\chapter{Finite-dimensional integration and concentration inequalities}\label{ch02:chapter} This section collects the finite-dimensional tools used throughout the renormalization group analysis. Each statement concerns a fixed finite lattice, a finite set of coordinates, or a fixed finite-dimensional measure; where we say so, the constants do not depend on the dimension. The results fall into three groups. The first group concerns gauge integration and Gaussian algebra. \Cref{ch02:lem:haar-and-forests} records the normalization of Haar measure on $\SU$, exact integration in a spanning-tree gauge, and analytic estimates for products of group elements near the identity, and \cref{ch02:lem:hodge-schur} decomposes closed periodic one-cochains and eliminates positive definite quadratic blocks. The second group is the calculus of the analytic coefficient norms of \cref{ch01:def:analytic-norms}: compositions, localized inverses and square roots, logarithms of determinants, and implicit functions (\cref{ch02:lem:coefficient-algebra}). The third group is probabilistic. \Cref{ch02:thm:reflecting-concentration} proves logarithmic Sobolev, covariance, and quadratic concentration inequalities for uniformly log-concave measures on products of balls, \cref{ch02:lem:bounded-transfer} controls conditioning, parameter derivatives, and gauge-invariance identities, and \cref{ch02:lem:seven-traces} shows that seven traces determine a triple of $\SU$ matrices up to simultaneous conjugation. The only inputs are the definitions of \cref{ch01:chapter}: the Wilson measure and the normalization of the Lie algebra (\cref{ch01:def:ensemble}), the coefficient norms (\cref{ch01:def:analytic-norms}), and the convention on constants (\cref{ch01:conv:constants}). Spanning-tree gauges and the estimates for products of group elements enter the construction of the block averaging map and the gauge fixing in \cref{ch03:chapter,ch04:chapter}, and the local Wilson flow in \cref{ch21:chapter}. The Gaussian elimination is used for the linearized theory in \cref{ch05:chapter}. The coefficient calculus underlies all analytic estimates in \cref{ch03:chapter,ch05:chapter,ch06:chapter,ch07:chapter,ch09:chapter,ch15:chapter}. The concentration theorem is the probabilistic input for the conditional fluctuation kernels in \cref{ch10:chapter,ch13:chapter,ch14:chapter,ch17:chapter,ch23:chapter,ch24:chapter,ch25:chapter,ch30:chapter}. The conditioning lemma is used in \cref{ch12:chapter,ch25:chapter,ch26:chapter,ch29:chapter}, and the seven traces in \cref{ch27:chapter,ch31:chapter,ch32:chapter}. \section{Haar measure, spanning-tree gauges, and products of group elements}\label{ch02:sec:haar} The following facts are standard for the compact group $\SU$ and for lattice gauge theory; we record them with the normalizations of \cref{ch01:sec:model}, which fix the constants used later. Gauge fixing along a spanning tree is the maximal-tree gauge, which we call the spanning-tree gauge. Bałaban used such gauges systematically in the renormalization group context \cite{Balaban1985Regular}. \begin{lemma}\label{ch02:lem:haar-and-forests} Let $\mathcal K$ be a finite oriented cubical complex, with vertex set $\mathcal V$, set of positive links $\mathcal E$, and set of positive plaquettes $\mathcal P$. Then the following statements hold. \begin{enumerate} \item For every $\beta\ge0$, \[ Z_{\mathcal K}(\beta)=\int e^{-\beta\sum_{p\in\mathcal P}q(U_p)}\prod_{e\in\mathcal E}dH(U_e) \quad\text{satisfies}\quad e^{-2\beta\abs{\mathcal P}}\le Z_{\mathcal K}(\beta)\le1, \] and the normalized density defines a gauge-invariant Borel probability measure. \item The measure $H$ is the Haar probability measure, and for $X=x^a\tau_a$ with $0\le r=\abs X<2\pi$, \[ \frac12\tr e^X=\cos(r/2),\qquad dH(e^X)=c_{\mathrm{Haar}}j_H(X)\,d^3x, \] where $c_{\mathrm{Haar}}=1/(16\pi^2)$ and $j_H(X)=\sinc^2(r/2)$. For $0\le u\le2\pi$ and $U,V\in\SU$, \[ H\{\rho_G0$, the Cauchy integral in each coordinate bounds a derivative of multi-index $\alpha$ by $B\alpha!\delta^{-\abs\alpha}$. To see this, we insert the absolutely convergent power series into the circle integrals; the integral of $z^k/z^{j+1}$ over a circle around zero vanishes for $k\ne j$ and equals $2\pi i$ for $k=j$. Summing all ordered derivatives of total order $s$ introduces at most a factor $(3\ell)^s$, which gives the bound $s!C^s$ for the absolute derivative tensor. Where the branches are defined, $\Log(kWk^{-1})=k(\Log W)k^{-1}$ and $\Log(W^{-1})=-\Log W$, by the same formula and analytic continuation. \emph{The expansion with a removable singularity.} Let $B(\epsilon)=\Log(e^ue^{\epsilon a}e^{-u-\epsilon v})$. For small fixed complex $u,v,a$ it is holomorphic in $\abs\epsilon<1+\delta_0$, with uniform bounds, and $B(0)=0$. Differentiating the exponential series, we obtain \[ \frac{d}{d\epsilon}e^{-u-\epsilon v}\Big|_{\epsilon=0}=-\int_0^1e^{-(1-t)u}ve^{-tu}\,dt. \] Since the derivative of $\Log$ at $I$ is the identity, $B'(0)=\Ad(e^u)a-\int_0^1\Ad(e^{tu})v\,dt$. The product depends on $a,v$ only through $(\epsilon a,\epsilon v)$, and the coefficient of $\epsilon^k$ in $B$ is therefore homogeneous of degree $k$ in $(a,v)$. Hence \[ \mathscr R_\epsilon=\int_0^1(1-t)B''(t\epsilon)\,dt \] is holomorphic, vanishes to order at least two in $(a,v)$, and satisfies the bounds just proved. Using Taylor's formula along the segment $t(a,v)$ and the bound on second derivatives, we find on a fixed smaller complex ball \begin{equation}\label{ch02:eq:removable-remainder-order} \norm{\mathscr R_\epsilon(u,v,a)}\le C(\abs a+\abs v)^2,\qquad 0\le\epsilon\le1, \end{equation} with $C$ depending only on the product, the larger complex domain, and its margin. At $v=0$ the product is $\exp(\epsilon\Ad(e^u)a)$, its logarithm is $\epsilon\Ad(e^u)a$, and the remainder vanishes. Every argument concerns a fixed product or an exact identity, and the stated dependence of the constants follows. \end{proof} \section{Closed one-cochains and Gaussian elimination}\label{ch02:sec:hodge} Parts (1) and (2) of the next lemma are the discrete Poincaré lemma and the fact that the first real cohomology of the three-torus is $\R^3$; part (3) is Gaussian elimination by Schur complements \cite{Bhatia1997}. The harmonic part $c$ is kept as data throughout. \begin{lemma}\label{ch02:lem:hodge-schur} The following statements hold. \begin{enumerate} \item On every finite contractible rectangular box in $\Z^3$, a real closed one-cochain is the coboundary of a function on the vertices. \item On the periodic cubic lattice of side $n\ge4$, every real closed one-cochain $\alpha$ has a unique decomposition $\alpha=\nabla f+c$, where $f$ is periodic with mean zero and $c$ is constant in each link direction. Moreover $c_i=n^{-3}\sum_x\alpha_i(x)$. The decomposition holds in each color separately and after rescaling to physical units. Thus requiring the spatial mean of $\alpha$ to vanish removes exactly $c$, and not a gauge degree of freedom. \item Let a real symmetric nonnegative quadratic form on a finite-dimensional space of fixed coordinates $c$ and integration coordinates $z\in\R^f$ have matrix \[ \mathsf A=\begin{pmatrix}A&B\\B^{\mathsf T}&K\end{pmatrix},\qquad K>0. \] For every $c$, including all specified harmonic coordinates, \[ \frac12(c,z)^{\mathsf T}\mathsf A(c,z) =\frac12c^{\mathsf T}(A-BK^{-1}B^{\mathsf T})c +\frac12(z+K^{-1}B^{\mathsf T}c)^{\mathsf T}K(z+K^{-1}B^{\mathsf T}c), \] and \[ \int_{\R^f}e^{-\frac12(c,z)^{\mathsf T}\mathsf A(c,z)}\,dz =(2\pi)^{f/2}(\det K)^{-1/2}e^{-\frac12c^{\mathsf T}(A-BK^{-1}B^{\mathsf T})c}. \] Successive minimization, or successive Gaussian integration, over the blocks of any ordered decomposition of $z$ gives the same Schur complement and the same product of determinants as simultaneous elimination. Conditionally on $c$, the Gaussian measure in $z$ has mean $-K^{-1}B^{\mathsf T}c$ and covariance $K^{-1}$. A Gaussian measure defined on a specified orthogonal complement of a null space, by a quadratic form positive there, has covariance equal to the inverse on that complement and zero on the null space. \end{enumerate} These are exact finite-dimensional statements; in particular, no lower bound on eigenvalues uniform in the side $n$ is asserted. \end{lemma} \begin{proof} \emph{Boxes.} We regard a one-cochain $\alpha$ as an antisymmetric function on oriented links; closedness means that its signed sum around every unit square vanishes. On $\Z^3$, a finite path can be sorted by coordinate direction by adjacent interchanges of steps in different directions. Each interchange crosses exactly one unit square and preserves the integral of $\alpha$, and adjacent opposite steps in one direction cancel. Then every closed path has integral zero, since after sorting its total displacement in each direction is zero and all steps cancel. Inside a rectangular box we integrate from a fixed corner, first in direction $1$, then in direction $2$, then in direction $3$. Interchanging an additional final step in direction $i$ with the later coordinate segments uses only the squares of the intervening rectangle, which lies in the box, and closedness makes the two path integrals equal. The resulting vertex function has coboundary $\alpha$ on every link. Boxes with a side of length zero are covered by omitting the absent direction. \emph{The periodic lattice.} We lift a periodic closed cochain to $\Z^3$. The argument just given produces a function $f_0$ on $\Z^3$ with $f_0(x+e_i)-f_0(x)=\alpha_i(x)$. Since $\alpha$ is periodic, $f_0(x+ne_i)-f_0(x)=P_i$ does not depend on $x$, because its difference across every link is zero; equivalently, $P_i$ is the integral of $\alpha$ along any straight cycle winding once in direction $i$. Set $c_i=P_i/n$. The function $f(x)=f_0(x)-\sum_ic_ix_i$ is periodic and satisfies $\alpha=\nabla f+c$; subtracting its average makes its mean zero. If $\nabla f+c=0$, then summing over a winding cycle gives $nc_i=0$. Then all differences of $f$ vanish, and the mean-zero condition implies $f=0$, which proves uniqueness. Because the sum of $\nabla_if$ over all periodic vertices telescopes to zero, we have $c_i=n^{-3}\sum_x\alpha_i(x)$, and the mean condition removes exactly the three constants. For cochains with values in $\su$, the argument applies to each real component. For physical potentials we use $\nabla_a=a^{-1}\nabla$ and integrate $a\alpha$ along paths; the conclusion has the same form, with $c_i=P_i/(na)$. \emph{Gaussian elimination.} If we expand the right side of the completion of the square, its cross term is $c^{\mathsf T}Bz$, and the added constant $c^{\mathsf T}BK^{-1}B^{\mathsf T}c/2$ cancels the subtracted one. After the translation $z\mapsto z-K^{-1}B^{\mathsf T}c$, an orthogonal diagonalization of the positive matrix $K$ reduces the integral to products of $\int_\R e^{-kt^2/2}dt=\sqrt{2\pi/k}$. To diagonalize $K$, we maximize its Rayleigh quotient on the unit sphere to obtain an eigenvector, restrict to the orthogonal complement of this eigenvector, and induct on the dimension. The value of the one-dimensional integral follows by squaring it and using polar coordinates in the plane. We obtain the Gaussian formula. For $f=0$, the centered Gaussian integral and the empty determinant $\det K$ are one; the full integral is $\exp(-c^{\mathsf T}Ac/2)$. The same completion of the square with a linear term in $z$, differentiated at zero, gives the conditional mean and covariance. For successive elimination, write $z=(z_1,z_2)$ and $K=\left(\begin{smallmatrix}K_{11}&K_{12}\\K_{21}&K_{22}\end{smallmatrix}\right)$. Both $K_{11}$ and $K_{22}-K_{21}K_{11}^{-1}K_{12}$ are positive definite, since their quadratic forms are the restriction of the form of $K$ and the minimum of that form over $z_1$. The change of variables $z_1\mapsto z_1+K_{11}^{-1}K_{12}z_2$ has determinant one, and by block multiplication we have \[ \det K=\det K_{11}\,\det\bigl(K_{22}-K_{21}K_{11}^{-1}K_{12}\bigr). \] The unique joint minimizer in $z$ solves $Kz=-B^{\mathsf T}c$; solving the first block and substituting gives the successive minimizers. The minimum value is therefore the same Schur form. By the determinant identity and the Gaussian formula, successive integration gives the same normalization. Induction over the blocks proves the statement for every ordered decomposition; for nonnegative integrands the repeated integrals are justified directly. No harmonic coordinate is discarded, since the vector $c$ was arbitrary throughout. On the given orthogonal complement of a null space, we diagonalize the restricted positive form. In the resulting coordinates the Gaussian measure is a product of independent one-dimensional Gaussian measures with variances equal to the inverse eigenvalues; embedding this measure by setting the null coordinates equal to zero, we obtain the stated covariance. This construction lives on the complement; it is not a Lebesgue integral along null directions. \end{proof} \section{Analytic coefficient bounds and implicit functions}\label{ch02:sec:coefficients} The following lemma is the calculus of the coefficient norms of \cref{ch01:def:analytic-norms}. Its first parts are combinatorial. A Taylor coefficient of a composite map is a finite or absolutely convergent sum of products of Taylor coefficients of the constituents. We bound the absolute value of the sum by the sum of absolute values. The spatial weight survives because the connecting networks of the factors can be joined at the contracted indices, and the volume factors in the column sums telescope. The simplest instance is a product of two kernels, $T$ from mesh $a_{\mathrm{mid}}$ to mesh $a_{\mathrm{out}}$ and $S$ from mesh $a_{\mathrm{in}}$ to mesh $a_{\mathrm{mid}}$. In counting coordinates the kernel of $TS$ is $\sum_jT_{ij}S_{jk}$, without any volume factor, and its column sum at a fixed input $k$ carries