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\title{Non-Uniqueness of Leray--Hopf Solutions to the Unforced\\
Three-Dimensional Navier--Stokes Equations}
\author{Tristan Buckmaster \and Matei P. Coiculescu \and Chen Li}
\date{November 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that the incompressible Navier--Stokes equations in $\R^3$,
without external force, admit a continuum of distinct suitable weak
solutions in the Leray--Hopf class with the same initial datum. The datum is compactly supported,
axisymmetric without swirl, and smooth away from the origin; it lies in
$L^{3,\infty}$ and in $L^q$ for every $q<3$, but not in $L^3$. The
solutions are axisymmetric without swirl and smooth at positive times,
and they satisfy the local and global energy equalities. Any two
of them separate in $L^2$ at the rate $t^{1/4+\sigma}$, for a fixed
$0<\sigma<1/4$. The proof follows the program of Jia and \v Sver\'ak,
with a discretely self-similar background in place of a self-similar one.
The background is an exact unforced solution built from a train of thin
vortex rings, which we obtain by lifting a periodic row of planar vortices
with Gaussian stream function. We construct a growing Floquet mode of its
linearization directly, from the angular-two instability of the planar
vortex, and conclude by a nonlinear correction and a localization of the
datum.
\end{abstract}

\tableofcontents

\section{Introduction}
\label{sec:intro}

We consider the incompressible Navier--Stokes equations in $\R^3$
without external force:
\begin{equation}\label{eq:ns}
 \partial_tu+\operatorname{div}(u\otimes u)-\Delta u+\nabla p=0,
 \qquad \operatorname{div}u=0,\qquad u(0)=u_0.
\end{equation}
Here $u$ is the velocity, $p$ is the pressure, and the viscosity is
normalized to one. For every divergence-free $u_0\in L^2(\R^3)$,
Leray~\cite{leray1934} constructed a global weak solution that satisfies
the energy inequality
\[
 \frac12\|u(t)\|_2^2+\int_0^t\|\nabla u(s)\|_2^2\,ds
 \le\frac12\|u_0\|_2^2,
\]
and Hopf~\cite{hopf1951} gave the corresponding construction in bounded
domains. We call such solutions Leray--Hopf solutions. Whether they are
uniquely determined by their datum is a basic question that goes back to
Leray's work. In this paper we prove that, without force, uniqueness fails
in the axisymmetric class without swirl: Theorem~A below gives a compactly
supported datum from which a continuum of distinct Leray--Hopf solutions
emanates. The proof is analytic. Recent results on the same question are
discussed in Section~\ref{sec:intro-nonuniqueness}.

\subsection{Uniqueness and regularity}
\label{sec:intro-uniqueness}

Uniqueness holds as long as one of the solutions is regular. If a
Leray--Hopf solution lies in $L^r(0,T;L^q)$ with $2/r+3/q=1$ and
$3<q\le\infty$, then every Leray--Hopf solution with the same datum
coincides with it on $[0,T]$. This weak--strong uniqueness theorem is due
to Prodi~\cite{prodi1959}, Serrin~\cite{serrin1963} and
Ladyzhenskaya~\cite{ladyzhenskaya1967}; Kozono and
Sohr~\cite{kozono-sohr1996} treated the endpoint $L^\infty(0,T;L^3)$.
Such regular solutions exist for data in the critical space $L^3$:
Kato~\cite{kato1984} constructed solutions for data in $L^3$, local in
time for large data and global for small data. Koch and Tataru~\cite{koch-tataru2001} obtained global
solutions for small data in $BMO^{-1}$. Escauriaza, Seregin and
\v Sver\'ak~\cite{ess2003} proved that Leray--Hopf solutions bounded in
$L^\infty(0,T;L^3)$ are smooth. For a general Leray--Hopf solution only
partial regularity is known. Scheffer~\cite{scheffer1976} bounded the
size of the singular set at the first singular time, and Caffarelli, Kohn
and Nirenberg~\cite{ckn1982} proved that suitable weak solutions, those
satisfying the local energy inequality, are smooth outside a closed set of
zero one-dimensional parabolic Hausdorff measure.
We refer to the monograph~\cite{lemarie2016} for these results.

The critical spaces are those invariant under the scaling
$u\mapsto\lambda u(\lambda x,\lambda^2t)$ of~\eqref{eq:ns}. The datum of
Theorem~A lies in the weak Lebesgue space $L^{3,\infty}$ but not in
$L^3$; near the origin it is of size $|x|^{-1}$. For such data the theory
is incomplete. Seregin and \v Sver\'ak~\cite{seregin-sverak2017}
studied global weak solutions with large data in $L^3$, and Barker,
Seregin and \v Sver\'ak~\cite{bss2018} constructed, for data in
$L^{3,\infty}$, global weak solutions that are stable under weak-$*$
convergence of the data; their uniqueness is not known.
For a scale-invariant datum it is natural to look for a self-similar
solution $u(x,t)=t^{-1/2}U(x/\sqrt t)$. Jia and \v Sver\'ak~\cite{jia-sverak2014} proved
that every scale-invariant datum that is smooth away from the origin,
of any size, gives rise to at least one forward self-similar solution.
Discretely self-similar data, invariant only under
$u_0(x)\mapsto\lambda u_0(\lambda x)$ for one $\lambda>1$, also admit
discretely self-similar solutions: Tsai~\cite{tsai2014} constructed them
for $\lambda$ close to one, and Bradshaw and Tsai~\cite{bradshaw-tsai2017}
and Chae and Wolf~\cite{chae-wolf2018} for every $\lambda>1$. Our solutions are built around a
discretely self-similar solution of this kind.

In the axisymmetric class without swirl the theory is more complete.
Ladyzhenskaya~\cite{ladyzhenskaya1968} and Ukhovskii and
Yudovich~\cite{ukhovskii-yudovich1968} proved that smooth, sufficiently
decaying axisymmetric data without swirl give global smooth solutions, and
Leonardi, M\'alek, Ne\v cas and Pokorn\'y~\cite{lmnp1999} relaxed the
regularity of the data. Feng and \v Sver\'ak~\cite{feng-sverak2015}
constructed solutions whose initial vorticity is a vortex ring, a measure
supported on circles, and Gallay and \v Sver\'ak~\cite{gallay-sverak2019}
proved that the solution emanating from a circular vortex filament is
unique. The datum of Theorem~A is axisymmetric without swirl and smooth
away from the origin. Theorem~A therefore shows that a singularity at one
point, of the critical size $|x|^{-1}$, is enough for uniqueness to fail
in this class.

\subsection{Non-uniqueness}
\label{sec:intro-nonuniqueness}

Two methods have produced non-unique solutions of the equations of fluid
mechanics. The first is convex integration.
Buckmaster and Vicol~\cite{buckmaster-vicol2019} proved that weak solutions of the
Navier--Stokes equations with finite kinetic energy are not unique.
Buckmaster, Colombo and Vicol~\cite{bcv2022} constructed non-unique weak
solutions that are smooth outside a set of singular times of Hausdorff
dimension less than one, and Cheskidov and Luo~\cite{cheskidov-luo2022}
proved non-uniqueness on the torus in $L^p_tL^\infty_x$ for every $p<2$,
which is sharp in view of the Ladyzhenskaya--Prodi--Serrin criterion; we
refer to the survey~\cite{buckmaster-vicol2021}. These solutions do not satisfy the energy inequality, so they are not
Leray--Hopf solutions, and convex integration leaves the question open for
Leray--Hopf solutions.

The second method is instability. In self-similar variables, an unstable
eigenvalue of the linearization around a self-similar solution becomes,
after the change back to physical variables, a second solution with the
same datum. Vishik~\cite{vishik2018a,vishik2018b} carried this out for the
two-dimensional Euler equations with vorticity in $L^p$, with a force in
$L^1_t(L^1\cap L^p)_x$; a complete account is given by Albritton, Bru\'e,
Colombo, De~Lellis, Giri, Janisch and Kwon~\cite{abcdgjk2024}. Albritton,
Bru\'e and Colombo~\cite{abc2022} adapted the method to the
three-dimensional Navier--Stokes equations and proved non-uniqueness of
Leray--Hopf solutions with a force; their unstable profile is a smooth
compactly supported vortex ring whose cross-section is a modification of
Vishik's unstable vortex. Castro, Faraco, Mengual and
Solera~\cite{cfms2026} obtained sharp non-uniqueness with forcing for the
$\alpha$-SQG equations, which include the Euler equations, using smooth
compactly supported unstable vortices. In all of these results the force is chosen so that an unstable profile
is an exact solution of the forced equation. Without force, the
background must be an exact solution, and constructing an exact solution
whose linearization is provably unstable is the main difficulty; see the
discussion in~\cite{coiculescu2026a}. For the unforced two-dimensional
Euler equations, Coiculescu~\cite{coiculescu2026b} constructed non-unique
solutions with vorticity in $L^1\cap L^p$ by perturbing a stationary
radial solution, with the spectral input verified by rigorous interval
arithmetic. Choi~\cite{choi2026} constructed two solutions with the same
initial vorticity in every finite $L^p$, using a cascade of quadrupoles
across infinitely many scales. For the Navier--Stokes
equations, Coiculescu and Palasek~\cite{coiculescu-palasek2026}
constructed data in $BMO^{-1}$ with two distinct global solutions, both
smooth at positive times, which shows that the small-data theorem of Koch
and Tataru is sharp.

For the unforced Navier--Stokes equations, Jia and
\v Sver\'ak~\cite{jia-sverak2015} proposed the following program. Take a
scale-invariant datum and a forward self-similar solution emanating from
it, which exists by~\cite{jia-sverak2014}. If the linearization around
this solution in self-similar variables has an unstable eigenvalue, then,
under a further spectral condition, there are two distinct Leray--Hopf
solutions with the same datum, and a localization of the datum makes it
of finite energy. The spectral condition concerns a non-self-adjoint
operator around a large solution that is not explicit. Guillod and
\v Sver\'ak~\cite{guillod-sverak2023} computed self-similar solutions for
a one-parameter family of axisymmetric scale-invariant data and found a
symmetry-breaking bifurcation at large data, which would give
non-uniqueness through this program.

Ionescu, Jia and Palasek~\cite{ijp2026} constructed
numerically, with pointwise residuals of order $10^{-10}$, unstable
self-similar solutions in the axisymmetric class without swirl. Binz and
Coiculescu~\cite{binz-coiculescu2026} proved that, for sufficiently
regular data, there are no nontrivial homothetic self-similar solutions in
three dimensions.

Theorem~A carries out a version of the program of Jia and \v Sver\'ak in
which the self-similar solution is replaced by a discretely self-similar
one and the unstable eigenvalue by a Floquet exponent; both are
constructed here. The proof uses no computer assistance, and the solutions
have no swirl.

\subsection{The main result}
\label{sec:intro-result}

Write $L^2_{\rm div}=\{u\in L^2(\R^3;\R^3):\operatorname{div}u=0\}$ and
$H^k_{\rm div}=H^k\cap L^2_{\rm div}$. The weak Lebesgue space
$L^{3,\infty}$ has the quasinorm
$\|f\|_{3,\infty}=\sup_{s>0}s|\{x:|f(x)|>s\}|^{1/3}$.
We include the initial condition in the weak formulation: for a viscosity
$\nu>0$, a pair $(u,p)$ solves the equation with datum $u_0$ on $[0,T)$ if,
for every real $\varphi\in C_c^\infty(\R^3\times[0,T);\R^3)$,
\begin{equation}\label{eq:initial-weak-form}
 \int_0^T\!\int_{\R^3}
 \bigl(u\cdot\partial_t\varphi+(u\otimes u):\nabla\varphi
       +\nu u\cdot\Delta\varphi+p\operatorname{div}\varphi\bigr)\,dx\,dt
 +\int_{\R^3}u_0\cdot\varphi(x,0)\,dx=0.
\end{equation}
Equation~\eqref{eq:ns} is the case $\nu=1$; a general viscosity is used
for intermediate constructions.
The main theorem of this paper is the following.

\begin{namedtheorem}{A}\label{thm:finite-energy-target}
There are
\(T>0\), a nonzero compactly supported datum \(u_0\in L^2_{\rm div}\),
and real solutions \((u^{(\alpha)},p^{(\alpha)})\), \(0\le\alpha\le1\),
of the unforced equation with viscosity one and the same datum. The datum
is smooth away from zero and satisfies
\[
 u_0\in L^{3,\infty}(\R^3)\cap\bigcap_{1\le q<3}L^q(\R^3),
 \qquad u_0\notin L^3(\R^3).
\]
Each solution is smooth at positive times and satisfies
\[
 \begin{aligned}
 u^{(\alpha)}&\in C([0,T];L^2_{\rm div})\cap L^2(0,T;H^1),\\
 u^{(\alpha)}&\in C([\delta,T];H^2_{\rm div})\quad(0<\delta<T),\qquad
 p^{(\alpha)}\in L^{5/3}(\R^3\times(0,T)).
 \end{aligned}
\]
All solutions have the strong \(L^2\) trace \(u_0\) at zero and
satisfy \eqref{eq:initial-weak-form} with \(\nu=1\). For every
\(0\le t_0\le t_1\le T\),
\[
 \frac12\|u^{(\alpha)}(t_1)\|_2^2+
 \int_{t_0}^{t_1}\|\nabla u^{(\alpha)}(t)\|_2^2\,dt
 =\frac12\|u^{(\alpha)}(t_0)\|_2^2.
\]
They also satisfy the local energy equality. Writing
$u=u^{(\alpha)}$, $p=p^{(\alpha)}$ and $e=|u|^2/2$, for every real
$\phi\in C_c^\infty(\R^3\times[0,T))$,
\begin{equation}\label{short-loc:local-energy}
 \begin{aligned}
 \int_0^T\!\int_{\R^3}|\nabla u|^2\phi\,dx\,dt
 ={}&\int_0^T\!\int_{\R^3}
       \bigl[e(\partial_t\phi+\Delta\phi)
                    +(e+p)u\cdot\nabla\phi\bigr]\,dx\,dt\\
 &+\frac12\int_{\R^3}|u_0|^2\phi(x,0)\,dx.
 \end{aligned}
\end{equation}
In particular, the solutions are suitable. Moreover,
\[
 \sup_{0\le\alpha\le1}\|u^{(\alpha)}\|_{L^\infty(0,T;L^{3,\infty})}<\infty,
 \qquad
 \sup_{0\le\alpha\le1}\|u^{(\alpha)}\|_{L^r(0,T;L^q)}<\infty
\]
for every $1\le r\le\infty$, $2\le q\le\infty$ with $2/r+3/q>1$.
There are constants $0<c_*\le C_*<\infty$ and $0<\sigma<1/4$ such that,
for every $\alpha,\alpha'\in[0,1]$ and $0<t\le T$,
\[
 c_*|\alpha-\alpha'|t^{\sigma+1/4}
 \le\|u^{(\alpha)}(t)-u^{(\alpha')}(t)\|_2
 \le C_*|\alpha-\alpha'|t^{\sigma+1/4}.
\]
In particular,
\[
 \int_0^T\|u^{(\alpha)}-u^{(\alpha')}\|_2^2\,dt
 \ge\frac{c_*^2|\alpha-\alpha'|^2T^{2\sigma+3/2}}{2\sigma+3/2}.
\]
Thus the family consists of pairwise distinct Leray--Hopf solutions.
The datum and all solutions are axisymmetric without swirl: they are
invariant under rotations about the \(x_3\)-axis, and their azimuthal
components vanish.
\end{namedtheorem}

The integrability in Theorem~A is sharp on the scale of mixed Lebesgue
spaces. Every solution of the family is bounded in $L^r(0,T;L^q)$ for
all $1\le r\le\infty$ and $2\le q\le\infty$ with $2/r+3/q>1$. By the
weak--strong uniqueness theorem~\cite{prodi1959,serrin1963,ladyzhenskaya1967},
none of them lies in $L^r(0,T;L^q)$ with $2/r+3/q=1$ and
$3<q\le\infty$, since otherwise all of them would coincide. Similarly,
the uniform bound in $L^\infty(0,T;L^{3,\infty})$ cannot be improved to
$L^\infty(0,T;L^3)$: the strong $L^2$ trace would then put $u_0$ in $L^3$.

The rate $t^{\sigma+1/4}$ has a simple origin. To leading order the
difference of two solutions is a multiple of
$t^{\sigma-1/2}b_*(x/\sqrt t,\log t)$, where $b_*$ is the periodic
profile of a Floquet mode with exponent $\sigma$ (see
Section~\ref{sec:intro-construction}); its $L^2$ norm is a multiple of
$t^{\sigma+1/4}$.

\subsection{The construction}
\label{sec:intro-construction}

We follow the program of Jia and \v Sver\'ak~\cite{jia-sverak2015}.
In the similarity variables $y=x/\sqrt t$ and $\tau=\log t$, write
$u(x,t)=t^{-1/2}U(x/\sqrt t,\log t)$. Equation~\eqref{eq:ns} becomes
\[
 \partial_\tau U=\Delta U+\tfrac12y\cdot\nabla U+\tfrac12U
 -\mathbb P\operatorname{div}(U\otimes U),
\]
where $\mathbb P$ is the Leray projection, and the initial time $t=0$
becomes $\tau=-\infty$. Self-similar solutions are stationary in $\tau$.
Discretely self-similar solutions, those with
$u(\lambda x,\lambda^2t)=\lambda^{-1}u(x,t)$ for one $\lambda>1$, are
periodic in $\tau$ with period $2\log\lambda$. Suppose that $U_*$ is a
periodic solution and that its linearization has a Floquet solution
$e^{\sigma\tau}b_*$, with $\sigma>0$ and $b_*$ periodic. Since
$e^{\sigma\tau}=t^\sigma$ vanishes at $t=0$, the fields
$U_*+\alpha e^{\sigma\tau}b_*$, $\alpha\in\R$, all carry the datum of
$U_*$, and one expects to correct each of them to an exact solution. The
difficulty is to find an exact unforced periodic solution whose
linearization provably has such a mode. Proposition~\ref{prop:unstable-background}
in Section~\ref{sec:reduction} states the background and the mode we
construct. The construction has four steps.

\paragraph{Step 1: the planar vortex train and its Gaussian instability
(Sections~\ref{sct:section}--\ref{sec:gaussian-evolution}).}
The building block is the planar vortex with Gaussian stream function
$\psi_0(x)=-2e^{-|x|^2/4}$, $x\in\R^2$. Its velocity
$\nabla^\perp\psi_0=e^{-|x|^2/4}x^\perp$ rotates, and its vorticity
$(2-|x|^2/2)e^{-|x|^2/4}$ changes sign and has zero integral: it is a
shielded vortex. The instability of shielded vortices is classical in
geophysical fluid dynamics~\cite{gent-mcwilliams1986}. We prove that the
Euler equations linearized around $\psi_0$ have exactly one unstable pair
of eigenvalues $\lambda,\bar\lambda$, with $\lambda=\gamma+i\omega$,
$\gamma>0$, and eigenfunctions in angular mode two
(Proposition~\ref{prop:short-gaussian-seed}). The count uses the index
theorem of Lin and Zeng~\cite{lin-zeng2021} together with an explicit
negative direction. Lin~\cite{lin2003} developed related criteria for
instability. With a small viscosity
$\nu>0$ the vortex diffuses: at time $s$ its vorticity is
$a^{-2}E_0(x/\sqrt a)$, $a=1+\nu s$, where $E_0$ is the vorticity of
$\psi_0$. The turnover clock $s/(1+\nu s)$ has total length
$\nu^{-1}$. In the heat variables the unstable evolution has the canonical
normalization $N_\nu=e^{\lambda/\nu}\nu^{-\kappa}$, for a constant
$\kappa\in\C$ (Proposition~\ref{prop:short-canonical}). A fixed canonical
age $\theta=[\nu(1+\nu s)]^{-1}>0$ corresponds to physical time
$s=(\nu^2\theta)^{-1}-\nu^{-1}$, of order $\nu^{-2}$. The train is the
exact solution of the planar Navier--Stokes equations with viscosity
$\nu$ on the cylinder $(\R/c\mathbb Z)\times\R$, $c=\nu^{-2}$, whose
initial stream function is the periodic sum
$\sum_{n\in\mathbb Z}\psi_0(q-nc,\eta)$. It exists globally by the
theorem of Gallay and Slijep\v cevi\'c~\cite{gallay-slijepcevic2015}.
Until the diffusing vortices overlap the train is close to the sum of
independently evolving vortices, and afterwards it is a decaying shear
flow plus a part that decays exponentially (Lemma~\ref{sct:train}).

\paragraph{Step 2: thin rings and the exact discretely self-similar
background (Sections~\ref{sec:ring-lift}--\ref{short-bg:section}).}
In cylindrical coordinates $(r,z)$ and for $R\ge2$, use the
logarithmic ring chart
\[
 q=R\log(r/R),\qquad \eta=Rz/r,\qquad s=R^2t/r^2.
\]
The stream function $r\psi(q,\eta,s)$ of the train defines an
axisymmetric field without swirl. The planar vortex centered at $q=nc$
becomes a vortex ring of radius $r_n=Re^{nc/R}$ whose cross-section has
size comparable to $r_n/R$; the rings accumulate at the origin and at
infinity. The dilation $(r,z,t)\mapsto(\Lambda r,\Lambda z,\Lambda^2t)$
with $\Lambda=e^{c/R}$ sends $q$ to $q+c$ and fixes $\eta$ and $s$, so
the periodicity of the train becomes exact discrete self-similarity:
translation becomes discrete scaling. The lifted field solves the
Navier--Stokes equations up to a curvature residual, which is small when
the rings are thin, $\epsilon=R^{-1}\to0$
(Proposition~\ref{sct:lift}). In similarity variables the residual is a
small periodic force. Since the background must be an exact unforced
solution, we remove it: a right inverse for the linearized periodic
problem and a nonlinear correction give an exact periodic solution
$U_*$. Its datum $u_{\rm dss}$ is discretely self-similar and of size
$|x|^{-1}$, and $U_*$ differs from the heat evolution of $u_{\rm dss}$
by a remainder in $H^2$ (Section~\ref{short-bg:section}). The three
small parameters are tied together: $c=\nu^{-2}$ and $\epsilon=e^{-d/\nu}$
with $d$ in a fixed compact interval.

\paragraph{Step 3: the direct three-dimensional Floquet construction
(Sections~\ref{sec:stress-spaces-short}--\ref{sec:spectral-assembly}).}
We fix the exact background and look for $0<\sigma<1/4$ and a nonzero
periodic profile $b_*$ solving~\eqref{loc:eq:mode}. The mode comes from
a loop through the rings. In similarity variables the rings move toward
the origin and age as $\tau$ increases: rings far from the origin are
young and still close to the inviscid Gaussian vortex, and rings near
the origin are old and have diffused. A strain acting on a young ring
excites its unstable angular-two mode, which grows by the factor
$N_\nu$ during the life of the ring. Once the ring is old, the stress of
the amplified perturbation generates, through the three-dimensional
pressure, a strain on the younger rings. By discrete self-similarity this
loop repeats in every period, and a Floquet mode is a loop that
reproduces its input up to the factor $e^{-\sigma\mathcal T}$, where
$\mathcal T$ is the period.

We find the exponent in the spirit of the Birman--Schwinger
principle~\cite{birman1961,schwinger1961}: instead of studying the
spectrum of the full linearized operator, we reduce the Floquet problem to
a scalar equation for the gain of one loop. We solve the forced
linearized equation along the train with zero initial data and bounds polynomial in
$\nu^{-1}$ (Section~\ref{sec:planar-response}), and rewrite the Floquet
problem as a fixed point problem for the return operator, which maps the
strain received by young rings to the strain they return. In the joint
limit $\epsilon,\nu\to0$, and on a fixed space of smooth sources, this
operator is a rank-one operator plus a remainder that tends to zero. Its
nonzero eigenvalue, the gain of one loop for an exponent $\mu$, is
\[
 2\epsilon^3\nu^{5+2\mu}\operatorname{Re}\{\mathcal C(\mu)N_\nu\}+o(1),
\]
where $\mathcal C$ is an explicit holomorphic function that does not
vanish identically. The factor $\epsilon^3$ measures the weak
three-dimensional coupling of thin rings, and $N_\nu$ is the planar
amplification. We choose $\nu_n\to0$ so that the phase of
$\mathcal C N_{\nu_n}$ is fixed and $\epsilon_n$ so that the gain is
close to one, construct the background once at these parameters, and vary
only $\mu$ in a window of width $|\log\nu_n|^{-1}$. Across the window the
gain moves from about $e$ to about $e^{-1}$. Its crossing of one gives a
Floquet exponent $\sigma_n$ of the exact three-dimensional operator, with
its full pressure and the correction of Step~2
(Section~\ref{sec:spectral-assembly}). The planar response only supplies
the limit of the return operator; the mode is constructed for the exact
operator.

\paragraph{Step 4: nonlinear correction and localization
(Section~\ref{sec:localization}).}
With the background and the mode in hand we follow the instability and
localization argument of Jia and \v Sver\'ak~\cite{jia-sverak2015},
adapted to a profile periodic in $\tau$. We replace the infinite-energy
datum $u_{\rm dss}$ by a compactly supported divergence-free truncation
that equals $u_{\rm dss}$ near the origin. The resulting common heat
correction $K$ satisfies $\|K(\tau)\|_{W^{2,\infty}}=O(e^{\tau/2})$.
For each amplitude $\alpha\in[0,1]$ we seek a profile
$U_*+K+\alpha e^{\sigma\tau}b_*+z_\alpha$ with
$\|z_\alpha(\tau)\|_{H^2}=O(e^{\beta\tau})$ as $\tau\to-\infty$,
where $\sigma<\beta<2\sigma<1/2$. We choose $\beta$ so that no Floquet
multiplier has modulus $e^{\beta\mathcal T}$ and split the periodic
evolution at this rate. The resulting right inverse accounts for every
faster unstable mode, so no maximal-growth assumption on $\sigma$ is
needed. Both forcing rates $2\sigma$ and $1/2$ exceed $\beta$, and a
contraction gives the nonlinear corrections. Their Lipschitz dependence
on $\alpha$, uniformly on one interval $[0,T]$, gives the two-sided
separation estimate of Theorem~A.

\paragraph{Organization.}
Section~\ref{sec:reduction} fixes conventions and states
Proposition~\ref{prop:unstable-background}. Sections~\ref{sct:section}--\ref{sec:localization} carry out the four steps above, and the proof of
Theorem~A is completed at the end of Section~\ref{sec:localization}.

\paragraph{Formalization.}
A Lean formalization of this work is forthcoming at
\url{https://github.com/mcoiculescu/unforced-leray-hopf-public}.

\subsection*{Declaration on the use of large language models}
\addcontentsline{toc}{subsection}{Declaration on the use of large language models}

This work was carried out with extensive use of the large language models
Claude (Anthropic) and Codex (OpenAI). The models were used to search for
the construction, to write and revise the proofs, and to check them.

While exploring the instability of an exact self-similar profile as a
mechanism, among others, for nonuniqueness, GPT-6 Astra proposed using a
planar unstable vortex as a building block to lift to an unstable
profile. We found this approach promising and pushed the model to
complete the construction of an exact discretely self-similar solution
with a growing mode.

The argument first appeared in a draft of about 200 pages, written largely
by the models, whose exposition was poor and highly redundant. We then
realized that the construction does not need swirl; working in the
axisymmetric class without swirl reduced the draft to about 150 pages.
Finally, we replaced the original proof of the existence of the Floquet
exponent by a more standard argument in the spirit of the Birman--Schwinger
principle, which reduces the Floquet problem to a scalar equation for the
gain of one loop (Section~\ref{sec:intro-construction}). This cut the
length of the proof in half, to about 70 pages.

The authors take full responsibility for the content of this paper.

\section{Conventions and the reduction to a growing mode}
\label{sec:reduction}

We use $(u\otimes v)_{ij}=u_iv_j$,
$(\operatorname{div}T)_i=\partial_jT_{ij}$,
$R_j=\partial_j(-\Delta)^{-1/2}$ and the Cartesian Leray projection
$\mathfrak P=\mathbb P$. The pressure is the global choice
$p=\sum_{i,k}R_iR_k(u_i u_k)$. A vector field is axisymmetric without
swirl if it is invariant under rotations about the $x_3$-axis and its
azimuthal component vanishes.
For intermediate constructions the viscosity is $\nu>0$, and the initial
condition is included through~\eqref{eq:initial-weak-form}.

The construction uses the similarity variables $y=x/\sqrt t$,
$\tau=\log t$. For a period $\mathcal T>0$, set
\[
 \mathcal Y_q=L^q([0,\mathcal T],d\tau/\mathcal T;L^q(\mathbb R^3)),
 \qquad \mathcal T_0=\partial_\tau-\tfrac12y\cdot\nabla.
\]
Periodic spaces identify their endpoints. The analytic construction
needed for localization is stated below.

\begin{proposition}\label{prop:unstable-background}
There exist \(\nu>0\), \(\mathcal T>0\), and a nonzero real divergence-free
field \(u_{\rm dss}\) on \(\mathbb R^3\), smooth away from zero, satisfying
\begin{equation}
 u_{\rm dss}(e^{\mathcal T/2}x)=e^{-\mathcal T/2}u_{\rm dss}(x),\qquad
 |D^\alpha u_{\rm dss}(x)|\le C_\alpha |x|^{-1-|\alpha|}
 \quad (x\ne0,\ |\alpha|\le4).
 \label{loc:eq:dss}
\end{equation}
Set
\[
 H_\nu[u_{\rm dss}](y,\tau)=e^{\tau/2}
       [e^{\nu e^\tau\Delta}u_{\rm dss}](e^{\tau/2}y).
\]
There is a real divergence-free \(\mathcal T\)-periodic distributional
solution \(U_*\) of
\begin{equation}
 \partial_\tau U_*=\nu\Delta U_*+\tfrac12 y\cdot\nabla U_*+\tfrac12 U_*-\mathfrak P\operatorname{div}(U_*\otimes U_*),
 \qquad U_*=H_\nu[u_{\rm dss}]+r_{\rm heat},\quad r_{\rm heat}\in C_{\rm per}H^2(\mathbb R^3).
 \label{loc:eq:background}
\end{equation}

There are \(0<\sigma<1/4\) and a nonzero real \(\mathcal T\)-periodic
divergence-free field \(b_*\) satisfying
\begin{equation}
 b_*\in\mathcal Y_{2,\rm div},\qquad \nabla b_*\in\mathcal Y_2.
 \label{loc:eq:energy-mode}
\end{equation}
The second space is tensor-valued. The periodic field \(b_*\) is the
Floquet profile: \(e^{\sigma\tau}b_*\) solves the linearized equation
and has multiplier \(e^{\sigma\mathcal T}>1\) over one period.
The equation for the profile is
\begin{equation}
 \begin{gathered}
 \sigma b_*+\mathcal T_0 b_*-\nu\Delta b_*-\tfrac12 b_*
 +\mathfrak P\operatorname{div}(U_*\otimes b_*+b_*\otimes U_*)=0\\
 \hbox{in }\mathcal D'(\R^3\times\R/\mathcal T\mathbb Z).
 \end{gathered}
 \label{loc:eq:mode}
\end{equation}
The fields \(u_{\rm dss}\), \(U_*\) and \(b_*\) are axisymmetric
without swirl.

\end{proposition}

Proposition~\ref{short-loc:reduction} in Section~\ref{sec:localization}
shows that Proposition~\ref{prop:unstable-background} implies Theorem~A.
Constants denoted by $P_\nu$ or $\mathcal P_\nu$ are fixed polynomials
in $\nu^{-1}$ and the specified polynomial cell parameters. Each
exponential loss is kept separately until the final parameter
inequalities are imposed.

\section{The exact periodic planar train}
\label{sct:section}

We start with an exact nonlinear solution on the periodic cylinder
$(\mathbb R/c\mathbb Z)\times\mathbb R$. Its initial stream is a train
of translated Gaussians; the independently evolving Gaussian sum is an
approximation whose interactions must be estimated. The train bounds proved here and the Gaussian instability proved in
Section~\ref{sec:gaussian-evolution} are used to construct the background
and its growing mode.

Fix a finite collection of derivative orders, transverse weights and
Lebesgue exponents, and fix $\rho>1/2$. Throughout this section
\[
 c=\nu^{-\rho},\qquad e_c=(\nu c^2)^{-1},\qquad
 C_c=(\mathbb R/c\mathbb Z)\times\mathbb R,
 \qquad a(s)=1+\nu s,\qquad s_c=(c^2-1)/\nu.
\]
Cell norms are unnormalized. A bar denotes the normalized horizontal
average and a tilde its complement. The letter $P$ denotes a polynomial
in $c,\nu^{-1}$, with degree depending only on the fixed collection;
$P_S$ may also depend polynomially on $1+S$. All the smallness
conditions below are satisfied by first fixing that collection and then
decreasing $e_c$. In particular they hold for $c=\nu^{-2}$ and small $\nu$.

Let $v=\nabla^\perp\psi$ be the exact cylinder Navier--Stokes solution
with viscosity $\nu$ and initial stream
\[
 \psi(q,\eta,0)=\sum_{n\in\mathbb Z}-2
                  e^{-((q-nc)^2+\eta^2)/4}.
\]
This specifies the nonlinear train, rather than the sum of independently
evolving vortices. The datum is divergence-free and belongs to
$\operatorname{BUC}\cap H^k$ for every $k$, where $\operatorname{BUC}$
denotes the bounded uniformly continuous functions on $C_c$; its vorticity is in
$L^1\cap L^\infty$ and has zero integral. The cylinder existence theorem
of Gallay and Slijep\v cevi\'c~\cite{gallay-slijepcevic2015} therefore
applies after interchanging the two coordinates and taking density one.
It gives a unique global mild solution in $\operatorname{BUC}$, smooth for positive
time, with the bounded periodic pressure choice.
It is also the finite-energy solution: on each finite interval the mild
equation in $\operatorname{BUC}\cap L^2$ gives
\[
 \|v(t)\|_2\leq\|v(0)\|_2+
 C\nu^{-1/2}\sup_{[0,t]}\|v\|_\infty
       \int_0^t(t-r)^{-1/2}\|v(r)\|_2\,dr,
\]
which continues $L^2$ membership. Approximation and transverse cutoffs
then give the velocity and vorticity energy identities. The scalar
vorticity equation preserves zero circulation and its $L^1,L^\infty$
bounds. Uniqueness gives $v(-q,-\eta,s)=-v(q,\eta,s)$.

\begin{lemma}
\label{sct:train}
Put
\[
 \psi^p(x,s)=-2a^{-1}e^{-|x|^2/(4a)},\qquad
 V_c^p=\sum_n\nabla^\perp\psi^p(q-nc,\eta,s).
\]
For every fixed $j$, with constants independent of $c,\nu$,
\begin{align}
 \|D^j(v-V_c^p)(s)\|_2
 &\leq C_j\nu^{-1}a^{-2-j/2}e^{-b_jc^2/a},\\
 \|D^j(v-V_c^p)(s)\|_\infty
 &\leq C_j\nu^{-1}a^{-(5+j)/2}e^{-b_jc^2/a},
 &&1\leq a\leq c^2. \label{sct:comparison}
\end{align}
After overlap, set $X=q/c$, $Y=\eta/c$,
$T=\nu(s-s_c)/c^2$, and write
$c^3v=b(Y,T)e_1+W(X,Y,T)$, $\overline W=0$. Then
\begin{align}
 \|\partial_Y^j b(T)\|_p
 &\leq C_{j,p}(1+T)^{-1-j/2+1/(2p)}, &&p=2,\infty,\\
 \|D^jW(T)\|_{2\cap\infty}&\leq C_j e^{-\lambda_jT}.
 \label{sct:overlap}
\end{align}
In particular, whenever $a(s_0)\leq c^2$,
\begin{align}
 \|v(s)\|_2&\leq Ca(s)^{-1} &&(s\leq s_c),\\
 \|v(s)\|_2&\leq Cc^{-2}(s/s_c)^{-3/4} &&(s\geq s_c),\\
 \int_{s_0}^\infty\sum_{j=1}^{k+1}\|D^jv(s)\|_\infty\,ds
 &\leq \frac{C_k}{\nu a(s_0)}+C_ke_c,\\
 \int_{s_0}^\infty\|\widetilde v(s)\|_2\,ds
 &\leq \frac C\nu\left(1+\log\frac{c^2}{a(s_0)}\right).
 \label{sct:integrated}
\end{align}
\end{lemma}

\begin{proof}
The isolated vortex is an exact radial heat solution. Its periodization
has residual $\operatorname{div}Q$, where
$Q=\sum_{i\ne j}V_i\otimes V_j$. Completing squares in the Gaussian
products and summing their separation index gives
\[
 \|D^jV_c^p\|_\infty\leq C_ja^{-(3+j)/2},\quad
 \|V_c^p\|_2\leq Ca^{-1},\quad
 \|D^jQ\|_2\leq C_ja^{-(5+j)/2}e^{-b_jc^2/a}.
\]
Indeed the product is bounded by $Ca^{-3}e^{-b(i-j)^2c^2/a}$
times a Gaussian of width $\sqrt a$ centered at the midpoint; its
cell $L^2$ norm contributes $C\sqrt a$. Fixed weights in units of
$\sqrt a$ obey the same estimate.

Write $e=v-V_c^p$, use slow time $\sigma=\nu s$, and choose an integer
$N\geq3$ at least two above the required supremum order. For
$0<\delta\leq(4N)^{-1}$ set
$E_N^2=\sum_{j=0}^N(\delta a)^j\|D^je\|_2^2$.
On the bootstrap interval $\|\nabla e\|_\infty\leq a^{-2}$,
differentiated energy and one integration by parts on the stress give
\[
 (E_N^2)' +\sum_{j=0}^N(\delta a)^j\|\nabla D^je\|_2^2
 \leq \frac{C_N}{\nu a^2}E_N^2+
                 C_N\nu^{-2}a^{-5}e^{-b_Nc^2/a}.
\]
Here differentiated transport is controlled by
$\|[D^j,e\cdot\nabla]e\|_2
 \leq C_j\|\nabla e\|_\infty\|D^je\|_2$.
Its constants are independent of the period: integration by parts gives
$\|D^lf\|_{2m/l}\leq C_m\|f\|_\infty^{1-l/m}
\|D^mf\|_2^{l/m}$, and applying this to $f=\nabla e$ bounds
each commutator product. The differentiated weights are absorbed by
the preceding derivative's dissipation. Time-ordered Gronwall yields
\[
 E_N^2(a)\leq C_N\nu^{-2}\int_1^a
 r^{-5}e^{-b_Nc^2/r+C_N/(\nu r)}\,dr
 \leq C_N\nu^{-2}a^{-4}e^{-b'_Nc^2/a},
\]
because $e_c$ is small. In the last integral use $u=c^2/r$.
The cylinder interpolation inequality
\[
 \|f\|_\infty\leq
 C\|f\|_2^{1/2}\|D^2f\|_2^{1/2}
 +Cc^{-1/2}\|f\|_2^{1/2}\|\nabla f\|_2^{1/2}
\]
includes the one-dimensional mean term; its other term follows by
Fourier inversion on nonzero horizontal modes. It proves the second
estimate in \eqref{sct:comparison}, since $a^{1/4}c^{-1/2}\leq1$.
Consequently $\|\nabla e\|_\infty\leq
 Ce_c(c^2/a)e^{-bc^2/a}a^{-2}\leq a^{-2}/2$.
Continuity from the smooth datum closes the bootstrap to $s_c$.

At overlap, $U=c^3v$ solves
$U_T+e_cU\cdot\nabla U+\nabla\Pi=\Delta U$ on the unit cylinder,
and \eqref{sct:comparison} gives $\|U(0)\|_{H^N}\leq C_N$.
Fix $N\geq3$ sufficiently above the entire required derivative budget
for the Sobolev interpolations below. Differentiated
transport gives
\[
 \frac12\frac d{dT}\|U\|_{H^N}^2+
 (1-C_Ne_c\|U\|_{H^N})\|\nabla U\|_{H^N}^2\leq0.
\]
Small $e_c$ thus gives the uniform high-order norm directly. The exact
fluctuation energy identity is
\[
 \frac12\frac d{dT}\|W\|_2^2+\|\nabla W\|_2^2
       =-e_c\int W_1W_2 b_Y.
\]
The horizontal Poincare gap gives exponential $L^2$ decay; interpolation
with the uniform high-order bound gives all required smooth decay.

The mean stream $S=\overline\psi$ is fixed by
\begin{equation}
 S_s-\nu S_{\eta\eta}=g:=\overline{v_1v_2},\qquad
 \overline v_1=-S_\eta,\qquad\overline v_2=0.
 \label{sct:mean}
\end{equation}
The last equality follows from divergence and finite energy; uniqueness
of the averaged velocity equation proves the preceding equality.
Separate reflection symmetries of $V_c^p$ imply
$\overline{(V_c^p)_1(V_c^p)_2}=0$. Consequently
\[
 \|S(s_c)-\overline{\psi_c^p}(s_c)\|_1
 \leq C\nu^{-2}c^{-5}+C\nu^{-3}c^{-7}.
\]
For example, the mixed term is bounded by
$C\nu^{-2}c^{-1}\int_1^{c^2}a^{-3}e^{-bc^2/a}\,da$;
the quadratic error has $\nu^{-3}a^{-4}$ in its place.
Thus the rescaled mean stream function $\mathsf S=c^2S$ satisfies
$\|\mathsf S(0)\|_{L^1_Y}\leq C$. It solves
\[
 \mathsf S_T-\mathsf S_{YY}=e_c\overline{W_1W_2},
 \qquad b=-\mathsf S_Y.
\]
Split Duhamel at $T/2$. On its early half use the source $L^1$ bound
and $\|\partial_Y^{j+1}G_T\|_p
\leq C T^{-1-j/2+1/(2p)}$; on its late half move derivatives
onto the exponentially small source. This proves \eqref{sct:overlap}.
Integrating the two regimes proves \eqref{sct:integrated}; in particular
the pre-overlap error contributes at most
$C_j\nu^{-2}c^{-j-3}$ to $\int\|D^je\|_\infty$,
and the post-overlap contribution is $C_j\nu^{-1}c^{-j-1}$.
For smoothness statements at fixed parameters there is no restriction
to this quantitative derivative budget. The resulting integrable
strain and the tame higher-order inequality
$d\|U\|_{H^k}/dT\leq C_ke_c\|\nabla U\|_\infty\|U\|_{H^k}$
propagate every higher norm from its finite smooth overlap datum.
All higher-order versions of the argument therefore hold at fixed
parameters, without requiring a new smallness condition for each order.
\end{proof}

\begin{lemma}[One weighted heat description at all ages]
\label{sct:heat}
Every fixed mixed age/spatial derivative of
$(\widetilde\psi,\widetilde v,p)$ obeys
\begin{equation}
 \|\langle\eta\rangle^m\partial_s^\ell D^j
       (\widetilde\psi,\widetilde v,p)(s)\|_{2\cap\infty}
       \leq P e^{-\kappa\nu s/c^2}. \label{sct:fluctuation}
\end{equation}
The decaying pressure gauge is used here. The mean stream is even and,
for every fixed $N$, has the following expansion and remainder for
nonnegative integers $m,b,\ell$ and $1\leq p\leq\infty$:
\begin{align}
 S(\eta,s)&=\sum_{j=0}^Na_{2j}\partial_\eta^{2j}G_{\nu s}(\eta)
                         +\mathcal R_N(\eta,s),\\
 a_{2j}&=\frac{\int\eta^{2j}S(\eta,0)\,d\eta}{(2j)!}
 +\int_0^\infty\sum_{k+\ell=j}
 \frac{(-\nu t)^\ell}{\ell!(2k)!}
          \int\eta^{2k}g(\eta,t)\,d\eta\,dt,\\
 \|\eta^m\partial_\eta^b\partial_s^\ell\mathcal R_N(s)\|_p
 &\leq P s^{(m-b-2N-3)/2-\ell+1/(2p)},\qquad s\geq1.
 \label{sct:expansion}
\end{align}
In particular, replacing the remainder power by
$(m-b-1)/2-\ell+1/(2p)$ bounds $S$ itself with $s$ replaced by $1+s$.
On each $0\leq s\leq S_*$, all these fields and fixed derivatives have
transverse tails
\begin{equation}
 P_{S_*}\{e^{-b l/c}+e^{-b l^2/(C_0+\nu S_*)}\},\qquad
 |\eta|\geq C/\nu+2+l. \label{sct:transverse}
\end{equation}
On every fixed slow-age interval $0\leq s\leq\Sigma/\nu$, put
$\sigma=\nu s$ and let $p_c^p$ be the sum of the isolated radial
pressures in the decaying gauge. For
$d=(v-V_c^p,\omega-\operatorname{curl}V_c^p,\nabla(p-p_c^p))$,
\begin{equation}
 \sum_{\ell\leq L,j\leq k}
 \left(\|\langle\eta\rangle^m\partial_\sigma^\ell D^jd\|_2
              +\|\partial_\sigma^\ell D^jd\|_\infty\right)
 \leq P e^{C_\Sigma/\nu-b_\Sigma c^2}.
 \label{sct:fixed-band}
\end{equation}
\end{lemma}

\begin{proof}
Lemma~\ref{sct:train} gives
$\int_0^\infty\|v_2(s)\|_\infty\,ds\leq C/\nu$:
integrate $a^{-3/2}$ and the comparison error before overlap, and
$c^{-3}W_2$ afterwards. For the vorticity drift--diffusion equation,
the Gaussian datum and the supersolutions
\[
 C\exp\{-\lambda(\eta-A(s))+(C_0+\nu s)\lambda^2\},\qquad
 A(s)=\int_0^s\|v_2(t)\|_\infty\,dt,
\]
with both transverse signs, imply
$|\omega|\leq C\exp[-b(|\eta|-C/\nu)_+^2/(C_0+\nu s)]$.
The comparison is justified by exhaustion with a positive quadratic
barrier. Unit-patch interpolation with the fixed higher smooth bounds
from Lemma~\ref{sct:train} gives the same tail, with reduced exponent,
for every required spatial derivative of $\omega$.

