Proof of The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split

Each result cited is universally quantified over the data in its own statement. Real order and arithmetic are those of The Real Numbers: Standing Notation and Background; finite sums over finite index sets are manipulated (agreement with indexed sums, reindexing along a bijection, additivity, homogeneity, dropping vanishing terms, interchange) by claims 1 to 4 of Properties of a Sum over a Finite Index Set and claims 1, 4 and 5 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, and this is not repeated below.

Step 0 (Conventions). Write $d=d_{M}$, $\chi=\chi_{M}$, $\Gamma=\Gamma_{M}$ and $\mathbb{L}=\mathbb{L}_{M}$. For $j\in\mathbb{N}$, $a_{j}>0$, since $a$ is a weight sequence by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §weights, whose terms are positive by Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §weights, and $c_{j}>0$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §variances and Variance Sequences and Their Truncations §variances; $a_{j}^{1/2}$ is the positive square root of $a_{j}$ and $a_{j}^{-1/2}$ its inverse, so $a_{j}(a_{j}^{-1/2})^{2}=1$ and $a_{j}a_{j}^{-1/2}=a_{j}^{1/2}$. Likewise $L^{-n/2}$ is the inverse of the positive square root $L^{n/2}$ of $L^{n}$, so $(L^{-n/2})^{2}=L^{-n}$ and $L^{-n/2}L^{n}=L^{n/2}$. By The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension, $\kappa(j)\notin\Gamma$ for $j>d$, and for $j\in[d]$, $\chi(j)\in\{0,1\}$ with $\chi(j)=1$ exactly when $\kappa(j)\in\Gamma$; hence, for $j\in\mathbb{N}$, $\kappa(j)\in\Gamma$ if and only if $j\le d$ and $\chi(j)=1$, and $\chi(j)^{2}=\chi(j)$. Let $J_{M}$ be the set of $j\in[d]$ with $\chi(j)=1$, that is, with $\kappa(j)\in\Gamma$: the set of active indices of The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension at cutoff $M$.

For $j\in[d]$ and $z\in\mathbb{L}$ put $\omega_{j}(z)=L^{-n/2}\chi(j)\,a_{j}^{-1/2}\psi_{\kappa(j)}(z)$, and for $u\in\mathbb{R}^{d}$ let $\ell_{u}$ be the lattice field $\ell_{u}(z)=\sum_{j=1}^{d}\omega_{j}(z)\,u_{j}$. Then $v_{\varsigma}(u)=\sum_{z\in\mathbb{L}}f_{\varsigma}(\ell_{u}(z))$ by the definition of $v_{\varsigma}$, and for $x\in X$, $\varphi_{x}=\ell_{p_{d}(x)}$ by The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §coordinates, since $p_{d}(x)=(x_{1},\dots,x_{d})$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. For a lattice field $F$ and $k\in\Gamma$, The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §coefficients gives

$$L^{-n/2}\sum_{z\in\mathbb{L}}F(z)\,\psi_{k}(z)=L^{-n/2}L^{n}\hat{F}(k)=L^{n/2}\hat{F}(k).\tag{F}$$

Step 1 (Lattice identities). (R) For every map $g:\Gamma\to\mathbb{R}$, $\sum_{j=1}^{d}\chi(j)\,g(\kappa(j))=\sum_{k\in\Gamma}g(k)$. Indeed, $\kappa$ is a bijection $\mathbb{N}\to\mathbb{Z}^{n}$; for $k\in\Gamma$ the index $j=\kappa^{-1}(k)$ has $\kappa(j)\in\Gamma$, so $j\in J_{M}$ by Step 0. Hence the restriction of $\kappa$ to $J_{M}$ is a bijection $J_{M}\to\Gamma$, and $J_{M}$ is nonempty as $\Gamma$ is nonempty (The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite). The summand $\chi(j)g(\kappa(j))$ vanishes for $j\in[d]\setminus J_{M}$ and equals $g(\kappa(j))$ for $j\in J_{M}$, so the sum is $\sum_{j\in J_{M}}g(\kappa(j))=\sum_{k\in\Gamma}g(k)$.

(G) For $j,k\in[d]$, $\sum_{z\in\mathbb{L}}\omega_{j}(z)\omega_{k}(z)=\chi(j)\,\delta_{jk}/a_{j}$, with $\delta_{jk}=1$ if $j=k$ and $0$ otherwise. Indeed, the left side is $\chi(j)\chi(k)a_{j}^{-1/2}a_{k}^{-1/2}\,L^{-n}\sum_{z}\psi_{\kappa(j)}(z)\psi_{\kappa(k)}(z)$. If $\chi(j)\chi(k)=0$ both sides vanish (if $j=k$ then $\chi(j)=0$). Otherwise $\kappa(j),\kappa(k)\in\Gamma$, and by Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §orthogonality the last factor is $1$ if $\kappa(j)=\kappa(k)$, that is $j=k$ ($\kappa$ being injective), and $0$ otherwise; and $a_{j}^{-1/2}a_{j}^{-1/2}=1/a_{j}$.

(Q) For $u,w\in\mathbb{R}^{d}$, expanding both factors and using (G),

$$\sum_{z\in\mathbb{L}}\ell_{u}(z)\ell_{w}(z)=\sum_{j=1}^{d}\sum_{k=1}^{d}u_{j}w_{k}\sum_{z\in\mathbb{L}}\omega_{j}(z)\omega_{k}(z)=\sum_{j=1}^{d}\chi(j)\frac{u_{j}w_{j}}{a_{j}} .$$

In particular $\sum_{z}\ell_{u}(z)^{2}=\sum_{j=1}^{d}\chi(j)u_{j}^{2}/a_{j}$, and, taking for $w$ the $k$-th unit vector (for which $\ell_{w}=\omega_{k}$), $\sum_{z}\ell_{u}(z)\omega_{k}(z)=\chi(k)u_{k}/a_{k}$ for $k\in[d]$. With $u=p_{d}(x)$, $x\in X$, Step 0 gives $\sum_{z}\varphi_{x}(z)^{2}=\sum_{j=1}^{d}\chi(j)x_{j}^{2}/a_{j}$.