the factor \[ \frac{a_{\mathrm{out}}^3}{a_{\mathrm{in}}^3}=\frac{a_{\mathrm{out}}^3}{a_{\mathrm{mid}}^3}\cdot\frac{a_{\mathrm{mid}}^3}{a_{\mathrm{in}}^3}, \] so that column norms are submultiplicative. For a composition with one fixed input, we use column sums along the chain of factors that joins this input to the output, and row sums for all other factors. The later parts concern inverses and square roots of matrices with a spectral gap. The exponential decay of an inverse follows from the conjugation argument of Combes and Thomas \cite{CombesThomas1973}, and square roots are represented by the standard resolvent integral \cite{Bhatia1997}; the proof keeps track of the spatial weights and of all fixed indices. Bounds by coefficientwise majorant series are a classical tool of constructive renormalization group methods; the volume-normalized columns, the arbitrary fixed inputs, and the traces with a fixed external index are specific to our use. Throughout this section, all indices are scalar indices and all sums are counting sums. A contraction identifies two indices of the same kind (same lattice and scale of introduction); an identification of indices of different kinds must first be represented by an explicitly bounded transfer kernel. We fix a rate $\asp\ge0$ and write $\abs T=\abs T_{\asp}$ for the norm \eqref{ch01:eq:tensor-norm}, $R(T)$ for its row norm, and $C_r(T)$ for its volume-normalized column sums. For a scalar tensor we use the sums \eqref{ch01:eq:scalar-root} with one index fixed; on several meshes, a scalar tensor may instead be normalized by a reference volume $v_*$, which multiplies each summand with the index $i_r$ fixed by $v_*/v_{i_r}$. Distances and network lengths are those of \cref{ch01:def:analytic-norms}, measured in units of the current output lattice spacing. For a matrix $B=(B_{xy})$ of blocks indexed by a finite metric space $(X,d)$, we write \[ \norm B_c=\max\Bigl\{\sup_x\sum_ye^{cd(x,y)}\norm{B_{xy}},\ \sup_y\sum_xe^{cd(x,y)}\norm{B_{xy}}\Bigr\},\qquad c\ge0, \] where $\norm{B_{xy}}$ is the operator norm of the block. \begin{lemma}\label{ch02:lem:coefficient-algebra} The following statements hold. \begin{enumerate} \item \emph{Substitution.} For tensors $T(o;j_1,\dots,j_k)$ and $S_r(j_r;\mathbf l_r)$, $1\le r\le k$, the tensor \[ U(o;\mathbf l_1,\dots,\mathbf l_k)=\sum_{j_1,\dots,j_k}T(o;j_1,\dots,j_k)\prod_{r=1}^kS_r(j_r;\mathbf l_r) \] has every output-row norm and every volume-normalized input-column norm bounded by $\abs T\prod_r\abs{S_r}$. In particular compatible matrix products are submultiplicative in the row norm and in the volume-normalized column norm; for counting-coordinate kernels from mesh $a_{\mathrm{in}}$ to mesh $a_{\mathrm{out}}$ the column factor is exactly $a_{\mathrm{out}}^3/a_{\mathrm{in}}^3$, and a contraction introduces no further volume factor. The same bound holds for a scalar outer tensor normalized by a reference volume. For a scalar outer tensor with counting sums on mesh $a_{\mathrm{old}}$, the sum with a new input $l$ fixed acquires the factor $v_l/v_{\mathrm{old}}$, maximized over the possible kinds of the replaced inputs. Pointwise products with a common output index satisfy the product bound, and a shared index may be represented by the equality tensor $\one_{o=i_1=\cdots=i_r}$, whose norm is one when all its ports are of the same kind. \item \emph{Contraction diagrams.} Consider a finite connected diagram of tensors, in which shared indices are represented by equality tensors and every contracted index identifies two ports, and fix one external index. Choose a spanning tree of the graph of factors, rooted at the factor containing the fixed index, and attach every other factor $V$ to its parent through a port $j(V)$. For a vector factor $V$ with output $o(V)$, let $A_V=\abs V$ if $j(V)=o(V)$ and $A_V=(v_{j(V)}/v_{o(V)})\abs V$ if $j(V)$ is an input; for the factor containing the fixed index use the same rule at that index; for a scalar factor use its sum with the corresponding port fixed. Then the sum over all indices except the fixed one of the absolute value of the diagram, weighted by $e^{\asp\Tree}$ of the locations of the external indices, is at most $\prod_VA_V$. If the fixed index is an external input $l$, the volume-normalized column of the diagram is obtained by multiplying by $v_o/v_l$, where $o$ is the output of the diagram; for the substitution trees of part (1) the volume ratios cancel exactly. A diagram without a fixed external index is not covered; for example $\tr I$ equals the number of indices. An externally differentiated trace is covered if the differentiated tensor has the bound with the input fixed and \emph{all} of its matrix ports and other Taylor inputs summed; a row bound alone does not suffice. \item \emph{Majorant series.} Products, substitutions, and summable connected diagrams of analytic maps with absolute majorants (in the sense of \cref{ch01:def:analytic-norms}) are bounded coefficientwise by the corresponding products, compositions, and diagram sums of the nonnegative scalar majorants, with the bounds of parts (1) and (2). The domain of the conclusion is the common product of polydiscs on which these scalar series converge absolutely. For $F\circ G$ it suffices that the scalar majorant of $G$ maps the domain strictly inside the coefficient domain of $F$; for formal substitution take $G(0)=0$, and otherwise require this absolute image condition, so that the substitutions into the constant terms also converge. The coefficients with one fixed input are bounded by the derivative of the same scalar majorant; no second Taylor factorial occurs. \item \emph{Smaller domains.} If $A(s)=\sum_nA_ns^n$ with $A_n\ge0$ converges at $R>0$, then for $00$ satisfy, for some $a>0$, \[ \max\Bigl\{\sup_x\sum_{y\ne x}e^{ad(x,y)}\norm{K_{xy}},\ \sup_y\sum_{x\ne y}e^{ad(x,y)}\norm{K_{xy}}\Bigr\}\le H. \] Let $b=\min\{a/2,a\lambda/(4H)\}$ if $H>0$ and $b=a/2$ if $H=0$; then the same maximum with weights $e^{bd(x,y)}-1$ is at most $\lambda/2$. For every $00$, depending only on $a,H,\lambda,\Lambda,D,v,b'$, such that for $\norm E_a<\eta_0$ the branches $(K+E)^{1/2}$ and $(K+E)^{-1/2}$ continuing the positive roots exist and are bounded in $\norm\cdot_{b'/2}$ by constants depending only on these data. If $E=E(\theta)$ has an absolute majorant with all fixed indices on a fixed analytic ball, the coefficients of these roots in $\theta$ have absolute bounds with every fixed index on a smaller coefficient ball. For volume-normalized coefficient columns the constants and the smaller radius may also depend on $\Gamma_X=\max_{x,y\in X}v_x/v_y$ and on the stated majorant. The coefficient ball is contained in the root domain $\norm{E(\theta)}_a<\eta_0$, and its output spatial rate is no larger than the supplied coefficient rate. The counting-norm bounds for the inverse and roots, and the preceding choice of $\eta_0$, are unchanged. \item \emph{Logarithms of determinants.