\chapter{Comparison of reconstructions on block regions}\label{ch16:chapter} This section compares one renormalization group step on a block region with the same step on the whole torus. Fix a composition $T_M^s$ of $s\le J$ averaging steps and a union $\Omega$ of blocks of its output lattice. In \cref{ch15:chapter} we reconstructed the fine configuration on $\Omega$ from the coarse field on $\Omega$ and free logarithmic coordinates, with the coordinates outside $\Omega$ held fixed. We also constructed the stationary point of the effective classical action in these coordinates and proved a gap for its free Hessian. Here we prove that the regional construction differs from the torus construction with the coarse field extended by zero outside $\Omega$ only by terms which decay exponentially with the distance to the boundary of $\Omega$. We also prove that the regional construction is stable under small analytic perturbations of the incoming action. All estimates are bounds on absolute Taylor coefficients. Their constants do not depend on the shape of $\Omega$, the number of its components, the depth, the number of preceding renormalization group steps, or the values of previously frozen coordinates. We use these estimates in the localized renormalization group analysis behind the exponential-moment bound for blocked actions (Step~1 in \cref{s3:largefield}). There the configurations are decomposed according to the regions where some averaged plaquette has large curvature, and the renormalization group is run outside these regions with the link variables inside them held fixed. \Cref{ch17:chapter} integrates the fluctuations on block regions using the reconstructions and perturbation estimates proved here. \Cref{ch18:chapter} uses the gauge transformation of \cref{ch16:lem:strict-hybrid-gauge} and the perturbation theorem \cref{ch16:thm:controlled-perturbation} to construct large-field decompositions, and \cref{ch20:chapter} uses the domain conditions of \cref{ch16:lem:admitted-products,ch16:lem:strict-two-radius-images,ch16:thm:controlled-perturbation} to choose the parameters. The results, in order, are as follows: \cref{ch16:lem:admitted-products} treats successive minimization; \cref{ch16:thm:nonflat-force-comparison} compares the regional and torus stationary points, together with the minimized actions, the Hessians, and the Haar disintegration densities; \cref{ch16:lem:strict-hybrid-gauge} constructs a gauge transformation which is the identity at prescribed vertices; \cref{ch16:thm:hybrid-columns} compares the gauge-transformed reconstructions and bounds their input columns; \cref{ch16:lem:strict-two-radius-images} collects the conditions on the radii; and \cref{ch16:thm:controlled-perturbation} proves stability under perturbations of the incoming action. The inputs are \cref{ch15:thm:regional-gap-and-center} (regional coordinates, the gap of the free Hessian, and the regional stationary point), \cref{ch15:lem:wall-networks} (coefficient bounds for connected networks which reach the fixed complement), \cref{ch15:lem:minimum-calculus} (derivatives of constrained stationary values), \cref{ch15:lem:freezing} (freezing of coordinates), the coefficient algebra of \cref{ch15:def:regional-norm}, the target-dependent gauge transformation of \cref{ch10:thm:whole-image-gauge}, the classical coefficient bounds of \cref{ch07:thm:all-root-classical}, and the equality of effective classical actions on compact gauge overlaps, \cref{ch08:thm:compact-overlap}. We compare constrained minimizers through the implicit equation for their response, and we control the response by an exponentially decaying inverse of the constrained Hessian. This approach follows Ba\l aban's analysis of the variational problem for background fields \cite[(182)--(190), pp.~307--308, and Proposition~9, p.~309]{Balaban1985Variational}, although our constraints, coordinate domains, and coefficient norms differ from his. The comparison between the regional and the torus problems and the gauge transformation fixing the prescribed endpoints are constructed here. The exponentially weighted resolvent bound, proved in \cref{ch15:lem:boundary-algebra}, is the finite-dimensional version of the exponential-conjugation method of Combes and Thomas \cite{CombesThomas1973}. \paragraph{The mechanism.} The regional problem and the torus problem with coarse field extended by zero have the same constraints away from a fixed neighborhood of the boundary of $\Omega$. If we restrict the torus stationary point to the regional coordinates, the gradient of the regional objective at this point is therefore concentrated near the boundary, up to exponentially decaying tails. This residual gradient has two sources, which are the two terms of the formula for $r_R$ in \cref{ch16:thm:nonflat-force-comparison}. First, the reconstructed fine field changes. Second, the tangent map of the constraints changes, since the regional problem omits the coarse constraints on links leaving $\Omega$. Writing $p$ for the restricted torus stationary point and $z_R$ for the regional stationary point, we have $z_R-p=-\mathcal A_R^{-1}r_R$, where $r_R$ is the residual gradient and $\mathcal A_R$, the regional Hessian averaged along the segment from $p$ to $z_R$, has a positive gap. Since this inverse decays exponentially in the distance, the two stationary points differ by an amount which decays exponentially away from the boundary. The remaining work is to make this mechanism stable under differentiation, composition, and successive minimization. This requires bounds on all Taylor coefficients, with any one external index fixed. A bound on the values of a holomorphic map does not give the spatial weights and the fixed-index sums which are needed when these functions are composed later. \Cref{ch16:fig:geometry} shows the sets which appear below. Two of them play different roles: the prescribed vertices, where the gauge transformation must be the identity, and the complement of the incoming list, which controls the spatial decay of the perturbation in \cref{ch16:thm:controlled-perturbation}. \begin{figure}[ht] \centering \setlength{\unitlength}{1pt} \begin{picture}(360,200) \put(0,0){\framebox(360,200){}} \put(10,182){$\Omega$} \put(95,30){\dashbox{1.5}(160,140){}} \put(107,42){\dashbox{1.5}(136,116){}} \put(120,55){\dashbox{5}(110,90){}} \put(135,70){\framebox(80,60){}} \put(196,113){$H$} \put(143,78){\dashbox{2}(42,30){}} \put(148,90){\scriptsize$D_{\mathrm{in}}^c$} \put(114,100){\circle*{4}} \put(114,100){\line(1,0){21}} \put(160,151){\circle*{4}} \put(160,151){\line(0,-1){21}} \put(265,150){\vector(-2,-1){48}} \put(268,150){\small$W$} \put(272,120){\vector(-1,0){40}} \put(276,117){\small$H_{L_J}$} \put(278,90){\vector(-1,0){33}} \put(282,87){\small$\chi=0$ inside} \put(278,60){\vector(-1,0){22}} \put(282,57){\small$\chi=1$ outside} \put(10,114){\small protected vertices} \put(60,108){\vector(4,-1){48}} \end{picture} \caption{The geometry in the output lattice (schematic). The solid inner rectangle is the fixed complement $H=\Omega^c$, and its boundary is the interface $W$. The dashed rectangle bounds the enlargement $H_{L_J}$ of $H$ by the reconstruction stencil. The cutoff $\chi$ of \cref{ch16:lem:strict-hybrid-gauge} vanishes inside the inner dotted rectangle and equals one outside the outer dotted rectangle. The black dots are protected vertices, here the endpoints of the paths of two based transports of frozen coordinates. The small dotted rectangle inside $H$ is the complement of the incoming list $D_{\mathrm{in}}$ of \cref{ch16:thm:controlled-perturbation}, which lies within a fixed distance of $H$.} \label{ch16:fig:geometry} \end{figure} \section{Setting and conventions}\label{ch16:sec:setting} \paragraph{Geometry.} Throughout this section $M\ge3$ is odd, $J\ge1$ is fixed, $1\le s\le J$, and $B_s=M^s$. The composition $T_M^s$ of $s$ averaging steps maps configurations of side $M^sm$ to configurations of side $m\ge4$, and the prior depth $d_0\ge0$ is an arbitrary integer. When $s=J$ the composition is one renormalization group step, with blocking factor $B=M^J$. Positions are measured in units of the output lattice, of side $m$. Let $\Omega$ be a union of blocks of the output lattice, let $H=\Omega^c$ be its fixed complement, let $W$ be the interface between $\Omega$ and $H$, and let $H_{L_J}$ be the enlargement of $H$ by the fixed reconstruction stencil $L_J$ of \cref{ch15:thm:regional-gap-and-center}. A coordinate is a label carrying a position, a direction, a color, and a scale index; its position is taken at the lattice where the coordinate was introduced and embedded in output units. In particular the newly frozen coordinates are counted with the density of the lattice on which they were introduced, not as coordinates of the output lattice. The density constant $v_J$ of \cref{ch15:def:regional-norm} includes the colors, the directions, and the at most $J$ scale indices. For a label $i$ in $\Omega$ we write $\delta(i)=\dist(i,W)$. Here $\dist$ and $\Tree$ are the graph distance and the $\ell^1$ network length of the refined position graph fixed before \cref{ch15:def:regional-norm}; coefficient bounds imported at a Euclidean rate $b$ are used at rate $b/\sqrt3$. If $\Omega$ is empty, every statement below with a fixed index in $\Omega$ is void. If $H$ is empty, every boundary difference considered below is zero; we never use an expression of the form $0\cdot e^{\infty}$. \paragraph{Coordinates.