(D) For every $y\in\mathbb{L}$, $\sum_{k\in\Gamma}\psi_{k}(y)^{2}=L^{n}$. Let $\iota_{y}$ be the lattice field equal to $1$ at $y$ and $0$ elsewhere. By The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §coefficients, $\hat{\iota}_{y}(k)=L^{-n}\psi_{k}(y)$ (only the summand at $z=y$ survives), and Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §inversion-formula at $y$ gives $1=\iota_{y}(y)=\sum_{k\in\Gamma}L^{-n}\psi_{k}(y)^{2}$.

(P) For every lattice field $F$, $\sum_{k=1}^{d}a_{k}\bigl(\sum_{z\in\mathbb{L}}F(z)\omega_{k}(z)\bigr)^{2}=\sum_{z\in\mathbb{L}}F(z)^{2}$. Indeed, by (F), $\sum_{z}F(z)\omega_{k}(z)=\chi(k)a_{k}^{-1/2}L^{n/2}\hat{F}(\kappa(k))$, the right side read as $0$ when $\chi(k)=0$; so $a_{k}(\sum_{z}F(z)\omega_{k}(z))^{2}=\chi(k)L^{n}\hat{F}(\kappa(k))^{2}$, and by (R) and Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §parseval (with $f=g=F$) the sum over $k$ is $L^{n}\sum_{l\in\Gamma}\hat{F}(l)^{2}=L^{n}L^{-n}\sum_{z}F(z)^{2}$.

Step 2 (Claim 1). Let $\varsigma$ be an admissible split and $k\in\Gamma$; write $r_{k}=r^{(\varsigma)}_{k}$. As $0<\mathfrak{q}_{k}<\varsigma$, the numbers $\mathfrak{q}_{k}\varsigma$ and $\varsigma-\mathfrak{q}_{k}$ are positive, so $r_{k}>0$ and $1/r_{k}=(\varsigma-\mathfrak{q}_{k})/(\mathfrak{q}_{k}\varsigma)=1/\mathfrak{q}_{k}-1/\varsigma$. Hence every $c^{(\varsigma)}_{j}$ is positive (Step 0). For $j\in\mathbb{N}$, $a_{j}=\mu_{\kappa(j)}c_{j}$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §ratio, so $\mu_{\kappa(j)}=a_{j}/c_{j}>0$ and $c^{(\varsigma)}_{j}=c_{j}w_{j}$ with $w_{j}=\mu_{\kappa(j)}r_{\kappa(j)}$ if $\kappa(j)\in\Gamma$ and $w_{j}=1$ otherwise. Let $D$ be the greatest entry of the tuple consisting of $1$ followed by the numbers $\mu_{k}r_{k}$ along an enumeration of $\Gamma$ (Greatest Element of a Finite Family in a Totally Ordered Set); then $0\le w_{j}\le D$, and $\sum_{j=1}^{\infty}c^{(\varsigma)}_{j}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound, applied to the nonnegative terms $c_{j}$, whose series converges by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §variances, and to the factors $w_{j}$ with the bound $D$. So $c^{(\varsigma)}$ is a variance sequence (Variance Sequences and Their Truncations §variances). Finally $1\le\bar{r}_{\varsigma}$ and $r_{k}\le\bar{r}_{\varsigma}$ for $k\in\Gamma$; if $\kappa(j)\in\Gamma$ then $c^{(\varsigma)}_{j}=a_{j}r_{\kappa(j)}\le\bar{r}_{\varsigma}a_{j}$, and otherwise $c^{(\varsigma)}_{j}=c_{j}\le a_{j}\le\bar{r}_{\varsigma}a_{j}$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §ratio. This proves claim 1. We record that, for $j\in\mathbb{N}$, $\kappa(j)\in\Gamma$ gives

$$\frac{1}{c^{(\varsigma)}_{j}}=\frac{1}{a_{j}r_{\kappa(j)}}=\frac{1}{a_{j}\mathfrak{q}_{\kappa(j)}}-\frac{1}{a_{j}\varsigma}=\frac{1}{c^{J,\eta}_{j}}-\frac{1}{a_{j}\varsigma},\tag{S}$$

by The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §variances, while $c^{(\varsigma)}_{j}=c_{j}=c^{J,\eta}_{j}$ if $\kappa(j)\notin\Gamma$.

Step 3 (One-variable calculus). Fix an admissible split $\varsigma$ and write $f=f_{\varsigma}$. Every point of $\mathbb{R}$ is an interior point of the interval $\mathbb{R}$ by Basic Facts about Intervals of the Real Line and Their Interior Points §whole-line; derivatives are those of Derivative at an Interior Point. For $t\in\mathbb{R}$ put $E(t)=\exp(t)$, $\bar{E}(t)=\exp(-t)$, $h=E+\bar{E}$, $g=E-\bar{E}$ and $\tau=g/h$; $E$ and $\bar{E}$ are positive by claim 2 of Basic Properties of the Exponential Function, so $h>0$, $\operatorname{lch}=\log\circ h$, and $-h<g<h$, whence $|\tau(t)|<1$ and $0\le\tau(t)^{2}<1$.

(i) $E'=E$ by claim 3 of Basic Properties of the Exponential Function; $t\mapsto-t=(-1)t^{1}$ has derivative $-1$ by claim 1 of Derivative of a Polynomial Function on the Real Line and claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, so $\bar{E}'=-\bar{E}$ by Chain Rule for One-Dimensional Derivatives. By claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $h'=g$.

