemma~\ref{l.cmp.conjugation}} (W); \draw[ar] (W) -- node[left] {$\Corr_m$\ } node[right] {\ Lemmas~\ref{l.cmp.cancel}, \ref{l.cmp.tail}--\ref{l.cmp.coarse}} (L); \end{tikzpicture} \caption{The three steps of the lift~\eqref{e.lift}, $Y^\sharp=\Corr_m[\expp(\Gamma_m:\nabla^2)\Upsilon_mY]$, from top to bottom. Each box names a field and the equation it solves, up to error terms; $\J_m^{\kappa_m}$ is the averaged flux~\eqref{e.flux}.} \label{f.cmp.route} \end{figure}} \paragraph{Setting.} Throughout, $C$ and~$c$ denote positive constants depending only on~$\deps$. In Subsection~\ref{ss.cmp.lift}, \emph{the setting of the lift} is: $M\ge1$, $\kappa\in\Kset_M$, $2\le m\le M$, and a family~$G$ of level~$m$ that satisfies the invariant (Definition~\ref{d.admissible.invariant}) on an interval $[\tA,\tf]$, whose sources have current order at most~$\Nsrc$ (the sums of~\eqref{e.cell.system} stop at the current order), whose paths of sources have at most~$\Npath$ sources, and whose sources of birth order at least~$3$ have birth radius at most~$\eps_m$. An admissible family has these properties (item~(1) of Subsection~\ref{ss.admissible.parameters}, Definition~\ref{d.admissible.seed}(i), and Step~5 of the proof of Lemma~\ref{l.admissible.cycle}). Its constants are those of Section~\ref{s.admissible}, or other constants fixed depending only on~$\deps$ (for the terminal family of Section~\ref{s.entry}, with the terminal orders and path bound). Above a level that depends only on~$\deps$ and is at least~$\mc$, such a family has the finite lifts of Section~\ref{s.cell} at every order~$N$, with constants depending only on~$N$ and~$\deps$. Indeed, the majorant matrix is at most the priced one, so its row and column sums are at most~$1$ by~(b3). By~(b1) and the bound on the birth radii, $\|\mathcal A_{\sigma,d}\|_{\mathrm{an},v}$ is at most a constant times~$\maj(\sigma)$. Cauchy's estimate in the base label on~$\mathcal D_v$, whose supremum runs over the whole window of~$k$, and the exponential weight in~$z$ give~\eqref{e.cell.hyp} for the forms, at every $j_1,j_2$; and~\eqref{e.fl.charts} gives it for the conductivity, as the window of~$k$ lies in $|t-l_{m,k}h_m|\le h_m$. Below, Proposition~\ref{p.cell.sizes} is used at the order $N=\Nret$. Clauses (a1)--(d) are those of Definition~\ref{d.admissible.invariant}. For a field~$F$ of a family of level~$m$ we write \begin{equation}\label{e.cmp.sources} (\mathbf A_mF)_v\coloneqq\sum_{v'\in V}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}A_{\sigma,d}\nabla^{d-1}F_{v'}\,,\qquad(\mathbf A_m^SF)_v\coloneqq\sum_{v'\in V}\sum_{\sigma\colon v'\to v}\sum_{d=2}^{\Nsrc}A_{\sigma,d}\nabla^{d-1}(S_\sigma F_{v'}) \end{equation} for the flux of its sources without and with the smoothing, so that the system~\eqref{e.family.system} of the family reads $\partial_tU+\b_m\cdot\nabla U-\kappa_m\Delta U=\nabla\cdot\mathbf A_m^SU$. Subsections~\ref{ss.cmp.successor}--\ref{ss.cmp.proof} work in the setting of Proposition~\ref{p.cycle}: $m_00$ and $\nu>0$, the \emph{component energy} of~$F$ of order~$N$ at the scale~$\rho$, with the diffusivity~$\nu$, on $[t_1,t_2]$ is \begin{equation}\label{e.cmp.energy} \Bigl(\sum_{v\in V}\frac1{\omega_v}\Bigl(\|F_v\|_{L^\infty([t_1,t_2];H^N_\rho(\T^2))}^2+\nu\,\|\nabla F_v\|_{L^2((t_1,t_2);H^N_\rho(\T^2))}^2\Bigr)\Bigr)^{\nf12}, \end{equation} the square root of the left side of Proposition~\ref{p.ft.energy}(c), with the supremum in time taken in each component. In this section, \emph{the energy of~$F$ of order~$N$} is its component energy~\eqref{e.cmp.energy} of order~$N$ at the scale~$\rho(m)$, with the diffusivity~$\kappa_{m-1}$, on the interval at hand. For a forcing $(F^{\mathrm f},F^{\mathrm f,0})$ of the system~\eqref{e.ft.system} of a family of level~$m$ on $[t_1,t_2]$, its \emph{norm} is \begin{equation}\label{e.cmp.forcing} \kappa_m^{-\nf12}\|F^{\mathrm f}\|_{L^2((t_1,t_2);L^2_\omega(\T^2))}+\|F^{\mathrm f,0}\|_{L^1((t_1,t_2);L^2_\omega(\T^2))}\,. \end{equation} \paragraph{Orders.} Besides the parameters of Subsection~\ref{ss.parameters} and Sections~\ref{s.admissible} and~\ref{s.passage}, the comparison uses the number~$\Ncl$ of antidivergence steps in the reconstruction of the cell tail (Lemma~\ref{l.cmp.tail}; ``cl'' for cleanup), and fixes the moment order~$\Nmom$ for the smoothings of the sources of~$G'$ at the scale~$\rho(m)$ and of~$G$ at the scale~$\rhomol(m)$ (Step~3 of the proof of Proposition~\ref{p.cmp.residual}). They are fixed in the order of item~(1) of Subsection~\ref{ss.admissible.parameters} (Table~\ref{t.admissible.schedule}), as follows. Given~$Q$ (Section~\ref{s.passage}) and the seed's bounds on its orders and path lengths (Section~\ref{s.entry}), we choose $\Ngen$, $\Ncor$, $p$ and~$\Ncl$; then $\Nsrc$ and~$\Theta$, with $(\Nsrc-2)\Theta\le\delta/8$; then~$\Nlen$; then~$\Nmom$, such that \begin{equation}\label{e.cmp.schedule} \begin{gathered} \frac{\eta(\Ngen+1)}8,\quad \frac{\delta(\Nlen+1)}2,\quad 3\delta\Ncor,\quad \delta p,\quad \Theta\Nmom\ \ >\ Q+q\gamma+4\,,\\ (q\gamma+3\delta)\Ncl\ge2-\beta+8\delta\,; \end{gathered} \end{equation} these give $q\delta\Nmom>12\Theta\Nmom>Q+4$, since $q\delta>\delta>12\Theta$ by~\eqref{e.II8.widths}; the kernels then have vanishing moments up to~$\Nmom$ (clause~(c1)), and the third bound of Proposition~\ref{p.paired} applies with $N_1=\Nmom$. Last, we choose~$\Nret$ and set $\Nout\coloneqq\Nret+4p$. With $\Nrec\coloneqq4(\Ncor+\Nsrc)+2\Ncl+16$, the order of the reconstruction, and the path bound~$\Npath$ of Section~\ref{s.admissible}, $\Nret$ is at least the passage floor of~\eqref{e.II8.orders} and \begin{equation}\label{e.cmp.floor} \begin{aligned} \Nret&\ge\max\bigl\{\Nrec+\Nsrc+\Nmom+6p,\ \Nsrc-2+\Nmom+\Ncor+2p+1\bigr\}\,,\\ \Nret&\ge\Nrec+2\Nsrc\Npath\\ &\qquad+\max\bigl\{\Nsrc+\Nmom+\Ncor+6p+4,\ (\Npath+3)(\Ncor+\Nsrc+12)+2p\bigr\}\,. \end{aligned} \end{equation} The second floor bounds the number of derivatives in the fast variable that a cycle takes along the paths of sources, which have at most~$\Npath$ sources (proof of Lemma~\ref{l.admissible.cycle}); it depends on~$\Nlen$ through~$\Npath$. \subsection{The two-scale expansion} \label{ss.cmp.lift} For a pulse~$k$ of level~$m$, a field~$F$ of~$G$ is read in the chart of its block as~$\widetilde F_k(t,y)$, with the label $y=X_{m-1}(l_{m,k}h_m,x,t)$ and the fast variable $z=y/\eps_m$. The parts $\mathcal L_0$, $\mathcal L_1$, $\mathcal L_2$ and $\mathcal L^{\mathrm{src}}_{j'}[\sigma]$ of the operator of the system are those of~\eqref{e.cell.cellop} and~\eqref{e.cell.split}. Written without a path, $\mathcal V_j$ and~$\mathcal M_{j,d}$ are the correctors and means of the system, the sums over the paths of Proposition~\ref{p.cell.means}(f), with $\mathcal V_0=I$. The operators $\Corr_{m,k}$, $\Corr_m$ and the mean flux~$\mathcal M_m$ are those of Definition~\ref{d.cell.corr}, the part of~$\mathcal M_m$ of degree~$d$ along the paths of~$i$ sources being the sum of the means of degree~$d$ along them. \paragraph{The cost of a degree.} Since $\rho(m)^2=\kappa_{m-1}h_m$, the corrector amplitude~$\ebar$ of Subsection~\ref{ss.idea}, divided by~$\rho(m)$, is the square root of the ratio of the cell time $\eps_m^2/\kappa_m$ to the renewal time: \begin{equation}\label{e.cmp.ratios} \frac{\ebar}{\rho(m)}=\Bigl(\frac{\eps_m^2}{\kappa_mh_m}\Bigr)^{\nf12}\le C\eps_{m-1}^{3\delta}\,,\qquad \frac{\eps_m}{\rho(m)}=\Bigl(\frac{\kappa_m}{\kappa_{m-1}}\Bigr)^{\nf12}\Bigl(\frac{\eps_m^2}{\kappa_mh_m}\Bigr)^{\nf12}\le C\eps_{m-1}^{q\gamma+3\delta}\,. \end{equation} The first bound is Lemma~\ref{l.cell.corrector}(c); the second holds at every $2\le m\le M$, since $\eps_m\le2\eps_{m-1}^q$ (Lemma~\ref{l.fl.scales}(a)), $\rho(m)=(\kappa_{m-1}h_m)^{\nf12}\ge c\eps_{m-1}^{1+\gamma/2+\delta}$ (Proposition~\ref{p.scales}(c)), and $q-1-\frac\gamma2-\delta=q\gamma+3\delta$ by~\eqref{e.exponents}. The degree is the number of derivatives in~$y$, and every derivative that falls on the field is one of them, so the order never exceeds the degree. By Proposition~\ref{p.cell.sizes}, the coefficients of order~$d$ of a corrector of degree~$j$ are bounded, with their derivatives (each slow derivative costing~$\eps_{m-1}^{-1}$, each fast one a constant), by \begin{equation}\label{e.cmp.sizes} C\ebar^{\,j}\eps_{m-1}^{d-j}=C\rho(m)^d\Bigl(\frac{\eps_m^2}{\kappa_mh_m}\Bigr)^{j/2}\Bigl(\frac{\rho(m)}{\eps_{m-1}}\Bigr)^{j-d}\,, \end{equation} in row and column sums of the matrices with the entries multiplied by $(\omega_{v'}/\omega_v)^{\nf12}$, as in~\eqref{e.admissible.majorant}. The excess of the degree over the order costs $\rho(m)/\eps_{m-1}\le C\eps_{m-1}^{\gamma/2+\delta}$. The coefficients of the means of degree~$j$ satisfy the same bound times~$\kappa_m/\eps_m^2$. We also use $c\eps_m\le\rho(m)\le\eps_{m-1}$, so that $(\rho(m)/\eps_{m-1})^{j-d}\le1$. The correctors of a pulse start from zero when its window opens, and stay zero while the shear is off and the forms do not depend on the fast variable (Proposition~\ref{p.cell.means}(e)). So at a time of~$[\tA+3\tau_m,\tf]$ with no pulse of level~$m$ active within~$\tau'_m$, such as~$\tpr$ (Proposition~\ref{p.prep.further}(i)), the lift adds no corrector. \begin{lemma}[Quiet entrance]\label{l.cmp.quiet} The following holds in the setting of the lift, with~$m$ above a level depending only on~$\deps$. Let $t\in[\tA+3\tau_m,\tf]$ be such that the velocity added at level~$m$ vanishes on $[t-\tau'_m,t+\tau'_m]$. Then, for every pulse~$k$ with $\Hc_{m,k}(t)\ne0$, every corrector~$\mathcal V_j$ of~$k$ with $j\ge1$ vanishes at~$t$. In particular, for every field~$F$ of~$G$, $\Corr_mF=F$ at the time~$t$. \end{lemma} \begin{proof} Let $\Hc_{m,k}(t)\ne0$, so $|t-k\tau_m|<\frac54\tau_m$. The window of~$k$ starts at $(k-\frac{61}{48})\tau_m>t-\frac{121}{48}\tau_m$. Since $\tau'_m/\tau_m=4\lceil\eps_{m-1}^{-\delta}\rceil+1\ge5$, every $t'$ between the start of the window and~$t$ has $[t'-\frac{\tau_m}{24},t'+\frac{\tau_m}{24}]\subset[t-\tau'_m,t+\tau'_m]$, and $t'\ge t-3\tau_m\ge\tA$. So the shear of~$k$ vanishes there, and, by~(a4), the two-scale forms of~$G$ do not depend on~$z$. For $m$ large, the conductivity of the chart lies between $\frac34I$ and $\frac54I$, by~\eqref{e.fl.charts}. So, by Proposition~\ref{p.cell.means}(e), the correctors of positive degree vanish at~$t$. \end{proof} When we apply the operator of the system to the field with its correctors and sort the terms by degree, each corrector of degree $j\le\Ncor$ removes the part of mean zero in~$z$ of the terms of degree~$j$, and their means move into the averaged equation of~$F$ (Definition~\ref{d.cell.lift}). The cell tail is what is left, of degrees above~$\Ncor$, each degree carrying the small factor~$\ebar/\rho(m)$ of~\eqref{e.cmp.ratios}; in Lemma~\ref{l.cmp.tail} one of them pays for the cell rate~$\kappa_m/\eps_m^2$ that multiplies its coefficients. \begin{lemma}[The correctors cancel the leading terms]\label{l.cmp.cancel} There exists $C(\deps)<\infty$ such that the following holds in the setting of the lift, with~$m$ above a level depending only on~$\deps$. Let~$k$ be a pulse, let~$I$ be an interval in the window of~$k$, and let~$\widetilde F(t,y)$ be a field on~$I$, in the chart of~$k$, with $\widetilde F(t,\cdot)\in C^{2\Ncor+\Nsrc+1}(\R^2)$ with bounded derivatives, and with $\nabla_y^a\widetilde F$, $|a|\le\Ncor+1$, Lipschitz in~$t$, uniformly in~$y$. Then there exists a differential operator~$\mathcal T_k$ in~$y$, the \emph{cell tail} on the pulse~$k$, depending on no field, of orders~$1$ to $2\Ncor+\Nsrc$, whose coefficients have mean zero in~$z$ and whose terms have degrees $\Ncor+1$ to $2\Ncor+\Nsrc$, such that, off a null set of~$t$ independent of $(y,z)$, \begin{align} & \Bigl(\mathcal L_0+\mathcal L_1+\mathcal L_2-\sum_\sigma\sum_{j'\le\Nsrc}\mathcal L^{\mathrm{src}}_{j'}[\sigma]\Bigr)\sum_{j''=0}^{\Ncor}\mathcal V_{j''}\widetilde F \notag \\ & \qquad -\sum_{j''=0}^{\Ncor}\mathcal V_{j''}\Bigl(\partial_t\widetilde F-\sum_{j=2}^{\Ncor+\Nsrc}\sum_{d=2}^{j}\nabla_y\cdot\bigl(\mathcal M_{j,d}\nabla_y^{d-1}\widetilde F\bigr)\Bigr)=\mathcal T_k\widetilde F\,.\label{e.cmp.cancel} \end{align} Its coefficients of order~$d$ and degree~$j$ satisfy~\eqref{e.cmp.sizes} times $\kappa_m/\eps_m^2$, in row and column sums as there, each slow derivative up to the order~$\Ncl$ costing~$\eps_{m-1}^{-1}$. \end{lemma} \begin{proof} Off a null set of~$t$, the derivatives in~$t$ of the $\nabla_y^a\widetilde F$, $|a|\le\Ncor$, and of the coefficients~$\mathcal V_{j,d}$ exist at every~$(y,z)$: their difference quotients are Lipschitz in~$(y,z)$, uniformly in the increment (by the Lipschitz bound one derivative higher, and by~\eqref{e.cell.mode} with the Wiener norm of Proposition~\ref{p.cell.sizes}), and they converge on a countable dense set, at the Lebesgue points of the data of the cell equations. So the limits for the $\nabla_y^a\widetilde F$ commute with~$\nabla_y$, and the limit for~$\mathcal V_{j,d}$ is the right side of its cell equation at every~$(y,z)$, by continuity. Thus, off this null set, the product rule holds for $\mathcal V_{j,d}\nabla_y^d\widetilde F$, Proposition~\ref{p.cell.means}(b) holds at every~$(y,z)$, and the left side of~\eqref{e.cmp.cancel} is computed at the time~$t$ as an identity between differential operators applied to~$\widetilde F$. We sort its terms by degree. The cell operator~$\mathcal L_0$ contains~$\partial_t$, so on $\mathcal V_j\widetilde F$ it acts on the coefficients and gives $\mathcal V_j\partial_t\widetilde F$, which cancels. The remaining terms act on~$\widetilde F$ through its derivatives in~$y$, with coefficients built from the correctors, the means and the parts of the operator, which depend on no field. At the degree~$0$ only $\sum_\sigma\mathcal L^{\mathrm{src}}_0[\sigma]\widetilde F$ remains, and it vanishes, as~$\widetilde F$ does not depend on~$z$. The terms of degree~$j\ge1$ are then~$\mathcal L_0$ acting on the coefficients of~$\mathcal V_j\widetilde F$, plus the bracket of degree~$j$ of Definition~\ref{d.cell.lift} (for the system, by Proposition~\ref{p.cell.means}(f)), plus $\sum_{j''=0}^{j-2}\mathcal V_{j''}\sum_d\nabla_y\cdot(\mathcal M_{j-j'',d}\nabla_y^{d-1}\widetilde F)$, with $\mathcal V_{j''}=0$ for $j''>\Ncor$ and no term for $j-j''>\Ncor+\Nsrc$. For $1\le j\le\Ncor$ this is $\mathcal L_0\mathcal V_j\widetilde F$ minus~\eqref{e.cell.recursion}, which vanishes by Proposition~\ref{p.cell.means}(b). For $j>\Ncor$ no corrector is added, and the terms of degree~$j$ are the bracket plus the sum. The term $j''=0$ of the sum cancels the mean of the bracket (Proposition~\ref{p.cell.means}(a); the bracket and the means of Definition~\ref{d.cell.lift} are both formed with $\mathcal V_j=0$ for $j>\Ncor$), and the terms $j''\ge1$ have mean zero, as the correctors do. For $j>\Ncor+\Nsrc$ the bracket and the term $j''=0$ vanish, since a source has at most~$\Nsrc$ slow derivatives. So the degrees are $\Ncor+1$ to $2\Ncor+\Nsrc$, and every term has order at least~$1$, since the correctors of positive degree annihilate constants and~$\mathcal L_1$, $\mathcal L_2$, the source parts and the means