} In the setting of part (8), assume the coefficient majorant for $E(\theta)$ is available at a positive spatial rate. Then $D\log\det K=\tr(K^{-1}DK)$, and the derivatives of positive order of $\log\det(K+E(\theta))$ in $\theta$ have absolute bounds with any fixed external index, with one reduction of the rate independent of the order. The undifferentiated $\log\det K$ is not bounded independently of $\abs X$. \item \emph{Implicit functions.} Let $\Phi(\theta,u)$ be an analytic map of input variables $\theta=(\theta_j)$ and unknowns $u=(u_o)$, with values indexed like $u$, defined on a product of polydiscs, with $\Phi(0,0)=0$. Assume that for all $k,l\ge0$ the output-row norm and every input-column norm, at any fixed input $\theta_j$ or $u_o$, of the coefficient tensor $D^k_\theta D^l_u\Phi(0,0)/(k!\,l!)$ are at most the coefficient of $t^kY^l$ in \[ \Phi^+(t,Y)=At+\kappa Y+\frac{A(t+Y)^2}{1-C(t+Y)},\qquad A,C>0,\quad0\le\kappa<1. \] In particular the linear map $D_u\Phi(0,0)$ has row and column norms at most $\kappa$. Let $\delta=1-\kappa$, $r=\min\{1,(4C)^{-1},\delta/(32A)\}$, and $t_0=\min\{r,\delta r/(4A)\}$, and assume that the domain of $\Phi$ contains a neighborhood of the closed product $\{\max_j\abs{\theta_j}\le t_0,\ \max_o\abs{u_o}\le r\}$. Then the scalar equation $Y=\Phi^+(t,Y)$ has a unique solution $Y(t)$ with $\abs{Y(t)}\le r$ for $\abs t0$, decomposing into the shells $k\le d(x,y)\norm M_{2\to2}^2/\lambda$, \[ \norm{(I-M/d)u}^2\le\bigl(1-\lambda/d+\norm M_{2\to2}^2/d^2\bigr)\abs u^2<\abs u^2. \] With $T=I-M/d$, expanding $(M+tI)^{-1}$ in powers of $T$ and integrating termwise in \eqref{ch02:eq:inverse-square-root}, which is justified by $\norm T_{2\to2}<1$, identifies $S(E)$ with $d^{-1/2}\sum_{r\ge0}c_rT^r$, where $c_r=4^{-r}\binom{2r}r$ are the coefficients of $(1-z)^{-1/2}$; indeed, differentiating $(1-z)^{-1/2}$ at zero gives $c_0=1$ and $c_{r+1}=(r+1/2)c_r/(r+1)$. The square of this scalar series is $(1-z)^{-1}$, which gives $S(E)^2M=I$; all factors commute with $M$, being limits of its resolvents. This proves the root identity without a symmetry assumption on $E$. Multiplying by $K+E$ gives the square root; since $K\le\Lambda I$ implies $\norm K_{c_2}\le\Lambda+H$, this multiplication is bounded uniformly. All choices depend only on $a,H,\lambda,\Lambda,D,v,b'$. Suppose now that $E=E(\theta)$ has an absolute majorant with all fixed indices on a fixed analytic ball inside the stated root domain. Write $E_0=E(0)$ and $\Delta E(\theta)=E(\theta)-E_0$. The preceding perturbation bound gives $R_t=(K+E_0+tI)^{-1}$ with counting row and column bounds $2C_0/(\lambda+t)$ at rate $c_2$. Its volume-normalized matrix columns are bounded by $2\Gamma_XC_0/(\lambda+t)$. Work at a rate $c\le c_2$ no larger than the supplied coefficient rate of $\Delta E$; if that rate is smaller, decrease $c$ accordingly. Let $A(r)$ be the common full absolute coefficient majorant of $\Delta E$ there. Since $\Delta E(0)=0$, $A(r)\to0$ as $r\downarrow0$. Choose a smaller coefficient ball, still in the root domain, with $2\Gamma_XC_0A(r)<\lambda/2$, and expand \[ (K+E(\theta)+tI)^{-1} =\sum_{j\ge0}(-R_t\Delta E(\theta))^jR_t. \] The substitution and contraction rules apply with these row and column bounds, retaining their volume factors. For $j$ insertions, integration against $t^{-1/2}/\pi$ is bounded by \[ \frac1\pi\int_0^\infty t^{-1/2} \Bigl(\frac{2\Gamma_XC_0}{\lambda+t}\Bigr)^{j+1}dt \le(2\Gamma_XC_0)^{j+1}\lambda^{-j-1/2}. \] Indeed, the same integral without the constant equals $\lambda^{-j-1/2}\binom{2j}{j}/4^j$. The resulting scalar coefficient series is geometrically summable on this ball, uniformly in the number of matrix labels. It bounds every fixed-index coefficient of the inverse square root. Its constant matrix coefficient has the preceding counting bound and the corresponding volume-normalized bound with factor $\Gamma_X$. Multiplication by $K+E(\theta)$ gives the positive square root; the coefficient norm of the constant matrix $K$ costs at most the same mesh-ratio factor. This proves part (8), with no identification of counting and volume-normalized columns. \paragraph{Step 7: logarithms of determinants.} Expanding the determinant in columns gives $D\det K=(\det K)\tr(K^{-1}DK)$, since the linear term of $\det(I+tB)$ is $t\tr B$. On the smaller coefficient ball just chosen, put $R_0=(K+E_0)^{-1}$. The centered identity is \[ \log\det(K+E(\theta))=\log\det(K+E_0) +\sum_{j\ge1}\frac{(-1)^{j+1}}j \tr\bigl((R_0\Delta E(\theta))^j\bigr). \] To see it, differentiate in a scalar multiple of $\Delta E$, use the convergent geometric inverse series, and integrate from zero. The logarithm is the branch continuing the value at $\theta=0$. The same coefficient gate as in Step~6 ensures absolute convergence before the trace is closed. Only positive-degree coefficients are bounded with a fixed external input; the constant logarithmic determinant is retained separately. For these coefficients, keep the two matrix ports $x,y$ open. Steps~1 and~2 and the centered series give an output-row majorant $B_s$ for the degree-$s$ tensor $T_s(x,y;u_1,\dots,u_s)$ at some positive rate $c$, no larger than the supplied coefficient rate. Fix an external input $u_j=u$ and close the trace by setting $y=x$. The unclosed network length is at least both the connecting length of the external inputs and $d(x,u)$. Splitting the rate in half bounds the sum over the other external inputs, with their external weight at rate $c/2$, by $B_se^{-cd(x,u)/2}$ for fixed $x$. If $u\notin X$, choose a nearest $x_0\in X$, so $d(x,x_0)\le2d(x,u)$; otherwise set $x_0=u$. Equation \eqref{ch02:eq:exponential-count} sums over $x$ with a factor at most $\mathcal S(c/4;D,v)$. This proves the counting fixed-external-index bound with one rate reduction independent of $s$. For a scalar reference-volume normalization, apply the distinguished-input column and diagram rules, keeping their attachment and external-normalization ratios, including the stated conversion from matrix volumes to the reference volume. No count of all internal labels is inserted. Without an external input the bound fails: $K=2I$ gives $\log\det K=\abs X\log2$ times the block dimension. This proves part (9). \paragraph{Step 8: the scalar equation.