} We use the following coordinates. \begin{itemize} \item The coarse field $c$ is the logarithm of the output links internal to $\Omega$, and $c\oplus0$ is its extension by zero to all output links of the torus. \item The free logarithmic coordinates $z=(z_i)_{0\le i0$. A product or contraction joins the connecting networks of its factors at the contracted indices. We root the joined network at any external index and sum over the internal indices, using the corresponding rooted bounds of the factors. The boundary-localized estimate applies if one factor is boundary-localized, or if the joined network visits $H_{L_J}$. An index deep in $H$ is bounded there, and the decay comes from its connection to the fixed index in $\Omega$. In particular, for decay exponents $b>a>0$, the sum over such indices is bounded by \begin{equation}\label{ch16:eq:hole-sum} e^{a\delta(e)}\sum_{h\in H_{L_J}}e^{-b\dist(e,h)} \le e^{aL'_J}S_J(b-a),\qquad S_J(u)=C v_J\sum_{r\ge0}(r+1)^3e^{-ur}, \end{equation} where $L'_J$ also includes the fixed convention for link endpoints. This is the case of a fixed index in $\Omega$ in \cref{ch15:lem:wall-networks}; there is no factor of the volume of $H$. \paragraph{Implicit series.} The implicit equations below are solved in the absolute coefficient algebra. After the construction constant $C_J$ is increased, the scalar majorant of the solution without boundary weight can be taken to satisfy \begin{equation}\label{ch16:eq:implicit-majorant} Y=C_Jt+\frac12Y+ \frac{C_J(t+Y)^2}{1-C_J(t+Y)}. \end{equation} Fix $t_0>0$ such that, on $0\le t\le t_0$ and $0\le Y\le4C_Jt_0$, the rational term has $Y$-derivative at most $1/4$ and denominator at least $1/2$, and its value at $(t,4C_Jt)$ is at most $C_Jt$. Such a $t_0$ exists and depends only on $C_J$: the derivative and denominator conditions have strict margins at $(t,Y)=(0,0)$, while the rational term at $(t,4C_Jt)$ equals $C_J(1+4C_J)^2t^2/[1-C_J(1+4C_J)t]=O(t^2)$ as $t\downarrow0$. Iteration from zero produces series with nonnegative coefficients which stay below $4C_Jt$. The iteration converges uniformly on every smaller disc, since the derivative in $Y$ of the right side of \eqref{ch16:eq:implicit-majorant} is at most $3/4$ on its majorant disc. The degree-$l$ coefficients of the solution are bounded by $4C_Jt_0^{1-l}$, and dividing Taylor tensors by $l!$ before this comparison gives factorial tensor bounds. The argument works in the absolute multivariate coefficient algebra because each product satisfies the rooted contraction bound just described. Uniform convergence of the power series implies holomorphy. A derivative which inserts one difference factor, localized near $H$, has the majorant \begin{equation}\label{ch16:eq:marked-majorant} Z\preceq H(t,Y(t))+A(t)Z,\qquad A(t)=\frac12+ \partial_Y\frac{C_J(t+Y)^2}{1-C_J(t+Y)}\bigg|_{Y=Y(t)}, \end{equation} where $\preceq$ denotes coefficientwise domination and $H(t,Y)$ is the series with the difference factor inserted. On a smaller fixed radius $A(t)\le3/4$, and the solution is majorized coefficientwise by $H(t,Y(t))/(1-A(t))$; this is a convergent majorant series with one inserted difference factor (compare \eqref{ch15:eq:marked-majorant}). Fixing an analytic input differentiates this same series and keeps the index of that input. By \cref{ch15:lem:wall-networks}, the connecting network and the route to the boundary are bounded once for the whole network, and their exponent is not divided by the Taylor order. \paragraph{Matrix functions.} We use matrix functions only on free coordinate spaces where a gap has been proved. The next lemma collects the facts we need. \begin{lemma}\label{ch16:lem:matrix-functions} Let $A^b\ge\lambda I$ be a real symmetric free matrix whose two-sided weighted Schur norm at exponent $b$ is at most $Q$, and set \[ \mu=\frac{b\lambda}{8(Q+\lambda)},\qquad C_R=2S_J(\mu/2). \] \begin{enumerate} \item For every $t\ge0$, the entries of $(A^b+t)^{-1}$ are bounded by $2e^{-\mu\dist(x,y)}/(\lambda+t)$, and its weighted Schur norm at exponent $\mu/2$ is at most $C_R/(\lambda+t)$; the constant $C_R$ is the same for every subset of the label list. \item If $L$ is a complex coefficient perturbation with $C_R\|L\|_{\mu/2}/\lambda\le1/4$, then the series \begin{equation}\label{ch16:eq:resolvent-series} (A^b+L+t)^{-1} =\sum_{k\ge0}[-(A^b+t)^{-1}L]^k(A^b+t)^{-1} \end{equation} converges normally, and its coefficient Schur norm is at most $4C_R/[3(\lambda+t)]$. \item For positive definite real matrices $A,B$, \begin{align} A^{-1/2}&=\frac1\pi\int_0^\infty t^{-1/2}(A+t)^{-1}\,dt, \label{ch16:eq:sqrt}\\ A^{-1/2}-B^{-1/2} &=-\frac1\pi\int_0^\infty t^{-1/2} (A+t)^{-1}(A-B)(B+t)^{-1}\,dt, \label{ch16:eq:sqrt-difference}\\ \log A-\log B &=\int_0^\infty (A+t)^{-1}(A-B)(B+t)^{-1}\,dt, \label{ch16:eq:log-difference} \end{align} and these identities extend holomorphically to the complex perturbations in (2). \item If $T=\begin{pmatrix}A&B\\ C&D_0\end{pmatrix}$ is invertible with invertible blocks $A$ and $D_0-CA^{-1}B$, then \begin{equation}\label{ch16:eq:block-inverse} (T^{-1})_{LL}-A^{-1} =A^{-1}B(D_0-CA^{-1}B)^{-1}CA^{-1}. \end{equation} \end{enumerate} \end{lemma} \begin{proof} Parts (1) and (2) are the Combes--Thomas bounds proved in \cref{ch15:lem:boundary-algebra}(3); see \eqref{ch15:eq:coefficient-resolvent}. We recall the choice of constants. For a real one-Lipschitz function $f$ on the label metric, conjugation by $\operatorname{diag}(e^{\mu f})$ changes $A^b$ by a matrix of Schur norm at most $Q\mu/(b-\mu)\le\lambda/8$; here we used $e^{\mu r}-1\le\mu re^{\mu r}$ and $\sup_{r\ge0}re^{-(b-\mu)r}\le(b-\mu)^{-1}$. By the geometric resolvent series, the conjugated resolvent has operator norm at most $2/(\lambda+t)$ for $t\ge0$. Taking $f(x)=\dist(x,y)$, we obtain the entry bound, and summing at exponent $\mu/2$ gives $C_R/(\lambda+t)$, uniformly over every subset of the label list. Under the condition in (2) each term of \eqref{ch16:eq:resolvent-series} after the first is smaller than the preceding one by the factor $1/4$, which gives the bound $4C_R/[3(\lambda+t)]$. For (3), we diagonalize the positive matrices and substitute $t=\lambda u^2$ in each eigenvalue integral, which gives \eqref{ch16:eq:sqrt}. The resolvent identity $(A+t)^{-1}-(B+t)^{-1}=-(A+t)^{-1}(A-B)(B+t)^{-1}$ fixes the order of the three factors and gives \eqref{ch16:eq:sqrt-difference} after integration against $\pi^{-1}t^{-1/2}\,dt$. For \eqref{ch16:eq:log-difference}, we integrate $(1+t)^{-1}I-(A+t)^{-1}$ to obtain $\log A$ and subtract the corresponding formula for $B$. The series \eqref{ch16:eq:resolvent-series} extends these identities holomorphically to the complex perturbations in (2). Their majorants are integrable: they are bounded by $C_Jt^{-1/2}(\lambda+t)^{-2}$ and $C_J(\lambda+t)^{-2}$ for the two differences, whose integrals are $\pi/(2\lambda^{3/2})$ and $1/\lambda$. Thus a boundary-localized middle factor $A-B$ gives boundary-localized differences of these functions, rooted at every index, and differentiation under the integral is justified by the same normally convergent majorants. Restricted diagonal sums are taken after this calculation; only a complete scalar trace permits a cyclic permutation of the factors. Part (4) follows by block multiplication. \end{proof} \paragraph{Additive constants.} The coefficient seminorm with boundary decay $\mathcal X$ of \cref{ch15:def:regional-norm} vanishes on constants, so it never estimates additive constants. The constant term of a difference of minimized actions remains in the full action. Scalar determinants, Haar disintegration densities, and traces over $H$, which have no index in $\Omega$, are kept as their exact finite values. The difference between two exterior extensions of the classical action is kept with all its constants (\cref{ch16:thm:controlled-perturbation}). After freezing at $c$, replacing an effective interaction $R(v,c)$ by $R(v,c)-R(0,c)$ changes the normalization constant from $\Lambda$ to $\Lambda-R(0,c)$ and preserves the density, as in \cref{ch15:lem:freezing}. \section{Conditional minimization and successive substitution}\label{ch16:sec:conditional} The reconstruction of the fine field proceeds from the coarsest intermediate scale to the finest. At scale $i$ the free coordinates are $a_i(c,z_{>i})+K_i^{-1/2}q_i$, where $a_i$ is a conditional stationary point which depends on the coarser coordinates $z_{>i}$ already reconstructed. We must check two things. First, for every fixed value of the coarser coordinates in some domain, the conditional stationary point exists. Second, the coarser coordinates produced by the earlier substitutions lie in that domain. The first does not imply the second, as the following example shows. \begin{example}\label{ch16:ex:conditional-domain} Let $F_0(x,y)=(x^2+y^2)/2$ and $F_1(x,y)=F_0(x,y)-\delta y$ with $\delta>0$, and eliminate $x$ first, so that $y$ is the coarser coordinate. For every fixed $y$ the conditional stationary point of $F_1$ in $x$ is $x=0$, and the joint Hessian is the identity. The joint stationary point of $F_1$ is $(0,\delta)$. If the conditional stationary points are constructed only for $\abs y<\eta$ with $\eta<\delta$, the successive substitution never reaches the joint stationary point, however small the fluctuation radius in $x$ is chosen. The quantity $\delta_V$ of \cref{ch16:lem:admitted-products} detects this: for the initial segment consisting of both coordinates it equals $\delta$, so the condition in part~(3) of the lemma fails. \end{example} In the next lemma, $I$ is an initial segment of the finer scale coordinates eliminated in the successive minimization, and $y=z_{I^c}$ denotes the coarser coordinates held as parameters. The segment containing all free coordinates is included. \begin{lemma}\label{ch16:lem:admitted-products} Fix the larger construction domains of \cref{ch15:thm:regional-gap-and-center}, within \eqref{ch16:eq:torus-reserve}, and the margins of their coefficient, inverse, and image bounds and of the lower bound for the real constrained Hessian. Let $F_0(c,z;\xi)$ be the unperturbed regional objective and $z^0(c;\xi)$ its joint stationary point. For each initial segment $I$ write $y=z_{I^c}$, and let $a_I(c,y;\xi)$ be the conditional stationary point of $F_0$ in the variables $z_I$. \begin{enumerate} \item One can first choose a solution radius $r_I>0$, and then a strictly smaller radius $\eta_I$ for the deviation $y-z^0_{I^c}(c;\xi)$, uniformly over the coarse fields in the domain, such that \[ a_I(c,z^0_{I^c};\xi)=z_I^0, \qquad D_ya_I=-F_{0,II}^{-1}F_{0,II^c}, \qquad \|a_I-z_I^0\|_{\mathrm{coeff}} \le C_J\|y-z^0_{I^c}\|_{\mathrm{coeff}}. \] The derivative identity is evaluated at the conditional stationary point. We write $\mathcal T_I$ for the resulting domain of $y$. \item Let $F_\lambda=F_0+\lambda V$, $0\le\lambda\le1$, be holomorphic and real at real arguments, with the same coefficient bounds on the larger domains. Set $A_I=F_{0,II}(c,a_I,y;\xi)$ and assume that, throughout the domains of part~(1), in the operator norm of the coefficientwise majorant, \[ \|A_I^{-1}F_{\lambda,I}(c,a_I,y)\|\le r_I/4, \qquad \|A_I^{-1}(F_{\lambda,II}-A_I)\|\le1/2. \] Assume also that the real Hessian is perturbed by less than half of the lower bound for the constrained Hessian, and that the complex resolvent condition of \cref{ch16:lem:matrix-functions}(2) holds. Then for each $I$ and each $y\in\mathcal T_I$ there is a unique holomorphic stationary point $a_{\lambda,I}(c,y;\xi)$ of $F_\lambda$ in $z_I$ within distance $r_I/2$ of $a_I$. It is real at real data and a local real minimum there, and it is unique in the convex domain on which the second inequality holds. \item Put $V=\partial_\lambda F_\lambda$ and \[ \delta_V=\max_I\sup_{y\in\mathcal T_I} \|A_I^{-1}V_I(c,a_I(c,y;\xi),y)\|_{\mathrm{coeff}}, \] where the coefficient norm includes the common product of coarse fields, uniformly at fixed real frozen coordinates, and let $\varepsilon_{\mathrm{tube}}$ be the minimum over $I$ of the solution and parameter margins at $I$. Then, for every $I$ and every $y\in\mathcal T_I$, \[ \|a_{\lambda,I}(c,y;\xi)-a_I(c,y;\xi)\|_{\mathrm{coeff}}\le2\lambda\delta_V. \] Assume further that $\delta_V<\infty$, that the mixed coefficient bounds hold uniformly on the larger domains, and that $r_q>0$ satisfies \[ C_{\mathrm{seq},J}(2\delta_V+r_q)<\frac12\varepsilon_{\mathrm{tube}}, \] where $C_{\mathrm{seq},J}$ is the constant given in the proof. For $I_i$ consisting of scales $0,\ldots,i$, let $a_{\lambda,i}$ be the scale-$i$ component of the conditional stationary point $a_{\lambda,I_i}$ of $F_\lambda$, and let $K_{\lambda,i}$ be $M^i$ times the Schur block of $F_\lambda$ at scale $i$ in input units. Then, whenever $\max_i\|q_i\|_\infty\le r_q$, the successive substitutions in the original free coordinates \[ z_i=a_{\lambda,i}(c,z_{>i};\xi) +K_{\lambda,i}^{-1/2}(c,z_{>i};\xi)q_i \] stay in every parameter domain and every solution domain, with half of each margin unused. At $q=0$ they give the joint stationary point. \end{enumerate} The radii are chosen in the following order: first the unperturbed product, for which $\delta_V=0$; then a bound on the size of the perturbation on the fixed larger unperturbed products, and a smaller perturbed product; only afterwards the smaller enclosing and fluctuation radii. Reducing $r_q$ does not reduce the radius of the coarse field, since the chain at $q=0$ is the joint stationary point. All radii and constants have the dependence of \cref{ch16:sec:setting}, independently of the later widths, the depth, and the geometry of $\Omega$. \end{lemma} \begin{proof} Part~(1) is a quantitative implicit function argument about the joint stationary point, and part~(2) is the same argument about $a_I$. The new point is part~(3), which first bounds the displacement of the conditional stationary points at every fixed $y$ and then propagates these bounds through the successive substitution by a finite recurrence. \paragraph{Step 1: the unperturbed conditional stationary points.} Fix $I$ and write $y=z^0_{I^c}(c;\xi)+v$. This shift is an analytic change of coordinates with the coefficient bounds of \cref{ch15:thm:regional-gap-and-center}. In these coordinates we set $u=z_I-z_I^0(c;\xi)$ and \[ B_I(c;\xi)=F_{0,II}(c,z^0(c;\xi);\xi). \] By the bounds on the inverse and on the joint third derivative, there exist constants $I_0,C_0$ on the fixed larger products such that \begin{align} \|B_I^{-1}F_{0,I}(c,z_I^0,z_{I^c}^0+v;\xi)\|_{\mathrm{coeff}} &\le I_0C_0\|v\|_{\mathrm{coeff}}, \label{ch16:eq:conditional-force}\\ \|B_I^{-1}(F_{0,II}(c,z_I^0+u,z_{I^c}^0+v;\xi)-B_I)\|_{\mathrm{coeff}} &\le I_0C_0(\|u\|_{\mathrm{coeff}}+\|v\|_{\mathrm{coeff}}). \label{ch16:eq:conditional-variation} \end{align} For the first inequality we integrate $F_{0,II^c}$ in $v$, starting from the joint stationarity $F_{0,I}(c,z^0;\xi)=0$; for the second we integrate the third derivative tensor in $(u,v)$. These are coefficient inequalities, since integration along a segment from the origin multiplies each term of positive degree by a number at most one, and the rooted bounds on the larger domains control the products. Fix $r_I>0$ within a fixed fraction of the solution margin of the chart, with $I_0C_0r_I<1/8$. Then fix $0<\eta_I0$ such that \[ \mathcal X_{\Omega,a_W,\alpha_W}^{\rho_0,\mathrm{cl}} \bigl(V_R-E_{d_0+s}(c\oplus0)\bigr)\le A_J^{\mathrm{cl}}. \] This is the seminorm of \cref{ch15:def:regional-norm} with the displayed constant classical radius and rates; it includes all degrees $l\ge1$. The constant term of the difference is kept in the full action. \item (Newly frozen coordinates.) For the newly frozen coordinates $\xi$, at their positions and with their density, \[ \begin{gathered} D_\xi z_R=-F_{R,zz}^{-1}F_{R,z\xi},\qquad \sum_f|(D_{\xi_f}z_R)_e|\le C_Je^{-a_W\dist(e,W)},\\ \|F_{R,z\xi}\|_{a_W}+\|F_{R,zc}\|_{a_W}\le C_J. \end{gathered} \] \item (Hessians.) The constrained Hessians at the two stationary points satisfy \[ \bigl|\bigl[F_{R,zz}(c,z_R;\xi) -PF_{T,zz}(c\oplus0,z_T)P^{\mathsf T}\bigr]_{ef}\bigr| \le C_Je^{-a_W[\dist(e,f)+\dist(e,W)+\dist(f,W)]}. \] The order-$l$ coefficient tensors of this difference, with every coarse-field and free argument in the connecting network and any root in $\Omega$ fixed, are bounded by $(l+2)!C_J^{l+2}$ at fixed exponents. \item (Successive minima, inverse square roots, and Haar disintegration densities.) Use a common smaller full torus product of higher coordinates $y=(y_L,y_H)$ for each initial segment $I$. On this product compare the regional conditional center $a_I^R(c,y_L;\xi)$ with the regional components of $a_I^T(c\oplus0,y_L,y_H)$, and compare the Hessian of the regional partial minimum with the $LL$ block of the torus partial-minimum Hessian at that same full argument. The conditional-center differences have the boundary-localized coefficient bounds of part~(1), with every parameter input fixed in turn. The Hessian differences have the bounds of part~(5), with the exact normalization $M^i$ at scale $i$. The common product can be taken as $\|y\|_{\mathrm{coeff}}<\eta_I/2$ after the coarse and frozen radii are reduced as in Step~10 of the proof; it is not a slice at an unspecified $y_H$. The inverse square roots of these Hessians and the nonconstant coefficients of the logarithms of the Haar disintegration densities satisfy the corresponding boundary-localized comparisons, with any index fixed. Scalar determinants and Haar disintegration densities which are not rooted at a label in $\Omega$ are kept as exact quantities. \end{enumerate} All constants have the dependence of \cref{ch16:sec:setting} and do not depend on the later widths. \end{theorem} The theorem gives no inverse on fixed boundary coordinates and no rank estimate for the boundary terms at a nonzero coarse field. \begin{proof} We pose both problems on the torus free list by interpolating between the torus objective and the augmented regional objective. The two objectives then differ by one inserted factor localized near $H$, and all estimates follow from the response of the interpolated stationary point to this factor and from the exponential decay of the inverse Hessian. \paragraph{Step 1: one coordinate list.} We write $z=(z_L,z_H)$ for the torus free list and set \begin{equation}\label{ch16:eq:augmented-objectives} \widetilde F_R(c,z;\xi)=F_R(c,z_L;\xi)+\frac12|z_H|^2, \qquad F_t=(1-t)F_T+t\widetilde F_R,\quad 0\le t\le1, \qquad D=\widetilde F_R-F_T. \end{equation} Since its minimum is zero, the added auxiliary quadratic does not change the regional value. \Cref{ch15:thm:regional-gap-and-center} gives a positive gap at the identity for the Hessian of this interpolation, for all its principal blocks, and for its Schur complements. We reduce the construction product by a fixed amount so that the coefficient variation of each of these Hessians satisfies the condition of \cref{ch16:lem:matrix-functions}(2), uniformly in $t$. This is possible since the variation is at most $Q_J(\|c\|+\|z\|+\|\xi\|)$, and the gap at the identity of the interpolation is at least the minimum of the regional gap, the torus gap, and $1$. All straight segments of fine fields used below then lie strictly inside the larger classical chart. We retain a split resolvent margin for the later perturbation argument. Choose a common intrinsic exponent $b_{\mathrm{ref}}>0$ and a constant $C_{0,J}$ such that every flat principal and scale Schur reference $K_i^b$ has resolvent norm at most $C_{0,J}/(\lambda_J+t)$ at this exponent, including the factors $M^i$ at the scale stages. When the larger unperturbed products are fixed, reduce them once so that their conditional matrices $K_{0,i}$ satisfy \[ C_{0,J}\|K_{0,i}-K_i^b\|_{b_{\mathrm{ref}}}\le\lambda_J/8 \] on those products, uniformly over the finitely many stages. We decrease the unperturbed reserve $c_J$ if necessary so that $c_J\le b_{\mathrm{ref}}$. These choices use only the unperturbed constructions at fixed $J$ and precede the incoming rates, perturbation threshold, and later widths. \paragraph{Step 2: localization of the difference of the reconstructions.