differentiate. Each term of degree~$j$ is a lower corrector times one of the four products~\eqref{e.cell.factors} divided by~$\eps_m^2/\kappa_m$, or a corrector times a mean. By Proposition~\ref{p.cell.sizes}, its coefficients obey~\eqref{e.cmp.sizes} times $\kappa_m/\eps_m^2$, in row and column sums, which are submultiplicative, those of the majorant matrix being at most~$1$ by~(b3). For the slow derivatives up to the order~$\Ncl$, we use Proposition~\ref{p.cell.sizes} at an order $N\ge\Ncl+\Ncor+\Nsrc$ (such as~$\Nret$, by~\eqref{e.cmp.floor}). In a coefficient of the tail, a corrector carries at most~$\Nsrc$ slow derivatives and a mean at most~$\Ncor+1$ (one from the divergence and at most~$\Ncor$ from the corrector applied to it), so that after~$\Ncl$ further slow derivatives each carries at most~$N$. Each slow derivative of a corrector, a mean, the conductivity or a form costs~$\eps_{m-1}^{-1}$ (Proposition~\ref{p.cell.sizes}, \eqref{e.cell.hyp}). \end{proof} At one time, the correctors change a field by at most a constant times~$\ebar/\rho(m)$ times its derivatives at the scale~$\rho(m)$, and, as they oscillate at the wavelength~$\eps_m$, each derivative of the change costs~$\eps_m^{-1}$ (Lemma~\ref{l.cmp.corrector}). On a field smooth at the wavelength~$\eps_{m-1}$, this gives the cost~\eqref{e.mu} of the correctors in Step~4 of the proof of Proposition~\ref{p.passage}, where the condition $s<\frac12(1-\alpha)$ is used. \begin{lemma}[The correctors on a field]\label{l.cmp.corrector} There exists $C(\deps)<\infty$ such that, in the setting of the lift with $m_00$ such that the following holds in the setting of the lift, with~$m$ above a level depending only on~$\deps$. Let $[t_1,t_2]\subset[\tA+3\tau_m,\tf]$ with $t_2-t_1\le T$, and let~$F$ be a field of~$G$ on $[t_1,t_2]$, of class $C^{\Nrec+1}$ in~$x$ at each time, with derivatives measurable in $(t,x)$, with $\nabla_x^aF$, $|a|\le\Ncor+2$, bounded on $[t_1,t_2]\times\T^2$, and with $\nabla_x^aF$, $|a|\le\Ncor+1$, Lipschitz in~$t$, uniformly in~$x$. Then, with a forcing $(F^{\mathrm{tl}},F^{\mathrm{tl},0})$ as in Lemma~\ref{l.cmp.tail}, at almost every $t\in(t_1,t_2)$, \begin{align}\label{e.cmp.lift} & (\partial_t+\b_m\cdot\nabla-\kappa_m\Delta)\Corr_mF-\nabla\cdot(\mathbf A_m\Corr_mF)-\Corr_m\bigl(\partial_tF+\b_{m-1}\cdot\nabla F-\nabla\cdot(\mathcal M_mF)\bigr) \notag \\ & \qquad =\nabla\cdot F^{\mathrm{tl}}+F^{\mathrm{tl},0}+\sum_{k\in2\Z+1}\Hc'_{m,k}(t)\,(\Corr_{m,k}F-F)\,, \end{align} and the norm of the last sum in $L^1((t_1,t_2);L^2_\omega(\T^2))$ is at most the energy of~$F$ of order~$\Nrec$ on $[t_1,t_2]$ times \begin{equation}\label{e.cmp.switching} C\Bigl(\frac{\eps_m^2}{\kappa_m}\Bigr)^{\nf12}\Bigl(\frac{1+(t_2-t_1)/\tau_m}{\tau_m}\Bigr)^{\nf12}\exp\Bigl(-c\,\frac{\kappa_m\tau_m}{\eps_m^2}\Bigr). \end{equation} Moreover $\kappa_m\tau_m/\eps_m^2\ge c\eps_{m-1}^{-4\delta}$, so the exponential is smaller than any power of~$\eps_{m-1}$. The field~$\Corr_mF$ is Lipschitz in time in~$L^2(\T^2)$, so~\eqref{e.cmp.lift} holds in the form~\eqref{e.ft.weak}. \end{lemma} \begin{proof} \emph{Step 1: One pulse.} Where $\Hc_{m,k}>0$, the velocity added at level~$m$ is the shear of~$k$, and $\mathbf A_m$ is the transport of the coefficients of~$k$. Where $\Hc_{m,k}=1$, this holds as the other cutoffs vanish; where a neighbouring cutoff is also positive, the velocity added at level~$m$ vanishes, and by~(a6) the transports of the coefficients of the two pulses agree. By Lemma~\ref{l.fl.equation}, the chart turns $\partial_t+\b_m\cdot\nabla-\kappa_m\Delta-\nabla\cdot\mathbf A_m$ into the operator on the left of~\eqref{e.cmp.cancel} in the labels, and $\partial_t$ at fixed label into $\partial_t+\b_{m-1}\cdot\nabla$ at fixed~$x$. On the part of the window outside the block of~$l_{m,k}$, the velocity added at level~$m$ vanishes, and the proof of Lemma~\ref{l.fl.equation} applies verbatim. Lemma~\ref{l.cmp.cancel} applies to~$\widetilde F_k$ on the interval~$I$ where $[t_1,t_2]$ meets the window of~$k$: its class follows from that of~$F$, by $\partial_tX_{m-1}=\b_{m-1}(t,X_{m-1})$ and~\eqref{e.fl.charts}. So Lemma~\ref{l.cmp.cancel}, carried to~$x$, is~\eqref{e.cmp.lift} with~$\Corr_{m,k}$ in place of~$\Corr_m$, the means of the pulse~$k$ carried to~$x$ in place of~$\mathcal M_m$, the cell tail of~$k$, and without the last sum. \emph{Step 2: Gluing.} The cutoffs depend only on~$t$, $\sum_k\Hc_{m,k}=1$ and $\sum_k\Hc'_{m,k}=0$. So the operator of~$G$ without smoothing, applied to $\Corr_mF=\sum_k\Hc_{m,k}\Corr_{m,k}F$, is $\sum_k\Hc_{m,k}$ times its application to~$\Corr_{m,k}F$, plus $\sum_k\Hc'_{m,k}(\Corr_{m,k}F-F)$. By Definition~\ref{d.cell.corr}, $\mathcal M_m$ is the sum over~$k$ of~$\Hc_{m,k}$ times the means of the pulse~$k$ carried to~$x$. By Step~1, the first sum gives~\eqref{e.cmp.lift} when, in the term of~$k$, the means of~$k$ are replaced by~$\mathcal M_m$. Their difference is nonzero only where the cutoffs of~$k$ and of a neighbour~$k'$ overlap, where $\frac34\tau_m\le|t-k\tau_m|\le\frac54\tau_m$, inside the quiet intervals. There the means of each pulse are the operator of~$G$ itself, read in its chart (Proposition~\ref{p.cell.means}(d)). Their part $\kappa_m\K$ carries to the flux $\kappa_m\nabla F$ in every chart, as $\nabla_yX_{m-1}\,\K_{m,l}\,(\nabla_yX_{m-1})^{\mathsf T}=I$ by~\eqref{e.fl.metric}, and consecutive charts describe the same coefficients of the sources, by~(a6); so the replacement costs nothing. The field $\Corr_mF$ is a finite sum of products of $\Hc_{m,k}(t)$, the coefficients~$\Vc_{j,d}$, derivatives of the chart and $\nabla_y^d\widetilde F_k$, each Lipschitz in~$t$ uniformly in~$x$, which proves the last assertion of the lemma. \emph{Step 3: The switching term.} The derivative~$\Hc'_{m,k}$ is supported where $\frac34\tau_m\le|t-k\tau_m|\le\frac54\tau_m$, and $|\Hc'_{m,k}|\le C/\tau_m$. There the shear of~$k$ vanishes, since its envelope is supported where $|t-k\tau_m|<\frac23\tau_m$. On the side before~$k$, the positive correctors of~$k$ vanish, as in Lemma~\ref{l.cmp.quiet}, since the stretch from the start $(k-\frac{61}{48})\tau_m$ of the window to $(k-\frac34)\tau_m$, enlarged by~$\frac{\tau_m}{24}$, lies in the quiet interval between $k-2$ and~$k$. On the side after~$k$, for $t'\ge(k+\frac{17}{24})\tau_m$, $[t'-\frac{\tau_m}{24},t'+\frac{\tau_m}{24}]$ lies in the quiet interval $[(k+\frac23)\tau_m,(k+\frac43)\tau_m]$, on which the velocity added at level~$m$ vanishes, as~$\zeta$ has compact support in $(-\frac23,\frac23)$. So, at these times, the shear of~$k$ vanishes, the forms do not depend on~$z$, by~(a4), and neither do the conductivity and the means. As in the proof of Proposition~\ref{p.cell.means}(e), the terms of the brackets~\eqref{e.cell.recursion} with $\Vc_0[\emptyset]=I$ then do not depend on~$z$ and give nothing, and the positive correctors solve a triangular system of homogeneous cell equations, with no datum except the lower correctors, the mode~$\xi$ in $z\cdot\eperp$ of each forced only by the modes~$\xi$ of the lower ones. On every transverse Fourier mode the cell inverse decays at a rate at least $c\kappa_m/\eps_m^2$, by the ellipticity of the conductivity (Lemma~\ref{l.cell.inverse}). After the time $\frac{\tau_m}{24}$ the positive correctors are therefore bounded by $C\exp(-c\kappa_m\tau_m/\eps_m^2)$ times their sizes~\eqref{e.cmp.sizes}; the polynomial factors of the triangular system are absorbed by lowering~$c$. The supports of the~$\Hc'_{m,k}$ in $(t_1,t_2)$ are at most $C(1+(t_2-t_1)/\tau_m)$ intervals of length at most~$\tau_m$. So, by the Cauchy--Schwarz inequality in time, the norm of the last sum of~\eqref{e.cmp.lift} in $L^1((t_1,t_2);L^2_\omega(\T^2))$ is at most $C((1+(t_2-t_1)/\tau_m)/\tau_m)^{\nf12}$ times the norm of the fields~$\Corr_{m,k}F-F$ in $L^2((t_1,t_2);L^2_\omega(\T^2))$ over these supports. As in Lemma~\ref{l.cmp.tail}, for $j\ge1$, \eqref{e.cmp.sizes} times $\rho(m)^{1-d}\kappa_{m-1}^{-\nf12}$ is at most $\rho(m)\kappa_{m-1}^{-\nf12}(\eps_m^2/(\kappa_mh_m))^{\nf12}=(\eps_m^2/\kappa_m)^{\nf12}$, which gives~\eqref{e.cmp.switching}. Last, $\kappa_m\tau_m/\eps_m^2\ge c\eps_{m-1}^{-4\delta}$ by Proposition~\ref{p.scales}(c). \end{proof} In Proposition~\ref{p.cmp.residual}, the field~$Y^\flat$ of~\eqref{e.cmp.flat} solves the averaged equation up to the divergence of a small flux~$F^{\mathrm{av}}$, which the corrector operator carries into the system of~$G$. The correctors of positive degree give the second, small term of~\eqref{e.cmp.coarse}, as in Lemma~\ref{l.cmp.tail}. The first term is the cost of the identity part: the flux is bounded against the dissipation of level $m-1$, but the norm~\eqref{e.cmp.forcing} uses~$\kappa_m$. For $m0$, \begin{equation*} \|(\Gamma_m:\nabla^2)^jF\|_{H^N_\rho(\T^2)}\le\Bigl(\frac{4|\Gamma_m|}{\rho^2}\Bigr)^j\frac{(N+2j)!}{N!}\|F\|_{H^{N+2j}_\rho(\T^2)}\,. \end{equation*} In $\expp(\Gamma_m:\nabla^2)\expp(-\Gamma_m:\nabla^2)-I$ the terms of powers~$1$ to~$p$ of $\Gamma_m:\nabla^2$ cancel, since they are those of $\exp(\Gamma_m:\nabla^2)\exp(-\Gamma_m:\nabla^2)$, which is the identity, and the powers $p+1$ to~$2p$ remain. Lemma~\ref{l.cell.covariance}(b) gives $|\Gamma_m(t)|\le C\eps_{m-1}^\delta\rho(m)^2$. \end{proof} \subsection{The coupled system at the coarser level} \label{ss.cmp.successor} \paragraph{The sources of the successor.} Let~$G$ be as in Definition~\ref{d.admissible.successor}. Its successor~$G'$ has the components and weights of Definition~\ref{d.admissible.successor}(ii). Its sources carry the pieces of the operator \begin{align}\label{e.cmp.successorflux} F & \mapsto\nabla\cdot\Bigl(\expp(-\Gamma_m:\nabla^2)\bigl((\mathcal M_m-\J_m^{\kappa_m}\nabla)\expp(\Gamma_m:\nabla^2)F\bigr)\Bigr) \notag \\ & \qquad -\expp(-\Gamma_m:\nabla^2)\,[\b_{m-1}\cdot\nabla,\expp(\Gamma_m:\nabla^2)]F \end{align} on fields~$F$ of~$G$, which is in divergence form by~\eqref{e.cmp.commflux}. The pieces are the following. \begin{itemize} \item \emph{The defect letter:} the part of~$\mathcal M_m$ on the paths without sources, of all orders, less~$\J_m^{\kappa_m}\nabla$ (the defect of the averaged flux), conjugated as in~\eqref{e.cmp.successorflux}, together with the commutator term of~\eqref{e.cmp.successorflux}, whose flux is $-\expp(-\Gamma_m:\nabla^2)$ applied to the flux of the commutator at~$F$. \item \emph{Old sources:} the part of~$\mathcal M_m$ of degree~$2$ along the nonempty paths of sources (Definition~\ref{d.admissible.successor}(iii)); and the cell averages of the sources of~$G$ in their orders $d\ge3$, carried to the charts of level $m-1$, which are the part of~$\mathcal M_m$ along the paths of one source that involves no corrector. \item \emph{Mean letters:} for $d\ge3$ and $i\ge1$, the part of~$\mathcal M_m$ of degree~$d$ along the paths of~$i$ sources; for $i=1$, less the cell average of the source in the order~$d$, which the old source carries. \item \emph{Covariance letters:} for $d\ge2$, $i\ge1$, and $j,j'\le p$ with $j+j'\ge1$, the part of~$\mathcal M_m$ of degree~$d$ along the paths of~$i$ sources, times $(-\Gamma_m:\nabla^2)^j/j!$ on the left and $(\Gamma_m:\nabla^2)^{j'}/j'!$ on the right. \end{itemize} Their sum is~\eqref{e.cmp.successorflux}, since the sum over $j,j'\le p$ of a part multiplied in this way is that part multiplied by $\expp(-\Gamma_m:\nabla^2)$ on the left and by $\expp(\Gamma_m:\nabla^2)$ on the right. By item~(1) of Subsection~\ref{ss.admissible.parameters}, the sources of~$G$ have current orders at most $\Nsrc-2\Ncor-4p-4$, so by~\eqref{e.cell.means} the means along their paths vanish above the degree $\Nsrc-\Ncor-4p-4$; as these paths have at most~$\Npath$ sources, every piece above has its letter in Definition~\ref{d.admissible.successor}(i). They are carried by the sources of Definition~\ref{d.admissible.successor}(iii)--(iv), with its words, smoothings and the power~$\eps_{m-1}^{-\eta}$ of the coefficients of the new sources; the components of generation above~$\Ngen$ or word length above~$\Nlen$ are omitted, with the sources into them. In the charts of level $m-1$ these fluxes are the transports~\eqref{p.admissible.transport} of their two-scale forms. So the successor exists for every such~$G$: its sources are explicit in the correctors and the means of Section~\ref{s.cell}, and no further cell problem is solved. \paragraph{The successor at the scale~$\rho(m)$.} In the setting of Proposition~\ref{p.cycle}, for~$m$ above a level that depends only on~$\deps$, $\rho(m)\le\rhomol(m-1)$, as $\rho(m)$ is below $\cmol$ times each entry of the minimum~\eqref{e.admissible.rhomol}: \[ \rho(m)\le C\eps_{m-1}^{1+\gamma/2+\delta}\le\cmol\eps_{m-1}\,,\qquad\rho(m)^2=\kappa_{m-1}h_m\le2^{-25}\eps_{m-1}^{2\delta}\kappa_{m-1}\eps_{m-1}^{2-\beta}\le\cmol^2\kappa_{m-1}\eps_{m-1}^{2-\beta}\,, \] by Proposition~\ref{p.scales}(c) and~\eqref{e.renewal}, as~$\cmol$ is fixed before this level (Section~\ref{s.preparation}). So Proposition~\ref{p.ft.energy} applies to~$G'$ on $[\tpr,\tf]$, at every order $N\le\Nret$ and the scale~$\rho(m)$, with the masses $\omega_v/\Psi_v^2$ (the marked norm) and with the masses~$\omega_v$: its hypothesis~(iii) holds by clause~(d) of~$G'$ ($\Nret\le\Ncoef$), since the coupling matrix~\eqref{e.ft.coupling} increases with the scale, and~(i) by~\eqref{e.prep.cmol} at the level $m-1$; hypothesis~(ii) holds by clause~(b1), whose analytic norm runs over every time of the window, through Cauchy's estimate on the polydiscs of Step~6 of Subsection~\ref{ss.admissible.cycle}, the chains being smooth (Lemma~\ref{l.admissible.chains}). The datum of~$Y$ is the retained trace, with the weights of~$G$ on the components of~$G$ (Lemma~\ref{l.admissible.successors}(i)), so Proposition~\ref{p.prep.further}(iv) gives $\|Y(\tpr)\|_{H^{\Nret}_{\rho(m),G'}(\T^2)}\le C\|g\|_{L^2_G(\T^2)}$. Lemma~\ref{l.cmp.frontier} and Proposition~\ref{p.cmp.energy} use these facts, as~$m_0$ is at least this level. Lemma~\ref{l.cmp.readout} checks the construction: summed with the readout weights, the equations of the components of~$G'$ give the equation of level $m-1$ for~$\Upsilon_mY$, forced by~\eqref{e.cmp.successorflux}, up to the fluxes of the smoothings and of the omitted components. \begin{lemma}[The successor read out]\label{l.cmp.readout} In the setting of Proposition~\ref{p.cycle}, let~$Y$ be a solution of the system of~$G'$ on $[\tpr,\tf]$ in $C([\tpr,\tf];H^{\Nret+3}(\T^2)^{V'})$. Then \begin{align}\label{e.cmp.readout} (\partial_t+\b_{m-1}\cdot\nabla-\kappa_{m-1}\Delta)\Upsilon_mY & =\nabla\cdot\Bigl(\expp(-\Gamma_m:\nabla^2)\bigl((\mathcal M_m-\J_m^{\kappa_m}\nabla)\expp(\Gamma_m:\nabla^2)\Upsilon_mY\bigr)\Bigr) \notag \\ & \qquad -\expp(-\Gamma_m:\nabla^2)\,[\b_{m-1}\cdot\nabla,\expp(\Gamma_m:\nabla^2)]\Upsilon_mY \notag \\ & \qquad -\nabla\cdot(\Fsm+\Ffr) \end{align} in the weak sense of~\eqref{e.ft.weak}. Here~$\Fsm$ is the \emph{smoothing defect}: the sum, over the pieces of~\eqref{e.cmp.successorflux}, of the flux of each piece acting on $(I-S_\sigma)\Upsilon_mY$, with~$S_\sigma$ the smoothing of the source of the construction that carries it, before the omission. And $\Ffr$ is the \emph{frontier