(ii) By (i) and claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $h$ and $g$ are differentiable at every point with $h'=g$ and $g'=E+\bar{E}=h$; so $h$ is smooth on $\mathbb{R}^{1}$ by claim 3 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line (with $\{h,g\}$ as the set of functions there), and $\partial_{1}h=h'=g$ by claim 2 there. By The Natural Logarithm, $\log$ is smooth on $(0,\infty)$, an open subset of $\mathbb{R}^{1}$ (for $s>0$, $|r-s|<s/2$ forces $r>0$), and its derivative $\log'(s)=1/s$ is the one of the smooth inverse function theorem with $n=1$, that is, $\partial_{1}\log(s)=1/s$ for $s>0$; and $h$ takes values in $(0,\infty)$. Smooth maps being of class $C^{1}$ (Smooth Map on a Euclidean Open Set), claim 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ (with $n=m=p=1$, $U=\mathbb{R}$, open in $\mathbb{R}^{1}$ by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $V=(0,\infty)$, $F=h$ and $G=\log$) gives $\partial_{1}\operatorname{lch}(t)=\partial_{1}\log(h(t))\,\partial_{1}h(t)=g(t)/h(t)=\tau(t)$. So, by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line applied to $\operatorname{lch}:\mathbb{R}\to\mathbb{R}$, $\operatorname{lch}$ is differentiable at every $t\in\mathbb{R}$ with $\operatorname{lch}'(t)=\tau(t)$.

(iii) Since $\exp(\log s)=s$ (The Natural Logarithm) and by claim 2 of Basic Properties of the Exponential Function, $\exp(-\operatorname{lch}(t))=1/h(t)$, so $\tau=g\cdot(\exp\circ(-\operatorname{lch}))$. By (ii), claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives and Chain Rule for One-Dimensional Derivatives, $\exp\circ(-\operatorname{lch})$ has derivative $-\tau(t)/h(t)$ at $t$; $g'=h$ by (ii); so by claim 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\tau'(t)=h(t)/h(t)-g(t)\tau(t)/h(t)=1-\tau(t)^{2}$.

(iv) By claim 1 of Derivative of a Polynomial Function on the Real Line ($t\mapsto t^{2}$ has derivative $2t$), claims 1 to 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, (ii) and (iii): $f$ is differentiable at every $t$ with derivative $f'(t)=t/\varsigma-\tau(t)$, which is the map $f'_{\varsigma}$ of claim 3; $f'$ is differentiable with derivative $f''(t)=1/\varsigma-1+\tau(t)^{2}$; and $f''$ is differentiable (product rule for $\tau^{2}$).

(v) $f$ is of class $C^{2}$ on $\mathbb{R}=\mathbb{R}^{1}$, open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, with $\partial_{1}f=f'$ and $\partial_{1}f'=f''$. Indeed, for a map $P:\mathbb{R}\to\mathbb{R}$ differentiable at every point with derivative $P'$, claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, with $\mathbb{R}$ regarded as $\mathbb{R}^{1}$, shows that for every $t\in\mathbb{R}$ the partial derivative $\partial_{1}P(t)$ of Partial Derivative on a Euclidean Open Set exists and equals $P'(t)$; and $P$ is continuous at every point in the sense of Continuity at a Point for Maps Between Euclidean Spaces, because by Differentiability at an Interior Point Implies Continuity There it is continuous for the metric $|s-t|$, and for $n=m=1$ and positive $\delta,\varepsilon$ the conditions $(s-t)^{2}<\delta^{2}$ and $(P(s)-P(t))^{2}<\varepsilon^{2}$ are equivalent to $|s-t|<\delta$ and $|P(s)-P(t)|<\varepsilon$. Applying this to $P=f,f',f''$ (by (iv)) and using clauses 1 to 3 of C^k Maps on a Euclidean Open Set: $f'$ is of class $C^{1}$, and $f$ is of class $C^{1}$ with $\partial_{1}f=f'$ of class $C^{1}$, that is, of class $C^{2}$.

(vi) Bounds. Let $K_{\varsigma}$ be the greater of $0$ and $1-\varsigma^{-1}$; then $0\le K_{\varsigma}$ and, by (iv) and $0\le\tau^{2}<1$,

$$-K_{\varsigma}\le\frac{1}{\varsigma}-1\le f''(t)<\frac{1}{\varsigma},\qquad f'(t)^{2}\le\frac{2t^{2}}{\varsigma^{2}}+2\tau(t)^{2}\le\frac{2t^{2}}{\varsigma^{2}}+2,$$

the second from $(p+q)^{2}\le2p^{2}+2q^{2}$. The map $\log$ is increasing on $(0,\infty)$: if $0<s\le s'$ and $\log s'<\log s$, then $s'=\exp(\log s')<\exp(\log s)=s$ by claim 4 of Basic Properties of the Exponential Function, a contradiction. As $t\le|t|$ and $-t\le|t|$, claim 4 of Basic Properties of the Exponential Function gives $h(t)\le2\exp(|t|)$, so, by The Natural Logarithm ($\log(st)=\log s+\log t$ and $\log\exp(|t|)=|t|$), $\operatorname{lch}(t)\le|t|+\log2$; and $\log2\ge\log1=0$, since $\exp(0)=1$ by claim 1 of Basic Properties of the Exponential Function. From $(|t|-\varsigma)^{2}\ge0$ and $(|t|-2\varsigma)^{2}\ge0$, $|t|\le t^{2}/(2\varsigma)+\varsigma/2$ and $|t|\le t^{2}/(4\varsigma)+\varsigma$; hence

$$f(t)\ge\frac{t^{2}}{2\varsigma}-|t|-\log2\ge-\Bigl(\frac{\varsigma}{2}+\log2\Bigr),\qquad f(t)\ge\frac{t^{2}}{4\varsigma}-\varsigma-\log2 .$$

Step 4 (Claim 2). Keep $\varsigma$, $f$ and $K_{\varsigma}$ of Step 3, and write $v=v_{\varsigma}$, $V=V_{\varsigma}$. By Step 0, $V(x)=\sum_{z}f(\ell_{p_{d}(x)}(z))=v(p_{d}(x))$, so $V=v\circ p_{d}$.