} We use the comparison of the nonlinear words in \cref{ch15:thm:regional-gap-and-center}. At common internal coarse values and common regional free values, we set \[ d(c,z;\xi)=\Psi_R(c,z_L;\xi)-\Psi_T(c\oplus0,z_L,z_H). \] On fine links in $\Omega$, its defining words vanish outside the fixed interface band. The prescribed values on $H$ can differ, and their difference is a bounded source on $H$ with the separate coefficient bounds of the reconstruction charts. The difference is supported in $H$ together with that band, hence in $H_{L_J}$; the two extensions need not agree on $H$. Its coefficient tensors have the corresponding finite-range bounds, rooted at any index. To check the statement about fine links in $\Omega$ at nonzero arguments, consider an internal eliminated-edge equation at the coarsest intermediate scale. On the two sides it has the same endpoint blocks, the same coarse value, and the same free values, so by uniqueness on the common local branch its solution is the same. We repeat this down the $s$ intermediate scales. Only equations meeting a block of $H$ can differ, and the union of their at most $J$ stencils is contained in $H_{L_J}$. This argument keeps the values of the coarse links across the boundary of the block region as functions of the common free values; it imposes no value on an omitted coarse constraint. \paragraph{Step 3: the inserted difference.} The difference in \eqref{ch16:eq:augmented-objectives} is \begin{equation}\label{ch16:eq:action-insertion} D=\int_0^1 E_{d_0}'(\Psi_T+v d)[d]\,dv +\frac12|z_H|^2. \end{equation} Every derivative of the integral either differentiates $d$, whose fine position stays in $H_{L_J}$, or leaves such a factor undifferentiated. In the latter case the classical tensor of \cref{ch07:thm:all-root-classical}, which is bounded at every root, connects that position to every differentiated chart argument. A derivative in a coarse field or in a free variable in $\Omega$ therefore cannot disconnect the difference factor from the network. The auxiliary quadratic has only indices in $H$, and any connected contribution from it to an external index in $\Omega$ passes through an index in $H$ and is bounded by \eqref{ch16:eq:hole-sum}. This verifies the hypotheses of \cref{ch15:lem:wall-networks} for the whole inserted difference, at every order and with any external root. \paragraph{Step 4: the gradient formula and the bounds on values.} Torus stationarity says that $J_T^{\mathsf T}E_{d_0}'(a_T^0)=0$. Adding and subtracting $J_R^{\mathsf T}E_{d_0}'(a_T^0)$ in $F_{R,z}(c,p;\xi)=J_R^{\mathsf T}E_{d_0}'(\widehat a_R)$, and applying the fundamental theorem of calculus on the complex segment from $a_T^0$ to $\widehat a_R$, we obtain the formula for $r_R$. The difference $J_R-J_T$ is supported in the construction band, in its regional input indices and its fine output indices. Its product with the gradient of the action on the fine links is kept: torus stationarity makes only the projection of this gradient onto the torus free directions vanish. By the classical bounds, we have \[ \|E_{d_0}'(a_T^0)\|_\infty +\|d_a\|_\infty\le C_J(\|c\|_\infty+\|\xi\|_\infty); \] for the first summand we used the bound on the torus stationary point, and for the second the word comparison of Step~2 and the same bound for $(p,h)$. Contracting the two terms of $r_R$ with their localized sources and using \eqref{ch16:eq:hole-sum}, we obtain \begin{equation}\label{ch16:eq:force-wall} |(r_R)_i|\le C_J(\|c\|_\infty+\|\xi\|_\infty) e^{-c_J\delta(i)}. \end{equation} Subtracting the regional gradient equations at $z_R$ and at $p$ and integrating along the segment between them, we obtain $\mathcal A_R(z_R-p)=-r_R$. The segment lies in the convex larger free box, and its averaged Hessian is the positive regional Hessian at the identity plus a coefficient perturbation satisfying the condition of \cref{ch16:lem:matrix-functions}(2). By that lemma, the inverse has entries bounded by $C_Je^{-b_J\dist(e,i)}$. Convolving this bound with \eqref{ch16:eq:force-wall} and using $\delta(e)\le\delta(i)+\dist(e,i)+L'_J$, we obtain the bound on $z_R-p$, at a fixed smaller exponent. For the fine field we have the identity \begin{equation}\label{ch16:eq:physical-response} a_R-a_T^0=d_a+ \int_0^1\Psi_{R,z}(c,p+v(z_R-p);\xi)(z_R-p)\,dv. \end{equation} The first term is supported in $H_{L_J}$, and the second is the product of a finite-range chart kernel, without boundary weight, with $z_R-p$. This proves the bound on fine links in $\Omega$. No decay is asserted at a fine link deep in $H$. \paragraph{Step 5: the coefficient bounds.} We now prove the coefficient bounds directly, using the interpolation \eqref{ch16:eq:augmented-objectives}. On a common smaller product of parameters, the stationary point $z_t$ of $F_t$ is obtained by the implicit iteration with the inverse of the free Hessian at the identity, which converges coefficientwise. The radius condition and the bound on the inverse times the gradient follow from \cref{ch15:thm:regional-gap-and-center}, and the scalar comparison equation is \eqref{ch16:eq:implicit-majorant}. Differentiating in $t$, we obtain \begin{equation}\label{ch16:eq:unperturbed-center-response} \dot z_t=-F_{t,zz}^{-1}D_z(c,z_t;\xi). \end{equation} This expression contains exactly one inserted difference from \eqref{ch16:eq:action-insertion}, and its convergent majorant is \eqref{ch16:eq:marked-majorant}. A further derivative in an input may hit the inverse, the inserted difference, or its argument $z_t$; in each case the joined network still contains the inserted difference and the index of the new input. We apply \cref{ch15:lem:wall-networks} and integrate from $t=0$ to $t=1$. Since the endpoints are $z_T$ and $(z_R,0)$, their difference on the regional list has factorial boundary-localized coefficient bounds, rooted at every index. Reading \eqref{ch16:eq:physical-response} as an identity of convergent majorant series, we obtain the corresponding bounds for the fine field, and the same argument applied to the formula for $r_R$ gives its bounds. Keeping one input $x$ fixed and summing over the remaining indices, we obtain the decay with a fixed input stated in part~(1). \paragraph{Step 6: the full output.} The composition $T_M^s\exp\Psi_R$ is holomorphic on the common larger chart, by the construction in \cref{ch15:thm:regional-gap-and-center}. At the stationary point it satisfies \[ \|\Log u_*\|_\infty\le C_J(\|c\|_\infty+\|\xi\|_\infty). \] The internal coordinates of $\Log u_*$ are $c$, and its other coordinates are the values of the coarse links which are not constrained in $\Omega$. These are analytic functions of $(c,z;\xi)$, whose differentiated series compose the coefficient series of the chart words with the series of the stationary point from Step~5. For instance, for any analytic direction $v$ the first derivative is \[ D_v\Log u_*=D_c\mathcal T\,D_vc +D_z\mathcal T\,D_vz_R+D_\xi\mathcal T\,D_v\xi, \qquad \mathcal T=\Log(T_M^s\exp\Psi_R). \] Since higher derivatives are compositions of these factors, all derivatives through the stationary point are included. We reduce the parameter radius so that the segment from $\Log u_*$ to $c\oplus0$ lies in the common torus target chart. If $\mathcal A_T(v)$ denotes the torus stationary fine field at the target $v$, then \begin{equation}\label{ch16:eq:target-change} a_T^*-a_T^0=\int_0^1 D\mathcal A_T(c\oplus0+t(\Log u_*-(c\oplus0))) [\Log u_*-(c\oplus0)]\,dt. \end{equation} The difference of targets is supported on coarse links which are not internal to $\Omega$. By the same implicit equation on the torus list, the lower bound for the torus constrained Hessian, the chart bounds, and the classical tensors give coefficient bounds without boundary weight for the response of $\mathcal A_T$. Each term of \eqref{ch16:eq:target-change} with a root in $\Omega$ therefore passes through a coarse link outside $\Omega$, and its composed network, including the derivatives of $u_*$, visits $H_{L_J}$. \Cref{ch15:lem:wall-networks} and the finite word bounds then give the same amplitude and fixed-input bounds as before. The same integral with $\mathcal A_T$ replaced by the torus free stationary point $z_T$ bounds $p^*-p$ on the regional free labels, where $(p^*,h^*)=z_T(\Log u_*)$. Thus $z_R-p^*=(z_R-p)-(p^*-p)$ satisfies the same bounds. For the new residual gradient, we subtract its value at $p$ and integrate $F_{R,zz}$ along the segment from $p$ to $p^*$. A Hessian without boundary weight multiplies this boundary-localized displacement, so the residual satisfies the same bounds. Its formula follows again from torus stationarity, now at the fixed target $\Log u_*$. These arguments are identities of convergent series in all variables, so they also give the fixed-input bounds at the full output. A representative in complementary forest coordinates, if needed, can use forest words of fixed depth based at the regional vertices and the coarse roots. These words change only the construction stencil outside $\Omega$, their coefficient series are among the chart bounds, and the comparison in $\Omega$ is unchanged. \paragraph{Step 7: the minimized action.