flux}: the readout of the flux of the construction into the omitted components, from the retained ones. \end{lemma} \begin{proof} By the readout formula, the readout weight of a component~$v'$ over a component~$v$ of~$G$ is~$\eps_{m-1}^{\eta\#v'}$ ($\#v'=0$ for $v'=v$). We multiply the equation of each component over~$v$ by its readout weight and sum. Old sources act between components with the same word, with the same coefficient and smoothing, so they sum to the old pieces acting on $S_\sigma\Upsilon_mY$. A letter source (a new source of Definition~\ref{d.admissible.successor}(iv)), with the same coefficients and smoothing for every word of its origin, runs from a component over a component~$v''$ of~$G$ to the component~$v'$ over~$v$ whose word is that of the first followed by the letter, so the word of its origin has length $\#v'-1$. Its coefficients carry~$\eps_{m-1}^{-\eta}$, so \begin{equation*} \eps_{m-1}^{\eta\#v'}\,\eps_{m-1}^{-\eta}=\eps_{m-1}^{\eta(\#v'-1)}\,, \end{equation*} the readout weight of its origin, and the letter sources sum to the letter pieces acting on $S_\sigma\Upsilon_mY$, as long as their targets are retained. The sources into omitted components are missing from the sum, and their readout is~$\Ffr$. Writing $S_\sigma\Upsilon_mY=\Upsilon_mY-(I-S_\sigma)\Upsilon_mY$ and summing the pieces into~\eqref{e.cmp.successorflux} gives~\eqref{e.cmp.readout}. As $Y\in C([\tpr,\tf];H^{\Nret+3}(\T^2)^{V'})$, the identities~\eqref{e.ft.weak} of the components, combined with the readout weights, give~\eqref{e.cmp.readout} in that sense. \end{proof} Lemma~\ref{l.cmp.frontier} bounds the flux into the omitted components, in the norm in which Lemma~\ref{l.cmp.coarse} reads a coarse flux. Those of generation $\Ngen+1$ have markers at least~$\eps_{m-1}^{-\eta(\Ngen+1)/8}$, and the solution is bounded in the marked norm: the first term of~\eqref{e.cmp.frontier}. Those with words of length $\Nlen+1$ are reached through $\Nlen+1$ letters, each gaining~$\eps_{m-1}^{\delta/2}$ in the coupling: the second. \begin{lemma}[The omitted components]\label{l.cmp.frontier} There exists $C(\deps)<\infty$ such that, in the setting of Proposition~\ref{p.cycle}, for every datum~$g$ and every integer $0\le N\le\Nret$, \begin{equation}\label{e.cmp.frontier} \kappa_{m-1}^{-\nf12}\|\Ffr\|_{L^2((\tpr,\tf);H^N_{\rho(m),\omega}(\T^2))} \le C\bigl(\eps_{m-1}^{\eta(\Ngen+1)/8}+\eps_{m-1}^{\delta(\Nlen+1)/2}\bigr)\|g\|_{L^2_G(\T^2)}\,. \end{equation} \end{lemma} \begin{proof} The datum of~$Y$ satisfies $\|Y(\tpr)\|_{H^N_{\rho(m),G'}(\T^2)}\le C\|g\|_{L^2_G(\T^2)}$ for $0\le N\le\Nret$, by the paragraph on the successor at the scale~$\rho(m)$, as the norm increases with~$N$. \emph{Omitted generation.} A child~$v'$ of generation $\Ngen+1$, an omitted component, was created by $\Ngen+1$ cycles, at levels $m'\ge m$. By~\eqref{e.admissible.weights}, applied before the omission, its marker is that of its root, at least~$1$, times $\eps_{m'-1}^{-\eta/8}\ge\eps_{m-1}^{-\eta/8}$ for each of these cycles, so at least $\eps_{m-1}^{-\eta(\Ngen+1)/8}$. The sources of the construction into~$v'$, before the omission, have row and column sums at most~$C$, at the order~$N$ and the scale~$\rho(m)$, in the coupling matrix for the masses $\omega_v/\Psi_v^2$. Indeed, Steps~3 and~8 of the proof of Lemma~\ref{l.admissible.cycle} use Definition~\ref{d.admissible.successor} only through the masses and markers of the copies and the bound of a new source by a fixed number times the product along its path, which hold before the omission. So do Steps~4 and~5, as a child of generation $\Ngen+1$ has reserve $1/(\Ngen+2+\Noff)>0$; so~(b1) holds for the sources into it, and Step~8 reads their derivatives by Cauchy's estimate on the margin of this reserve. Lemma~\ref{l.ft.source}, with $\vartheta=C$, bounds their flux, measured with the mass and the marker of~$v'$, by $C\kappa_{m-1}$ times the norm of~$\nabla Y$ in $L^2((\tpr,\tf);H^N_{\rho(m),G'}(\T^2))$. Dividing by the marker of~$v'$ gives the first term of~\eqref{e.cmp.frontier}; the readout weight and the ratio of the masses of~$v'$ and of the component of~$G$ whose child it is cost at most~$C$. Proposition~\ref{p.energy} for~$G'$ with the masses $\omega_v/\Psi_v^2$ bounds the norm of~$\nabla Y$ in the marked norm by $\kappa_{m-1}^{-\nf12}\|Y(\tpr)\|_{L^2_{G'}(\T^2)}$, and at order~$N$ Proposition~\ref{p.ft.energy}(b) does so, by the paragraph on the successor at the scale~$\rho(m)$. \emph{Omitted length.} The components of~$G'$ whose words have length~$j$ form a family, the \emph{layer of length~$j$}, whose coupling matrix is a principal submatrix of that of~$G'$, with sums at most~$\frac18$; the sources within a word are old sources, and those from the layer of length $j-1$ letter sources. For $j\ge1$ its datum is zero, and its forcing is the flux of the letter sources from the layer of length $j-1$. By Steps~3 and~8 of the proof of Lemma~\ref{l.admissible.cycle}, for each letter the entries of the coupling matrix of these sources into a component, or out of one, add up to at most $C\eps_{m-1}^{\delta-11\eta/8-(\Nsrc-2)\Theta}\le C\eps_{m-1}^{45\delta/64}\le C\eps_{m-1}^{\delta/2}$ (as $\eta\le\delta/8$ by~\eqref{e.exponents}, and $(\Nsrc-2)\Theta\le\delta/8$), the factor~$\eps_{m-1}^{-\eta}$ and the mass and marker ratios included. So Proposition~\ref{p.ft.energy}(a), with Lemma~\ref{l.ft.source} at $\vartheta=C\eps_{m-1}^{\delta/2}$, bounds the dissipation $\kappa_{m-1}^{\nf12}\|\nabla Y\|_{L^2((\tpr,\tf);H^N_{\rho(m),G'}(\T^2))}$ of the components of~$Y$ in the layer of length $j\ge1$ by $C\eps_{m-1}^{\delta/2}$ times that of the layer of length $j-1$, with the marked norms~\eqref{e.norms}. One more letter, from the layer of length~$\Nlen$, bounds the flux into the omitted layer of length $\Nlen+1$: in all, $\Nlen+1$ letters. The readout weight costs at most~$C$. \end{proof} \subsection{Bounds on the residual of the two-scale expansion} \label{ss.cmp.residual} Lemmas~\ref{l.cmp.tail}--\ref{l.cmp.coarse} bound the residual of the lift by energies of the fields they act on; Proposition~\ref{p.cmp.energy}, with Lemma~\ref{l.cmp.conjugation}(iii), bounds these and the energy of the lift. By item~(i) of Proposition~\ref{p.cmp.energy}, the solution of the successor keeps the $\Nret$ derivatives at the scale~$\rho(m)$ of its datum. By item~(ii), the lift has derivatives at the scale~$\rhomol(m)$, with the diffusivity~$\kappa_m$: a fast derivative of a corrector costs~$\eps_m^{-1}$, against~$\kappa_m^{\nf12}$ in the dissipation, and one degree offsets it. \begin{proposition}[Energy of the successor and of the lift]\label{p.cmp.energy} There exists $C(\deps)<\infty$ such that, in the setting of Proposition~\ref{p.cycle}, for every datum~$g$ of~$G$ the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item The solution~$Y$ lies in $C([\tpr,\tf];H^{\Nret+3}(\T^2)^{V'})$, and \begin{equation*} \|Y\|_{L^\infty([\tpr,\tf];H^{\Nret}_{\rho(m),G'}(\T^2))}\le C\|g\|_{L^2_G(\T^2)}\,; \end{equation*} and the energy of~$\Upsilon_mY$ of order~$\Nret$ on $[\tpr,\tf]$ is at most $C\|g\|_{L^2_G(\T^2)}$. \item For every integer $0\le N\le\Nret-\Ncor-2p$, the component energy of~$Y^\sharp$ of order~$N$ at the scale~$\rhomol(m)$, with the diffusivity~$\kappa_m$, on $[\tpr,\tf]$, is at most $C\|g\|_{L^2_G(\T^2)}$. In particular, for every $t\in[\tpr,\tf]$, $\|Y^\sharp(t)\|_{H^{2\Nrep+4}_{\rhomol(m),\omega}(\T^2)}\le C\|g\|_{L^2_G(\T^2)}$. \end{enumerate} \end{proposition} \begin{proof} (i) By the paragraph on the successor at the scale~$\rho(m)$, Proposition~\ref{p.ft.energy}(b) with the marked masses gives the bound on~$Y$ and the continuity in $H^{\Nret}(\T^2)^{V'}$, and~(c) with the masses~$\omega_v$ gives the component energy of~$Y$, since the markers are at least~$1$. By Lemma~\ref{l.admissible.successors}(iii), the readout has norm at most $1+\Nword^{\nf12}\eps_{m-1}^{5\eta/4}\le2$ in the coordinates scaled by~$\omega_v^{-\nf12}$, with nonnegative entries; applied to the component norms, it bounds the component energy of~$\Upsilon_mY$ by twice that of~$Y$. The solution~$Y$ lies in $C([\tpr,\tf];H^{\Nret+3}(\T^2)^{V'})$, by Proposition~\ref{p.ft.energy} at the order $\Nret+3$ and a scale $\rho'\le\rho(m)$ small enough, and the uniqueness at the order~$0$. Indeed, the datum lies in $H^{\Nout-2p}(\T^2)^{V'}$, with $\Nout-2p=\Nret+2p\ge\Nret+3$ ($p\ge2$ by~\eqref{e.cmp.schedule}); hypothesis~(i) holds at~$\rho'$ as at~$\rho(m)$; (ii) holds as the coefficients of~$G'$ have bounded derivatives of every order, by Cauchy's estimate on its forms at positive reserve (Steps~4 and~6 of Subsection~\ref{ss.admissible.cycle}) and Lemma~\ref{l.admissible.chains}; and (iii) holds as, when $\rho'\to0$, the coupling sums of order $\Nret+3$ tend to those at the order~$0$, below~$\frac18$. (ii) By Lemma~\ref{l.cmp.conjugation}(iii), the energy of~$Y^\flat$ of order $N+\Ncor$ is at most~$C$ times that of~$\Upsilon_mY$ of order $N+\Ncor+2p$. In $Y^\sharp=\Corr_mY^\flat$, we consider the part of a corrector of degree $j\ge1$ on the paths without sources, acting on derivatives of order~$d$ of~$Y^\flat$, with $1\le d\le j$ (the order~$0$ vanishes, as these correctors annihilate constants). In the dissipation, the costly derivative is the fast one, $\eps_m^{-1}$, against $\kappa_m^{\nf12}$. Its contribution to $\kappa_m^{\nf12}\|\nabla Y^\sharp\|_{L^2((\tpr,\tf);H^N_{\rhomol(m),\omega}(\T^2))}$ is at most \begin{equation*} C\kappa_m^{\nf12}\eps_m^{-1}\rho(m)\kappa_{m-1}^{-\nf12}\Bigl(\frac{\eps_m^2}{\kappa_mh_m}\Bigr)^{\frac j2}\Bigl(\frac{\rho(m)}{\eps_{m-1}}\Bigr)^{j-d}=C\Bigl(\frac{\eps_m^2}{\kappa_mh_m}\Bigr)^{\frac{j-1}2}\Bigl(\frac{\rho(m)}{\eps_{m-1}}\Bigr)^{j-d}\le C \end{equation*} times the energy of~$Y^\flat$, by~\eqref{e.cmp.sizes} and the exact relation $\kappa_m^{\nf12}\eps_m^{-1}\rho(m)\kappa_{m-1}^{-\nf12}=\rho(m)/\ebar=(\kappa_mh_m/\eps_m^2)^{\nf12}$ of~\eqref{e.cmp.ratios}. A derivative that falls on~$\nabla^dY^\flat$ in place of the fast variable costs instead $(\kappa_m/\kappa_{m-1})^{\nf12}(\eps_m^2/(\kappa_mh_m))^{j/2}(\rho(m)/\eps_{m-1})^{j-d}\le C$ times the energy; the transport of the coefficients to~$x$ by~\eqref{e.fl.charts} lowers the order~$d$ at the cost~$\eps_{m-1}^{-1}$ per order, which keeps the form of~\eqref{e.cmp.sizes}; and $\rhomol(m)\le\cmol\eps_m\le C\rho(m)$. The supremum in time is at most $C(\eps_m^2/(\kappa_mh_m))^{j/2}$ times the energy. Derivatives at the scale~$\rhomol(m)$ cost at most~$C$, and the identity part of~$\Corr_m$ costs $(\kappa_m/\kappa_{m-1})^{\nf12}\le1$ in the dissipation. The paths with sources are bounded in the same way, the sums over the paths by~(b3). With $2\Nrep+4\le\Nret-\Ncor-2p$ by~\eqref{e.II8.orders}, this gives the last bound. \end{proof} In Proposition~\ref{p.cmp.residual}, the lift starts from the preparation, up to the two truncated exponentials~\eqref{e.cmp.initial}, since it adds no corrector at the preparation time (Lemma~\ref{l.cmp.quiet}). Its residual has four terms: the coarse flux (Lemma~\ref{l.cmp.coarse}), the cell tail, the switching term, and the fine smoothing defect (the error of the smoothings of~$G$); Table~\ref{t.cmp.bill} lists their bounds and the origin of their smallness. \begin{proposition}[The residual of the lift]\label{p.cmp.residual} There exists $C(\deps)<\infty$ such that, in the setting of Proposition~\ref{p.cycle}, for every datum~$g$ of~$G$, the lift~$Y^\sharp$ solves the system~\eqref{e.ft.system} of~$G$ on $[\tpr,\tf]$ from \begin{equation}\label{e.cmp.initial} Y^\sharp(\tpr)=\expp(\Gamma_m(\tpr):\nabla^2)\expp(-\Gamma_m(\tpr):\nabla^2)\,\Pi g\,, \end{equation} with a forcing $(F,F^0)$ such that \begin{equation}\label{e.cmp.residual} \kappa_m^{-\nf12}\|F\|_{L^2((\tpr,\tf);L^2_\omega(\T^2))}+\|F^0\|_{L^1((\tpr,\tf);L^2_\omega(\T^2))}\le C\eps_{m-1}^Q\|g\|_{L^2_G(\T^2)}\,. \end{equation} \end{proposition} \begin{proof} The top level~$m_0$ is at least the levels of the lemmas above, and so large that the finitely many inequalities below between powers of~$\eps_{m-1}$ hold for $m>m_0$. \emph{Step 1: The initial value.} The velocity added at level~$m$ vanishes on $[\tpr-\tau'_m,\tpr+\tau'_m]$ (Proposition~\ref{p.prep.further}(i)), and $\tpr\ge\tc\ge\tA+3\tau_m$, so Lemma~\ref{l.cmp.quiet} gives $Y^\sharp(\tpr)=Y^\flat(\tpr)$. As~$Y(\tpr)$ vanishes on the new components, $\Upsilon_mY(\tpr)=\expp(-\Gamma_m(\tpr):\nabla^2)\Pi g$, and~\eqref{e.cmp.initial} follows. \emph{Step 2: The exact identity.} By Proposition~\ref{p.cmp.energy}(i) and~\eqref{e.ft.weak}, $\Upsilon_mY$ is Lipschitz in~$t$ with values in $H^{\Nret+3-\Nsrc}(\T^2)$: the drift costs one order, $\kappa_{m-1}\Delta$ two, a source at most~$\Nsrc$, and the coefficients of~$G'$ have bounded derivatives of every order. As~$\Gamma_m$ is Lipschitz in~$t$, $Y^\flat$ of~\eqref{e.cmp.flat} is Lipschitz in~$t$ with values in $H^{\Nret+3-\Nsrc-2p}(\T^2)$ and bounded in $H^{\Nret+3-2p}(\T^2)$, and by~\eqref{e.cmp.floor} and Sobolev's embedding these fields have the classes of Lemmas~\ref{l.cmp.conjugation}(ii) and~\ref{l.cmp.switching}. So~\eqref{e.cmp.readout} holds at almost every~$t$, and, with Lemma~\ref{l.cmp.conjugation}(ii), it gives the averaged equation of~$Y^\flat$, with the \emph{flux of the averaged equation}~$F^{\mathrm{av}}$: \begin{equation}\label{e.cmp.coarseflux} \begin{aligned} &\partial_tY^\flat+\b_{m-1}\cdot\nabla Y^\flat-\nabla\cdot(\mathcal M_mY^\flat)=\nabla\cdot F^{\mathrm{av}}\,,\qquad\text{where}\\ &F^{\mathrm{av}}\coloneqq\bigl(\expp(\Gamma_m:\nabla^2)\expp(-\Gamma_m:\nabla^2)-I\bigr)\bigl((\mathcal M_m-\J_m^{\kappa_m}\nabla)Y^\flat-\Fcom\bigr)\\ &\qquad\qquad-\expp(\Gamma_m:\nabla^2)(\Fsm+\Ffr)-(\J_m^{\kappa_m}-\kappa_{m-1}I)\nabla\frac{(\Gamma_m:\nabla^2)^p}{p!}\Upsilon_mY\,. \end{aligned} \end{equation} Here~$\Fcom$ is the flux of the commutator at~$\Upsilon_mY$; the first line of~$F^{\mathrm{av}}$ is the \emph{truncation part}, and its last term the \emph{Taylor part}. Indeed, $[\b_{m-1}\cdot\nabla,\expp(\Gamma_m:\nabla^2)]\Upsilon_mY=\nabla\cdot\Fcom$ by~\eqref{e.cmp.commflux}, $\J_m^{\kappa_m}:\nabla^2=\nabla\cdot(\J_m^{\kappa_m}\nabla)$, and $\expp(\pm\Gamma_m:\nabla^2)$ commute with~$\nabla\cdot$. Lemma~\ref{l.cmp.switching}, applied to the field~$Y^\flat$, then gives, with $\mathbf A_m=\mathbf A_m^S+(\mathbf A_m-\mathbf A_m^S)$, \begin{align*} \lefteqn{(\partial_t+\b_m\cdot\nabla-\kappa_m\Delta)Y^\sharp-\nabla\cdot(\mathbf A_m^SY^\sharp)} \qquad & \notag \\ & =\Corr_m(\nabla\cdot F^{\mathrm{av}})+\nabla\cdot F^{\mathrm{tl}}+F^{\mathrm{tl},0}+\sum_k\Hc'_{m,k}(\Corr_{m,k}Y^\flat-Y^\flat)+\nabla\cdot F^{\mathrm{fine}}\,, \end{align*} with the \emph{fine smoothing defect} $F^{\mathrm{fine}}\coloneqq(\mathbf A_m-\mathbf A_m^S)Y^\sharp$, the sources of~$G$ acting on $(I-S_\sigma)Y^\sharp$. The forcing $(F,F^0)$ of~\eqref{e.cmp.residual} collects the four terms on the right side, the coarse flux, the cell tail, the switching term and the fine smoothing defect: with the forcing $(F^{\mathrm{co}},F^{\mathrm{co},0})$ of Lemma~\ref{l.cmp.coarse} for~$F^{\mathrm{av}}$, it is $F=F^{\mathrm{co}}+F^{\mathrm{tl}}+F^{\mathrm{fine}}$ and $F^0=F^{\mathrm{co},0}+F^{\mathrm{tl},0}+\sum_k\Hc'_{m,k}(\Corr_{m,k}Y^\flat-Y^\flat)$. And~$Y^\sharp$ lies in the class of solutions: by Lemma~\ref{l.cmp.switching} it is Lipschitz with values in $L^2(\T^2)^V$, the display holds in the form~\eqref{e.ft.weak}, and Proposition~\ref{p.cmp.energy}(ii) gives the rest. \emph{Step 3: The four terms.