Regularity and derivatives. $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. For $z\in\mathbb{L}$ let $L_{z}:\mathbb{R}^{d}\to\mathbb{R}$, $L_{z}(u)=\ell_{u}(z)=\sum_{j=1}^{d}\omega_{j}(z)u_{j}$. For $m\in[d]$ let $L^{[m]}_{z}:\mathbb{R}^{d}\to\mathbb{R}$, $L^{[m]}_{z}(u)=\sum_{j=1}^{m}\omega_{j}(z)u_{j}$, and let $\pi_{l}$ be the $l$-th coordinate function on $\mathbb{R}^{d}$. Let $I_{z}$ be the set of $m\in\mathbb{N}$ such that, if $m\le d$, then $L^{[m]}_{z}$ is smooth on $\mathbb{R}^{d}$. Then $1\in I_{z}$: if $1\le d$, claim 1 of Properties of Finite Sums gives $L^{[1]}_{z}=\omega_{1}(z)\pi_{1}$, which is smooth by claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. And $S(m)\in I_{z}$ for $m\in I_{z}$: if $S(m)\le d$, then $m<S(m)$ by Properties of the Order on the Natural Numbers §successor, so $m\le d$ by Properties of the Order on the Natural Numbers §basic and $L^{[m]}_{z}$ is smooth; as $S(m)\in[d]$, claim 1 of Properties of Finite Sums gives $L^{[S(m)]}_{z}=L^{[m]}_{z}+\omega_{S(m)}(z)\pi_{S(m)}$, which is smooth by claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. By Principle of Induction for the Natural Numbers, $I_{z}=\mathbb{N}$; since $d\le d$ (Properties of the Order on the Natural Numbers §basic), $L_{z}=L^{[d]}_{z}$ is smooth, hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set; and $\partial_{i}L_{z}(u)=\omega_{i}(z)$ by claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, because the difference quotient of Partial Derivative on a Euclidean Open Set for the $l$-th coordinate function in the $i$-th variable is constantly $1$ if $l=i$ and $0$ otherwise. By claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ (with $k=2$, $F=L_{z}$ and $G=f$, of class $C^{2}$ on $\mathbb{R}$ by Step 3(v)), $f\circ L_{z}$ is of class $C^{2}$ on $\mathbb{R}^{d}$; by claim 1 there, applied with $G=f$ and with $G=f'$ (of class $C^{1}$ by Step 3(v)), $\partial_{i}(f\circ L_{z})(u)=f'(\ell_{u}(z))\omega_{i}(z)$ and $\partial_{k}(f'\circ L_{z})(u)=f''(\ell_{u}(z))\omega_{k}(z)$. As $\partial_{i}(f\circ L_{z})$ is the scalar multiple $\omega_{i}(z)\,(f'\circ L_{z})$, claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set gives $\partial_{k}\partial_{i}(f\circ L_{z})(u)=f''(\ell_{u}(z))\,\omega_{k}(z)\,\omega_{i}(z)$. Now sum over the finite set $\mathbb{L}$ along an enumeration $z_{1},\dots,z_{m_{0}}$ of $\mathbb{L}$, and for $m\in[m_{0}]$ let $v^{[m]}=\sum_{q=1}^{m}f\circ L_{z_{q}}$. Say that $m\in[m_{0}]$ has property (C$_{m}$) if $v^{[m]}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with $\partial_{i}v^{[m]}(u)=\sum_{q=1}^{m}f'(\ell_{u}(z_{q}))\,\omega_{i}(z_{q})$ and $\partial_{k}\partial_{i}v^{[m]}(u)=\sum_{q=1}^{m}f''(\ell_{u}(z_{q}))\,\omega_{k}(z_{q})\,\omega_{i}(z_{q})$ for all $u\in\mathbb{R}^{d}$ and $i,k\in[d]$. Let $I$ be the set of $m\in\mathbb{N}$ such that, if $m\le m_{0}$, then (C$_{m}$) holds. Then $1\in I$: if $1\le m_{0}$, claim 1 of Properties of Finite Sums gives $v^{[1]}=f\circ L_{z_{1}}$ and reduces the two sums to their single terms, so (C$_{1}$) is what was just shown for $z=z_{1}$. And $S(m)\in I$ for $m\in I$: if $S(m)\le m_{0}$, then $m<S(m)$ by Properties of the Order on the Natural Numbers §successor, so $m\le m_{0}$ by Properties of the Order on the Natural Numbers §basic and (C$_{m}$) holds; as $S(m)\in[m_{0}]$, claim 1 of Properties of Finite Sums gives $v^{[S(m)]}=v^{[m]}+f\circ L_{z_{S(m)}}$, and likewise splits off the last term of the two derivative sums; so $v^{[S(m)]}$ is of class $C^{2}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, and claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to the sum and then to the sum of the first partial derivatives, gives the two formulas, that is, (C$_{S(m)}$). By Principle of Induction for the Natural Numbers, $I=\mathbb{N}$; since $m_{0}\le m_{0}$ (Properties of the Order on the Natural Numbers §basic), (C$_{m_{0}}$) holds, and $v=v^{[m_{0}]}$. Hence $v$ is of class $C^{2}$ on $\mathbb{R}^{d}$, and for $u\in\mathbb{R}^{d}$ and $i,k\in[d]$

$$\partial_{i}v(u)=\sum_{z\in\mathbb{L}}f'(\ell_{u}(z))\,\omega_{i}(z),\qquad\partial_{k}\partial_{i}v(u)=\sum_{z\in\mathbb{L}}f''(\ell_{u}(z))\,\omega_{k}(z)\,\omega_{i}(z).\tag{V}$$

In particular $\partial_{i}v=0$ when $\chi(i)=0$, since then $\omega_{i}=0$.