} We write $\varphi(s)=s^2/(1-Cs)$, so that $\Phi^+(t,Y)=At+\kappa Y+A\varphi(t+Y)$. For $\abs t\le t_0$ and $\abs Y\le r$ we have $\abs{t+Y}\le2r$, $C\abs{t+Y}\le1/2$, and \[ \abs{At}+A\abs{\varphi(t+Y)}\le\frac{\delta r}4+8Ar^2\le\frac{\delta r}2. \] Moreover $\abs{\varphi'(s)}\le2\abs s/(1-C\abs s)+C\abs s^2/(1-C\abs s)^2\le12r$ for $\abs s\le2r$. The map $Y\mapsto\Phi^+(t,Y)$ therefore maps the closed disc of radius $r$ into itself, since $\kappa r+\delta r/20$. Let \[ \Omega=\prod_{x\in X}\{w_x\in\R^{q_x}:\abs{w_x}\le p_x\},\qquad d\mu=Z^{-1}e^{-V(w)}\,dw\quad\text{on }\Omega, \] where $V$ is real and of class $C^2$ on a neighborhood of $\Omega$, and $D^2V\ge mI$ for some $m>0$. Let $q=\sum_xq_x$ and $\xi=w-\E_\mu w$. Then the following statements hold. \begin{enumerate} \item For every real $f\in H^1(\mu)$ and every real function $F$ that is Lipschitz for the Euclidean norm, \[ \Ent_\mu(f^2)\le\frac2m\E_\mu\abs{\nabla f}^2,\qquad \Var_\mu(f)\le\frac1m\E_\mu\abs{\nabla f}^2, \] \[ \log\E_\mu e^{t(F-\E_\mu F)}\le\frac{t^2\Lip(F)^2}{2m}\qquad(t\in\R), \] where $\Ent_\mu(h)=\E_\mu[h\log h]-\E_\mu h\log\E_\mu h$. These constants do not depend on $\abs X$, on the $q_x$ and $p_x$, or on upper bounds for the diagonal blocks of the Hessian. \item Let $d$ be a metric on $X$ and $b>0$, and write $V_{xy}=D_{w_x}D_{w_y}V$. If \[ \max\Bigl\{\sup_x\sum_{y\ne x}(e^{bd(x,y)}-1)\sup_\Omega\norm{V_{xy}},\ \sup_y\sum_{x\ne y}(e^{bd(x,y)}-1)\sup_\Omega\norm{V_{xy}}\Bigr\}\le\frac m2, \] then for all real or complex $F,G\in H^1(\mu)$, \[ \abs{\Cov_\mu(F,G)}\le\frac2m\sum_{x,y}e^{-bd(x,y)}\norm{\nabla_xF}_{L^2(\mu)}\norm{\nabla_yG}_{L^2(\mu)}, \] where the covariance is bilinear for complex functions. In particular, if $V_{xy}=0$ for $d(x,y)>r_0$ and the mixed blocks have row and column sums at most $H_0$, the hypothesis holds whenever $H_0(e^{br_0}-1)\le m/2$; exponentially decaying mixed Hessians are covered by the displayed condition. \item For every real symmetric $q\times q$ matrix $A$, the quadratic form $Q_A=\xi^{\mathsf T}A\xi$ satisfies \[ \log\E_\mu e^{t(Q_A-\E_\mu Q_A)}\le\frac{8t^2}{m^2}\norm A_{\mathrm{HS}}^2,\qquad\abs t\le\frac m{4\norm A_{\mathrm{op}}}, \] and for every $\zeta\in\R^q$, \[ \log\E_\mu e^{\zeta\cdot\xi+t(Q_A-\E_\mu Q_A)}\le\frac{\abs\zeta^2}m+\frac{16t^2}{m^2}\norm A_{\mathrm{HS}}^2,\qquad\abs t\le\frac m{8\norm A_{\mathrm{op}}}, \] where the range of $t$ is all of $\R$ if $A=0$. \item Suppose $q_x=3$ for all $x$. Let $\mathcal F$ be a family of distinct pairs $(e,f)$ of blocks such that every block belongs to at most $D_0$ pairs of $\mathcal F$, counting both orientations if both occur, and let $j_{ef}\in\R^3$. Then $\sum_{(e,f)\in\mathcal F}j_{ef}\cdot(\xi_e\times\xi_f)=\xi^{\mathsf T}A_j\xi$ for a real symmetric matrix $A_j$ with \[ \norm{A_j}_{\mathrm{op}}\le\frac{D_0}2\norm j_\infty,\qquad \norm{A_j}_{\mathrm{HS}}^2\le D_0\sum_{(e,f)\in\mathcal F}\abs{j_{ef}}^2. \] If $\mu$ is invariant under the simultaneous rotation $w\mapsto(Rw_x)_x$ for every $R\in\mathrm{SO}(3)$, then this quadratic form has mean zero. \end{enumerate} All statements concern the measure $\mu$ on the fixed convex set $\Omega$, with its full potential $V$, and they are centered at the mean under $\mu$. They give no convexity statement for marginals of $\mu$ or for measures on nonconvex sets. \end{theorem} Steps 1--6 construct the Neumann realization $\mathcal L$ of $-\Delta+\nabla V\cdot\nabla$ and a realization $\mathcal A_1$ of the corresponding operator on vector fields, whose form contains the Hessian of $V$ and a nonnegative curvature term from the spheres. They also prove the commutation $\nabla\mathcal L^{-1}=\mathcal A_1^{-1}\nabla$, and establish the semigroup bound $\abs{\nabla P_th}\le e^{-mt}P_t\abs{\nabla h}$. Steps 7 and 8 derive the entropy bound and, from $\Cov(F,G)=\ip{\nabla F}{\mathcal A_1^{-1}\nabla G}$ and exponential conjugation, the covariance bound. Step 9 derives the quadratic moments by comparison with an auxiliary Gaussian vector and absorption of the relative entropy, without independent coordinates or a Hessian bound for the tilted measure, and Step 10 treats cross products. The construction of the operators in Steps 1 and 2 uses constants that may depend on the dimension, the radii, and the bounds of $e^{-V}$ on the fixed compact set $\Omega$; these constants do not enter the inequalities. \begin{proof}[Proof of \cref{ch02:thm:reflecting-concentration}] All Hilbert spaces in this proof use the measure $\mu$ unless indicated otherwise. We write $B_x=\{\abs{w_x}0$, the spectral expansion puts $P_th$ and $\mathcal LP_th$ in the required domains, and \eqref{ch02:eq:gradient-core-identity} shows that $u_t=\nabla P_th$ solves $\partial_tu_t=-\mathcal A_1u_t$ weakly, with initial value $\nabla h$. Uniqueness follows by differentiating the squared norm of the difference of two solutions and using \eqref{ch02:eq:oneform-gap}. We conclude that $\nabla P_th=e^{-t\mathcal A_1}\nabla h$. We claim that \begin{equation}\label{ch02:eq:semigroup-domination} \abs{e^{-t\mathcal A_1}u}\le e^{-mt}P_t\abs u. \end{equation} Let $u_t=e^{-t\mathcal A_1}u$ and $h_\varepsilon=(\abs{u_t}^2+\varepsilon^2)^{1/2}$. In the weak equation for $u_t$ we use the test field $\psi u_t/h_\varepsilon$, with $\psi\in H^1$ bounded and nonnegative; it belongs to $\mathcal H_1$, since multiplication by a scalar preserves the face conditions. The kinetic term equals \[ \nabla h_\varepsilon\cdot\nabla\psi+\psi\Bigl(\frac{\abs{\nabla u_t}^2}{h_\varepsilon}-\frac{\abs{\nabla h_\varepsilon}^2}{h_\varepsilon}\Bigr), \] whose second summand is nonnegative by Cauchy--Schwarz. The Hessian term is at least $m\psi\abs{u_t}^2/h_\varepsilon\ge m\psi(h_\varepsilon-\varepsilon)$, and every boundary curvature term is nonnegative. Hence $(\partial_t+\mathcal L+m)(h_\varepsilon-\varepsilon)\le0$ in the weak scalar sense. We compare with $e^{-mt}P_t\abs u$ and test the difference with its positive part; the squared $L^2$ norm of this positive part does not increase from zero. This comparison is justified first on intervals $[\delta,T]$, where the spectral solutions have time derivatives in $L^2$ and lie in the form domains, and then by continuity in $L^2$ as $\delta\downarrow0$; truncation of the tests and approximation in the form norm justify the chain rule, whose derivatives are bounded for fixed $\varepsilon$. Letting $\varepsilon\downarrow0$ proves \eqref{ch02:eq:semigroup-domination}. Combined with the previous paragraph, this gives \[ \abs{\nabla P_th}\le e^{-mt}P_t\abs{\nabla h}. \] \paragraph{Step 7: entropy, variance, and exponential moments.