} By Proposition~\ref{p.cmp.energy}(i), the energy of~$\Upsilon_mY$ of order~$\Nret$ is at most $C\|g\|_{L^2_G(\T^2)}$, and so, by Lemma~\ref{l.cmp.conjugation}(iii), is that of~$Y^\flat$ of order $\Nret-2p$. The truncation part reads the energy of~$\Upsilon_mY$ of order at most $\Nrec+6p+\Nsrc-2$, and the smoothing defect that of order at most $\Nrec+2p+\Nsrc+\Nmom-2$, both at most~$\Nret$ by the first line of~\eqref{e.cmp.floor}. The fine smoothing defect reads the component energy of~$Y^\sharp$ of Proposition~\ref{p.cmp.energy}(ii) at the order $\Nmom+\Nsrc-2\le\Nret-\Ncor-2p$, by the same line. Moreover, at each time~$Y^\flat$ and~$F^{\mathrm{av}}$ have $\Nrec+3$ derivatives in~$L^2$, by Proposition~\ref{p.cmp.energy}(i) and~\eqref{e.cmp.floor}, hence $\Nrec+1$ continuous ones (measurable in $(t,x)$, being continuous in~$x$ and measurable in~$t$, the smoothings by clause~(c3) and Lemma~\ref{l.ft.moving}), as Lemmas~\ref{l.cmp.tail}--\ref{l.cmp.coarse} require. Table~\ref{t.cmp.bill} lists the terms. \begin{itemize} \item \emph{The coarse flux}, by Lemma~\ref{l.cmp.coarse}: its cost is $C(\kappa_{m-1}/\kappa_m)^{\nf12}\le C\eps_{m-1}^{-q\gamma}$, since its second term is at most~$C$, by~\eqref{e.cmp.ratios} and the condition on~$\Ncl$ in~\eqref{e.cmp.schedule} (see the cell tail below). Then $\kappa_{m-1}^{-\nf12}\|F^{\mathrm{av}}\|_{L^2((\tpr,\tf);H^{\Nrec}_{\rho(m),\omega}(\T^2))}$ has four parts. The \emph{truncation part} is at most $C\eps_{m-1}^{\delta(p+1)}\|g\|_{L^2_G(\T^2)}$, by~\eqref{e.cmp.truncation}: the coefficients of $\mathcal M_m-\J_m^{\kappa_m}\nabla$ and of the flux of the commutator are at most $C\kappa_{m-1}$ at the scale~$\rho(m)$, by Proposition~\ref{p.cell.sizes} and, for the flux of the commutator, by \[ \rho(m)^l\|\nabla^{l+1}\b_{m-1}\|_{L^\infty(\R\times\T^2)}\,|\Gamma_m(t)|\le C(l)\,\kappa_{m-1}\eps_{m-1}^\delta\,,\qquad l\ge0\,, \] which holds by~\eqref{e.cell.covariance}, with \eqref{e.fl.hessian} and~\eqref{e.renewal} at $l=0$, and with the first line of~\eqref{e.prep.cmol} at the level $m-1$ and $\rho(m)\le\rhomol(m-1)$ at $l\ge1$. The \emph{Taylor part} is at most $C\eps_{m-1}^{\delta p}\|g\|_{L^2_G(\T^2)}$, since $|\J_m^{\kappa_m}(t)|\le\kappa_m+\eps_m^{2\beta}/(2\kappa_m)\le C\kappa_{m-1}$. The \emph{smoothing defect}~$\Fsm$ appears under the truncated exponential, so by the second bound of~\eqref{e.cmp.truncation} we bound it in $H^{\Nrec+2p}_{\rho(m),\omega}(\T^2)$. It is at most $C\eps_{m-1}^{\Theta\Nmom}\|g\|_{L^2_G(\T^2)}$: by the third bound of Proposition~\ref{p.paired} at each time (which needs no transversality and has a constant independent of~$\ell$), with $N_1=\Nmom$ and the bound~$C_\Omega$ of~(c1), each piece acting on $(I-S_\sigma)\Upsilon_mY$ costs $(r_\sigma/\rho(m))^{\Nmom}$, with $r_\sigma\le C\rho(m)\eps_{m-1}^\Theta$ by~(c2) for~$G$, whose old sources~$G'$ copies, by~\eqref{e.II8.width} for the letters, and $4\delta>\Theta$ (by~\eqref{e.II8.widths}). The coefficients are bounded by clauses (b1)--(b3) of the admissible family~$G'$; a letter piece into a component of~$G$ of generation~$\Ngen$ has no source in~$G'$, and its coefficients obey the same bounds before the omission, as in the proof of Lemma~\ref{l.cmp.frontier}. The \emph{frontier part} is Lemma~\ref{l.cmp.frontier} at the order $\Nrec+2p$, under the truncated exponential as for~$\Fsm$. \item \emph{The cell tail}, by Lemma~\ref{l.cmp.tail}: at most $C\eps_{m-1}^{3\delta\Ncor}$ by~\eqref{e.cmp.ratios}. Its term in parentheses is bounded: \begin{equation*} (\tf-\tpr)^{\nf12}\,\frac{\kappa_m^{\nf12}}{\eps_m}\Bigl(\frac{\eps_m}{\rho(m)}\Bigr)^{\Ncl}\le CT^{\nf12}\eps_{m-1}^{-(2-\beta+8\delta)/2+(q\gamma+3\delta)\Ncl}\le C\,, \end{equation*} by~\eqref{e.cmp.ratios}, the condition on~$\Ncl$ in~\eqref{e.cmp.schedule}, and $\kappa_m/\eps_m^2\le C\eps_m^{\beta+\gamma-2}\le C\eps_{m-1}^{-(2-\beta+8\delta)}$ (Proposition~\ref{p.scales}(b) and Lemma~\ref{l.fl.scales}(c)). \item \emph{Switching}, by Lemma~\ref{l.cmp.switching}: at most $C\eps_{m-1}^{-C}\exp(-c\eps_{m-1}^{-4\delta})\le C\eps_{m-1}^{Q}$, since $\tf-\tpr\le T$, $\tau_m\asymp\eps_{m-1}^{2-\beta+4\delta}$ by~\eqref{e.times}, and $\eps_m^2/\kappa_m\le C\tau_m$ by Proposition~\ref{p.scales}(c). \item \emph{The fine smoothing defect.} The sources of~$G$ at the scale $\rhomol(m)$ are bounded by clause~(b2$'$). By the third bound of Proposition~\ref{p.paired} at this scale, at each time, as for~$\Fsm$, each costs $(r_\sigma/\rhomol(m))^{\Nmom}\le C\eps_m^{\delta\Nmom}\le C\eps_{m-1}^{q\delta\Nmom}$, by~(c2). The remaining norm of~$Y^\sharp$, of order $\Nmom+\Nsrc-2$ at the scale~$\rhomol(m)$, is bounded by Proposition~\ref{p.cmp.energy}(ii). \end{itemize} \emph{Step 4: The orders.} By~\eqref{e.cmp.schedule}, whose margin~$q\gamma$ absorbs the cost~$\eps_{m-1}^{-q\gamma}$ of the coarse flux, the norm~\eqref{e.cmp.forcing} of each term is at most $C\eps_{m-1}^{Q+4}\|g\|_{L^2_G(\T^2)}$, and~\eqref{e.cmp.residual} follows. \end{proof} \paperfigure{% \begin{table}[!htbp] \centering \renewcommand{\arraystretch}{1.3} \begin{tabular}{llll} \hline term & bound (times $\|g\|_{L^2_G(\T^2)}$) & smallness from & statement\\ \hline cell tail & $C\eps_{m-1}^{3\delta\Ncor}$ & correctors up to degree~$\Ncor$ & \ref{l.cmp.cancel}, \ref{l.cmp.tail}\\ switching & $C\eps_{m-1}^{-C}\exp(-c\eps_{m-1}^{-4\delta})$ & decay in the quiet intervals & \ref{l.cmp.switching}\\ coarse smoothing & $C\eps_{m-1}^{\Theta\Nmom-q\gamma}$ & moments of the smoothings of~$G'$ & \ref{p.paired}, \ref{l.cmp.coarse}\\ fine smoothing & $C\eps_{m-1}^{q\delta\Nmom}$ & moments of the smoothings of~$G$ & \ref{p.paired}, \ref{p.cmp.energy}\\ truncation, Taylor & $C\eps_{m-1}^{\delta p-q\gamma}$ & truncation of the exponential & \ref{l.cmp.conjugation}, \ref{l.cmp.coarse}\\ frontier, generation & $C\eps_{m-1}^{\eta(\Ngen+1)/8-q\gamma}$ & markers & \ref{l.cmp.frontier}, \ref{l.cmp.coarse}\\ frontier, length & $C\eps_{m-1}^{\delta(\Nlen+1)/2-q\gamma}$ & one letter per word length & \ref{l.cmp.frontier}, \ref{l.cmp.coarse}\\ \hline \end{tabular} \caption{The residual of the lift (Proposition~\ref{p.cmp.residual}) in the norm~\eqref{e.cmp.forcing}, the coarse flux split as in Step~3, each part with its cost $\eps_{m-1}^{-q\gamma}$ (Lemma~\ref{l.cmp.coarse}). Each bound is at most $C\eps_{m-1}^Q$ by~\eqref{e.cmp.schedule}.} \label{t.cmp.bill} \end{table}} \subsection{Proof of Proposition~\ref{p.cycle}(b)} \label{ss.cmp.proof} Here~$m_0$ is the top level of item~(4) of Subsection~\ref{ss.admissible.parameters}, at least every threshold below (Table~\ref{t.admissible.schedule}). \emph{Admissibility.} By Definition~\ref{d.admissible.admissible}, $G$ descends from an admissible seed by successors, with~\eqref{e.admissible.windows} and $\tA=\tA^{(m)}$; we append~$G'$ to it. The new condition~\eqref{e.admissible.windows} is $\tA^{(m-1)}=\tA+3\tau_m\le\tf-4h_m$, which holds since $\tA+3\tau_m\le\tc\le\tf-4h_m$, and $m-1\ge m_0$. So~$G'$ is admissible on $[\tA+3\tau_m,\tf]$. \emph{The marked amplitude.} So~$G'$ satisfies clause~(d) on $[\tpr,\tf]$ (Proposition~\ref{p.admissible.bounds}), whose matrix at the order~$0$ is the coupling matrix of Proposition~\ref{p.energy}; with the masses $\omega_v/\Psi_v^2$, this proposition gives $\|Y\|_{L^\infty([\tpr,\tf];L^2_{G'}(\T^2))}\le\|Y(\tpr)\|_{L^2_{G'}(\T^2)}$. The datum, the retained trace, has the masses and markers of~$G$ on the components of~$G$ (Lemma~\ref{l.admissible.successors}(i)), so Lemma~\ref{l.cmp.conjugation}(iii) and Proposition~\ref{p.cycle}(a), with $\Nout\ge2p$, give the first bound of Proposition~\ref{p.cycle}(b): \begin{align*} \|Y(\tpr)\|_{L^2_{G'}(\T^2)} & \leq \|\Pi g\|_{L^2_G(\T^2)}+C\eps_{m-1}^\delta\|\Pi g\|_{H^{\Nout}_{\rho(m),G}(\T^2)} \notag \\ & \le\bigl(1+C\eps_{m-1}^Q\bigr)\|g\|_{L^2_G(\T^2)}+C\eps_{m-1}^\delta\|g\|_{L^2_G(\T^2)}\,. \end{align*} \emph{The comparison.} The restriction of~$U$ to $[\tpr,\tf]$ solves the system of~$G$ from~$U(\tpr)$ (Section~\ref{s.familytools}). By Proposition~\ref{p.cmp.residual}, $U-Y^\sharp$ solves~\eqref{e.ft.system} with the forcing $-(F,F^0)$ from \begin{equation*} U(\tpr)-Y^\sharp(\tpr)=\bigl(U(\tpr)-\Pi g\bigr)-\bigl(\expp(\Gamma_m(\tpr):\nabla^2)\expp(-\Gamma_m(\tpr):\nabla^2)-I\bigr)\Pi g\,. \end{equation*} We bound both terms in $L^2_\omega(\T^2)$ by their norms with the markers, which are at least~$1$. The first term is at most $C\eps_{m-1}^Q\|g\|_{L^2_G(\T^2)}$ by the first bound of Proposition~\ref{p.cycle}(a). By~\eqref{e.cmp.truncation} at $N=0$, the third bound of Proposition~\ref{p.cycle}(a) with $\Nout\ge4p$, and $\delta p>Q$, the second term is at most \begin{equation*} C\eps_{m-1}^{\delta(p+1)}\|\Pi g\|_{H^{4p}_{\rho(m),G}(\T^2)}\le C\eps_{m-1}^{Q}\|g\|_{L^2_G(\T^2)}\,. \end{equation*} As~$G$ satisfies~(d) with sums at most~$\frac18$ for the masses~$\omega_v$ (Proposition~\ref{p.admissible.bounds}), by linearity and uniqueness (Proposition~\ref{p.ft.energy}, $N=0$), $U-Y^\sharp$ is the solution given there for this datum and the forcing $-(F,F^0)$, and Proposition~\ref{p.ft.energy}(a) bounds $\|U-Y^\sharp\|_{L^\infty([\tpr,\tf];L^2_\omega(\T^2))}+\kappa_m^{\nf12}\|\nabla(U-Y^\sharp)\|_{L^2((\tpr,\tf);L^2_\omega(\T^2))}$ by~$C$ times the norms of this datum in $L^2_\omega(\T^2)$ and of the forcing in~\eqref{e.cmp.forcing}, hence by $C\eps_{m-1}^Q\|g\|_{L^2_G(\T^2)}$ by~\eqref{e.cmp.residual}: the second bound of Proposition~\ref{p.cycle}(b). The third is Proposition~\ref{p.cmp.energy}(i). \qed \paragraph{The terminal cycle.} Section~\ref{s.entry} compares the terminal cycle, at level~$M$, in the same way: Lemmas~\ref{l.cmp.quiet}, \ref{l.cmp.cancel} and~\ref{l.cmp.tail}--\ref{l.cmp.coarse} hold at $m=M$, where the cost $(\kappa_{M-1}/\kappa_M)^{\nf12}$ in Lemma~\ref{l.cmp.coarse} is bounded; Lemma~\ref{l.cmp.corrector} is for $m0$ of the width \begin{equation}\label{e.II8.width} r_m\coloneqq\rho(m)\,\eps_{m-1}^{\Theta} \end{equation} of the fresh smoothing~\eqref{e.II8.S} of the new sources of a successor created at level~$m$, and the order~$\Nmom$ of its vanishing moments (Section~\ref{s.admissible}); \item the constant $\cmol\in(0,1]$ of the molecular radius~$\rhomol(m)$ of~\eqref{e.admissible.rhomol}, the scale at which a solution of a family of level~$m$ has derivatives after the time $\rhomol(m)^2/\kappa_m$ (Section~\ref{s.preparation}); \item the power-loss exponent \begin{equation}\label{e.II8.loss} Z\coloneqq1+s+\Bigl(\frac\gamma2+\delta\Bigr)(\Nrep+1)\,, \end{equation} lost when a Sobolev bound is read as a profile. \end{itemize} These parameters and~$\Nret$ satisfy \begin{equation} \label{e.II8.orders} \Nret\ge2\Nrep+4+\Ncor+2p+\Nmom\,,\qquad Q\ge4+2qZ\,, \end{equation} and the width exponent is so small that \begin{equation} \label{e.II8.widths} 12\Theta<\delta\,,\qquad \Theta\Ncor\le\Bigl(\frac\gamma2+\delta\Bigr)(\Nrep-1)\,. \end{equation} \subsection{Estimates from the homogenization cycle} \label{ss.II8.inputs} For a field~$F$ of a family~$G$ and $\rho>0$, the norm~\eqref{e.ft.norm} of $H^N_{\rho,\omega}(\T^2)$ is the norm~\eqref{e.norms} of $H^N_{\rho,G}(\T^2)$ with all markers replaced by~$1$. Since the markers are at least~$1$, \begin{equation}\label{e.II8.marked} \|F\|_{H^N_{\rho,\omega}(\T^2)}\le\|F\|_{H^N_{\rho,G}(\T^2)}\,. \end{equation} A passage works at one time. Proposition~\ref{p.II8.complement} bounds the factor by which the readout multiplies the size of a representation (Lemma~\ref{l.II8.rep}(b)), states the comparison at every time, and gives~$U(t)$ the $2\Nrep+4$ derivatives at the scale~$\rhomol(m)$ that the lift has (Proposition~\ref{p.cmp.energy}(ii)); Step~6 of the proof of Proposition~\ref{p.passage} interpolates between the comparison and these derivatives. \begin{proposition}[Further properties of a cycle] \label{p.II8.complement} There exists $C(\deps)<\infty$ such that, for every $M$, $\kappa$, $m$, $\tA$, $\tc$, $\tf$, $G$ and~$G'$ as in Proposition~\ref{p.cycle}, there exist~$\tpr$ and~$\Pi$ satisfying \textup{(a)} and~\textup{(b)} of Proposition~\ref{p.cycle} with this~$C$, such that the following holds for every datum~$g$ of~$G$, with~$U$ and~$Y$ as there. \begin{enumerate}[label=\textup{(\roman*)}] \item The readout satisfies $|\Upsilon_m|\le1+C\eps_{m-1}^{5\eta/4}$. \item For every $t\in[\tpr,\tf]$, \begin{align*} \|U(t)-Y^\sharp(t)\|_{L^2_\omega(\T^2)} & \le C\eps_{m-1}^Q\|g\|_{L^2_G(\T^2)}\,,\\ \|U(t)\|_{H^{2\Nrep+4}_{\rhomol(m),\omega}(\T^2)} & \le C\|g\|_{L^2_G(\T^2)}\,. \end{align*} \end{enumerate} \end{proposition} Proposition~\ref{p.II8.complement} holds for the time~$\tpr$ and the map~$\Pi$ constructed in Section~\ref{s.preparation}: part~(i) is Lemma~\ref{l.admissible.successors}(iii). The first bound of~(ii) is Proposition~\ref{p.cycle}(b), and the second is Proposition~\ref{p.prep.further}(iii), each at every time, as~$U$ (Proposition~\ref{p.energy}) and~$Y^\sharp$ (Proposition~\ref{p.cmp.residual}) are continuous in time with values in~$L^2(\T^2)^V$. For the same~$\tpr$ and~$\Pi$, Proposition~\ref{p.cmp.energy} bounds~$Y$ and~$Y^\sharp(t)$. Proposition~\ref{p.passage} is proved below with these~$\tpr$ and~$\Pi$. Lemma~\ref{l.cmp.corrector} applies to the family~$G$ of Proposition~\ref{p.cycle} at every time $t\in[\tc,\tf]$, since~$G$ is admissible on $[\tA,\tf]$, hence in the setting of the lift of Section~\ref{s.comparison} (Proposition~\ref{p.admissible.bounds} gives the invariant), and $\tc\ge\tA+3\tau_m$. By Lemma~\ref{l.cell.covariance}(b) and~\eqref{e.II8.width}, the covariance is small against the square of the width: \begin{equation}\label{e.II8.covwidth} |\Gamma_m(t)|\le C\eps_{m-1}^{\delta-2\Theta}\,r_m^2\,. \end{equation} By Proposition~\ref{p.scales}(c), for $m\ge2$, $\rho(m)\asymp\eps_{m-1}^{1+\gamma/2+\delta}$, so that \begin{equation}\label{e.II8.scales} c\,\eps_{m-1}^{1+\gamma/2+\delta}\le\rho(m)\le C\eps_{m-1}^{1+\gamma/2+\delta}\le C\eps_{m-1}\,,\qquad r_m\le\rho(m)\,. \end{equation} \subsection{Representations and the smoothing} \label{ss.II8.rep} By Lemma~\ref{l.II8.rep}(a), the size of a representation bounds the field, its derivatives at the wavelength and its H\"older norm. By~(b), the readout carries a representation to a finer wavelength at the cost of the norm of the readout, as each profile keeps its wavelength and the base, read at a finer wavelength, does not grow; added profiles add their norms. \begin{lemma}[Bounds from a representation, and copies] \label{l.II8.rep} Let $0<\lambda\le\lambda'\le1$. \begin{enumerate}[label=\textup{(\alph*)}] \item For every field~$U$ of a family and every representation~$R$ of~$U$ at wavelength~$\lambda$, \begin{equation*} \|U\|_{L^\infty(\T^2)}+\lambda^{-s}[U]_{C^{\Nrep}_\lambda(\T^2)}\le\size_\lambda(R)\,,\qquad \|U\|_{C^{0,s}(\T^2)}\le3\size_\lambda(R)\,. \end{equation*} \item Let~$R'$ be a