For $k_n=2\pi n/c\ne0$, the stream inverse has kernel
$-e^{-|k_n||\eta|}/(2|k_n|)$. Move higher derivatives to the source
and sum modes using a fixed extra horizontal derivative. Splitting the
convolution at distance $l/2$ proves \eqref{sct:transverse} for the
nonzero stream and velocity modes, with only polynomial costs in $c$.
The zero mode has two decaying primitives:
\[
 \overline v_1(\eta)=\int_\eta^\infty\overline\omega(\xi)\,d\xi,
 \qquad S(\eta)=\int_\eta^\infty\overline v_1(\xi)\,d\xi.
\]
Zero circulation makes the first primitive agree at both ends.
Central oddness makes $\overline v_1$ odd and integrable, so its
integral is zero and the second primitive also decays at both ends.
It agrees with \eqref{sct:mean} and is even. These formulas prove the
mean tails and polynomial weighted bounds for all fields at every age.

Interpolate a doubled polynomial weight with the unweighted exponential
fluctuation bounds in \eqref{sct:overlap}; the resulting polynomial in
$1+T$ is absorbed in a smaller exponential. For pressure,
$\overline p=-\overline{v_2^2}$, and the pure mean tensor
$\overline v_1^2e_1\otimes e_1$ has zero double divergence.
Every other term in $-\Delta p=\partial_i\partial_j(v_iv_j)$
contains a fluctuation. The same inverse kernels prove
\eqref{sct:fluctuation} and \eqref{sct:transverse} for pressure.
The equations now give the age derivatives by finite induction:
each fluctuation equation retains a fluctuation factor, and each
differentiated vorticity equation retains a Gaussian vorticity factor.

It follows that $S(0)$ has every fixed spatial moment and $g$ has
every fixed mixed space/time moment, all bounded by $P$. Taylor-expand
$G_{\nu s}(\eta-\xi)$ through order $2N+1$; evenness removes odd
terms. In the source Duhamel integral, for $t\leq s/2$, Taylor-expand
jointly in $(\xi,t)$ and retain $k+\ell\leq N$. The identity
$\partial_sG_{\nu s}=\nu\partial_\eta^2G_{\nu s}$ gives exactly
the coefficients in \eqref{sct:expansion}. Integral Taylor remainders
are controlled by the next even derivative and a fixed higher moment.
For $t\geq s/2$, transfer derivatives to the exponentially decreasing
source. Its tail, and the coefficient-integral tails, are exponential
times a polynomial and obey the stated remainder. The same heat proof
without subtraction gives the asserted bound for $S$.

Finally \eqref{sct:comparison}, interpolated with the just-proved
doubled weights, proves fixed-band comparison for velocity and
vorticity. The pressure difference has source tensor
$Q+V_c^p\otimes e+e\otimes V_c^p+e\otimes e$.
On each nonzero mode the pressure kernels are $k_n^2G_{k_n}$,
$k_nG_{k_n}'$, and $G_{k_n}''=k_n^2G_{k_n}-\delta_0$ up to signs;
their weighted $L^1$ norms cost only powers of $c$. The zero Fourier mode of the pressure difference is
$-\overline{(Q+V_c^p\otimes e+e\otimes V_c^p+e\otimes e)_{22}}$. Weighted convolution proves its
estimate, and the difference equations inductively give slow-time
derivatives at polynomial cost in $\nu^{-1}$.
\end{proof}


\section{The Gaussian instability and canonical evolution}
\label{sec:gaussian-evolution}
\subsection{The Gaussian seed, its two couplings, and the complementary evolution}
\label{sec:short-gaussian-seed}

Use $u=\nabla^\perp\psi$, $\operatorname{curl}u=\Delta\psi$ and put
\[
 x=r^2/4,\quad \Omega=e^{-x},\quad
 V_0(y)=\Omega y^\perp,\quad E_0=(2-2x)e^{-x},\quad
 k=E_0'/r=(x-2)e^{-x},\quad G=(4\pi)^{-1}e^{-x}.
\]
The Euler linearization is
\[
 \mathscr Lf=-\Omega\partial_\vartheta f
       -(\nabla^\perp\Delta^{-1}f)\cdot\nabla E_0.
\]
For $j\ge0$, $M>1$, let $H^j_M$ be the space with norm
$\sum_{|\alpha|\le j}\|\langle y\rangle^M\partial^\alpha f\|_2$ and let
$\mathcal X_{j,M}=\{f\in H^j_M:\int f=0\}$.
In angular sector two use
\[
 X_G=L^2((0,\infty),G^{-1}r\,dr)e^{2i\vartheta},\qquad
 L_0=-2i\Omega+2ikR,\quad R=\Delta_2^{-1},
\]
where the pairing is linear in its first argument and omits $2\pi$.
For a general planar function, $f_2(r)=(2\pi)^{-1}\int_0^{2\pi}
f(r,\vartheta)e^{-2i\vartheta}d\vartheta$.
The regular decaying Green inverse in angular sector $n\ge1$ is
\begin{equation}\label{eq:short-seed-green}
 R_nw=-\frac1{2n}\left[
 r^{-n}\int_0^r s^{n+1}w(s)\,ds+
 r^n\int_r^\infty s^{1-n}w(s)\,ds\right].
\end{equation}

\begin{proposition}\label{prop:short-gaussian-seed}
On every $\mathcal X_{j,M}$ the entire spectrum of $\mathscr L$ in
$\{\Re z>0\}$ is one algebraically simple conjugate pair
$\lambda=\gamma+i\omega$, $\bar\lambda$, where $\gamma>0$ and
$-2<\omega<0$. Its eigenvectors are $q=h(r)e^{2i\vartheta}$ and
$\bar q$. Every derivative of $q$ is bounded by a polynomial times
$e^{-r^2/4}$, $\int q=0$, its velocity belongs to $L^2$, and
\[
 M_q:=\int_0^\infty r^3h(r)\,dr\ne0.
\]
The same $\lambda$ is the sole growing spectral value of $L_0$ on $X_G$.
Its projection is $P=q\ell$, with
\begin{equation}\label{eq:short-seed-functional}
 \ell(f)=\frac1{J_q}\int_0^\infty
       \frac{h(r)f_2(r)}{k(r)}r\,dr,\qquad
 J_q=\int_0^\infty\frac{h(r)^2}{k(r)}r\,dr\ne0.
\end{equation}
The quotients at $k=0$ are removable. In particular, the harmonic-strain
input $H=r^2k(r)e^{2i\vartheta}$ satisfies $\ell(H)=M_q/J_q\ne0$.

With $\mathscr P$ the projection onto the pair and $Q=I-P$,
\begin{equation}\label{eq:short-seed-complement}
 \|e^{t\mathscr L}(I-\mathscr P)\|_{H^j_M\to H^j_M}
       \le C_{j,M,\eta}e^{\eta t},\qquad
 \|e^{t(L_0-\lambda)}Q\|_{X_G\to X_G}
       \le C_\eta e^{-(\gamma-\eta)t}
\end{equation}
for every $\eta>0$ and $t\ge0$. Finally
$D=\Delta+\frac12y\cdot\nabla+2$ is selfadjoint and nonpositive on
$X_G$, with spectrum $\{0,-1,-2,\ldots\}$. Both $q$ and the normalized
Gaussian adjoint vector $q^\dagger$ belong to every $\mathcal D(D^m)$;
thus $DP$ and $PD$ extend boundedly to $X_G$.
\end{proposition}

\begin{proof}
\emph{One negative direction.}
Set $a_*=e^{-2}$, $t=\log r$, and
\[
 v=-\frac{k}{\Omega-a_*}
       =\frac{x-2}{e^{x-2}-1}>0,\qquad
 W=r^2v=\frac{4x(x-2)}{e^{x-2}-1},\qquad
 \mathcal H_n=-\partial_t^2+n^2-W.
\]
At $x=2$, define $v=1$ and $W=8$ by continuity. The bounded function $W$ tends to
zero at both ends, so $\mathcal H_n$ is selfadjoint on $H^2(\mathbb R)$
and has essential spectrum $[n^2,\infty)$: multiplication by $W$ is
relatively compact by truncation and Rellich's theorem. Also $\sup W<9$: for $y=x-2$, the function
$F(y)=9(e^y-1)-8y-4y^2$ satisfies $F(0)=0$ and
$F'(y)\ge-8\log(8/9)>0$, so $9-W=F(y)/(e^y-1)>0$;
at $y=0$ the inequality follows from $W=8$.

There is a negative trial in sector two. For
$\phi=r^2e^{-r^2/8}$ direct integration gives
\[
 \int_0^\infty(|\phi'|^2+4r^{-2}|\phi|^2-v|\phi|^2)r\,dr
 =48-32\int_0^\infty\frac{x-2}{e^{x-2}-1}x^2e^{-x}\,dx.
\]
For all real $y$,
\begin{equation}\label{eq:short-seed-rational}
 \frac y{e^y-1}\ge1-\frac y2+\frac{5y^2}{60+y^2}.
\end{equation}
Indeed, with $z=y/2$, this is equivalent to
\[
 \frac{(z^3+15z)\cosh z-(6z^2+15)\sinh z}{\sinh z}\ge0.
\]
At $z=0$ take the continuous extension. The numerator has coefficients
$(2m)(2m-2)(2m-4)/(2m+1)!\ge0$ at $z^{2m+1}$.
For $d\rho=x^2e^{-x}dx$, $y=x-2$, and $p=y(16-y)/930$,
completion of the square and $\int_0^\infty x^ke^{-x}dx=k!$ give
\[
 \int\frac{y^2}{60+y^2}\,d\rho
 \ge\int\bigl(2yp-(60+y^2)p^2\bigr)\,d\rho=\frac{16}{155}.
\]
Thus the trial form is at most $-16/31$.

There cannot be a second nonpositive direction. The function
$h_*(t)=e^t(\Omega-a_*)$ has one simple zero
$t_*=\log\sqrt8$ and $h_*''/h_*=1-W$. Ground-state factorization
on the two half-lines gives
\[
 \int(|F'|^2+(1-W)|F|^2)\,dt
    =\int|F'-(h_*'/h_*)F|^2\,dt\ge0
 \quad\text{when }F(t_*)=0.
\]
Prove it first for compact support away from $t_*$ and then use
$H^1_0$ density. Hence
$\langle F,\mathcal H_nF\rangle\ge(n^2-1)\|F\|_2^2$ on a
codimension-one subspace. Min--max implies that $\mathcal H_2$ has
exactly one negative eigenvalue, counting multiplicity, and no kernel.
For $n\ge3$ the bound $W<9$ directly gives
\begin{equation}\label{eq:short-seed-higher}
 \langle F,\mathcal H_nF\rangle\ge
 (1-\|W\|_\infty/n^2)(\|F'\|_2^2+n^2\|F\|_2^2).
\end{equation}

\emph{A bounded Hamiltonian realization and its neutral modes.}
On $Y=L^2((0,\infty),r\,dr)$ set
\[
 K_n=\sqrt v(-\Delta_n)^{-1}\sqrt v,\quad B_n=I-K_n,
 \quad J=\Omega-a_*,\quad \widetilde T_n=a_*+JB_n.
\]
The kernel of $K_n$, relative to $s\,ds$, is
$\frac1{2n}\sqrt{v(r)v(s)}(\min(r,s)/\max(r,s))^n$.
Since $\int rv(r)dr<\infty$, it is Hilbert--Schmidt; Green's identity
makes it nonnegative and selfadjoint. More explicitly, on the energy
space $\mathcal E_n$ with norm
$\int(|\phi'|^2+n^2r^{-2}|\phi|^2)r\,dr$, the compact map
$S\phi=\sqrt v\phi$ satisfies $SS^*=K_n$.
The nonzero eigenvalues of $SS^*$ and $S^*S$ agree with multiplicity.
The preceding form computation therefore says that $B_2$ is invertible
with one complex negative direction, whereas $B_n$ is strictly positive
for $n\ge3$. The change $w=\sqrt v f$ intertwines
$T_n=\Omega-k\Delta_n^{-1}$ with $\widetilde T_n$.

If $\widetilde T_nf=cf$ and $\psi=S^*f$, then
$F(t)=\psi(e^t)\in H^2(\mathbb R)$ and
\begin{equation}\label{eq:short-seed-rayleigh}
 (\Omega-c)(F''-n^2F)-AF=0,\qquad
 A=r^2k=\partial_t^2\Omega+2\partial_t\Omega.
\end{equation}
Indeed the Green source $\rho=r^2\sqrt v f$ has squared logarithmic
$L^2$ norm $\int W|f|^2r\,dr$, and
$-F''+n^2F=\rho$. Also $F=0$ forces $f=0$.

The operator $\widetilde T_2$ has no real eigenvalues. At $c=a_*$ the equation
$JB_2f=0$ gives $f=0$, since $J$ vanishes at one point and $B_2$ is
invertible. At $c=0$, \eqref{eq:short-seed-rayleigh} becomes
$-F''+4(x-1)^2F=0$, whose cutoff energy identity gives $F=0$.
For the other real $c$, put $h_c=e^t(\Omega-c)$; then
$h_c''/h_c=1+A/(\Omega-c)$ and the equation is
$-F''+(3+h_c''/h_c)F=0$.
When $0<c<1$, $c\ne a_*$, its only critical point $t_c$ has
$A(t_c)\ne0$. The equation and $F''-4F\in L^2$ force
$F(t_c)=0$, since otherwise $AF/(\Omega-c)$ has a nonintegrable
simple pole. As $F\in H^2$, $F=O(|t-t_c|)$.
For $c<0$ or $c\ge1$ there is no finite zero. In every case,
factorization on the intervals avoiding $t_c$ gives
\[
 0=\int\left(|F'-(h_c'/h_c)F|^2+3|F|^2\right)dt.
\]
The critical boundary term is $O(|t-t_c|)$; the terms at infinity
vanish because $F,F'\to0$ and $h_c'/h_c$ is bounded there
(its limit at $-\infty$ is three if $c=1$). Thus $F=0$.

Regard $Y$ as a real Hilbert space. The operator
$\mathcal A=-2iJB_2$ has bounded skew-adjoint factor $-2iJ$
and bounded selfadjoint energy $B_2$. Its negative index is two
over the reals, its kernel is zero, and its positive space is
uniformly positive since $B_2$ is an invertible compact perturbation
of identity. These verify the hypotheses of the index theorem of Lin and
Zeng~\cite{lin-zeng2021}, which gives
\[
 k_r+2k_c+2k_i^{\le0}+k_0^{\le0}=2.
\]
The last two counts vanish by the preceding absence of imaginary
eigenvectors; continuous spectrum creates no additional term in this
theorem. The complexification of the real operator is unitarily
equivalent to $\mathcal A\oplus\overline{\mathcal A}$ on $Y\oplus Y$,
so its total unstable algebraic multiplicity is twice that of
$\mathcal A$ on complex $Y$. Hence it is one on $Y$.
Equivalently, $\widetilde T_2$ has exactly one upper-half-plane
eigenvalue $c$, algebraically simple.

\emph{Frequency, Gaussian regularity, and the quadrupole.}
For this nonreal $c$, substitute $F=(\Omega-c)\Phi$ into
\eqref{eq:short-seed-rayleigh} and integrate against $\bar\Phi$:
\[
 \int(\Omega-c)^2\Theta\,dt=0,\qquad
 \Theta=|\Phi'|^2+4|\Phi|^2-2\Re(\Phi'\bar\Phi)
          \ge3|\Phi|^2.
\]
Here $\Phi\in H^2$, so integration by parts is valid.
The imaginary and real parts imply respectively
$\int\Omega\Theta=(\Re c)\int\Theta$ and
$\int\Omega^2\Theta=|c|^2\int\Theta$.
Since $0<\Omega<1$, $|c-1/2|<1/2$. Thus
$\lambda=-2ic$ has the asserted growth and frequency.

Write $\phi(r)=F(\log r)$. The coefficient
$B=A/(\Omega-c)$ in $F''-4F=BF$ is $O(e^{2t})$ at $-\infty$
and decreases faster than every exponential at $+\infty$, with
the same differentiated bounds. At $+\infty$ the Volterra equation
\[
 F_+(t)=e^{-2t}+\frac12\int_t^\infty
                   \sinh(2(s-t))B(s)F_+(s)\,ds
\]
is a contraction on $\sup_{t\ge T}e^{2t}|F(t)|$ for large $T$.
It gives $F_+=e^{-2t}(1+o(1))$ with differentiated asymptotics;
reduction of order supplies an independent solution growing as
$e^{2t}$. The analogous forward Volterra construction at $-\infty$
selects $e^{2t}(1+O(e^{2t}))$. Thus every nonzero $H^1$ solution has
\[
 \phi(r)=B_0r^2(1+O(r^2))\ (r\downarrow0),\qquad
 \phi(r)=C_\infty r^{-2}(1+o(1))\ (r\to\infty),
 \qquad B_0C_\infty\ne0.
\]
The Cartesian streamfunction $\Psi=\phi e^{2i\vartheta}$ is in $L^2$.
These endpoint estimates remove the inner-circle boundary terms in
Green's identity, so
$\Delta\Psi=[k/(\Omega-c)]\Psi$ on the whole plane.
The coefficient is smooth with bounded derivatives. The estimate
$\|\Psi\|_{H^2}\le C(\|\Delta\Psi\|_2+\|\Psi\|_2)$,
differentiated and iterated, gives all Sobolev norms and bounded
Cartesian derivatives. Hence
$q=\Delta\Psi=h e^{2i\vartheta}$, with
$h=k\phi/(\Omega-c)$, has the claimed Gaussian bounds.
Its velocity is $O(r)$ at zero and $O(r^{-3})$ at infinity.
Finally
\begin{equation}\label{eq:short-seed-quadrupole}
 M_q=\int_0^\infty r^3\Delta_2\phi\,dr
       =[r^3\phi'-2r^2\phi]_0^\infty=-4C_\infty\ne0.
\end{equation}
Its circulation vanishes by angular integration.

\emph{Excluding other angular sectors and transferring rootspaces.}
For $n\ge3$, $B_n>0$ and
$\widetilde T_n^*B_n=B_n\widetilde T_n$ imply every phase-speed
eigenvalue is real. For $n=1$, $\phi_1=r(\Omega-c)$ is an exact
solution for every nonreal $c$. Its independent solution is
$\phi_1(r)\int^r ds/(s\phi_1(s)^2)
=-[2(1-c)r]^{-1}+O(r|\log r|)$ at zero, and is not locally
square integrable. The regular solution $\phi_1$ grows as $-cr$
at infinity, excluding a regular decaying mode. Radial vorticities
are stationary. Negative sectors are the conjugates of positive ones.

This classification applies to the actual weighted spaces without
claiming a bounded similarity $w\mapsto w/\sqrt v$ there.
Cauchy--Schwarz in \eqref{eq:short-seed-green} gives, for $r\ge2$,
\[
 |R_nw(r)|\le C_{n,M}\|w\|_{L^2_M}
       \left(r^{1-M}+r^{-n}(1+\log r)^{1/2}\right);
\]
at zero it is $O(r)$, with an extra $|\log r|^{1/2}$ only for $n=1$.
For a nonreal rootvector the equation
$w=kR_nw/(\Omega-c)$, and for each subsequent Jordan vector the
same equation with its preceding vector divided by $\Omega-c$,
show inductively that $w/\sqrt v\in Y$.
Indeed $k/\sqrt v$ is Gaussian at infinity and bounded at zero.
The Green inverse agrees with $-S^*(w/\sqrt v)$: their difference
is $ar^n+br^{-n}$, and both vanish at the two endpoints.
Thus whole Jordan chains transfer injectively to $Y$.
Local elliptic regularity selects precisely the regular solutions
used above. In $X_G$ membership in $Y$ is immediate from $v\ge cG$.
Conversely the constructed Gaussian eigenmode belongs to all these
spaces, so the growing eigenvalue and its algebraic multiplicity agree.

\emph{Generators, projections, and complementary bounds.}
The free transport is
$T(t)f(r,\vartheta)=f(r,\vartheta-\Omega(r)t)$.
The chain rule gives $\|T(t)\|_{H^j_M\to H^j_M}\le C_j(1+|t|)^j$.
Density and differentiation of the flow give its generator domain
$\{f:\Omega\partial_\vartheta f\in H^j_M\}$.
For the converse domain inclusion use the distributional identity
$T(t)f-f=-\int_0^tT(s)\Omega\partial_\vartheta f\,ds$ and strong continuity.
The interaction $(\nabla^\perp\Delta^{-1}f)\cdot\nabla E_0$ is
compact on $H^j_M$: splitting the Biot--Savart kernel at distance
one controls its undifferentiated near part in $L^2$ and far part
in $L^\infty$ by $\|f\|_{L^2}+\|f\|_1$; derivatives have Fourier
order at most $j$. Multiplication by the Schwartz coefficient
$\nabla E_0$ therefore maps $H^j_M$ into $H^{j+1}_M$.
Truncation to a disk, Rellich compactness, and the vanishing weighted
coefficient tails prove compactness on $H^j_M$.
The bounded perturbation theorem now realizes $\mathscr L$ on the
stated domain. Both terms have zero integral, so $\mathcal X_{j,M}$
is invariant.

The Duhamel difference $e^{t\mathscr L}-T(t)$ is compact for each
$t>0$: its integrand consists of the compact interaction between
strongly continuous evolutions, and is norm continuous because the
adjoint evolutions are strongly continuous on this Hilbert space.
Thus the essential growth bound is at most zero. The standard
semigroup spectral decomposition outside the essential growth bound
\cite[Corollary~IV.2.11 and Theorem~V.3.1]{engel-nagel2000} makes all growing spectrum discrete of finite
algebraic multiplicity. Each such finite-dimensional rootspace is
rotation invariant, so the circle representation splits into integer
angular characters. The preceding transfer and exclusions leave
precisely $\lambda,\bar\lambda$, each simple. The same theorem gives
the first estimate in \eqref{eq:short-seed-complement}.
This statement includes the full $H^j_M$: splitting off a fixed radial
function of integral one merely adds a stationary direction.
Resolvents, hence Riesz projections, agree on intersections of
weighted spaces by uniqueness in a common weaker realization.

On $X_G$, the Green estimate
$|Rf(r)|\le C\|f\|_{X_G}\min(r,r^{-2})$ shows that $L_0$ is bounded.
After the unitary map $f\mapsto f/\sqrt G$, its interaction kernel is
\[
 -\frac i2\frac{k(r)}{\sqrt{G(r)}}
       \left(\frac{\min(r,s)}{\max(r,s)}\right)^2\sqrt{G(s)},
\]
which is Hilbert--Schmidt. Hence the same compact-perturbation and
spectral-decomposition argument gives the second estimate in
\eqref{eq:short-seed-complement}.

\emph{Actual adjoint, strain coupling, and diffusion domains.}
Unweighted selfadjointness of $R$ and $k=\Omega(x-2)$ give
\[
 L_0^*g=2i\Omega[g-R((x-2)g)].
\]
The explicit adjoint eigenvector is
\[
 h_X(r,\vartheta)=G(r)\frac{\overline{\phi(r)}}{
                    \Omega(r)-\bar c}e^{2i\vartheta},
 \qquad L_0^*h_X=\bar\lambda h_X.
\]
Indeed the conjugate Rayleigh equation gives
$R(k\bar\phi/(\Omega-\bar c))=\bar\phi$, with no homogeneous
ambiguity by its endpoints. Every derivative of $h_X$ is a polynomial
times $G$. The Fredholm operator $L_0-\lambda$ has index zero,
one-dimensional kernel, and adjoint kernel spanned by $h_X$.
Its pairing with $q$ cannot vanish, since otherwise $q$ would lie
in the range and produce a Jordan vector. This pairing is exactly
$J_q$ in \eqref{eq:short-seed-functional}. Setting
$q^\dagger=h_X/\overline{J_q}$
therefore gives that formula and
$\ell(H)=M_q/J_q\ne0$. The kernel $h/k=\phi/(\Omega-c)$ is smooth
at zero and $O(r^{-2})$ at infinity, so the same functional is bounded
on each polynomially weighted space. Its adjoint identity extends from
smooth data by density and gives the corresponding Riesz projection.

Finally, conjugation by $U_Gf=f/\sqrt G$ gives
\[
 U_GDU_G^{-1}=\Delta-|y|^2/16+3/2.
\]
The selfadjoint harmonic oscillator has Hermite levels
$1-N/2$; angular weight two consists exactly of $N=2+2m$,
$m\ge0$. This proves the domain and spectral assertion about $D$.
The Gaussian derivative bounds make $U_Gq$ and $U_Gh_X$ Schwartz,
so both belong to every oscillator power domain. With
$\ell(f)=\langle f,q^\dagger\rangle$, selfadjointness gives
$\ell(Df)=\langle f,Dq^\dagger\rangle$. Consequently
$PD=q\langle\,\cdot\,,Dq^\dagger\rangle$ and
$DP=(Dq)\ell$ are bounded.
\end{proof}

\begin{corollary}[Frozen heat profiles]\label{cor:short-seed-dilation}
Set
\[
 J_af(y)=a^{-2}f(y/\sqrt a),\qquad
 E_a=J_aE_0,\qquad V_a(y)=a^{-3/2}V_0(y/\sqrt a).
\]
Their Euler generators and unstable projections satisfy
\[
 \mathscr L_a=a^{-2}J_a\mathscr LJ_a^{-1},\qquad
 \mathscr P_a=J_a\mathscr PJ_a^{-1}.
\]
Thus the unstable eigenvalues are $a^{-2}\lambda,a^{-2}\bar\lambda$;
the complementary estimates are uniform for $1\le a\le A<\infty$,
and the projections depend continuously on $a$ in operator norm.
\end{corollary}
\begin{proof}
The generator identity is the Biot--Savart scaling identity.
$J_a,J_a^{-1}$ are uniformly bounded on $H^j_M$ on each compact
$a$ interval, so conjugating \eqref{eq:short-seed-complement} proves
the bounds. The finite-rank projection consists of the Gaussian
eigenvectors and the dual functional \eqref{eq:short-seed-functional};
their dilates are norm continuous by dominated convergence.
\end{proof}


\subsection{A compact-perturbation lemma and the canonical heat branch}
\label{sec:short-seed-canonical}

We use the Gaussian angular-two Hilbert space $X_G$, its Euler operator
$L=T+A$ with $T=-2i\Omega$ and compact $A=2ik\Delta_2^{-1}$,
the simple eigenpair $(\lambda,q)$, and $P=q\ell$, $Q=I-P$ established
above. Write $\gamma=\Re\lambda>0$ and
\[
 D=\Delta+\tfrac12y\cdot\nabla+2,\qquad \kappa=\ell(Dq).
\]
On $X_G$, $D$ is selfadjoint and nonpositive; $q,q^\dagger$ belong to
every power domain of $D$. In particular $DP$ and $PD$ extend boundedly.
These statements concern the actual Gaussian adjoint, not an unweighted
formal adjoint.

\begin{lemma}[Compact Duhamel comparison]\label{lem:short-compact-duhamel}
Fix a finite time length $B$. Suppose uniformly bounded free evolutions
$F_n$ converge strongly to $F_0$, and their adjoints converge strongly,
uniformly in both times and any auxiliary parameters. Suppose the bounded
strongly measurable interactions $A_n(t)$ converge uniformly in operator
norm to $A_0(t)$,
and all the latter form a norm-precompact family of compact operators.
The full evolutions $U_n$ and their adjoints converge strongly, and
\[
 \sup_{0\le t-s\le B}
 \|(U_n-F_n)(t,s)-(U_0-F_0)(t,s)\|\longrightarrow0.
\]
\end{lemma}
\begin{proof}
The Dyson series have a uniform exponential majorant, so termwise
convergence proves the two strong assertions. For finite-rank
$K=\sum_{j=1}^m u_j\langle\,\cdot\,,v_j\rangle$, strong convergence
on $u_j$ and adjoint-strong convergence on $v_j$ imply norm convergence
of $F_n(t,r)KU_n(r,s)$. A finite norm-net and finite-rank approximation
of the compact family $A_0(r)$ make the conclusion uniform. Integrate
\[
 (U_n-F_n)(t,s)=\int_s^tF_n(t,r)A_n(r)U_n(r,s)\,dr.
\]
Uniform bounds absorb $A_n-A_0$. This proves norm convergence only for
the compact remainder, as required.
\end{proof}

\begin{lemma}[Small diffusion on the complementary space]
\label{lem:short-diffusion-complement}
There are $\beta_*,g,C>0$ such that, for every measurable
$0\le\beta(t)\le\beta_*$, the forward evolution on $QX_G$ generated by
\[
 Q\{L-\lambda+\beta(t)(D-\kappa)\}Q
\]
has norm at most $Ce^{-g(t-s)}$ for $t\ge s$.
\end{lemma}
\begin{proof}
Extend the generator to
$G_\beta=L-\lambda-P+\beta Q(D-\kappa)Q$ on $X_G$. Since
\[
 Q(D-\kappa)Q=D-\kappa+R_f,
 \quad R_f=-DP-PD+PDP+\kappa P,
\]
$R_f$ is bounded and finite rank. Split $G_\beta$ into the free
generator $T-\lambda-\beta\kappa+\beta D$ and interaction
$A-P+\beta R_f$. The evolution exists for measurable $\beta$ by the
Dyson series based on $\exp(D\int_s^t\beta)$; its short-interval
bounds are uniform. Dissipativity of $D$ and skew-adjointness of $T$
give
\[
 \|F_\beta(t,s)\|
 \le\exp\{-\gamma(t-s)+|\Re\kappa|\beta_*(t-s)\}.
\]
This inequality can first be proved for bounded nonpositive Yosida
approximants of $D$ and then passed to the strong limit.

On a fixed block, $\exp(D\int_s^t\beta)\to I$ strongly as
$\beta_*\downarrow0$, uniformly over all admissible coefficients.
The Dyson series give the corresponding direct and adjoint strong
limits for the free and full evolutions. For the adjoints, reverse
time on the block; the class of admissible coefficients is unchanged.
Lemma~\ref{lem:short-compact-duhamel} therefore gives
\[
 U_\beta(s+B,s)-F_\beta(s+B,s)
 \longrightarrow e^{B(L-\lambda-P)}-e^{B(T-\lambda)}
\]
in norm uniformly in $s$ and $\beta$. The inviscid complementary
semigroup estimate, with any rate below $\gamma$, makes
$L-\lambda-P$ exponentially stable. Consequently
\[
 \limsup_{\beta_*\downarrow0}\sup_{s,\beta}
 \|U_\beta(s+B,s)\|
 \le \|e^{B(L-\lambda-P)}\|+2e^{-\gamma B}<\tfrac14
\]
when $B$ is sufficiently large. Fixing $\beta_*$ then makes every
full block contract by $1/2$; iteration and the short-block bound
prove exponential decay. Finally $G_\beta$ commutes with $P$ and
equals $-1$ on its range. In the mild equation this follows by testing
against $q^\dagger\in\mathcal D(D)$ and uniqueness from the datum
$q$. Thus $QX_G$ is invariant and restriction proves the claim.
\end{proof}

The radial Gaussian evolves exactly by heat diffusion:
\[
 a=1+\nu s,\quad E(s,x)=a^{-2}E_0(x/\sqrt a),\quad
 V(s,x)=a^{-3/2}V_0(x/\sqrt a).
\]
It is also an unforced planar Navier--Stokes solution, since its
vorticity is radial and its velocity tangential. Substitution of
$w(s,x)=a^{-2}W(z,x/\sqrt a)$ gives the exact linearized equation
\begin{equation}\label{eq:short-heat-clock}
 W_z=(L+\theta^{-1}D)W,\qquad
 z=\frac{s}{1+\nu s}=\nu^{-1}-\theta,
 \qquad \theta=\frac1{\nu(1+\nu s)}.
\end{equation}
Here $dz/ds=a^{-2}$; the Biot--Savart velocity scales as $a^{-3/2}$.

\begin{proposition}[Canonical branch]
\label{prop:short-canonical}
There is a unique solution of
\[
 -V_{\rm can}'=(L+\theta^{-1}D)V_{\rm can},\qquad \theta>0,
\]
with
\begin{equation}\label{eq:short-canonical-asymptotic}
 V_{\rm can}(\theta)=e^{-\lambda\theta}\theta^{-\kappa}
     \{q+O_{X_G}(\theta^{-1})\}\quad(\theta\to\infty).
\end{equation}
It is bounded in $X_G$ as $\theta\downarrow0$ and smooth for positive
$\theta$. For $X_{G,j}=\mathcal D((I-D)^{j/2})$, fixed $j$, and
$\theta\ge b$ with a fixed sufficiently large $b$,
\[
 \|e^{\lambda\theta}\theta^\kappa V_{\rm can}(\theta)\|_{X_{G,j}}
       \le C_j(1+\theta)^{j/2}.
\]
\end{proposition}
\begin{proof}
Write $e^{\lambda\theta}\theta^\kappa V=Aq+r$, $r\in QX_G$.
The equations are
\[
 -A'=\theta^{-1}\ell(Dr),\qquad
 -r'=Q\{L-\lambda+\theta^{-1}(D-\kappa)\}Qr
                   +\theta^{-1}A\,QDq.
\]
For $b\ge\beta_*^{-1}$ let $S(\theta,s)$, $b\le\theta\le s$,
be the forward complementary evolution. It satisfies
$\|S(\theta,s)\|\le Ce^{-g(s-\theta)}$. The normalization at infinity
is imposed by
\[
 r_A(\theta)=\int_\theta^\infty
 S(\theta,s)QDq\,\frac{A(s)}s\,ds,\qquad
 (\mathcal TA)(\theta)=1+\int_\theta^\infty
                  \frac{\ell(Dr_A(s))}s\,ds.
\]
Because $\ell D$ is bounded,
$\|r_A(\theta)\|+|\mathcal TA(\theta)-1|
\le C\theta^{-1}\|A\|_\infty$. The same estimate for differences
makes $\mathcal T$ a contraction on $C_b([b,\infty))$ after enlarging
$b$. Its fixed point gives \eqref{eq:short-canonical-asymptotic}.
Splitting both integrals at any finite $s_0$ verifies the coupled
mild initial-value equations, whose uniqueness follows from the
bounded coupling maps and Gronwall. Conversely every normalized
solution satisfies the integrals: the term
$S(\theta,s_0)r(s_0)$ tends to zero as $s_0\to\infty$.
This also proves uniqueness at infinity.

Set $\tau=\log(b/\theta)$. Then
\[
 \partial_\tau V=DV+be^{-\tau}LV.
\]
The bounded-perturbation formula and $D\le0$ give continuation for
all $\tau\ge0$ with
$\sup_{0<\theta\le b}\|V(\theta)\|
\le e^{b\|L\|}\|V(b)\|$. This is forward parabolic continuation;
no backward heat inverse occurs.

For completeness, conjugation by $G^{1/2}$ turns $D$ into a shifted
harmonic oscillator. Its Hermite decomposition shows that the squared
$X_{G,j}$ norm is equivalent to
$\sum_{|\alpha|\le j}\|\partial^\alpha f\|_{L^2(G^{-1})}^2$
(Cartesian derivatives may leave angular sector two). Multiplication
by $\Omega$ is bounded on these spaces. The radial Green formula for
$\Delta_2^{-1}$ first gives $\|\Delta_2^{-1}f\|_2\le C\|f\|_{X_G}$;
elliptic regularity and boundedness of
$G^{-1/2}\partial^\alpha k$ then show $L:X_{G,j}\to X_{G,j}$ is
bounded. Spectral calculus gives a one-derivative heat gain
$\|e^{aD}\|_{X_{G,j}\to X_{G,j+1}}\le C_ja^{-1/2}$ for $0<a\le1$.
On the unit turnover interval from $\theta+1$ down to $\theta$,
diffusion is comparable to $(1+\theta)^{-1}$. Duhamel therefore gains
one derivative at elapsed time $h$ at cost
$C_j((1+\theta)/h)^{1/2}$; its integral kernel is integrable.
Divide the interval into $j$ equal pieces to obtain the stated
$C_j(1+\theta)^{j/2}$ bound from the bounded $X_G$ norm at its inlet.
The $\tau$ equation propagates every fixed graph norm to zero because
$\int_0^\infty be^{-\tau}\,d\tau=b$.
\end{proof}

\begin{corollary}[Nontrivial full-age output]\label{cor:short-mellin}
Let $\mathfrak M(f)=\int_0^\infty r^3f_2(r)\,dr$ and
$M_q=\mathfrak M(q)\ne0$. Then
\[
 \mathcal J(\mu)=\int_0^\infty
       \theta^{\mu+1/2}\mathfrak M(V_{\rm can}(\theta))\,d\theta
\]
is holomorphic and not identically zero on $\Re\mu>-3/2$.
\end{corollary}
\begin{proof}
Weighted Cauchy--Schwarz bounds $\mathfrak M$ on $X_G$. Thus its
canonical value is $O(1)$ at zero and
$e^{-\lambda\theta}\theta^{-\kappa}(M_q+O(\theta^{-1}))$ at infinity.
These bounds prove absolute convergence and permit differentiation
under the integral, including every logarithmic factor on compact
subsets of the half-plane. If $\mathcal J$ vanished identically, then
for fixed real $b>-3/2$ the $L^1(\mathbb R)$ function
$e^{(b+3/2)x}\mathfrak M(V_{\rm can}(e^x))$ would have identically zero
Fourier transform. Fourier uniqueness would contradict $M_q\ne0$.
\end{proof}


\section{The logarithmic ring lift and its residual}
\label{sec:ring-lift}

The planar shift becomes a physical scale change under the ring chart.
For $q=R\log(r/R)$ and $\Lambda=e^{c/R}$, the map
$(r,z,t)\mapsto(\Lambda r,\Lambda z,\Lambda^2t)$ sends $q$ to $q+c$
and leaves $\eta=Rz/r$ and $s=R^2t/r^2$ fixed. Thus the raw lift obeys
\[
 u_{\rm raw}(\Lambda x,\Lambda^2t)=\Lambda^{-1}u_{\rm raw}(x,t).
\]
This is exact DSS symmetry; the Navier--Stokes equation still has a
small curvature residual. We estimate it and construct an exact unforced
background $U$ with its DSS datum and common $H^2$ heat remainder.
The physical free-operator estimates proved here also serve Sections~\ref{sec:stress-spaces-short}--\ref{sec:spectral-assembly}.

Let $v=\nabla^\perp\psi$ and $p$ be the exact periodic planar train
and its bounded pressure constructed in Section~\ref{sct:section}.
The following stream construction converts its period into discrete scaling.

\begin{proposition}[Exact lift, residual and trace]
\label{sct:lift}
Let $R\geq2$, $\epsilon=R^{-1}$, and set
\[
 q=R\log(r/R),\quad\eta=Rz/r,\quad s=R^2t/r^2,
 \quad K=\eta\partial_\eta+2s\partial_s,\quad H=(1-K)\psi.
\]
The physical stream $r\psi(q,\eta,s)$ defines the exactly divergence-free
axisymmetric field without swirl
\[
 u_r=\frac Rrv_1,\qquad
 u_z=\frac Rr(v_2+\epsilon H),\qquad
 p_{\rm raw}=(\frac Rr)^2p.
\]
It is smooth through the axis for positive time and discretely
self-similar with ratio $e^{c/R}$. Its similarity field $U_R$ has
period $2c/R$, globally bounded fixed spatial derivatives, and residual
\begin{equation}
 \|\mathcal F_R\|_{C_{\rm per}W^{1,p}(\mathbb R^3)}
      \leq P_p\epsilon^{1-2/p},\qquad 2\leq p<\infty.
 \label{sct:residual-bound}
\end{equation}
This includes the projected residual. In particular $p=11/4$ gives
$P\epsilon^{3/11}$.
Its nonzero initial datum $a_{\rm raw}$ is divergence-free, has the same discrete
scaling, and satisfies
\begin{equation}
 |D^ja_{\rm raw}(x)|\leq C_j|x|^{-1-j}.
 \label{sct:raw-trace}
\end{equation}
\end{proposition}

\begin{proof}
Put $D_\epsilon=\partial_q-\epsilon K$.
The differentiation rules are
$\partial_r=(R/r)D_\epsilon$, $\partial_z=(R/r)\partial_\eta$,
$\partial_t=(R/r)^2\partial_s$. Differentiating the stream proves
the divergence-free condition, including its geometric correction $\epsilon H$.
After dividing the physical residual by $(R/r)^3$, direct expansion
of the cylindrical equations gives
\[
 r_r=\epsilon A_1+\epsilon^2B_1,\qquad
 r_z=\epsilon A_2+\epsilon^2B_2+\epsilon^3B_3,
\]
where the complete coefficients are
\begin{align*}
 A_1&=-v_1Kv_1+H(v_1)_\eta-v_1^2
             +2\nu(K+1)(v_1)_q-(K+2)p,\\
 A_2&=(\partial_s+v\cdot\nabla-\nu\Delta)H
             +H(v_2)_\eta-v_1(K+1)v_2+2\nu(K+1)(v_2)_q,\\
 B_1&=-\nu K(K+2)v_1,\\
 B_2&=-v_1(K+1)H+HH_\eta+2\nu(K+1)H_q-\nu(K+1)^2v_2,\\
 B_3&=-\nu(K+1)^2H.
\end{align*}
For verification, the normalized diffusion operators on the radial
and axial components respectively are
$D_\epsilon^2-2\epsilon D_\epsilon+\partial_\eta^2$ and
$D_\epsilon^2-2\epsilon D_\epsilon+\epsilon^2+\partial_\eta^2$;
the radial pressure operator is $D_\epsilon-2\epsilon$.
Thus the formulas include all age derivatives and are identities at
every age, with no truncation.

The heat identity $K\partial_\eta^jG_{\nu s}=-(j+1)
\partial_\eta^jG_{\nu s}$ and Lemma~\ref{sct:heat} give the following
complete late residual table. Each entry denotes its
$L^\infty_\eta L^\infty_q$ bound with normalized weight
$\langle\eta/\sqrt{s}\rangle^m$; an $\eta$ derivative gains
$s^{-1/2}$ and an $L^p_\eta$ norm costs $s^{1/(2p)}$:
\[
\begin{array}{c|c|c}
 \text{coefficient}&\text{size}&(K+3)\text{ size}\\ \hline
 A_1,B_1&Ps^{-2}&Ps^{-2}\\
 B_2,B_3&Ps^{-3/2}&Ps^{-5/2}\\
 A_2&Pe^{-\kappa\nu s/c^2}&Pe^{-\kappa'\nu s/c^2}.
\end{array}
\]
Every $q$ derivative is exponentially small. Indeed the leading means
of $B_2,B_3$ are $4a_0^2G_{\nu s}\partial_\eta G_{\nu s}$ and
$-16\nu a_2\partial_\eta^2G_{\nu s}$; both have $K$ degree $-3$.
Also $(\partial_s-\nu\partial_\eta^2)(S-KS)=-(K+1)g$,
and every remaining term of $A_2$ has a fluctuation factor.

In similarity variables $\varrho=r/\sqrt t$, $z_y=z/\sqrt t$,
\[
 \varrho=R/\sqrt s,\quad z_y=\eta/\sqrt s,\quad
 q=R\tau/2-(R/2)\log s,\quad dy=\pi R^2s^{-5/2}\,ds\,d\eta,
\]
and $U_R=\sqrt s(v_1e_r+(v_2+\epsilon H)e_z)$,
$\mathcal F_R=s^{3/2}r$. Its derivatives in $\varrho$ and $z_y$ are
\[
 \partial_{z_y}\mathcal F_R=s^2\partial_\eta r,
 \qquad\partial_\varrho\mathcal F_R
       =s^2[\partial_q-\epsilon(K+3)]r;
\]
the angular frame derivative costs only $\epsilon s^2|r_r|$.
For the field norm the late integrands are $s^{-p/2-2}$ and $s^{-2}$;
for its radial derivative they are $s^{-2}$ and $s^{-p/2-2}$.
The axial and angular derivatives are integrable by the same table.
At zero age the integrands are bounded by $Ps^{3p/2-5/2}$ or
$Ps^{2p-5/2}$, integrable for $p>1$. The Jacobian contributes
$R^{2/p}$ and every residual term contributes at least $\epsilon$.
This proves \eqref{sct:residual-bound}, including continuity by dominated
convergence. The Leray projection is bounded on $W^{1,p}$ by the
Euclidean Riesz-transform theorem.