We verify the four conditions of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §admissible with head dimension $d$, profile $v$ and the nonnegative constant $K_{\varsigma}$. Let $u\in\mathbb{R}^{d}$; by Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §count-sites, a sum over $z\in\mathbb{L}$ of a constant $s$ equals $L^{n}s$.

(a) By Step 3(vi), $v(u)=\sum_{z}f(\ell_{u}(z))\ge-L^{n}(\varsigma/2+\log2)=-b$; this is Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below.

(b) Let $\xi\in\mathbb{R}^{d}$. By (V), expanding as in (Q),

$$\sum_{j=1}^{d}\sum_{k=1}^{d}\partial_{j}\partial_{k}v(u)\,\xi_{j}\xi_{k}=\sum_{z\in\mathbb{L}}f''(\ell_{u}(z))\,\ell_{\xi}(z)^{2}\ge-K_{\varsigma}\sum_{z\in\mathbb{L}}\ell_{\xi}(z)^{2}=-K_{\varsigma}\sum_{j=1}^{d}\chi(j)\frac{\xi_{j}^{2}}{a_{j}}\ge-K_{\varsigma}\sum_{j=1}^{d}\frac{\xi_{j}^{2}}{a_{j}},$$

by Step 3(vi) with $\ell_{\xi}(z)^{2}\ge0$, by (Q), and since $0\le\chi(j)\le1$, $\xi_{j}^{2}/a_{j}\ge0$ and $K_{\varsigma}\ge0$. This is Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §semiconvex.

(c) By (V) and (P) with $F=f'\circ\ell_{u}$, then Step 3(vi),

$$\sum_{k=1}^{d}a_{k}\bigl(\partial_{k}v(u)\bigr)^{2}=\sum_{z\in\mathbb{L}}f'(\ell_{u}(z))^{2}\le\frac{2}{\varsigma^{2}}\sum_{z\in\mathbb{L}}\ell_{u}(z)^{2}+2L^{n}.$$

The second bound on $f$ in Step 3(vi) gives $t^{2}\le4\varsigma(f(t)+\varsigma+\log2)$, so $\sum_{z}\ell_{u}(z)^{2}\le4\varsigma\bigl(v(u)+L^{n}(\varsigma+\log2)\bigr)$. Hence, with $A=8L^{n}(\varsigma+\log2)/\varsigma+2L^{n}>0$ and $C=8/\varsigma+A$,

$$\sum_{k=1}^{d}a_{k}\bigl(\partial_{k}v(u)\bigr)^{2}\le\frac{8}{\varsigma}v(u)+A\le\frac{8}{\varsigma}|v(u)|+A\le C\bigl(1+|v(u)|\bigr)\le C\bigl(1+|v(u)|\bigr)^{2},$$

the last since $1\le1+|v(u)|$. This is Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope.

(d) For $k\in[d]$ and $z\in\mathbb{L}$, $a_{k}\omega_{k}(z)^{2}=L^{-n}\chi(k)\psi_{\kappa(k)}(z)^{2}$ (Step 0), so by (R) and (D), $\sum_{k=1}^{d}a_{k}\omega_{k}(z)^{2}=L^{-n}\sum_{l\in\Gamma}\psi_{l}(z)^{2}=1$. Hence by (V) and Step 3(vi),

$$\sum_{k=1}^{d}a_{k}\,\partial_{k}\partial_{k}v(u)=\sum_{z\in\mathbb{L}}f''(\ell_{u}(z))\sum_{k=1}^{d}a_{k}\omega_{k}(z)^{2}=\sum_{z\in\mathbb{L}}f''(\ell_{u}(z))\le\frac{L^{n}}{\varsigma}.$$

For positive $\varepsilon$ put $C_{\varepsilon}=L^{n}/\varsigma>0$; since $\varepsilon\sum_{k}a_{k}(\partial_{k}v(u))^{2}\ge0$ and $1+\sum_{k}u_{k}^{2}\ge1$, the right side of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §curvature is at least $C_{\varepsilon}$, which proves (d).

So $V$ is an admissible cylindrical potential with head dimension $d$, profile $v$ and semiconvexity constant $K_{\varsigma}$, and $b$ serves in (a). This proves claim 2.

Step 5 (Claim 3). Let $x\in X$, $u=p_{d}(x)$ and $F=f'\circ\varphi_{x}=f'\circ\ell_{u}$ (Step 0), and let $w=\nabla_{a}V(x)=\sum_{k=1}^{d}a_{k}\partial_{k}v(u)\,e_{k}$ (Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient) and $y=\mathcal{E}_{M}F\in X$ (The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §embedding). For $i\in\mathbb{N}$, the orthonormality of $(e_{j})$ (White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §basis) gives $w_{i}=\langle w,e_{i}\rangle_{H^{-m}}=a_{i}\partial_{i}v(u)$ if $i\le d$ and $w_{i}=0$ if $i>d$. By (V), (F) and Step 0, for $i\le d$ with $\chi(i)=1$,

$$a_{i}\partial_{i}v(u)=a_{i}a_{i}^{-1/2}L^{-n/2}\sum_{z\in\mathbb{L}}F(z)\psi_{\kappa(i)}(z)=a_{i}^{1/2}L^{n/2}\hat{F}(\kappa(i))=a_{i}^{1/2}\,y(\kappa(i)),$$

while $a_{i}\partial_{i}v(u)=0$ if $\chi(i)=0$. On the other hand $y_{i}=a_{i}^{1/2}y(\kappa(i))$ by The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §coordinates, and $y(\kappa(i))=0$ whenever $\kappa(i)\notin\Gamma$. By Step 0, therefore, $w_{i}=y_{i}$ for every $i\in\mathbb{N}$. Applying The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §coordinates to $w$ and $y$, $w(\kappa(i))=a_{i}^{-1/2}w_{i}=a_{i}^{-1/2}y_{i}=y(\kappa(i))$ for every $i$; as $\kappa$ is onto $\mathbb{Z}^{n}$, the coefficient families $w$ and $y$ agree at every $k\in\mathbb{Z}^{n}$ (The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data), so $\nabla_{a}V(x)=\mathcal{E}_{M}(f'\circ\varphi_{x})$. With Step 3(iv) this proves claim 3.