} Let $f_t(c;\xi)=F_t(c,z_t;\xi)$. By stationarity and \cref{ch15:lem:minimum-calculus}, we have \begin{equation}\label{ch16:eq:value-envelope} \partial_tf_t=D(c,z_t;\xi). \end{equation} At the endpoints, $B_sf_1=V_R$ and $B_sf_0=E_{d_0+s}(c\oplus0)$. The second identity follows by iterating the recurrence $E_{d+1}(c)=M\min_zE_d(b(c,z))$ which defines the effective classical action: by \cref{ch15:lem:minimum-calculus} the successive minima equal the joint minimum, and the product of the $s$ factors $M$ is $B_s$. A nonconstant derivative of the right side of \eqref{ch16:eq:value-envelope} has the network structure checked in Step~3. Therefore, for $k\ge1$ and a root $i$ in $\Omega$, \begin{equation}\label{ch16:eq:value-tensors} \sum_{i_2,\ldots,i_k} e^{c_J\Tree(i,i_2,\ldots,i_k)} |D^k[B_s(f_1-f_0)](0;\xi)_{i,i_2,\ldots,i_k}| \le k!C_J^{k+1}e^{-c_J\delta(i)}, \end{equation} where the factor $B_s\le M^J$ is included in $C_J$. For a scalar series $f(c)=\sum_\nu f_\nu c^\nu$ we have, as in \eqref{ch15:eq:root-fraction}, \begin{equation}\label{ch16:eq:root-fraction} \sum_{|\nu|=k}\frac{\nu_i}{k}|f_\nu|e^{a\Tree(\nu)} =\frac1{k!}\sum_{i_2,\ldots,i_k} |D^kf(0)_{i,i_2,\ldots,i_k}|e^{a\Tree(i,i_2,\ldots,i_k)}. \end{equation} Indeed, the multiset obtained from $\nu$ by removing one occurrence of $i$ has $(k-1)!/[(\nu_i-1)!\prod_{j\ne i}\nu_j!]$ ordered listings, and $D^kf(0)$ evaluated on any of them is $\nu!f_\nu$. Fix $a_W,\alpha_W0$ in the smaller construction domain with $C_J\rho_0<1/4$. Inserting \eqref{ch16:eq:value-tensors} into \eqref{ch16:eq:root-fraction}, we bound the regional seminorm by \[ \sum_{k\ge1}\rho_0^{k-2}C_J^{k+1}<\infty. \] This proves part~(3), including degree one. The value $B_s(f_1-f_0)(0;\xi)$ is never subtracted in \eqref{ch16:eq:value-envelope}; it remains in the full action. \paragraph{Step 8: derivatives in the newly frozen coordinates.} Differentiating the regional gradient equation in $\xi$ and using invertibility of the Hessian, we obtain $D_\xi z_R=-F_{R,zz}^{-1}F_{R,z\xi}$. The mixed derivative has exactly two terms, \begin{equation}\label{ch16:eq:full-mixed-hessian} F_{R,uv}=\Psi_{R,u}^{\mathsf T}E_{d_0}''(\Psi_R)\Psi_{R,v} +E_{d_0}'(\Psi_R)\Psi_{R,uv}, \end{equation} and the bounds of \cref{ch15:thm:regional-gap-and-center} on the larger products control both terms for $(u,v)=(z,\xi)$ and $(u,v)=(z,c)$. An index of $\xi$ is at a fine position outside $\Omega$ or in its fixed interface stencil. Contracting it with the free inverse and summing over the labels of $\xi$, we obtain the row sum $C_Je^{-a_W\delta(e)}$ by \eqref{ch16:eq:hole-sum}. This proves both bounds on the mixed matrices and the bound on $D_\xi z_R$, with no change of the density of the inputs. \paragraph{Step 9: the Hessians.} We compare the endpoints of \eqref{ch16:eq:augmented-objectives} at their stationary points. Before any elimination, the total derivative is \begin{equation}\label{ch16:eq:hessian-interpolation} \frac{d}{dt}F_{t,zz}(c,z_t;\xi) =D_{zz}(c,z_t;\xi)+F_t^{(3)}(c,z_t;\xi)[\dot z_t]. \end{equation} Both terms contain the inserted difference, by \eqref{ch16:eq:action-insertion} and \eqref{ch16:eq:unperturbed-center-response}, and all terms of \eqref{ch16:eq:full-mixed-hessian}, including the one with the gradient of the action, are part of these tensors. For two matrix roots in $\Omega$, \cref{ch15:lem:wall-networks} gives the three distances in the Hessian bound of part~(5), and at degree $l$ in additional coarse-field and free arguments it gives the bound $(l+2)!C_J^{l+2}$ after $C_J$ is increased. For example, the corresponding coefficient series of the matrix is dominated on radius $R$ by \[ \sum_{l\ge0}\frac{(l+2)!}{l!}C_J^{l+2}R^l =\frac{2C_J^2}{(1-C_JR)^3},\qquad C_JR<1. \] The same majorant with a fixed input gives every version rooted at an input. \paragraph{Step 10: successive minima.} We restrict to smaller products for the initial segments before this comparison is used, following the construction of \cref{ch16:lem:admitted-products}(1) uniformly in $t$. For an initial segment $I$, we take a solution radius $r_I$ about the joint stationary point and a smaller radius $\eta_I$ for the deviation of the coarser coordinates. The principal inverse has bound $I_J$ and the mixed tensors have bound $C_J$, so at the joint stationary point the inverse times the conditional gradient is at most $I_JC_J\eta_I$. We require $I_JC_Jr_I<1/8$ and $I_JC_J\eta_I0$ such that the following statements hold on $\|c\|_\infty0$ with $a+2a_W0$ be a network exponent in the range just fixed, and let $\Delta_{e,\nu}$ be the Taylor coefficients of $\Delta_e$ in its analytic variables, namely the coarse field, the increments $q$, and the newly frozen coordinates $\xi$. For a fixed input $x$, the differentiated series \[ \mathcal D_{e,x}(\boldsymbol R) =\sum_{\nu:\nu_x\ge1}\nu_x|\Delta_{e,\nu}| \boldsymbol R^{\nu-\mathbf 1_x} e^{a\Tree(\{e\}\cup\supp\nu)}, \] with the measure conventions of \cref{ch16:sec:setting}, satisfies \[ \mathcal D_{e,x}(\boldsymbol R)\le C_Je^{-a_W[\dist(e,W)+\dist(e,x)]}. \] Unlike $C_{e,x}$ in part~(4), this series does not contain the factor $R_x$. \item At $q=0$, the dependence of $\Delta$ on the newly frozen coordinates satisfies, in the coefficient norm in $c$, \[ \|\Delta_e(\cdot,0;\xi)-\Delta_e(\cdot,0;0)\|_{\mathrm{coeff}} \le C_F\|\xi\|_\infty e^{-a_W\dist(e,W)}, \qquad \Delta(0,0;0)=0. \] \item At radius $s_q$ for the fluctuation increments, the absolute coefficient row sum of the terms of $\Delta$ of positive degree in $q$ is at most $C_{\Delta,J}s_q$. The rows of the coarse-field terms near the boundary and all rows in the free variables are only bounded by finite constants depending on $J$; no contraction factor $M^{-J}$ is asserted for them. \item (Input columns.) Let $R_y\le K_*\rho$ be a construction profile, and choose $a_1,b_1>0$ with $b_1>\awall$ and with $a_1+b_1$ smaller than a decay exponent for which the full reconstruction has coefficient bounds without boundary weight. Define \[ C_{e,x}(\boldsymbol R)=\sum_\nu\nu_x|\Phi_{R,e,\nu}(\xi)| \boldsymbol R^\nu e^{a_1\Tree(\{e\}\cup\supp\nu)}, \] where $\Phi_{R,e,\nu}(\xi)$ are the Taylor coefficients of $\Phi_R$ at the fine output $e$. Then $C_{e,x}(\boldsymbol R)\le D_JR_xe^{-b_1\dist(e,x)}$, and \[ \begin{gathered} \sum_e e^{-\awall d_{\mathrm{old}}(e)}C_{e,x}(\boldsymbol R) \le K_J\rho\,e^{-\awall d_{\mathrm{out}}(x)},\\ K_J\le C_{M,J}D_JK_*e^{\awall C_J} \sum_{l\ge0}(l+1)^3e^{-(b_1-\awall)l}. \end{gathered} \] Here $d_{\mathrm{old}}$ and $d_{\mathrm{out}}$ are the distances to the complements of the specified old and output position lists, respectively, both expressed in output units, and we assume that, for every summation label $e$ and every input $x$, \[ d_{\mathrm{out}}(x)\le d_{\mathrm{old}}(e)+\dist(e,x)+C_J, \] with a buffer $C_J$ depending only on the fixed construction geometry. The sum runs over the full list of old output labels (the fine labels), at their creation-scale density. For an input among the free coordinates the same statement holds with its own radius $R_x$ in place of $K_*\rho$ in the last bound. For a kernel from inputs to outputs, the column of a fixed input carries the volume of the output cells divided by the volume of the cell of that input. The same calculation applies to the profile $R_y=2\rho$ within the auxiliary construction ball and to the smaller spatially varying radius profiles of \cref{ch15:def:regional-norm}. These column bounds are different from the bounds on the image under the reconstruction majorant in \cref{ch16:lem:strict-two-radius-images}. \end{enumerate} The constants have the dependence of \cref{ch16:sec:setting}, are uniform in the depth and in the shape of $\Omega$, and do not depend on the later widths. No equality with a perturbed incoming constrained problem is asserted. \end{theorem} \begin{proof} We compare the successive regional and torus reconstructions before applying the gauge transformation with cutoff, and then estimate the differentiated coefficient series. \paragraph{Step 1: the domain of the successive substitutions.} We take a common larger construction product as in \cref{ch16:thm:nonflat-force-comparison}, with radii for coarse fields and fluctuation increments at most $U_T$ and $Q_T$ from \eqref{ch16:eq:torus-reserve}. Then the torus reconstruction, its gauge transformation, and all column series below are evaluated with $u+v0$ be a free construction radius, distinct from the density constant $v_J$, with \[ Q_Jv_{\mathrm{free},J}<\lambda_J/64, \qquad I_JQ_Jv_{\mathrm{free},J}0$, inside both the chart of the block averaging constraint and the chart of \cref{ch08:thm:compact-overlap}, which may be chosen below \[ \min\Bigl\{\frac{\min(r_{\Gamma,J},r_E)}{64G_J}, \frac{\lambda_J}{64Q_J},\frac{r_{z,J}}{64I_JQ_J}, \frac{v_{\mathrm{free},J}}{64C_{\mathrm{lift},J}},16\min(\rho_J,\beta_{\mathrm{rec},J})\Bigr\}. \] Here $\rho_J$ and $\beta_{\mathrm{rec},J}$ are the coarse and newly frozen parameter radii of \cref{ch15:thm:regional-gap-and-center}; thus the lifted arguments of size less than $\Rraw/8$ stay inside its parameter product. The quantities $G_J$ and $r_{\Gamma,J}$ are the bound and the radius of the finite constraint words, $r_E$ is the larger radius of the torus action, and $C_{\mathrm{lift},J}$ bounds the block-constant lift and the coarse-field lift, including all embeddings for the initial segments. If \[ C_{\mathrm{lift},J} [K_*\max(\rho,\rho_{\mathrm{cl}})+r+\rrecord+q_{\mathrm{raw}}] <\Rraw/8, \] where $r$ is the radius of the small-link domain $B_r$ and $q_{\mathrm{raw}}$ is the radius allowed for the fluctuation increments before gauge transformation, then all images of the reconstruction before gauge transformation lie in these fixed larger charts, and their real free Hessians have gap at least $\lambda_J/2$, with the complex inverse margins. No factor $M^{-J}$ is required of the lift. \item (Zero fluctuations.) For each $\rho_\bullet\in\{\rho,\rho_{\mathrm{cl}}\}$, use its own spatially varying radius profile $\phi_\bullet$, equal to the boundary radius $\rho_{\partial,\bullet}$ at labels within distance $w_{\mathrm{bd}}$ of $H$ and to $\rho_\bullet$ elsewhere, the