Smoothness at the axis follows term by term from the full heat expansion:
\[
 r a_{2j}\partial_\eta^{2j}G_{\nu s}
       =\frac{a_{2j}r^{2j+2}}{R^{2j+1}}
                    \partial_z^{2j}G_{\nu t}(z).
\]
After $a$ radial derivatives the remainder after order $N$ is
$O(r^{2N+4-a})$ on compact positive-time sets; fluctuations are flat
because $e^{-\kappa\nu R^2t/(c^2r^2)}$ dominates all radial derivative
costs. Thus $u_r/r$ and $u_z$ are smooth functions of $(r^2,z,t)$.
The same argument with the finite needed expansion order proves
$\|U_R\|_{W^{k,\infty}}\leq P_k$, uniformly for $R\geq2$.
For example the leading mean is
$(-\varrho R^{-1}\partial_{z_y}(a_0G_\nu),
2R^{-1}a_0G_\nu)$; its Cartesian second derivatives are regular at the axis. Shifting $q$ by $c$ proves the stated DSS period.

At $t=0$ use the same stream formula with the Gaussian inlet. Each
derivative costs $C/r$; all powers of $|x|/r$ are absorbed by the
Gaussian decay in $\eta=Rz/r$. This proves the datum bound away from
the origin, including smoothness at nonzero axis points. Flux across
a sphere of radius $h$ is $O(h)$, proving that the divergence is zero in distributions.
The datum is nonzero on the central ring core.

\end{proof}


\section{The common free equation and local comparison}
\subsection{The free Cartesian equation}

Put \(\mathcal T=\partial_\tau-y\cdot\nabla/2\),
\(L=\mathcal T-\nu\Delta-1/2\), and let \(Y_r\) denote
\(L^r((\mathbb R/P\mathbb Z)\times\mathbb R^3)\), with normalized
phase measure. Its maximal distributional graph is
\[
 D_r=\{w\in Y_r:\nu\Delta w,\mathcal Tw\in Y_r\},\qquad
 \|w\|_{D_r}=\|w\|_{Y_r}+\|\nu\Delta w\|_{Y_r}
                                      +\|\mathcal Tw\|_{Y_r}.
\]
The derivatives in this definition are Cartesian; there is no axis
condition. Write \(\gamma_r=(3/r-1)/2\), \(a_r=\operatorname{Re}\mu+\gamma_r\).

\begin{lemma}\label{bg:common-body-graph}
For \(1<r<3\), \(a_r>0\), \(\nu>0\), and \(0<P\le1\),
the componentwise operator \(\mu+L:D_r\to Y_r\) is bijective and
\begin{equation}\label{bg:body-graph}
 \|w\|_{D_r}+\sqrt\nu\|\nabla w\|_{Y_r}
 \le C_r[1+(1+|\mu|)/a_r]\|f\|_{Y_r},
 \qquad (\mu+L)w=f.
\end{equation}
Its projected inverse is obtained by applying Cartesian Leray projection
to \(f\). At \(r=2\) the adjoint has the same maximal graph and the
same bound. All these estimates are independent of \(P\).
\end{lemma}

\begin{proof}
The heat-dilation formula gives a distributional inverse
\[
 S(t)h(y)=e^{t/2}[e^{\nu(e^t-1)\Delta}h](e^{t/2}y),\qquad
 \mathcal R_\mu f(\tau)=\int_0^\infty e^{-\mu t}S(t)f(\tau-t)\,dt.
\]
Its integrand has \(Y_r\) norm at most
\(e^{-a_rt}\|f\|_{Y_r}\); hence \(\|\mathcal R_\mu\|\le a_r^{-1}\).
The local heat estimate needed here is
\begin{equation}\label{free:heat-estimate}
 \|\nu D^2u\|_r+\|u_t\|_r\le C_{d,r}\|F\|_r,\qquad
 u_t-\nu\Delta u=F,
\end{equation}
for compactly time-supported distributional solutions \(u,F\in
L^r(\mathbb R\times\mathbb R^d)\). To justify its precise forcing
class, scale to \(\nu=1\). The multiplier
\(-\xi_i\xi_j/(i\omega+|\xi|^2)\) has \(L^2\) norm at most one.
Its kernel away from the origin is
\(K(t,x)=1_{t>0}\partial_{ij}H_t(x)\). For
\(\rho(t,x)=\max\{|x|,\sqrt{|t|}\}\), Gaussian differentiation gives
\[
 |K|\le C\rho^{-d-2},\quad
 |\nabla_xK|\le C\rho^{-d-3},\quad
 |\partial_tK|\le C\rho^{-d-4}.
\]
The kernel extends smoothly by zero through \(t=0\), away from
\(x=0\). Integrating these derivatives in parabolic annuli gives
\[
 \int_{\rho(z)>4\rho(h)}|K(z-h)-K(z)|\,dz\le C_d.
\]
The spatial and temporal differences are bounded, respectively, by
\[
 C|h_x|\int_{4\rho(h)}^\infty R^{-2}dR,\qquad
 C|h_t|\int_{4\rho(h)}^\infty R^{-3}dR,
\]
which are uniformly bounded.
For completeness, decompose an \(L^1\cap L^2\) function at height
\(\alpha\) using maximal dyadic rectangles with spatial side \(2^k\)
and time side \(2^{2k}\). The good part has
\(\|g\|_2^2\le C_d\alpha\|f\|_1\). The bad parts \(b_Q\) have
zero integral, total \(L^1\) norm at most \(2\|f\|_1\), and their
rectangles have total measure at most \(\alpha^{-1}\|f\|_1\).
Outside fixed enlargements of these rectangles, cancellation and
the displayed kernel integral bound the \(L^1\) norm of each
\(Tb_Q\) by \(C_d\|b_Q\|_1\). The \(L^2\) estimate for the good
part and Chebyshev's inequality for the bad part prove weak type
\((1,1)\). Integrating this distribution bound for the portions
\(|f|>\alpha\) and \(|f|\le\alpha\), together with the \(L^2\)
bound, gives \(L^r\) boundedness for \(1<r<2\). Duality gives the
remaining exponents since the reflected kernel has the same bounds.
Mollify the compact-time equation. The mollified solution vanishes
in the distant past, so its causal heat formula identifies its
second derivatives with these bounded operators applied to the
mollified \(F\). Their \(L^r\) limit equals the distributional
derivatives of \(u\); the equation gives \(u_t\). This proves
\eqref{free:heat-estimate}, with no viscosity loss.

On \(|t|<2\), set
\(v(t,x)=e^{-t/2}w(\tau_0+t,e^{-t/2}x)\).
Then \(v_t-\nu e^t\Delta v\) is the same pullback of \(f-\mu w\).
A fixed cutoff equal to one on \(|t|<1\), followed by
\(\theta=e^t\), reduces this equation to the preceding heat estimate.
All Jacobians on the window are bounded. Consequently
\[
 \int_{\tau_0-1}^{\tau_0+1}\|\nu\Delta w\|_r^r\,d\tau
 \le C_r\int_{\tau_0-2}^{\tau_0+2}
       [\|f\|_r^r+(1+|\mu|)^r\|w\|_r^r]\,d\tau.
\]
For periodic \(h\),
\(\int_0^P\int_Ih(\tau_0+t)\,dt\,d\tau_0
=|I|\int_0^Ph\).
Averaging the last inequality over the centers proves the diffusion
bound with no inverse-period factor. The equation gives the drift
bound; spatial elliptic interpolation gives the gradient bound.
The same reasoning applies to any distributional \(Y_r\) solution.

The adjoint semigroup is
\(S(t)^*h=e^{-t}[e^{\nu(1-e^{-t})\Delta}h](e^{-t/2}y)\).
For a smooth compact spatial test, its adjoint resolvent has itself and
its first two spatial derivatives in \(Y_{r'}\), by the gap and the
extra factor \(e^{-t/2}\) per derivative. Test a homogeneous solution
against this resolvent times a radial cutoff at radius \(N\).
The drift cutoff is bounded, and the diffusion cutoffs cost
\(N^{-1},N^{-2}\); all terms are bounded by the \(Y_r\) tail of the
solution times fixed \(Y_{r'}\) norms and tend to zero. Thus the
solution vanishes. The inverse therefore has precisely the displayed
maximal distributional domain. To exhibit a core directly, approximate
the right side by smooth periodic Schwartz functions and truncate the
inverse integral at \(M\). Such a truncated integral is Schwartz and
its equation has right side
\[
 f-e^{-\mu M}S(M)f(\tau-M).
\]
The remainder tends to zero in \(Y_r\); the graph estimate gives
graph convergence of the truncations. Thus the smooth periodic
Schwartz class is a core. Radial cutoffs approximate it in this graph,
since their drift commutator is bounded on an escaping annulus and
their diffusion commutators tend to zero. At \(r=2\), reverse
the phase window and use \(\theta=1-e^{-t}\) for
\(L^*=-\mathcal T-\nu\Delta+1\). Testing against the compact core
identifies the distributional equation of every vector in the actual
adjoint domain; this reversed-window estimate places it in \(D_2\).
Conversely, core integration by parts puts every element of \(D_2\)
in the adjoint domain. Its resolvent norm is at most \(a_2^{-1}\)
by duality, so the same graph bound follows. Leray commutes with
heat, dilation and phase translation, so projection preserves all
identities. Restriction to the real axisymmetric no-swirl subspace is
also legitimate since its symmetry projections commute with these maps.
On the Hilbert core, direct integration by parts gives
\[
 \operatorname{Re}\langle(\mu+L)w,w\rangle
       =(\operatorname{Re}\mu+1/4)\|w\|_2^2
                                      +\nu\|\nabla w\|_2^2.
\]
The adjoint has the same real part. For a tensor source
\(\mathbb P\operatorname{div}T\), its right side is at most
\(\|T\|_2\|\nabla w\|_2\). Completing the square gives
\(\|\nabla w\|_2\le\nu^{-1}\|T\|_2\) and
\(\|w\|_2\le[2\sqrt{\nu(\operatorname{Re}\mu+1/4)}]^{-1}\|T\|_2\).
Approximation of tensors completes this weak-stress map and the
inequality without assigning a full graph to its rough outputs.
\end{proof}

\begin{lemma}\label{bg:graph-embeddings}
Every \(w\in D_r\) has a representative in
\(C(\mathbb R/P\mathbb Z;L^r)\), and
\begin{equation}\label{bg:phase-embedding}
 \sup_\tau\|w(\tau)\|_r
 \le(1+3P/(2r))\|w\|_{Y_r}+P\|\mathcal Tw\|_{Y_r}.
\end{equation}
For \(r>5/2\),
\begin{equation}\label{bg:supremum-embedding}
 \|w\|_{L^\infty_{\tau,y}}
       \le C_r\nu^{-3/(2r)}\|w\|_{D_r}.
\end{equation}
\end{lemma}

\begin{proof}
On a fixed phase interval the distributional chain rule gives
\(V(t,x)=w(\tau_0+t,e^{-t/2}x)\in W^{1,r}_tL^r_x\), with
\(V_t\) the pullback of \(\mathcal Tw\). Undoing the strongly
continuous dilation gives the continuous phase representative;
distributional periodicity matches its endpoints. For
\(f(\tau)=\|w(\tau)\|_r\), the norm chain rule gives
\(\lvert f'+3f/(2r)\rvert\le\|\mathcal Tw(\tau)\|_r\).
Start at a phase with \(f\le\|w\|_{Y_r}\), integrate for at most
one period, and apply H\"older to obtain \eqref{bg:phase-embedding}.

For the supremum fix \((\tau_0,y_0)\) and put
\(W(t,z)=w(\tau_0+t,e^{-t/2}(y_0+\sqrt\nu z))\) on
\(|t|<1/2\). Then \(W_t=(\mathcal Tw)\circ\),
\(\Delta_zW=\nu e^{-t}(\Delta_yw)\circ\), and
\(dz=\nu^{-3/2}e^{3t/2}dy\). The integral of a nonnegative
\(P\)-periodic function over a unit interval is at most
\((1+P)\) times its phase average. Thus
\[
 \|W\|_{L^r}+\|W_t\|_{L^r}+\|\Delta_zW\|_{L^r}
       \le C_r\nu^{-3/(2r)}\|w\|_{D_r}.
\]
Choose a fixed time cutoff \(\chi=1\) near zero. The right side
\(F=(\partial_t-\Delta_z+1)(\chi W)\), after zero extension,
has this same bound. Its causal inverse kernel
\(k(t,z)=1_{t>0}e^{-t}H_t(z)\) satisfies
\[
 \|k\|_{L^{r'}}^{r'}
   =C_r\int_0^\infty e^{-r't}t^{-3(r'-1)/2}\,dt<\infty
       \quad\Longleftrightarrow\quad r>5/2.
\]
The identity \(\chi W=k*F\) follows distributionally; uniqueness
follows from the nonvanishing symbol \(i\omega+|\xi|^2+1\).
H\"older bounds its value at \((0,0)\), proving
\eqref{bg:supremum-embedding}. Translation continuity of \(k\) in
\(L^{r'}\) supplies a continuous local representative. No external
Sobolev embedding or inverse-period factor is used.
\end{proof}

\subsection{Tensor forces}

Write \(R_\mu=(\mu+L)^{-1}\mathbb P\). For tensor inputs its
divergence is interpreted by completion from smooth tensors, rather
than as an ordinary \(Y_2\) body force. Let
\[
 B=\mathcal T-\tfrac34,\qquad A=-\nu\Delta.
\]
The group \(u(\tau,y)\mapsto e^{-3t/4}u(\tau+t,e^{-t/2}y)\) is
unitary on \(Y_2\) and has generator \(B\); hence \(B\) is skew-adjoint.
The operator \(A\) is nonnegative self-adjoint.

\begin{lemma}\label{free:weak-stress}
For \(a=\operatorname{Re}\mu+1/4>0\), the map
\(S_\mu=R_\mu\operatorname{div}\) extends uniquely to
\(Y_2(\operatorname{Sym}_3)\), with
\begin{equation}\label{free:stress-energy}
 \|S_\mu T\|_{Y_2}\le\frac{\|T\|_{Y_2}}{2\sqrt{a\nu}},\qquad
 \|\nabla S_\mu T\|_{Y_2}\le\nu^{-1}\|T\|_{Y_2}.
\end{equation}
Uniformly on compact subsets of this half-plane,
\begin{equation}\label{free:half-drift}
 \|(1+|B|)^{1/2}S_\mu T\|_{Y_2}
       \le C_\mu\nu^{-1/2}\|T\|_{Y_2}.
\end{equation}
The weak solution is unique among distributional \(Y_2\) solutions.
Bounded measurable tensor partitions are allowed before divergence.
The body and tensor inverses depend
holomorphically on \(\mu\) in their respective half-planes.
\end{lemma}

\begin{proof}
The energy identity and completion proved above give
\eqref{free:stress-energy}. Differences of smooth tensors give strong
convergence of both velocities and first gradients, so the equation
and energy identity pass to the limit. Distributional uniqueness
follows by subtracting two solutions and applying the homogeneous
uniqueness argument in Lemma~\ref{bg:common-body-graph}.
In particular a bounded measurable multiplier may be applied to a
tensor before this completion; the distributional divergence includes
all interface terms.

Put \(C=1+|B|\), \(D=1+A\), and let \(\mathcal R_\mu\) be the
inverse of \(\mu+L\), acting componentwise. Its direct graph bounds
\(C\mathcal R_\mu\); the actual adjoint graph bounds
\(D\mathcal R_\mu^*\), and duality bounds the extension of
\(\mathcal R_\mu D\). The imaginary powers of \(C,D\) are unitary.
Insert the respective spectral cutoffs on the two sides of
\[
 C^z\mathcal R_\mu D^{1-z},\qquad
                 0\le\operatorname{Re}z\le1.
\]
At the two boundary lines the preceding graph estimates give
uniform bounds. The scalar three-lines theorem applied to pairings,
followed by density and closedness, yields
\(\|C^{1/2}\mathcal R_\mu D^{1/2}\|\le C_\mu\).
No commutation between \(A\) and \(B\) is asserted. Spatial Fourier
transformation gives
\(\operatorname{div}=\nu^{-1/2}A^{1/2}\mathcal Q\), where
\(\mathcal Q\) is a bounded tensor-to-vector combination of Riesz
transforms. Since \(D^{-1/2}A^{1/2}\) is bounded and Leray commutes
with \(A,B\), the last estimate proves \eqref{free:half-drift}.

The inverse of \(\mu+L\) is holomorphic by its convergent semigroup integral.
For the stress inverse, distributional uniqueness gives
\[
 S_\mu-S_\zeta=(\zeta-\mu)R_\mu S_\zeta.
\]
The body and stress bounds make this the resolvent identity in
operator norm, proving the asserted holomorphy.
\end{proof}

At \(r=11/4\) the body gap at \(\mu=0\) is \(1/22\), and at
\(r=23/8\) it is \(1/46\). In particular all shifts in the
background and spectral constructions lie in the stipulated ranges.
The graph embeddings also give, at \(r=11/4\),
\[
 \|\mathbb P[(u\cdot\nabla)v]\|_{Y_r}
 \le C\nu^{-23/22}\|u\|_{D_r}\|v\|_{D_r}.
\]
For any unitary velocity and tensor lifts \(J_v:H\to Y_2\) and
\(J_T:H_T\to Y_2\), the transported stress map is exactly
\(J_v^{-1}S_\mu J_T:H_T\to H\), with the same energy bounds.
This formula transports the completed tensor inverse; it does not
assign an ordinary body-force norm to \(\operatorname{div}J_TT\).


\subsection{Free comparison with global tensor inputs}

Let \(R=\epsilon^{-1}\), \(P=2\epsilon c\le1\), and use
\[
 s=R^2/\rho^2,\qquad q=R\log(\rho/R)+R\tau/2,
 \qquad \eta=Rz/\rho.
\]
The velocity and tensor measures are
\(H=L^2(s^{-3/2}ds\,dq/c\,d\eta)\) and
\(H_T=L^2(s^{-1/2}ds\,dq/c\,d\eta;\operatorname{Sym}_3)\).
The natural lifts \(b\mapsto\sqrt{s}\,b\) and \(T\mapsto sT\)
both multiply norms by \(\sqrt\pi R\). All vector fields are
axisymmetric without swirl, and tensors have no component with exactly
one azimuthal index. Set
\[
 S=s\partial_s,\quad K=2S+\eta\partial_\eta,\quad
 Q_\epsilon=\partial_q-\epsilon(K+1),\quad E=\operatorname{diag}(1,1,0).
\]
Direct Cartesian differentiation gives
\begin{align*}
 A_\epsilon&=-S+\nu s[(Q_\epsilon^2+\partial_\eta^2)I-\epsilon^2E],
 &A_0&=-S+\nu s\Delta_{q,\eta},\\
 \operatorname{div}(\sqrt{s}\,b)
 &=s[(\partial_q-\epsilon K)b_r+\partial_\eta b_z],\\
 (\operatorname{div}_\epsilon T)_r
 &=Q_\epsilon T_{rr}+\partial_\eta T_{rz}-\epsilon T_{\phi\phi},
 &(\operatorname{div}_\epsilon T)_z
 &=Q_\epsilon T_{zr}+\partial_\eta T_{zz}.
\end{align*}
Thus the full tensor divergence, including the hoop traction, is retained.
The domains of \(A_\epsilon\) and its adjoint are the exact Cartesian
domains under these lifts. No boundary is introduced at either age end.
Let \(P_\epsilon\) be the conjugated Cartesian Leray projection and
\(P_0\) the ordinary planar cylinder projection. Write
\[
 T_\epsilon=(\mu-A_\epsilon)^{-1}P_\epsilon,
 \qquad \mathcal S_\epsilon=T_\epsilon s\operatorname{div}_\epsilon.
\]
The tensor map means completion from smooth tensors. The free Cartesian
energy estimate, and its flat age-energy counterpart, give uniformly
for \(0\le\epsilon\le1\) and \(\mu\) in a compact subset of
\(\operatorname{Re}\mu>-1/4\),
\begin{equation}\label{near:free-energy}
 \|b\|_H+\sqrt\nu\|(Q_\epsilon b,\partial_\eta b,
                    \epsilon b_r)\|_{H_T}
 \le C\nu^{-1/2}\|M\|_{H_T},\qquad b=\mathcal S_\epsilon M.
\end{equation}
The flat equation is
\((\partial_s+\mu/s-\nu\Delta)b=P_0\operatorname{div}_0M\).
Pairing it with \(s^{-1/2}\bar b\) gives the coercive constant
\(\operatorname{Re}\mu+1/4\) and proves its completion independently
of comparison. Bounded measurable partitions of \(M\) are therefore
allowed before divergence; their distributional interface terms remain.

\begin{proposition}\label{near:compact-free-comparison}
Let \(\chi\) be supported in \(\sigma\le s\le H_1\),
\(|\eta|\le L_1\), and let \(\chi_1\) equal one on a larger
neighborhood. There is a fixed integer \(k\) such that
\begin{equation}\label{near:compact-free-rates}
 \|\chi(T_\epsilon-T_0)\|_{H\to H}
       \le C\Gamma^k\epsilon^{1/14},\qquad
 \|\chi(\mathcal S_\epsilon-\mathcal S_0)\|_{H_T\to H}
       \le C\Gamma^k\epsilon^{1/50},
\end{equation}
where \(\gamma_*=\inf(\operatorname{Re}\mu+1/4)>0\) over the fixed
parameter set and
\[
 \Gamma=2+\nu+\nu^{-1}+c+c^{-1}+\sigma^{-1}+H_1+L_1
 +\sup|\mu|+\gamma_*^{-1}
 +\|\chi\|_{C^8}+\|\chi_1\|_{C^8}.
\]
The integer is fixed before any window is allowed to depend on
\(\epsilon\). Inputs are global and arbitrary in their indicated spaces.
\end{proposition}

\begin{proof}
First compare the projections on smooth tests. On each nonzero
\(q\)-frequency, a flat divergence-free field and gradient have forms
\((-\psi_\eta,\psi_q)\), \((\varphi_q,\varphi_\eta)\).
Their exact counterparts are
\[
 (-\psi_\eta,(\partial_q-\epsilon K)\psi+\epsilon\psi),
 \qquad ((\partial_q-\epsilon K)\varphi,\varphi_\eta).
\]
The first has zero exact divergence; the second lifts to a Cartesian
gradient. The inverse \(q\)-derivative costs at most \(c/(2\pi)\).
Also, for the flat Fourier multiplier,
\(|\partial_\xi^jP_0(k,\xi)|\le C_j|k|^{-j}\), \(j=1,2\),
so one or two transverse moments of the projected field cost only
fixed powers of \(c\). All projection operations commute with \(S\).

For the zero frequency, the radial component is divergence-free and the
axial component is a gradient. Take the corresponding primitives to
vanish at \(\eta=0\), and cut them off at \(|\eta|\sim L_0\).
Their suprema, including one \(S\) derivative, are bounded by
\(C\|\langle\eta\rangle f\|_2\) and
\(C\|\langle\eta\rangle Sf\|_2\).
Cutoff differentiation costs \(L_0^{-1/2}\), geometric correction
costs \(\epsilon L_0^{1/2}\), and the omitted field tail costs
\(L_0^{-1}\). Choose \(L_0=\epsilon^{-1}\).
Distances to the exact gradient and divergence-free subspaces, both closed
orthogonal subspaces of \(H\), now give
\begin{equation}\label{near:leray-test}
 \|(P_\epsilon-P_0)f\|_H
 \le p(1+c+c^{-1})\sqrt\epsilon
              \sum_{a,j=0}^1\|\eta^jS^af\|_H.
\end{equation}
Initially use finite Fourier sums, compact log-age support and Schwartz
transverse profiles. The exact corrections lift to annular fields.
Cutoff, mollification and Fourier truncation are dense in the displayed
norm, after \(x=\log s\), \(u=e^{-x/4}f(e^x)\).
Thus \eqref{near:leray-test} holds on its full stated domain and retains
the zero mode.

Next obtain uniform local regularity of the outputs.
The unitary lift conjugates \(\mathcal T-3/4\) to \(S-1/4\).
The Cartesian estimate \eqref{free:half-drift} therefore gives
\[
 \||S-1/4|^{1/2}\mathcal S_\epsilon M\|_H
           \le C\nu^{-1/2}\|M\|_{H_T}\qquad(\epsilon>0).
\]
At \(\epsilon=0\), a compact-age cutoff \(u\) of the flat solution
satisfies \(u\in L_s^2H^1\), \(u_s\in L_s^2H^{-1}\) by
\eqref{near:free-energy} and its equation. Fourier Cauchy--Schwarz gives
\(\||\partial_s|^{1/2}u\|_2^2\le
\|u_s\|_{L_s^2H^{-1}}\|u\|_{L_s^2H^1}\).
All local constants are polynomial in \(\Gamma\).

In log age the conjugate of \(Q_\epsilon\), shifted by \(\epsilon\),
is the skew operator
\(\widetilde Q_\epsilon=\partial_q-2\epsilon\partial_x
                     -\epsilon(\eta\partial_\eta+1/2)\).
Its three constituent flows commute. Multiplication by a smooth cutoff
preserves the half age derivative: its difference quotient splits into
the cutoff times the quotient of the function and its first derivative
times a translated function. Both terms have the stated polynomial bound.
For a compactly supported
cutoff \(f\) of \(e^{-x/4}b(e^x)\), their unitary flow identities give
\[
 \|f(q+h)-f(q)\|_2
 \le |h|\|\widetilde Q_\epsilon f\|_2
 +C|\epsilon h|^{1/2}\||\partial_x|^{1/2}f\|_2
 +|\epsilon h|\|(\eta\partial_\eta+1/2)f\|_2.
\]
For \(|h|\le1\), square and integrate against \(|h|^{-3/2}\);
for larger \(h\), use \(2\|f\|_2\). The half age derivative
and full transverse derivative already have bounds. Directional Fourier
inequalities and the smooth log-age change prove
\begin{equation}\label{near:local-smoothing}
 \|\chi\mathcal S_\epsilon M\|_{H^{1/4}(s,q,\eta)}
       \le C\Gamma^k\|M\|_{H_T}.
\end{equation}
For a body source the graph controls \(Sb\), and the gradient controls
\(Q_\epsilon b,\partial_\eta b\). Since
\(\partial_s=s^{-1}S\) and
\(\partial_q=Q_\epsilon+\epsilon(2S+\eta\partial_\eta+1)\),
the body output has a full local \(H^1\) bound.
The flat body graph follows directly by cutting off its heat equation
on dyadic intervals \([a/2,4a]\): the Fourier heat estimate controls
\(\nu s\Delta b\), the equation controls \(Sb\), and finite overlap
sums these bounds. Periodic extension over many cells followed by division
by their number gives constants independent of the number of cells.

It remains to compare smooth adjoint observations globally. For a smooth
test \(z\) in the observation box, put \(h=P_0z\) and \(v=T_0^*z\).
Exactly,
\[
 v(s)=\int_s^\infty(s/t)^{\bar\mu+1/2}
                e^{\nu(t-s)\Delta}h(t)\,\frac{dt}{t}.
\]
It vanishes for \(s>H_1\). With \(a=\bar\mu+1/2\),
\[
 Sv=av-\nu s\Delta v-h,\qquad
 S^2v=a^2v-\nu s(2a+1)\Delta v+\nu^2s^2\Delta^2v
                         -ah+\nu s\Delta h-Sh.
\]
In log age the integral is a heat contraction followed by translation,
with scalar kernel \(e^{-(\bar\mu+1/4)r}\), \(r>0\).
Minkowski bounds it by the inverse coercivity gap. The identities
\[
 \eta e^{t\Delta}=e^{t\Delta}(\eta-2t\partial_\eta),\qquad
 \eta^2e^{t\Delta}=e^{t\Delta}
       (\eta^2-4t\eta\partial_\eta-2t+4t^2\partial_\eta^2)
\]
control two transverse moments since the heat time is at most \(\nu H_1\).
The multiplier estimate used in \eqref{near:leray-test} transfers these
moments through \(P_0\). The preceding two age identities therefore give,
with \(\Lambda=K+1\),
\begin{equation}\label{near:adjoint-polynomial}
 \|\nu s\Lambda\partial_qv\|_H+\|\nu s\Lambda^2v\|_H+\|\nu sEv\|_H
 +\|\eta\partial_\eta v\|_H
 +\sum_{a,j=0}^1\|\eta^jS^av\|_H
 \le C\Gamma^k\|z\|_{H^6}.
\end{equation}
Indeed \(\Lambda^2=4S^2+4S\eta\partial_\eta+4S+
\eta^2\partial_\eta^2+3\eta\partial_\eta+1\).
At most two age differentiations, two moment commutations and six test
derivatives occur. Below the test support all norms are integrable by
\(\int_0^\sigma s^{2\operatorname{Re}\mu-1/2}ds<\infty\).
Thus the degree \(k\) is fixed independently of later windows.

The exact generator difference is
\(A_\epsilon-A_0=\nu s[-2\epsilon\Lambda\partial_q+
\epsilon^2\Lambda^2-\epsilon^2E]\), also for adjoints. If
\(w=T_\epsilon^*z-v\), then
\[
 (\bar\mu-A_\epsilon^*)w
 =(P_\epsilon-P_0)z+(A_\epsilon^*-A_0^*)v.
\]
The right side has \(H\) norm at most
\(C\Gamma^k\sqrt\epsilon\|z\|_{H^6}\).
The exact componentwise free adjoint energy estimate bounds \(w\)
and its physical gradient by this quantity (with a polynomial viscosity
cost). There is no projection before the second source term.
The domain application is legitimate because
\eqref{near:adjoint-polynomial} puts \(v\) in the actual maximal adjoint
graph; its lift vanishes near the axis, without any assumption that
\(v\) is divergence-free for finite \(\epsilon\).

Let \(G_\epsilon\) be the negative weighted symmetric gradient adjoint
to \(s\operatorname{div}_\epsilon\). Then
\[
 (\mathcal S_\epsilon^*-\mathcal S_0^*)z
   =G_\epsilon w+(G_\epsilon-G_0)v.
\]
The divergence formula at the start shows that the second term is
\(\epsilon\) times \(Sv,\eta\partial_\eta v,v\) and frame terms.
Consequently
\begin{equation}\label{near:smooth-adjoint-comparison}
 \|(T_\epsilon^*-T_0^*)z\|_H
 +\|(\mathcal S_\epsilon^*-\mathcal S_0^*)z\|_{H_T}
 \le C\Gamma^k\sqrt\epsilon\|z\|_{H^6}.
\end{equation}

Place the output support in a larger periodic box in \(s,\eta\),
retaining the original normalized \(q\) measure. Fourier projection
to frequencies at most \(N\), and the local \(H^1\) and
\(H^{1/4}\) bounds, give high-frequency errors \(C\Gamma^kN^{-1}\)
and \(C\Gamma^kN^{-1/4}\). A unit low-frequency dual vector corresponds
to the compact smooth \(H\)-test \(s^{3/2}\chi F_Nh\), with
\(H^6\) norm at most \(C\Gamma^kN^6\). Thus the full operator
differences are at most
\[
 C\Gamma^k(N^{-1}+\sqrt\epsilon N^6),\qquad
 C\Gamma^k(N^{-1/4}+\sqrt\epsilon N^6).
\]
Choose \(N=\epsilon^{-1/14}\) and \(N=\epsilon^{-2/25}\), respectively.
No Fourier mode-count factor is used. This proves
\eqref{near:compact-free-rates}.
\end{proof}

The argument uses only normalized phase averages. The imported Cartesian
graph bound is obtained by averaging fixed-size heat windows over their
centers, so it has no inverse-period loss. Every remaining operation
above is a spatial or age identity. Hence the comparison remains valid
when \(P=2\epsilon\nu^{-2}\) tends to zero. Exact Cartesian divergence,
the pressure projection, the zero cylinder mode and tensor interface
terms have all been retained.

\subsection{The background free comparison}

Use the chart, operators and cutoffs of
Proposition~\ref{near:compact-free-comparison}. For \(1<r<3\), set
\[
 H_r=L^r(s^{r/2-5/2}\,ds\,dq/c\,d\eta),\qquad
 dm=s^{-5/2}\,ds\,dq/c\,d\eta,\qquad Mb=\sqrt{s}\,b.
\]
The common multiplication \(M:H_r\to L^r(dm)\) is an isometry for
every \(r\). Both the physical body source and velocity have lift
\(\sqrt{s}\) and norm factor \((\pi R^2)^{1/r}\); this factor
cancels from every source-to-state estimate. In particular the
conjugated Leray bound is uniform in \(R\).

The exact graph gives
\begin{equation}\label{bg:profile-graph}
 \|T_\epsilon F\|_{H_r}+\|ST_\epsilon F\|_{H_r}
 +\sqrt\nu\|\sqrt{s}(Q_\epsilon,\partial_\eta)T_\epsilon F\|_{H_r}
 \le C_{r,\mu}\|F\|_{H_r}.
\end{equation}
Indeed \(\mathcal T(\sqrt{s}b)=\sqrt{s}(S+1/2)b\),
\(\partial_\rho(\sqrt{s}b)=sQ_\epsilon b\), and
\(\partial_z(\sqrt{s}b)=s\partial_\eta b\).
For \(\epsilon=0\), the same estimate follows without a limiting
argument. The causal formula
\[
 T_0F(s)=\int_0^s(t/s)^\mu e^{\nu(s-t)\Delta}P_0F(t)\,dt/t
\]
and weighted dilation give \(\|T_0\|_{H_r\to H_r}\le C_r/a_r\).
On each dyadic window \([a,2a]\), cut its heat equation off inside
\([a/2,4a]\) and apply the heat estimate used above. The cutoff costs
\(a^{-1}\); multiplication by \(s\) cancels it. Finite overlap and
comparability of \(s^{r/2-5/2}\) on each window prove the global bounds
for \(ST_0\) and \(\nu sD^2T_0\), and interpolation gives its gradient
bound. The cylinder heat estimate follows by applying the Euclidean
estimate to a cutoff of \(N\) repeated cells, dividing its \(r\)-th
power by \(N\), and letting \(N\to\infty\). There is no period loss.
Consequently \(\partial_s=s^{-1}S\) and
\(\partial_q=Q_\epsilon+\epsilon(2S+\eta\partial_\eta+1)\)
give uniform local first-derivative bounds at both endpoints.

\begin{proposition}\label{bg:lp-free-comparison}
Fix \(p=11/4\) and \(r=23/8\). On any compact set of shifts with
\(\inf(\operatorname{Re}\mu+1/46)>0\), let \(B\) be a first-order
operator in \(q,\eta\), with smooth matrix coefficients supported
in the observation box. There is a fixed integer \(k\) such that
\begin{equation}\label{bg:lp-comparison-rate}
 \|B(T_\epsilon-T_0)\|_{H_p\to H_p}
       \le C\Gamma_r^k\epsilon^{2/539}.
\end{equation}
Here \(\Gamma_r\) is \(\Gamma\) of the shared Hilbert proposition,
augmented by the reciprocal \(r\)-gap and the finite coefficient
seminorms of \(B\). The inputs may have arbitrary age support.
\end{proposition}

\begin{proof}
We first upgrade the shared body \(H_2\) estimate to spatial
derivatives. For \(\|F\|_{H_2}=1\), put
\(u=T_\epsilon F\), \(v=T_0F\), \(w=u-v\), and
\(\omega=\|\chi_1(T_\epsilon-T_0)\|\).
Subtract their componentwise equations and test against \(\chi^2w\).
This compact test is allowed by the local first-derivative graph.
Writing \(L_1=2S+\eta\partial_\eta+1\), the principal diffusion
difference has exactly the two terms
\[
 \nu\epsilon\langle\sqrt{s}L_1v,
                    \sqrt{s}Q_\epsilon(\chi^2w)\rangle
 +\nu\epsilon\langle\sqrt{s}\partial_qv,
                    \sqrt{s}L_1(\chi^2w)\rangle.
\]
Absorb one quarter of the gradient energy from the first term;
the second is \(O(\Gamma_r^k\epsilon)\) by the separate bounds for
\(Sv,Sw\) on this box. Diffusion cutoff terms cost another quarter
plus \(C\Gamma_r^k\omega^2\). The drift identity is
\[
 \operatorname{Re}\langle Sw,\chi^2w\rangle_{H_2}
 =\tfrac14\|\chi w\|_{H_2}^2
          -\int s\chi\chi_s|w|^2\,dH_2.
\]
The projection-source difference \((P_\epsilon-P_0)F\) has norm
at most two, so its pairing costs \(2\omega\). The radial frame
discrepancy costs \(C\Gamma_r^k\epsilon^2\); retain its nonnegative
energy on \(w\). It follows that
\[
 \|\chi(Q_\epsilon,\partial_\eta)w\|_{H_2}^2
 \le C\Gamma_r^k(\omega+\omega^2+\epsilon).
\]
By \eqref{near:compact-free-rates}, \(\omega\le
C\Gamma_r^k\epsilon^{1/14}\). Since
\(\partial_qw=Q_\epsilon w+\epsilon L_1w\), the localized
first-derivative difference is \(O(\Gamma_r^k\epsilon^{1/28})\).
This proves the Hilbert version of \eqref{bg:lp-comparison-rate}.

At exponent \(r=23/8\), \eqref{bg:profile-graph} and the local
coordinate identities give a uniform polynomial bound for the same
operator. Interpolate the single consistent operator
\(MB(T_\epsilon-T_0)M^{-1}\) on \(L^r(dm)\), rather than
interpolating unequal weighted spaces. The weight on its Hilbert
endpoint is
\[
 \lambda=\frac{1/p-1/r}{1/2-1/r}=\frac8{77},\qquad
 \frac{\lambda}{28}=\frac2{539}.
\]
The semigroup formulas agree on the dense intersection of the endpoint
spaces, so this is an interpolation of one operator.

\end{proof}

For the actual localized interaction write
\(K_\epsilon=sB_\epsilon T_\epsilon\), \(K_0=sB_0T_0\).
Then
\[
 K_\epsilon-K_0=sB_0(T_\epsilon-T_0)
                       +s(B_\epsilon-B_0)T_\epsilon.
\]
The first term is covered by \eqref{bg:lp-comparison-rate}. Exact
chart substitution writes the second as \(\epsilon\) times
polynomially bounded coefficients multiplying
\(b,\partial_qb,\partial_\eta b,Sb\); these are bounded by
\eqref{bg:profile-graph} and its local consequence. Thus the same
\(\epsilon^{2/539}\) bound applies. The interaction remains
unprojected in force coordinates: Leray is already inside \(T_\epsilon\).
No pressure-cutoff commutator or derivative of a rough input has been
introduced. Output tails outside the box require the separate train
coefficient estimates; none are inferred from this local proposition.

\section{The exact unforced background}
\label{short-bg:section}

We use the real axisymmetric class without swirl.  Set
\[
 p=11/4,\quad c=\nu^{-2},\quad R=\epsilon^{-1},\quad
 \epsilon=e^{-d/\nu},\quad P=2\epsilon c,
 \qquad 0<d_-\le d\le d_+.
\]
Powers of $\nu^{-1},c$ and fixed polynomial cutoff lengths are denoted
by $\mathcal P_\nu$. Constants may depend on $d_\pm$ and on a fixed
derivative budget. The phase measure is normalized by $d\tau/P$.
Use the spaces $Y_p,D_p$ of Lemma~\ref{bg:common-body-graph}, and write
$A_0=L$ and $X_p=D_p\cap\{\operatorname{div}w=0\}$ with the same graph
norm. Let $v_c$ be the train of Proposition~\ref{sct:train} and
$V_0=U_{\rm raw}$ its divergence-free lift from Proposition~\ref{sct:lift},
with residual $F_R$. In particular,
\eqref{sct:residual-bound} gives
$\|F_R\|_{C_\tau W^{1,p}}\le\mathcal P_\nu\epsilon^{3/11}$.

For $m>8$, write $X_m=L^2(\mathbb R^2,\langle x\rangle^{2m})$.
The frozen growing pair and complementary bound are given by
Corollary~\ref{cor:short-seed-dilation}. The train estimates give
\begin{align}
 &\sup_{s\ge0}\sum_{j=0}^3\|D^jv_c(s)\|_\infty\le C,
 \label{short-bg:train-uniform}\\
 &\int_{\Sigma/\nu}^\infty
       (\|Dv_c(s)\|_\infty+\|D^2v_c(s)\|_\infty)\,ds
 \le {C\over\nu(1+\Sigma)}+C\nu^3.
 \label{short-bg:train-late}
\end{align}
The weighted coefficient and polynomial spatial $L^{22/3}$ bounds follow
from Lemma~\ref{sct:heat}. In normalized cylinder coordinates set
\begin{equation}
 H_p=L^p(s^{-9/8}\,ds\,dq/c\,d\eta),\qquad
 \|\sqrt{s}\,b\|_{Y_p}=(\pi R^2)^{1/p}\|b\|_{H_p},
 \quad
 T_0F(s)=\int_0^s e^{\nu(s-t)\Delta}\mathbb P_0 F(t){dt\over t}.
 \label{short-bg:measure}
\end{equation}
The global free graph and compact-output comparison are
\eqref{bg:profile-graph} and Proposition~\ref{bg:lp-free-comparison};
their inputs have arbitrary age support.

\subsection{One mixed construction for the finite age interval}

We first verify the common actual-cylinder comparison. On the centered
cell let $X_{m,c}=L^2(C_c,(1+q^2+\eta^2)^m)$ and let $J_c,R_c$ be
zero extension and restriction. Extend the free cylinder evolution by
the identity on the exterior summand of $X_m$ and the interaction by
zero there. The vorticity equation is
\begin{equation}
 \omega_s=(\nu\Delta-v_c(s)\cdot\nabla+A_c(s))\omega,
 \qquad A_c(s)\omega=-(K_c\omega)\cdot\nabla E_c(s).
 \label{short-bg:actual-cylinder}
\end{equation}
Here $E_c=\operatorname{curl}v_c$ and $K_c$ includes the horizontal
mean mode. On $s\le\Sigma/\nu$, the train comparison gives, in any
fixed finite weighted derivative budget,
\[
 \|v_c-V_c^p\|+\|E_c-E_c^p\|
 \le\delta_c:=\mathcal P_\nu
                  e^{C_\Sigma/\nu-b_\Sigma c^2}.
\]
The central radial Gaussian transports the radial weight exactly;
the velocities induced by the other periodic copies, the region near the
cell boundary, and the displayed error contribute at most
$C(\delta_c+e^{-b_\Sigma c^2})$ to its logarithmic derivative.
The weight is continuous across the cell boundary and its normal derivative has
a negative jump. Its distributional Laplacian is therefore bounded
above by $C_m$ times the weight. Weighted integration by parts gives
\begin{equation}
 \|S_{\nu,c}(t,s)\|_{X_{m,c}\to X_{m,c}}
 \le\exp[(C_m\nu+C\delta_c+Ce^{-b_\Sigma c^2})(t-s)].
 \label{short-bg:free-cylinder}
\end{equation}
One may smooth the weight, preserving uniform comparability, to justify
this calculation. On a fixed block, Duhamel for a smooth compactly
supported datum against the frozen radial transport gives convergence:
the transported support remains a fixed disk, the viscosity error is
$\nu\Delta$ applied only to that smooth datum, and the coefficient
change is $O(\nu L)+o(1)$ there. Density and
\eqref{short-bg:free-cylinder} give strong convergence on $X_m$,
uniformly for block lengths in a compact interval. The frozen radial
transport is unitary in this radial weighted space. Multiplying by its
inverse reduces the free convergence to $W_\nu\to I$ strongly with
$\|W_\nu\|\le1+o(1)$. The identity for
$\|W_\nu^*f-f\|^2$ then gives strong adjoint convergence too. This
argument makes no operator-norm assertion about the Laplacian.

The cylinder kernel is uniformly bounded away from its common local
$1/|x-y|$ singularity. On $|x|,|y|\le c/4$, paired periodic images
give
$|K_c(x-y)-K_{\mathbb R^2}(x-y)|
\le Cc^{-2}(1+|x|+|y|)$.
Multiplication by the Gaussian observation coefficient $\nabla E$
and weighted Cauchy--Schwarz control this part by $Cc^{-2}\|f\|_{X_m}$.
For $|y|>c/4$, the same Cauchy--Schwarz estimate gives a factor
$c^{1-m}$, with an additional harmless source-moment power if needed;
for $|x|>c/4$ the coefficient is Gaussian-small. The coefficient error
contributes $C\delta_c$. Thus
\begin{equation}
 \|J_cA_c(\sigma/\nu)R_c-A_{\rm pl,\sigma}\|_{X_m\to X_m}
 \le C_\Sigma(c^{-2}+\delta_c),\qquad 0\le\sigma\le\Sigma.
 \label{short-bg:interaction-cylinder}
\end{equation}
The near kernel gains one local derivative, by the Fourier multiplier
bound for $DK$, and the coefficient is localized. Truncate source and
observation, use local Rellich compactness, and then the same tail
estimates: every interaction is compact and their family is
collectively compact. The artificial exterior is invariant and has no
growing component. Restriction removes it. To implement zero
circulation in the comparison one may adjoin a radial mass-one
neutral vector to the plane zero-circulation space; radial Euler
linearization annihilates it. Cylinder curl sources and periodized
angular-two entrance vectors have zero circulation, so their actual
solutions remain in the desired subspace.