Step 6 (Claim 4). Write $c'=c^{J,\eta}$; $c^{(\varsigma)}$ and $c'$ are variance sequences (Step 2 and The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §variances), with truncations $(c^{(\varsigma)})^{(N)}$ and $c'^{(N)}$ (Variance Sequences and Their Truncations §truncations); push-forwards, image measures and measures with densities are those of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward and Image Measures, Measures with Densities, and Change of Variables. A map between Euclidean spaces or $X$ that is continuous is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space; $\exp$ is continuous by claim 3 of Basic Properties of the Exponential Function and Differentiability at an Interior Point Implies Continuity There, and the coordinate maps $p_{N}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity.

(i) The density. Let $G:X\to\mathbb{R}$, $G(x)=\exp\bigl(-\frac{1}{2\varsigma}\sum_{z\in\mathbb{L}}\varphi_{x}(z)^{2}\bigr)$. It is positive (claim 2 of Basic Properties of the Exponential Function), at most $1$ (claim 4 there), and continuous, hence Borel: each $x\mapsto\varphi_{x}(z)$ is continuous by The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §site-continuous, so is its square by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, the finite sum $x\mapsto\sum_{z\in\mathbb{L}}\varphi_{x}(z)^{2}$ over the nonempty finite set $\mathbb{L}$ is continuous by Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous §continuous (all coefficients $1$), its multiple by $-1/(2\varsigma)$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, and its composite with $\exp$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map. For $N\in\mathbb{N}$ with $d\le N$, (Q) gives $G=G_{N}\circ p_{N}$ with $G_{N}:\mathbb{R}^{N}\to\mathbb{R}$, $G_{N}(u)=\exp\bigl(-\sum_{i=1}^{N}\epsilon_{i}u_{i}^{2}/(2\varsigma a_{i})\bigr)$, where $\epsilon_{i}=1$ if $\kappa(i)\in\Gamma$ and $\epsilon_{i}=0$ otherwise (so $\epsilon_{i}=\chi(i)$ for $i\le d$ and $\epsilon_{i}=0$ for $i>d$, Step 0); $G_{N}$ is continuous for the Euclidean distance, hence Borel: each coordinate function $u\mapsto u_{i}$ is $1$-Lipschitz ($|u_{i}-u'_{i}|\le\lVert u-u'\rVert$), hence continuous, so $u\mapsto u_{i}^{2}$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, the map $u\mapsto\sum_{i=1}^{N}\bigl(-\epsilon_{i}/(2\varsigma a_{i})\bigr)u_{i}^{2}$, a finite sum over the nonempty finite set $[N]$, is continuous by Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous §continuous (with coefficients $b_{i}=-\epsilon_{i}/(2\varsigma a_{i})$), and its composite with $\exp$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map; so $\mathbf{1}_{B}G_{N}$ is Borel for $B\in\mathcal{B}(\mathbb{R}^{N})$. By (S) and Step 2, $1/c'_{i}=1/c^{(\varsigma)}_{i}+\epsilon_{i}/(\varsigma a_{i})$ for every $i\in\mathbb{N}$. Hence, with the diagonal Gaussian densities $\rho_{(c^{(\varsigma)})^{(N)}}$ and the numbers $Z_{(c^{(\varsigma)})^{(N)}}$ of The Diagonal Gaussian Density on Euclidean Space and Its Notation §density and claim 1 of Basic Properties of the Exponential Function,

$$G_{N}(u)\,\rho_{(c^{(\varsigma)})^{(N)}}(u)=\exp\Bigl(-\frac{1}{2}\sum_{i=1}^{N}\frac{u_{i}^{2}}{c'_{i}}-Z_{(c^{(\varsigma)})^{(N)}}\Bigr)=\theta_{N}\,\rho_{c'^{(N)}}(u),\qquad\theta_{N}=\exp\bigl(Z_{c'^{(N)}}-Z_{(c^{(\varsigma)})^{(N)}}\bigr)>0 .$$

Let $B\in\mathcal{B}(\mathbb{R}^{N})$. By claim 2 of Image Measures, Measures with Densities, and Change of Variables, $(p_{N})_{\#}\gamma_{c^{(\varsigma)}}=\gamma_{(c^{(\varsigma)})^{(N)}}$ (Diagonal Gaussian Measures on a Hilbert Space §measure), claim 3 of Image Measures, Measures with Densities, and Change of Variables for $\gamma_{(c^{(\varsigma)})^{(N)}}$, the measure with density $\rho_{(c^{(\varsigma)})^{(N)}}$ with respect to Lebesgue measure $\lambda_{N}$ (Diagonal Gaussian Measures on Euclidean Space §measure), and Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\int_{X}\mathbf{1}_{p_{N}^{-1}(B)}G\,d\gamma_{c^{(\varsigma)}}=\int_{X}(\mathbf{1}_{B}G_{N})\circ p_{N}\,d\gamma_{c^{(\varsigma)}}=\int_{\mathbb{R}^{N}}\mathbf{1}_{B}G_{N}\,\rho_{(c^{(\varsigma)})^{(N)}}\,d\lambda_{N}=\theta_{N}\,\gamma_{c'^{(N)}}(B).\tag{H}$$

With $B=\mathbb{R}^{N}$, $Z_{G}=\int_{X}G\,d\gamma_{c^{(\varsigma)}}=\theta_{N}$, a positive real number, $\gamma_{c'^{(N)}}$ being a probability measure.