representation, with base~$E$ and profiles~$\mathcal P$, of a field~$Y$ of a successor~$G'$ of a family~$G$ of level~$m$ at wavelength~$\lambda'$. Let finitely many fields $P_i\in C^{\Nrep}(\T^2)^V$ of~$G$ be given, with wavelengths $\lambda_i\in[\lambda,1]$. Then the base~$\Upsilon_mE$, the profiles~$\Upsilon_mP$, $P\in\mathcal P$, with the wavelengths~$\lambda_P$, and the profiles~$P_i$ with the wavelengths~$\lambda_i$ form a representation~$R$ of $\Upsilon_mY+\sum_iP_i$ at wavelength~$\lambda$, and \begin{equation*} \size_\lambda(R)\le|\Upsilon_m|\,\size_{\lambda'}(R')+\sum_i\lambda_i^{-s}\|P_i\|_{C^{\Nrep}_{\lambda_i}(\T^2)}\,. \end{equation*} \end{enumerate} \end{lemma} \begin{proof} (a) For a profile~$P$ and $|j|\ge1>s$, $\lambda\le\lambda_P\le1$ gives $\lambda^{|j|-s}\le\lambda_P^{|j|-s}$ and $1\le\lambda_P^{-s}$; adding the definitions~\eqref{e.seminorms} over the base and the profiles gives the first bound. For the second, for $x,y\in\T^2$, \begin{align*} |P(x)-P(y)|_\omega&\le\min\bigl\{2\|P\|_{L^\infty(\T^2)},\,\bigl(\|\partial_1P\|_{L^\infty(\T^2)}+\|\partial_2P\|_{L^\infty(\T^2)}\bigr)|x-y|_{\T^2}\bigr\}\\ &\le2\lambda_P^{-s}\|P\|_{C^{\Nrep}_{\lambda_P}(\T^2)}|x-y|_{\T^2}^s\,, \end{align*} since $\lambda_P(\|\partial_1P\|_{L^\infty(\T^2)}+\|\partial_2P\|_{L^\infty(\T^2)})\le[P]_{C^{\Nrep}_{\lambda_P}(\T^2)}$ and $s<1$, and adding $[E]_{C^{0,s}(\T^2)}$ and the first bound gives $\|U\|_{C^{0,s}(\T^2)}\le3\size_\lambda(R)$. (b) The readout, a constant linear map, commutes with derivatives and at every point costs at most its operator norm~$|\Upsilon_m|$ for~$|\cdot|_\omega$, hence at most~$|\Upsilon_m|$ in each term of~\eqref{e.representation}. The base, measured at $\lambda\le\lambda'$, only decreases by the monotonicity of~(a), and the profiles~$P_i$ add their terms. \end{proof} Lemma~\ref{l.II8.sobolev} turns a Sobolev bound into a profile: $\Nrep+2$ derivatives in~$L^2$ at a scale~$\rho$, read at a wavelength~$\lambda$, cost the fixed power~$\lambda^{-Z}$ where the maximum in~\eqref{e.II8.sobolev} is bounded. \begin{lemma}[Sobolev bounds read at a wavelength] \label{l.II8.sobolev} There exists $C(\Nrep)<\infty$ such that, for every $0<\lambda\le1$, $0<\rho\le1$ and every field $F\in H^{\Nrep+2}(\T^2)^V$ of a family, \begin{equation}\label{e.II8.sobolev} \lambda^{-s}\|F\|_{C^{\Nrep}_\lambda(\T^2)}\le C\max\Bigl\{1,\frac{\lambda^{1+\gamma/2+\delta}}\rho\Bigr\}^{\Nrep+1}\lambda^{-Z}\,\|F\|_{H^{\Nrep+2}_{\rho,\omega}(\T^2)}\,. \end{equation} The pair $(F,\emptyset)$, with base~$F$ and no profile, is a representation of~$F$ at wavelength~$\lambda$ with $\size_\lambda((F,\emptyset))\le4\lambda^{-s}\|F\|_{C^{\Nrep}_\lambda(\T^2)}$. \end{lemma} \begin{proof} For a scalar function~$g$, a multi-index~$j$ and $\rho\le1$, the Cauchy--Schwarz inequality in Fourier gives $\|\partial^jg\|_{L^\infty(\T^2)}\le C\rho^{-|j|-1}\|g\|_{H^{|j|+2}_\rho(\T^2)}$. Applied to each component, it bounds the left side of~\eqref{e.II8.sobolev} by $C\sum_{i=0}^{\Nrep}\lambda^{i-s}\rho^{-i-1}\|F\|_{H^{\Nrep+2}_{\rho,\omega}(\T^2)}$, and for $0\le i\le\Nrep$, since $\lambda\le1$, \begin{equation*} \lambda^{i-s}\rho^{-i-1}=\Bigl(\frac{\lambda^{1+\gamma/2+\delta}}\rho\Bigr)^{i+1}\lambda^{-(1+s+(\gamma/2+\delta)(i+1))}\le\max\Bigl\{1,\frac{\lambda^{1+\gamma/2+\delta}}\rho\Bigr\}^{\Nrep+1}\lambda^{-Z}\,. \end{equation*} Finally, $F\in C^{\Nrep}(\T^2)^V$ by the Sobolev embedding, and the size of $(F,\emptyset)$ is bounded as in the proof of Lemma~\ref{l.II8.rep}(a). \end{proof} At the time~$t$, the fresh smoothing of the new sources of a successor created at level~$m$ is the product of two averages~\eqref{e.Srh} along ellipses of width~$r_m$, \begin{equation}\label{e.II8.S} S(t)\coloneqq S_{\ell,r_m,\Omega_1(t)}\,S_{\ell,r_m,\Omega_2(t)}\,, \end{equation} where~$\ell$ is a fixed integer with $2\ell>\Nmom$, depending only on~$\deps$, and $\Omega_1(t)$, $\Omega_2(t)$ are the maps of the ellipses~\eqref{e.admissible.ellipses}. By Step~7 of the proof of Lemma~\ref{l.admissible.cycle}, they are invertible with $|\Omega_i(t)|_\infty\le C_\Omega$, and, at the levels finer than the top level of Subsection~\ref{ss.admissible.parameters}, their long axes $\Omega_1(t)e_1$, $\Omega_2(t)e_2$ are uniformly transverse in the sense of Proposition~\ref{p.paired}. A passage splits~$Y(t)$ with this smoothing. By Lemma~\ref{l.II8.smoothing}(a), $S(t)$ moves derivatives onto its kernel at the cost~$r_m^{-1}$ each, uniformly in the eccentricity, as every derivative combines derivatives along the two long axes; by~(b), the vanishing moments of the kernel make $(I-S)Y(t)$ small. \begin{lemma}[The smoothing] \label{l.II8.smoothing} There exists $C(\deps)<\infty$ such that the following holds for every $M\ge1$, $\kappa\in\Kset_M$, $m_0m_0$. \end{proof} \subsection{Proof of Proposition~\ref{p.passage}} \label{ss.II8.passage} \begin{proof}[Proof of Proposition~\ref{p.passage}] The top level $m_0\ge2$ (so that $m\ge2$, as Lemma~\ref{l.fl.scales}(a) and~\eqref{e.II8.scales} require) is so large that, for $m>m_0$, $\rho(m)\le1$ and $C\eps_{m-1}^{5\eta/4}\le1$, with the constant~$C$ of Proposition~\ref{p.II8.complement}(i); then $|\Upsilon_m|\le2$. Let~$\tpr$ and~$\Pi$ be those of the paragraph after Proposition~\ref{p.II8.complement}. Fix a datum~$g$, a time $t\in[\tpr,\tf]$ and a representation~$R'$ of~$Y(t)$ at wavelength~$\eps_{m-1}$, and write~$S$ for~$S(t)$. The proof writes~$U(t)$ as $\Upsilon_mY(t)$ plus five profiles (Step~2), bounds their profile norms (Steps~3--6), and adds them (Step~7). \emph{Step 1: Regularity.} By Proposition~\ref{p.cmp.energy}(i), Proposition~\ref{p.II8.complement}(ii) and the Sobolev embedding, $Y(t)$ is of class $C^{\Nret-2}$ on~$\T^2$, with $\Nret-2\ge\Nrep+\Ncor+2p+2$ by~\eqref{e.II8.orders}, and $U(t)$ is of class $C^{2\Nrep+2}$ on~$\T^2$; this is the first claim of Proposition~\ref{p.passage}. Since~$S$ keeps regularity, $\expp(\Gamma_m(t):\nabla^2)$ costs $2p$ derivatives and~$\Corr_m$ maps fields of class $C^{\Nrep+\Ncor}$ to fields of class~$C^{\Nrep}$, the five profiles defined in Step~2 are of class~$C^{\Nrep}$ on~$\T^2$. \emph{Step 2: The exact split.} The operator $\expp(\Gamma_m(t):\nabla^2)$ acts on every component by the same scalar differential operator with constant coefficients, so it commutes with the constant matrix~$\Upsilon_m$. For a field~$F$ of~$G'$ of class $C^{\Nrep+\Ncor+2p+2}$ on~$\T^2$, define, as in~\eqref{e.cmp.flat}, $F^\flat\coloneqq\expp(\Gamma_m(t):\nabla^2)\,\Upsilon_mF$, and its exponential and corrector parts \begin{equation}\label{e.II8.parts} F^{\mathrm{exp}}\coloneqq\Upsilon_m\bigl(\expp(\Gamma_m(t):\nabla^2)-I\bigr)F\,,\qquad F^{\mathrm{cor}}\coloneqq(\Corr_m-I)\,F^\flat\,. \end{equation} Then the lift~\eqref{e.lift} of~$F$ is $F^\sharp=\Corr_mF^\flat=\Upsilon_mF+F^{\mathrm{exp}}+F^{\mathrm{cor}}$, since $F^\flat=\Upsilon_mF+F^{\mathrm{exp}}$. Both parts are linear in~$F$, so the splitting $Y(t)=SY(t)+(I-S)Y(t)$ gives the exact identity \begin{align} U(t)& = \Upsilon_mY(t)+(SY(t))^{\mathrm{exp}}+(SY(t))^{\mathrm{cor}} \notag \\ & \qquad +((I-S)Y(t))^{\mathrm{exp}}+((I-S)Y(t))^{\mathrm{cor}}+\bigl(U(t)-Y^\sharp(t)\bigr).\label{e.II8.split} \end{align} We give the two exponential parts the wavelength~$\eps_{m-1}$, and the two corrector parts and $U(t)-Y^\sharp(t)$ the wavelength~$\eps_m$. Lemma~\ref{l.II8.rep}(b), with $\lambda'=\eps_{m-1}$ and $\lambda=\eps_m$, turns~$R'$ and these five profiles into a representation~$R$ of~$U(t)$ at wavelength~$\eps_m$, whose size is at most $|\Upsilon_m|\size_{\eps_{m-1}}(R')$ plus the five profile norms bounded in Steps~3--6. \emph{Step 3: The exponential part of~$SY(t)$.} We show \begin{equation}\label{e.II8.P1} \eps_{m-1}^{-s}\bigl\|(SY(t))^{\mathrm{exp}}\bigr\|_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}\le C\eps_{m-1}^{\delta-2\Theta}\,\eps_{m-1}^{-s}[Y(t)]_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}\,. \end{equation} Every term carries $\Gamma_m(t)$ at least once, which is small compared with~$r_m^2$ by~\eqref{e.II8.covwidth}, and~$r_m^2$ absorbs the two derivatives that come with it: on~$S$ each costs~$r_m^{-1}$, and on~$Y(t)$ each costs $\eps_{m-1}^{-1}\le Cr_m^{-1}$. In detail, every term of $\expp(\Gamma_m(t):\nabla^2)-I$ is $(\Gamma_m(t):\nabla^2)^l/l!$ with $1\le l\le p$, and $(\Gamma_m(t):\nabla^2)^l$ is a sum of $4^l$ derivatives of order~$2l$ with coefficients of absolute value at most $|\Gamma_m(t)|^l$. Let~$j$ be a multi-index with $|j|\le\Nrep$, and put $N_2\coloneqq|j|+2l$ and $N_1\coloneqq\min\{\Nrep,N_2\}\ge1$. Lemma~\ref{l.II8.smoothing}(a) keeps~$N_1$ derivatives on~$Y(t)$ and puts the other $N_2-N_1$ on~$S$: \begin{align*} \bigl\|\partial^j(\Gamma_m(t):\nabla^2)^lSY(t)\bigr\|_{L^\infty(\T^2)}&\le4^l|\Gamma_m(t)|^l\sum_{|i|=N_2}\|\partial^iSY(t)\|_{L^\infty(\T^2)}\\ &\le C|\Gamma_m(t)|^l\,r_m^{N_1-N_2}\sum_{|i|=N_1}\|\partial^iY(t)\|_{L^\infty(\T^2)}\,. \end{align*} We multiply by $\eps_{m-1}^{|j|-s}$ and write $|\Gamma_m(t)|^lr_m^{N_1-N_2}\eps_{m-1}^{|j|}=(|\Gamma_m(t)|/r_m^2)^l(r_m/\eps_{m-1})^{N_1-|j|}\eps_{m-1}^{N_1}$. Here $(|\Gamma_m(t)|/r_m^2)^l\le C\eps_{m-1}^{\delta-2\Theta}$ by~\eqref{e.II8.covwidth} and $2\Theta<\delta$, and $(r_m/\eps_{m-1})^{N_1-|j|}\le C$ by~\eqref{e.II8.scales}, since $N_1\ge|j|$. As $1\le N_1\le\Nrep$, for $|i|=N_1$ we have $\eps_{m-1}^{N_1}\|\partial^iY(t)\|_{L^\infty(\T^2)}\le i![Y(t)]_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}$, so \begin{equation*} \eps_{m-1}^{|j|-s}\bigl\|\partial^j(\Gamma_m(t):\nabla^2)^lSY(t)\bigr\|_{L^\infty(\T^2)}\le C\eps_{m-1}^{\delta-2\Theta}\,\eps_{m-1}^{-s}[Y(t)]_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}\,. \end{equation*} The readout commutes with derivatives and $|\Upsilon_m|\le2$; dividing by~$j!$ and summing over $|j|\le\Nrep$ and $1\le l\le p$ gives~\eqref{e.II8.P1}. \emph{Step 4: The corrector part of~$SY(t)$.} We show \begin{equation}\label{e.II8.P2} \eps_m^{-s}\bigl\|(SY(t))^{\mathrm{cor}}\bigr\|_{C^{\Nrep}_{\eps_m}(\T^2)}\le C\eps_{m-1}^{\mu}\,\eps_{m-1}^{-s}[Y(t)]_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}\,. \end{equation} The correction costs the corrector amplitude times one derivative of~$Y(t)$ at the wavelength~$\eps_{m-1}$; the length~$\rho(m)$ cancels, and~\eqref{e.mu} says that this is small when measured at the wavelength~$\eps_m$. This is where $s<\frac12(1-\alpha)$ is used. We apply Lemma~\ref{l.cmp.corrector} to the field $(SY(t))^\flat$ of~$G$. Its derivatives at the scale~$\rho(m)$ are controlled by the seminorm of~$Y(t)$ at the wavelength~$\eps_{m-1}$: \begin{equation}\label{e.II8.input} \sum_{1\le|i|\le\Nrep+\Ncor}\rho(m)^{|i|}\bigl\|\partial^i(SY(t))^\flat\bigr\|_{L^\infty(\T^2)}\le C\,\frac{\rho(m)}{\eps_{m-1}}\,[Y(t)]_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}\,. \end{equation} Indeed, we expand $\expp(\Gamma_m(t):\nabla^2)$ as in Step~3, now with $0\le l\le p$; the readout commutes with derivatives, $|\Upsilon_m|\le2$, and $|\Gamma_m(t)|\le Cr_m^2$ by~\eqref{e.II8.covwidth}, as $2\Theta<\delta$. For a multi-index~$i$ with $1\le|i|\le\Nrep+\Ncor$, Lemma~\ref{l.II8.smoothing}(a) with $N_1\coloneqq\min\{\Nrep,|i|+2l\}\ge1$ bounds the term of degree~$l$ by $C\rho(m)^{|i|}r_m^{N_1-|i|}\sum_{|j|=N_1}\|\partial^jY(t)\|_{L^\infty(\T^2)}$. If $|i|\le\Nrep$, then $N_1\ge|i|$ and $\rho(m)^{|i|}r_m^{N_1-|i|}\le\rho(m)^{N_1}\le C\rho(m)\,\eps_{m-1}^{N_1-1}$ by~\eqref{e.II8.scales}. If $|i|>\Nrep$, then $N_1=\Nrep$, and the $|i|-\Nrep\le\Ncor$ derivatives beyond~$\Nrep$ fall on~$S$, each at the cost $\rho(m)/r_m=\eps_{m-1}^{-\Theta}$. The power $\rho(m)^{\Nrep-1}\le C\eps_{m-1}^{\Nrep-1}\eps_{m-1}^{(\gamma/2+\delta)(\Nrep-1)}$ (by~\eqref{e.II8.scales}) absorbs them: \begin{equation*} \rho(m)^{|i|}r_m^{\Nrep-|i|}=\rho(m)^{\Nrep}\Bigl(\frac{\rho(m)}{r_m}\Bigr)^{|i|-\Nrep}\le C\rho(m)\,\eps_{m-1}^{\Nrep-1}\,\eps_{m-1}^{(\gamma/2+\delta)(\Nrep-1)-\Theta\Ncor}\,, \end{equation*} and the last exponent is nonnegative by~\eqref{e.II8.widths}. In both cases the term is at most \begin{equation*} C\,\frac{\rho(m)}{\eps_{m-1}}\,\eps_{m-1}^{N_1}\sum_{|j|=N_1}\|\partial^jY(t)\|_{L^\infty(\T^2)}\le C\,\frac{\rho(m)}{\eps_{m-1}}\,[Y(t)]_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}\,, \end{equation*} which proves~\eqref{e.II8.input}. Now Lemma~\ref{l.cmp.corrector} and~\eqref{e.II8.input} give, for $|j|\le\Nrep$, \begin{align*} \eps_m^{|j|-s}\bigl\|\partial^j(SY(t))^{\mathrm{cor}}\bigr\|_{L^\infty(\T^2)}&\le C\eps_m^{-s}\,\frac{\ebar}{\rho(m)}\,\frac{\rho(m)}{\eps_{m-1}}\,[Y(t)]_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}\\ &=C\,\bigl(\eps_m^{-s}\,\ebar\,\eps_{m-1}^{s-1}\bigr)\,\eps_{m-1}^{-s}[Y(t)]_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}\,, \end{align*} and the first factor is at most $C\eps_{m-1}^\mu$ by~\eqref{e.mu}. Summing over~$j$ proves~\eqref{e.II8.P2}. \emph{Step 5: The parts of $(I-S)Y(t)$.} We show \begin{equation}\label{e.II8.P3} \eps_{m-1}^{-s}\bigl\|((I-S)Y(t))^{\mathrm{exp}}\bigr\|_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}+\eps_m^{-s}\bigl\|((I-S)Y(t))^{\mathrm{cor}}\bigr\|_{C^{\Nrep}_{\eps_m}(\T^2)}\le C\eps_{m-1}^2\|g\|_{L^2_G(\T^2)}\,. \end{equation} By Lemma~\ref{l.II8.smoothing}(b) with $\rho=\rho(m)$, the bound $r_m/\rho(m)\le\eps_{m-1}^\Theta$, \eqref{e.II8.orders} and Proposition~\ref{p.cmp.energy}(i), \begin{equation}\label{e.II8.moments} \|(I-S)Y(t)\|_{H^{\Nrep+\Ncor+2p+2}_{\rho(m),G'}(\T^2)}\le C\eps_{m-1}^{\Theta\Nmom}\|Y(t)\|_{H^{\Nret}_{\rho(m),G'}(\T^2)}\le C\eps_{m-1}^{\Theta\Nmom}\|g\|_{L^2_G(\T^2)}\,. \end{equation} For the exponential part, in the norms $H^N_{\rho(m),\omega}(\T^2)$ each operator $\Gamma_m(t):\nabla^2$ costs two orders (Lemma~\ref{l.ft.product}(b)) and the factor $C|\Gamma_m(t)|/\rho(m)^2\le C\eps_{m-1}^\delta$ (Lemma~\ref{l.cell.covariance}(b)), and the readout costs $|\Upsilon_m|\le2$. Lemma~\ref{l.II8.sobolev} with $\lambda=\eps_{m-1}$ and $\rho=\rho(m)$, where $\eps_{m-1}^{1+\gamma/2+\delta}/\rho(m)\le C$ by~\eqref{e.II8.scales}, the bound~\eqref{e.II8.marked} and~\eqref{e.II8.moments} give \begin{align*} \eps_{m-1}^{-s}\bigl\|((I-S)Y(t))^{\mathrm{exp}}\bigr\|_{C^{\Nrep}_{\eps_{m-1}}(\T^2)}&\le C\eps_{m-1}^{-Z}\bigl\|((I-S)Y(t))^{\mathrm{exp}}\bigr\|_{H^{\Nrep+2}_{\rho(m),\omega}(\T^2)}\\ &\le C\eps_{m-1}^{\delta-Z}\|(I-S)Y(t)\|_{H^{\Nrep+2p+2}_{\rho(m),G'}(\T^2)}\le C\eps_{m-1}^{\Theta\Nmom-Z}\|g\|_{L^2_G(\T^2)}\,. \end{align*} For the corrector part, the inequality of the proof of Lemma~\ref{l.II8.sobolev}, with the costs of $\expp(\Gamma_m(t):\nabla^2)$ and~$\Upsilon_m$ just used and~\eqref{e.II8.marked}, bounds $\rho(m)^{|i|}\|\partial^i((I-S)Y(t))^\flat\|_{L^\infty(\T^2)}$, for $|i|\le\Nrep+\Ncor$, by $C\rho(m)^{-1}$ times the left side of~\eqref{e.II8.moments}. Moreover, $\ebar\le C\rho(m)$ and $\rho(m)\ge c\eps_m$ by~\eqref{e.cmp.ratios}, and $\eps_m\ge c\eps_{m-1}^q$ by Lemma~\ref{l.fl.scales}(a). Lemma~\ref{l.cmp.corrector} and~\eqref{e.II8.moments} then give \begin{equation*} \eps_m^{-s}\bigl\|((I-S)Y(t))^{\mathrm{cor}}\bigr\|_{C^{\Nrep}_{\eps_m}(\T^2)}\le C\eps_m^{-s}\rho(m)^{-1}\eps_{m-1}^{\Theta\Nmom}\|g\|_{L^2_G(\T^2)}\le C\eps_{m-1}^{\Theta\Nmom-q(1+s)}\|g\|_{L^2_G(\T^2)}\,. \end{equation*} The moments of~$S$ beat both losses: by~\eqref{e.cmp.schedule}, and as $Q\ge4+2qZ$ by~\eqref{e.II8.orders}, $Z\ge1+s$ and $q>1$, \begin{equation*} \Theta\Nmom>Q+q\gamma+4\ge2+\max\{Z,q(1+s)\}\,, \end{equation*} which proves~\eqref{e.II8.P3}. \emph{Step 6: The comparison error.