outer profile $K_*\phi_\bullet$, and the outer boundary radius $R_{\partial,\bullet}=K_*\rho_{\partial,\bullet}$. For a full renormalization group step, $s=J$ and $B=M^J$, assume the torus contraction inequality (the first inequality below) and impose the four inequalities \[ \begin{aligned} B^{-1}Z_\bullet(K_*\rho_\bullet)&\le\rho_\bullet/16,& C_{\Delta,\bullet}C_{a,\bullet}R_{\partial,\bullet} &\le\rho_\bullet/16,\\ C_{\Delta,\bullet}C_{a,\bullet}e^{-a_Ww_{\mathrm{bd}}/2} K_*\rho_\bullet&\le\rho_\bullet/16,& C_{F,\bullet}\rrecord&\le\rho_\bullet/16, \end{aligned} \] where $Z_\bullet$ is the classical row majorant of \cref{ch10:thm:whole-image-gauge}, $C_{\Delta,\bullet}$ and $C_{F,\bullet}$ are the constants of \cref{ch16:thm:hybrid-columns}, and $C_{a,\bullet}$ is a summation constant fixed in the proof. These constants are evaluated on the larger auxiliary construction ball, before the radii and widths are chosen. The sum of the four left sides bounds the row sum $S_0$ at $q=0$ by $\rho_\bullet/4$. \item (Fluctuation increments.) For a radius $s_q$ of the fluctuation increments, the increment of $S_0$ is bounded by \[ W_\bullet(s_q)= \frac{C_{T,\bullet}s_q} {1-C_{T,\bullet}(K_*\rho_\bullet+s_q)} +C_{\Delta,\bullet}s_q. \] Require the denominator to be at least $1/2$ and $W_\bullet(s_q)\le\rho_\bullet/8$ for both radii. The same rule applies to the product in the jointly normalized coordinates $u$, with its finite-composition free radius and its own denominator, which includes $\rrecord$ if it occurs in that majorant. The products for the successive substitution must lie in this joint product. \item (Spatial weight.) The resulting unweighted absolute row sum satisfies $S_0\le3\rho_\bullet/8$. If $S_c\le C_J\rho_\bullet$ for a fixed $c>0$, then there is a fixed $0<\vartheta<1$, common to both radii and to the finitely many initial segments, such that \[ S_{\vartheta c}\le S_0^{1-\vartheta}S_c^\vartheta \le\rho_\bullet(3/8)^{1-\vartheta}C_J^\vartheta <\rho_\bullet/2. \] Thus, for both radii, the image under the reconstruction majorant, with connecting-network weight at one positive exponent, is less than half of the radius. \end{enumerate} All constants and thresholds have the dependence of \cref{ch16:sec:setting} and do not depend on the later widths. For a composition of $si};\xi)^{1/2} \bigl(z_i-a_i(c,z_{>i};\xi)\bigr), \] which inverts the unperturbed insertion at scale $i$. The factor $K_i^{1/2}=K_iK_i^{-1/2}$ has the same larger coefficient bounds. At $u=0$ all these increments vanish, since the chain at zero fluctuation is the joint stationary point. By finite composition of majorants, there exists a joint radius for which the resulting majorant of the increments is at most $Q_T$. We take the product of $u$ inside this radius and inside the solution domains of the initial segments. Its target argument is at most $U_T$, so \eqref{ch16:eq:torus-reserve} also verifies the torus domain and the denominator for this composition. The finite composition has a rational majorant of the same kind for its increment, with its own constants and argument; in particular, a contribution of the newly frozen coordinates to that argument remains in its denominator. Requiring this majorant to be at most $\rho_\bullet/8$ gives \eqref{ch16:eq:zero-exponent-image} for the joint product as well. The products for the successive substitution are restricted to the preimage of this product and to the domains of \cref{ch16:lem:admitted-products}. This restricts a coefficient series which is already bounded. \paragraph{Step 5: the spatial weight.} For one output row, we write its terms as $b_\nu\ge0$ and let $\ell_\nu$ be the length of the attached network, including the output root. For finite partial sums, by Hölder's inequality with exponents $1/(1-\vartheta)$ and $1/\vartheta$, we have \[ \sum_\nu b_\nu e^{\vartheta c\ell_\nu} =\sum_\nu b_\nu^{1-\vartheta} (b_\nu e^{c\ell_\nu})^\vartheta \le\Bigl(\sum_\nu b_\nu\Bigr)^{1-\vartheta} \Bigl(\sum_\nu b_\nu e^{c\ell_\nu}\Bigr)^\vartheta, \] and passing to increasing partial sums gives the inequality for the convergent series. Using $S_c\le C_J\rho_\bullet$ and \eqref{ch16:eq:zero-exponent-image}, we obtain the bound in part~(4). The function $\vartheta\mapsto(3/8)^{1-\vartheta}C_J^\vartheta$ is continuous at $\vartheta=0$, with value $3/8<1/2$. We fix $\vartheta\in(0,1)$ at which its value is below $1/2$ for both radii and for the finitely many initial segments. This proves both weighted bounds on the image under the reconstruction majorant at the same positive exponent. Because the weighted row bound is a bound of the unperturbed construction, this choice of exponent uses no later perturbative estimate. \paragraph{Step 6: shorter compositions.} For $s0$, and $r_c=\rho_{\mathrm{cl}}$. Let $b=(\xi,\Rec)$, where the newly frozen coordinates $\xi$ are small and the previously frozen coordinates $\Rec$ are arbitrary in their allowed real set. \emph{The incoming action.} Let $E_{\mathrm{in}}(a;b)=E_{d_0}(\iota_0a)+P(a;b)$ be holomorphic and real at real arguments in its variable input $a$, with $\Rec$ fixed, where $\iota_0$ extends $a$ by zero. Fix its incoming list $D_{\mathrm{in}}$ of variable labels, the positions of these labels at the lattices where they were introduced, and their cell volumes. The incoming seminorm is the seminorm $\mathcal X_{D_{\mathrm{in}},a_{\mathrm{cl}},\alpha_{\mathrm{cl}}}^{r_c}$ of \cref{ch15:def:regional-norm}, with the constant profile $\phi_{\mathrm{in}}\equiv r_c$ on every label of $D_{\mathrm{in}}$ at which $\Phi_0$ below is substituted; the labels of $\xi$ are fixed parameters of this seminorm. The spatially varying radius profiles are used only for the output. Assume that the complement of $D_{\mathrm{in}}$ lies in $H$ up to the fixed stencil, as in \cref{ch16:lem:input-wall-distance} ($H$ may contain additional blocks), and that, in input units, for every position $e$ of a label in $\Omega$ and every incoming position $i$, \[ d_{D_{\mathrm{in}}}(i)\ge B_s d_{\mathrm{current}}(e) -B_s\dist(e,i/B_s)-C_{\mathrm{buf},J}, \qquad B_s=M^s, \] where the positions on the right are in output units and the fixed buffer includes the chart stencils. If $D_{\mathrm{in}}$ has no complement, the nonconstant boundary part of the incoming perturbation is zero. All images of $\Phi_0$ and all connecting segments lie strictly inside the ball of the incoming seminorm. A difference of two perturbations uses the same incoming list, weights, and profile. \emph{The exterior extension.} If the unperturbed regional objective uses the extension $\iota_\xi$, which inserts the newly frozen coordinates into their slots while all previously frozen torus slots remain zero, define the classical boundary-completion difference and the adjusted perturbation \[ S_\xi^{\mathrm{cl}}(a)=E_{d_0}(\iota_\xi a)-E_{d_0}(\iota_0a), \qquad\overline P=P-S_\xi^{\mathrm{cl}}, \] and assume $\mathcal X_{D_{\mathrm{in}},a_{\mathrm{cl}},\alpha_{\mathrm{cl}}}^{r_c}(\overline P)\le2\sigma_*$. If the reference objective already uses the extension by zero, take $S_\xi^{\mathrm{cl}}=0$. All constants of $S_\xi^{\mathrm{cl}}$ remain in the full classical action. The corresponding difference for the effective interaction, if the reference extension is changed there, is kept with the exact sign of the constant term in \cref{ch15:lem:freezing}. \emph{The objective.} Let $z_0$ be the unperturbed joint stationary point, $K_0=F_{0,zz}(c,z_0;\xi)$, and $S_0=K_0^{-1/2}$, and put \[ \begin{split} F_0(c,z;\xi)&=E_{d_0}(\iota_\xi\Psi_R(c,z;\xi)),\\ A_0(c,u;\xi)&=\Psi_R(c,z_0+S_0u;\xi),\\ \Phi_0(c,u;\xi)&=\Log[(\exp A_0(c,u;\xi))^{G^\Omega(c)}],\\ \mathcal F_\lambda(c,u;b) &=F_0(c,z_0+S_0u;\xi) +\lambda\overline P(\Phi_0(c,u;\xi);b), \qquad0\le\lambda\le1, \end{split} \] and $\mathcal Q_\lambda(c,z;b)=\mathcal F_\lambda(c,S_0^{-1}(z-z_0);b)$. The perturbation is evaluated on the gauge-transformed reconstruction $\Phi_0$, and the unperturbed Hessian of $\mathcal F_\lambda$ in $u$ at $u=0$ is $I$. Initial segments, the conditional stationary points $a_{\lambda,I}$, and the Hessians $K_{\lambda,i}$ refer to $\mathcal Q_\lambda$, in the original free coordinates. Then the following statements hold. \begin{enumerate} \item (Incoming derivatives.) If the inputs at which $\overline P$ is evaluated have absolute coefficient row sums at most $\kappa\rho_{\mathrm{cl}}$, with $0<\kappa<1$, then for every $j\ge1$, \[ \|D^j\overline P\|_{\mathrm{root}} \le\frac{j!