(ii) Identification of $\gamma_{c'}$. Let $\nu$ be the measure with density $G/Z_{G}$ with respect to $\gamma_{c^{(\varsigma)}}$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables); by Linearity and Monotonicity of the Lebesgue Integral §nonnegative, $\nu(A)=Z_{G}^{-1}\int_{X}\mathbf{1}_{A}G\,d\gamma_{c^{(\varsigma)}}$, so $\nu(X)=1$ and $\nu\in\mathcal{P}(X)$. For $N\ge d$, (H) and $Z_{G}=\theta_{N}$ give $(p_{N})_{\#}\nu=\gamma_{c'^{(N)}}$. For $N<d$, let $\pi:\mathbb{R}^{d}\to\mathbb{R}^{N}$ keep the first $N$ coordinates; it is $1$-Lipschitz for the Euclidean distances, hence Borel, and $p_{N}=\pi\circ p_{d}$; so, push-forwards composing since preimages do, $(p_{N})_{\#}\nu=\pi_{\#}(p_{d})_{\#}\nu=\pi_{\#}\gamma_{c'^{(d)}}=\pi_{\#}(p_{d})_{\#}\gamma_{c'}=(p_{N})_{\#}\gamma_{c'}=\gamma_{c'^{(N)}}$, by Diagonal Gaussian Measures on a Hilbert Space §measure twice. By the uniqueness in Diagonal Gaussian Measures on a Hilbert Space §measure, $\nu=\gamma_{c'}$. Consequently, by claim 3 of Image Measures, Measures with Densities, and Change of Variables and Linearity and Monotonicity of the Lebesgue Integral §nonnegative, $\int_{X}\Phi\,d\gamma_{c'}=Z_{G}^{-1}\int_{X}\Phi\,G\,d\gamma_{c^{(\varsigma)}}$ for every Borel $\Phi:X\to[0,\infty)$.

(iii) The Gibbs measure. Let $\varrho=\varrho^{\mathrm{HS}}_{J,\eta}$ be the density of The Hubbard-Stratonovich Transform Couples the Ising Measure and the Field Law: the Density of the Field Law, the Disintegration of the Joint Law, and Entropy Contraction under Both Kernels §density; it is positive and continuous, hence Borel. By that clause and claim 1 of Basic Properties of the Exponential Function, $\varrho(x)=\vartheta\exp\bigl(\sum_{z}\operatorname{lch}(\varphi_{x}(z))\bigr)$ with $\vartheta=Z_{J}^{-1}\exp(-\eta L^{n}/2)$, which is positive because $\varrho$ and $\exp$ are; so, by claim 1 of Basic Properties of the Exponential Function and the definition of $f$,

$$\varrho(x)\,G(x)=\vartheta\exp\Bigl(\sum_{z\in\mathbb{L}}\Bigl(\operatorname{lch}(\varphi_{x}(z))-\frac{\varphi_{x}(z)^{2}}{2\varsigma}\Bigr)\Bigr)=\vartheta\exp\bigl(-V(x)\bigr)=\vartheta\,w_{V,1}(x),$$

with the weight $w_{V,1}$ of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §weight. For $B\in\mathcal{B}(X)$, by The Hubbard-Stratonovich Transform Couples the Ising Measure and the Field Law: the Density of the Field Law, the Disintegration of the Joint Law, and Entropy Contraction under Both Kernels §density, (ii) with $\Phi=\mathbf{1}_{B}\varrho$, and Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\gamma^{\mathrm{HS}}_{J,\eta}(B)=\int_{X}\mathbf{1}_{B}\varrho\,d\gamma_{c'}=\frac{\vartheta}{Z_{G}}\int_{X}\mathbf{1}_{B}\,w_{V,1}\,d\gamma_{c^{(\varsigma)}}.$$

For $B=X$ this reads $1=\vartheta Z_{G}^{-1}Z_{V,1}$, with the normaliser $Z_{V,1}$ of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §normaliser; so $\vartheta/Z_{G}=1/Z_{V,1}$ and $\gamma^{\mathrm{HS}}_{J,\eta}(B)=\gamma^{V}_{1}(B)$ by The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §gibbs, $V$ being admissible by claim 2 with $b$ as there. This proves claim 4.

Step 7 (Claim 5). Let $\varsigma,\varsigma'$ be admissible splits; for $s\in\{\varsigma,\varsigma'\}$ write $r^{(s)}_{k}$, $c^{(s)}$, $V_{s}=v_{s}\circ p_{d}$ (Step 4) and $\rho_{s}=\gamma_{c^{(s)}}$, and use the reading of the statement with $s$. By claim 1, $c^{(s)}_{k}\le\bar{r}_{s}a_{k}$ for every $k$, with $\bar{r}_{s}\ge1>0$; so the hypotheses of The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §hypothesis and The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §hypothesis hold with the constant $\bar{r}_{s}$, and the Gibbs entropy pair $(\mathcal{D}^{s},\mathcal{D}^{s}_{\Sigma},\mathcal{E}^{s},\Sigma^{s})$ with potential $V_{s}$ (admissible by claim 2) and temperature $1$ is defined (The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair).

(i) The noise-connected measures. Apply Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile in the reading with $\varsigma$, with $\beta=1$, its constant $\bar{r}_{\varsigma}$, and the sequence written $d$ there taken to be $\delta=(\delta_{k})_{k\in\mathbb{N}}$, $\delta_{k}=\epsilon_{k}(1/\varsigma-1/\varsigma')/a_{k}$, with $\epsilon_{k}$ as in Step 6(i). It is admissible in the sense of Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §admissible: the terms $|\delta_{k}|c^{(\varsigma)}_{k}$ vanish for $k>d$ (Step 0), so the partial sums of $\sum_{k}|\delta_{k}|c^{(\varsigma)}_{k}$ are constant from $d$ on and the series converges (Series of Real Numbers §convergent); and $|\delta_{k}|a_{k}\le\Delta$ with $\Delta=|1/\varsigma-1/\varsigma'|\ge0$. By (S), if $\kappa(k)\in\Gamma$ then $1/c^{(\varsigma)}_{k}+\delta_{k}=\bigl(1/\mathfrak{q}_{\kappa(k)}-1/\varsigma+1/\varsigma-1/\varsigma'\bigr)/a_{k}=1/c^{(\varsigma')}_{k}$, and otherwise $\delta_{k}=0$ and $c^{(\varsigma)}_{k}=c_{k}=c^{(\varsigma')}_{k}$. So $1/c^{(\varsigma)}_{k}+\delta_{k}=1/c^{(\varsigma')}_{k}>0$, the sequence written $c'$ there is $c^{(\varsigma')}$, and Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §space gives $\mathcal{P}^{a}_{\rho_{\varsigma'}}=\mathcal{P}^{a}_{\rho_{\varsigma}}$.