} Let $h$ be smooth, positive, and bounded above and below. Applying the positive operator $P_t$ to $(f-ch)^2/h\ge0$ for every rational $c$ and minimizing over $c$, we obtain the pointwise inequality $(P_tf)^2\le P_t(f^2/h)\,P_th$. With $f=\abs{\nabla h}$ and the gradient bound of Step 6, we obtain \begin{equation}\label{ch02:eq:entropy-gradient} \frac{\abs{\nabla P_th}^2}{P_th}\le e^{-2mt}P_t\Bigl(\frac{\abs{\nabla h}^2}{h}\Bigr). \end{equation} By the weak equation and the chain rule for $\log P_th$, we have \[ -\frac d{dt}\E_\mu[(P_th)\log P_th]=\E_\mu\frac{\abs{\nabla P_th}^2}{P_th}. \] Since the bounds on $h$ persist under $P_t$, convergence of $P_th$ to its mean in $L^2$ implies convergence of the entropy integral to $(\E h)\log\E h$. Integrating in time and using \eqref{ch02:eq:entropy-gradient} and the invariance of $\mu$, we find $\Ent_\mu(h)\le(2m)^{-1}\E_\mu\abs{\nabla h}^2/h$. For bounded smooth $f$, substituting $h=f^2+\varepsilon$ and letting $\varepsilon\downarrow0$, we obtain $\Ent_\mu(f^2)\le(2/m)\E_\mu\abs{\nabla f}^2$. For general $f\in H^1$, we truncate, approximate by smooth functions, and use that the entropy is lower semicontinuous under convergence in $L^2$. Indeed, $s\log s$ is bounded below, and Fatou's lemma applies along an almost everywhere convergent subsequence, while the mean term converges in $L^1$. This proves the logarithmic Sobolev inequality. Expanding at $f=1+\varepsilon g$ with $g$ bounded, we find $\Ent((1+\varepsilon g)^2)=2\varepsilon^2\Var(g)+o(\varepsilon^2)$. Then the logarithmic Sobolev inequality implies the Poincaré inequality with constant $1/m$; truncation and density extend it to $H^1$. For a real Lipschitz $F$ let $\varphi(t)=\log\E_\mu e^{tF}$. Applying the logarithmic Sobolev inequality to $e^{tF/2}$, we find $t\varphi'(t)-\varphi(t)\le t^2\Lip(F)^2/(2m)$. Integrating $(\varphi(t)/t)'$ from zero proves the exponential moment bound for $t>0$, and applying it to $-F$ gives $t<0$. To justify the chain rule, we extend $F$ to a neighborhood by $\inf_{y\in\Omega}(F(y)+\Lip(F)\abs{x-y})$ and smooth; this preserves the Lipschitz constant and converges uniformly on $\Omega$; no differentiability of $F$ at the boundary is needed. This proves part (1). \paragraph{Step 8: covariances.} For real $G\in H^1$, we solve $\mathcal Lh=G-\E G$ with $\E h=0$, using the preliminary gap. By weak integration by parts and \eqref{ch02:eq:gradient-commutation}, we have \begin{equation}\label{ch02:eq:covariance-oneform} \Cov_\mu(F,G)=\ip{\nabla F}{\mathcal A_1^{-1}\nabla G}_{L^2(\mu)}, \end{equation} first for smooth $F$ and then for $F\in H^1$ by density. Complexifying gives the bilinear identity for complex $F,G$, with neither function conjugated; its absolute value is bounded by the Cauchy--Schwarz inequality in the complex Hilbert space. Fix $y_0\in X$ and set $W_x=e^{bd(x,y_0)}I_{q_x}$ on the components of the block $x$. This weight does not depend on $w$; it preserves the face conditions and commutes with the kinetic and boundary forms and with every diagonal block of the Hessian. In the closed-form realization, we obtain \[ W\mathcal A_1W^{-1}=\mathcal A_1+B,\qquad B_{xz}=\bigl(e^{b(d(x,y_0)-d(z,y_0))}-1\bigr)V_{xz}(w). \] Each block multiplier has norm at most $(e^{bd(x,z)}-1)\sup_\Omega\norm{V_{xz}}$. By the row-column bound of Step 5 in the proof of \cref{ch02:lem:coefficient-algebra} and the hypothesis of part (2), we have $\norm B_{2\to2}\le m/2$ on the direct sum of the block spaces $L^2(\mu)$. By \eqref{ch02:eq:oneform-gap} and a Neumann series, $\norm{(W\mathcal A_1W^{-1})^{-1}}\le2/m$, and the $(x,y_0)$ block satisfies \[ \norm{(\mathcal A_1^{-1})_{xy_0}}_{L^2\to L^2}\le\frac2me^{-bd(x,y_0)}. \] Inserting this bound into \eqref{ch02:eq:covariance-oneform} and applying Cauchy--Schwarz to each block pairing, we obtain the covariance bound of part (2), for real and complex functions. For an interaction of range $r_0$, every off-diagonal weight is at most $e^{br_0}-1$, which gives the stated criterion. Neither the individual $p_x,q_x$ nor an upper bound on the diagonal Hessian blocks entered. \paragraph{Step 9: quadratic moments.} The classical Hanson--Wright inequality concerns independent coordinates \cite{HansonWright1971}, and extensions based on concentration allow dependent vectors \cite[Theorem~2.5, p.~4]{Adamczak2015}; here we derive the required bounds directly from the logarithmic Sobolev inequality for $\mu$. Applying part (1) to the linear function $v\cdot w$, we obtain \begin{equation}\label{ch02:eq:linear-subgaussian} \E_\mu e^{v\cdot\xi}\le e^{\abs v^2/(2m)}. \end{equation} If $A=0$, all quadratic statements follow from this. Let $A\ne0$, and let $Z_0$ be a standard Gaussian vector in $\R^q$, independent of $w$. Since $e^{\alpha\abs y^2}=\E e^{\sqrt{2\alpha}Z_0\cdot y}$ (completion of the square, as in \cref{ch02:lem:hodge-schur}) and all integrands are positive, we have for $\alpha\ge0$ \[ \E_\mu e^{\alpha\abs{A\xi}^2}=\E_{Z_0}\E_\mu e^{\sqrt{2\alpha}(AZ_0)\cdot\xi}\le\E_{Z_0}e^{(\alpha/m)\abs{AZ_0}^2}=\det\bigl(I-(2\alpha/m)A^2\bigr)^{-1/2}, \] provided $2\alpha\norm A_{\mathrm{op}}^20$, integrating $(\psi(t)/t)'=D_t/t^2$ from zero, where $\psi(t)/t\to\E_\mu Q$, proves the first bound of part (3); replacing $A$ by $-A$ gives $t<0$. For the joint bound, by the Cauchy--Schwarz inequality and \eqref{ch02:eq:linear-subgaussian}, we have \[ \log\E_\mu e^{\zeta\cdot\xi+t(Q-\E Q)}\le\frac12\log\E_\mu e^{2\zeta\cdot\xi}+\frac12\log\E_\mu e^{2t(Q-\E Q)}\le\frac{\abs\zeta^2}m+\frac{16t^2}{m^2}\norm A_{\mathrm{HS}}^2, \] for $\abs t\le m/(8\norm A_{\mathrm{op}})$. This proves part (3). \paragraph{Step 10: cross products.} We write $J(j)v=j\times v$. A pair $(e,f)$ contributes the symmetric matrix with blocks $A_{ef}=-J(j_{ef})/2$ and $A_{fe}=A_{ef}^{\mathsf T}$, because $\xi_e^{\mathsf T}[-J(j_{ef})]\xi_f=j_{ef}\cdot(\xi_e\times\xi_f)$. Since $\norm{J(j)}_{\mathrm{op}}=\abs j$ and $\norm{J(j)}_{\mathrm{HS}}^2=2\abs j^2$, a single pair has block row sums $\abs{j_{ef}}/2$ and squared Hilbert--Schmidt norm $\abs{j_{ef}}^2$. Since at most $D_0$ pairs meet a block, the row-column bound implies $\norm{A_j}_{\mathrm{op}}\le D_0\norm j_\infty/2$; using $\abs{\sum_{k=1}^rb_k}^2\le r\sum_k\abs{b_k}^2$ with $r\le D_0$ in each repeated matrix entry, we obtain $\norm{A_j}_{\mathrm{HS}}^2\le D_0\sum\abs{j_{ef}}^2$. Pairs with $e=f$ contribute zero. If $\mu$ is invariant under simultaneous rotations, then $\E_\mu w$ is invariant and $\xi$ transforms as a vector. The law of $\xi$ therefore has the same invariance. The vector $\E(\xi_e\times\xi_f)$ is then fixed by every rotation, hence zero (a rotation by $\pi$ about an axis perpendicular to a nonzero vector reverses it). This proves part (4). \end{proof} \section{Conditioning, exponential moments, and gauge identities}\label{ch02:sec:conditioning} The next lemma gives conditioning bounds and identities for parameter derivatives and gauge transformations; the bound on large fields enters the proof through the comparison of a measure with its conditioning (see \eqref{s3:eq:transfer}). For a link $e$ and $a\in\{1,2,3\}$ we write $L_e^a$ and $R_e^a$ for the derivatives of a function of $U$ along $s\mapsto e^{s\tau_a}U_e$ and $s\mapsto U_ee^{s\tau_a}$ at $s=0$, all other link variables being fixed. \begin{lemma}\label{ch02:lem:bounded-transfer} Let $\mu$ be a probability measure on a measurable space, and let $S$ be a measurable event with $p_{\mathrm{fail}}=\mu(S^c)<1$. Put $\nu=\mu(\cdot\mid S)$. Then the following statements hold. \begin{enumerate} \item With $\norm{\mu-\nu}_{\TV}=\sup_A\abs{\mu(A)-\nu(A)}$, we have $\norm{\mu-\nu}_{\TV}=p_{\mathrm{fail}}$, and \[ \abs{\E_\mu F-\E_\nu F}\le2p_{\mathrm{fail}}\norm F_\infty \] for every bounded complex measurable $F$; if $0\le F\le A$, the bound is $p_{\mathrm{fail}}A$. The image measures of $\mu$ and $\nu$ under a common measurable map satisfy the