}{(1-\kappa)^{j+1}} (2\sigma_*)\rho_{\mathrm{cl}}^{2-j}, \] with the incoming network and boundary weights, and with any differentiated input as root. \item (Stationary points.) There exist $\sigma_{0,J},Q_{0,J}>0$ and a fixed smaller domain of $u$ such that the following holds for every $0\le\sigma_*\le\sigma_{0,J}$, every $0\le\lambda\le1$, and every product of increments with $\max_i\|q_i\|_\infty\le Q_{0,J}$ whose substitutions stay in that domain. Every initial segment of the original free coordinates has a unique holomorphic stationary point of $\mathcal Q_\lambda$ in the domains of \cref{ch16:lem:admitted-products}, and the inequalities of \cref{ch16:lem:admitted-products}(2) on the preconditioned gradient and Hessian hold, not only a spectral bound. The stationary points are real at real data, the real free Hessians satisfy \[ \frac12\lambda_J I\le K_{\lambda,i}\le\Lambda_J I, \qquad\|u_\lambda\|_{\mathrm{coeff}} \le C_J\sigma_*\rho_{\mathrm{cl}}, \] with the exact factors $M^i$ included in $K_{\lambda,i}$, and for every initial segment $I$ and every $y\in\mathcal T_I$, \[ \|a_{\lambda,I}(c,y;b)-a_I(c,y;\xi)\|_{\mathrm{coeff}} \le C_J\sigma_*\rho_{\mathrm{cl}}. \] The quantity $\delta_V$ of \cref{ch16:lem:admitted-products} satisfies $\delta_V\le I_0C_P\sigma_*\rho_{\mathrm{cl}}$, with constants $I_0,C_P$ depending only on the fixed data, and $\sigma_{0,J},Q_{0,J}$ are chosen so that the condition of \cref{ch16:lem:admitted-products}(3) holds for all initial segments. The weighted complex resolvents are bounded by $C_J/(\lambda_J+t)$ for $t\ge0$ at the working rate $b_{\mathrm w}$ fixed in part~(4). \item (Differences.) The differences from $\lambda=0$ of the stationary points, the Hessians, their inverses and inverse square roots, and the successive reconstructions have convergent boundary-localized coefficient bounds in all variables, including every coarse-field input in $\Omega$ and every fluctuation increment; their networks carry the incoming boundary dependence. These differences are $O_J(\sigma_*)$, and the displacements of stationary points are $O_J(\sigma_*\rho_{\mathrm{cl}})$ on the fixed product. For example, at fixed coarser coordinates $y$, \[ \partial_\lambda z_{I,\lambda} =-\mathcal Q_{\lambda,II}^{-1}V_I, \] where $V_I$ is the $I$-component of the gradient of $z\mapsto\overline P(\Phi_0(c,S_0^{-1}(z-z_0);\xi);b)$. Along the successive chain this derivative contains the additional term $-\mathcal Q_{\lambda,II}^{-1}\mathcal Q_{\lambda,II^c}\partial_\lambda z_{I^c}$ and the full derivative $(\partial_\lambda K_{\lambda,i}^{-1/2})q_i$. The function $V=\overline P\circ\Phi_0$ satisfies $\|V_u\|\le C_J\sigma_*\rho_{\mathrm{cl}}$ and $\|V_{uu}\|\le C_J\sigma_*$; the term $\overline P'\Phi_{0,uu}$ is kept in the second derivative. \item (Output rates.) The estimates of part~(3) hold at any fixed positive output rates $a_{\mathrm{out}},\alpha_{\mathrm{out}}$ with \[ a_{\mathrm{out}}+\alpha_{\mathrm{out}}0$, a fixed multiple of $\min(r_c,1)$, with $\mathfrak b_J(t_*)\le(1-\kappa)r_c/2$. By nonnegativity of its coefficients, the coefficient of degree $j$ of \eqref{ch16:eq:composed-jet-majorant} is at most $C_J\sigma_*r_c^2t_*^{-j}$, and restoring $j!$ gives the factorial bounds $j!C_J^{j+1}\sigma_*r_c^{2-j}$, with constants depending on the fixed radius. Fixing a coarse-field or free input uses the column version of the same composition, with the cell volume of that input. Thus all derivatives used here satisfy coefficient bounds on the product. \paragraph{Step 4: transport of the incoming boundary dependence.} A rooted input monomial carries its incoming network and a path from its chosen input root to the complement of $D_{\mathrm{in}}$. The incoming list and its complement are those used in $A_k(b)$. By hypothesis the complement of $D_{\mathrm{in}}$ lies in $H$ up to the fixed stencil, so \cref{ch16:lem:input-wall-distance} implies \eqref{ch16:eq:input-wall-comparison} with its fixed buffer, and hence the comparison assumed in the statement. We scale these paths by $B^{-1}$ to output units and join every network of the chart at its fine input label. The union is connected and contains every external coarse-field and free argument. If the input root lies in $\Omega$, its path to the complement of $D_{\mathrm{in}}$ reaches $H$ after at most one additional leg of fixed length. If it lies in a newly removed region, the chart branch joining it to a root in $\Omega$ already reaches the boundary. Additional components of $H$ can only shorten the required distance to the boundary. To make this precise, let $\ell_0$ be the length of the incoming network in input units, $d_0^{\mathrm{in}}$ the distance of the chosen root to the complement of $D_{\mathrm{in}}$, and $\ell_\beta$ the lengths of the branches of the chart and of the free inverse in output units. Let $A$ be the set of external positions and $e$ any root in $\Omega$. There exists a constant $C_{\mathrm{dim}}$, depending only on the dimension and on the comparison of elementary paths, and fixed before $J$, such that \begin{equation}\label{ch16:eq:network-transport} \begin{split} a_{\mathrm{out}}\Tree(A)+\alpha_{\mathrm{out}}\delta(e) \le{}&C_{\mathrm{dim}}(a_{\mathrm{out}}+\alpha_{\mathrm{out}}) \Bigl(\ell_0/B+\sum_\beta\ell_\beta\Bigr)\\ &+C_{\mathrm{dim}}\alpha_{\mathrm{out}}d_0^{\mathrm{in}}/B +(a_{\mathrm{out}}+\alpha_{\mathrm{out}})L'_J(k+1). \end{split} \end{equation} Indeed, the scaled incoming network and the chart branches alone join all external positions, and adding the scaled path to the complement of $D_{\mathrm{in}}$ also reaches the boundary. The first network bounds $\Tree(A)$, and the second bounds the distance of $e$ to the boundary; this gives the separate coefficients of $\ell_0$ and $d_0^{\mathrm{in}}$ in \eqref{ch16:eq:network-transport}. Here $k$ is the incoming degree, equivalently the number of attached input factors, including factors evaluated at a nonzero frozen value. The last term accounts for their fixed displacements by the stencil. To see why $C_{\mathrm{dim}}$ depends only on the dimension, we take shortest paths in the graph metric and their connected union. Using a tree on the external vertices increases the length by at most a factor two, by traversing each edge of a Steiner tree twice. Replacing segments by the fixed paths in coordinate order increases it only by the fixed comparison constant of the three coordinate directions. Scaling is exact, and neither operation depends on the number of scales. We sum the stencil offsets using the strict image bound, not by shrinking the external Taylor variables. Reduce the intrinsic rate $c_J$ once, before output rates are chosen, so that $e^{c_JL'_J}\le4/3$. Put $\beta=(a_{\mathrm{out}}+\alpha_{\mathrm{out}})L'_J$ and $\kappa'=e^\beta\kappa$. The reconstruction image has $\kappa\le1/2$, so the allowed output rates give $\kappa'\le2/3$. Multiplying the degree-$k$ derivative term of Step~2 by $e^{\beta(k+1)}$ gives \[ e^{\beta(k+1)}\kappa^{k-j} =e^{\beta(j+1)}(\kappa')^{k-j}. \] Thus its degree sum is bounded by \[ (2\sigma_*)r_c^{2-j} \frac{e^{\beta(j+1)}j!}{(1-\kappa')^{j+1}}. \] In \eqref{ch16:eq:composed-jet-majorant} the same calculation replaces $\kappa$ by $\kappa'$ and $\mathfrak b_J(t)$ by $e^\beta \mathfrak b_J(t)$, with one additional fixed factor $e^\beta$. A smaller fixed $t_*$ leaves a positive denominator. These changes alter only the fixed-$J$ amplitudes. They cover degree-zero contributions from frozen coordinates as well, without assuming geometric decay of the incoming sequence $A_k$. The intrinsic-rate reduction precedes all later profiles and widths and does not change $C_{\mathrm{geom}}$. We take $C_{\mathrm{geom}}\ge8C_{\mathrm{dim}}$, increased according to the fixed allocation of exponents among the external network, the route to the boundary, and the internal sums, and we keep part of the remaining exponent in each chart branch for its density sum $S_J(\varepsilon)$. Under the three strict inequalities of part~(4), the incoming factors $e^{a_{\mathrm{cl}}\ell_0}$ and $e^{\alpha_{\mathrm{cl}}d_0^{\mathrm{in}}}$, together with the exponent $c_J$ of the unperturbed branches, dominate the output network and boundary weights in \eqref{ch16:eq:network-transport}, and a positive part of the exponent is left for the summations. The exponent in the bound on the image under the reconstruction majorant from \cref{ch16:lem:strict-two-radius-images} is included in the choice of $c_J$ before these rates are specified. The allocation is applied to the whole network, so the exponent is not divided by the Taylor order or by the number of scales. This proves the boundary-localized versions of \eqref{ch16:eq:perturbing-bounds} and \eqref{ch16:eq:composed-jet-majorant} at every allowed output rate. \paragraph{Step 5: the joint stationary point.} We now verify the domain conditions, working first in $u$ near zero. By the unperturbed bound on the third derivative and \eqref{ch16:eq:unit-joint-hessian}, we have \[ \|D_u^2F_0(c,z_0+S_0u;\xi)-I\|\le C_1\|u\|_{\mathrm{coeff}}. \] Fix $u_0>0$ inside the joint product with $C_1u_0<1/8$, and fix $\sigma_{0,J}>0$ with $C_P\sigma_{0,J}<1/8$ and $C_P\sigma_{0,J}r_c0$ and, if necessary, reduce $\sigma_{0,J}$ further, so that \[ C_{\mathrm{seq},J}(2I_0C_P\sigma_{0,J}r_c+Q_{0,J}) <\frac12\varepsilon_{\mathrm{tube}}, \qquad Q_{0,J}\le r_c, \] where the conversion between the original and the normalized coordinates is included in the minimal margin. The bound \eqref{ch16:eq:controlled-conditional-force} verifies the condition of \cref{ch16:lem:admitted-products}(3) for every $y$. The recurrence bounds the substituted higher arguments and the constructed conditional centers. At each such argument, the Picard iterations stay in the fixed solution balls of \cref{ch16:lem:admitted-products}(2); these balls were chosen inside the larger reconstruction product. Thus the hypothesis on the image under the reconstruction majorant holds throughout the construction. This proves the statements on stationary points in part~(2), for all initial segments simultaneously, on the product of increments. \paragraph{Step 8: the Hessians.