Fix $\Sigma$ and put $T=\Sigma/\nu$. Choose
$0<a<a'<b'<b<\gamma/(1+\Sigma)^2$. On a bounded block of length $L$,
the common compact-Duhamel lemma gives
\[
 U(t+L,t)=W_{\nu,t}e^{L L_{\nu t}}+o_{\nu\to0}(1)
 \quad\hbox{in operator norm},
\]
where $\|W_{\nu,t}\|\le1+o(1)$ and $W_{\nu,t},W_{\nu,t}^*\to I$
strongly. The error is uniform for block lengths in $[0,L]$.
Indeed the interaction Duhamel term converges in norm; its limit is
compact, so multiplying it by $W_{\nu,t}-I$ gives a norm-small error.
Let $E_t,F_t$ be the frozen growing and complementary spaces. Finite
rank and strong adjoint convergence make both off-diagonal blocks
small. For $L$ fixed sufficiently large, each full block has matrix
\[
 \begin{pmatrix}a_n&b_n\\c_n&d_n\end{pmatrix},\qquad
 \|a_n^{-1}\|\le A,\quad \|d_n\|\le D,\quad
 \|b_n\|+\|c_n\|\le\eta_\nu\longrightarrow0,
 \qquad AD<1.
\]
Here $A,D$ can be chosen below $e^{-b'L},e^{a'L}$ with strict
margins. Frozen projections use the periodized plane direct vectors,
which have zero circulation, and the restricted plane dual vectors,
with their finite pairing matrix normalized. Weighted-dual convergence
suffices; no far-field expansion of
the dual vector is used.

For completeness, construct the invariant subspaces using
\begin{equation}
 \xi_{n+1}=(c_n+d_n\xi_n)(a_n+b_n\xi_n)^{-1},\qquad
 \zeta_n=(a_n-\zeta_{n+1}c_n)^{-1}(\zeta_{n+1}d_n-b_n),
 \quad \xi_0=0,\quad\zeta_N=0.
 \label{short-bg:graphs}
\end{equation}
On a ball of radius $r=C\eta_\nu$, these maps have Lipschitz constant
at most $A(D+\eta_\nu r)/(1-A\eta_\nu r)<1$. They preserve the ball
if $r(1-AD)\ge A\eta_\nu(1+r^2)$. Hence the graphs are complementary,
their projections are uniformly bounded, and the products of their
coordinate maps have bounds $e^{a'(t-s)}$ forward on the center and
$e^{-b'(s-t)}$ backward on the growing space. Coordinate equivalence
is used only at the two endpoints of a product. An appended block of
length at most $L$ has uniformly bounded growing inverse and forward
map and changes only the prefactor.

At intermediate times in a block $[t_n,t_{n+1}]$, put
$J_t=U(t,t_n)|_{E_n^u}$ and
$F_t=\Pi^u_{n+1}U(t_{n+1},t)$. Their product is the invertible
two-dimensional block $B_n=U(t_{n+1},t_n)|_{E_n^u}$, so
$\Pi^u(t)=J_tB_n^{-1}F_t$ and $\Pi^c(t)=I-\Pi^u(t)$.
This proves invariance and bounds without inverting a parabolic map
on the infinite-dimensional space. Consequently
\begin{equation}
 z(s)=\int_0^s U(s,t)\Pi^c(t)f(t)\,dt
       -\int_s^T U(s,t)\Pi^u(t)f(t)\,dt
 \label{short-bg:green}
\end{equation}
satisfies the equation, has initial value in the fixed entrance pair,
has vanishing terminal growing coordinate, and obeys
\begin{equation}
 \sup_{[0,T]}\|z(s)\|_{X_m}
       \le C e^{a'T}\|f\|_{L^1(0,T;X_m)}.
 \label{short-bg:green-bound}
\end{equation}
Only a rank-two evolution was inverted. All constructions preserve
zero circulation and reality.

\subsection{Completion for arbitrary body forces}

Let $F\in H_p$, set $b_0=T_0F$, and solve for the correction in the $(r,z)$-components
\begin{equation}
 z_s-\nu\Delta z+\mathbb P_0(v\cdot\nabla z+z\cdot\nabla v)
     =g:=-\mathbb P_0(v\cdot\nabla b_0+b_0\cdot\nabla v).
 \label{short-bg:body-eq}
\end{equation}
Use the fixed Green selector \eqref{short-bg:green} for the vorticity
$\operatorname{curl}z$: its initial value belongs to the fixed entrance
pair and its terminal growing coordinate vanishes. No condition is
imposed on $b_0(T)$ or on a velocity observation. The train coefficient
and source remain present at every age. The global free graph gives
\[
 \|b_0\|_{H_p}+\|\sqrt{\nu s}Db_0\|_{H_p}
                    +\|\nu sD^2b_0\|_{H_p}\le C\|F\|_{H_p}.
\]
Put $b=13/8$, so $b-p=-9/8$. Spatial H\"older with $22/3$,
followed by one derivative, bounds both $g$ in $H^1$ and
$\operatorname{curl}g$ in $X_m$ on a finite slow-age band by
$\mathcal P_\nu(\|D^2b_0\|_p+\|Db_0\|_p+\|b_0\|_p)$.
Thus
\begin{equation}
 \|g\|_{L^p((0,T),s^bds;H^1)}
 +\|\operatorname{curl}g\|_{L^p((0,T),s^bds;X_m)}
 \le\mathcal P_\nu\|F\|_{H_p}.
 \label{short-bg:source}
\end{equation}
The circulation of the curl source is zero. The elementary estimates
\begin{align}
 \|f\|_{L^1(0,T)}&\le14^{7/11}T^{1/22}
                        \|f\|_{L^p(s^{13/8}ds)},\\
 \left\|\int_0^s f(t)\,dt\right\|_{L^p(s^{-9/8}ds)}
          &\le22\|f\|_{L^p(s^{13/8}ds)},\\
 \left\|\int_0^s(s-t)^{-1/2}f(t)\,dt\right\|_{L^p(s^{13/8}ds)}
          &\le C\sqrt T\|f\|_{L^p(s^{13/8}ds)}
 \label{short-bg:hardy}
\end{align}
follow respectively by H\"older and by $t=su$, Minkowski, and the
integrable factors $u^{-21/22}$ and
$(1-u)^{-1/2}u^{-21/22}$. These apply to norms of Banach-valued
functions.

The zero-circulation cylinder Biot--Savart map sends $X_m$ to $H^1$
with polynomial $c$ cost. On nonzero horizontal modes use the gap
$2\pi/c$; on the mean mode use
$|\widehat\omega(0,\xi)|\le|\xi|\int|\eta\bar\omega(\eta)|d\eta$
before dividing by $\xi$. Thus \eqref{short-bg:green-bound} and
\eqref{short-bg:source} give
\begin{equation}
 \sup_{[0,T]}\|z(s)\|_{H^1}+|\alpha|
 \le\mathcal P_\nu e^{a'T}\|F\|_{H_p},
 \label{short-bg:early}
\end{equation}
where $\alpha\in\mathbb R^2$ is the entrance coefficient of the
Green-selected correction. It is determined linearly by $F$; no force
trace is used.

For the global continuation, the train estimates imply, with normalized
spatial measure and $r=22/3$,
\[
 \|D^jv(s)\|_r\le\mathcal P_\nu s^{-(41+22j)/44},\quad
 \|v(s)\|_\infty\le\mathcal P_\nu s^{-1},\quad
 \|Dv(s)\|_\infty\le\mathcal P_\nu s^{-3/2},\qquad s\ge1.
\]
Write $B_j(s)=(\nu s)^{j/2}\|D^jb_0(s)\|_p$. H\"older gives
\begin{equation}
 \|g(s)\|_{H^1}\le\mathcal P_\nu
 [\nu^{-1}s^{-85/44}B_2(s)
       +s^{-63/44}(\nu^{-1/2}B_1(s)+B_0(s))].
 \label{short-bg:tail-source}
\end{equation}
Each $B_j$ has $L^p(s^{-9/8}ds)$ norm at most $C\|F\|_{H_p}$.
For a factor $s^{-k}B_j$, weighted H\"older on $(T,\infty)$ gives
$T^{23/22-k}$; hence the exponents in its $L^1$ bound are
$-39/44$ and $-17/44$, respectively. The same expression belongs to
$L^p(s^{-9/8}ds;H^1)$. Differentiated velocity energy and
\eqref{short-bg:train-late} therefore give
\begin{equation}
 M_z:=\sup_{s\ge0}\|z(s)\|_{H^1}
 \le\mathcal P_\nu
 \exp\!\left({a'\Sigma+C/(1+\Sigma)\over\nu}+C\nu^3\right)
 \|F\|_{H_p}.
 \label{short-bg:global-energy}
\end{equation}
Local smoothing introduces no further exponential. To check this,
differentiate \eqref{short-bg:body-eq} homogeneously twice. Incompressibility,
orthogonality of $\mathbb P_0$, and \eqref{short-bg:train-uniform} give
$E_j'+\nu E_{j+1}\le C\sum_{i\le j}E_i$, $j\le2$.
Multiplication by successive powers of $\nu(t-s)$ and summation
gives $H^1\to H^2$ norm $C[\nu(t-s)]^{-1/2}$ for $t-s\le1$.
On $0<s<2$, Duhamel and \eqref{short-bg:hardy} give the weighted
$L^p(s^{13/8}ds;H^2)$ estimate. For $s\ge2$ use Duhamel only on
$[s-1,s]$; weights at $s$ and $s-r$, $0<r<1$, are comparable.
Consequently
\[
 \|z\|_{L^p((T,\infty),s^{-9/8}ds;H^2)}
 \le C\nu^{-1/2}\bigl[T^{-1/22}M_z
             +\|g\|_{L^p((T-1,\infty),s^{-9/8}ds;H^1)}\bigr].
\]
Since $s\|v(s)\|_\infty+s\|Dv(s)\|_\infty\le\mathcal P_\nu$
for $s\ge1$, these estimates bound $sB_vz$ in $H_p$ with the right
side of \eqref{short-bg:global-energy}. Here
$B_vz=v\cdot\nabla z+z\cdot\nabla v$.

Let $E(s)\alpha$ be the fixed inlet stream with an age cutoff equal
to one near zero and zero before the terminal time $T$. Its homogeneous
solution differs from $E\alpha$ by $O(s)|\alpha|$ in the velocity
state. The first Hardy estimate controls the forced difference, so
$z-E\alpha\in H_p$ at zero; the global $H^1$ bound and
$\int_T^\infty s^{-9/8}ds=8T^{-1/8}$ give the tail.
Thus the exact global identity is
\begin{equation}
 b_0+z=E\alpha+T_0 f,\qquad
 f=F-sB_v(b_0+z)-s(\partial_s-\nu\Delta)E\alpha.
 \label{short-bg:flat-reconstruction}
\end{equation}
It follows from the differential equation and free uniqueness; each
term defining $f$ belongs to $H_p$. For the remaining free interaction
term this is explicit: on $s<T$,
$\|sB_vb_0\|_{H_p}\le\mathcal P_\nu(\nu^{-1/2}\sqrt T+T)\|F\|_{H_p}$;
on $s>T$, the bounds $v=O(s^{-1})$, $Dv=O(s^{-3/2})$ and the free
weighted gradient give
$\|sB_vb_0\|_{H_p}\le\mathcal P_\nu T^{-1/2}(\nu^{-1/2}+1)\|F\|_{H_p}$.
Approximation in
$L^p(s^{13/8}ds;X_m)$ and the difference estimates above justify all
identities for the original arbitrary force. Choosing $\Sigma$ large
and then $a'$ small proves, for every fixed $m>0$,
\begin{equation}
 \|f\|_{H_p}+|\alpha|\le\mathcal P_\nu e^{m/\nu}
                                \|F\|_{H_p}.
 \label{short-bg:flat-inverse}
\end{equation}

\subsection{Exact geometry and the nonlinear correction}

The exact stream lift $E_\epsilon\alpha$ is affine: its nonzero
inlet is not in $Y_p$. Its complete equation source is in $Y_p$.
The Gaussian angular-two eigenstream is a harmonic quadrupole plus
a Gaussian remainder outside a fixed core. Periodizing its harmonic
part uses only
\[
 \sum_{n\in\mathbb Z}(q+nc+i\eta)^{-2}
       =(\pi/c)^2\csc^2(\pi(q+i\eta)/c).
\]
Its apparent poles are replaced by the smooth eigenstream core. Thus
the actual periodized stream and every fixed derivative have
exponential transverse decay, with polynomial $c$ costs. This proves
the inlet bounds needed below without any estimate on a dual tail.
Indeed write $K=2s\partial_s+\eta\partial_\eta$, and for one inlet
stream $\psi$ write its normalized lift as
\[
 e_\epsilon=e_0+\epsilon e_1,\qquad
 e_0=\chi(-\psi_\eta,\psi_q),\qquad
 e_1=(0,(\chi-2s\chi')\psi-\chi\eta\psi_\eta).
\]
The exact vector diffusion is
$L_\epsilon=\Delta-2\epsilon(K+1)\partial_q
+\epsilon^2[(K+1)^2-\operatorname{diag}(1,0)]$.
Substitution in the complete unprojected source column
\[
 \mathcal C_\epsilon\alpha
 =s[(\partial_s-\nu L_\epsilon)e_\epsilon+B_\epsilon e_\epsilon]\alpha
\]
gives $\mathcal C_\epsilon-\mathcal C_0=\epsilon s r_\epsilon$,
where $r_\epsilon$ is smooth at zero, compact in upper age, and has
polynomial weighted bounds. Hence
\begin{equation}
 \|\mathcal C_\epsilon-\mathcal C_0\|_{H_p}\le\epsilon\mathcal P_\nu,
 \qquad
 \|A_0E_\epsilon\alpha+\mathbb PB_{V_0}E_\epsilon\alpha\|_{W^{1,Y_p}}
 \le\mathcal P_\nu R^{2/p}|\alpha|.
 \label{short-bg:affine}
\end{equation}
The integrability exponent is $-9/8+p=13/8>-1$. The quadratic
inlet source has the same bound with $|\alpha|^2$, and
$\|E_\epsilon\alpha\|_{W^{j,\infty}}\le\mathcal P_{\nu,j}|\alpha|$.
These assertions compare sources, never the nonintegrable inlet states.

The global flat completion now permits one all-age perturbation. Put
$X=H_p$, $K_\epsilon=sB_\epsilon T_\epsilon$,
$K_0=sB_vT_0$. The flat train is retained at every age. Give
$X\times\mathbb R^2$ the norm $\|(f,\alpha)\|=\|f\|_X+|\alpha|$.
The equation row is
\[
 \mathfrak A_0:X\times\mathbb R^2\longrightarrow X,\qquad
 \mathfrak A_0(f,\alpha)=(I+K_0)f+\mathcal C_0\alpha.
\]
The all-age reconstruction \eqref{short-bg:flat-reconstruction}
and the complete inlet source defining $\mathcal C_0$ give
$\mathfrak A_0(f,\alpha)=F$. The Green selector, Biot--Savart map
and reconstruction are linear, so \eqref{short-bg:flat-inverse}
constructs a bounded linear right inverse
\[
 R_0:X\longrightarrow X\times\mathbb R^2,\qquad
 \mathfrak A_0R_0=I_X,\qquad
 \|R_0\|\le M=\mathcal P_\nu e^{m/\nu}.
\]
We compare operator outputs while inputs remain arbitrary global
elements of $H_p$.

Here the all-age mean expansion gives a useful stronger coefficient
bound than a fixed small late constant:
\begin{equation}
 \sup_{s>S}(|V_0|+|DV_0|)
       \le\mathcal P_\nu(S^{-1/2}+\epsilon).
 \label{short-bg:strong-late-coeff}
\end{equation}
To check the Cartesian gradient, write $b=(v_1,v_2+\epsilon H)$.
Its three derivative types are $s\partial_\eta b$,
$s[\partial_q-\epsilon(K+1)]b$, and the frame term
$\epsilon s v_1$. The mean stream is
$S_0=a_0G_{\nu s}+a_2\partial_\eta^2G_{\nu s}
+O_{\rm weighted}(\mathcal P_\nu s^{-5/2})$.
Thus $v_1=-\partial_\eta S_0$ and
$H=2a_0G_{\nu s}+4a_2\partial_\eta^2G_{\nu s}+\cdots$.
The potentially nondecaying radial derivative cancels because
$(K+1)G_{\nu s}=0$, giving
$(K+1)H=-8a_2\partial_\eta^2G_{\nu s}
+O_{\rm weighted}(\mathcal P_\nu s^{-5/2})$.
Every fluctuation derivative is exponentially small. These identities
prove \eqref{short-bg:strong-late-coeff}, including the full frame
term; the axial mean velocity itself has the genuine $O(\epsilon)$
floor. The physical free graph and the exact norm identity then give
\[
 \|1_{s>S}K_\epsilon\|_{X\to X}
 \le\mathcal P_\nu(S^{-1/2}+\epsilon).
\]
For the flat operator, $v=O(\mathcal P_\nu/s)$ and
$Dv=O(\mathcal P_\nu/s^{3/2})$, paired with the free
$\sqrt{s}Db_0$ graph bound, give
$\|1_{s>S}K_0\|\le\mathcal P_\nu S^{-1/2}$.
At the young end, smooth inlet coefficients give
$\|1_{s<\sigma}K_\epsilon\|+\|1_{s<\sigma}K_0\|
\le\mathcal P_\nu\sqrt\sigma$ by the same free graph.

Take $\sigma=\epsilon^{a}$ and $S=\epsilon^{-b}$, where $a,b>0$
will be small fixed numbers. In \eqref{sct:transverse}, choose
$l=C_D(c/\nu+\sqrt S+1)$. Both negative exponents dominate any
prescribed multiple of $\nu^{-1}$, while $P_S\le\mathcal P_\nu
e^{kb d_+/\nu}$ for fixed $k$. Increasing $C_D$ makes all required
coefficient tails at most $e^{-D/\nu}$; fixed weights and $K$
derivatives change only the polynomial cost. The prefactor in the
localized comparison is polynomial in this width, $S$ and
$\sigma^{-1}$. With $\theta=2/539$ from
Proposition~\ref{bg:lp-free-comparison}, for some fixed $N<\infty$,
\begin{equation}
 \|K_\epsilon-K_0\|_{X\to X}
 +\|\mathcal C_\epsilon-\mathcal C_0\|
 \le\mathcal P_\nu
 [\epsilon^{a/2}+\epsilon^{b/2}+\epsilon
       +\epsilon^{\theta-N(a+b)}+e^{-D/\nu}]
 =:\Delta_\nu.
 \label{short-bg:global-operator-comparison}
\end{equation}
The affine column is supported
in a fixed polynomial age interval and has the already proved
$O(\epsilon)$ source difference. No equation or source is cut off.
Choose $a,b$ so $\theta-N(a+b)>0$, and put
$\zeta=\min\{a/2,b/2,1,\theta-N(a+b)\}>0$.
Choose $m<d_-\zeta/2$ and $D>2m$. Then $M\Delta_\nu<1/2$ for
small $\nu$. The exact equation row
\[
 \mathfrak A_\epsilon:X\times\mathbb R^2\longrightarrow X,\qquad
 \mathfrak A_\epsilon(f,\alpha)
       =(I+K_\epsilon)f+\mathcal C_\epsilon\alpha
\]
has the bounded linear right inverse
\[
 R_\epsilon=R_0[I+(\mathfrak A_\epsilon-\mathfrak A_0)R_0]^{-1},
 \qquad \|R_\epsilon\|\le2M.
\]
Indeed $(\mathfrak A_\epsilon-\mathfrak A_0)R_0:X\to X$
has norm at most $M\Delta_\nu<1/2$, and multiplication verifies
$\mathfrak A_\epsilon R_\epsilon=I_X$.
This compares all global inputs, including those supported arbitrarily
late; it does not assume that the exact inverse is causal in age.
Returning to physical normalization proves
\begin{equation}
 \|w\|_{X_p}+R^{2/p}|\alpha|
 \le\mathcal P_\nu e^{m/\nu}
                \|f\|_{Y_p}
 \label{short-bg:exact-inverse}
\end{equation}
for
$A_0w+\mathbb PB_{V_0}w+(A_0E_\epsilon+\mathbb PB_{V_0}E_\epsilon)\alpha
=\mathbb Pf$. Here the physical $w$ is the free response to the
first coordinate selected by $R_\epsilon$, and the free graph supplies
its $X_p$ norm. The same normalized measure multiplies
source, state, and finite coordinate, so no additional power of $R$
is lost.

Set $A=R^{2/p}\alpha$ and $\|(w,A)\|_Z=\|w\|_{X_p}+|A|$.
By Lemmas~\ref{bg:common-body-graph}--\ref{bg:graph-embeddings} and
\eqref{short-bg:affine}, the quadratic
terms in $w+E_\epsilon\alpha$ and their polarized differences are
bounded by $\mathcal P_\nu\|(w,A)\|_Z^2$ in $Y_p$. Apply the fixed
linear right-inverse selection to minus these terms and minus $F_R$.
With $\delta=\mathcal P_\nu\epsilon^{3/11}$ and
$M=\mathcal P_\nu e^{m/\nu}$, this is a contraction on the radius
$2M\delta$ ball if $4\mathcal P_\nu M^2\delta<1$. This holds after
decreasing $m$ so $2m<3d_-/11$. We obtain an exact smooth unforced
profile
\begin{equation}
 U=V_0+E_\epsilon\alpha+w,\qquad
 \|w\|_{X_p}+R^{8/11}|\alpha|
       \le\mathcal P_\nu e^{m/\nu}\epsilon^{3/11},\qquad
 |\alpha|\le\mathcal P_\nu e^{m/\nu}\epsilon.
 \label{short-bg:result}
\end{equation}
The projected equation is
$A_0U+\mathbb P(U\cdot\nabla U)=0$ with zero residual. Reality,
no-swirl symmetry, and periodicity are preserved at each iteration.

No second unstable inverse is needed for spatial regularity. Write
$V=V_0+E_\epsilon\alpha$ and
$A_0w=f-\mathbb P(V\cdot\nabla w+w\cdot\nabla V+w\cdot\nabla w)$.
The complete source has $W^{1,Y_p}$ norm bounded by the right side
of \eqref{short-bg:result}, up to a polynomial. Since
$\partial_jA_0w=(A_0-1/2)\partial_jw$ and
$\|Dw\|_{Y_{2p}}^2\le C\|w\|_\infty\|D^2w\|_{Y_p}$,
every differentiated term is in $Y_p$. Also $Dw\in Y_p$ is already
known. Rewrite the differentiated equation as
$A_0D_jw=f_j+\tfrac12D_jw$, with the entire right side in $Y_p$,
and apply the maximal graph at shift zero. No inverse of
$A_0-\tfrac12$ is used. Thus $Dw\in\widetilde X_p$, and
\begin{equation}
 \sup_\tau\|w(\tau)\|_p+
 \|w\|_{W^{1,\infty}}+\|E_\epsilon\alpha\|_{W^{1,\infty}}
 \le\mathcal P_\nu e^{m/\nu}\epsilon^{3/11}.
 \label{short-bg:strong}
\end{equation}
For any fixed derivative order, the train and inlet complete sources
belong to the corresponding $W^{k,Y_p}$. Explicitly, in the
large-radius region their unprojected derivatives are bounded by
$C_{k,N}\varrho^{-3-k}(1+|y_3|/\varrho)^{-N}$, whose $p$th integral
reduces to $C\int_R^\infty\varrho^{2-p(3+k)}d\varrho<\infty$.
On $\varrho<R$, take the all-age heat expansion to a sufficiently
high finite order: its even radial powers extend smoothly through the
axis and its weighted remainder gives arbitrary axial decay. The
inlet cutoff vanishes near the axis. These estimates are uniform in
phase; the bounded Cartesian Leray map preserves them in $W^{k,p}$.
Inductively, if derivatives
through order $k$ are in the graph, all factors through order $k$ are
bounded and derivatives through order $k+2$ are in $Y_p$. In each
order-$(k+1)$ derivative of $w\cdot\nabla w$, one factor has order at
most $k$. Put it in $L^\infty$ and the other in $Y_p$, then apply the
maximal graph to the differentiated equation. This proves all spatial
orders. On positive physical-time compact intervals the resulting
$C_\tau W^{k,4}$ regularity and the projected equation give all time
derivatives, so $U$ is jointly smooth through the axis.

\subsection{The datum, heat remainder, and pressure}

Let $\psi_{\rm in}$ be the raw inlet plus its single entrance
adjustment. Define, with $\varrho=|x'|$,
\[
 q=R\log(\varrho/R),\quad\eta=Rz/\varrho,\qquad
 a_r={R\over\varrho}(-\psi_{{\rm in},\eta}),\quad
 a_z={R\over\varrho}
 [\psi_{{\rm in},q}+\epsilon(\psi_{\rm in}
                          -\eta\psi_{{\rm in},\eta})].
\]
The stream is periodic in $q$ and has arbitrary fixed transverse
polynomial decay. Differentiation and this decay give
$|D^ja(x)|\le C_j|x|^{-1-j}$, including smooth extension at each
nonzero axis point. The omitted divergence flux through a radius-$r$
sphere is $O(r)$, so $a$ is distributionally divergence-free on all of
$\mathbb R^3$. Thus $a\in L^{3,\infty}\cap L^2_{\rm loc}$.
The shift $q\mapsto q+c$ proves its DSS identity. The raw inlet is
nonzero on a fixed core and \eqref{short-bg:result} makes its finite
adjustment exponentially small there, so $a\ne0$.

Put $a_\tau(y)=e^{\tau/2}a(e^{\tau/2}y)$ and
$H_a(\tau)=e^{\nu\Delta}a_\tau$. Taylor expansion against the Gaussian,
whose first moment vanishes, gives
\[
 |D^j(e^{\nu\Delta}a_\tau-a_\tau)(y)|
       \le C_{j,\nu}|y|^{-3-j},\qquad |y|\ge1.
\]
Use the expansion only in a ball of radius $|y|/4$ about $y$;
differentiate the Gaussian on its complement, which is exponentially
small despite the singularity of $a$ at zero. The $O(s)$ inlet
expansion gives the same inverse-cubic bound for $V-a_\tau$ in the
large-radius region, with arbitrary transverse decay. The all-age
train expansion gives arbitrary axial decay in the remaining bounded
radial region. Therefore
\begin{equation}
 V-H_a\in C_{\rm per}H^k\cap X_p
 \quad\hbox{for every fixed required }k.
 \label{short-bg:raw-heat}
\end{equation}
The graph membership includes the mixed phase/dilation derivative;
it follows from the same source expansion. In particular this
conclusion is stronger than an $X_p$ estimate alone.

The inverse-distance bounds and the all-age expansion give
$D^jV\in C_{\rm per}L^4$. For $w$, interpolation between its
$C_{\rm per}L^p$ derivatives and their uniform supremum bounds gives
$D^jw\in C_{\rm per}L^4$. Therefore
$D^jU\in C_{\rm per}L^4$ for $j\le2$, and
$\mathbb T=U\otimes U\in C_{\rm per}H^2$. The field
$Q=U-H_a\in X_p$ solves $A_0Q=-\mathbb P\operatorname{div}\mathbb T$.
The exact free semigroup is
\[
 S_0(t)f(y)=e^{t/2}[e^{\nu(e^t-1)\Delta}f](e^{t/2}y).
\]
Free uniqueness identifies $Q$ with
$-\int_0^\infty S_0(t)\mathbb P\operatorname{div}\mathbb T(\tau-t)dt$.
For $j\le2$, move derivatives to $\mathbb T$ when $t\le1$ and to
the heat kernel when $t\ge1$. The resulting integrable majorants are
\[
 \|D^jS_0(t)\mathbb P\operatorname{div}\mathbb T\|_2
 \le C_{j,\nu}
 \begin{cases}t^{-1/2}\|\mathbb T\|_{H^j},&0<t\le1,\\
 e^{-3t/4}\|\mathbb T\|_2,&t\ge1.
 \end{cases}
\]
Thus $Q\in C_{\rm per}H^2$. Also
$(\partial_\tau-y\cdot\nabla/2)Q
=\nu\Delta Q+Q/2-\mathbb P\operatorname{div}\mathbb T
\in C_{\rm per}L^2$. Scaling back gives
\[
 \|u_\nu(t)-e^{\nu t\Delta}a\|_2
       =t^{1/4}\|Q(\log t)\|_2\le C t^{1/4}.
\]
A compact part of $a$ is in $L^2$; its complement is bounded and
uniformly continuous. The Gaussian approximate identity therefore
shows $e^{\nu t\Delta}a\to a$ locally in $L^2$, and hence the same
is true for $u_\nu$. Finally
$p_U=\sum_{i,j}R_iR_j(U_iU_j)\in C_{\rm per}L^2$ recovers the
discarded gradient part of the exact projected equation.


\section{Thin and repair stress spaces}
\label{sec:stress-spaces-short}

Fix the exact background from Section~\ref{short-bg:section}. The goal is
its nonzero periodic energy mode $b$ with exponent $0<\sigma<1/4$.
The planar Gaussian eigenpair supplies transient train amplification
and a signed integrated stress response. The three-dimensional pressure
interaction returns a strain to the young rings and closes the loop.
The singular limit compares these response operators as curvature and
viscosity decrease together; the final characteristic zero belongs to
the exact three-dimensional operator, with its full pressure and repair.

We first establish the causal near inverse and all-age moments, then
identify the returned strain and control its errors, and finally solve
the rank-one scalar return equation. The background and geometry remain
fixed as we vary the real Floquet exponent.

Use the exact background
$U=V+r_b$, where $V=U_{\rm raw}+E_\epsilon\alpha_{\rm bg}$
and $r_b$ is the energy repair. Set $c=\nu^{-2}$, $R=\epsilon^{-1}$,
and retain component order $(r,\phi,z)$. In the cylinder chart
\[
 s=R^2/\rho^2,\qquad q=R\log(\rho/R)+R\tau/2,\qquad
 \eta=Rz/\rho,
\]
put $d\varsigma=ds\,dq/c\,d\eta$ and
\[
 H=L^2(s^{-3/2}d\varsigma),\qquad
 H_T=L^2(s^{-1/2}d\varsigma;\operatorname{Sym}_3).
\]
The natural velocity and tensor lifts are
$\mathcal N_vb=\sqrt{s}\,b$ and $\mathcal N_TT=sT$.
Changing variables in the normalized phase/spatial integral gives
\begin{equation}\label{stress-short:lifts}
 \|\mathcal N_vb\|_{Y_2}^2=\pi R^2\|b\|_H^2,\qquad
 \|\mathcal N_TT\|_{Y_2}^2=\pi R^2\|T\|_{H_T}^2.
\end{equation}
Indeed the physical space Jacobian after azimuthal integration is
$\pi R^2s^{-5/2}ds\,d\eta$ and normalized phase becomes $dq/c$.
The physical divergence and Leray projection remain in the free inverse;
these formulas only identify its Hilbert spaces.

Fix $m_Z>3/2$, for example $m_Z=2$, and define
\[
 W=\left\langle\eta/\sqrt{1+s}\right\rangle^{m_Z},\qquad
 Z_1=L^2(s^{-1/2}W^2d\varsigma),\qquad
 Z=Z_1\oplus\rho_b^{-1/2}H_T.
\]
The norm on $Z$ is the sum norm
$\|(T_1,T_2)\|_Z=\|T_1\|_{Z_1}+\rho_b^{-1/2}\|T_2\|_{H_T}$.
Here
\[
 \rho_b=P_\nu\epsilon^{3/11}e^{m_0/\nu}
\]
is a fixed upper bound from the exact-background construction,
chosen so that
$\sup_\tau(\|r_b(\tau)\|_{11/4}+\|r_b(\tau)\|_\infty)\le\rho_b$.
It is not the radial coordinate. The stress inverse acts on
$T_1+T_2$; the second entry is not assigned a transverse weight.

Write $v=s^{-1/2}V$ and $v_b=s^{-1/2}r_b$. The train heat expansion,
transverse bounds, and compact-age inlet construction imply
\begin{equation}\label{stress-short:coefficients}
 \begin{aligned}
 \sup\sqrt{s}\,W|v|&\le P_\nu,&
 \sup_{s>S}\sqrt{s}\,W|v|&\le P_\nu(S^{-1/2}+\epsilon),\\
 \sup_{s<\sigma}\sqrt{s}\,W|v|&\le P_\nu\sqrt{\sigma},
       &&0<\sigma\le1,\\
 \|Wv(s)\|_{L^2_\eta L^\infty_q}
       &\le P_\nu[(1+s)^{-3/4}
                         +\epsilon(1+s)^{-1/4}].
 \end{aligned}
\end{equation}
Here $S$ is larger than the compact-age inlet support. To check the
age powers, the mean stream and its fixed weighted derivatives have
size $(1+s)^{-1/2}$ on transverse scale $\sqrt{1+s}$.
One transverse derivative gives the velocity size $(1+s)^{-1}$.
The stream completion
$\epsilon(1-\eta\partial_\eta-2s\partial_s)\psi$ has size
$P_\nu\epsilon(1+s)^{-1/2}$. Its slower decay is retained.
The $L^2_\eta$ norm adds the factor $(1+s)^{1/4}$.
Fluctuations have exponential decay after overlap; before overlap
the Gaussian comparison gives the same polynomial bounds.
The finite inlet is absorbed by $P_\nu$, and early smoothness gives
the third bound. Multiplying by any further fixed power of
$\langle\eta/\sqrt{1+s}\rangle$ leaves these conclusions unchanged,
by the weighted train expansion.

If $S=\epsilon^{-b}$ and $D_\perp>0$ is fixed, the coefficient can
also be localized to
\[
 |\eta|\le L_\perp=C_\perp(c/\nu+\sqrt S+1)
\]
with
\begin{equation}\label{stress-short:transverse}
 \sup_{s<S,\ |\eta|>L_\perp}
 \sqrt{s}\,W(|v|+|v_{\rm flat}|)
       \le e^{-D_\perp/\nu}.
\end{equation}
The Gaussian vorticity tail controls the heat-width part.
The nonzero horizontal stream kernels
$e^{-|k_n||\eta|}/(2|k_n|)$, $|k_n|\ge2\pi/c$, control the
$c/\nu$ part. The zero mode has decaying heat primitives.
Increasing $C_\perp$ absorbs the fixed weights, derivative budgets,
and the finite inlet's stream tail. No such localization is claimed
for $r_b$.

Define the exact split multiplier by
\[
 \mathcal M_Ub=(v\otimes b+b\otimes v,\
                         v_b\otimes b+b\otimes v_b).
\]
The flat multiplier uses the actual planar train in its first entry
and zero in its second entry. Since the ratio of stress and velocity
age densities is $s$, \eqref{stress-short:coefficients} gives
\begin{equation}\label{stress-short:multiplier}
 \|\mathcal M_Ub\|_Z\le P_\nu\|b\|_H,\qquad
 \|(\mathcal M_Ub)_2\|_{\rho_b^{-1/2}H_T}
                         \le2\sqrt{\rho_b}\|b\|_H.
\end{equation}
Its thin old and young restrictions have respectively norms
$P_\nu(S^{-1/2}+\epsilon)$ and $P_\nu\sqrt{\sigma}$.
Indeed $|a\otimes b+b\otimes a|\le2|a||b|$, so the thin
multiplier norm is at most $2\sup\sqrt{s}W|v|$.
For the repair, $\sqrt{s}v_b=r_b$ and multiplication followed by
the factor $\rho_b^{-1/2}$ gives the second estimate.
These bounds also hold after multiplication of both sides by any
common positive age weight, in particular the gain $g(s)^{-1}$.

For $A\ge1$ the thin source has the absolute old moment bound
\begin{equation}\label{stress-short:moment}
 \mathcal N_1(1_{s>A}T):=
 \int_A^\infty s^{-3/2}\int
 (1+|\eta|/\sqrt{s})\|T(s,\cdot,\eta)\|_{L^2(dq/c)}\,d\eta\,ds
 \le C_{m_Z}A^{-1/2}\|1_{s>A}T\|_{Z_1}.
\end{equation}
Transverse Cauchy--Schwarz gives
\[
 \int(1+|\eta|/\sqrt{s})\|T(s,\cdot,\eta)\|_{L^2_q}\,d\eta
       \le C_{m_Z}s^{1/4}\|WT(s)\|_{L^2_{q,\eta}},
\]
because $\int\langle x\rangle^{2-2m_Z}dx<\infty$.
The remaining time integral is
$\int_A^\infty s^{-5/4}\|WT(s)\|_2ds$.
Cauchy--Schwarz against $s^{-1/2}ds$ leaves
$\int_A^\infty s^{-2}ds=A^{-1}$, proving
\eqref{stress-short:moment}.

In outer coordinates $x'=y'/R$, $u=|x'|^{-2}$ and
$Y=\eta/\sqrt u$, the thin physical tensor is represented by
$F=uT$. Directly,
\[
 \int(1+|Y|)\|F(x',Y,\cdot)\|_{L^2(dq/c)}\,dx'\,dY
       =\pi\mathcal N_1(T).
\]
Thus \eqref{stress-short:moment} supplies exactly the first moment
used in the separated-kernel approximation. It also justifies
extending an old annulus to the axis by moment-norm approximation.
All cutoffs multiply the tensor before its distributional divergence;
no artificial boundary force is removed. The broad repair is instead
estimated by its physical $L^{11/4}$ coefficient and energy bounds.


\section{Planar amplification, the full age tail, and the causal inverse}
\label{sec:planar-response}
\subsection{The full moving evolution and weak sources}
\label{sec:short-full-clock}

Let $C_c=(\mathbb R/c\mathbb Z)\times\mathbb R$, $c=\nu^{-2}$,
and let $v=v_c$ be the actual planar train. Fix $m>8$. Write
\[
 X_{m,c}=L^2(C_c,\langle x\rangle^{2m}dx),\qquad
 Z^{-1}_{m,c}=\{f\in(H^1_{-m,c})^*:\langle f,1\rangle=0\},
 \qquad g(s)=\exp\frac{\gamma s}{1+\nu s}.
\]
The test $1$ belongs to $H^1_{-m,c}$. The centered radial weight is
used for evolution norms. Smooth periodic weights are used only for
local smoothing, with equivalence applied at its two endpoints.

\begin{proposition}[Full forward train estimate]
\label{prop:short-full-clock}
For a fixed sufficiently large $b$ and all sufficiently small $\nu$,
the moving full complement has the exponential relative bound
\eqref{eq:short-causal-stable-moving}, the full moving evolution
satisfies \eqref{eq:short-causal-long-forward}, and on
$0<t<s\le H=\nu^{-1}((\nu b)^{-1}-1)$ its physical weak-source
extension satisfies
\[
 \left\|\frac{g(t)}{g(s)}U(s,t)\right\|_{Z^{-1}_{m,c}\to X_{m,c}}
 \le C_b\nu^{-M_s}\{1+(s-t)^{-1/2}\},\qquad
 M_s=\lceil m/2+|\Re\kappa|+2\rceil.
\]
These bounds hold on the entire zero-circulation space, with no
input parity restriction.
\end{proposition}
\begin{proof}
\emph{The full moving complement.}
Set
\[
 a=1+\nu s,\quad \theta=(\nu a)^{-1},\quad
 z=s/a=\nu^{-1}-\theta,\quad e_c=(\nu c^2)^{-1}=\nu^3,
 \quad d=c/\sqrt a=\sqrt{\theta/e_c}.
\]
The exact substitution $w(s,x)=a^{-2}W(z,x/\sqrt a)$ gives
\begin{equation}\label{eq:short-causal-moving}
 W_z=G(z)W,\qquad G=\theta^{-1}D+L_{\nu,c}(z),\qquad
 D=\Delta+\tfrac12 y\cdot\nabla+2,\qquad
 d_z/d=-1/(2\theta).
\end{equation}
The rescaled train differs from the periodized Gaussian by
$C_{j,m}e_c e^{-b_{j,m}\theta/e_c}$ in every fixed weighted derivative
norm on $\theta\ge b>0$. This follows by rescaling the train error,
interpolating its Gaussian-small unweighted norm with a larger fixed
weighted norm, and absorbing the resulting fixed powers into the
Gaussian. The required region is contained in $1\le a\le c^2$.

The common actual-cylinder comparison
\eqref{short-bg:free-cylinder}--\eqref{short-bg:interaction-cylinder}
applies in moving coordinates after including Reynolds' formula.
Here is the quantitative part needed for all parities. The paired-image
kernel satisfies, for $|x|\le d/8$, $|y|\le d/4$,
\[
 K_d(x-y)-K_\infty(x-y)
 =d^{-2}B_1(x-y)+O(d^{-4}|x-y|^3).
\]
For zero-mass input the $x$ times mass term disappears; the first
moment is retained and bounded by $C_m\|f\|_{X_m}$. The source tail
has $L^1$ norm at most $C_md^{1-m}\|f\|_{X_m}$; the separated
far kernel costs $O(d^{-1})$. The observation outside $d/8$ is
Gaussian-small. Hence the localized interaction error, also after
$d\partial_d$, is $O(d^{-2})=O(e_c/\theta)$ on the entire space.
No first-moment cancellation or parity restriction is used. The same
$O(d^{-2})$ bound holds for auxiliary nonzero-mass inputs: retain
$x\int f$ and bound it against the localized observation coefficient.
Thus restricting an ambient input to a cell creates no mass-related
gap in the compact comparison.

Let $E(z)$ be the two periodized growing vectors from
Proposition~\ref{prop:short-gaussian-seed}, and normalize their dual
rows $\Lambda(z)$ by $\Lambda E=I_2$. The planar dual kernels and
their fixed derivatives have the $O(|y|^{-2-j})$ decay obtained from
\eqref{eq:short-seed-green} and \eqref{eq:short-seed-functional}.
Paired translates therefore give dual error $O(d^{-2})$ in the
weighted dual norm, with the same bound after $d\partial_d$; direct
columns have Gaussian tails. The finite Gram inverse exists uniformly
and its differentiated error is $O(\theta^{-1}d^{-2})$.
The identities
\[
 \theta^{-1}D(P_dq)-\partial_z(P_dq)=\theta^{-1}P_dDq,
 \quad
 \theta^{-1}D^\#(P_d\ell)+\partial_z(P_d\ell)
       =\theta^{-1}P_dD^\#\ell,
 \quad D^\#=\Delta-\tfrac12y\cdot\nabla+1
\]
follow term by term using $d_z/d=-1/(2\theta)$. They are identities
for the unweighted dual kernels, with Gram normalization then applied.
In particular the combined direct and dual rows are bounded; neither
$\Lambda D$ alone nor a derivative of sharp zero extension is used.

Put $K_0=GE-E_z$. Define $H_0$ by differentiating the actual moving
pairing: $(\Lambda W)'=H_0W$ whenever $W'=GW$. Then
$M_0=H_0E=\Lambda K_0$. For $\beta_0=\gamma+\Re\kappa/\theta$ set
\begin{equation}\label{eq:short-causal-damped-complement}
 \mathcal B=G-\beta_0-EH_0+
       [-K_0+(\beta_0-1)E+EM_0]\Lambda.
\end{equation}
All added terms are bounded finite rank. Direct substitution gives
$(\Lambda W)'=-\Lambda W$ for $W'=\mathcal BW$, and
$\mathcal BE=E_z-E$. Thus $\ker\Lambda(z)$ is invariant and the
artificial pair decays at rate one. The planar limit is
$\mathscr Q(\mathscr L-\gamma)\mathscr Q-\mathscr P$, where
$\mathscr Q=I-\mathscr P$ removes both growing modes. Its full-space
semigroup is
exponentially stable by \eqref{eq:short-seed-complement}.
This statement includes the horizontal mean and every angular sector.
For ambient identifications a radial mass-one neutral vector can be
adjoined; $\mathscr L$ annihilates it and the relative damping makes
it stable. Restriction to zero circulation is preserved by the actual
equation.

To compare on a fixed Hilbert space let $I_z,R_z,\Pi_z=I_zR_z$ be
zero extension, restriction and cell projection. Extend the free
moving evolution $F$ by the actual cocycle
\[
 \overline F(z,t)=I_zF(z,t)R_t+S(z,t)(I-\Pi_t),\qquad
 Sf(y)=\rho^4f(\rho y),\quad \rho=d(t)/d(z).
\]
The two terms have disjoint entrance and exit supports. The exterior
term has norm at most $\rho^3$. For the interior, Reynolds' boundary
term cancels the dilation seam term. With $p=(1+|y|^2)^m$,
\[
 \frac d{dz}\|W\|_p^2\le
 \left[\theta^{-1}\sup\left(\frac{\Delta p}p+3
                   -\tfrac12y\cdot\nabla\log p\right)+\delta_\nu\right]
                   \|W\|_p^2.
\]
The two seam weights agree and the remaining diffusion seam term is
nonpositive. Radial Euler transport preserves the weight; the image
and actual-base velocity errors give $\sup\delta_\nu\to0$ on
$\theta\ge b$ for fixed $b$. One can take
$C_m=(m^2+3m+3)/2$ for the norm-growth contribution of $D$.
Consequently the relative free evolution satisfies
\begin{equation}\label{eq:short-causal-free-moving}
 \|\overline F_{\rm rel}(z,t)\|
 \le\exp\{[-\gamma+(C_m+|\Re\kappa|)/b+\sup\delta_\nu](z-t)\}.
\end{equation}

On each fixed block of length $B$, compact smooth tests and Duhamel
give direct strong convergence to radial Euler transport in the
successive limit $\nu\to0$, then $b\to\infty$, uniformly in the
entrance time. Density extends the convergence. The undamped norms
have limiting upper bound one, and radial transport $T$ is unitary;
the identity
\[
 \|F_n^*f-T^*f\|^2\le\|F_n\|^2\|f\|^2
                 -2\Re\langle f,F_nT^*f\rangle+\|f\|^2
\]
gives adjoint-strong convergence. The localized interaction is compact
by source and observation truncation and local Rellich compactness;
the image and dual estimates above give norm convergence of every
compact and finite-rank term in
\eqref{eq:short-causal-damped-complement}. Lemma~\ref{lem:short-compact-duhamel}
therefore compares the full-minus-free remainders in norm. If the
limiting full relative bound is $C_0e^{-hB}$, choose $B$ so that
$C_0e^{-hB}+2e^{-\gamma B}<1/4$, then $b$ large, then $\nu$ small.
Each full block contracts by $1/2$. Uniform bounds on the remaining
short block yield
\begin{equation}\label{eq:short-causal-stable-moving}
 \|S_{\rm rel}(z,t)\|\le Ce^{-h_0(z-t)},\qquad
 b\le\theta(z)\le\theta(t)\le\nu^{-1}.
\end{equation}
All block products use the same ambient radial weighted norm.