} We show \begin{equation}\label{e.II8.P4} \eps_m^{-s}\bigl\|U(t)-Y^\sharp(t)\bigr\|_{C^{\Nrep}_{\eps_m}(\T^2)}\le C\eps_{m-1}^2\|g\|_{L^2_G(\T^2)}\,. \end{equation} Proposition~\ref{p.II8.complement}(ii) bounds $U(t)-Y^\sharp(t)$ in~$L^2_\omega(\T^2)$ by $C\eps_{m-1}^Q\|g\|_{L^2_G(\T^2)}$, and, with Proposition~\ref{p.cmp.energy}(ii), in $H^{2\Nrep+4}_{\rhomol(m),\omega}(\T^2)$ by $C\|g\|_{L^2_G(\T^2)}$. Interpolation between these two bounds (in Fourier, by the Cauchy--Schwarz inequality) gives $\Nrep+2$ derivatives with half the smallness: \begin{equation}\label{e.II8.interp} \bigl\|U(t)-Y^\sharp(t)\bigr\|_{H^{\Nrep+2}_{\rhomol(m),\omega}(\T^2)}\le C\eps_{m-1}^{Q/2}\|g\|_{L^2_G(\T^2)}\,. \end{equation} Since $m\frac56\delta$, so each of $5\eta/4$, $\delta-2\Theta$ and~$\mu$ is at least~$\eta$. Since $\eps_{m-1}\le1$, this proves Proposition~\ref{p.passage}. \end{proof} \subsection{Proof of Proposition~\ref{p.cascade}} \label{ss.II8.cascade} \begin{proof}[Proof of Proposition~\ref{p.cascade}] Each cycle starts once the previous preparation is complete and the admissibility window has moved forward by~$3\tau_j$. \emph{Step 1: Levels finer than~$m_0$.} By the choice of~$m_0$ (item~(4) of Subsection~\ref{ss.admissible.parameters}), $\rho(j)\le1$ for every level $j>m_0$. An admissible seed of level $j>m_0$ on a window $[\tS,\tf]$ with $\tS+\frac12t_0<\tf$ is an admissible family of level~$j$ on $[\tS,\tf]$ (Definition~\ref{d.admissible.admissible}, with no successor). For every $j\ge m_0$, the row and column sums of the coupling matrices of an admissible family of level~$j$, for the masses~$\omega_v$ and $\omega_v/\Psi_v^2$, are at most~$\frac18$ (Proposition~\ref{p.admissible.bounds}). By the bounds~\eqref{e.times} on~$\tau_j$ and~$h_j$ (Lemma~\ref{l.fl.scales}(b)), and since $\eps_j\le\Lambda^{-j}$ (Lemma~\ref{l.fl.scales}(a)), the series of the $3\tau_j+4h_j$ converges, and item~(4) takes~$m_0$ so large that, for every $m>m_0$, \begin{equation}\label{e.II8.wait} \sum_{j=m_0+1}^{m}(3\tau_j+4h_j)<\tfrac12t_0\,. \end{equation} \emph{Step 2: The homogenization cascade.} Let~$m$, $\kappa$, $G$, $\tS$, $\tf$, $g$ and~$U$ be as in Proposition~\ref{p.cascade}. Set $G_m\coloneqq G$, $U_m\coloneqq U$, $t_{(m)}\coloneqq\tS$ and $\tA^{(m)}\coloneqq\tS$; by Step~1, $G_m$ is an admissible family of level~$m$ on $[\tA^{(m)},\tf]$. For $j=m,m-1,\dots,m_0+1$ in turn, given an admissible family~$G_j$ of level~$j$ on $[\tA^{(j)},\tf]$ and a solution~$U_j$ of~$G_j$ on $[t_{(j)},\tf]$, put \begin{equation}\label{e.II8.restart} \tc^{(j)}\coloneqq\max\bigl\{t_{(j)},\,\tA^{(j)}+3\tau_j\bigr\}\,. \end{equation} Let~$G_j'$ be the successor of~$G_j$ in the descent of the seed~$G$ given by Lemma~\ref{l.admissible.successors}(iv), and let $\tpr^{(j)}\in[\tc^{(j)},\tc^{(j)}+4h_j]$ and~$\Pi_j$ be the time and the map constructed in Section~\ref{s.preparation} for $G_j$ and~$G_j'$, with $\tA=\tA^{(j)}$ and $\tc=\tc^{(j)}$. By the paragraph after Proposition~\ref{p.II8.complement}, Propositions~\ref{p.cycle}, \ref{p.II8.complement}, \ref{p.cmp.energy} and~\ref{p.passage} hold for this pair. The hypothesis $\tc^{(j)}+4h_j\le\tf$ of Proposition~\ref{p.cycle} follows from~\eqref{e.II8.timing} at level~$j$, whose proof below uses only the times $\tpr^{(i)}$ of the levels $i>j$, constructed before. By Proposition~\ref{p.cycle}(b), $G_{j-1}\coloneqq G_j'$ is an admissible family of level $j-1$ on $[\tA^{(j-1)},\tf]$, where $\tA^{(j-1)}\coloneqq\tA^{(j)}+3\tau_j$. Let~$U_{j-1}$ be the solution of~$G_{j-1}$ from the retained trace of $\Pi_jU_j(\tc^{(j)})$ at time $t_{(j-1)}\coloneqq\tpr^{(j)}$. By the uniqueness in Proposition~\ref{p.energy} and~\eqref{e.ft.weak}, subtracted at the times $\tc^{(j)}$ and~$t\ge\tc^{(j)}$, the restriction of~$U_j$ to $[\tc^{(j)},\tf]$ is the solution of~$G_j$ from $U_j(\tc^{(j)})$ at time~$\tc^{(j)}$. Hence these four propositions apply at level~$j$ to the datum $U_j(\tc^{(j)})$, with $U=U_j$ and $Y=U_{j-1}$. The times satisfy, for $m_05+2q^n\,,\qquad 2(q+2)\gamma\,(\Nlw+1)>4+3q^n\,. \end{equation} They absorb the cell tail and the frontier of a weak passage (Lemma~\ref{l.entry.passage}). With them, $\theta$ is recovered from a field of~$G_M$ at the order $n\Ncw+5$ (Lemma~\ref{l.entry.weakrep}(b)). \item \emph{The terminal orders}: the accuracy $\Qe\coloneqq2\bigl(5+q^{n+1}(1+s)\bigr)$ of the terminal cycle; then its corrector order~$\Nce$, truncation order~$\pe$, word length~$\Nle$ and number~$\Ncle$ of antidivergence steps, with \begin{equation}\label{e.entry.termorders} \begin{gathered} (q\gamma+3\delta)\Nce>\Qe+2\,,\qquad \delta\pe>\Qe+2\,,\\ \tfrac12\delta(\Nle+1)>\Qe+2\,,\qquad (q\gamma+3\delta)\Ncle>3\,; \end{gathered} \end{equation} then the largest birth order of the sources the terminal cycle creates and the width exponent of their smoothing, \begin{equation}\label{e.entry.termdefs} \Nse\coloneqq\Nce+n\Ncw+4\pe+1\,,\qquad \The\coloneqq\frac{\delta}{16(\Nse-1)}\,. \end{equation} The accuracy absorbs the loss $\eps_{M+n}^{-1-s}\le C\eps_{M-1}^{-q^{n+1}(1+s)}$ of the recovery of~$\theta$ (Lemma~\ref{l.entry.identity}), and~\eqref{e.entry.termorders} the residual of the terminal cycle (Table~\ref{t.entry.bill}). \item \emph{The representation order} \begin{equation}\label{e.entry.Nrep} \Nrep\coloneqq n\Ncw+\Nce+2\pe+4\,. \end{equation} This is Part~I's~$\Nrep$: the terminal lift spends $\Nce+2\pe+1$ derivatives and the weak lifts $n\Ncw$, which leaves three. The orders of items~(1) and~(2) fix the seed's bounds: birth orders at most~$\Nse$, at most $N_{\mathrm{path,e}}\coloneqq(\Nle+2)(n\Nlw+1)$ components along a path of sources, that is, at most $B\coloneqq N_{\mathrm{path,e}}-1$ sources on a path, the bound~$B$ of item~(1) of Subsection~\ref{ss.admissible.parameters}. Section~\ref{s.admissible} then fixes the remaining parameters of its item~(1). \item After these: the moment orders~$\Nmw$ of the smoothing at level~$M$ and~$\Nme$ of the smoothing of the sources the terminal cycle creates, with \begin{equation}\label{e.entry.moments} \begin{gathered} \delta-\tfrac{\delta}{16}-\tfrac{\delta}{64}-\The(\Nse-2)+\The\Nme>\Qe+2\,,\\ q\Bigl(2(q+2)\gamma-\tfrac{5\delta}{64}+\bigl(\tfrac\gamma4+\tfrac\delta2\bigr)\Nmw\Bigr)>2\Qe+4\,; \end{gathered} \end{equation} they absorb the smoothings in Table~\ref{t.entry.bill}. Since $2(q+2)\gamma<1$ and $q<\frac{17}{16}$ by~\eqref{e.exponents} and~\eqref{e.q}, the second condition gives $(\frac\gamma4+\frac\delta2)\Nmw>(2\Qe+4)/q-1>2+2q^n$, which Lemma~\ref{l.entry.identity} uses. Then the recovery order~$\Nrde\ge n\Ncw+5$ of the terminal comparison (Proposition~\ref{p.entry.terminal}(c)), which, with $N_{\mathrm{rec,e}}\coloneqq4(\Nce+\Nse)+2\Ncle+16$ and $\bar N_{\mathrm{mom}}\coloneqq\max\{\Nmw,\Nme,\Nmom\}$, is also at least the floor~\eqref{e.cmp.floor} of the comparison at $m=M$, with the terminal orders, \begin{align}\label{e.entry.outsharp} \Nrde&\ge\max\bigl\{N_{\mathrm{rec,e}}+\Nse+\bar N_{\mathrm{mom}}+6\pe,\,\Nse-2+\bar N_{\mathrm{mom}}+\Nce+2\pe+1\bigr\}\,,\notag\\ \Nrde&\ge N_{\mathrm{rec,e}}+2\Nse N_{\mathrm{path,e}}\\ &\quad+\max\bigl\{\Nse+\bar N_{\mathrm{mom}}+\Nce+6\pe+4,\,(N_{\mathrm{path,e}}+3)(\Nce+\Nse+12)+2\pe\bigr\}\,;\notag \end{align} then the output order~$\Noe$ of the terminal preparation and the trace order~$\Ntw$ of a weak passage, \begin{equation}\label{e.entry.outorders} \begin{gathered} \Noe\coloneqq\max\{\Nrep,\,2\Nrde+2,\,\Nce+2\pe+\Nrde+2\}\,,\\ \Ntw\ge\max\{\Nmw+n\Ncw+1+\Nrde,\,2\Nrde+2\}\,. \end{gathered} \end{equation} As $\Noe\ge2\Nrde+2$, the proof of Proposition~\ref{p.entry.terminal} runs at the order~$\Noe$, above this floor. The condition on~$\Ntw$ in~\eqref{e.entry.outorders} leaves room for the smoothing and the sources of~$G_M$ in Lemma~\ref{l.entry.smoothing}, above~$\Nrde$. The order~$\Ntw$ is also at least the order~$\Nbuf$ of~\eqref{e.prep.Npr} that the preparation reads at the terminal level to produce~$\Pi$ at~$\Noe$, above the order $2\Nrde$ at which the proof of Proposition~\ref{p.entry.terminal}(a) interpolates. Then the seed's constants (Section~\ref{s.admissible}, item~(2)). \item The threshold~$M_*$, last. \end{enumerate} Lemma~\ref{l.entry.scales} separates the weak levels from the terminal one. At the levels finer than~$M$, by~(a), the P\'eclet number is small, the strain of~$\b_{i-1}$ is small over the time diffusion needs to reach~$\eps_{i-1}$, the quiet margins are long enough for item~1 of the route (as $\gamma>3\delta$), and a pulse lasts many cell diffusion times. At level~$M$, by~(b), the margins are too short, and the contrast is at most~$2$. \begin{lemma}[Scales of the entry]\label{l.entry.scales} There exist $\Lambda_0(\alpha,s,q,\zeta)\ge128$ and $C(\alpha,s,q,\zeta)$ with the following property. For every $\Lambda\ge\Lambda_0$ there exists an integer $M_2(\alpha,s,q,\zeta,\Lambda)\ge2$ such that, for every $M\ge2$ and $\kappa\in\Kset_M$, the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item For every $MM$, the square root of the mass ratio and the factor give \begin{equation*} (4n\Ncw)^{\nf12}\eps_{i-1}^{-\delta/64}\cdot\eps_{i-1}^{-\delta/16}\Bigl(\frac{\eps_i^\beta}\kappa\Bigr)^2\le C\eps_{i-1}^{2(q+2)\gamma-5\delta/64}\le C\eps_M^{2(q+2)\gamma-5\delta/64}\,, \end{equation*} by Lemma~\ref{l.entry.scales}(a). For an inherited source, the factor is the product of the factors along its path, the square roots of the mass ratios telescope, and the word multiplies the masses at both ends by the same number; so its entry is the product of the entries of the letters in its history. Since~$G_{M+n}$ has no source, every source of~$G_M$ has at least one letter, and at most $n\Nlw$, in its history, and every entry is at most $C\eps_M^{2(q+2)\gamma-5\delta/64}$. A row or a column has a number of entries bounded by the numbers of words, paths and degrees; this gives the sums, and $2(q+2)\gamma-\frac{5\delta}{64}\ge q-1$ since $(q+2)\gamma\ge q-1$ (as $\beta>1$) and $\frac{5}{64}\delta\tin$, by parabolic regularity, so it lies in the class of Proposition~\ref{p.ft.plain} at every order on $[t_{(M+n)},\tf]$. Let $M0$. \emph{Step 4: energy.} Proposition~\ref{p.ft.plain} for~$G_i$ at the scale~$\eps_i$ and the orders~$\widetilde N_v$, which decrease along the sources of~$G_i$ as it asks ($d_\sigma-2\le k\Ncw-1\le n\Ncw-1$), bounds the energy of the difference by~$C$ times its forcing; its hypotheses hold by Lemmas~\ref{l.entry.scales}(a) and~\ref{l.entry.coefficients}, and its uniqueness identifies the difference with the solution from zero, as the difference lies in its class: the lift by the count of Step~2, and~$U_i$ by the proof of Proposition~\ref{p.entry.reduction}, as $\Ntw\ge N_i$ (for $i=M+n$, $\theta$ is smooth). With Steps~1 and~3 this gives the lemma, as $\widetilde N_v\ge N_i$, with one power of~$\eps_M$ to spare. \end{proof} \subsection{Level~\texorpdfstring{$M$}{M}: smoothing and a homogenization cycle} \label{ss.entry.terminal} \paragraph{The smoothing.} The weak families have unsmoothed sources: Proposition~\ref{p.ft.plain} solves their systems at orders that decrease along the paths of sources, and the derivatives are regained on each quiet margin. The terminal cycle uses Sections~\ref{s.preparation} and~\ref{s.comparison}, which need the clauses of the invariant, and clause~(c) asks for paired smoothings with vanishing moments and widths below~$\rhomol(M)$ and~$\rho(M)$; so the sources are smoothed once, at level~$M$. Let $\Sw\coloneqq S_{\ell_{\mathrm w},\rw,2^{-\nf12}I}\,S_{\ell_{\mathrm w},\rw,2^{-\nf12}I}$, the product of two averages~\eqref{e.Srh} over round discs of radius $2^{-\nf12}\rw$, with $2\ell_{\mathrm w}-1\ge\max\{\Nmw,\Nmom\}$. Its matrix of long axes, $2^{-\nf12}I$, has the inverse $2^{\nf12}I$, of $|\cdot|_\infty$-norm $2^{\nf12}\le2$, so Proposition~\ref{p.paired} applies. Let~$\GM$ be the family~$G_M$ with every source smoothed by~$\Sw$, and let~$\UM$ be the solution of~$\GM$ from~$U_M(t_{(M)})$ at time~$t_{(M)}$. By Lemma~\ref{l.entry.smoothing}, $\UM$ keeps the derivatives of~$U_M$ at the scale~$\eps_M$, and, at a lower order, differs from it by at most a constant times the gain of the moments of~$\Sw$ times the small coupling of Lemma~\ref{l.entry.coefficients}(b). \begin{lemma}[The smoothing at level~$M$]\label{l.entry.smoothing} There exists $C(\deps)$ such that, for every $M\ge M_*$, $\thetain$ and~$\theta$, the component energy~\eqref{e.cmp.energy} of~$\UM$ of order~$\Ntw$ at the scale~$\eps_M$, with the diffusivity~$\kappa$, on $[t_{(M)},\tf]$ is at most $C\|\thetain\|_{L^2(\T^2)}$, and, with $N\coloneqq\Ntw-\Nmw-n\Ncw-1$, \begin{equation*} \|U_M-\UM\|_{L^\infty([t_{(M)},\tf];H^{N}_{\eps_M,\omega}(\T^2))}\le C\eps_M^{q-1+(\gamma/4+\delta/2)\Nmw}\|\thetain\|_{L^2(\T^2)}\,. \end{equation*} \end{lemma} \begin{proof} Proposition~\ref{p.ft.energy} applies to~$\GM$ at every order up to~$\Ntw$ and the scale~$\eps_M$: its condition~(i) holds by the bound on~$\nabla^j\b_M$ stated after~\eqref{e.prep.cmol} and $\eps_M^\beta/\kappa\le C\eps_M^\gamma$ (Lemma~\ref{l.entry.scales}(b)), and~(ii) and~(iii) by Lemma~\ref{l.entry.coefficients}(a) at $i=M+1$ and the order~$\Ntw$, with the sums of the proof of Lemma~\ref{l.entry.coefficients}(b). The solution~$\UM$ has no forcing, and its datum $U_M(t_{(M)})$ has norm at most $C\|\thetain\|_{L^2(\T^2)}$ in $H^{\Ntw}_{\eps_M,\omega}(\T^2)$ by Proposition~\ref{p.entry.reduction}(iv); so part~(c) gives the first bound. The difference $U_M-\UM$ vanishes at~$t_{(M)}$ and solves the system of~$\GM$ with the forcing $\nabla\cdot\sum_\sigma\sum_dA_{\sigma,d}\nabla^{d-2}(I-\Sw)\nabla U_{M,v'}$, the sums over the sources $\sigma\colon v'\to v$. By the third bound of Proposition~\ref{p.paired} at the scale~$\eps_M$ and the order $\Nmw<2\ell_{\mathrm w}$, $I-\Sw$ costs $C(\rw/\eps_M)^{\Nmw}=C\eps_M^{(\gamma/4+\delta/2)\Nmw}$ and~$\Nmw$ more derivatives at the scale~$\eps_M$, and $d-2\le n\Ncw-1$, as the birth orders of~$G_M$ are at most $1+n\Ncw$. By Lemma~\ref{l.entry.coefficients}, $\kappa^{-1}\eps_M^{2-d}$ times the coefficients, with the mass ratios, has row and column sums at most $C\eps_M^{q-1}$. So the forcing, measured as in~\eqref{e.cmp.forcing} at the order~$N$, is at most $C\eps_M^{q-1+(\gamma/4+\delta/2)\Nmw}$ times $\kappa^{\nf12}\|\nabla U_M\|_{L^2((t_{(M)},\tf);H^{\Ntw}_{\eps_M,\omega}(\T^2))}$, which Proposition~\ref{p.entry.reduction}(iv) bounds by $C\|\thetain\|_{L^2(\T^2)}$. Proposition~\ref{p.ft.energy}(a) at the order~$N$ then gives the second bound. \end{proof} The terminal successor of Definition~\ref{d.entry.successor} is the successor of Section~\ref{s.comparison} at the level~$M$, with four changes: the orders; one generation of children, with the marker~$1$, so that it can be declared a seed without change (Proposition~\ref{p.entry.seed}); the coefficients, majorant and smoothing of the new