} The unperturbed real Hessian in the original coordinates is at least $7\lambda_J/8$ on these products, since its variation from the Hessian at the identity is at most $\lambda_J/8$ by \cref{ch15:thm:regional-gap-and-center}, and the perturbation changes it by less than $\lambda_J/8$. Its principal blocks and Schur complements have the same lower bound, since for a symmetric matrix $H\ge\lambda I$ \[ v^{\mathsf T}(H_{yy}-H_{yI}H_{II}^{-1}H_{Iy})v =\min_x(x,v)^{\mathsf T}H(x,v)\ge\lambda|v|^2, \] as in \eqref{ch15:eq:schur-lower}. After the normalization by $M^i\ge1$, we obtain $K_{\lambda,i}\ge\lambda_JI/2$. The upper bounds on the tensors and on the inverses give a common upper bound $\Lambda_J$ on the same product. The complex resolvent condition is used at the working exponent $b_{\mathrm w}$ defined in part~(4), with strict reserve for the output network and boundary weights. Its defining inequalities put it below $b_{\mathrm{ref}}$ and inside the transported incoming-rate allocation of Step~4, uniformly over every allowed output pair. For each flat reference $K_i^b$, including the factor $M^i$ when present, the unperturbed bounds give \[ \|(K_i^b+t)^{-1}\|_{b_{\mathrm w}} \le\frac{C_{0,J}}{\lambda_J+t}\qquad(t\ge0). \] Put $L_i=K_{\lambda,i}-K_i^b$. At each fixed coarser argument on the conditional products, the fixed-reference Picard bounds of Steps~5 and~6 give the coefficient displacement of the conditional center. For this bulk matrix estimate the offset calculation of Step~4 is used with $\beta_{\mathrm w}=b_{\mathrm w}L'_J$: since $b_{\mathrm w}\le c_J$, one still has $e^{\beta_{\mathrm w}}\kappa\le2/3$, and $C_{\mathrm{dim}}b_{\mathrm w}\le\min(c_J,Ba_{\mathrm{cl}})/2$ pays the transported network. Together with the unperturbed mixed tensor bounds, this bounds the perturbative matrix difference at $b_{\mathrm w}$ by $C_{P,b_{\mathrm w},J}\sigma_*$. For a Schur block, telescope its defining product and use the principal inverses already constructed. This estimate is obtained before successive substitution. Since $b_{\mathrm w}i})+K_{\lambda,i}(c,z_{>i})^{-1/2}q_i] =\partial_\lambda a_{\lambda,i} +D_ya_{\lambda,i}\dot z_{>i} +\frac{d}{d\lambda}K_{\lambda,i}^{-1/2}q_i, \] where the last derivative includes both the explicit change of the objective and the change of the coarser argument. We start at the coarsest conditional stationary point and descend. Each direct derivative carries the perturbation, and each new contribution from the coarser arguments contains a factor which is already known to carry it. By \eqref{ch16:eq:schur-response} and \eqref{ch16:eq:sqrt-difference} this remains true for every inverse square root used at the next scale. Since $\|q_i\|_\infty\le Q_{0,J}\le r_c$, the finite recurrence gives $O_J(\sigma_*r_c)$ for the stationary points and the reconstructed displacements, and $O_J(\sigma_*)$ for their matrix coefficients. Differentiating in a coarse-field or free input fixes the index of that input in these convergent series, and the allocation of Step~4 keeps the perturbation and every external argument in one connected network. This proves part~(3) on the product, and together with Step~4 it proves part~(4). \paragraph{Step 10: two perturbations.} For two perturbations we interpolate between them. Since the coefficient bounds are convex and the fixed bound $\sigma_{0,J}$ holds for both, every interpolated objective stays in the same domains. Its derivative in the interpolation parameter is given by \eqref{ch16:eq:perturbed-center-response} with $V=(\overline P_1-\overline P_0)\circ\Phi_0$. We set \begin{equation}\label{ch16:eq:input-difference-norm} \Delta_{\mathrm{in}}= \mathcal X_{D_{\mathrm{in}},a_{\mathrm{cl}},\alpha_{\mathrm{cl}}}^{r_c} (\overline P_1-\overline P_0), \qquad \phi_{\mathrm{in}}\equiv r_c. \end{equation} Since both perturbations use the same $D_{\mathrm{in}}$, the same positions and cell volumes, and the same incoming boundary, all the preceding difference estimates are linear in $\Delta_{\mathrm{in}}$. We write $\mathcal Q_\lambda$ also for the interpolated objective. The derivative of the outgoing values in the interpolation parameter is \[ \partial_\lambda[M^s\mathcal Q_\lambda(c,z_\lambda;b)] =M^s(\overline P_1-\overline P_0) (\Phi_0(c,u_\lambda;\xi);b). \] Its nonconstant coefficient networks contain the same incoming difference, also when a derivative hits the stationary point. To display the degree sum, denote the right side by $G_\lambda$ and use a fixed outer profile $\psi\ge\zeta\phi_{\mathrm{cl}}$, $\zeta>1$, within the already chosen construction product. The distinguished reconstruction column and the transport estimate of Step~4 give \[ \sup_{i\in\Omega}e^{\alpha_{\mathrm{out}}d_\Omega(i)} \sum_{\nu\ne0}\nu_i|(G_\lambda)_\nu| e^{a_{\mathrm{out}}\Tree(\nu)}\psi^\nu \le C_Jr_c^2\Delta_{\mathrm{in}}. \] At degree $n$, restriction to $\phi_{\mathrm{cl}}$ supplies $\zeta^{-n}$ and the root fraction in \eqref{ch16:eq:root-fraction} supplies $1/n$. Thus, uniformly in $\lambda$ and the frozen values, \[ \mathcal X_{\Omega,a_{\mathrm{out}},\alpha_{\mathrm{out}}}^{\phi_{\mathrm{cl}}, \mathrm{cl}}(G_\lambda) \le C_J\Delta_{\mathrm{in}}\sum_{n\ge1}\frac{\zeta^{-n}}n =-C_J\Delta_{\mathrm{in}}\log(1-\zeta^{-1}). \] The fixed profile ratio is used once for this scalar sum. Integrating in $\lambda$ gives \begin{equation}\label{ch16:eq:output-lipschitz} \mathcal X_{\Omega,a_{\mathrm{out}},\alpha_{\mathrm{out}}}^{\phi_{\mathrm{cl}}, \mathrm{cl}} (V^{\mathrm{out}}_1-V^{\mathrm{out}}_0) \le K_J^{\mathrm{cl}}\Delta_{\mathrm{in}}. \end{equation} Here the outgoing classical seminorm uses the output profile $\phi_{\mathrm{cl}}$ with interior radius $r_c$. The seminorm in \eqref{ch16:eq:input-difference-norm} is the incoming seminorm with constant profile. The output seminorm includes degree one. A difference of incoming constants stays in the full value and has zero seminorm. This proves part~(5). \paragraph{Step 11: the full action.} We prove the identity of part~(6), with its constants. When $\Psi_R$ is written on its regional list, $\iota_\xi$ inserts the newly frozen coordinates once; when it is written as a complete fine field, the same insertion is already included. In either notation, the exact identities of \cref{ch16:lem:strict-hybrid-gauge} at the endpoints imply the equality of complete group fields \[ \exp(\iota_\xi\Phi_0) =(\exp(\iota_\xi A_0))^{G^\Omega(c)}, \] since the basepoint and the fine endpoints of every based transport of a newly frozen coordinate are protected, and the gauge transformation is the identity on the previously frozen extension. For real data both complete fields lie in the common chart of \cref{ch08:thm:compact-overlap}: the field before gauge transformation by the choice of $\Rraw$, and the gauge-transformed field by the bound on the image under the reconstruction majorant and the bound on the newly frozen coordinates. The gauge transformation is periodic and compact there. Thus \cref{ch08:thm:compact-overlap} applies to the torus classical action, and only to it, and we obtain \begin{align*} \mathcal F_1 &=E_{d_0}(\iota_\xi\Phi_0) +P(\Phi_0;b)-S_\xi^{\mathrm{cl}}(\Phi_0)\\ &=E_{d_0}(\iota_0\Phi_0)+P(\Phi_0;b) =E_{\mathrm{in}}(\Phi_0;b). \end{align*} These equalities keep $S_\xi^{\mathrm{cl}}(0)$ as well as all nonconstant coefficients of $S_\xi^{\mathrm{cl}}$; no constant Taylor coefficient has been subtracted. Both sides are holomorphic on the connected product, and they agree on its real open part. Expanding at a real interior point and differentiating the real identity in each coordinate, we find that all Taylor coefficients of the difference vanish. Continuation through overlapping polydiscs in the connected product gives the equality everywhere on it. If the reference extension of the effective interaction is changed, the corresponding difference is treated by \cref{ch15:lem:freezing}. In particular, replacing a resulting interaction $R(v,c)$ by $R(v,c)-R(0,c)$ changes the normalization constant from $\Lambda$ to $\Lambda-R(0,c)$ and preserves the density; the constant of the extension is not deleted. The perturbation has been evaluated on $\Phi_0$ throughout. \paragraph{Step 12: dependence on the frozen coordinates.} Fix the geometry, the finite lists of labels, and a strictly smaller analytic product, and suppose that the incoming coefficients and the charts are Borel functions of the real parameter $b$. A finite Taylor partial sum of any chart or incoming function is then jointly Borel in its analytic arguments and $b$. Since the common coefficient bounds on the larger radius give a geometric tail on the smaller product, uniformly in $b$, the limits of the series are jointly Borel. For the scalar incoming function this argument is applied to its part of positive degree, and the unrestricted coefficient $p_0(b)$ is kept as a separate Borel summand. Every finite-dimensional matrix inverse used in a Picard iterate is a quotient of polynomials in its entries with nonzero determinant on the domain, hence Borel there, and each Picard iterate is a composition of such operations and of the jointly Borel functions just obtained. The uniform contraction ratio bounds the distance of the iterates to the stationary point by a geometric sequence, uniformly in $b$, which implies that the stationary point is jointly Borel. This applies first to the joint problem and then to every conditional problem, at arbitrary coarser arguments in the domains, and the finite successive substitution of these maps is Borel. For matrix functions we use \eqref{ch16:eq:resolvent-series} and \eqref{ch16:eq:sqrt}. Finite Neumann sums are Borel, and their uniform geometric limit gives the resolvent. On a compact interval of integration inside $(0,\infty)$ the Riemann sums of the integral for the inverse square root are Borel, and the uniform bounds on the gap and the resolvent make the tails at zero and at infinity small, uniformly in the parameters; hence the integral is Borel. The same argument applies to the differences and to the logarithmic factors on their fixed nonvanishing branches. Evaluating the full objective at the stationary point gives Borel minimized values, with the incoming constant restored exactly. If the coefficients and charts are continuous in $b$, every finite partial sum, inverse, and Picard iterate in this argument is continuous. The common geometric bounds and contraction ratios make all limits uniform on the smaller products, and the tails of the integrals satisfy the same uniform bounds. The limits and the finite successive compositions are therefore continuous in $b$. This proves part~(7) without any derivative in $\xi$ or $\Rec$. \paragraph{Step 13: shorter compositions.} For $s