Write $W=E\alpha+r$, $\Lambda r=0$. The amplitude diagonals are
$\lambda_i+\kappa_i/\theta$; angular orthogonality makes the planar
off-diagonal $D$ pairing zero. The direct errors are bounded by
\[
 \delta_*(\theta)=C\{e_c/\theta+e_c e^{-b_2\theta/e_c}\},
\]
and both amplitude--complement couplings by
$C(\theta^{-1}+\delta_*)$. Divide each amplitude by
$e^{\lambda_i(r_0-\theta)}(r_0/\theta)^{\kappa_i}$ and the
complement by their common modulus. Equation
\eqref{eq:short-causal-stable-moving} bounds its stable convolution.
For small $\nu$, enlarge constants to bound the coupling by
$\eta(z)=C(1+e_c)/\theta(z)$ and the direct error by
$\delta(z)=Ce_c/\theta(z)$. The Gaussian error is absorbed in
these bounds. If $A_*(z)$ is the running maximum of the normalized
amplitude, the stable convolution gives
$\|r(z)\|\le Ce^{-h_0(z-t)}\|r(t)\|+C\eta(z)A_*(z)$.
The initial-complement term in the amplitude is integrable with
bound $C\|r(t)\|/b$. Equivalently, for the double convolution,
$\eta'/\eta=1/\theta\le1/b$ and $b\ge2/h_0$ give
$\int_u^z\eta(v)e^{-h_0(v-u)}dv\le(2/h_0)\eta(u)$.
Substitution into the amplitude equation and scalar Gronwall give
an exponential of
\[
 C\int_b^{1/\nu}[\delta_*(r)+(r^{-1}+\delta_*(r))^2]dr
 \le C/b+Ce_c\log(1/(\nu b))+Ce_c/b+Ce_c^2/b+o(1).
\]
The initial complement contributes only a uniformly bounded decaying
convolution. The displayed budget is uniformly bounded. We obtain,
on the entire zero-mass space and with no input symmetry restriction,
\begin{equation}\label{eq:short-causal-long-forward}
 \|U_{\rm mov}(\theta,r_0)\|_{X_{m,d(r_0)}\to X_{m,d(\theta)}}
 \le C_b e^{\gamma(r_0-\theta)}(r_0/\theta)^{\Re\kappa}.
\end{equation}

\emph{Weak sources and the polynomial loss.}
Choose the core endpoint
\[
 H=s_b=\nu^{-1}((\nu b)^{-1}-1),\qquad a_b=(\nu b)^{-1}.
\]
For $J_aW(x)=a^{-2}W(x/\sqrt a)$, change of variables gives
\begin{equation}\label{eq:short-causal-scales}
 \|J_a\|_{X_{m,d}\to X_{m,c}}\le a^{(m-3)/2},\qquad
 \|J_a^{-1}\|_{X_{m,c}\to X_{m,d}}\le a^{3/2},\qquad
 \|J_a^{-1}\|_{(H^1_{-m,c})^*\to(H^1_{-m,d})^*}\le a^{3/2}.
\end{equation}
For the last inequality,
$\langle J_a^{-1}f,\phi\rangle=a\langle f,\phi(\cdot/\sqrt a)\rangle$
and $\|\phi(\cdot/\sqrt a)\|_{H^1_{-m,c}}
\le\sqrt a\|\phi\|_{H^1_{-m,d}}$. At two endpoints
$1\le a_t\le a_s\le a_b$, the product of the first two factors is
at most $a_b^{m/2}$. Normalizing both physical cell norms by $c^{-1}$
does not change this operator estimate.

For short weak smoothing use only here the equivalent periodic weight
\[
 p_d=[1+y_2^2+(d/\pi)^2\sin^2(\pi y_1/d)]^m.
\]
Its first two logarithmic spatial derivatives are uniformly bounded,
$|\partial_z\log p_d|\le C_m/\theta$, and
$|y\cdot\nabla\log p_d|\le C_m$. The rescaled train and its first
two derivatives are uniformly bounded for $\theta\ge b$.
The adjoint localized interaction maps $X_{-m,d}$ to $H^1_{-m,d}$
uniformly: its localized source has uniformly bounded $L^1$ and
$L^2$ norms by weighted Cauchy--Schwarz; split the Biot--Savart kernel
into its integrable near part and bounded far part, and use the
uniform order-zero multiplier bound for its derivative, including
the horizontal mean. This also proves the stated bound for the
actual train perturbation.

For the reverse-time adjoint let $E_j$ be its squared weighted
$H^j_{-m}$ energy. On $\Delta z\le1$, with
$\beta=\theta^{-1}$, Reynolds cancellation and differentiated
weighted energy give
\[
 E_0'+\beta_{\min}E_1\le C_bE_0,\qquad
 E_1'+\beta_{\min}E_2\le C_b(E_1+E_0).
\]
The diffusion-weight errors have coefficient $C_m\beta$ and the
transport and compact terms have coefficient $C_b$; no negative
power of $\nu$ occurs in a Gronwall coefficient. Moreover
$\beta_{\max}/\beta_{\min}\le1+1/b$ on such an interval.
Integrate the first inequality and then the inequality for $rE_1(r)$
to get $E_1(\Delta z)\le C_b(\beta_{\min}\Delta z)^{-1}E_0(0)$.
By duality, the moving weak-to-strong norm is at most
$C_b(\beta_{\min}\Delta z)^{-1/2}$.

For $0<s-t\le1$, the clock identities are exactly
\[
 \Delta z=\frac{s-t}{a_sa_t},\qquad
 \beta_{\min}=\nu a_t,\qquad
 \beta_{\min}\Delta z=\frac{\nu(s-t)}{a_s}.
\]
Combining with \eqref{eq:short-causal-scales} gives the physical
weak-to-strong cost $C_b\nu^{-(m+2)/2}(s-t)^{-1/2}$.
For a longer gap factor at $t+1/2$ in moving coordinates and apply
\eqref{eq:short-causal-long-forward} after the short smoothing step.
The two intermediate maps $J_{a_{t+1/2}}^{-1}J_{a_{t+1/2}}$
cancel \emph{before} taking norms. Thus only one endpoint cost
$\nu^{-m/2}$ occurs, the smoothing costs at most $C_b\nu^{-1}$,
and the long-clock algebraic factor costs $\nu^{-|\Re\kappa|}$.
For the fixed integer
\[
 M_s=\left\lceil m/2+|\Re\kappa|+2\right\rceil
\]
we have on $0<t<s\le H$
\begin{equation}\label{eq:short-causal-weak-forward}
 \left\|\frac{g(t)}{g(s)}U(s,t)\right\|_{Z^{-1}_{m,c}\to X_{m,c}}
 \le C_b\nu^{-M_s}\{1+(s-t)^{-1/2}\}.
\end{equation}
Zero mass is preserved by the conservative physical equation and
then by continuous extension in the actual weighted dual source norm.

\end{proof}

\subsection{From the plane branch to the actual train output}
\label{sec:canonical-transfer}

All cell measures in this subsection are unnormalized. Put
\[
\begin{gathered}
 \Theta=\nu^{-1},\quad c=\nu^{-2},\quad e=(\nu c^2)^{-1}=\nu^3,\quad
 a=1+\nu s,\\
 \theta=(\nu a)^{-1},\quad d=c/\sqrt a=\sqrt{\theta/e},\quad
 N=e^{\lambda/\nu}\nu^{-\kappa}.
\end{gathered}
\]
Write $P_d$ for periodization, $K_d$ for cylindrical Biot--Savart,
and $X_{m,d}$ for the centrally even zero-circulation class with
weight $\langle y\rangle^{2m}$ on the centered cell, $m>8$.
Its first moments vanish. The canonical plane branch is that of
Proposition~\ref{prop:short-canonical}; set
$M(\theta)=\mathfrak M(V_{\rm can}(\theta))$. Angular integration
and cell summation give $Q_dP_dV_{\rm can}(\theta)=M(\theta)$.

\paragraph{The actual residual, including moving images.}
For $z=\Theta-\theta$, the identity $d_z/d=-1/(2\theta)$ gives
\begin{equation}\label{eq:ct-moving-period}
 \theta^{-1}D(P_df)-\partial_z(P_df)
       =\theta^{-1}P_dDf
\end{equation}
when $f$ is independent of $z$. Thus periodization commutes with
the combined diffusion/dilation/moving-cell derivative, not with
each term separately. Reynolds' boundary term cancels the dilation
seam term. All calculations below may first be made on the fixed
physical cylinder and then rescaled.

For completeness the improved even image error is elementary.
The symmetrically summed planar kernels satisfy
\[
 K_d(\xi)-K_\infty(\xi)
      =d^{-2}B_1\xi+O(d^{-4}|\xi|^3),\qquad |\xi|\le3d/8.
\]
Indeed the opposite translates cancel the constant and quadratic
Taylor coefficients, while their $j$th derivatives are
$O((|n|d)^{-1-j})$. The affine convolution vanishes because both
mass and first moments are zero. On $|y|\le d/4$, $|x|\le d/8$,
the remaining convolution is bounded by
$Cd^{-4}(1+|x|^3)\|f\|_{X_m}$. The excluded source tail is at most
$Cd^{1-m}\|f\|_{X_m}$ in $L^1$ and its far kernel costs $Cd^{-1}$.
Multiplication by the Gaussian observation coefficient controls
the polynomial in $x$; its complement $|x|>d/8$ is exponentially
small. The same reasoning after $d\partial_d$ gives the same bound.
Consequently the localized linearized interaction error is
$O(d^{-4})$ on this even class. For a periodized Gaussian input,
cross transport between different cells is exponentially small,
using its fixed Gaussian derivative norms.

Let $E_c$ be the actual train vorticity and
$h=a^2E_c(s,\sqrt a\,y)-P_dE_0$.
The train comparison \eqref{sct:comparison}, after rescaling, gives
\[
 \|\partial^j h\|_2+
 \|\partial^jK_dh\|_{2\cap\infty}
       \le C_j\theta e^{-b_jd^2}.
\]
Here the curl estimate uses one additional derivative. Since
$\theta=ed^2$, the right side is at most $C_je e^{-b'_jd^2}$.
There is no reliance on smallness of an unbounded differential
operator. To add fixed polynomial weights, the heat-coordinate
vorticity energy with weight $\langle y\rangle^{4m}$ and finitely
many derivatives gives the crude bound $P_\nu e^{C_{m,j}/\nu}$:
transport coefficients and their selected derivatives are uniformly
bounded by the preceding unweighted comparison, their clock interval
has length at most $\nu^{-1}$, and the diffusion/dilation contribution
integrates $C_{m,j}\theta^{-1}$, costing only a power of $\nu$.
The periodic seam diffusion term is nonpositive and the moving seam
cancels as in \eqref{eq:ct-moving-period}. Interpolating that bound
with the unweighted error, and using $d^2\ge r/e$ on any fixed
$[r,\Theta]$, absorbs $e^{C/\nu}$ and every fixed polynomial into
a smaller Gaussian for sufficiently small $\nu$:
\begin{equation}\label{eq:ct-base-error}
 \|h\|_{H_m^j}+\|K_dh\|_{W^{j,\infty}}
       \le C_{m,j}e\,e^{-b_{m,j}\theta/e},
 \qquad r\le\theta\le\Theta.
\end{equation}

Let $w=\mathcal U_{\nu,c}(s,0)P_cq$ be the actual bare column and
write $w(s,x)=Na^{-2}Z(\theta,x/\sqrt a)$.
The canonical asymptotic gives
\[
 \|e^{\lambda\Theta}\Theta^\kappa
       [Z(\Theta)-P_cV_{\rm can}(\Theta)]\|_{X_{m,c}}\le C\nu.
\]
Applying
\eqref{eq:ct-moving-period}, the image estimate, and
\eqref{eq:ct-base-error} to $V_d=P_dV_{\rm can}$ gives the following
residual in the equation $-\partial_\theta Z=\mathcal L_\theta Z$:
\begin{align}
 \|e^{\lambda\theta}\theta^\kappa
       (-\partial_\theta-\mathcal L_\theta)V_d\|_{X_{m,d}}
 &\le C_b(e^2\theta^{-2}+e\,e^{-a_0\theta/e}),
 &&b\le\theta\le\Theta, \label{eq:ct-long-residual}\\
 \|(-\partial_\theta-\mathcal L_\theta)V_d\|_{X_{m,d}}
 &\le C_b(e^2\theta^{-2}+e\,e^{-a_0\theta/e}),
 &&r\le\theta\le b. \label{eq:ct-late-residual}
\end{align}
The image term needs only the Gaussian zeroth norm of the branch;
cross transport and the base-error terms use its fixed derivative
budgets. Their polynomial factors in $1+\theta$ are absorbed into
the displayed Gaussian. This explicitly accounts for the actual
nonlinear train, rather than substituting its independent-vortex sum.

The full-input forward bound \eqref{eq:short-causal-long-forward}
and Duhamel, including this normalized inlet error, imply
\begin{equation}\label{eq:ct-long-comparison}
 \|e^{\lambda\theta}\theta^\kappa(Z-V_d)(\theta)\|_{X_{m,d}}
 \le C_b\{\nu+e^2/\theta+e^{-a_1\theta/e}\},\qquad b\le\theta\le\Theta.
\end{equation}
In particular the error at $b$ is $O_b(\nu)$, and both columns
have an exponentially decaying, polynomially weighted majorant
for $\theta\ge b$. No terminal projection or backward parabolic
propagator has been used.

\paragraph{Kinetic and tested moment continuation through overlap.}
The embeddings
\begin{equation}\label{eq:ct-embeddings}
 |Q_d f|+\|K_df\|_2\le C_m\|f\|_{X_{m,d}},\qquad
 Q_df=-\frac2\pi\int\eta^2f,\qquad d\ge1
\end{equation}
are uniform in $d$. The moment bound is weighted Cauchy--Schwarz.
For velocity, zero mass gives
$|\widehat f(\xi)|\le|\xi|\int|y||f(y)|dy$ at low cylindrical
frequencies. In Plancherel, the $O(d)$ horizontal frequencies
of modulus at most one cancel the measure factor $d^{-1}$;
the high frequencies are bounded by $\|f\|_2$. This proves
\eqref{eq:ct-embeddings} without a factor $1+d$.

Let $\mathcal U_Z(\theta,r)$ be the exact heat-coordinate evolution.
The velocity energy inequality and \eqref{sct:integrated} give,
for $0<\theta\le r\le b$,
\begin{equation}\label{eq:ct-kinetic-impulse}
 \|K_{d(\theta)}\mathcal U_Z(\theta,r)h\|_2
       \le C_b(r/\theta)\|K_{d(r)}h\|_2.
\end{equation}
Indeed the input physical kinetic norm is
$\nu r\|K_{d(r)}h\|_2$ and the integrated strain costs at most
$C(r+e)$. The exact physical quadrupole law is
\[
 \frac{d}{ds}Q_c(w)
    =-\frac4\pi\int(v_1u_2+v_2u_1)
    =-\frac4\pi\int(\widetilde v_1\widetilde u_2+
                              \widetilde v_2\widetilde u_1).
\]
The mean terms vanish because the mean vertical velocities vanish.
Since $Q$ is invariant under heat scaling, the same identity gives
a tested bound uniform in $\nu$. Before overlap
$\|\widetilde v(s(\theta))\|_2\le C\nu\theta$;
after overlap it is at most
$C\nu e\exp[-a(e/\theta-1)]$. Thus at every age
$\nu^{-1}\theta^{-2}\|\widetilde v(s(\theta))\|_2\le C/\theta$,
and
\begin{align}
 |Q\mathcal U_Z(\theta,r)h|
 &\le |Qh|+C_b r\log(r/\theta)\|K_{d(r)}h\|_2,
                      \label{eq:ct-partial-moment}\\
 \int_0^r\theta^{p-1}|Q\mathcal U_Z(\theta,r)h|\,d\theta
 &\le C_{b,p}r^p(|Qh|+r\|K_{d(r)}h\|_2),\quad p>0.
                      \label{eq:ct-tested-impulse}
\end{align}
The second follows from $\int_0^r\theta^{p-1}\log(r/\theta)d\theta=r^p/p^2$.
Similarly the absolute stress, divided by its scale
$\nu^{3+2\Re\mu}/c$, for an impulse at $r$ is bounded by
\begin{equation}\label{eq:ct-absolute-impulse}
 C_{K,b}r^{p+1}\max\{1,(e/r)^{1/4}\}\|K_{d(r)}h\|_2,
 \qquad p=\Re\mu+3/2.
\end{equation}
For $\theta\ge e$ insert $\|v\|_2\le C\nu\theta$ and
$\|u\|_2\le C_b\nu r\|Kh\|_2$. For $\theta\le e$ insert
$\|v\|_2\le C\nu e^{1/4}\theta^{3/4}$ instead. Its integral is
$Cr e^{1/4}\min(e,r)^{p-1/4}\|Kh\|_2$, proving the assertion.
All these identities first hold for smooth data and sources.
Duhamel and the integrable displayed bounds give the stated
completion on the fixed physical cylinder; the actual decaying
mean stream is retained when identifying its quadrupole.

For each fixed $r>0$, apply \eqref{eq:ct-kinetic-impulse} and
\eqref{eq:ct-partial-moment} to the residual and the error at $b$.
The kinetic residual integral is bounded by
$C_be^2\theta^{-1}\log(b/\theta)$; the partial-moment factors are
bounded on $r\le\theta\le b$. Thus
\begin{equation}\label{eq:ct-compact-comparison}
 \sup_{r\le\theta\le b}
 \bigl(\|K_d(Z-V_d)(\theta)\|_2+
                |Q_d(Z-V_d)(\theta)|\bigr)\longrightarrow0.
\end{equation}
The plane bounds and periodization imply
$\limsup_{\nu\to0}(|Q_dZ(r)|+\|K_dZ(r)\|_2)\le C_b$ uniformly
in $0<r\le b$. Apply the impulse estimates to the homogeneous actual
column with data $Z(r)$, not to its discrepancy from $V_d$.
For $p$ in a compact subset of $(1/4,\infty)$,
\[
 \limsup_{\nu\to0}\int_0^r\theta^{p-1}|Q_dZ(\theta)|d\theta
       \le C_br^p,
\]
and \eqref{eq:ct-absolute-impulse}, with $e<r$, bounds its normalized
absolute stress tail by $C_br^{p+1}$. Letting first $\nu\to0$ and
then $r\downarrow0$ therefore gives the required tested convergence
through the entire overlap tail.

\paragraph{Mellin output and stress defects.}
Fix $K\Subset\{0<\Re\mu<1/4\}$, put $\sigma=\Re\mu$,
$u_o=\nu^{-3/2}$, and $\theta_o=(\sqrt\nu+\nu)^{-1}$.
For the actual column define
$A_+(s)=-\pi s^{-\mu}Q_c(w(s))/(4c)$.
The exact substitution
$s=(1-\nu\theta)/(\nu^2\theta)$ yields
\begin{align}
 I_A:=\int_{u_o}^{\infty}A_+(s)s^{-5/2}ds
 &=-\frac{\pi N}{4c}\nu^{3+2\mu}
   \int_0^{\theta_o}\theta^{\mu+1/2}
       (1-\nu\theta)^{-\mu-5/2}QZ(\theta)d\theta \notag\\
 &=-\frac{\pi N}{4c}\nu^{3+2\mu}
       [\mathcal J(\mu)+o_K(1)].
                                      \label{eq:ct-mellin}
\end{align}
Use \eqref{eq:ct-long-comparison} for the exponentially integrable
large-clock part and the compact-clock convergence and tested tail
bound above for the rest. The error is locally uniform and holomorphic.
An extra $s^{-k/2}$ costs
$\nu^k\theta^{k/2}(1-\nu\theta)^{-k/2}$, $k=0,1$.
At $u_o$, every selected state or stress observation divided by $N$
is $P_\nu e^{-\gamma'/\sqrt\nu}$ for any $\gamma'<\gamma$.
The partial-moment bound gives $QZ(\theta)=O_\nu(1+|\log\theta|)$
as $\theta\downarrow0$, so the upper weighted moment endpoint is zero.
If the old cutoff
starts at $\sigma_{\rm in}/\nu<u_o$, its entire preceding part
has the same exponentially small coefficient after polynomial
weights; that source and cutoff transition have not been removed.

For $u=K_cw$, $u_\mu=s^{-\mu}u$, set
\[
\begin{gathered}
 E_\eta=\frac2c\int v_1u_{\mu,1},\quad
 E_q=\frac2c\int v_2u_{\mu,2},\quad
 C=\frac1c\int(v_1u_{\mu,2}+v_2u_{\mu,1}),\\
 D_c=E_\eta+E_q,\quad R_c=E_q-iC,\quad
 \mathcal A_\nu=|N|\nu^{3+2\sigma}/c.
\end{gathered}
\]
These are bilinear pairings. Define the stronger absolute component
quantity
\[
 S_{\rm abs}(s)=c^{-1}\int
       |v\otimes u_\mu+u_\mu\otimes v|\,dq\,d\eta.
\]
For $k=0,1$,
\begin{equation}\label{eq:ct-absolute-stress}
 \int_{u_o}^\infty s^{-3/2-k/2}
       S_{\rm abs}(s)ds\le C_{K,k}\mathcal A_\nu\nu^k.
\end{equation}
This controls each absolute tensor component before cell integration,
and hence also $|E_\eta|+|E_q|+|C|$, up to an absolute constant.
Indeed the integrand before its time weight is at most
$C|N|c^{-1}s^{-\sigma}\|v\|_2\|u/N\|_2$.
For $\theta\ge b$, $\|u/N\|_2\le Ca^{-1}H(\theta)$ with
$H$ exponentially decreasing times a polynomial. Changing variables
gives the integrable majorant
$C\mathcal A_\nu\nu^k\theta^{\sigma+3/2+k/2}H(\theta)$.
For $\theta\le b$, energy gives $\|u/N\|_2\le C_b\nu$.
Before overlap $\|v\|_2\le C/(\nu s)$; after overlap
$\|v\|_2\le Cc^{-2}(s/s_c)^{-3/4}$.
The two integrals are bounded by
$C\mathcal A_\nu\nu^k$ and
$C\mathcal A_\nu\nu^ke^{\sigma+3/2+k/2}$ respectively.

For a plane radial background in positive angular sector two,
$E_\eta=-E_q$ and $E_q=iC$ by direct angular integration.
On each fixed positive clock interval,
\eqref{eq:ct-compact-comparison} and the actual background comparison
therefore make the scaled $D_c,R_c$ tend to zero. Their absolute
small-clock tails are bounded by \eqref{eq:ct-absolute-impulse} with
data $Z(r)$, hence by $C_K\mathcal A_\nu r^{\sigma+5/2}$ in the
$\nu\to0$ limit. Large-clock domination, followed by $r\downarrow0$,
proves
\begin{equation}\label{eq:ct-stress-defect}
 \int_{u_o}^{\infty}(D_c-3R_c)s^{-3/2}ds=o_K(\mathcal A_\nu).
\end{equation}
The unforced moment law is exactly $C=A_+'+\mu A_+/s$; hence
\begin{equation}\label{eq:ct-anisotropy}
 \int_{u_o}^{\infty}(2E_q-E_\eta)s^{-3/2}ds
   =3i(\mu+3/2)I_A+o_K(\mathcal A_\nu),
\end{equation}
including the boundary term
$3i[A_+s^{-3/2}]_{u_o}^{\infty}$, just shown negligible.
For the negative channel the sign of $i$ reverses and
$N\mathcal J(\mu)$ becomes
$\bar N\,\overline{\mathcal J(\bar\mu)}$.

\subsection{A polynomial causal inverse in the gain norm}
\label{sec:short-polynomial-causal}

Retain the fixed spaces, actual train, and gain from
Proposition~\ref{prop:short-full-clock}. Put
\[
 \mathcal H=L^2(s^{-3/2}ds;L^2_{\rm div}(dq/c\,d\eta)),\quad
 \mathcal T=L^2(s^{-1/2}ds;L^2(dq/c\,d\eta)).
\]
Let $K$ be a fixed compact subset of $\{0<\Re\mu<1/4\}$ and
$k_*:=\inf_K(\Re\mu+1/4)>0$. Constants below are uniform on $K$.

\begin{proposition}\label{prop:short-polynomial-causal}
For every planar symmetric tensor $F:C_c\times(0,\infty)\to
\operatorname{Sym}_2$ with $g^{-1}F\in\mathcal T$, there is
a unique causal zero-entrance energy solution of
\begin{equation}\label{eq:short-causal-equation}
 b_s+\frac\mu s b-\nu\Delta b+\mathbb P_0\operatorname{div}
       (v\otimes b+b\otimes v)=\mathbb P_0\operatorname{div}F.
\end{equation}
Here zero entrance means $b\in\mathcal H$ near zero; no homogeneous
$s^{-\mu}$ entrance is adjoined. The energy class also requires
$\nabla b\in L^2_{\rm loc}((0,\infty);L^2)$. There is a finite
$M$ such that
\begin{equation}\label{eq:short-causal-bound}
 \|g^{-1}b\|_{\mathcal H}
 +\sqrt\nu\|\nabla(g^{-1}b)\|_{\mathcal T}
 \le C\nu^{-M}\|g^{-1}F\|_{\mathcal T}.
\end{equation}
The response is locally continuous in $\mu$ with the geometry fixed.
This is a bound after conjugation by $g$; its ordinary norm can contain
$e^{\gamma/\nu}$.
\end{proposition}

\begin{proof}
Use Proposition~\ref{prop:short-full-clock} with its fixed $b$,
$H=\nu^{-1}((\nu b)^{-1}-1)$, and weak-source degree $M_s$.

\emph{The zero-entrance construction and all ages.}
Put $T=g^{-1}F$, $u=g^{-1}b$ and
$h=g'/g=\gamma/(1+\nu s)^2\ge0$. Solve the free equation for $u_0$
with scalar term $\mu/s+h$ and tensor $T$ by starting from zero at
$s=\delta$ and taking $\delta\downarrow0$. The weighted energy is
coercive and gives
\[
 \|u_0\|_{\mathcal H}\le(2\sqrt{k_*\nu})^{-1}\|T\|_{\mathcal T},
 \qquad \|\nabla u_0\|_{\mathcal T}\le\nu^{-1}\|T\|_{\mathcal T}.
\]
Spatial coefficient estimates may first be taken with unnormalized
cell measure. Multiplying velocity, tensor, $X_m$, and weighted-dual
source norms by the same $c^{-1/2}$ gives the normalized convention
used in $\mathcal H$ and $\mathcal T$; all inequalities below are
unchanged. Equivalently define the source duality using normalized
cell pairing and normalized test norm.
The localized vorticity source
$f=-\operatorname{curl}(v\cdot\nabla u_0+u_0\cdot\nabla v)$
has zero mass and satisfies
\[
 \|f(s)\|_{Z^{-1}_{m,c}}
 \le A_m(\|\nabla u_0(s)\|_2+\|u_0(s)\|_2),\qquad
 A_m\le C\nu^{-p_m}\quad(0<s<H),
\]
for one fixed finite $p_m$. Indeed $A_m$ is bounded by a constant
times $\sup_{s\le H}\|\langle x\rangle^m(v,\nabla v)(s)\|_\infty$.
The actual train tail \eqref{sct:transverse}, with $S_*=H$, bounds
this by a fixed polynomial in $c,\nu^{-1},H$: the nonzero modes
have exponential transverse kernels, and the zero mode has decaying
primitives at both ends. Here $c=\nu^{-2}$ and $H=O(\nu^{-2})$.
Thus this is a polynomial coefficient bound, with no exponential
weighted-energy loss. Testing the outer curl against $H^1_{-m}$
then gives the displayed negative-order estimate. Hence
$\|f\|_{L^2(s^{-1/2};Z^{-1}_m)}
\le C\nu^{-p_m-3/2}\|T\|_{\mathcal T}$.

For $\sigma=\Re\mu$, the substitution $t=su$ and Minkowski give
the exact Hardy inequality
\begin{equation}\label{eq:short-causal-hardy}
 \left\|\int_0^s(t/s)^\sigma(s-t)^{-1/2}f(t)dt\right\|_{L^2(s^{-3/2})}
 \le B(\sigma+3/4,1/2)\|f\|_{L^2(s^{-1/2})}.
\end{equation}
The kernel $1$ on $s<H$ costs an additional $\sqrt H\le C_b\nu^{-1}$.
Define using only forward propagation
\[
 \omega(s)=\int_0^s(t/s)^\mu\frac{g(t)}{g(s)}U(s,t)f(t)dt,
 \qquad u=u_0+K_c\omega.
\]
Equations \eqref{eq:short-causal-weak-forward}--\eqref{eq:short-causal-hardy}
give $\|\omega\|_{L^2(s^{-3/2};X_m)}
\le C\nu^{-M_s-p_m-5/2}\|T\|_{\mathcal T}$.
Zero-circulation Biot--Savart reconstruction gives
$\|K_c\omega\|_{H^1}\le C_m(1+c)\|\omega\|_{X_m}$:
nonzero horizontal modes cost $c$, while the mean is the
zero-endpoint primitive controlled by one-dimensional Hardy:
$\|\int_\eta^\infty\bar\omega(\zeta)d\zeta\|_2
\le2\|\eta\bar\omega(\eta)\|_2$. This follows by integrating
$(\eta|\int_\eta^\infty\bar\omega|^2)'$; zero circulation
removes both endpoint terms.
Since $\sqrt H=O(\nu^{-1})$, we obtain on the core
\[
 \|u\|_{\mathcal H(0,H)}
 +\sqrt\nu\|\nabla u\|_{\mathcal T(0,H)}
 \le C\nu^{-M_s-p_m-5}\|T\|_{\mathcal T(0,H)}.
\]
The velocity equation follows from its curl and divergence equations:
after testing in age the discrepancy is spatial $H^{-1}$, curl-free
and divergence-free; its Fourier transform is supported only at zero
and must vanish. This retains the horizontal mean. Local Hilbert
triple regularity gives a strong $L^2$ trace at every positive age.

For $E=s^{-1/2}\|u\|_2^2$, $L=\|\operatorname{sym}\nabla v\|_\infty$
and $A(s)=\exp(-2\int_{t_*}^sL)$, the exact energy inequality is
\begin{equation}\label{eq:short-causal-tail-energy}
 (AE)'+2k_*AE/s+2hAE+\nu A s^{-1/2}\|\nabla u\|_2^2
 \le\nu^{-1}A s^{-1/2}\|T\|_2^2.
\end{equation}
Selecting a time in $[H-1,H-1/2]$ and applying this inequality to $H$
gives
$H^{-1/4}\|u(H)\|_2
\le C_b[\sqrt H\|u\|_{\mathcal H(H-1,H)}
+\nu^{-1/2}\|T\|_{\mathcal T(H-1,H)}]$.
With $a_b=1+\nu H=(\nu b)^{-1}$, the actual train estimate
\eqref{short-bg:train-late} yields
\[
 I_H:=\int_H^\infty L(s)ds
 \le C/(\nu a_b)+Ce_c=Cb+C\nu^3.
\]
Thus continuation from the exact strong trace, with the entire
prescribed tensor after $H$, costs only the fixed factor $e^{Cb+C\nu^3}$.
It crosses the overlap regime and reaches infinite age without a
vorticity moment assumption on the late response. Combining the
core and tail gives \eqref{eq:short-causal-bound}, for example with
any $M\ge M_s+p_m+6$.

For uniqueness, a homogeneous difference in the energy class has
a sequence $\delta_j\downarrow0$ with
$\delta_j^{-1/2}\|u(\delta_j)\|_2^2\to0$, because its integral
against $ds/s$ is finite. Apply \eqref{eq:short-causal-tail-energy}
on $[\delta_j,B]$ and let $j\to\infty$. This also proves causality.
Continuity in $\mu$ on the core follows from the same Hardy estimate
with one factor $|\log(t/s)|$; its beta integral is finite uniformly
on $K$. In the free and tail equations the difference contributes
$-(\mu-\mu')u_{\mu'}/s$, whose weighted pairing is bounded by
$|\mu-\mu'|\|u_{\mu'}\|_{\mathcal H}\|u_\mu-u_{\mu'}\|_{\mathcal H}$
and is absorbed by the $k_*$ coercivity. The local energy estimate
also gives trace continuity. All geometric parameters stay fixed.
\end{proof}

\begin{corollary}\label{cor:short-polynomial-stress}
Let $S_{0,\mu}$ and $S_{\rm lin,\mu}$ denote the free and actual
causal tensor responses, acting on the sum of the two components of
the split stress space $\mathcal Z=\mathcal Z_1\oplus
\rho_b^{-1/2}\mathcal T$, with its specified norm and $0<\rho_b\le1$.
For a three-component cylindrical tensor these flat maps act on its
$(r,z)$ block; its hoop component is retained in the exact curved
comparison.
The first-component weight is at least one, so
$\|F_1+F_2\|_{\mathcal T}\le\|F_1\|_{\mathcal Z_1}
+\sqrt{\rho_b}\|F_2\|_{\rho_b^{-1/2}\mathcal T}$. Let
$M_0u=(v\otimes u+u\otimes v,0)$ and suppose its actual train
multiplier bound is $\|M_0u\|_{\mathcal Z_1}
\le C\nu^{-p_V}\|u\|_{\mathcal H}$. Then
\[
 D_0^{-1}=(I+M_0S_{0,\mu})^{-1}=I-M_0S_{\rm lin,\mu},\qquad
 \|g^{-1}D_0^{-1}g\|_{\mathcal Z\to\mathcal Z}\le C\nu^{-M},
 \quad M=\lceil p_m+p_V+M_s+10\rceil.
\]
Indeed $b=S_{\rm lin,\mu}F$ satisfies
$b=S_{0,\mu}(F-M_0b)$ by free uniqueness, giving both inverse
identities. The second split input component is retained. The degree
has slack for the fixed norm equivalences and trace estimate, and
depends only on the fixed weights and coefficient budgets.
\end{corollary}

\section{Separated stress kernels and one phase average}
\label{sec:separated-short}

All phase norms use normalized measure on $\mathbb T_P$.
Write $B=\partial_\tau-y\cdot\nabla/2$,
$D_\mu=B-\nu\Delta+\mu-1/2$, and
$S_\mu=D_\mu^{-1}\mathbb P\operatorname{div}$.
The exponent $\mu$ ranges over a fixed compact subset $K$ of
$\{\operatorname{Re}\mu>-1/4\}$. Let $T$ be supported in the
radial disk $|z'|\le r_o$, and restrict receivers to $|y'|\ge r_i>r_o$.
Put $D=r_i-r_o$. The support condition is imposed on the tensor before
its divergence. We begin with smooth tensors and then complete in the
norms appearing below.

Let $G_t$ be the three-dimensional heat kernel. The heat--Leray kernel is
\[
 (K_t)_{ij}=\delta_{ij}G_t+
   \partial_i\partial_j\int_t^\infty G_a\,da.
\]
Differentiation and the substitution $a=|x|^2r$ give, off the origin,
\begin{equation}\label{sep:kernel}
 |D^j\nabla K_t(x)|\le C_j|x|^{-4-j},\qquad
 |D^j(K_t-K_0)(x)|\le C_jt|x|^{-5-j}.
\end{equation}
The second bound also follows by integrating
$\partial_tK_t=\Delta K_t$ from zero to $t$. Thus it remains valid
as $t\downarrow0$ at every separated observation point. At $t=0$,
$K_0=D^2(4\pi|x|)^{-1}$ off the origin; its local distributional
part is retained by the operator, but vanishes at a separated receiver.

The free semigroup is
\[
 e^{h/2}\bigl[e^{\nu(e^h-1)\Delta}f(\tau-h)\bigr](e^{h/2}y).
\]
In its Laplace integral set $\lambda=e^{-h/2}$ and integrate the
tensor divergence by parts. Heat scaling contributes $\lambda^3$,
the divergence contributes $\lambda$, and the semigroup amplitude
and differential contribute $\lambda^{-1}$ and $-2d\lambda/\lambda$.
Consequently the negative response $W_\mu=-S_\mu T$ is exactly
\begin{equation}\label{sep:resolvent}
 W_\mu(y,\tau)=-2\int_0^1\lambda^{2\mu+2}
  \int\nabla K_{\nu(1-\lambda^2)}(y-\lambda z)
       :T(z,\tau+2\log\lambda)\,dz\,d\lambda.
\end{equation}
The radial gap persists under every dilation:
$|y-\lambda z|\ge |y'|-\lambda|z'|\ge D$.
No temporal derivative of $T$ occurs.

\subsection{Thin sources, repair sources, and the stream cutoff}

For $b>3/(4+j)$, integrating \eqref{sep:kernel} outside the gap gives
\[
 \|1_{|x|>D}D^j\nabla K_t\|_{L^b}
       \le C_{j,b}D^{-4-j+3/b}.
\]
In Young's inequality we use $b\ge1$. Dilation of an $L^p$ source
contributes $\lambda^{-3+3/p}$, leaving the integrable power
$\lambda^{2\operatorname{Re}\mu-1+3/p}$ in
\eqref{sep:resolvent}. Minkowski and phase-translation invariance
therefore prove, whenever $1+1/q=1/p+1/b$ and
$\inf_{\mu\in K}(2\operatorname{Re}\mu+3/p)>0$,
\begin{equation}\label{sep:young}
 \|1_{\rm young}D^jW_\mu\|_{L^2_\tau L^q_y}
 \le C_{K,j,p,q}D^{-4-j+3/b}
                   \|T\|_{L^2_\tau L^p_y}.
\end{equation}
The pointwise version has $q=\infty$, $b=p'$.
In particular, for $p=22/19$,
\begin{align}
 \|\nabla W_\mu(y,\cdot)\|_{L^2_\tau}
   &\le C_KD^{-101/22}\|T\|_{L^2_\tau L^{22/19}},\label{sep:repair-point}\\
 \|1_{\rm young}\nabla W_\mu\|_{L^2_\tau L^{22/3}}
   &\le C_KD^{-46/11}\|T\|_{L^2_\tau L^{22/19}},\label{sep:repair-global}\\
 \|1_{\rm young}D^jW_\mu\|_{L^2_\tau L^2}
   &\le C_{K,j}D^{-23/11-j}\|T\|_{L^2_\tau L^{22/19}}.
 \label{sep:repair-energy}
\end{align}
Here the kernel exponents are respectively $22/3$, $11/3$, and
$11/7$. A repair tensor $T=r_b\otimes w+w\otimes r_b$ obeys
\[
 \|T\|_{L^2_\tau L^{22/19}}
 \le2\|r_b\|_{L^\infty_\tau L^{11/4}}\|w\|_{Y_2}.
\]
These estimates include the Cartesian pressure projection.

For a thin source with
$\int\|T(z,\cdot)\|_{L^2_\tau}\,dz\le R^2N_0$ and
$D\ge c_0R\sqrt\nu$, the same argument with $p=1$ gives, for
fixed $q>1$ and fixed $j$,
\begin{equation}\label{sep:thin-global}
 \|D^jW_\mu\|_{L^2_\tau L^q(\rm young)}
       \le P_\nu R^{-2-j+3/q}N_0.
\end{equation}
The polynomial $P_\nu$ depends on the fixed derivative and exponent
budgets and on the fixed fractional radial gap.

For axisymmetric no-swirl divergence-free $W$, choose a smooth radial
cutoff $\chi$ whose transition lies outside the source disk, on an
annulus of radius comparable to $D$. Its divergence-free stream cutoff is
\[
 (\mathcal C_\chi W)_r=\chi W_r,\qquad
 (\mathcal C_\chi W)_z=\chi W_z-
  \frac{\chi'(\rho)}\rho\int_\rho^\infty\rho'W_z(\rho',z)\,d\rho'.
\]
The stream primitive converges by the separated kernel bound.
The cutoff correction has a translation-invariant majorant of the form
\[
 C D^{-2}(D^2+|z-z_{\rm src}|^2)^{-1}
       1_{|y'-z'_{\rm src}|\le CD}.
\]
Indeed the radial integral of the order-four kernel is bounded by
$C(D^2+|z-z_{\rm src}|^2)^{-1}$, while $\chi'/\rho=O(D^{-2})$.
After $j$ derivatives the same estimate costs $D^{-j}$.
Its $L^b$ norm is $CD^{-4-j+3/b}$ for $b>1/2$, as follows by
scaling the two horizontal variables and the vertical integral.
Thus \eqref{sep:young}--\eqref{sep:thin-global} remain valid after
this cutoff in all exponents used here. The estimate is an inequality
for the full tensor convolution; it does not discard a cutoff source.

Put
\[
 X_0=W_\mu-W_{\mu+1},\quad X=\mathcal C_\chi X_0,
 \quad \mathcal Q=[B,\mathcal C_\chi]-\nu[\Delta,\mathcal C_\chi],
 \quad Y=\mathcal C_\chi W_\mu+\mathcal QX_0.
\]
The difference in $X_0$ inserts $1-\lambda^2$ in
\eqref{sep:resolvent}. It costs no inverse. Derivatives of $\chi$
cost $D^{-1}$, $B\chi$ is bounded, and each extra Laplacian costs
$\nu D^{-2}$. For $D\ge c_0R\sqrt\nu$ and $R\ge1$ this last
factor is bounded. Therefore the displayed estimates also hold for
$X,Y$, with finitely many increased derivative budgets.

\subsection{One phase primitive gives an extra curvature factor}

Set $\epsilon=R^{-1}$, $P=2\epsilon c$, and $x=y/R$.
Write the thin physical stress as
\[
 T(Rx,\tau)=F(x',Y,q_\epsilon(|x'|,\tau)),\qquad
 Y=x_3/\epsilon,\quad
 q_\epsilon(r,\tau)=\epsilon^{-1}\log r+\tau/(2\epsilon),
\]
where $F$ has $q$-period $c$. Define
\[
 N_j=\int(1+|Y|)^j
       \|F(x',Y,\cdot)\|_{L^2(dq/c)}\,dx'\,dY,
       \qquad j=0,1.
\]
Then its physical phase-averaged mass is $R^2N_0$.
Changing variables in \eqref{sep:resolvent} gives
\begin{equation}\label{sep:outer}
 W_\mu=-2\epsilon^2\int_0^1\lambda^{2\mu+2}
  \int\nabla K_{\nu\epsilon^2(1-\lambda^2)}
       (x-\lambda(x',\epsilon Y))
 F(x',Y,q_\epsilon(r',\tau+2\log\lambda))\,dx'\,dY\,d\lambda.
\end{equation}
Let $\bar F$ be its normalized phase mean, and set
\[
 V_{\mu,0}(x)=-2\int_0^1\lambda^{2\mu+2}
  \int\nabla K_0(x-\lambda(x',0))\bar F(x',Y)
                                  \,dx'\,dY\,d\lambda.
\]
Taylor expansion in $\epsilon Y$ and \eqref{sep:kernel} control
the mean collapse and heat errors by first transverse moments. The
strict radial gap remains valid along the entire Taylor segment.

For the oscillatory part let $J=\partial_q^{-1}$ on zero-mean
periodic functions. Fourier series gives
$\|J\|_{L^2(dq/c)\to L^2(dq/c)}\le c/(2\pi)$. Exactly in
distributions in $\lambda$,
\[
 F_\perp(q_\epsilon(r',\tau+2\log\lambda))
 =\epsilon\lambda\partial_\lambda
   JF_\perp(q_\epsilon(r',\tau+2\log\lambda)).
\]
Integrate once by parts in \eqref{sep:outer}. The boundary at
$\lambda=1$ is retained. The boundary at zero vanishes in normalized
$L^2_\tau$, since $2\operatorname{Re}\mu+3>0$ and translations of
$JF_\perp$ preserve its norm. The derivative of the kernel is
\[
 -(x',\epsilon Y)\cdot\nabla(\nabla K_t)
       -2\nu\epsilon^2\lambda\Delta(\nabla K_t),
       \qquad t=\nu\epsilon^2(1-\lambda^2).
\]
The factor $x'$ is bounded by the source radial endpoint and the
factor $Y$ by $N_1$. Every fixed spatial observation derivative is
estimated by \eqref{sep:kernel}. This proves
\begin{equation}\label{sep:mean-comparison}
 \|\nabla_yW_\mu-\epsilon^3\nabla_xV_{\mu,0}\|_{L^2_\tau}
       \le P_\nu\epsilon^4N_1
\end{equation}
on compact separated observation sets, with the stated polynomial
dependence on radial endpoints, their gap, and $c$.
No age or phase derivative of $F$ has been used.