sources; and the covariance, anchored at the preparation time so that the retained trace there is the identity on the components of~$\GM$. The preparation time depends on~$t_{(M)}$ alone: with $l\in\Z$ such that $(l-\frac12)h_M\le t_{(M)}<(l+\frac12)h_M$, it is $\tpM\coloneqq(l+\frac32)h_M$, the time~\eqref{e.prep.tpr} at $m=M$ with $t_1=t_{(M)}$. \begin{definition}[The terminal successor]\label{d.entry.successor} The \emph{terminal successor}~$G'$ of~$\GM$ is the family of level $M-1$ that the comparison of Section~\ref{s.comparison} constructs at $m=M$ (Definition~\ref{d.admissible.successor}), with these changes. \begin{enumerate}[label=(\roman*)] \item The orders are~$\Nce$ and~$\pe$; the mean and covariance letters run over the degrees up to $\Nce+1+n\Ncw$ and the paths of sources of~$\GM$, so that the birth orders are at most~$\Nse$. \item One generation: the components of~$\GM$, with their masses, and their children~$v'$, with nonempty words of length $\#v'\le\Nle$, each with the mass $\ce\eps_{M-1}^{\delta/32}\omega_v$ divided by the number of these words, for a constant $\ce(\deps)>0$, the readout weight $\eps_{M-1}^{\delta\#v'/16}$ in~\eqref{e.lift}, and the marker~$1$. \item The coefficients of the new sources are multiplied by $\eps_{M-1}^{-\delta/16}$. The new sources have the birth radius~$\rho(M)$, the majorant $\Cmaj\eps_{M-1}^{\delta-\delta/16}$ times the product of the majorants along the path, and the round smoothing of~$\Sw$, with the width $\rho(M)\eps_{M-1}^{\The}$ in place of~$\rw$ and with the kernel order~$\ell_{\mathrm e}$ in place of~$\ell_{\mathrm w}$, where $2\ell_{\mathrm e}-1\ge\max\{\Nme,\Nmom\}$. \item The covariance is anchored at the preparation time~$\tpM$: $\Gamma_M(t)-\Gamma_M(\tpM)$ replaces~$\Gamma_M(t)$ in Definition~\ref{d.admissible.successor}, in~\eqref{e.cmp.successorflux} (the defect and covariance letters) and in the lift. \end{enumerate} The \emph{terminal lift} of a field~$Y$ of~$G'$ is \begin{equation}\label{e.entry.lift} Y^\sharp(t)\coloneqq\Ce(t)\Bigl[\exp_{\pe}\bigl((\Gamma_M(t)-\Gamma_M(\tpM)):\nabla^2\bigr)\,\Ue Y(t)\Bigr], \end{equation} where~$\Ue$ is the readout~\eqref{e.lift} of~$G'$, $\exp_{\pe}$ is the truncated exponential~\eqref{e.covariance} with~$\pe$ in place of~$p$, and~$\Ce$ is the corrector operator~$\Corr_M$ of Definition~\ref{d.cell.corr} for~$\GM$ with the corrector order~$\Nce$. \end{definition} The anchored covariance is small: by Lemmas~\ref{l.cell.covariance}(b) and~\ref{l.entry.scales}(b), \begin{equation*} |\Gamma_M(t)-\Gamma_M(\tpM)|\le C\kappa_{M-1}\tau'_M\le C\eps_{M-1}^\delta\rho(M)^2\le C\eps_{M-1}^{\gamma+3\delta}\eps_{M-1}^2\,. \end{equation*} Proposition~\ref{p.entry.terminal} is Proposition~\ref{p.cycle} at the terminal level, with a preparation for~$\GM$, which is not admissible but satisfies the clauses of the invariant (Lemma~\ref{l.entry.coefficients}(b)); the contrast $\kappa_{M-1}/\kappa$ is at most~$2$, and the datum already has derivatives at the scale $\rhomol(M)=\cmol\eps_M$. The last bound of~(a), by interpolation, and the comparison~(c) carry $\Nrde$ derivatives at the scale~$\eps_M$ at half the accuracy of the $L^2$ bound; the weak lifts read~(c) in Lemma~\ref{l.entry.identity}. Parts~(b) and~(c) use sums of~$G'$ bounded in the proof of Proposition~\ref{p.entry.seed}, which does not use~(b) or~(c). \begin{proposition}[The terminal cycle]\label{p.entry.terminal} There exist $C(\deps)$ and $c(\deps)>0$ such that, for every $M\ge M_*$, there exist~$\tpM$ and a linear map $\Pi\colon H^{\Ntw}(\T^2)^V\to H^{\Noe}(\T^2)^V$, $V$ the set of components of~$\GM$, both depending only on~$M$, $\kappa$ and~$\tin$, with the following properties. For every~$\thetain$ and~$\theta$, let~$Y$ be the solution of~$G'$ on $[\tpM,\tf]$ from the datum equal to $\Pi\UM(t_{(M)})$ on the components of~$\GM$ and to~$0$ on their children. \begin{enumerate}[label=\textup{(\alph*)}] \item $t_{(M)}+ch_M\le\tpM\le t_{(M)}+Ch_M$, $\b_M=\b_{M-1}$ on $[\tpM-\tau'_M,\tpM+\tau'_M]$, and \begin{align*} \|\Pi\UM(t_{(M)})\|_{H^{\Noe}_{\rho(M),\omega}(\T^2)} & \le C\|\thetain\|_{L^2(\T^2)}\,,\\ \|\UM(\tpM)-\Pi\UM(t_{(M)})\|_{L^2_\omega(\T^2)} & \le C\eps_{M-1}^{\Qe}\|\thetain\|_{L^2(\T^2)}\,,\\ \|\UM(\tpM)-\Pi\UM(t_{(M)})\|_{H^{\Nrde}_{\eps_M,\omega}(\T^2)} & \le C\eps_{M-1}^{\Qe/2}\|\thetain\|_{L^2(\T^2)}\,. \end{align*} \item The component energy~\eqref{e.cmp.energy} of~$Y$ of order~$\Noe$ at the scale~$\rho(M)$, with the diffusivity~$\kappa_{M-1}$, on $[\tpM,\tf]$ is at most $C\|\thetain\|_{L^2(\T^2)}$. \item For every $t\in[\tpM,\tf]$, $Y^\sharp(t)\in H^{\Nrde}(\T^2)^V$, and \begin{equation*} \|\UM-Y^\sharp\|_{L^\infty([\tpM,\tf];H^{\Nrde}_{\eps_M,\omega}(\T^2))}\le C\eps_{M-1}^{\Qe/2}\|\thetain\|_{L^2(\T^2)}\,. \end{equation*} \end{enumerate} \end{proposition} \begin{table}[!htbp] \centering \footnotesize \renewcommand{\arraystretch}{1.15} \begin{tabular}{llll} \hline part of the residual & size, over $\|\thetain\|_{L^2(\T^2)}$ & bounded by & coarse flux\\ \hline cell tail & $\bigl(\bar\eps_M/\rho(M)\bigr)^{\Nce}\times{}$cleanup factor & Lemma~\ref{l.cmp.tail} & \\ frontier & $\eps_{M-1}^{\delta(\Nle+1)/2}$ & as Lemma~\ref{l.cmp.frontier} & yes\\ truncation of $\exp_{\pe}$ & $\eps_{M-1}^{\delta\pe}$ & \eqref{e.cmp.truncation}, Lemma~\ref{l.cell.covariance}(b) & yes\\ new smoothings & $\eps_{M-1}^{\delta-\delta/16-\delta/64-\The(\Nse-2)+\The\Nme}$ & Proposition~\ref{p.paired} & yes\\ old smoothings at $\rho(M)$ & $\eps_M^{2(q+2)\gamma-5\delta/64}\bigl(\rw/\rho(M)\bigr)^{\Nmw}$ & Proposition~\ref{p.paired} & yes\\ old smoothings at $\cmol\eps_M$ & $\eps_M^{2(q+2)\gamma-5\delta/64}\bigl(\rw/\eps_M\bigr)^{\Nmw}$ & Proposition~\ref{p.paired} & \\ switching & $\eps_{M-1}^{-C}\exp(-c\kappa\tau_M/\eps_M^2)$ & Lemma~\ref{l.cmp.switching} & \\ \hline \end{tabular} \caption{The residual of the terminal lift~$Y^\sharp$ in the system of~$\GM$ on $[\tpM,\tf]$, measured as in~\eqref{e.cmp.forcing} at the order~$\Nrde$ and the scale~$\eps_M$. The cell tail has the degrees above~$\Nce$; the frontier the words of length $\Nle+1$, each letter gaining $\eps_{M-1}^{\delta/2}$ (proof of Proposition~\ref{p.entry.seed}); the new smoothings are bounded by the priced majorant entry of that proof times the gain $\eps_{M-1}^{\The\Nme}$ of the moments. The cleanup factor $1+(\kappa T)^{\nf12}\eps_M^{-1}(\eps_M/\rho(M))^{\Ncle}$ is at most~$C$, as $(q\gamma+3\delta)\Ncle>3$, and so is the cost of a coarse flux under the lift (Lemma~\ref{l.cmp.coarse}; fourth column), as $\kappa_{M-1}/\kappa\le2$. The truncation row includes the Taylor part, as in the proof of Proposition~\ref{p.cmp.residual}.} \label{t.entry.bill} \end{table} \begin{proof} (a) The preparation of Section~\ref{s.preparation} runs at the terminal level for~$\GM$ from $t_1=t_{(M)}$, the start of the interval $[t_{(M)},\tf]$ on which~$\GM$ satisfies the invariant (Lemma~\ref{l.entry.coefficients}(b)), so that $\tA=t_{(M)}$ and, for $M\ge M_*$, $t_1+2h_M\le \tin+6nh_{M+1}+2h_M\le\tf$, as Proposition~\ref{p.prep.joint} asks. It uses that proposition at $m=M$ for~$\GM$, with~$\Noe$ in place of~$\Nout$ and the constants of Lemma~\ref{l.entry.coefficients}(b), and Propositions~\ref{p.prep.branch}--\ref{p.prep.high} and Lemmas~\ref{l.prep.fibre}--\ref{l.prep.seam}, which hold at $m=M$ (Remark~\ref{r.prep.terminal}), where the contrast $\kappa_{M-1}/\kappa$ is at most~$2$ (Lemma~\ref{l.entry.scales}(b)), so that their powers of the contrast factor are constants. Assembled as in Subsection~\ref{ss.prep.proof}, with the bound $\|\UM(t_{(M)})\|_{H^{\Nbuf}_{\rhomol(M),\omega}(\T^2)}\le C\|\thetain\|_{L^2(\T^2)}$ of Proposition~\ref{p.entry.reduction}(iv) ($\UM(t_{(M)})=U_M(t_{(M)})$, $\rhomol(M)=\cmol\eps_M$, $\Nbuf\le\Ntw$) in place of the molecular smoothing, they give~$\Pi$, depending only on~$\GM$ and~$t_{(M)}$, the first two bounds, the second as $Q\ge4+2qZ>\Qe$ by~\eqref{e.II8.loss}, \eqref{e.II8.orders}, \eqref{e.entry.Nrep} and~\eqref{e.entry.termorders}, and the time~$\tpM$, a block boundary between two quiet margins, with $t_{(M)}+h_M<\tpM\le t_{(M)}+2h_M$. Here~$\GM$ has paths of at most $n\Nlw\le\Npath$ sources (the remark after Definition~\ref{d.entry.weakfamily}, and item~(1) of Subsection~\ref{ss.admissible.parameters}), as Proposition~\ref{p.prep.joint} asks; and that proposition holds with~$\Noe$ in place of~$\Nout$, as its proof uses~$\Nout$ only as the order~$J$ of Lemma~\ref{l.prep.seam}, which holds for every~$J$, in Step~4, and as the order of the norms of Step~5, which the chart at~$\tpr$ changes by at most a constant. The third bound follows by interpolation in Fourier between the second bound and the bounds of $\UM(\tpM)$ in $H^{\Ntw}_{\eps_M,\omega}(\T^2)$ (Lemma~\ref{l.entry.smoothing}) and of $\Pi\UM(t_{(M)})$ in $H^{\Noe}_{\rho(M),\omega}(\T^2)$ (the first bound), as $2\Nrde\le\min\{\Ntw,\Noe\}$ and $\eps_M\le\rho(M)$. (b) Proposition~\ref{p.ft.energy}(c) applies to~$G'$ at the order~$\Noe$, the scale~$\rho(M)$ and the diffusivity~$\kappa_{M-1}$. Its condition~(i) holds since $\rho(M)\le\eps_{M-1}$ and $\rho(M)^2\eps_{M-1}^{\beta-2}=\kappa_{M-1}h_M\eps_{M-1}^{\beta-2}\le C\eps_{M-1}^{2\delta}\kappa_{M-1}$ by~\eqref{e.times}; (ii) as in the proof of Proposition~\ref{p.cmp.energy}(i); and the coupling sums of~$G'$ are at most $C\eps_{M-1}^{\delta/2}$ by clauses~(b3) and~(d) in the proof of Proposition~\ref{p.entry.seed}, which bound the matrix~\eqref{e.ft.coupling} at the order~$\Noe$ and the scale~$\rho(M)$, as it increases with both, $\rho(M)\le\rhomol(M-1)$ and the seed's derivative order is at least~$\Noe$ (item~(2) of Subsection~\ref{ss.admissible.parameters}). That proof uses Lemma~\ref{l.entry.coefficients}(b), on $[t_{(M)},\tf]$, and bounds that hold at every time, so these clauses hold on $[t_{(M)}+3\tau_M,\tf]\supset[\tpM,\tf]$, as $\tpM>t_{(M)}+h_M\ge t_{(M)}+3\tau_M$ (proof of~(a), \eqref{e.scales}). The datum is bounded by~(a). (c) \emph{The identities.} Lemmas~\ref{l.cmp.quiet}, \ref{l.cmp.cancel} and~\ref{l.cmp.tail}--\ref{l.cmp.coarse} hold at $m=M$ for~$\GM$, with $\Nce$, $\Nse$, $\Ncle$, $N_{\mathrm{rec,e}}$, $n\Nlw$ in place of $\Ncor$, $\Nsrc$, $\Ncl$, $\Nrec$, $\Npath$: $\GM$ is in the setting of the lift (Section~\ref{s.comparison}) with these orders, as it satisfies the clauses of the invariant (Lemma~\ref{l.entry.coefficients}(b)), its birth orders are at most $1+n\Ncw\le\Nse$, its paths have at most $n\Nlw$ sources (proof of~(a)), and its birth radii, those of~$G_M$, are $\eps_{M+1}\le2\eps_M^{q-1}\eps_M\le\eps_M$ for $M\ge M_*$ (Definition~\ref{d.entry.weakfamily}, Lemma~\ref{l.fl.scales}(a)). The identity of Lemma~\ref{l.cmp.conjugation} is exact, and the anchor does not change it, as only the derivative $\J_M^{\kappa}-\kappa_{M-1}I$ of the covariance appears in it; the readout identity of~$G'$ is that of Lemma~\ref{l.cmp.readout}, with the same proof. By~(b), $\Ue Y$ and its anchored exponential have the classes of Lemmas~\ref{l.cmp.conjugation}(ii) and~\ref{l.cmp.switching}, as in Step~2 of the proof of Proposition~\ref{p.cmp.residual} with $\Noe$, $\Nse$, $\pe$ in place of $\Nret+3$, $\Nsrc$, $p$: by~\eqref{e.entry.outorders} and the first line of~\eqref{e.entry.outsharp}, $\Noe-\Nse-2\pe-2\ge N_{\mathrm{rec,e}}+1$, which gives the orders $2\pe+1$ and $\Nce+1$ of these lemmas. As in Step~2 of the proof of Lemma~\ref{l.entry.passage}, with the cost $C\eps_M/\rho(M)$ of a derivative at the scale~$\eps_M$ on slow factors, the proofs of Lemmas~\ref{l.cmp.tail}--\ref{l.cmp.coarse} give, for $N\le\Nrde$, the same bounds with $H^N_{\eps_M,\omega}(\T^2)$ in place of $L^2_\omega(\T^2)$ in the norm~\eqref{e.cmp.forcing}, reading~$N$ more orders. At $N=\Nrde$ the residual reads~$Y$ at most at the order $2\Nrde\le\Noe-2$, and the row of the old smoothings at $\cmol\eps_M$ in Table~\ref{t.entry.bill} reads~$Y^\sharp$ at the order $\Nrde+\Nmw+n\Ncw-1\le\Noe-\Nce-2\pe$ (first line of~\eqref{e.entry.outsharp}), up to which the component energy of~$Y^\sharp$ at the scale~$\rhomol(M)$, with the diffusivity~$\kappa$, is at most $C\|\thetain\|_{L^2(\T^2)}$, by the proof of Proposition~\ref{p.cmp.energy}(ii) with the sizes of Lemma~\ref{l.entry.scales}(b). So~$Y^\sharp$ solves the system of~$\GM$ on $[\tpM,\tf]$ with a forcing, the residual, listed in Table~\ref{t.entry.bill}; by~\eqref{e.entry.termorders}, \eqref{e.entry.moments} and Lemma~\ref{l.entry.scales}(b), each is at most $C\eps_{M-1}^{\Qe+2}\|\thetain\|_{L^2(\T^2)}$, the energy of~$Y$ being bounded by~(b). \emph{The initial difference.} At~$\tpM$ the anchored exponential is the identity, $Y$ vanishes on the children, and the correctors of the pulses whose cutoff is positive vanish, by Lemma~\ref{l.cmp.quiet}, as the velocity added at level~$M$ vanishes on $[\tpM-\tau'_M,\tpM+\tau'_M]$ and $\tpM>t_{(M)}+3\tau_M$. So $Y^\sharp(\tpM)=\Pi\UM(t_{(M)})$, and by~(a) the difference $\UM-Y^\sharp$ at~$\tpM$ is at most $C\eps_{M-1}^{\Qe/2}\|\thetain\|_{L^2(\T^2)}$ in $H^{\Nrde}_{\eps_M,\omega}(\T^2)$. \emph{Energy.} Proposition~\ref{p.ft.energy}(a) for~$\GM$ at the order~$\Nrde$ and the scale~$\eps_M$, whose hypotheses hold as in the proof of Lemma~\ref{l.entry.smoothing}, bounds the left side of~(c) by~$C$ times the sum of the initial difference and the residual, and $\eps_{M-1}^{\Qe+2}\le\eps_{M-1}^{\Qe/2}$. By~(b), $Y^\sharp(t)\in H^{\Nrde}(\T^2)^V$, as $\Noe\ge\Nce+2\pe+\Nrde+2$ and the lift spends $\Nce+2\pe$ derivatives, and two more in the Sobolev embedding. \end{proof} \subsection{The first family of the induction} \label{ss.entry.seed} Proposition~\ref{p.entry.seed} makes the terminal successor, restarted at~$\te$, an admissible seed $G=G_{M-1}$. The reserve of generation~$0$ is the one the terminal cycle reaches from~$\res_M$, and the markers of~$G'$ are~$1$, so declaring its components of generation~$0$ changes no coefficient, smoothing or mass. As~$G'$ does not descend from a seed, Lemma~\ref{l.admissible.cycle} does not apply; the main estimate is the recursion for~(b3): a new source gains $\eps_{M-1}^\delta$ from its majorant and loses less than $\eps_{M-1}^{-\delta/2}$. \begin{proposition}[The seed]\label{p.entry.seed} Let $M\ge M_*$ and $\te\coloneqq\tpM+h_M$. Then $\te\le \tin+\frac14t_0$, and the family~$G$ of level $M-1$ obtained from~$G'$ by declaring all its components of generation~$0$ is an admissible seed of level $M-1$ on $[\te,\tf]$ (Definition~\ref{d.admissible.seed}), with the bounds of item~(3) of Subsection~\ref{ss.entry.parameters} and constants depending only on~$\deps$. \end{proposition} \begin{proof} By Propositions~\ref{p.entry.reduction} and~\ref{p.entry.terminal}(a), and as $M\ge M_*$, $\te\le \tin+6nh_{M+1}+(C+1)h_M\le \tin+\frac14t_0$, and $\te\ge\tpM\ge t_{(M)}\ge \tin$, so $\tf-\te\le T$; we check Definition~\ref{d.admissible.seed} with $\tS=\te$. \emph{(i).