same bound on the total variation distance. If $d\mu=Z^{-1}f\,d\omega$ with $f\ge0$, then the restricted partition function $Z_S=\int_Sf\,d\omega$ satisfies $Z_S/Z=1-p_{\mathrm{fail}}$, and $\abs{\log(Z_S/Z)}\le2p_{\mathrm{fail}}$ if $p_{\mathrm{fail}}\le1/2$. \item For nested events $S_0\supset S_1\supset\cdots\supset S_k$ of positive probability and $0\le i\le k$, \[ \mu(S_k)=\mu(S_i)\prod_{j=i}^{k-1}\mu(S_{j+1}\mid S_j). \] With $r_j=\mu(S_j\setminus S_{j+1}\mid S_j)$, it follows that $1-\mu(S_k)\le1-\mu(S_i)+\sum_{j=i}^{k-1}r_j$; the successive conditional failure probabilities are added without dividing by $\mu(S_i)$. \item For a real function $J$ with $\abs J\le R$, the tilted probability measure $\mu_J=e^J\mu/\E_\mu e^J$ satisfies $\mu_J(A)\le \exp(2R)\mu(A)$ for every event $A$. \item Let $O_1,\dots,O_r$ be real measurable functions with $\abs{O_i}\le1$, and set $M_\mu(z)=\E_\mu e^{\sum_iz_iO_i}$ and $M_\nu(z)=\E_\nu e^{\sum_iz_iO_i}$ for $z\in\C^r$. If $\sum_i\abs{z_i}\le R$, then \[ \abs{M_\mu(z)-M_\nu(z)}\le2\exp(R)p_{\mathrm{fail}}. \] If moreover $R<\log2$, the branches of $\log M_\mu$ and $\log M_\nu$ vanishing at $z=0$ satisfy \[ \abs{\log M_\mu(z)-\log M_\nu(z)}\le\frac{2\exp(R)}{2-\exp(R)}\,p_{\mathrm{fail}}, \] and their derivatives satisfy the corresponding Cauchy bounds on every smaller polydisc, with constants depending only on the two polydiscs and the order of the derivative. \item Let $\omega$ be a measure on a fixed domain, let $V(\theta,w)$ be real with $00$, there is only the exact identity \[ \E_\mu F-\E_\nu F=p_{\mathrm{fail}}\bigl(\E_{\mu(\cdot\mid S^c)}F-\E_\nu F\bigr); \] an error tending to zero requires additional control of these integrals. \end{enumerate} All constants are explicit as displayed. \end{lemma} \begin{proof} \emph{Parts (1)--(3).} If $p_{\mathrm{fail}}=0$, then $\mu=\nu$ and there is nothing to prove. For $p_{\mathrm{fail}}>0$ let $\nu^c=\mu(\cdot\mid S^c)$. Splitting every integral over $S$ and $S^c$, we obtain \begin{equation}\label{ch02:eq:conditioning-mixture} \mu=(1-p_{\mathrm{fail}})\nu+p_{\mathrm{fail}}\nu^c. \end{equation} For an event $A$, we have $\mu(A)-\nu(A)=p_{\mathrm{fail}}(\nu^c(A)-\nu(A))$, where both $\nu(A)$ and $\nu^c(A)$ lie in $[0,1]$. This bounds the total variation distance by $p_{\mathrm{fail}}$, and $A=S$ attains it. For complex $F$, the two conditional expectations have absolute value at most $\norm F_\infty$, which gives $2p_{\mathrm{fail}}\norm F_\infty$; if $0\le F\le A$, their difference is at most $A$. A common measurable map replaces each test event by its preimage and therefore does not increase the total variation distance. In the density notation, $Z_S=\int_Sf\,d\omega=Z\mu(S)$, and for $0\le p_{\mathrm{fail}}\le1/2$ we have $-\log(1-p_{\mathrm{fail}})=\int_0^{p_{\mathrm{fail}}}(1-t)^{-1}dt\le2p_{\mathrm{fail}}$. For nested events, consecutive conditional probabilities cancel: \[ \prod_{j=i}^{k-1}\mu(S_{j+1}\mid S_j)=\prod_{j=i}^{k-1}\frac{\mu(S_{j+1})}{\mu(S_j)}=\frac{\mu(S_k)}{\mu(S_i)}. \] Induction on the number of factors gives $1-\prod_j(1-r_j)\le\sum_jr_j$, the induction step being $1-(1-a)(1-b)=a+b-ab\le a+b$. Since $\mu(S_k)=\mu(S_i)\prod_j(1-r_j)\ge\mu(S_i)-(1-\prod_j(1-r_j))$, we obtain $1-\mu(S_k)\le1-\mu(S_i)+\sum_jr_j$. For $\abs J\le R$, the density of $\mu_J$ with respect to $\mu$ is at most $\exp(R)/\exp(-R)=\exp(2R)$. \emph{Part (4).} Let $T(z)=\sum_iz_iO_i$; on the ball $\sum_i\abs{z_i}\le R$ we have $\abs{T(z)}\le R$ and $\abs{e^{T(z)}}\le \exp(R)$. Applying part (1) to $e^{T(z)}$, we obtain $\abs{M_\mu-M_\nu}\le2\exp(R)p_{\mathrm{fail}}$. By the exponential series, we have $\abs{M_\mu(z)-1}\le \exp(R)-1$ and $\abs{M_\nu(z)-1}\le \exp(R)-1$. If $R<\log2$, these values lie in the open disc of center $1$ and radius $\exp(R)-1<1$, where the series of $\log(1+w)$ defines the branch vanishing at the origin. Integrating its derivative $1/u$ along the segment between the two values, on which $\abs u\ge2-\exp(R)$, we find \[ \abs{\log M_\mu(z)-\log M_\nu(z)}\le(2-\exp(R))^{-1}\abs{M_\mu(z)-M_\nu(z)}. \] The functions are holomorphic, since bounded observables allow termwise integration of the exponential series. If a smaller polydisc has coordinate distances $\delta_i>0$ to a larger one inside the ball, the Cauchy formula gives \[ \abs{\partial^\alpha(\log M_\mu-\log M_\nu)}\le\alpha!\prod_i\delta_i^{-\alpha_i}\frac{2\exp(R)}{2-\exp(R)}\,p_{\mathrm{fail}} \] at every point of the smaller polydisc. \emph{Part (5).} Write $g_\theta=e^{-V(\theta,\cdot)}$, $h_\theta=F(\theta,\cdot)e^{-V(\theta,\cdot)}$, and $N(\theta)=\int h_\theta\,d\omega$. Their pointwise derivatives are $\partial_\theta g_\theta=-(\partial_\theta V)e^{-V}$ and $\partial_\theta h_\theta=(\partial_\theta F-F\partial_\theta V)e^{-V}$. The assumed interchanges give \begin{equation}\label{ch02:eq:score-dominated-derivatives} \partial_\theta Z=-\int(\partial_\theta V)e^{-V}\,d\omega,\qquad \partial_\theta N=\int(\partial_\theta F-F\partial_\theta V)e^{-V}\,d\omega. \end{equation} The added integrability assumption permits splitting the second integral into its two finite terms. Since $00$. Neither the regularity of $\partial S$ nor a derivative of $\one_S$ is needed, since the indicator is unchanged for every $s$. For the triangular change of variables, given the transformed variables of higher index, we solve for $w_i$ using $O_i^{-1}$, proceeding downward. This shows that the map is invertible, and it preserves the product of balls. Its Jacobian matrix is block triangular with orthogonal diagonal blocks $O_i$, and its determinant therefore has absolute value one. For a differentiable one-parameter family of such maps, the generator is $Y_i=B_i(\theta,w_{>i})w_i$ with $B_i^{\mathsf T}=-B_i$. Hence \[ \operatorname{div}_{w_i}Y_i=\tr B_i=0,\qquad w_i\cdot Y_i=0\quad\text{on }\abs{w_i}=p_i, \] and the flux through every face vanishes; intersections of faces carry no additional flux, since the map preserves each radius $\abs{w_i}$. Under the assumed invariance of the full joint probability measure, we obtain the gauge-invariance identity by differentiating the change-of-variables identity. If this measure is written with conditional factors $Z(\theta)^{-1}e^{-V(\theta,w)}$, their parameter derivatives are the derivatives of the normalized log-densities computed in part (5); a derivative of an unnormalized conditional density is never substituted into a joint identity. \emph{Part (7).