For the full young band $s\le1/\nu$, old support $t\ge1.2/\nu$,
and $r=s^{-1/2}$ in outer coordinates, the same calculation gives
the more explicit bounds
\begin{align}
 \|D_yW_{\mu,\perp}\|_{L^2_\tau}
     &\le P_\nu\epsilon^4s^{5/2}N_1,\label{sep:phase-age}\\
 \|D_yW_\mu-\epsilon^3D_xV_{\mu,0}\|_{L^2_\tau}
     &\le P_\nu\epsilon^4(s^{5/2}+s^3)N_1.
 \label{sep:mean-age}
\end{align}
The direct differentiated kernel contributes $r^{-5}$; a transverse
collapse costs one further derivative, $r^{-6}$. In the phase
calculation the extra factor $x'$ is at most a fixed multiple of $r$;
the $Y$ term has the harmless factor $\epsilon/r$, and the heat term
has $\nu\epsilon^2/r^2\le\epsilon^2$.
These estimates hold with any fixed number of spatial observation
derivatives, with the corresponding powers of $r^{-1}$.
The stream cutoff estimates proved above extend them to $X,Y$.
Moment-norm approximation proves the statements for all $N_1$ inputs,
without assuming smoothness of the source.

\subsection{Phase suprema use a separate ordinary bound}

The extra $\epsilon$ in \eqref{sep:phase-age} is not asserted after
arbitrary phase or conormal derivatives. Instead use the exact free
equations
\[
 BX_0=\nu\Delta X_0-(\mu+1/2)X_0+W_\mu,
 \qquad
 BW_\mu=\nu\Delta W_\mu-(\mu-1/2)W_\mu
                                  -\mathbb P\operatorname{div}T.
\]
At a separated receiver the last term has an order-four kernel, and
its gradient has order five. It is bounded by precisely the ordinary
kernel estimates already proved. It does not require differentiating
$T$, but need not retain the extra phase-average factor.
For the cutoff terms use
\[
 BT_{\rm rad}W=T_{\rm rad}BW-T_{\rm rad}W,
 \qquad [B,\Delta]=\Delta,
\]
and
\[
 [B,\mathcal Q]=[B,[B,\mathcal C_\chi]]
 -\nu[\Delta,\mathcal C_\chi]
 -\nu[\Delta,[B,\mathcal C_\chi]].
\]
Thus $BX,BY$ have the same ordinary separated bounds as $X,Y$,
with a fixed increased derivative budget and the factors
$\nu D^{-2}$ already controlled. Also
$B\partial_j=\partial_jB+\partial_j/2$.

For a global cutoff field $Z$ among these vectors or tensors and
$1<q<\infty$, spatial
integration by parts gives
\[
 \left|\frac{d}{d\tau}\|Z(\tau)\|_q\right|
 \le\|BZ(\tau)\|_q+\frac3{2q}\|Z(\tau)\|_q.
\]
Choose a phase where the norm is no greater than its normalized $L^2$ average,
then integrate over an interval of length at most $P\le1$. This yields
\begin{equation}\label{sep:phase-sup}
 \|Z\|_{L^\infty_\tau L^q}
 \le(1+3P/(2q))\|Z\|_{L^2_\tau L^q}
                         +P\|BZ\|_{L^2_\tau L^q}.
\end{equation}
In particular \eqref{sep:repair-global} for $\nabla X,\nabla Y$
holds also in phase supremum, with $D\ge c_0R\sqrt\nu$ and a
fixed viscosity polynomial. The thin bounds have the same upgrade.
For weighted spatial suprema, apply the transport identity along
$y(\tau)=e^{-\tau/2}y(0)$ to
$\rho^{4+j}D^jX$ and $\rho^{4+j}D^jY$; the weight has
$B\rho^{4+j}=-(4+j)\rho^{4+j}/2$. The same argument bounds the supremum in time by the $L^2$ average. Its input is the $L^2_\tau L^\infty_y$ bound:
take the weighted spatial supremum of the separated kernel first,
then apply Minkowski in the source and dilation variables. The thin
$L^1_yL^2_\tau$ moment supplies this bound. A pointwise-in-space
$L^2_\tau$ estimate alone would not suffice.
Approximation of the source in its stated norm
passes all these estimates to rough tensors.

\section{The direct zero-entrance near equation}

Use the normalized cylinder spaces
\[
 H=L^2(s^{-3/2}\,ds\,dq/c\,d\eta),\qquad
 H_T=L^2(s^{-1/2}\,ds\,dq/c\,d\eta),\qquad
 Z=Z_1\oplus\rho_b^{-1/2}H_T,
\]
where \(Z_1=L^2(s^{-1/2}W^2\,ds\,dq/c\,d\eta)\) and
\(W=\langle\eta/\sqrt{1+s}\rangle^{m_Z}\), \(m_Z>3/2\).
The stress inverse always acts on the sum of the components. Let
\(\mathcal M_0\) be multiplication by the actual flat train, with
zero second output component, and let \(\mathcal M_U\) be the
exact thin/repair multiplier. Set
\[
 \mathsf D_0=I+\mathcal M_0\mathcal S_0,\qquad
 g(s)=e^{\gamma s/(1+\nu s)},\qquad \mathsf G T=gT.
\]
The direct causal construction on the full moving clock gives
\begin{equation}\label{near:flat-bound}
 \|\mathsf G^{-1}\mathsf D_0^{-1}\mathsf G\|_{Z\to Z}
 \le Q:=P_\nu.
\end{equation}
There are no auxiliary entrance coordinates in this statement.
The ordinary norm of this inverse is not asserted to have the same
bound; it can contain the full planar amplification. The polynomial
degree is fixed by the velocity and tensor spaces.

The finite-curvature free inverse is the conjugate of the Cartesian
weak-stress inverse. In particular its divergence and pressure are
not replaced by planar ones. In the profile chart
\[
 s=R^2/\rho^2,\quad q=R\log(\rho/R)+R\tau/2,
 \quad\eta=Rz/\rho,\quad R=\epsilon^{-1},
\]
its exact divergence is
\[
 \operatorname{div}(\sqrt{s}\,b)
 =s\{[\partial_q-\epsilon(2s\partial_s+\eta\partial_\eta)]b_r
                   +\partial_\eta b_z\}.
\]
The geometric comparison below concerns these exact operators.

Fix a small positive logarithmic gap \(\delta\). For an output age
\(s\) and input age \(t\), define
\[
 \mathcal S_{\rm far}
   =1_{\{s<S,\ t>e^\delta s\}}\mathcal S_\epsilon,
 \quad \mathcal S_N=\mathcal S_\epsilon-\mathcal S_{\rm far},
 \quad \mathsf D_N=I+\mathcal M_U\mathcal S_N.
\]
The notation for the mask means its exact kernel realization with
tensor partitions before distributional divergence. Interface terms
are retained. A masked intermediate velocity need not satisfy the
whole-space divergence-free constraint by itself; the stress equation is
well defined independently of that issue. Only the recombined full
Cartesian inverse is used to recover the final physical mode.

This \(\mathsf D_N\) is distinct from subtracting the divergence-free
old-to-young velocity in the nilpotent translation transformation.
Their difference is retained in the transformed equation.

\begin{proposition}\label{near:equation-comparison}
Let \(\mu\) range over a fixed compact subset of \(0<\Re\mu<1/4\).
Put \(s_-=\epsilon^a\), \(S=\epsilon^{-b}\), \(0<b<1\).
There is a fixed integer \(K\), chosen before \(a,b\), such that
\begin{equation}\label{near:deltaN}
 \begin{split}
 \delta_N:=\|\mathsf G^{-1}(\mathsf D_N-\mathsf D_0)\mathsf G\|
 \le P_\nu\big[&\epsilon^{a/2}+\epsilon^{b/2}
       +\epsilon e^{(m_0+\eta_\delta)/\nu}
       +\epsilon^{1/50-K(a+b)}e^{\eta_\delta/\nu}\\
       &+\epsilon^{3/22}e^{(\eta_\delta+m_0/2)/\nu}
       +e^{-(D_\perp-\eta_\delta)/\nu}\big],
 \end{split}
\end{equation}
where \(\eta_\delta\downarrow0\) as \(\delta\downarrow0\).
The constant \(D_\perp>0\) specifies the transverse-tail accuracy.
If \(Q\delta_N<1/2\), then
\begin{equation}\label{near:inverse-comparison}
 \begin{split}
 \|\mathsf G^{-1}\mathsf D_N^{-1}\mathsf G\|&\le2Q,\\
 \|\mathsf G^{-1}(\mathsf D_N^{-1}-\mathsf D_0^{-1})\mathsf G\|
       &\le2Q^2\delta_N.
 \end{split}
\end{equation}
Both statements are equation-only estimates; no terminal functional
or affine source is added.
\end{proposition}

\begin{proof}
The required free comparison is global in its input and compact in
its output:
\[
 \|\chi(\mathcal S_\epsilon-\mathcal S_0)\|_{H_T\to H}
       \le C\Gamma^K\epsilon^{1/50}.
\]
Here \(\Gamma\) is a polynomial in inverse lower age, upper age,
transverse box size, viscosity inverse, cell width, and finitely many
cutoff derivatives. The exponent \(K\) is fixed. The transverse
cutoff may be chosen at \(L_\perp=C_\perp(c/\nu+\sqrt S+1)\),
so \(L_\perp\le C'_\perp S\) for small \(\nu\), and
\(\Gamma^K\le P_\nu\epsilon^{-K(a+b)}\).
The actual flat train coefficient outside this box is at most
\(P_\nu e^{-D_\perp/\nu}\). Its curved stream and entrance
adjustments are estimated separately by their global small multiplier
norm, so their tails need not be Gaussian. No transverse weight is
imposed on the broad repair coefficient.

Partition \([s_-,S]\) into output bins \([l,r]\) with
\(r/l<e^{\delta/2}\). Their number is a viscosity polynomial
in the distinguished regime. The retained input region on a bin
is the complete block \(t<e^\delta l\) and the partial strip
\(e^\delta l<t<e^\delta r\). On the complete block, multiplication
by \(g\) on inputs and \(g^{-1}\) on outputs costs at most
\(g(e^\delta l)/g(l)\). More generally
\[
 \sup_{s>0,\ s\le t\le Ms}
 \nu\left(\frac{t}{1+\nu t}-\frac{s}{1+\nu s}\right)
     =\frac{\sqrt M-1}{\sqrt M+1}.
\]
Indeed, writing \(t=ks\), \(x=\nu s\), the expression is
\((k-1)x/[(1+x)(1+kx)]\), maximized at \(x=k^{-1/2}\).
Thus all complete blocks and partial strips cost at most
\(e^{\eta_\delta/\nu}\), with
\(\eta_\delta=\gamma\tanh(3\delta/8)\) after enlarging the
overlapping bins if necessary. The polynomial bin count is included
in \(P_\nu\). Applying the compact-output comparison and the train
multiplier proves the fourth term of \eqref{near:deltaN}; the
transverse tail gives its last term.

On a partial strip the flat causal inverse is zero. Its source radius
is separated from the output radius by a fixed multiple of
\(R/\sqrt r\), also after the source contraction in the resolvent.
By \eqref{sep:kernel}, the heat--Leray stress kernel has the
uniform positive majorant $C|x|^{-4}$, whose spatial $L^1$ norm
outside distance $d$ is $C/d$.
The exact dilation integral on an \(L^2\) source has power
\(\lambda^{2\Re\mu+1/2}\), hence a uniform integrability margin.
Young's inequality, followed by normalized-phase Minkowski, gives
\(P_\nu\epsilon\sqrt r\) for this output-dependent strip.
After gain multiplication this is at most
\(P_\nu e^{\eta_\delta/\nu}\epsilon^{1-b/2}\), absorbed by
the fourth term. This direct positive-kernel estimate is needed;
boundedness of an arbitrary triangular operator truncation is not used.

For \(s<s_-\), the train multiplier has norm \(P_\nu\sqrt{s_-}\).
Retained inputs have \(t<e^\delta s_-\), where the gain ratio
is \(1+o(1)\). Subtracting the forbidden separated subblock costs
at most \(P_\nu\epsilon\sqrt{s_-}\), by the same positive-kernel
estimate. For \(s>S\), use the full free inverse and the train tail
\(P_\nu(S^{-1/2}+\epsilon)\); its gain ratio is bounded by
\[
 \exp\{\gamma/[\nu(1+\nu S)]\}=1+o(1).
\]
These give the first two terms.

The same complete blocks, using free energy in place of the difference
estimate, give a global bound without a coefficient multiplier. The
positive partial-strip bounds remain valid, the lower end has gain
ratio \(1+o(1)\), and the upper end has the ratio just displayed.
Consequently
\begin{equation}\label{near:masked-free-bound}
 \|g^{-1}\mathcal S_N g\|_{H_T\to H}
       \le P_\nu e^{\eta_\delta/\nu}.
\end{equation}

The exact thin coefficient differs from the flat train by its
\(\epsilon\) stream completion and fixed entrance adjustment,
whose multiplier norm is \(P_\nu\epsilon e^{m_0/\nu}\).
For its contribution use the exact decomposition
\[
 \mathcal M_{\rm thin}\mathcal S_N-\mathcal M_0\mathcal S_0
 = (\mathcal M_{\rm thin}-\mathcal M_0)\mathcal S_N
       +\mathcal M_0(\mathcal S_N-\mathcal S_0).
\]
The first uses the global near free bound and costs
\(P_\nu\epsilon e^{(m_0+\eta_\delta)/\nu}\), including
all tails of the entrance adjustment. The second is the geometric
difference with the Gaussian train multiplier estimated above.
Each broad input/output coupling is bounded by a polynomial times
\(\sqrt{\rho_b}\), using the unweighted free stress bound.
The same bin and end-region estimates apply without a transverse
weight; since \(\rho_b=P_\nu\epsilon^{3/11}e^{m_0/\nu}\),
this gives the fifth term. This proves the complete equation
difference, including every split component.

Conjugate by \(\mathsf G\) and apply the geometric inverse series
to \(\mathsf D_N=\mathsf D_0+(\mathsf D_N-\mathsf D_0)\).
Its norm is at most \(Q\delta_N<1/2\).
The resolvent identity then gives \eqref{near:inverse-comparison}.
\end{proof}

The parameters can be chosen without a terminal observation. Fix the
derivative budgets and \(K\), then choose \(a,b>0\) with
\(K(a+b)<1/100\). Let
\[
 \theta=\min\{a/2,b/2,1/100,3/22\},\qquad d_0=\gamma/3.
\]
Choose \(\delta,m_0\) so that
\(\eta_\delta<\theta d_0/8\),
\(m_0<\theta d_0/8\), and take
\(D_\perp>\eta_\delta+\theta d_0\).
The background construction realizes the prescribed \(m_0\).
All these choices precede \(\nu\downarrow0\).
For \(\epsilon=C_*\nu^r e^{-d_0/\nu}\), uniformly on bounded
\(r\) and positive compact \(C_*\)-ranges,
\[
 \delta_N\le P_\nu e^{-\theta d_0/(2\nu)},\qquad
 Q^2\delta_N\le P_\nu e^{-\theta d_0/(2\nu)}.
\]
Thus the inverse and its comparison close with a positive exponential
margin. The choice is uniform over the spectral interval while all
geometric parameters and the exact background are held fixed.

\subsection{The actual forced column}

Let \(f\) be a normalized body source, so that its physical
representative is \(\mathcal N_v f\), and set
\[
 u_\epsilon=\mathcal M_U\mathcal R_\epsilon f,\qquad
 T_\epsilon=\mathsf D_N^{-1}u_\epsilon,
 \qquad b_\epsilon=\mathcal R_\epsilon f-\mathcal S_NT_\epsilon.
\]
The first row of the equation gives exactly
\(T_\epsilon=\mathcal M_U b_\epsilon\). Hence both its thin and
repair components are actual polarized stresses, even before full
spectral closure. Free body coercivity and \(g^{-1}\le1\) give
\[
 \|g^{-1}T_\epsilon\|_Z
 \le P_\nu\|f\|_H.
\]
By \eqref{near:masked-free-bound},
\[
 \|g^{-1}b_\epsilon\|_H
 \le P_\nu e^{\eta_\delta/\nu}\|f\|_H.
\]
The remaining exponent can be made as small as required before viscosity
is chosen. These estimates do not assert that the intermediate masked
velocity is the final whole-space mode.
They apply to the prescribed body input. For an arbitrary equation
input of the form \(\mathcal M_Uv_{\rm in}\), the corresponding
identity is \(b=v_{\rm in}-\mathcal S_NT\), and its velocity bound
retains \(\|g^{-1}v_{\rm in}\|_H\). A small stress norm alone does
not control that velocity witness.

If a fixed smooth Gaussian affine force is used in both geometries,
write \(u_0=\mathcal M_0\mathcal R_0f\),
\(T_0=\mathsf D_0^{-1}u_0\). Suppose its \(H\) norm is a fixed
viscosity polynomial. The body version of the same output comparison,
with rate \(\epsilon^{1/14}\), and the coefficient/end-region
bounds above give
\(\|g^{-1}(u_\epsilon-u_0)\|_Z\le P_\nu e^{-c/\nu}\)
for some fixed \(c>0\). No gain factor is needed on the input here:
\(f\) is the prescribed unit force, not an amplified stress history.
The resolvent identity therefore yields
\[
 \|g^{-1}(T_\epsilon-T_0)\|_Z
 \le2Q\|g^{-1}(u_\epsilon-u_0)\|_Z
       +2Q^2\delta_N\|g^{-1}u_0\|_Z
 \le P_\nu e^{-c'/\nu}.
\]
In particular any old tensor
moment whose norm after gain multiplication is a viscosity polynomial
has the same limit on these two columns.


\section{The exact Ward change on the multiplier range}
\label{sec:short-ward}

Fix the exact real unforced profile $U$ and its full pressure $p$:
\[
 D_0U+U\cdot\nabla U+\nabla p=0,\qquad
 D_\mu=\partial_\tau-\tfrac12y\cdot\nabla-\nu\Delta
                       +\mu-\tfrac12,\qquad \operatorname{div}U=0.
\]
Use normalized phase measure $d\tau/P$, the prescribed orthogonal
symmetry, and the whole-space free inverses
$R_\mu=D_\mu^{-1}$ and $S_\mu=R_\mu\mathbb P\operatorname{div}$.
Here $R_\mu$ acts on divergence-free body forces. Fix
$K\Subset(0,1/4)$ and all geometric parameters. Put
\[
 M_Uw=U\otimes w+w\otimes U,\qquad
 \mathcal D_\mu^T=I+M_US_\mu.
\]
We use only physical stresses $T=M_Uw$ represented in the actual
pair graph of the final return. Its velocity witness $w\in Y_2$
has the stated symmetry and need not be divergence-free. Since $U$ is
bounded, $T\in Y_2$; its old thin part has finite first transverse
moment $N_1$, and its old repair part belongs to
$L^2_\tau L^{22/19}$. The multiplier, moment and repair-product
bounds establish these facts and are preserved by approximation.

\subsection{The full old source and its outward stream cutoff}

Let $r=|y'|$, $s=R^2/r^2$ and $R=\epsilon^{-1}$. Take smooth cutoffs
\[
 \chi_o=0\ (s\le1.2/\nu),\quad \chi_o=1\ (s\ge1.5/\nu),
 \qquad
 \chi_y=1\ (s\le0.8/\nu),\quad \chi_y=0\ (s\ge1/\nu).
\]
Thus the old source disk $r\le R\sqrt{\nu/1.2}$ is separated
from the young receiver $r\ge R\sqrt\nu$. Set
\begin{equation}\label{ward:displacement}
 W_\mu=-S_\mu(\chi_oT),\qquad
 X_0=W_\mu-W_{\mu+1},\qquad
 X=\mathcal C_yX_0,\quad Y=D_{\mu+1}X,\quad h=[U,X].
\end{equation}
Here $[U,X]=U\cdot\nabla X-X\cdot\nabla U$, and the tensor
$\chi_oT=\chi_oM_Vw+\chi_oM_{r_b}w$ is identical in both
resolvents. In particular the full repair stress enters before
inversion and before taking divergence.

For these radially separated divergence-free fields define directly
\[
 \Psi_Z(r,z)=-\int_r^\infty\rho Z_z(\rho,z)\,d\rho,
 \qquad
 \mathcal C_y Z=\chi_yZ+\frac{\chi_y'}r\Psi_Z e_z.
\]
The phase is suppressed. The separated-kernel estimates prove convergence
of this primitive and its differentiated versions for the thin
$N_1$ and repair $L^2_\tau L^{22/19}$ sources. Their decay makes
the boundary term $rZ_r$ at infinity vanish. Hence
$\partial_r\Psi_Z=rZ_z$ and $\partial_z\Psi_Z=-rZ_r$.
The cutoff is divergence-free, equals $Z$ on its plateau and vanishes
in the old region; define it by zero where $\chi_y=0$.
The same kernels control the stream correction and commutators,
so $X,DX,Y\in Y_2$ and, since $U,DU$ are bounded, $h\in Y_2$.
These are fixed-parameter bounds on the actual multiplier range.

The full free resolvent identity gives
\begin{equation}\label{ward:shifted-resolvent}
 X_0=R_{\mu+1}W_\mu,\qquad D_{\mu+1}X_0=W_\mu.
\end{equation}
Consequently the exact cutoff identity is
\begin{equation}\label{ward:cutoff-source}
 Y=\mathcal C_yW_\mu+[D_{\mu+1},\mathcal C_y]X_0.
\end{equation}
It follows first for smooth graph inputs and then by the moment,
repair-product and separated-kernel bounds. For example direct
integration of the outward primitive gives
$\Psi_{AZ}=A\Psi_Z+\Psi_Z$ for $A=-y\cdot\nabla/2$, and
$\Psi_{\Delta Z}=(\partial_r^2-r^{-1}\partial_r+\partial_z^2)\Psi_Z$;
the same decay removes the boundary terms. Thus this identity does
not require a choice of stream primitive near the axis.
Relative to a scalar-cutoff reference return, both
$(\mathcal C_y-\chi_y)W_\mu$ and
$[D_{\mu+1},\mathcal C_y]X_0$ remain. They are supported on the
transition annulus and are separately estimated below.

\subsection{Derivation with the exact pressure}

We derive the identity before projecting. In physical coordinates let
$u$ solve the unforced Navier--Stokes equation with pressure $p_{\rm phys}$,
let $\mathsf X$ be divergence-free, and set
\[
 \mathsf D=\partial_t-\nu\Delta,\quad k(t)=\mu/t,
 \quad \mathsf Y=(\mathsf D+k)\mathsf X,
 \quad \mathsf h=[u,\mathsf X].
\]
Repeated indices are summed, and $(D\mathsf X)_{ij}=\partial_j\mathsf X_i$.
The product rule yields
\[
 (\mathsf D+k)\mathsf h_i
 =[\mathsf Du,\mathsf X]_i+[u,\mathsf Y]_i-2\nu\mathsf C_i,
 \quad
 \mathsf C_i=\sum_{j,l}
 \bigl[(\partial_ju_l)\partial_{jl}\mathsf X_i
       -(\partial_j\mathsf X_l)\partial_{jl}u_i\bigr].
\]
For $N=u\cdot\nabla u$, expansion and cancellation give
\[
 -[N,\mathsf X]_i+u_j\partial_j\mathsf h_i
             +\mathsf h_j\partial_ju_i
       =u_ju_l\partial_{jl}\mathsf X_i.
\]
The pressure identity is
\[
 -[\nabla p_{\rm phys},\mathsf X]_i
 =\partial_i(\mathsf X_j\partial_jp_{\rm phys})
  -(\partial_j\mathsf X_i+\partial_i\mathsf X_j)
                                      \partial_jp_{\rm phys}.
\]
Using $\mathsf Du=-N-\nabla p_{\rm phys}$ and
$[u,\mathsf Y]=-(u\cdot\nabla\mathsf Y+
\mathsf Y\cdot\nabla u)+2u\cdot\nabla\mathsf Y$, these formulas
prove the desired identity. The global Leray projection removes
precisely $\nabla(\mathsf X\cdot\nabla p_{\rm phys})$.

Now substitute
\[
 u=t^{-1/2}U(y,\tau),\quad
 \mathsf X=\sqrt t\,X(y,\tau),\quad
 p_{\rm phys}=t^{-1}p(y,\tau),\quad
 y=x/\sqrt t,\quad\tau=\log t.
\]
Then $\mathsf Y=t^{-1/2}D_{\mu+1}X$ and
$\mathsf h=t^{-1/2}[U,X]$; every term in the differential identity
has factor $t^{-3/2}$. Consequently, in profile variables,
\begin{equation}\label{ward:physical-identity}
 D_\mu h+\mathbb P\operatorname{div}M_Uh
 =-\mathbb P\operatorname{div}M_UY+\mathbb P\widetilde E_\mu,
\end{equation}
where the full unprojected residual is
\begin{align}
 \widetilde E_\mu
   &=2U\cdot\nabla Y-2\nu C_{\rm diff}
       +(U\otimes U):D^2X-(DX+DX^{\mathsf T})\nabla p,
       \label{ward:residual}\\
 (C_{\rm diff})_i
   &=\sum_{j,l}\bigl[(\partial_jU_l)\partial_{jl}X_i
              -(\partial_jX_l)\partial_{jl}U_i\bigr].\notag
\end{align}
All four terms are retained. In particular the pressure is the exact
global pressure of $U$, including the repair. A spatially constant
displacement has zero residual; spatial derivatives of the returned
displacement and velocity produce the remaining force.

For actual graph inputs this identity holds distributionally by
symmetry-preserving smooth approximation of $w$, followed by the
multiplier, moment, repair-product and separated-kernel bounds.
The energy field $h+S_\mu M_U(Y+h)$ has distributional
$D_\mu$ equal to $\mathbb P\widetilde E_\mu$.
The actual-range body estimates below show
$\widetilde E_\mu\in Y_2$: the repair part is bounded directly,
and the thin part differs from its Gaussian affine body by a
$Y_2$ error. Free weak uniqueness therefore gives
\[
 h+S_\mu M_U(Y+h)=R_\mu\mathbb P\widetilde E_\mu.
\]
Multiplying by $M_U$ and adding $M_UY$ yields precisely the
stress identity used in the final return:
\begin{equation}\label{ward:stress-normal}
 \mathcal D_\mu^T M_U(Y+h)
       =M_UY+M_UR_\mu\mathbb P\widetilde E_\mu.
\end{equation}

\subsection{Support and invertibility on the actual range}

Both $Y$ and $h$ are divergence-free and supported in $s\le1/\nu$:
$D_{\mu+1}$ preserves the divergence-free condition, the bracket of divergence-free
fields is divergence-free, and these operations are local after the
stream cutoff. No projection is applied to $Y+h$. Thus
\begin{equation}\label{ward:support-zero}
 \chi_oM_U(Y+h)=0.
\end{equation}
The change $H_oT=M_U(Y[T]+h[T])$ maps the actual multiplier
range to itself by the multiplier bounds. It depends only on
$\chi_oT$ and has young support, so $H_o^2=0$ and its inverse
change is $I-H_o$. The same argument applies after adding the
later affine correction: it depends only on old data and is
supported in $s\le1.1/\nu$, still disjoint from the old source.
No smallness of these changes is required.

In particular a nonzero actual stress $T$ gives a nonzero
$T'=(I+H)T$. If the transformed equation gives
$\mathcal D_\mu^T T'=0$, then $b=-S_\mu T'$ satisfies
$T'=M_Ub$ and is a nonzero divergence-free energy field. This implication
uses neither injectivity of $M_U$ nor an isomorphism of full ambient
velocity and stress kernels. The final return below supplies the
nonzero $T$ and retains both stream transitions, the full pressure,
and all old repair terms in its exact error decomposition.

\section{A direct map from young forcing to the old stress}
\label{sec:young-old-direct}

This section turns the actual train's forward estimate and the
all-age outputs of Section~\ref{sec:canonical-transfer} into a
uniform map on young forcing. The physical pressure and curved-space
comparison are treated separately.

Put $a(s)=1+\nu s$, $z(s)=s/a(s)$,
$\theta(s)=1/[\nu a(s)]$, $e=(\nu c^2)^{-1}$, and $c=\nu^{-2}$.
Fix $K\Subset\{0<\operatorname{Re}\mu<1/4\}$, $1<\alpha<1.2$,
and a fixed $\Sigma>\alpha$.
Write $H=\Sigma/\nu$, $T_f=\alpha/\nu$, and
$u_o=\nu^{-3/2}$. The auxiliary time $s$ is distinct from physical
logarithmic time. All cell norms use the unnormalized spatial measure.
The weighted vorticity space is the centrally even, zero-circulation
subspace $X_{m,c}$, with a fixed $m>8$; its first moments vanish.

Let $\mathcal U(s,t)$ be the actual train evolution, and let
$q_{\pm,s}$ and $\ell_{\pm,s}$ be the heat-scaled, periodized frozen
Gaussian pair and its finite Gram-normalized dual. This Gram normalization
only makes $\ell_{i,s}q_{j,s}=\delta_{ij}$; it is not a terminal
measurement. Set
\[
 F_\pm(s,t)=
 \exp\!\left[\lambda_\pm(z(s)-z(t))\right]
 \left(\frac{a(s)}{a(t)}\right)^{\kappa_\pm},
 \quad(\lambda_+,\kappa_+)=(\lambda,\kappa),
 \quad(\lambda_-,\kappa_-)=(\bar\lambda,\bar\kappa).
\]
The two diagonal factors have equal modulus. On $0\le s\le H$,
the direct/dual residual is $\delta_c$ and the following are the
retained block estimates: relative complementary propagation is
$Ce^{-g(s-t)}$, the direct amplitude error is $C\delta_c$, and both
amplitude/complement couplings are $C(\nu+\delta_c)$. One has
\[
 \delta_c=C_\Sigma(c^{-2}+\delta_{\rm base}),\qquad
 \delta_{\rm base}=P_\nu e^{C_\Sigma/\nu-b_\Sigma c^2},\qquad
 \eta_\nu=\nu+\delta_c/\nu=O(\nu).
\]
These bounds follow from the block construction in
Proposition~\ref{prop:short-full-clock}: on $a\le1+\Sigma$,
$\theta^{-1}=\nu a$, $d^{-2}=e/\theta=O_\Sigma(c^{-2})$,
and conversion to physical auxiliary time divides the generator by
$a^2$. Its two diagonal entries are
$\lambda_\pm/a^2+\kappa_\pm\nu/a$. Moreover
$z(s)-z(t)\ge(s-t)/(1+\Sigma)^2$, and the endpoint heat-scaling
maps have uniform norms on this finite band. Thus the moving
complementary decay is $Ce^{-g_\Sigma(s-t)}$ and the stated
direct and coupling errors follow.

Divide the two amplitudes by $F_\pm(s,t)$ and the complement by
their common modulus. If $A$ and $R$ are the resulting amplitude
and complementary norms, variation of constants gives
\[
 R(s)\le Ce^{-g(s-t)}R(t)
       +C(\nu+\delta_c)\int_t^s e^{-g(s-r)}A(r)\,dr.
\]
Insert this estimate in the amplitude equation. Its deviation from
the initial amplitude is bounded by
\[
 C\left[\delta_c(s-t)+(\nu+\delta_c)
                 +(\nu+\delta_c)^2(s-t)\right]
       (|A(t)|+R(t))
\]
times a Gronwall factor bounded uniformly on $s-t\le\Sigma/\nu$.
Indeed the coefficient of that factor is
$C[\delta_c+(\nu+\delta_c)^2]$, whose integral is bounded and
tends to zero. Consequently, since $H-t\ge(\Sigma-\alpha)/\nu$ for
$0\le t\le T_f$,
\begin{equation}\label{eq:young-frozen-pair}
 \mathcal U(H,t)=
 \sum_{j=\pm}F_j(H,t)q_{j,H}\ell_{j,t}
       +|F_+(H,t)|\,O_{X_{m,c}\to X_{m,c}}(\eta_\nu).
\end{equation}
The exponentially decaying initial complement is absorbed in
$\eta_\nu$; the full error is not claimed exponentially small.
Keeping that complement in the same inequalities gives the full
gain bound on every subinterval.

Define the actual bare columns by
\[
 \phi_{\nu,+}(s)=\mathcal U(s,0)P_cq,\qquad
 \phi_{\nu,-}(s)=\overline{\phi_{\nu,+}(s)}.
\]
Using \eqref{eq:young-frozen-pair} at $t=0$ shows
\[
 \phi_{\nu,j}(H)=F_j(H,0)q_{j,H}
                          +|F_+(H,0)|O(\eta_\nu).
\]
Thus the same estimate may be written with these actual bare columns,
with coefficient $F_j(t,0)^{-1}\ell_{j,t}$ and the same relative
$O(\eta_\nu)$ error. This is an approximate covector derived directly
from the frozen pair; no exact transported observer is used.

\subsection{A uniform tested estimate for all later ages}

Fix the large clock $b$ used in the long-clock comparison.
The kinetic, tested-moment and absolute-stress impulse estimates
\eqref{eq:ct-kinetic-impulse}, \eqref{eq:ct-tested-impulse} and
\eqref{eq:ct-absolute-impulse} apply uniformly through overlap.
Only their value at $r=b$ is needed here.

For a state at $H$, \eqref{eq:short-causal-long-forward}
propagates it to $b$ with bound
\[
 \|a_b^2w(s_b,\sqrt{a_b}\,\cdot)/N_\nu\|_{X_{m,d(b)}}
   \le C_b |F_+(H,0)|^{-1}\|w(H)\|_{X_{m,c}},
 \qquad N_\nu=e^{\lambda/\nu}\nu^{-\kappa}.
\]
Above $b$ the same estimate has an exponentially decaying factor
$e^{-\gamma\theta}\theta^{-\operatorname{Re}\kappa}$.
At $b$, the uniform Biot--Savart and moment embedding applies since
$d(b)\ge1$. Equations \eqref{eq:ct-tested-impulse} and
\eqref{eq:ct-absolute-impulse} therefore show that every old
weighted moment and absolute stress functional used in the leading
strain calculation has norm at most
\begin{equation}\label{eq:young-later-output}
 C_K |F_+(H,0)|^{-1}\|w(H)\|_{X_{m,c}}
\end{equation}
after division by its canonical scale
$|N_\nu|\nu^{3+2\operatorname{Re}\mu}/c$.
No ordinary inverse factor or logarithmic loss occurs here.

\subsection{Young source theorem and angular signs}

Let $f$ be supported in $0<s\le T_f$ and valued in $X_{m,c}$.
Consider the zero-entrance causal equation
\[
 (\partial_s+\mu/s)w=L_{\nu,c}(s)w+f(s),
 \qquad s^\mu w(s)\longrightarrow0\quad(s\downarrow0).
\]
The tested source norm below defines a causal old-output theorem;
it alone does not imply a physical energy entrance. For the Gaussian
affine sources actually used here, $f(s)=O(s^{3/2})$ in the fixed
smooth weighted norms near zero. The local bounded forward evolution
then gives $w(s)=O(s^{5/2})$ by Duhamel, uniformly for $\mu\in K$.
Thus its planar velocity has finite entrance energy
$\int_0^1s^{-3/2}\|K_cw(s)\|_2^2ds<\infty$; the same fixed
derivative bounds give the required local derivative integrability.
The curved physical reconstruction still uses the separate free graph
and comparison theorem. A general source in the tested completion
must not be assigned these stronger properties without checking them.
Define the finite tested norm and explicit frozen covectors
\begin{align*}
 \|f\|_{\mathcal I_{\nu,\mu}}
  &=\int_0^{T_f}t^{\operatorname{Re}\mu}|F_+(t,0)|^{-1}
                         \|f(t)\|_{X_{m,c}}\,dt,\\
 a_{\nu,\pm}(f)
  &=\int_0^{T_f}t^\mu F_\pm(t,0)^{-1}\ell_{\pm,t}f(t)\,dt.
\end{align*}
Their bounds are viscosity-independent:
$|a_{\nu,+}|+|a_{\nu,-}|\le C_K\|f\|_{\mathcal I_{\nu,\mu}}$.
Since $a(t)\le1+\alpha$ on the source band, the norm is bounded by
$C_K\int_0^{T_f}t^{\operatorname{Re}\mu}e^{-\gamma t/(1+\alpha)}
\|f(t)\|\,dt$.
For a spectral scan, keep the source space fixed: choose a slightly
larger compact strip with endpoints $\sigma_-<\operatorname{Re}\mu
<\sigma_+$ and use $t^{\sigma_-}$ on $t\le1$ and
$t^{\sigma_+}$ on $t\ge1$ in the defining norm. The strict margins
absorb powers of $|\log t|$, so the displayed covectors and tested
outputs are holomorphic in $\mu$ for holomorphic source families.
All geometry, coefficients and cutoffs remain fixed during this scan.

Duhamel and \eqref{eq:young-frozen-pair} give exactly, at $H$,
\[
 H^\mu w(H)=
 a_{\nu,+}(f)\phi_{\nu,+}(H)
 +a_{\nu,-}(f)\phi_{\nu,-}(H)+r_H,
 \quad
 \|r_H\|\le C_K\eta_\nu|F_+(H,0)|
                                  \|f\|_{\mathcal I_{\nu,\mu}}.
\]
For $s\ge H$ both sides evolve by the same unforced equation.
Apply \eqref{eq:young-later-output}: the normalized old moment and
absolute stress errors are at most
$C_K\eta_\nu\|f\|_{\mathcal I_{\nu,\mu}}$.
This is an operator statement on the specified source class, not
only a calculation for one chosen forcing.

Put $J(\mu)=\mathcal J(\mu)$, $J_+(\mu)=J(\mu)$ and
$J_-(\mu)=\overline{J(\bar\mu)}$, and set
$\mathcal A_\nu=|N_\nu|\nu^{3+2\operatorname{Re}\mu}/c$.
The canonical Mellin limit and stress defect, with the opposite
angular identities of the two harmonics, give uniformly on the source class
\begin{align}
 I_T^\mu(w)
  &=-\frac{3\pi i}{4c}(\mu+3/2)\nu^{3+2\mu}
      \bigl[a_{\nu,+}(f)N_\nu J_+(\mu)
          -a_{\nu,-}(f)\overline{N_\nu}J_-(\mu)\bigr]
       +o_K(\mathcal A_\nu)\|f\|_{\mathcal I_{\nu,\mu}}.
       \label{eq:young-old-anisotropy}
\end{align}
Here $I_T^\mu$ is the integral of
$(2E_q-E_\eta)s^{-3/2}$, where, for the already Floquet-weighted
solution $w$ in this theorem,
\[
 E_\eta=\frac2c\int v_1(K_cw)_1,\qquad
 E_q=\frac2c\int v_2(K_cw)_2,\qquad
 C=\frac1c\int[v_1(K_cw)_2+v_2(K_cw)_1].
\]
These are bilinear pairings, without complex conjugation or another
factor $s^{-\mu}$.
For real $\mu$ and real forcing, $a_-=\bar a_+$, so the leading
term of \eqref{eq:young-old-anisotropy} is
$[3\pi/(2c)](\mu+3/2)\nu^{3+2\mu}
\operatorname{Im}(a_{\nu,+}N_\nu J(\mu))$.
The Floquet parameter is not conjugated in the negative channel
when it is complexified.

The source is zero before the old analytical tail begins, since
$T_f<u_o$. Thus the old moment law is homogeneous. Its lower endpoint
is exponentially small after division by $N_\nu$ and its upper
weighted endpoint vanishes; the old cutoff transition below $u_o$
is retained in its separate earlier-old contribution.

\section{The signed old-stress response to young affine forcing}

Fix a real compact interval $K\Subset(0,1/4)$ and $c=\nu^{-2}$.
Cell $L^2$ norms use the unnormalized measure $dq\,d\eta$ on
$(-c/2,c/2)\times\mathbb R$; stress averages contain $c^{-1}$.
Velocity components are ordered $(r,\phi,z)$, with zero azimuthal velocity.
The vorticity sources are centrally even and have zero circulation.
Let
\[
 V_G(x,s)=a(s)^{-2}e^{-|x|^2/(4a(s))}Jx,\qquad a(s)=1+\nu s,
 \qquad V_j=V_G(q-jc,\eta,s).
\]
Write $E_s=\operatorname{curl}V_G(\cdot,s)$, and let $H_{s,c}$ and
$R_{s,c}$ be the periodizations of $rE_s'(r)e^{2i\vartheta}$ and
$rE_s'(r)+2E_s(r)$ respectively. The Gaussian defines the source;
the evolution operator $L_{\nu,c}(s)$ uses the actual cylinder train.
Take $\chi_f(s)=\widehat\chi_f(\nu s)$ with $\widehat\chi_f=1$ on
$[0,1]$ and support in $[0,1.1)$, and take $\chi_o=0$ for
$s\le1.2/\nu$ and $\chi_o=1$ for $s\ge1.5/\nu$. Set
\[
\begin{gathered}
 I_T(M)=\int_0^\infty\chi_o(s)s^{-3/2}
       \frac1c\int_{\rm cell}(2M_{zz}-M_{rr})\,dq\,d\eta\,ds,
 \qquad \ell_\mu(M)=\frac{\epsilon^3I_T(M)}{8(\mu+3/2)},\\
 N_\nu=e^{\lambda/\nu}\nu^{-\kappa},\quad
 \gamma=\operatorname{Re}\lambda,\quad \eta_\nu=\nu+\delta_c/\nu,
 \quad \mathcal A_\nu=|N_\nu|\nu^{3+2\mu}/c,\quad
 A_*=\epsilon^3\mathcal A_\nu.
\end{gathered}
\]

Fix $0<a<\gamma/16$. Let $\mathscr B_a$ consist of measurable real
trace-free matrices $B(s)$, with no mixed azimuthal entries, supported
in $0<s<1.1/\nu$, with norm
\[
 \|B\|_a=\mathop{\rm ess\,sup}_{s>0}e^{-as}s^{-3/2}|B(s)|.
\]
For $x_j=(q-jc,\eta)$ define the full affine force
\[
 f_B=-P_0\sum_j[(B_{\rm mer}x_j)\cdot\nabla V_j+B V_j].
\]
Decompose
\[
 B_{\rm mer}=\beta I+\begin{pmatrix}d&e\\e&-d\end{pmatrix}
       +\omega\begin{pmatrix}0&-1\\1&0\end{pmatrix},
 \qquad B_{\phi\phi}=-2\beta.
\]
The antisymmetric contribution is a periodic gradient, annihilated
by $P_0$. The vorticity source is
\[
 g_B=-\tfrac12(d-ie)H_{s,c}-\tfrac12(d+ie)\overline{H_{s,c}}
          -\beta R_{s,c}.
\]
Let
\[
 \xi_B(s)=s^{-\mu}\int_0^s t^\mu\mathcal U_{\nu,c}(s,t)g_B(t)\,dt,
 \qquad b_B=K_c\xi_B,\qquad M_B=v\otimes b_B+b_B\otimes v.
\]
Since $g_B=O(s^{3/2})$ near zero, $\xi_B=O_{X_{m,c}}(s^{5/2})$;
this causal column has finite-energy entrance. Define its limiting amplitude
\[
 a_0(B)=-\tfrac12\ell_+(H_0)
              \int_0^\infty t^\mu e^{-\lambda t}(d(t)-ie(t))\,dt.
\]

\begin{proposition}[Uniform affine-source return]
There is a canonical modulus $\omega_{\rm can}(\nu)\to0$ such that,
uniformly on $K$ and $\mathscr B_a$,
\begin{equation}\label{flat-return:class-amplitude}
 |a_{\nu,+}(B)-a_0(B)|\le C_{K,a,\Sigma}\eta_\nu\|B\|_a,
\end{equation}
where $a_{\nu,+}$ is the frozen-source amplitude of the young-source theorem.
For every fixed $0<\gamma'<\gamma$,
\begin{align*}
 \biggl|\ell_\mu(M_B)
  +\frac{3\pi}{16}\frac{\epsilon^3\nu^{3+2\mu}}c
       \operatorname{Re}[i a_0(B)N_\nu J(\mu)]\biggr|
 \le C_{K,a,\Sigma}A_*\bigl[&\omega_{\rm can}+\eta_\nu
       +P_\nu e^{-\gamma'/\sqrt\nu}\bigr]\|B\|_a.
\end{align*}
For $k=0,1$ one also has
\begin{equation}\label{flat-return:all-moments}
 \int_0^\infty\chi_o(s)s^{-3/2-k/2}\frac1c
       \int_{\rm cell}|M_B(s,q,\eta)|\,dq\,d\eta\,ds
 \le C_{K,a,\Sigma,k}\mathcal A_\nu\nu^k\|B\|_a.
\end{equation}
\end{proposition}
\begin{proof}
The explicit amplitude is
\[
 a_{\nu,+}(B)=\int_0^{1.1/\nu}t^\mu F_+(t,0)^{-1}
                            \ell_{+,\nu t,c}g_B(t)\,dt,
 \qquad F_+(t,0)^{-1}=e^{-\lambda t/(1+\nu t)}(1+\nu t)^{-\kappa}.
\]
The direct/dual Gaussian comparisons and heat scaling give
\[
 \ell_{+,\nu t,c}H_{t,c}=\ell_+(H_0)+O(\nu t+c^{-2}),\qquad
 \ell_{+,\nu t,c}\overline{H_{t,c}},\ \ell_{+,\nu t,c}R_{t,c}=O(c^{-2}),
\]
\[
 |F_+(t,0)^{-1}-e^{-\lambda t}|
       \le C\nu(t+t^2)e^{-\gamma t/3},\qquad 0\le t\le1.1/\nu.
\]
The resulting error has integrable majorant
$C(\eta_\nu+\nu(t+t^2))t^{\mu+3/2}e^{-(\gamma/3-a)t}\|B\|_a$,
proving \eqref{flat-return:class-amplitude}. The radial and opposite
harmonic sources have zero limiting projection; their source norms
need not be small.