} The components of~$\GM$ have total mass at most~$\frac23$ (Lemma~\ref{l.entry.coefficients}(b)), their children at most $\ce\eps_{M-1}^{\delta/32}$ times that. The sources of~$\GM$ have birth orders at most $1+n\Ncw$, and the new ones at most~$\Nse$. Along a path, at most $\Nle+1$ new sources separate at most $\Nle+2$ runs of sources of~$\GM$, each of at most $n\Nlw$ sources, so a path has at most $(\Nle+2)(n\Nlw+1)$ components. The widths are $\rw\le\eps_M$ and $\rho(M)\eps_{M-1}^{\The}\le\eps_{M-1}$, and the birth radii of the sources from~$\GM$ are at most $\eps_{M+1}$, those of the new ones $\rho(M)\le\eps_{M-1}$. \emph{(ii).} The coefficients of~$G$ are those of~$G'$. By~(b1) of~(iv) and the transport, as in Steps~6 and~8 of Subsection~\ref{ss.admissible.cycle}, each derivative of a coefficient costs at most~$\eps_{M-1}^{-1}$ (the fast variable of level $M-1$; the slow ones cost~$\eps_{M-2}^{-1}$) and the transport costs $\eps_{M-2}^{d-d'}\le\rho_\sigma^{d-d'}$; so the sum in~(ii) at $\rho=\eps_{M-1}$, which increases with~$\rho$, is at most $C\fac(\sigma)\maj(\sigma)\kappa_{M-1}\rho_\sigma^{d-2}\le C\maj(\sigma)\kappa_{M-1}\rho_\sigma^{d-2}$, as $\fac(\sigma)\le1$: it is~$1$ or a product of factors of letters of the~$G_{i-1}$ (Definitions~\ref{d.admissible.successor}, \ref{d.entry.successor} and the proof of Lemma~\ref{l.entry.coefficients}(b)), each at most $C\eps_{i-1}^{2(q+2)\gamma-\delta/16}\le1$ for $M\ge M_*$ (Lemma~\ref{l.entry.scales}(a)). \emph{(iii).} The cell solutions are those of the weak finite lifts, finitely many two-dimensional cell problems per pulse (Lemma~\ref{l.entry.cellinverse}), each solved once, from zero, on the window of its pulse, in the cell time $\kappa t/\eps_i^2$ of its level $i>M$, depending, among the pulses of level~$i+1$, only on those whose charts are used there; the terminal cycle solves none. Their invariance when the phases move along the pulses and their determination on the window are proved as in Steps~1 and~2 of Subsection~\ref{ss.admissible.cycle}. \emph{(iv): clauses (a), (b1), (b2), (b2$'$).} As in Steps~1, 2 and~4--6 of Subsection~\ref{ss.admissible.cycle}, for~$\GM$ and~$G'$ at $m=M$, with Lemma~\ref{l.entry.coefficients}(b) in place of the invariant, with the reserve~$\res_M$ on all components of~$\GM$, one step $1/(\Ngen+2+\Noff)$ above generation~$0$, and with the supremum in time, as every bound used holds at every time. Three scale facts change: the correctors and the covariance gain $\bar\eps_M/\rho(M)\le C\eps_{M-1}^{q\gamma+3\delta}$ (Lemma~\ref{l.entry.scales}(b)) and $|\Gamma_M(t)-\Gamma_M(\tpM)|/\rho(M)^2\le C\eps_{M-1}^\delta$ (after~\eqref{e.entry.lift}); in Step~5, the cell average of a source~$\sigma$ of~$\GM$ subtracted by a mean letter along~$\sigma$ gains $(\rho_\sigma/\rho(M))^{d'-2}$ or $(\rho_\sigma/\eps_{M-1})^{d-d'}$, at most $C\eps_{M-1}^{q\gamma+3\delta}$ as $\rho_\sigma\le\eps_{M+1}$, in place of $\kappa_M/\kappa_{M-1}$, which is not small here; and in Step~6 the birth radii $\eps_{M+1}$ and $\rho(M)\le C\eps_{M-1}^{1+\gamma/2+\delta}$ are below $\rhomol(M-1)\ge c\eps_{M-1}^{1+\gamma/2}$, and the widths $\rw$, $\rho(M)\eps_{M-1}^{\The}$ below~$\eps_{M-1}$. \emph{(iv): the recursion for (b3).} The markers are~$1$, so the two choices $\omega_v$ and $\omega_v/\Psi_v^2$ of the masses agree. Step~3 of Subsection~\ref{ss.admissible.cycle} uses Definition~\ref{d.admissible.successor} only through two facts, which hold for~$\GM$ and~$G'$ by Definition~\ref{d.entry.successor}: the copies keep the masses of the family, and the entry of a new source is at most a fixed number times the product along its path. It uses the invariant only through the old priced sums, which must be at most~$\frac12$ and are at most $C\eps_M^{2(q+2)\gamma-5\delta/64}$ here, with prices~$1$ (Lemma~\ref{l.entry.coefficients}(b)). So the computation of~\eqref{e.admissible.recursion} applies, with~$C$ depending only on the numbers of words and letters and on~$\Cmaj$. A new source gains~$\eps_{M-1}^\delta$ from its majorant $\Cmaj\eps_{M-1}^{\delta-\delta/16}$, which clause~(b1) justifies as in Step~5 of Subsection~\ref{ss.admissible.cycle}: the mean letters gain powers of~$\bar\eps_M/\rho(M)$, and the covariance and defect letters~$\eps_{M-1}^\delta$, by the scale facts above (for the defect letter, the bound after~\eqref{e.entry.lift} replaces Lemma~\ref{l.cell.covariance}(b)). It costs the power~$\eps_{M-1}^{-\delta/16}$ of its coefficients, the square root $C\eps_{M-1}^{-\delta/64}$ of the mass ratio, and the price $\eps_{M-1}^{-\The(\Nse-2)}$ of its width $\rho(M)\eps_{M-1}^{\The}$. Since $\The(\Nse-2)<\frac1{16}\delta$ by~\eqref{e.entry.termdefs}, the exponent $\delta-\frac{\delta}{16}-\frac{\delta}{64}-\The(\Nse-2)$ exceeds $\frac{55}{64}\delta>\frac12\delta$, and, as $\eps_M^{q-1}\le C\eps_{M-1}^{q(q-1)}$ and $q(q-1)\ge4\delta$, every row and column sum of the priced majorant matrix of~$G'$ is at most \begin{equation*} C\eps_M^{2(q+2)\gamma-5\delta/64}+C\eps_{M-1}^{\delta-\delta/16-\delta/64-\The(\Nse-2)}\le C\eps_{M-1}^{\delta/2}\le\Cseed\eps_{M-1}^{\delta/4}\,. \end{equation*} \emph{(iv): clause (c).} (c1): the sources from~$\GM$ have the smoothing~$\Sw$, and the new ones the round smoothings of Definition~\ref{d.entry.successor}(iii); all have the inverse $2^{\nf12}I$ of the matrix of long axes, of $|\cdot|_\infty$-norm $2^{\nf12}\le2$, and the order~$\ell$ of the kernel~$W_\ell$ of their smoothings satisfies $2\ell-1\ge\Nmom$. (c2): by Lemma~\ref{l.fl.scales}(a), (c) and Proposition~\ref{p.scales}(c), $\rhomol(M-1)\ge c\eps_{M-1}^{1+\gamma/2}$ and $\rho(M-1)\ge c\eps_{M-2}^{1+\gamma/2+\delta}$, and, with $q-1-\frac\gamma2=4\delta+q\gamma$, \begin{align*} \frac{\rw}{\rhomol(M-1)} & \le C\eps_{M-1}^{q\gamma+4\delta+q(\gamma/4+\delta/2)}\,, & \frac{\rho(M)\eps_{M-1}^{\The}}{\rhomol(M-1)} & \le C\eps_{M-1}^{\delta+\The}\,,\\ \frac{\rw}{\rho(M-1)} & \le C\eps_{M-2}^{q^2(1+\gamma/4+\delta/2)-1-\gamma/2-\delta}\,, & \frac{\rho(M)\eps_{M-1}^{\The}}{\rho(M-1)} & \le C\eps_{M-2}^{(q-1)(1+\gamma/2+\delta)+q\The}\,. \end{align*} The exponents of the first line are at least~$\delta$, those of the second at least $q-1\ge4\delta$, as $\frac\gamma2+\delta<\frac{7}{12}(q-1)$. (c3): Lemma~\ref{l.ft.moving}, with the constant matrices $2^{-\nf12}I$. \emph{(iv): clause (d).} As in Step~8 of Subsection~\ref{ss.admissible.cycle}, the coupling matrices of~$G$ at the scale $\rhomol(M-1)$ are at most~$C$ times the priced majorant matrix, entry by entry, so their row and column sums are at most $C\eps_{M-1}^{\delta/2}\le C\eps_{M-2}^{q\delta/2}\le\Cseed\eps_{M-2}^{q\delta/4}$, by Lemma~\ref{l.fl.scales}(a). \end{proof} \subsection{From the first family to the solution~\texorpdfstring{$\theta$}{θ}} \label{ss.entry.readout} \paragraph{Representations of any order.} Each lift spends derivatives, so the representations on the way back down have the orders $n\Ncw+3$ (at~$\eps_M$) down to~$3$ (at~$\eps_{M+n}$): for $N\ge1$, a \emph{representation of order~$N$} at wavelength~$\lambda$ and its size $\size_{\lambda,N}(R)$ are defined by Subsection~\ref{ss.representations} and~\eqref{e.representation} with~$N$ in place of~$\Nrep$ ($\size_\lambda=\size_{\lambda,\Nrep}$), and Lemma~\ref{l.II8.rep} holds for them, with the same proof. \paragraph{The composed weak lift.} For $t\in\R$, let $\W_{M+n}(t)U\coloneqq U$ for a field~$U$ of~$G_{M+n}$, a function, and, for $Ms$. The profiles of Step~2 keep their wavelengths, and the base, read at the wavelength~$\eps_M$, has $\eps_M^{-s}[E]_{C^N_{\eps_M}(\T^2)}\le\eps_{M-1}^{-s}[E]_{C^N_{\eps_{M-1}}(\T^2)}$, as $(\eps_M/\eps_{M-1})^{1-s}\le1$. The three steps use at most $N+\Nce+2\pe+1=\Nrep$ derivatives, which gives the lemma. \end{proof} By Lemma~\ref{l.entry.weakrep}(a), the weak lifts do the same, each spending~$\Ncw$ derivatives, as their correctors take derivatives of the field with a factor~$\eps_i$ each. By~(b), a Sobolev bound at the wavelength~$\eps_i$ becomes, after the lifts, a H\"older bound with the loss~$\eps_{M+n}^{-1-s}$, offset by the accuracies of the weak passages, of the smoothing and of the terminal cycle. \begin{lemma}[The weak lifts of representations]\label{l.entry.weakrep} There exists $C(\deps)$ such that, for every $M\ge M_*$, $M\le i\le M+n$ and $t\in\R$, the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item For every representation~$R$ of a field~$U$ of~$G_i$ of order $(M+n-i)\Ncw+3$ at wavelength~$\eps_i$, there exists a representation~$R^\sharp$ of $\W_i(t)U$ of order~$3$ at wavelength~$\eps_{M+n}$ with $\size_{\eps_{M+n},3}(R^\sharp)\le C\size_{\eps_i,(M+n-i)\Ncw+3}(R)$. \item For every field~$U$ of~$G_i$ with components in $H^{N_i}(\T^2)$, \begin{equation*} \|\W_i(t)U\|_{C^{0,s}(\T^2)}\le C\eps_{M+n}^{-1-s}\|U\|_{H^{N_i}_{\eps_i,\omega}(\T^2)}\,. \end{equation*} \end{enumerate} \end{lemma} \begin{proof} (a) By induction on $M+n-i$: $|\Uw_i|\le2$ as in Step~1 of the proof of Lemma~\ref{l.entry.terminalrep}, and the coefficient of~$\nabla^dP$ in a corrector of degree $j\ge d\ge1$ of~$\Cw_i$ is at most $C\eps_i^j\eps_{i-1}^{d-j}\le C\eps_i^d$, each derivative costing $C\eps_i^{-1}$ (proof of Lemma~\ref{l.entry.coefficients}(a)); so Step~3 there, with~$\eps_i$ for~$\eps_M$ and no exponential, gives the passage, which spends~$\Ncw$ derivatives. (b) As in the proof of Lemma~\ref{l.II8.sobolev}, $U$, with no profile, is a representation of order $N_i-2=(M+n-i)\Ncw+3$ at wavelength~$\eps_i$ of size at most $C\eps_i^{-1-s}\|U\|_{H^{N_i}_{\eps_i,\omega}(\T^2)}$; (a) and Lemma~\ref{l.II8.rep}(a) with the order~$3$ give~(b), as $\eps_i\ge\eps_{M+n}$. \end{proof} In Lemma~\ref{l.entry.identity}, the difference between~$\theta$ and the solution of the terminal successor, lifted through the terminal cycle and the~$n$ weak passages, telescopes into the errors of Lemmas~\ref{l.entry.passage} and~\ref{l.entry.smoothing} and Proposition~\ref{p.entry.terminal}(c), each read in $C^{0,s}(\T^2)$ by Lemma~\ref{l.entry.weakrep}(b). \begin{lemma}[Recovering~$\theta$]\label{l.entry.identity} There exists $C(\deps)$ such that, for every $M\ge M_*$, $\thetain$ and~$\theta$, with $(\W_MY^\sharp)(t)\coloneqq\W_M(t)Y^\sharp(t)$, \begin{equation*} \|\theta-\W_MY^\sharp\|_{L^\infty([\tin+\frac14t_0,\tf];C^{0,s}(\T^2))}\le C\eps_{M-1}^2\|\thetain\|_{L^2(\T^2)}\,. \end{equation*} \end{lemma} \begin{proof} Let $t\in[\tin+\frac14t_0,\tf]$, so $t\ge\te\ge t_{(M)}$. As $\Ntw\ge\Nrde\ge n\Ncw+5$ (item~(4) of Subsection~\ref{ss.entry.parameters}), by the Sobolev embedding $\W_{i-1}(t)$ applies to~$U_{i-1}(t)$ and $\W_M(t)$ to~$Y^\sharp(t)$, whose components lie in $H^{\Ntw}(\T^2)$ and $H^{\Nrde}(\T^2)$ (proof of Lemma~\ref{l.entry.passage}, Proposition~\ref{p.entry.terminal}(c)). By Proposition~\ref{p.entry.reduction}(i) and the definition of the~$\W_i$, \begin{equation*} \theta(t)-\W_M(t)Y^\sharp(t)=\sum_{i=M+1}^{M+n}\W_i(t)\bigl[U_i(t)-\Cw_i(t)\Uw_iU_{i-1}(t)\bigr]+\W_M(t)\bigl[U_M(t)-Y^\sharp(t)\bigr], \end{equation*} and $U_M-Y^\sharp=(U_M-\UM)+(\UM-Y^\sharp)$. We bound each term by Lemma~\ref{l.entry.weakrep}(b), with $\eps_{M+n}^{-1-s}\le C\eps_M^{-2q^n}$ and $\eps_{M+n}^{-1-s}\le C\eps_{M-1}^{-q^{n+1}(1+s)}$ (Lemma~\ref{l.fl.scales}(a), $s<1$). By Lemma~\ref{l.entry.passage} and Proposition~\ref{p.entry.reduction}(iv), each term of the sum is at most $C\eps_M^{-2q^n}\eps_M^{3+2q^n}\|\thetain\|_{L^2(\T^2)}$; by Lemma~\ref{l.entry.smoothing}, whose order $\Ntw-\Nmw-n\Ncw-1$ is at least $\Nrde\ge N_M$, and as $(\frac\gamma4+\frac\delta2)\Nmw>2+2q^n$ (item~(4)), the term with $U_M-\UM$ is at most $C\eps_M^{-2q^n}\eps_M^{q+1+2q^n}\|\thetain\|_{L^2(\T^2)}$; by Proposition~\ref{p.entry.terminal}(c), the term with $\UM-Y^\sharp$ is at most $C\eps_{M-1}^{\Qe/2-q^{n+1}(1+s)}\|\thetain\|_{L^2(\T^2)}=C\eps_{M-1}^{5}\|\thetain\|_{L^2(\T^2)}$. As $\eps_M\le\eps_{M-1}$, the sum is at most $C\eps_{M-1}^2\|\thetain\|_{L^2(\T^2)}$. \end{proof} \subsection{Proof of Proposition~\ref{p.entry}} \label{ss.entry.proof} \begin{proof}[Proof of Proposition~\ref{p.entry}] Let $M\ge M_*$, $\kappa\in\Kset_M$ and $\tin\in\R$. The times $t_{(M)}$, $\tpM$, $\te\in[\tin,\tin+\frac14t_0]$, the map~$\Pi$ and the admissible seed~$G$ of level $M-1$ on $[\te,\tf]$ depend only on~$M$, $\kappa$ and~$\tin$ (Propositions~\ref{p.entry.reduction}, \ref{p.entry.terminal} and~\ref{p.entry.seed}). Given~$\thetain$ and~$\theta$, let~$Y$ be the solution of~$G'$ of Proposition~\ref{p.entry.terminal} and $g\coloneqq Y(\te)$, a datum of~$G$ with $\|g\|_{L^2_\omega(\T^2)}\le C\|\thetain\|_{L^2(\T^2)}$ by~(b) there. As the systems of~$G$ and~$G'$ coincide, the solution~$U$ of~$G$ from~$g$ at time~$\te$ is~$Y$ on $[\te,\tf]$ (Proposition~\ref{p.energy}). For~$[t_1,t_2]$ and~$\Xi$ as in the statement and $t\in[t_1,t_2]$ at which~$R(t)$ exists (so $t\ge\te\ge\tpM$), Lemma~\ref{l.entry.terminalrep}, applied to the representation~$R(t)$ of $U(t)=Y(t)$, and then Lemma~\ref{l.entry.weakrep}(a) at $i=M$ give a representation of $\W_M(t)Y^\sharp(t)$ of order~$3$ at wavelength~$\eps_{M+n}$ of size at most~$C\Xi$, so $\|\W_M(t)Y^\sharp(t)\|_{C^{0,s}(\T^2)}\le C\Xi$ by Lemma~\ref{l.II8.rep}(a). With Lemma~\ref{l.entry.identity}, $\|\theta(t)\|_{C^{0,s}(\T^2)}\le C\Xi+C\|\thetain\|_{L^2(\T^2)}$ for almost every $t\in[t_1,t_2]$. \end{proof} % ===== 15-notation ===== % Index of notation and terms: an unnumbered appendix, placed before the references. Symbols local % to one proof are left out. Each row is a line of three top-aligned boxes, so that the index % breaks between rows at any page boundary (no longtable package is loaded). \phantomsection \section*{Index of notation and terms} \addcontentsline{toc}{section}{Index of notation and terms} {\small \setlength{\parindent}{0pt} \setlength{\parskip}{0pt} \newcommand{\ntrow}[3]{\par\noindent \parbox[t]{0.20\linewidth}{\raggedright\strut#1\strut}\hfill \parbox[t]{0.66\linewidth}{\raggedright\strut#2\strut}\hfill \parbox[t]{0.13\linewidth}{\raggedright\strut#3\strut}\par} \newcommand{\ntgroup}[1]{\par\smallskip\noindent\textbf{#1}\par\nopagebreak\vspace{1pt}\hrule\nopagebreak\vspace{2pt}} The main symbols, by topic, then a glossary of terms, each with the place of its definition; local symbols are left out. Throughout, $k$ indexes pulses, $l$ renewal blocks and $m$ levels (a second level is~$i$: in a chain, in Definition~\ref{d.admissible.admissible} and in the entry; Subsection~\ref{ss.II8.cascade} writes~$j$ and~$i$); $M$ is the terminal level ($\kappa\in\Kset_M$). \ntgroup{General notation} \ntrow{$x\asymp y$}{$C^{-1}y\le x\le Cy$, with~$C$ depending only on $\alpha,q,\zeta$}{\S\ref{s.construction}} \ntrow{$x\ll y$, $y\gg x$}{for $m\ge2$: $x\le C\eps_{m-1}^cy$, with $c>0$ and~$C$ depending only on $\alpha,q,\zeta$}{\S\ref{s.construction}} \ntrow{$|\cdot|$, $|\cdot|_\infty$}{Euclidean norm; largest modulus of a coordinate; their operator norms}{\S\ref{s.construction}} \ntrow{$(g)_E$}{the average $\fint_Eg$ of~$g$ over~$E$}{\S\ref{s.construction}} \ntrow{$\|\cdot\|_{C^{0,s}}$, $[\cdot]_{C^{0,s}}$}{H\"older norm and seminorm}{\eqref{e.holdernorm}} \ntrow{$\nabla^\perp$; $\nabla^jg$}{$(-\partial_2,\partial_1)$; the array of the $2^j$ ordered derivatives of order~$j$ of~$g$}{\S\ref{s.construction}, \S\ref{s.familytools}} \ntgroup{Parameters and exponents} \ntrow{$\alpha$, $\beta$}{H\"older exponent of the velocity field~$\b$, $0<\alpha<\frac13$; $\beta\coloneqq1+\alpha$}{\S\ref{s.construction}} \ntrow{$s$}{H\"older exponent of Theorem~\ref{t.A}, $0