} The identity \eqref{ch02:eq:conditioning-mixture} applies to an integrable $F$ when $p_{\mathrm{fail}}>0$, and both conditional integrals are finite since both events have positive probability. Without further control there is no small error: on a two-point space, give $S^c$ probability $p_{\mathrm{fail}}$ and take $F=p_{\mathrm{fail}}^{-2}\one_{S^c}$; then $\E_\nu F=0$ and $\E_\mu F=p_{\mathrm{fail}}^{-1}$. \end{proof} \section{Invariant functions of three holonomies}\label{ch02:sec:traces} The weak-coupling results concern continuous functions of the three holonomies $H_1,H_2,H_3$ around the coordinate cycles of the torus, with a common basepoint, that are invariant under simultaneous conjugation. The next lemma shows that such functions are exactly the continuous functions of seven traces. If we write $H_i=a_iI+v_i\cdot t$, the traces of $H_i$ and $H_iH_j$ determine the scalar parts $a_i$ and the Gram matrix of the vectors $v_i$. The trace of $H_1H_2H_3$ determines in addition the orientation $\det(v_1,v_2,v_3)$, and a triple of vectors is determined up to rotation by its Gram matrix and its orientation. These seven coordinates are the case of three generators in the classical theory of trace coordinates of Horowitz \cite{Horowitz1972Characters}; the argument below proves the separation and the continuous factorization for all triples in $\SU$, including those of deficient rank. \begin{lemma}\label{ch02:lem:seven-traces} Define $\mathfrak p:\SU^3\to[-1,1]^7$ by the normalized traces of $H_1,H_2,H_3,H_1H_2,H_1H_3,H_2H_3,H_1H_2H_3$, in this order. For all $H,H'\in\SU^3$, \[ \mathfrak p(H)=\mathfrak p(H')\quad\Longleftrightarrow\quad\text{there exists }k\in\SU\text{ with }H_i'=kH_ik^{-1}\text{ for }i=1,2,3. \] This includes all triples of deficient rank and the central triples. Explicitly, if $H_i=a_iI+2v_i^a\tau_a$ with $a_i^2+\abs{v_i}^2=1$, the normalized trace of $H_iH_j$ is $a_ia_j-v_i\cdot v_j$, and that of $H_1H_2H_3$ is \[ a_1a_2a_3-a_3v_1\cdot v_2-a_1v_2\cdot v_3-a_2v_1\cdot v_3-\det(v_1,v_2,v_3). \] The map induced by $\mathfrak p$ from the quotient of $\SU^3$ by simultaneous conjugation onto $\mathfrak p(\SU^3)$ is a homeomorphism. Every continuous function $F:\SU^3\to\C$ invariant under simultaneous conjugation has the form $F=f\circ\mathfrak p$, where $f$ is continuous on $\mathfrak p(\SU^3)$ and extends to a continuous function on $[-1,1]^7$ with $\norm f_\infty\le\norm F_\infty$. In particular, if the trace vectors $\mathfrak p(H^{(k)})$ of random triples $H^{(k)}$ converge jointly in distribution, then for every such $F$ the expectations $\E F(H^{(k)})$ converge, to the expectation of $f$ under the limiting law, which is supported on $\mathfrak p(\SU^3)$. \end{lemma} The lemma concerns one triple of matrices and its seven traces; it does not by itself relate lattice holonomies to continuum observables. \begin{proof} By \eqref{ch02:eq:quaternion-product}, $H_i=a_iI+v_i\cdot t$ with $t_a=2\tau_a$; the scalar part $a_i$ is the normalized trace, and $a_i^2+\abs{v_i}^2=1$. The scalar part of $H_iH_j$ is $a_ia_j-v_i\cdot v_j$. Multiplying once more, we find that the scalar part of $H_1H_2H_3$ is \[ (a_1a_2-v_1\cdot v_2)a_3-(a_1v_2+a_2v_1+v_1\times v_2)\cdot v_3, \] which expands to the stated formula, with the minus sign in front of $\det(v_1,v_2,v_3)=(v_1\times v_2)\cdot v_3$ for our choice of $\tau_a$. Suppose $\mathfrak p(H)=\mathfrak p(H')$. Then $a_i=a_i'$, the Gram matrices $(v_i\cdot v_j)$ and $(v_i'\cdot v_j')$ agree (including the diagonal entries $1-a_i^2$), and $\det(v_1,v_2,v_3)=\det(v_1',v_2',v_3')$. We define a map between the spans by $T(\sum_ic_iv_i)=\sum_ic_iv_i'$. Equality of the Gram matrices shows that $T$ is well defined, since if $\sum_ic_iv_i=0$ then $\abs{\sum_ic_iv_i'}^2$ is the same quadratic form in $c$ and vanishes, and that $T$ is an isometry onto the span of the $v_i'$. Completing orthonormal bases of the two spans to bases of $\R^3$, we extend $T$ to an orthogonal map $R$. If the common rank is three, equality of the nonzero determinants implies $\det R=1$. If the rank is at most two and $\det R=-1$, we compose $R$ with the reflection in a direction perpendicular to the span of the $v_i'$; this fixes every $v_i'$ and changes the determinant to $+1$. Rank zero covers the central triples. In every case some $R\in\mathrm{SO}(3)$ maps each $v_i$ to $v_i'$. By the surjectivity of $\Ad$ (\cref{ch02:lem:haar-and-forests}) there exists $k\in\SU$ with $\Ad(k)=R$. Since conjugation does not change the scalar parts, we have $H_i'=kH_ik^{-1}$. Conversely, simultaneous conjugation preserves the trace of every product, by cyclicity of the trace. Let $Q_0$ be the quotient of $\SU^3$ by simultaneous conjugation, with the quotient topology, and let $A=\mathfrak p(\SU^3)$. Then $Q_0$ is compact, as a continuous image of a compact space, and so is $A$. Since the polynomial map $\mathfrak p$ is continuous and constant on equivalence classes, it induces a continuous map $Q_0\to A$, which is bijective by what we just proved. A closed subset of $Q_0$ is compact, and its image is compact and hence closed in the metric space $A$. Thus the inverse map is continuous, and the induced map is a homeomorphism. A continuous invariant function $F$ descends to a continuous function on $Q_0$, which defines a continuous function $f_A$ on $A$ with $\norm{f_A}_\infty=\norm F_\infty$. We construct a continuous extension to the cube. For a real continuous function $g$ on the compact set $A$ with $\abs g\le B$, set $g_T(x)=\inf_{y\in A}\{g(y)+T\abs{x-y}\}$ for $x\in[-1,1]^7$. By the triangle inequality $g_T$ is $T$-Lipschitz, and $g_T\le g$ on $A$. Given $\varepsilon>0$, choose $\delta>0$ from the uniform continuity of $g$, and $T$ with $T\delta\ge2B$. For $x\in A$, the terms with $\abs{x-y}\le\delta$ are at least $g(x)-\varepsilon$, and the remaining terms are at least $-B+T\delta\ge B\ge g(x)$. Thus $g-\varepsilon\le g_T\le g$ on $A$, and clipping $g_T$ to $[-B,B]$ preserves its continuity and this approximation, and it bounds the result by $B$ on the cube. Let $M=\norm{f_A}_\infty$. If $M=0$, extend by zero. Otherwise let $r_0=f_A$. If $\norm{r_k}_\infty\le B_k=2^{-k}M$, apply this construction to the real and imaginary parts of $r_k$, with errors at most $B_k/(2\sqrt2)$; this gives a continuous complex function $h_k$ on the cube with $\norm{h_k}_\infty\le\sqrt2B_k$ and $\norm{r_k-h_k|_A}_\infty\le B_k/2$, and we set $r_{k+1}=r_k-h_k|_A$. Since $\sum_k\sqrt2B_k<\infty$, the series $h=\sum_kh_k$ converges uniformly on the cube. Then $h$ is continuous, and $h=f_A$ on $A$ since the residuals tend to zero uniformly. Composing $h$ with the radial retraction of $\C$ onto the closed disc of radius $M$, \[ z\longmapsto\begin{cases}z,&\abs z\le M,\\ Mz/\abs z,&\abs z>M,\end{cases} \] which is continuous and fixes the values of $f_A$, gives a continuous extension $f$ with $\norm f_\infty\le M$. Finally, $\E F(H^{(k)})=\E f(\mathfrak p(H^{(k)}))$. Since $f$ is bounded and continuous on the cube, convergence in distribution of the traces implies convergence of the expectations. The limiting law, a probability measure on the cube, is supported on the closed set $A$, since the nonnegative continuous function $\dist(\cdot,A)$ has expectation zero under every approximating law and hence under the limit. The limit of the expectations is therefore the integral of the intrinsically defined function $f_A$. \end{proof}