Apply the young-source theorem with $\alpha=1.1$. Its tested
source norm is at most $C\|B\|_a$, since
$t/(1+\nu t)\ge t/2.1$ on its support. Thus
\eqref{eq:young-later-output} transports the finite-band error to the
whole old output without a polynomial loss. The real form of
\eqref{eq:young-old-anisotropy} gives
\[
 I_T(M_B; s\ge u_o)=\frac{3\pi}{2c}(\mu+3/2)\nu^{3+2\mu}
      \operatorname{Im}[a_{\nu,+}(B)N_\nu J(\mu)]
      +O_K(\mathcal A_\nu(\eta_\nu+\omega_{\rm can}))\|B\|_a,
 \quad u_o=\nu^{-3/2}.
\]
The earlier-old part has bound
$\mathcal A_\nu P_\nu e^{-\gamma'/\sqrt\nu}\|B\|_a$ by the full
forward gain estimate. The old cutoff acts on the full tensor.
The source ends before that cutoff begins, so the old moment law is
homogeneous; both weighted endpoints were retained in the canonical
calculation. Substitute \eqref{flat-return:class-amplitude} and divide
by $8(\mu+3/2)/\epsilon^3$ to obtain the scalar estimate.

For the absolute moments use
$s^{-k/2}=\nu^k\theta^{k/2}(1-\nu\theta)^{-k/2}$.
On $\theta\le b$, apply \eqref{eq:ct-absolute-impulse} with
$p=\mu+3/2+k/2$. On $b\le\theta\le\theta(u_o)$ the denominator
is bounded and the exponential long-clock factor absorbs
$\theta^{k/2}$. The earlier-old exponential bound absorbs its fixed
polynomial age factors. Apply these estimates to both bare columns
and to the uniform frozen-pair error, using the state-at-$H$ bound
\eqref{eq:young-later-output}. This proves the absolute estimates on
the whole source unit ball.
\end{proof}

\begin{corollary}[Unit signed coefficient]\label{flat-return:coefficient}
Put
\[
\begin{gathered}
 D_0=\operatorname{diag}(12,-3,-9),\qquad
 A_\mu(s)=\frac{\chi_f(s)s^{5/2}}{\mu+5/2}D_0,\qquad
 B_\mu=A_\mu'+\frac\mu s A_\mu=r_\mu(s)D_0,\\
 r_\mu(s)=\chi_f(s)s^{3/2}
      +\frac{\chi_f'(s)s^{5/2}}{\mu+5/2},\qquad
 L(\mu)=\ell_+(H_0)\frac{\Gamma(\mu+5/2)}{\lambda^{\mu+5/2}},
 \qquad \mathcal C(\mu)=\frac{63\pi i}{128}L(\mu)J(\mu).
\end{gathered}
\]
Let $(f_\mu,\xi_\mu,b_\mu,M_\mu)=(f_{B_\mu},\xi_{B_\mu},b_{B_\mu},M_{B_\mu})$.
For some $b>0$,
\[
 \left|\ell_\mu(M_\mu)-2\epsilon^3\nu^{5+2\mu}
                 \operatorname{Re}[\mathcal C(\mu)N_\nu]\right|
 \le C_KA_*[\omega_{\rm can}+\eta_\nu+
               P_\nu e^{-b/\nu}+P_\nu e^{-\gamma'/\sqrt\nu}].
\]
\end{corollary}
\begin{proof}
Here $\beta=3r_\mu/2$, $d=21r_\mu/2$ and $e=\omega=0$, so
\begin{equation}\label{flat-return:source}
 g_\mu=-\frac{21}{4}r_\mu(H_{s,c}+\overline{H_{s,c}})
              -\frac32r_\mu R_{s,c}.
\end{equation}
This retains the trace of the $(r,z)$ block $3r_\mu$ and hoop entry $-3r_\mu$.
The unit $\mathscr B_a$ norm is bounded. Its cutoff derivative is
supported above $1/\nu$, so the explicit Laplace integral gives
$a_0(B_\mu)=-(21/4)L(\mu)+O_K(P_\nu e^{-b/\nu})$.
Substitute this into the general scalar formula and use $c^{-1}=\nu^2$.
\end{proof}

\section{The return on a fixed space of young displacements}

Fix $K\Subset(0,1/4)$, an integer $k=6$, and $a_s>0$.
Choose $a_s<\gamma/16$ and smaller than the fixed exponential reserves
in the curved near comparison. All constants below may depend on $a_s,k,K$.
Let $\mathscr A$ be the real Banach space of $C^k$ matrix functions on
$(0,\infty)$, trace free and with no entry having exactly one azimuthal
index, with norm
\[
 \|A\|_{\mathscr A}
 =\sup_{s>0}e^{-a_ss}s^{-5/2}
       \sum_{j=0}^k|(s\partial_s)^jA(s)|.
\]
This space is independent of $\nu,\epsilon,\mu$. Its elements need not
be symmetric. Choose two fixed smooth cutoffs. The stream cutoff
$\chi_y(s)=\widehat\chi_y(\nu s)$ equals one below $4/(5\nu)$ and
zero above $1/\nu$. The source cutoff $\chi_f$ equals one through
$1/\nu$ and zero above $1.1/\nu$. Put
\[
 A^y=\chi_f A,\qquad
 B_{\mu,\nu}[A]=(A^y)'+\frac\mu s A^y.
\]
Then $\|B_{\mu,\nu}[A]\|_{\mathscr B_{a_s}}
\le C_K\|A\|_{\mathscr A}$, with the source norm of the preceding
section. The matrix cutoff derivative is included in this definition.
Let $\mathfrak T_{\mu,\nu}A$ denote the complete flat stress of the
zero-entrance forced column with Gaussian age force $f_{B_{\mu,\nu}[A]}$.
The actual cylinder train remains in this evolution.

We next define a return map which retains the entire mean-thin kernel.
For an old thin tensor $M$, use its normalized phase mean and set
\[
 E_{ij}(u)=\frac{\chi_o(u)}c
          \int_{\rm cell}M_{ij}(u,q,\eta)\,dq\,d\eta .
\]
For $x'=(\rho\cos\varphi,\rho\sin\varphi)$, $u=\rho^{-2}$,
let $\Sigma_M(x')$ be the Cartesian rotation of the cylindrical tensor
$\rho^{-1}E(u)$, embedded on $x_3=0$. The zero azimuthal row and column
are retained as zero entries; the $rz$ and $zr$ entries are retained.
Define the exact zero-width Newton field by
\[
 V_{\mu,M}(x)=-2\int_0^1t^{2\mu+2}
       \int_{\mathbb R^2}D^3N(x-t(x',0)):\Sigma_M(x')\,dx'\,dt,
 \qquad N(x)=\frac1{4\pi|x|}.
\]
The contraction is the velocity contraction of $\nabla K_0$, with
$K_0=D^2N$ away from the source. Put
\[
 W^{\rm mt}_{\mu,M}(y)=\epsilon^2V_{\mu,M}(\epsilon y),\qquad
 X^{\rm mt}_{\mu,M}
 =\mathcal C_{\chi_y}\bigl(W^{\rm mt}_{\mu,M}
                              -W^{\rm mt}_{\mu+1,M}\bigr),
\]
where the same tensor $M$ is used in the shifted field. The stream
cutoff uses the outward primitive, so the field is fixed without an
additional entrance constant. Define
\begin{equation}\label{source-class:L}
 (\mathcal L_{\mu,\nu}M)(s)
   =D_yX^{\rm mt}_{\mu,M}(R/\sqrt s,0,0)
       \quad\hbox{in the }(r,\phi,z)\hbox{ frame}.
\end{equation}
This is the full physical gradient, including its hoop entry and all
stream-cutoff terms. It is trace free, has the required azimuthal
symmetry, and vanishes for $s\ge1/\nu$. The uncut field is a harmonic
gradient, but the cutoff gradient need not be symmetric.
In particular $\chi_f\mathcal L M=\mathcal L M$ exactly. Thus the
source map applied to $\mathcal L M$ introduces no second cutoff.

The fixed-space flat return is
$\mathcal K^{\rm flat}_{\mu,\nu}
  =\mathcal L_{\mu,\nu}\mathfrak T_{\mu,\nu}$.
Define the unit displacement and real row by
\[
 a_\mu(s)=\frac{s^{5/2}}{\mu+5/2}D_0,
 \quad D_0=\operatorname{diag}(12,-3,-9),
\]
\[
 \Lambda_{\mu,\nu}(A)
 =-\frac{3\pi}{16}\epsilon^3\nu^{5+2\mu}
      \operatorname{Re}\bigl[iN_\nu J(\mu)
                       a_0(B_{\mu,\nu}[A])\bigr].
\]
Both real harmonic coordinates enter the complex amplitude $a_0$;
the matrix force still contains its radial and hoop components.

\begin{proposition}\label{source-class:flat-limit}
Use the direct young-source theorem and the Gaussian force calculation
of the preceding section.
Uniformly for $\mu\in K$,
\[
 \|\mathcal K^{\rm flat}_{\mu,\nu}
                 -a_\mu\Lambda_{\mu,\nu}\|_{\mathscr A\to\mathscr A}
 \le C A_*\left[\nu+\eta_\nu+\omega_{\rm can}(\nu)
      +P_\nu e^{-\gamma'/\sqrt\nu}
      +P_\nu e^{-4a_s/(5\nu)}\right].
\]
The constants multiplying $\nu,\eta_\nu,\omega_{\rm can}$ are fixed,
not viscosity polynomials. Also
$\|a_\mu\|_{\mathscr A}\le C$ and
$\|\Lambda_{\mu,\nu}\|\le C A_*$.
The operators are real and continuous in $\mu$ for each fixed geometry.
On every band where $A_*$ stays bounded, the displayed error tends to zero.
For the unit displacement,
\[
 \Lambda_{\mu,\nu}(a_\mu)
 =2\epsilon^3\nu^{5+2\mu}
       \operatorname{Re}\left[\frac{63\pi i}{128}
                               L(\mu)J(\mu)N_\nu\right]
       +O(A_*P_\nu e^{-b/\nu}).
\]
\end{proposition}
\begin{proof}
Write $M=\mathfrak T_{\mu,\nu}A$, with $\|A\|_{\mathscr A}\le1$.
The uniform source theorem gives, for $j=0,1$,
\[
 \int_0^\infty u^{-3/2-j/2}\frac{\chi_o(u)}c
        \int_{\rm cell}|M(u,q,\eta)|\,dq\,d\eta\,du
       \le C_j\mathcal A_\nu\nu^j.
\]
Its earlier-old part $1.2/\nu<u<u_o=\nu^{-3/2}$ instead has
$\mathcal A_\nu P_\nu e^{-\gamma'/\sqrt\nu}$ as a bound, including
any fixed extra age powers. All tensor components are estimated together.

For the analytical tail $u\ge u_o$, the source radius satisfies
$\rho/r\le\nu^{1/4}<1/2$ on $s\le1/\nu$.
The integrated Cartesian tensor has the form
$\operatorname{diag}(T_h,T_h,T_z)$, with
$T_z-T_h=(\pi/2)I_T(M;u\ge u_o)$. The cross tensor has zero
integrated Cartesian moment. The isotropic part contracts to zero,
since $\partial_i\Delta N=0$ off source. Direct differentiation gives
\[
 D^2\partial_{33}N(r,0,0)
     =(4\pi r^5)^{-1}\operatorname{diag}(-12,3,9).
\]
The negative sign in $V_{\mu,M}$ and the dilation integral therefore give
\[
 D_yW_{\mu,M}^{(0)}
    =\frac{\epsilon^3 I_T(M;u\ge u_o)}{8(\mu+3/2)}s^{5/2}D_0.
\]
This calculation includes the hoop derivative $W_r/|y'|$.
Subtracting the shifted field, with the same tensor, supplies the
additional factor $(\mu+5/2)^{-1}$.

Every point in the mean-value remainder lies at distance at least
$r/2$ from zero. The fifth Newton derivative is bounded by $Cr^{-6}$.
Its radial first moment is the $j=1$ moment above. Thus, for all
$0\le j\le k$,
\[
 |(s\partial_s)^jD_y(W^{\rm mt}_{\mu,M}-W^{\rm mt}_{\mu+1,M}
                     -W^{(0)}_{\mu,M}+W^{(0)}_{\mu+1,M})|
       \le C_k\epsilon^3\mathcal A_\nu\nu s^3.
\]
Receiver conormal derivatives act only on the smooth separated kernel;
no derivative of $M$ is taken. This bound includes all higher source
multipoles, without expanding them one at a time. Since
$\sup_{s>0}s^{1/2}e^{-a_ss}<\infty$, its $\mathscr A$ norm is
at most $C_{a_s,k}A_*\nu$.

The same separated-kernel bounds, on the outward primitive ray,
control the stream cutoff and its fixed conormal derivatives. The
analytical-tail transition contribution is bounded by
$P_\nu A_*e^{-4a_s/(5\nu)}$ in $\mathscr A$.
The uncut leading displacement beyond the cutoff has the same bound.
The earlier-old source has a fixed fractional gap even at
$u=1.2/\nu,s=1/\nu$; its entire exact kernel, including the stream
cutoff, is bounded by $P_\nu A_*e^{-\gamma'/\sqrt\nu}$.
Consequently
\[
 \|\mathcal L_{\mu,\nu}M-a_\mu\ell_\mu(M)\|_{\mathscr A}
 \le C A_*[\nu+P_\nu e^{-\gamma'/\sqrt\nu}
                      +P_\nu e^{-4a_s/(5\nu)}].
\]
The uniform scalar estimate in the preceding section replaces
$\ell_\mu(M)$ by $\Lambda_{\mu,\nu}(A)$ and proves the result.

The row is bounded because its Laplace integrand is bounded by
$C t^{\mu+3/2}e^{-(\gamma-a_s)t}$.
For $A=a_\mu$ its positive amplitude is
$-(21/4)L(\mu)+O(P_\nu e^{-b/\nu})$, with the cutoff derivative
included. This gives the stated coefficient. Continuity in $\mu$
follows by dominated convergence in the Laplace and dilation integrals
and in the exact causal formula. One may use a slightly larger compact
interval within $(0,1/4)$ for the dominating old-age powers.
The finite conormal norms are dominated by the same separated kernels,
so this gives operator-norm continuity, not merely pointwise continuity.
\end{proof}

For later use, the smooth extraction map has a bound on arbitrary
old thin tensors. If
\[
 N_0(M)=\int_{1.2/\nu}^\infty u^{-3/2}
                \frac1c\int_{\rm cell}|M|\,dq\,d\eta\,du,
\]
then the separated fourth-derivative kernel and its outward primitive give
\begin{equation}\label{source-class:ambient-L}
 \|\mathcal L_{\mu,\nu}M\|_{\mathscr A}
       \le C_{K,k,a_s}\epsilon^3N_0(M).
\end{equation}
Indeed each fixed $r\partial_r$ derivative costs the same power $r^{-5}$;
the fractional radial gap is fixed on the whole young--old rectangle.
Conormal derivatives of $\widehat\chi_y(\nu s)$ are uniformly bounded.
In the gain stress space, $N_0(M)\le P_\nu e^{\gamma/\nu}
\|g^{-1}M\|_Z$, hence the norm of $\mathcal L$ is polynomial on a
bounded loop-gain band. This estimate uses only the thin component.
It assigns no thin moment to the broad repair tensor.

The curved loop is $\mathcal L D_N^{-1}M_UR_\mu F_A$, with the
original cutoff to the near region and the exact physical source below. The proposition
does not identify that loop with $\mathcal K^{\rm flat}$. The equation
comparison supplies their difference as an exponentially small operator
on $\mathscr A$, after the factor $e^{1.1a_s/\nu}$ is paid. The remaining
velocity witnesses belong to the mixed ambient block. Thus no polynomial
inverse is applied to the $O(\nu)$ or canonical-modulus terms above.

\subsection{The physical affine lifting identity}

Use component order $(r,\phi,z)$ and the full matrix space $\mathscr A$.
Let $\chi_f(s)=\widehat\chi_f(\nu s)$ equal one on $s\le1/\nu$
and zero on $s\ge1.1/\nu$. For $A\in\mathscr A$ set
\[
 A^y=\chi_f A,\qquad B=(A^y)'+\mu A^y/s,\qquad
 M(A)=\sup_{0<s\le1.1/\nu}s^{-5/2}
                \sum_{j=0}^6|(s\partial_s)^jA^y(s)|.
\]
The cutoff derivative is included in $B$; the matrices are not
continued beyond their support. All assertions are uniform for
$\mu\in K\Subset(0,1/4)$ and the fixed cutoff.

Write $a=1+\nu s$, $x_j=(q-jc,\eta)$, $c=\nu^{-2}$, and
\[
 V_j=a^{-2}e^{-|x_j|^2/(4a)}Jx_j,\quad
 p_j=-a^{-3}e^{-|x_j|^2/(2a)},\quad
 \psi_j=-2a^{-1}e^{-|x_j|^2/(4a)},\qquad J(q,\eta)=(-\eta,q).
\]
Embed $V_j=J\nabla\psi_j$ with zero azimuthal component and define
\[
 \begin{split}
 f_A&=-\sum_j[(B_{\rm mer}x_j)\cdot\nabla V_j+BV_j],\\
 e_A&=\sum_j[2BV_j+2\nu A^y:D^2V_j
                         -(A^y+(A^y)^T)\nabla p_j],\\
 h_A^p&=\sum_j[V_j,A^y_{\rm mer}x_j],\\
 \psi_A^p&=\sum_j[({\rm tr}A^y_{\rm mer})\psi_j
                     -(A^y_{\rm mer}x_j)\cdot\nabla\psi_j].
 \end{split}
\]
The full matrix, including its hoop entry, is used before restriction
to the $(r,z)$-plane; symmetry of $A$ is not assumed. The identity
$A^y_{\rm mer}J=J[({\rm tr}A^y_{\rm mer})I-(A^y_{\rm mer})^T]$
gives $h_A^p=J\nabla\psi_A^p$.
The sums and their required derivatives converge by Gaussian decay.
With $\mathcal K=\eta\partial_\eta+2s\partial_s$, put
\[
 \widehat h_A=\sqrt s\,[(h_A^p)_r e_r+
       ((h_A^p)_z+\epsilon(1-\mathcal K)\psi_A^p)e_z],
 \qquad F_A=s^{3/2}f_A,\qquad E_A=s^{3/2}e_A.
\]
The physical cylindrical streamfunction is $r\psi_A^p$, with
$r=R/\sqrt s$, so $\widehat h_A$ is exactly divergence-free, vanishes
near the axis, and has young support and finite energy. The forces
are unprojected; the physical body inverse supplies the Cartesian
Leray projection. All these maps depend linearly on $A$.

\begin{lemma}\label{ret:physical-affine}
There is a real physical body force $\mathfrak r_A$, linear in $A$,
such that on the exact background $U$,
\[
 (L_{{\rm f},\mu}+\mathbb P\operatorname{div}M_U)\widehat h_A
       =\mathbb P(F_A+E_A+\mathfrak r_A).
\]
For $0<\epsilon,\nu\le1$, with $\delta_U=\|U-U_{\rm raw}\|_{W^{1,\infty}}$,
\begin{equation}\label{ret:general-affine-bounds}
 \begin{split}
 R^{-1}(\|F_A\|_{Y_2}+\|E_A\|_{Y_2}
       +\|\widehat h_A\|_{Y_2}+\|\nabla\widehat h_A\|_{Y_2})
       &\le P_\nu M(A),\\
 R^{-1}\|\mathfrak r_A\|_{Y_2}
       &\le P_\nu(\epsilon+\delta_{\rm base}+\delta_U)M(A).
 \end{split}
\end{equation}
Here $\delta_{\rm base}$ is the fixed envelope for the train-to-Gaussian
and cross-center norms on $0\le s\le1.1/\nu$, with the required
finite spatial and conormal derivative budget.
\end{lemma}
\begin{proof}
Write $v$ for the planar train in this calculation. The Cartesian
bracket calculation, with the full matrix retained, gives
\[
 (\partial_s+\mu/s-\nu\Delta)h_A^p
 +(v\cdot\nabla)h_A^p+(h_A^p\cdot\nabla)v
       =f_A+e_A+\nabla\Psi_A+r_{\rm per},\qquad
 \Psi_A=\sum_j(A^y_{\rm mer}x_j)\cdot\nabla p_j.
\]
Here $r_{\rm per}$ consists of distinct-center convection products
and $(v-\sum_jV_j)$ terms. Gaussian summation and the fixed-band
train comparison bound its unnormalized cell norms, with the finite
derivative budget, by $C\delta_{\rm base}M(A)s^{5/2}$.

The exact ring-chart rules are
\[
 \partial_r=\sqrt s(\partial_q-\epsilon\mathcal K),\qquad
 \partial_z=\sqrt s\partial_\eta,\qquad
 (\partial_\tau-y\cdot\nabla/2-1/2)(\sqrt s b)
       =s^{3/2}\partial_s b.
\]
The physical Laplacian of $\sqrt s b$ is $s^{3/2}$ times
\[
 [\Delta_{q,\eta}-2\epsilon(\mathcal K+1)\partial_q
       +\epsilon^2(\mathcal K+1)^2]b-\epsilon^2b_r e_r.
\]
Together with the stream completion, these identities put every
geometric difference into $\epsilon M(A)$ times a fixed Gaussian
polynomial of age size at most $C(s^{3/2}+s^{5/2})$ in age-force units.
The pressure is completed before projection:
\[
 s^{3/2}\nabla_{q,\eta}\Psi_A
 =\nabla_y(s\Psi_A)+\epsilon s^{3/2}(\mathcal K+2)\Psi_Ae_r.
\]
The first term is a global gradient with adequate decay; the second
belongs to the geometric error. Thus the physical pressure tail
remains accounted for without identifying the two Leray projections.
The remaining convection difference is exactly
\[
 (U-U_{\rm raw})\cdot\nabla\widehat h_A
            +\widehat h_A\cdot\nabla(U-U_{\rm raw}),
\]
which is bounded by $\delta_U$ times the asserted energy norm.

The Gaussian cell norms and fixed spatial derivatives are bounded by
$CM(A)s^{3/2}$ for $f_A$ and by $CM(A)s^{5/2}$ for both
$h_A^p$ and $\psi_A^p$.
The residual $e_A$ has size $CM(A)(s^{3/2}+s^{5/2})$.
For body fields the physical conversion is
\[
 R^{-2}\|s^{3/2}g\|_{Y_2}^2
   =\pi\int_0^{1.1/\nu}s^{1/2}
                   \|g(s)\|_{L^2(dq/c\,d\eta)}^2ds.
\]
Thus the relevant age powers, including $s^{7/2}$ and $s^{11/2}$,
are integrable at zero; their upper-end integrals and the normalization
$c^{-1/2}=\nu$ cost only $P_\nu$. The same calculation with the
physical spatial Jacobian $s^{-5/2}$ bounds the lift and its gradient.
The smooth cutoff introduces no boundary source.

For the derivative budget, the bracket uses $A^y$, the stream
completion uses one conormal derivative, and one physical spatial
derivative uses two. Its Laplacian uses three; one further spatial
derivative of the residual uses four. The declared six suffice.
These are body and spatial estimates; no bound for arbitrary pure
phase derivatives is asserted. The maps are continuous in $\mu\in K$.
\end{proof}

There are two uses of this single estimate. For generic $A\in\mathscr A$,
$M(A)\le C e^{1.1a_s/\nu}\|A\|_{\mathscr A}$. For the actual jet
$A=\mathcal LT$, the separated-kernel conormal bound gives instead
$M(A)\le P_\nu\epsilon^3N_1(\chi_oT_1)$, since $\chi_f A=A$.
Together with $\delta_U\le P_\nu\rho$ from the background bounds,
this gives the actual-range remainder without the generic exponential
envelope. For $a_\mu(s)=s^{5/2}\operatorname{diag}(12,-3,-9)/(\mu+5/2)$,
the formulas give the unit affine force used in the signed return.

The unprojected force also satisfies the stronger weighted estimate:
\[
 R^{-1}\|g^{-1}F_A\|_{Y_2}\le P_\nu\|A\|_{\mathscr A},\qquad
 g(s)=\exp[\gamma s/(1+\nu s)].
\]
Indeed $a_s<\gamma/16$ and $s\le1.1/\nu$ give
$g(s)^{-1}e^{a_ss}\le e^{-(\gamma/3-a_s)s}$; inserting this
pointwise bound in the same Gaussian integrals proves the estimate.
Multiplication by $g^{-1}$ is not commuted through the Cartesian
Leray projection; the near comparison keeps that projection explicit.

\section{Repair, pressure and transition errors on the generated range}

Write $R=\epsilon^{-1}$, $c=\nu^{-2}$,
$g(s)=\exp\{\gamma s/(1+\nu s)\}$ and
$N_\nu=e^{\lambda/\nu}\nu^{-\kappa}$, where $\Re\lambda=\gamma>0$.
The geometry satisfies
\[
 \epsilon=C_*\nu^r e^{-d_0/\nu},\qquad d_0=\gamma/3,
 \qquad \epsilon^3|N_\nu|\nu^{5+2\mu}\asymp1.
\]
All constants below are uniform for $\mu$ in a fixed compact subset
of $(0,1/4)$ and for the stipulated compact ranges of $C_*,r$.
The gain-band assertions refer to parameter triples satisfying the
displayed relation. At fixed $\nu,\epsilon$ the final spectral scan
is only an interval of width $O(|\log\nu|^{-1})$; no positive gain
bound is asserted on a nonshrinking spectral interval for that fixed
background.
$P_\nu$ denotes a fixed polynomial in $\nu^{-1}$. The exact
background is $U=V+r_b$, $V=U_{\rm raw}+E\alpha_{\rm bg}$, with the unprojected raw residual and
coefficient bounds of the background construction. In particular
\[
 \|r_b\|_{C_\tau L^{11/4}}+\|r_b\|_{W^{1,\infty}}
  +\sum_{j=1}^2\|D^jr_b\|_{Y_{11/4}}
 \le P_\nu\rho,\qquad
 \rho=\epsilon^{3/11}e^{m_0/\nu}.
\]
Fixed additional viscosity powers are included in $P_\nu$.
For an arbitrary axisymmetric physical velocity $w$ without swirl put
\[
 G(w)=R^{-1}\|g^{-1}w\|_{Y_2}.
\]

\subsection{Normalize the full old stress before inversion}

Let $\chi_o$ vanish on $s\le1.2/\nu$ and equal one on
$s\ge1.5/\nu$. The old source must be
\[
 T_o=\chi_o(U\otimes w+w\otimes U),
\]
including the repair. Define $W_\mu=-S_\mu T_o$,
$X=\mathcal C_\chi(W_\mu-W_{\mu+1})$,
$Y=L_{{\rm free},\mu+1}X$ and $h=[U,X]$, where
$\mathcal C_\chi$ is the divergence-free stream cutoff supported on
$s\le1/\nu$. All operators are applied in physical units here.
The tensor in the two shifted resolvents is identical.

The thin part is controlled by its first transverse moment. If
$w=\mathcal N_v b$ and $A=1.2/\nu$, weighted Cauchy--Schwarz gives
\[
 N_1(\chi_o M_vb)
 \le P_\nu A^{-1/2}\|b\|_H
 \le P_\nu |N_\nu|G(w).
\]
For the repair, use instead the global product estimate
\begin{equation}\label{repair:source}
 F_r:=\|\chi_o(r_b\otimes w+w\otimes r_b)\|_{L^2_\tau L^{22/19}}
 \le P_\nu\rho R|N_\nu|G(w).
\end{equation}
This follows from H\"older's inequality and $g\le e^{\gamma/\nu}$.
The fixed power relating $e^{\gamma/\nu}$ to $|N_\nu|$ is absorbed
in $P_\nu$. No moment is assigned to the repair source.

The general separated kernels give $X,DX,Y\in Y_2$ for both
source classes. For example the repair source-to-$L^2$ bound uses
the order-four kernel in $L^{11/7}$, with norm
$CD^{-23/11}$ outside distance $D$. Hence
$w\mapsto H_ow:=Y+h$ is bounded at fixed parameters.
Its output has young support. Consequently $H_o^2=0$ and
$(I+H_o)^{-1}=I-H_o$ by the disjoint old and young supports.

For $T=M_Uw$, the later full affine correction gives
$\widehat w=w+Y[T]+h[T]-h_{\mathcal LT}$. The added correction has
young support and depends only on old data. Hence
\begin{equation}\label{repair:old-invariance}
 \chi_oM_U\widehat w=\chi_oM_Uw.
\end{equation}
Thus every old-source estimate below applies literally to the new
unknown $w$. No gain norm of the possibly large young correction is
inserted into the old repair product.

\subsection{The repair source and exact pressure give a positive margin}

Let $\widetilde E(U;X,Y)$ be the full unprojected Ward residual,
\[
 2U\cdot\nabla Y-2\nu C_{\rm diff}
 +(U\otimes U):D^2X-(DX+DX^T)\nabla p.
\]
Here $p$ is the exact pressure. For the old repair source, the
pointwise gradient kernel has $L^{22/3}$ norm $CD^{-101/22}$.
At age $s\le1/\nu$ this gives
\[
 \|DX(s)\|_{L^2_\tau L^\infty_y}
 +\|DY(s)\|_{L^2_\tau L^\infty_y}
 \le P_\nu\epsilon^{101/22}s^{101/44}F_r,
\]
with the spatial supremum understood on that age surface. The
global gradient estimate into $L^{22/3}$ uses the kernel exponent
$11/3$ and is
\begin{equation}\label{repair:phase-gradient}
 \|DX\|_{L^\infty_\tau L^{22/3}}
 +\|DY\|_{L^\infty_\tau L^{22/3}}
 \le P_\nu\epsilon^{46/11}F_r.
\end{equation}
The phase supremum follows from the free equations for
$(\partial_\tau-y\cdot\nabla/2)DX$ and $DY$ and the normalized
phase estimate over $P\le1$. The instantaneous projected tensor
divergence is separated from the receiver and has the same kernel
bound. This argument differentiates no source in phase.

To estimate $V\cdot\nabla Y$, use $V=\sqrt{s}v$ and the physical
Jacobian $\pi R^2s^{-5/2}ds\,d\eta$. Its squared normalized norm
is bounded by
\[
 P_\nu\epsilon^{101/11}F_r^2
 \int_0^{1/\nu}s^{34/11}
             \|\sup_q|v(s,q,\cdot)|\|_2^2\,ds.
\]
The integral is finite at zero and polynomial in $\nu^{-1}$.
For the remaining thin-coefficient terms use
$R^{-1}(\|DV\|_2+\|D^2V\|_2+\|V\otimes V\|_2)\le P_\nu$.
Here their products with the returned derivatives use the ordinary
$L^\infty_\tau L^\infty_y$ bound on the cutoff fields, of size
$P_\nu\epsilon^{101/22}F_r$ for a first derivative. To obtain it,
apply the same phase norm argument at its $q=\infty$ endpoint:
dilation is an isometry on $L^\infty$, so along its characteristics
the norm difference is bounded by the integral of the $L^\infty$
norm of $BZ$. Young's inequality bounds both $Z$ and $BZ$ in
$L^2_\tau L^\infty_y$. Thus no product of two merely phase-$L^2$
factors is used for the global pressure term below.
Define the canonical pressure by
$\nabla p_V=-(I-\mathbb P)(V\cdot\nabla V)$. It obeys
$\|\nabla p_V\|_2\le C\|V\|_\infty\|DV\|_2$.
Therefore
\[
 R^{-1}\|\widetilde E(V;X,Y)\|_{Y_2}
 \le P_\nu\epsilon^{101/22}F_r
 \le P_\nu\epsilon^{85/22}|N_\nu|e^{m_0/\nu}G(w).
\]

The exact pressure correction is
\[
 \nabla(p-p_V)=-(I-\mathbb P)
 (V\cdot\nabla r_b+r_b\cdot\nabla V+r_b\cdot\nabla r_b),
\]
whose $Y_{11/4}$ norm is at most $P_\nu\rho$.
By H\"older's inequality, \eqref{repair:phase-gradient} bounds all
products with a phase-average coefficient, including
$D^2r_bDX$ and $DX\nabla(p-p_V)$. They contribute at most
\[
 P_\nu R^{-1}\rho\epsilon^{46/11}F_r
 \le P_\nu\epsilon^{52/11}|N_\nu|e^{2m_0/\nu}G(w).
\]
Terms with $D^2X$ have an additional curvature power. This proves
the old repair bound on the entire source interval, including every
large age and every spatial pressure tail.

For a thin old source, replacing the exact background pressure by
the flat lifted pressure has the same first error size. Indeed the
unprojected residual gives
\[
 \nabla(p_{\rm raw}-p_{\rm lift})
       =-(I-\mathbb P)F_{\rm raw}.
\]
The entrance convection terms and the explicit radial pressure
completion then give
$\|\nabla p-G_{\rm flat}\|_{Y_{11/4}}
 \le P_\nu\epsilon^{3/11}e^{m_0/\nu}$.
The general thin-source estimate is
$\|DX\|_{L^\infty_\tau L^{22/3}}
 \le P_\nu\epsilon^{57/22}N_1$.
H\"older's inequality and $R^{-1}=\epsilon$ give the factor
$\epsilon^{85/22}$. Nonpressure repair and entrance
coefficient changes are bounded in the same way or have a larger
curvature power. Thus the combined error $f_r$ satisfies
\begin{equation}\label{repair:body-error}
 R^{-1}\|f_r\|_{Y_2}
 \le P_\nu |N_\nu|G(w)
 \left(\epsilon^{85/22}e^{m_0/\nu}
       +\epsilon^{52/11}e^{2m_0/\nu}\right).
\end{equation}
All the unprojected forces here have young support, because every
term contains a derivative of $X$ or $Y$. Applying the Leray
projection does not change their curl, so these forces do not create
a direct vorticity-source moment at old ages.

\subsection{Both transition terms and the other age rectangles}

Let the near operator remove only the original far mask
$s<S$, $t>e^\delta s$, with $S=\epsilon^{-b}$ and
$0<\delta<\log(6/5)$. The young-old rectangle is contained in this
mask. Its complement is exactly the late-output part
$(1-\chi_y)F_{\rm far}$ and the early-source part
$\chi_yF_{\rm far}(1-\chi_o)$. Neither is identified with a local
causal inverse. The two differences produced by the stream cutoff
are also retained:
\[
 (\mathcal C_\chi-\chi)W_\mu,
 \qquad [L_{{\rm free},\mu+1},\mathcal C_\chi]
                   (W_\mu-W_{\mu+1}).
\]
They are supported on $0.8/\nu<s<1/\nu$.

On late output, $g(s)^{-1}\le e^{-4\gamma/(9\nu)}$; on early
source, $g(t)\le e^{3\gamma/(5\nu)}$. The general separated
kernel bounds, followed by the actual product estimate
\eqref{repair:source}, give the following weakest curvature powers
for the four source/receiver combinations, in the split stress norm:
\[
 \begin{array}{c|cc}
   &\text{thin receiver}&\text{scaled repair receiver}\\ \hline
 \text{thin source}&\epsilon^2&
      \epsilon^{49/22}e^{m_0/(2\nu)}\\
 \text{repair source}&\epsilon^{63/22}e^{m_0/\nu}&
      \epsilon^{79/22}e^{3m_0/(2\nu)}.
 \end{array}
\]
Each entry has a common fixed polynomial $P_\nu S^k$, and is
multiplied by $G(w)|N_\nu|e^{-4\gamma/(9\nu)}$ for late output,
or by $G(w)e^{3\gamma/(5\nu)}$ for early source. For example,
the thin-to-repair entry uses the order-four kernel from $L^2$ to
$L^{22/3}$, with off-diagonal norm $D^{-23/11}$, followed by
$\sqrt\rho$. The repair-to-thin entry uses the pointwise exponent
$79/22$, followed by $\rho R$. The repair-to-repair entry uses
the source $L^{22/19}$ norm and the $L^{11/3}$ kernel. The
outward stream primitive has the same norm powers, so both
transition terms have the late-output bound.
The thin-to-thin estimate at age zero does not use a first-moment
bound with a fixed positive lower source age. Use the mass estimate
$N_0(a)\le C(a^{-3/4}+a^{-1/2})\|T\|_{Z_1}$ proved below instead.
The separated pointwise velocity is at most
$P_\nu\epsilon^2s^2N_0(e^\delta s)$. After the normalized velocity
lift and thin multiplication, the square root of the output stress
age density is bounded by
\[
 P_\nu\epsilon^2s^{5/4}V_m(s)N_0(e^\delta s)
 \le P_\nu\epsilon^2(s^{1/2}+s^{3/4})V_m(s)\|T\|_{Z_1}.
\]
Its square is integrable at zero, and the integral up to $S$ is
polynomial in $S$. This supplies the first table entry on the full
age interval.

The two weakest exponential margins are
\[
 2d_0-\gamma+4\gamma/9=\gamma/9,
 \qquad 2d_0-3\gamma/5=\gamma/15.
\]
For completeness the thin-to-repair margins are
$37\gamma/198$ and $47\gamma/330$, before its $m_0/2$ loss;
the repair-to-thin margins are $79\gamma/198$ and
$39\gamma/110$, before its $m_0$ loss. Thus, if $\delta T_a$
is the total stress correction from these regions and both stream
transitions,
\begin{equation}\label{repair:age-error}
 \|g^{-1}\delta T_a\|_Z
 \le P_\nu
  \exp\!\left[-\frac{\gamma/15-kbd_0-3m_0/2}{\nu}\right]G(w).
\end{equation}

\subsection{The whole thin geometric residual and the actual affine lift}

We now include the terms remaining in the exact transformed equation,
not just the repair and age errors. Write $T=(T_1,T_2)=M_Uw$ in
the split stress space, so $T_1=M_Vw$ is the thin component and the
physical repair tensor represented by $T_2$ is $M_{r_b}w$.
Let $X_1,Y_1$ denote the response to $\chi_oT_1$ alone and let
$X^{\rm mt},Y^{\rm mt}=D_{\mu+1}X^{\rm mt}$ be its phase-mean,
zero-transverse-width Newton response, with the same stream cutoff.
The uncut Newton fields are harmonic outside the old source disk;
their shifted dilation identity therefore gives the same displayed
commutator formula for $Y^{\rm mt}$. In particular the Laplacian
of the stream-cutoff field is retained even though the uncut fields
are harmonic there.
Set
\[
 A_T(s)=\mathcal L T_1(s)
       =D_yX^{\rm mt}(R/\sqrt s,0,0),\qquad
 B_T=A_T'+\mu A_T/s.
\]
The full Cartesian matrix is retained, including its hoop entry.
The first stream cutoff $\chi_y$ is supported in $s\le1/\nu$;
the affine source cutoff $\chi_f$ equals one there and vanishes for
$s\ge1.1/\nu$. Hence $\chi_f A_T=A_T$ exactly. In particular the
source $F_{A_T}$ uses $B_T$, without a second cutoff derivative.

Put $N=N_1(\chi_oT_1)$. The explicit separated kernels imply,
for $0\le j\le6$,
\begin{equation}\label{repair:actual-jet}
 |(s\partial_s)^j A_T(s)|\le C_j\epsilon^3s^{5/2}N,
 \qquad A_T(s)=0\quad(s\ge1/\nu).
\end{equation}
Indeed the gradient Newton kernel has degree $-5$, and the ratio of
the old source radius to the young receiver radius is at most
$1/\sqrt{1.2}<1$. Each $s\partial_s=-r\partial_r/2$ derivative
keeps degree $-5$ after its factor $r$ is included. The outward
stream primitive has the same degree. Fixed conormal derivatives
of $\widehat\chi_y(\nu s)$ are bounded. This proves the displayed
bound directly, without first passing through the exponentially
weighted smooth-source norm.

For clarity we give the geometric comparison that is needed for
arbitrary thin inputs. One phase primitive in the dilation variable,
with its dilation-one boundary term retained, gives
\[
 \|D_yX_1-D_yX^{\rm mt}\|_{L^2_\tau}
 +\|D_yY_1-D_yY^{\rm mt}\|_{L^2_\tau}
       \le P_\nu\epsilon^4s^{5/2}N.
\]
The same bound includes the finite transverse width and the
heat-to-Newton error: Taylor's formula costs the first transverse
moment, and the strict radial gap holds along every Taylor segment.
The initially stronger age power $s^3$ is bounded by
$P_\nu s^{5/2}$ on the young interval. For $Y$ use the exact formula
\[
 Y=\mathcal C_{\chi_y}W_\mu+
 [D_{\mu+1},\mathcal C_{\chi_y}](W_\mu-W_{\mu+1}).
\]
Apply the spatial kernel comparison to these terms separately.
Thus no phase derivative of the improved bound, or of the source,
is taken. These are normalized phase $L^2$ estimates; the raw-train
coefficients multiplying them are bounded in phase with all the
finite transverse weights used below.

The mean transverse Taylor estimate and the Hessian estimate are
\[
 \begin{split}
 |D_yX^{\rm mt}(s,\eta)-A_T(s)|
 +|D_yY^{\rm mt}(s,\eta)-sB_j(s)|
   &\le P_\nu\epsilon^4s^{5/2}|\eta|N,\\
 \|D_y^2X_1(s,\eta)\|_{L^2_\tau}
   &\le P_\nu\epsilon^4s^3N,
 \qquad B_j=s^{-1}D_yY^{\rm mt}(R/\sqrt s,0,0).
 \end{split}
\]
There is an exact clock correction, including on the cutoff annulus:
\begin{equation}\label{repair:exact-jet-clock}
 B_T=B_j+\frac\nu sD_y\Delta_yX^{\rm mt}(R/\sqrt s,0,0),
 \qquad |B_T-B_j|\le P_\nu\nu\epsilon^5s^{5/2}N.
\end{equation}
To check its sign and constant, differentiate
$Y^{\rm mt}=(B-\nu\Delta+\mu+1/2)X^{\rm mt}$, use
$D_yB=BD_y-D_y/2$, and evaluate the phase-independent matrix at
$z=0$, where $B D_yX^{\rm mt}=sA_T'$. All stream-cutoff terms
are already in $Y^{\rm mt}$; none are omitted from this identity.

The exact ring-chart rules give
$U_{\rm raw}=\sqrt s(v+\epsilon H e_z)$,
$DU_{\rm raw}=s(D_{\rm flat}v+\epsilon C_1)$ and
$D^2U_{\rm raw}=s^{3/2}(D_{\rm flat}^2v+\epsilon C_2)$.
Here $H,C_1,C_2$ have the finite weighted train bounds supplied by
the explicit chart differentiation and the young train construction.
Substitution into the four terms of the Ward residual gives the
affine expression
\[
 s^{3/2}\{2B_Tv+2\nu A_T:D_{\rm flat}^2v
                      -(A_T+A_T^T)\nabla_{\rm flat}p_{\rm cyl}\}.
\]
The error in $2U\cdot\nabla Y$ has size
$P_\nu\epsilon^4s^3N$ times a weighted train coefficient.
The errors in $DXD^2U$, $DX\nabla p$, $DUD^2X$ and
$(U\otimes U):D^2X$ have size
$P_\nu\epsilon^4s^4N$. The clock error has the smaller size
$P_\nu\nu\epsilon^5s^4N$. In physical units the squared body
norm divided by $R^2$ uses the density $\pi s^{-5/2}ds\,d\eta$;
the remaining powers $s^{7/2}$ and $s^{11/2}$ are integrable at
zero and give only viscosity polynomials at $1/\nu$. This also
covers coordinate factors $q-jc$ in an alternative fixed-center
Taylor expansion, since they are Gaussian moments of fixed order.

Replacing the actual train and its decaying pressure by the summed
Gaussian profiles costs $P_\nu\epsilon^3\delta_{\rm base}N$.
Here the young-train comparison supplies the quantitative envelope
\[
 \delta_{\rm base}\le
       P_\nu\exp(C_G/\nu-b_G/\nu^4),\qquad b_G>0,
\]
for the fixed derivative and transverse-weight budget on
$0\le s\le1.1/\nu$, with $C_G,b_G$ independent of the small
parameters. It follows directly from the Gaussian train separation
$c=\nu^{-2}$. This is the bound used when absorbing its contribution
in an exponential curvature margin below.
The exact-pressure and repair-coefficient errors were estimated
above, using the ordinary phase-supremum gradient bound, not a
phase derivative of the extra-$\epsilon$ comparison. Consequently,
with $E_{A_T}$ the Gaussian residual of the physical affine identity,
\begin{equation}\label{repair:thin-complete}
 R^{-1}\|E(U;X_1,Y_1)-E_{A_T}\|_{Y_2}
 \le P_\nu N\bigl(\epsilon^4+
       \epsilon^{85/22}e^{m_0/\nu}
                         +\epsilon^3\delta_{\rm base}\bigr).
\end{equation}
The $85/22$ term retains the entire Cartesian pressure correction.