(ii) Penalty domain and penalty. By The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain and The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair, $\mathcal{D}^{s}$ is the set of $\mu\in\mathcal{P}(X)$ of finite relative entropy with respect to $\gamma^{V_{s}}_{1}$ and $\mathcal{E}^{s}(\mu)=H(\mu\,|\,\gamma^{V_{s}}_{1})$; since $\gamma^{V_{s}}_{1}=\gamma^{\mathrm{HS}}_{J,\eta}$ by claim 4, $\mathcal{D}^{\varsigma}=\mathcal{D}^{\varsigma'}$ is the set described in claim 5, and $\mathcal{E}^{\varsigma}=\mathcal{E}^{\varsigma'}=H(\cdot\,|\,\gamma^{\mathrm{HS}}_{J,\eta})$. Write $\mathcal{D}$ for this set.

(iii) A split-free identity. For $k\in\mathbb{N}$ and $x\in X$, with $\partial_{k}V_{s}$ as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient,

$$\frac{x_{k}}{c^{(\varsigma)}_{k}}+\partial_{k}V_{\varsigma}(x)=\frac{x_{k}}{c^{(\varsigma')}_{k}}+\partial_{k}V_{\varsigma'}(x).\tag{I}$$

Let $u=p_{d}(x)$ and, for $k\in[d]$, $T_{k}(u)=\sum_{z}\tau(\ell_{u}(z))\omega_{k}(z)$, which does not involve the split. If $\kappa(k)\in\Gamma$, then $k\le d$, $\chi(k)=1$ and $u_{k}=x_{k}$, and by (V), Step 3(iv) and (Q), $\partial_{k}v_{s}(u)=\frac{1}{s}\sum_{z}\ell_{u}(z)\omega_{k}(z)-T_{k}(u)=\frac{x_{k}}{s\,a_{k}}-T_{k}(u)$; so by (S) both sides of (I) equal $x_{k}/c^{J,\eta}_{k}-T_{k}(u)$. If $k\le d$ and $\chi(k)=0$, then $\partial_{k}V_{s}(x)=\partial_{k}v_{s}(u)=0$ (Step 4), and if $k>d$, $\partial_{k}V_{s}(x)=0$ by definition; in both cases $\kappa(k)\notin\Gamma$ and $c^{(s)}_{k}=c_{k}$, so both sides equal $x_{k}/c_{k}$.

(iv) Score domain. Let $\mu\in\mathcal{D}$. By Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain, in each reading, $\mu\in\mathcal{P}_{2}(X)$ and each $\partial_{k}V_{s}$ is integrable with respect to $\mu$, so The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential applies to $\mu$ for $s=\varsigma$ and $s=\varsigma'$. The class $\mathcal{F}C^{1}_{b}(X)$, the derivatives $\partial_{k}\varphi$ and $L^{2}(\mu)$ do not involve the variance sequence, and by (I) the integrands in The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential §score (with $\beta=1$) coincide for the two splits. Hence $\mu$ has a relative score with respect to $\gamma^{V_{\varsigma}}_{1}$ if and only if it has one with respect to $\gamma^{V_{\varsigma'}}_{1}$, and then, by the uniqueness in that clause, the components $\zeta^{V}_{k}$ are the same. The series of The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential §information involves only these components and $a$, so finite Fisher information relative to $\gamma^{V_{\varsigma}}_{1}$ and relative to $\gamma^{V_{\varsigma'}}_{1}$ with weights $a$ are the same condition. By The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, $\mathcal{D}^{\varsigma}_{\Sigma}=\mathcal{D}^{\varsigma'}_{\Sigma}$.

(v) Score. Let $\mu\in\mathcal{D}^{\varsigma}_{\Sigma}$ and $s\in\{\varsigma,\varsigma'\}$. In the reading with $s$, $V_{s}$ is integrable with respect to $\mu$ by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy, $\mu\in\mathcal{P}_{2}(X)$ by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain, and by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §splitting $\mu$ has a relative score with respect to $\rho_{s}$, finite Fisher information relative to $\rho_{s}$ with weights $a$, and $\int_{X}|\nabla_{a}V_{s}|_{a}^{2}\,d\mu<\infty$. So Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §field (with $\beta=1$) shows that $\Sigma^{s}(\mu)=Z^{a}_{\mu}+\nabla_{a}V_{s}$, an element of $L^{2}(\mu;X^{a})$, which does not involve the variance sequence, has coordinate $a_{k}^{1/2}\zeta^{V}_{k}$ along $f_{k}$ for every $k$, with the split-independent $\zeta^{V}_{k}$ of (iv). By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates the series $\sum_{k}\lVert a_{k}^{1/2}\zeta^{V}_{k}\rVert_{L^{2}(\mu)}^{2}$ converges, so by the uniqueness in The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §synthesis, $\Sigma^{\varsigma}(\mu)=\Sigma^{\varsigma'}(\mu)$.

By (i) to (v), $\mathcal{P}^{a}_{\rho_{\varsigma}}=\mathcal{P}^{a}_{\rho_{\varsigma'}}$ and the two Gibbs entropy pairs are the same quadruple, with $\mathcal{D}$ and $\mathcal{E}$ as stated. This proves claim 5.