Apply the explicit physical affine lifting identity to the actual
jet $A_T$. Its proof uses at most three conormal derivatives of the
matrix, so \eqref{repair:actual-jet} gives directly
\begin{equation}\label{repair:actual-affine-remainder}
 R^{-1}\|r_{A_T}\|_{Y_2}
 \le P_\nu\epsilon^3N
       (\epsilon+\rho+\delta_{\rm base}).
\end{equation}
This follows by inserting the displayed conormal bounds into each
Gaussian product in that identity. Geometric terms have one extra
$\epsilon$, the exact-background convection difference is
$(U-U_{\rm raw})\cdot\nabla\widehat h_{A_T}
+\widehat h_{A_T}\cdot\nabla(U-U_{\rm raw})$, and its bound is
$P_\nu\rho$ times the normalized $H^1$ bracket bound. The explicit
lifted pressure potential is a global gradient plus its
$\epsilon$ radial completion. Thus no projection tail is dropped.
There is no factor $e^{1.1a_s/\nu}$ in this actual-range estimate;
that factor is needed only for a generic source in $\mathscr A$.

Since $N\le P_\nu|N_\nu|\|g^{-1}T\|_Z$, the new thin and
affine body errors have exponential margins at least
$\gamma/3$, $19\gamma/66-m_0$ and $\gamma/11-m_0$.
All exceed the weakest age stress margin $\gamma/15$ when the
stipulated losses are small.

\subsection{The direct near solver and an actual-range pair norm}

The direct zero-entrance near inverse satisfies
\[
 \|g^{-1}D_L^{-1}g\|_{Z\to Z}\le P_\nu,
 \quad
 \|g^{-1}M_UR_\mu f\|_Z\le P_\nu R^{-1}\|f\|_{Y_2},
 \quad
 \|g^{-1}S_{\rm near}g\|_{H_T\to H}
       \le P_\nu e^{\eta_\delta/\nu}.
\]
For $D_LT=M_UR_\mu f$, the exact equation gives
\[
 T=M_Uw,\qquad w=R_\mu f-S_{\rm near}T.
\]
It follows that $G(w)\le P_\nu e^{\eta_\delta/\nu}
R^{-1}\|f\|_{Y_2}$. No homogeneous entrance appears. For a general
product datum $M_Uv$, however, the reconstruction is
$w=v-S_{\rm near}T$. Its velocity norm must retain the norm of $v$;
a small stress norm does not imply a small velocity witness.

On the distinguished gain band, \eqref{repair:body-error} has
operator size at most
\[
 P_\nu\left[e^{-(19\gamma/66-m_0)/\nu}
             +e^{-(19\gamma/33-2m_0)/\nu}\right]
\]
from the gain-energy norm to the normalized body norm. For the full
actual affine bracket, \eqref{repair:actual-affine-remainder} and
$\epsilon^3N\le P_\nu A$ give the background-error margin
$\gamma/11-m_0$. Geometric lift errors with an extra $\epsilon$
have the larger margin $\gamma/3$.

Here is the additional estimate needed to propagate actual-range
witnesses. Let $A=\|g^{-1}T\|_Z$, with $T=M_Uw$. For an unweighted
thin stress and $a>0$, its old physical mass divided by $\pi R^2$ is
\[
 N_0(a)=\int_a^\infty t^{-3/2}\int\|T(t,\cdot,\eta)\|_{L^2_q}
                  \,d\eta\,dt
 \le C(a^{-3/4}+a^{-1/2})\|T\|_{Z_1}.
\]
Indeed transverse Cauchy--Schwarz contributes $(1+t)^{1/4}$,
and the remaining squared age integral is
$\int_a^\infty t^{-5/2}(1+t)^{1/2}dt$. Partition the output ages
into logarithmic bins of width $h<\delta/2$. On a bin of upper age
$s_j$, all source ages exceed $e^{\delta-h}s_j$, so there is a
uniform radial gap $D_j\asymp R/\sqrt{s_j}$. The order-four
kernel has spatial $L^2$ norm $CD_j^{-5/2}$. Minkowski and the
mass estimate therefore bound the normalized velocity on that bin by
\[
 C\epsilon^{3/2}(s_j^{1/2}+s_j^{3/4})\|T\|_{Z_1}.
\]
Summing geometric bins down to age zero is convergent. Up to a fixed
polynomial in $S$, the far thin-source velocity operator has norm
$P_\nu S^k\epsilon^{3/2}$ from $Z_1$ to the normalized physical
$L^2$ space. The outward stream primitive obeys the same bound.
This proves the estimate for both stream transitions as well.

Thus the late-output thin velocity has possible growth
$e^{\gamma/(18\nu)}$ after gain weighting, and the early-source
thin velocity has possible growth $e^{\gamma/(10\nu)}$:
\[
 \tfrac32d_0-\gamma+4\gamma/9=-\gamma/18,
 \qquad \tfrac32d_0-3\gamma/5=-\gamma/10.
\]
For an actual repair source, Young's $L^{22/19}\to L^2$ bound
instead gives normalized velocity
$P_\nu S^k\rho\epsilon^{23/11}$ times its gain. Its late and early
margins are respectively $23\gamma/99$ and $31\gamma/165$, before
the $m_0$ loss. These bounds include the full old interval.

Put $\omega_b=kbd_0$, increasing the fixed $k$ if necessary.
The error families proved in this section have a physical velocity
witness $v_e$, with stress $M_Uv_e$, satisfying
\begin{align*}
 \|g^{-1}M_Uv_e\|_Z&\le\delta_A A+\delta_G G(w),\\
 R^{-1}\|g^{-1}v_e\|_{Y_2}&\le\chi_A A+\chi_G G(w),
\end{align*}
where it suffices to take
\begin{align*}
 \delta_A&=P_\nu e^{-(\gamma/15-\omega_b-3m_0/2)/\nu},&
 \delta_G&=P_\nu e^{-(19\gamma/66-\omega_b-3m_0/2)/\nu},\\
 \chi_A&=P_\nu e^{(\gamma/10+\omega_b+3m_0/2)/\nu},&
 \chi_G&=P_\nu e^{-(31\gamma/165-\omega_b-3m_0/2)/\nu}.
\end{align*}
The pressure and Ward body-force contributions use $v_e=R_\mu f_e$,
whose energy norm is bounded by the normalized body norm. Their thin
source bounds use $A$, whereas the repair source bounds use $G(w)$.
The full affine bracket's background error is bounded by
\eqref{repair:actual-affine-remainder} and is included in $\delta_A$.
The actual jet has the direct bound \eqref{repair:actual-jet}; on
the unit gain band $\epsilon^3N_1\le P_\nu A$.

Let $Q=P_\nu$ and $K_s=P_\nu e^{\eta_\delta/\nu}$
be the direct near bounds. For one correction solve
$D_LT_+=-M_Uv_e$ and set $w_+=-v_e-S_{\rm near}T_+$.
Then $T_+=M_Uw_+$ exactly, and
\[
 A_+\le Q(\delta_A A+\delta_G G),\qquad
 G_+\le\chi_A A+\chi_G G+K_s A_+.
\]
This proves closure on the generated range, with no replacement of
the velocity witness by its stress norm. More precisely, use the
closed graph space
\begin{equation}\label{repair:pair-space}
 \mathcal E_\nu=\{(T,w):g^{-1}T\in Z,\ g^{-1}w\in Y_2,
                       \ T=M_Uw\}.
\end{equation}
The velocities in this graph have the required orthogonal symmetry
but need not be divergence-free. Boundedness of the thin multiplier and
the repair multiplier proves that the graph is closed. Thus the pair
norm below makes it a Banach space. The displayed update preserves
this graph exactly; the divergence-free condition is recovered from the full
physical kernel equation, not from the masked near reconstruction.
Choose $a=\gamma/8$ and use the pair norm $A+e^{-a/\nu}G$.
The four matrix coefficients are
\[
 Q\delta_A,\quad Q\delta_Ge^{a/\nu},\quad
 e^{-a/\nu}\chi_A+e^{-a/\nu}K_sQ\delta_A,\quad
 \chi_G+K_sQ\delta_G.
\]
All tend to zero exponentially if
$\eta_\delta,\omega_b,m_0$ are sufficiently small. For example
each less than $\gamma/1000$ leaves a margin larger than
$\gamma/50$ before absorbing fixed viscosity powers. The narrowest
new margin is $a-\gamma/10=\gamma/40$. An initial body-forced
column has pair norm at most $P_\nu e^{\eta_\delta/\nu}$, so
the resulting correction series also tends to zero under the bounded
old smooth-jet observation.

\begin{proposition}\label{repair:complete-graph-error}
Let $\mathcal E_\nu$ have the pair norm just defined. Define the exact error
velocity and its graph correction by
\[
 v_Q(T)=S_{\rm far}T+Y[T]
       +R_\mu\mathbb P\{E[T]-E_{\mathcal LT}-r_{\mathcal LT}\},
 \qquad \mathcal Q(T,w)=-\mathcal J(v_Q(T)).
\]
Here $E[T]$ uses the full old source, $\mathcal LT$ uses its thin
part, and all projections and cutoffs are the ones specified above.
Every term of this exact $v_Q$ is covered by the preceding estimates.
If $\eta_\delta,\omega_b,m_0<\gamma/1000$ and the background
and train envelopes satisfy the stated bounds, then, for sufficiently
small $\nu$ on the distinguished gain band,
\[
 \|\mathcal Q\|_{\mathcal E_\nu\to\mathcal E_\nu}
           \le P_\nu e^{-\gamma/(100\nu)}.
\]
For each fixed geometry this is a real operator-norm-continuous
family in $\mu$ on the stipulated compact interval.
\end{proposition}
\begin{proof}
The decomposition of $S_{\rm far}T+Y[T]$ consists exactly of the
early-source and late-output regions and the two stream transitions.
For the body term, linearity splits the full old response into its
thin and repair responses. The repair response is
\eqref{repair:body-error}; its thin counterpart is
\eqref{repair:thin-complete}; the final subtraction is
\eqref{repair:actual-affine-remainder}. These exhaust the displayed
definition, including finite width, oscillatory phase, the exact
matrix clock, pressure and the physical affine lift. The graph
lift gives precisely the four pair-norm coefficients above.
Their smallest unperturbed exponent is $\gamma/40$. The specified
losses leave more than $\gamma/50$, hence the stated weaker
$\gamma/100$ bound absorbs every fixed viscosity polynomial.
Continuity follows from the resolvent identity, the dominated
separated dilation kernels and the polynomial near inverse; the
products and finite conormal affine construction are continuous.
No differentiation of a rough source in phase is required.
\end{proof}

\section{A return equation with a smooth source and a remainder}

The following real block lemma permits the leading return to be estimated
in its natural smooth-source norm. It does not send an algebraically
small error through an exponentially large general inverse.

\begin{lemma}\label{closure:mixed-return}
For each fixed $n$ and $z\in[-1,1]$, let $E_n,X_n$ be real Banach
spaces and consider the bounded, norm-continuous system
\begin{equation}\label{closure:block-system}
 x=K_n(z)x+A_n(z)y,\qquad
 y=B_n(z)x+C_n(z)y.
\end{equation}
The spaces and all auxiliary geometric parameters are fixed as $z$
varies. Suppose
\[
 K_n=a_n\ell_n+R_n,\qquad
 \sup_z\|a_n(z)\|\|\ell_n(z)\|\le M,
\]
where $a_n\in E_n$, $\ell_n\in E_n^*$ are real and continuous.
Write
\[
 c_n=\sup_z\|C_n(z)\|<1,\quad
 r_n=\sup_z\|R_n(z)\|,
 \quad
 h_n=\sup_z\frac{\|A_n(z)\|\|B_n(z)\|}{1-c_n}.
\]
If $r_n+h_n\to0$ and
\[
 \ell_n(z)a_n(z)\longrightarrow e^{-z}
 \quad\hbox{uniformly for }-1\le z\le1,
\]
then, for every sufficiently large $n$, there is $z_n\in(-1,1)$
for which \eqref{closure:block-system} has a nonzero solution.
Moreover $z_n\to0$. No convergence or compactness of the ambient
inverse on $X_n$ is required.
\end{lemma}

\begin{proof}
The second equation has the unique solution
$y=(I-C_n)^{-1}B_nx$. Substitution into the first gives
\[
 x=a_n\ell_nx+\widetilde R_nx,
 \qquad
 \widetilde R_n=R_n+A_n(I-C_n)^{-1}B_n,
 \qquad \|\widetilde R_n\|\le r_n+h_n.
\]
For large $n$ set $w_n=(I-\widetilde R_n)^{-1}a_n$ and
\[
 d_n(z)=1-\ell_n(z)w_n(z).
\]
The real function $d_n$ is continuous and
\[
 |d_n-(1-\ell_na_n)|
 \le \frac{M(r_n+h_n)}{1-r_n-h_n}.
\]
It therefore converges uniformly to $1-e^{-z}$, which has opposite
signs at $-1$ and $1$. At a zero $z_n$, take
$x=w_n$ and $y=(I-C_n)^{-1}B_nw_n$. Since $\ell_nw_n=1$,
$w_n\ne0$, and both equations hold. Uniform convergence implies
that every such zero tends to zero. This also proves the asserted
nonvanishing without a choice of norm for the large physical mode.
\end{proof}

\section{Closing the exact return and recovering the energy mode}
\label{sec:spectral-assembly}

We combine the preceding estimates on one fixed background. The tensor
inverse acts on the sum of the thin and repair components, and the
multiplier returns both. Write $R_\mu$ for the physical body inverse,
$S_\mu$ for the full tensor inverse, and $M$ for the exact multiplier,
including the prescribed normalization maps.

\subsection{The exact two-block equation}

Put $D=I+MS_\mu$, $D_N=I+MS_N$ and $S_{\rm far}=S_\mu-S_N$.
Use the closed graph $\mathcal E_\nu$ and pair norm from
\eqref{repair:pair-space}. The near graph lift is
\[
 \mathcal J(v)=
 \bigl(D_N^{-1}Mv,\ v-S_ND_N^{-1}Mv\bigr).
\]
Its components satisfy $T=Mw$ exactly.

Let $L$ be the complete old-thin-to-young-jet map
\eqref{source-class:L}. The full old-source Ward fields $X,Y,h,E$
and the physical affine fields, written $h_A=\widehat h_A$ and
$r_A=\mathfrak r_A$ as in Lemma~\ref{ret:physical-affine}, satisfy
\[
 DM(Y+h)=MY+MR_\mu E,\qquad
 DMh_A=MR_\mu(F_A+E_A+r_A).
\]
The nested cutoffs give $\chi_f LT=LT$. Therefore, with
\[
 HT=M(Y+h-h_{LT}),\qquad
 v_Q(T)=S_{\rm far}T+Y+R_\mu(E-E_{LT}-r_{LT}),
\]
the exact transformed equation is
\begin{equation}\label{closure:exact-normal-form}
 D(I+H)T=D_NT-MR_\mu F_{LT}+Mv_Q(T).
\end{equation}
Every correction in $H$ has young support and depends only on old
data. Thus $H^2=0$ and $(I+H)^{-1}=I-H$ at fixed parameters.

Define
\[
 \mathcal P A=\mathcal J(R_\mu F_A),\qquad
 \mathcal Q(T,w)=-\mathcal J(v_Q(T)),\qquad L(T,w)=LT.
\]
The force $F_A$ is unprojected; $R_\mu$ supplies the Cartesian Leray
projection. The return system
\begin{equation}\label{closure:exact-return-system}
 V=\mathcal P x+y,\qquad x=LV,\qquad y=\mathcal QV
\end{equation}
has the two block equations
\[
 x=(L\mathcal P)x+Ly,\qquad
 y=(\mathcal Q\mathcal P)x+\mathcal Qy.
\]
A solution gives zero on the right of
\eqref{closure:exact-normal-form}. These are exact operators, before
any asymptotic replacement.

\subsection{One parameter budget for the two blocks}

Choose $\sigma_0$ and the compact interval $K$ as in the phase-selection
paragraph below before fixing the following constants and budgets.
Write $J$ for the canonical Mellin function of
Corollary~\ref{cor:short-mellin}.
Fix the derivative budgets first, including $k=6$ in $\mathscr A$.
Let $K_0$ and $k_0$ be the fixed exponents in the compact-output and
separated-kernel window bounds. Put $d_0=\gamma/3$. Choose $a,b_{\rm win}>0$
so that
\[
 K_0(a+b_{\rm win})<1/100,\quad k_0b_{\rm win}d_0<\gamma/1000,\quad
 \vartheta=\min\{a/2,b_{\rm win}/2,1/100,3/22\}>0.
\]
Here $b_{\rm win}$ is the small exponent in $S=\epsilon^{-b_{\rm win}}$,
distinct from the large fixed heat-clock endpoint $b$.
Next choose $\delta,m_0>0$ with
\[
 \delta<\log(12/11),\qquad
 \eta_\delta,m_0<\min\{\vartheta d_0/16,\gamma/1000\}.
\]
Fix the transverse-tail accuracy larger than the required losses.
The background construction realizes this $m_0$; all its internal
choices are then fixed. There is no incompatible lower bound on
$a,b_{\rm win},\delta,m_0$ in the preceding estimates.

The polynomial flat inverse and
Proposition~\ref{near:equation-comparison} give, for some $c_N>0$,
\[
 \|g^{-1}D_N^{-1}g\|\le P_\nu,\qquad
 \|g^{-1}(D_N^{-1}-D_0^{-1})g\|\le P_\nu e^{-c_N/\nu}.
\]
Decrease $c_N$ to include the body-source comparison and coefficient
errors. Proposition~\ref{repair:complete-graph-error} gives
\begin{equation}\label{closure:Q-small}
 \|\mathcal Q\|_{\mathcal E_\nu\to\mathcal E_\nu}
       \le P_\nu e^{-c_Q/\nu},\qquad c_Q=\gamma/100.
\end{equation}
This estimate covers all of $v_Q$, including full old repair,
pressure, finite width, oscillatory phase and both stream transitions.
The actual extracted jet has a polynomial age bound, so the estimate
does not pay the exponential envelope of a general source.

Now fix $a_s>0$ so small that
\[
 a_s<\gamma/16,\qquad 1.1a_s<\tfrac14\min\{c_N,c_Q\}.
\]
The general affine bounds and near reconstruction imply
\begin{equation}\label{closure:source-and-extraction}
 \|\mathcal P\|_{\mathscr A\to\mathcal E_\nu}
      \le P_\nu e^{1.1a_s/\nu},\qquad
 \|L\|_{\mathcal E_\nu\to\mathscr A}\le P_\nu .
\end{equation}
For the velocity component use
$e^{-\gamma/(8\nu)}e^{\eta_\delta/\nu}\le1$; the second bound is
\eqref{source-class:ambient-L}. No velocity witness is estimated
solely by its stress.

The curved body-column comparison is uniform over the unit ball
of $\mathscr A$: the six conormal derivatives and the required fixed
Gaussian spatial seminorms of its sources are bounded by
$P_\nu e^{1.1a_s/\nu}$. The global-input body comparison followed
by the near resolvent identity therefore gives
\[
 \|L\mathcal P-\mathcal K^{\rm flat}_{\mu,\nu}\|
       \le P_\nu e^{-(c_N-1.1a_s)/\nu}.
\]
Here the inputs are prescribed smooth affine forces. No canonical
moment error is passed through this inverse comparison.
Proposition~\ref{source-class:flat-limit} yields
\begin{equation}\label{closure:K-rank-one}
 L\mathcal P=a_\mu\Lambda_{\mu,\nu}+\mathcal R_{\mu,\nu},
 \qquad \sup\|\mathcal R_{\mu,\nu}\|\longrightarrow0,\qquad
 \sup\|a_\mu\|\|\Lambda_{\mu,\nu}\|<\infty.
\end{equation}
The suprema are on the final bounded-gain spectral window. The
algebraic and canonical-modulus errors retain their fixed constants.
Moreover
\[
 \frac{\|L\|\|\mathcal Q\mathcal P\|}{1-\|\mathcal Q\|}
 \le P_\nu e^{-(c_Q-1.1a_s)/\nu}\longrightarrow0.
\]
This is the second error in Lemma~\ref{closure:mixed-return}.
All estimates are locally uniform in real $\mu$; kernels, resolvent
identities, the near inverse and affine formulas give operator-norm
continuity for each fixed geometry.

\subsection{Phase selection and the scalar crossing}

Put
\[
 \mathcal C(\mu)=\frac{63\pi i}{128}
 \ell_+(H_0)\frac{\Gamma(\mu+5/2)}{\lambda^{\mu+5/2}}J(\mu),
 \qquad N_\nu=e^{\lambda/\nu}\nu^{-\kappa},
 \qquad \lambda=\gamma+i\omega.
\]
The planar construction gives $\gamma>0$, $\omega\ne0$,
$\ell_+(H_0)\ne0$, and holomorphic, nonidentically zero $J$.
Choose $\sigma_0\in(0,1/4)$ with $\mathcal C(\sigma_0)\ne0$ and
fix a compact interval $K\Subset(0,1/4)$ around it.
Choose $\nu_n\downarrow0$ satisfying
\[
 \frac{\omega}{\nu_n}-\operatorname{Im}\kappa\log\nu_n
       +\arg\mathcal C(\sigma_0)
   =2\pi n\operatorname{sign}\omega.
\]
The left side diverges with sign $\omega$ and has derivative
$-\omega/\nu^2-\operatorname{Im}\kappa/\nu$ of fixed nonzero sign
for small $\nu$, proving existence of this sequence. Set
\[
 \epsilon_n^3=
 \frac{e^{-\gamma/\nu_n}
       \nu_n^{\operatorname{Re}\kappa-5-2\sigma_0}}
      {2|\mathcal C(\sigma_0)|},
 \qquad c_n=\nu_n^{-2},\qquad P_n=2\epsilon_nc_n.
\]
Construct the background once at these parameters. The polynomial
factor in $\epsilon_n$ is allowed throughout; equivalently
$-\nu_n\log\epsilon_n\to d_0$. The background and geometry stay fixed
as
\[
 \mu_n(z)=\sigma_0+\frac{z}{2|\log\nu_n|},\qquad -1\le z\le1,
\]
varies. The loop gain
$\epsilon_n^3|N_{\nu_n}|\nu_n^{5+2\mu_n(z)}$
stays between positive fixed constants. The signed unit response is
\begin{align*}
 \Lambda_{\mu_n(z),\nu_n}a_{\mu_n(z)}
 &=2\epsilon_n^3\nu_n^{5+2\mu_n(z)}
       \operatorname{Re}\{\mathcal C(\mu_n(z))N_{\nu_n}\}+o(1)\\
 &=e^{-z}\operatorname{Re}
       \frac{\mathcal C(\mu_n(z))}{\mathcal C(\sigma_0)}+o(1)
 \longrightarrow e^{-z}
\end{align*}
uniformly. Lemma~\ref{closure:mixed-return} supplies $z_n\to0$
and a real solution of \eqref{closure:exact-return-system} with
$x\ne0$ at $\sigma_n=\mu_n(z_n)\in(0,1/4)$.

Since $x=LT\ne0$, $T\ne0$. Nilpotency gives $T'=(I+H)T\ne0$,
and \eqref{closure:exact-normal-form} gives $DT'=0$. Thus
\[
 b=-S_{\sigma_n}T',\qquad T'=Mb
\]
is nonzero, real, periodic and divergence-free, with the required symmetry
and $b,\nabla b\in Y_2$. No injectivity of $M$ is needed: $b=0$
would imply $T'=0$. The exact identities also give
$b=w+Y+h-h_{LT}$, so the masked terms recombine into the physical
equation with its full pressure. Together with the background and its
heat remainder this proves Proposition~\ref{prop:unstable-background}.
The localization argument below then proves Theorem~A.

\section{From a growing periodic mode to finite-energy nonuniqueness}
\label{sec:localization}

We now use the exact background and growing mode from
Sections~\ref{short-bg:section} and~\ref{sec:spectral-assembly}. This is the instability
and localization mechanism of Jia and \v Sver\'ak~\cite{jia-sverak2015},
adapted here to a profile periodic in similarity time. A divergence-free
cutoff localizes the common DSS datum while preserving it near the
origin. A common heat correction and a family of nonlinear correctors then
produce the finite-energy solutions. The periodic right inverse includes
every faster unstable mode; no maximal-growth assumption is added.

We localize the initial datum and correct the background and its growing
linear perturbation directly. The correction of the common heat tail
decays as $e^{\tau/2}$, while the quadratic mode has size
$e^{2\sigma\tau}$. A periodic spectral splitting at a rate
$\sigma<\beta<2\sigma<1/2$ gives a right inverse below both forcing
rates, without an upper bound on other unstable exponents.

\begin{proposition}\label{short-loc:reduction}
The background and mode in Proposition~\ref{prop:unstable-background}
imply Theorem~\ref{thm:finite-energy-target}.
\end{proposition}

\begin{proof}[Proof of Proposition~\ref{short-loc:reduction} and Theorem~A]
The background construction and the return argument prove
Proposition~\ref{prop:unstable-background}. It remains to localize these
fields and construct the nonlinear solutions. Write $a=u_{\rm dss}$,
$U=U_*$, $Q=r_{\rm heat}$, $b=b_*$ and $P=\mathcal T$.
\paragraph{Periodic evolution and regularity of the mode.}
The map $D_\nu f(y)=\nu^{-1/2}f(\sqrt \nu\,y)$ changes the profile viscosity
to one: it commutes with the dilation drift and with $\mathbb P$,
$\Delta D_\nu f=D_\nu(\nu\Delta f)$, and
$\operatorname{div}(D_\nu f\otimes D_\nu g)
=D_\nu\operatorname{div}(f\otimes g)$. Also
$H_1[D_\nu a]=D_\nu H_\nu[a]$ and
\[
 \|\nabla^kD_\nu f\|_q=\nu^{(k-1)/2-3/(2q)}\|\nabla^kf\|_q.
\]
We apply this fixed normalization and retain the notation $a,U,Q,b$.
All constants below may depend on these fixed profiles and on the
spatial cutoff, but not on the terminal time $\tau_0\le0$ or on
$\alpha,\alpha'\in[0,1]$.

Let $X=H^1_{\rm div}(\mathbb R^3)$, $Z=H^2_{\rm div}(\mathbb R^3)$,
and write
\[
 \mathcal A=\Delta+\tfrac12y\cdot\nabla+\tfrac12,
 \qquad \mathcal B(\tau)f
 =-\mathbb P\operatorname{div}(U(\tau)\otimes f+f\otimes U(\tau)),
 \qquad \mathcal L=\mathcal A+\mathcal B.
\]
Gaussian convolution and the bounds on $a$ give
$|D^jH_1[a](y,\tau)|\le C_j\langle y\rangle^{-1-j}$ for $j\le2$,
uniformly in phase. Hence $U\in C_{\rm per}C_0$,
$\nabla U\in C_{\rm per}L^3$, and
\[
 \mathcal B\in C_{\rm per}\mathcal L(X,L^2_{\rm div})
                 \cap C_{\rm per}\mathcal L(Z,X).
\]
For the first bound use
$\|U\|_\infty\|\nabla f\|_2+\|\nabla U\|_3\|f\|_6$;
for the second use the $H^2$ product estimate on $Qf$ and the
bounded derivatives of $H_1[a]$. The Leray multiplier is bounded on
all the Sobolev spaces used here.

The free evolution is
\[
 S(s)f(y)=e^{s/2}[e^{(e^s-1)\Delta}f](e^{s/2}y).
\]
It has $X\to X$ norm at most $Ce^{-s/4}$, and $L^2\to X$ and
$X\to Z$ norms at most $Cs^{-1/2}$ for $0<s\le1$. The large-time
gradient estimate follows by applying the heat gradient bound to $f$,
rather than differentiating $f$. Volterra iteration therefore constructs
the periodic evolution $E(\tau,s)$ of $\mathcal L$, bounded on
$L^2,X,Z$ over bounded time strips, with these same short-time smoothing
bounds. The iteration in the smoothing norm uses the integrable
convolution of two kernels $r^{-1/2}$.

These evolutions also represent the given weak mode. On a finite time
interval set $g(x,\tau)=b(e^{-\tau/2}x,\tau)$ and
$\widetilde U(x,\tau)=U(e^{-\tau/2}x,\tau)$. Then
\[
 g_\tau-e^\tau\Delta g+(\sigma-\tfrac12)g
 +e^{\tau/2}\mathbb P\operatorname{div}
       (\widetilde U\otimes g+g\otimes\widetilde U)=0.
\]
The energy hypotheses imply $g\in L^2H^1$, and this equation gives
$g_\tau\in L^2H^{-1}$. Its form is bounded on $H^1\times H^1$;
Young's inequality makes it coercive after adding $C\|g\|_2^2$,
since $e^\tau$ has a positive minimum on the interval. The Hilbert
triple energy identity gives a continuous $L^2$ trace and uniqueness
from every initial time. Thus
$b(\tau)=e^{-\sigma(\tau-s)}E(\tau,s)b(s)$ whenever $b(s)\in H^1$.
Choosing such an earlier phase and using smoothing gives
\begin{equation}\label{short-loc:mode-regularity}
 b\in C_{\rm per}H^2,\qquad
 m_b:=\min_{0\le\tau\le P}\|b(\tau)\|_2>0.
\end{equation}
Indeed a zero phase would, by forward uniqueness and periodicity,
make the entire mode zero.

\paragraph{A right inverse on the ancient interval.}
We next construct an ancient right inverse, including every faster
unstable mode. Multiplication by $U(\tau)\in C_0$ is compact
$X\to L^2$, by compactness on balls and the small uniform spatial tail.
Consequently $\mathcal B(\tau):X\to H^{-1}$ is compact.
For positive $r$, $S(r)\mathcal B(\tau):X\to X$ is compact; its
alternative factorization through $L^2$ has norm $O(r^{-1/2})$.
The Duhamel formula applied to a weakly null sequence, followed by
dominated convergence with this integrable bound, proves
$E(t,s)-S(t-s)$ compact on $X$. For the monodromy
$M=E(P,0)$, periodicity and the free decay therefore give
\[
 r_{\rm ess}(M)\le e^{-P/4}<1.
\]
The spectrum outside this disk consists of isolated eigenvalues with
finite algebraic multiplicity. Choose
$\sigma<\beta<2\sigma<1/2$ so that $|z|=e^{\beta P}$ avoids the
spectrum. The Riesz projections $P_-$ inside that circle and
$P_+=I-P_-$ are real, $P_+X$ is finite dimensional, and for some
$\beta_-<\beta<\beta_+$,
\[
 \|M^nP_-\|\le Ce^{\beta_-nP},\qquad
 \|(M|_{P_+X})^{-n}\|\le Ce^{-\beta_+nP},\quad n\ge0.
\]
These are the spectral-radius estimates for the two restrictions;
no inverse of the full parabolic evolution is taken.

For $0\le\theta\le P$ put
\[
 F_\theta=E(\theta,0)|_{P_+X},\qquad
 G_\theta=(M|_{P_+X})^{-1}P_+E(P,\theta),\qquad
 \Pi_+(\theta)=F_\theta G_\theta,\quad\Pi_-=I-\Pi_+.
\]
Here $G_\theta F_\theta=I$, and the endpoints agree. Extend the
projections periodically. They intertwine the evolution. On the
finite-dimensional exterior fibers its backward extension is
\[
 E_+(kP+\theta,lP+\phi)\Pi_+(\phi)
       =F_\theta(M|_{P_+X})^{k-l}G_\phi.
\]
One-period smoothing shows $P_+X\subset Z$, since
$P_+=M(M|_{P_+X})^{-1}P_+$. Factoring general intervals into
periods and bounded phase strips now gives
\[
 \begin{aligned}
 \|E(\tau,s)\Pi_-(s)\|_{X\to Z}
   &\le C(1+(\tau-s)^{-1/2})e^{\beta_-(\tau-s)}&& (\tau>s),\\
 \|E_+(\tau,s)\Pi_+(s)\|_{X\to Z}
   &\le Ce^{\beta_+(\tau-s)}&& (\tau\le s).
 \end{aligned}
\]
If $P_+X=0$, the second expression is zero. For continuous functions
on $(-\infty,\tau_0]$ use the norms
$\|f\|_{X_\beta}=\sup e^{-\beta\tau}\|f(\tau)\|_X$ and
$\|f\|_{Z_\beta}=\sup e^{-\beta\tau}\|f(\tau)\|_Z$. Then
\begin{equation}\label{short-loc:right-inverse}
 (\mathcal R_\beta f)(\tau)
 =\int_{-\infty}^{\tau}E(\tau,s)\Pi_-(s)f(s)\,ds
  -\int_\tau^{\tau_0}E_+(\tau,s)\Pi_+(s)f(s)\,ds,
 \qquad
 \|\mathcal R_\beta f\|_{Z_\beta}\le C_\beta\|f\|_{X_\beta}.
\end{equation}
Both weighted kernels are integrable. The evolution law verifies
$(\partial_\tau-\mathcal L)\mathcal R_\beta f=f$ and the terminal
condition $\Pi_+(\tau_0)\mathcal R_\beta f(\tau_0)=0$.

\paragraph{Divergence-free truncation of the datum and the common heat correction.}
To obtain a compact datum, define
\[
 A(x)=-\int_0^1r\,x\times a(rx)\,dr,
 \qquad u_0=\operatorname{curl}(\chi A),\qquad d=u_0-a,
\]
where $\chi$ is a smooth compact radial cutoff equal to one near zero.
The derivative bounds give $|D^jA(x)|\le C_j|x|^{-j}$ and
\[
 \operatorname{curl}[r x\times a(rx)]
       =-\partial_r[r^2a(rx)],\qquad \operatorname{curl}A=a.
\]
The latter identity holds across zero: the omitted spherical boundary
term is $O(r^2)$. Thus $u_0$ is compact, divergence-free and in $L^2$.
It is nonzero, since it equals $a$ near zero and DSS rescales any
nonzero value of $a$ into that neighborhood. It is smooth off zero and
$|u_0(x)|\le C|x|^{-1}$; compact support therefore gives
$u_0\in L^{3,\infty}\cap\bigcap_{1\le q<3}L^q$.
To exclude $L^3$, put $\lambda=e^{P/2}>1$ and
$E=\{1<|x|<\lambda\}$. Nonvanishing, continuity and DSS imply
$0<\int_E|a|^3<\infty$. Each of the disjoint annuli $\lambda^{-n}E$
has this same integral, and $u_0=a$ on all sufficiently small such
annuli. Thus $\int|u_0|^3=\infty$.
The field $d$ is smooth,
vanishes near zero, and satisfies inverse-radius derivative bounds.
Set
\[
 H(x,t)=e^{t\Delta}d(x),\qquad
 K(y,\tau)=e^{\tau/2}H(e^{\tau/2}y,e^\tau).
\]
For $\tau\le0$, $\|K(\tau)\|_{W^{2,\infty}}\le Ce^{\tau/2}$.
Write
\[
 \mathcal N(f)=-\mathbb P\operatorname{div}(f\otimes f),\qquad
 \mathcal C(f,g)=\mathbb P\operatorname{div}(f\otimes g+g\otimes f),
 \qquad G=-\mathcal C(U,K)+\mathcal N(K).
\]
Then $\|G(\tau)\|_{H^1}\le Ce^{\tau/2}$. For the $Q$ contribution
this follows from the $H^2$ product bound. For the other contributions,
put $H_a=e^{t\Delta}a$. The physical force
\[
 -\mathbb P\operatorname{div}(H_a\otimes H+H\otimes H_a+H\otimes H)
\]
is uniformly bounded in $H^1$ for $0<t\le1$: on an inner ball the
heat evolution of $d$ and its required derivatives are exponentially
small, whereas derivatives of $H_a$ lose only powers of $t$; outside
that ball the unprojected divergence and its derivative decay as
$\langle x\rangle^{-3}$ and $\langle x\rangle^{-4}$. Apply
$\mathbb P$ after these bounds. Multiplication by $t^{3/2}$ and
evaluation at $\sqrt t\,y$ give $L^2$ and gradient norms bounded by
$Ct^{3/4}$ and $Ct^{5/4}$, which prove the claim for $G$.

\paragraph{Nonlinear correctors and their amplitude dependence.}
Let $v=e^{\sigma\tau}b$. For $\alpha\in[0,1]$ solve in $Z_\beta$
\begin{equation}\label{short-loc:fixed-point}
 z_\alpha=\mathcal R_\beta\{G+\alpha^2\mathcal N(v)
       -\alpha\mathcal C(K,v)
       -\mathcal C(K+\alpha v,z_\alpha)+\mathcal N(z_\alpha)\}.
\end{equation}
The constant, linear and bilinear parts of this map have respective
norms bounded, uniformly in $\alpha$, by
\[
 C_0(e^{(1/2-\beta)\tau_0}+e^{(2\sigma-\beta)\tau_0}),\qquad
 C(e^{\tau_0/2}+e^{\sigma\tau_0}),\qquad Ce^{\beta\tau_0}.
\]
These follow from \eqref{short-loc:right-inverse}, the $H^2$ product
estimate and the bound on $K$. The mixed constant term has the smaller
size $Ce^{(1/2+\sigma-\beta)\tau_0}$. For sufficiently negative
$\tau_0$, all these maps preserve the ball of radius
\[
 \rho=2C_0(e^{(1/2-\beta)\tau_0}+e^{(2\sigma-\beta)\tau_0})
\]
and have Lipschitz constants at most $1/2$. Indeed these constants are
bounded by the displayed linear bound plus $2C\rho e^{\beta\tau_0}$,
which tends to zero. The resulting real $z_\alpha$ satisfy
$\|z_\alpha\|_{Z_\beta}\le\rho$.
Expanding the quadratic terms in \eqref{short-loc:fixed-point} proves
that $U+K+\alpha v+z_\alpha$ solves the exact profile equation.

Subtract the fixed-point equations at $\alpha$ and $\alpha'$.
The terms involving $z_\alpha-z_{\alpha'}$ are absorbed by the
common contraction bound, while $|\alpha^2-\alpha'^2|\le2|\alpha-\alpha'|$.
The remaining terms give
\begin{equation}\label{short-loc:amplitude}
 \|z_\alpha-z_{\alpha'}\|_{Z_\beta}
 \le L_{\tau_0}|\alpha-\alpha'|,\qquad
 L_{\tau_0}=C\bigl(e^{(2\sigma-\beta)\tau_0}
       +e^{(1/2+\sigma-\beta)\tau_0}+\rho e^{\sigma\tau_0}\bigr).
\end{equation}
The three contributions come from $\mathcal N(v)$,
$\mathcal C(K,v)$ and $\mathcal C(v,z_{\alpha'})$, respectively.
Since
\[
 L_{\tau_0}e^{(\beta-\sigma)\tau_0}
 \le C\bigl(e^{\sigma\tau_0}+e^{\tau_0/2}
                    +\rho e^{\beta\tau_0}\bigr)\longrightarrow0,
\]
choose one $\tau_0$ so that this quantity is at most $m_b/2$ and
the preceding contraction bounds hold. This choice is independent
of both amplitudes.

\paragraph{Finite energy, the common trace, and separation.}
With $T=e^{\tau_0}$, the physical solutions are
\begin{equation}\label{short-loc:finite-energy}
 u^{(\alpha)}(x,t)=e^{t\Delta}u_0(x)
       +t^{-1/2}[Q+\alpha v+z_\alpha](x/\sqrt t,\log t).
\end{equation}
The $L^2$ and gradient norms of $t^{-1/2}f(x/\sqrt t)$ are
$t^{1/4}\|f\|_2$ and $t^{-1/4}\|\nabla f\|_2$. Thus the three
remainders have $L^2$ orders $t^{1/4}$, $t^{\sigma+1/4}$ and
$t^{\beta+1/4}$, and squared gradient orders $t^{-1/2}$,
$t^{2\sigma-1/2}$ and $t^{2\beta-1/2}$. All dissipation powers are
integrable. Heat energy and continuity yield the strong trace,
$C_tL^2\cap L_t^2H^1$ and positive-time $C_tH^2$.
The projected equation then gives $\partial_tu^{(\alpha)}\in C_tL^2$
on every positive closed interval. Pairing it with $u^{(\alpha)}$ gives
the energy equality there; strong trace and finite dissipation extend
it to $t_0=0$. For $s\ge2$, the nonlinearity maps $H^s$ to $H^{s-1}$,
and heat smoothing from $H^{s-1}$ to $H^{s+1/2}$ has the integrable
kernel $(t-r)^{-3/4}$. Iteration on smaller positive-time intervals
gives all spatial derivatives; the equation gives all time derivatives.

The interpolation inequality
$\|u\|_{10/3}^{10/3}\le C\|u\|_2^{4/3}\|\nabla u\|_2^2$
gives $u^{(\alpha)}\in L^{10/3}_{x,t}$. The Riesz-transform bounds yield
the stated $L^{5/3}$ pressure and recover the full equation.
Integrating from a positive time and using the strong trace gives
the distributional formulation including $u_0$.

Orthogonal maps fixing the axis commute with the heat and
Leray operators and the nonlinearity. They therefore commute with
the evolution, its spectral projections, and $\mathcal R_\beta$.
The potential $A$ is azimuthal, so $\operatorname{curl}(\chi A)$ is
axisymmetric without swirl. Uniqueness in the common contraction ball
preserves this symmetry for every $z_\alpha$ and
\eqref{short-loc:finite-energy}. By \eqref{short-loc:mode-regularity}
and \eqref{short-loc:amplitude}, for $\tau\le\tau_0$,
\[
 \begin{aligned}
 \| (\alpha-\alpha')v+z_\alpha-z_{\alpha'}\|_2
 &\ge |\alpha-\alpha'|e^{\sigma\tau}
     \bigl(m_b-L_{\tau_0}e^{(\beta-\sigma)\tau_0}\bigr),\\
 \| (\alpha-\alpha')v+z_\alpha-z_{\alpha'}\|_2
 &\le |\alpha-\alpha'|e^{\sigma\tau}
     \bigl(\max_{[0,P]}\|b\|_2+m_b/2\bigr).
 \end{aligned}
\]
Physical scaling gives the two-sided separation with $c_*=m_b/2$;
integrating its square gives the stated integral bound. In particular,
$\alpha\mapsto u^{(\alpha)}-u^{(0)}$ is Lipschitz for the norm
$\sup_{0<t\le T}t^{-\sigma-1/4}\|\cdot(t)\|_2$.

\paragraph{Local energy equality and mixed norms.}
For a fixed branch write $u=u^{(\alpha)}$, $p=p^{(\alpha)}$ and
$e=|u|^2/2$. Its positive-time smoothness gives the classical local
energy identity. For every real
$\phi\in C_c^\infty(\mathbb R^3\times[0,T))$, integration from
$\varepsilon>0$ and then passage to zero give
\eqref{short-loc:local-energy}.
Indeed $|\nabla u|^2$ is integrable; $u\in L^{10/3}_{x,t}$ and
$p\in L^{5/3}_{x,t}$ put both $|u|^3$ and $pu$ in $L^{10/9}_{x,t}$,
hence in $L^1$ on the support of $\phi$. Strong $L^2$
trace implies $|u(\varepsilon)|^2\to|u_0|^2$ in $L^1$.
These facts justify every limit in the identity and, with nonnegative
$\phi$, prove suitability.

Put $F_\alpha=Q+\alpha v+z_\alpha$ and $H_0=e^{t\Delta}u_0$.
The uniform $H^2$ bound on $F_\alpha$ gives, for $2\le q\le\infty$,
\[
 \|t^{-1/2}F_\alpha(x/\sqrt t,\log t)\|_q
 \le C_qt^{(3/q-1)/2}.
\]
The pointwise bound on $u_0$ and Gaussian convolution give
$|H_0(x,t)|\le C/(|x|+\sqrt t)$: rescale by $\sqrt t$ and split
the convolution into $|y-x|<|x|/2$ and its complement when
$|x|>2\sqrt t$. Hence $\|H_0(t)\|_{3,\infty}\le C$ and
$\|H_0(t)\|_\infty\le Ct^{-1/2}$. Since the scaled remainder has
uniform $L^3$ norm, \eqref{short-loc:finite-energy} proves the uniform
weak-$L^3$ assertion.

Heat contraction gives $\|H_0(t)\|_q\le\|u_0\|_q$ for $2\le q<3$.
For $q\ge3$, integrate the distribution function of $|H_0|$,
using its uniform $L^2$ and weak-$L^3$ bounds and its $Ct^{-1/2}$
supremum. Together with the remainder estimate this yields
\[
 \|u^{(\alpha)}(t)\|_q\le C_q
 \begin{cases}
  1,&2\le q<3,\\
  (1+|\log t|)^{1/3},&q=3,\\
  t^{(3/q-1)/2},&3<q\le\infty.
 \end{cases}
\]
For $q=3$ this is integrable to every finite time power. For $q>3$
and finite $r$, the displayed power is integrable exactly when
$r(1-3/q)/2<1$;
for $q<3$ it is uniformly bounded in time. These conditions give
every claimed strict mixed norm, with constants independent of $\alpha$.
\end{proof}


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\end{document}