Proof of The Galerkin Wick-Ordered $Phi^4$ Potentials on the Torus of Dimension at Most Two under the Free Field: Chaos Order, Conditional Expectations, and Increments with Their Cameron-Martin Derivatives

Each result cited is universally quantified over the data in its own statement.

Throughout, $\psi_{k}$, $\mu_{k}$, $\lambda_{Q}$, $a_{j}^{-1/2}=1/a_{j}^{1/2}$, $x(k)$, $\Gamma_{L}$, $d_{L}$, $J_{L}$, $\Phi_{L}[x]$, $C_{L}$ and $f_{L}$ ($L\in\mathbb{N}$) are as in The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus, $x_{j}=\langle x,e_{j}\rangle$, $\mathbb{E}[F]$ denotes $\int_{X}F\,d\gamma_{c}$, and $dz$ stands for $\lambda_{Q}(dz)$. Put $b_{4}=\varkappa/4$, $b_{2}=\mathfrak{m}/2$ and $f^{\sharp}_{L}(s)=\varkappa(s^{3}-3C_{L}s)+\mathfrak{m}s$. By Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §low-orders and The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §polynomial,

$$f_{L}=b_{4}H^{C_{L}}_{4}+b_{2}H^{C_{L}}_{2},\qquad f^{\sharp}_{L}=\varkappa H^{C_{L}}_{3}+\mathfrak{m}H^{C_{L}}_{1}.$$

Fix $N,M\in\mathbb{N}$ with $N\le M$, and let $L,L'$ range over $\{N,M\}$, $L\wedge L'$ being the smaller one.

Step 0 (Indices, and a finite interchange rule). By The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §nested, $\Gamma_{N}\subseteq\Gamma_{M}$; as $\kappa(d_{N})\in\Gamma_{N}\subseteq\Gamma_{M}$ and $d_{M}$ is the largest $j$ with $\kappa(j)\in\Gamma_{M}$ (The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension), $d_{N}\le d_{M}$ and $J_{N}\subseteq J_{M}$. Every $j$ with $\kappa(j)\in\Gamma_{L}$ satisfies $j\le d_{L}$, so $\kappa$ maps $J_{L}$ bijectively onto $\Gamma_{L}$; $J_{N}\neq\emptyset$ since the zero point lies in $\Gamma_{N}$ (The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite). Put $J_{N,M}=J_{M}\setminus J_{N}$. A sum $\sum_{j\in I}u_{j}$ over a subset $I$ of $[d_{M}]$, such as $J_{L}$ or $J_{N,M}$, means the finite sum $\sum_{j=1}^{d_{M}}\mathbf{1}_{I}(j)\,u_{j}$; in particular it is $0$ if $I=\emptyset$. By The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §noise-norm, $x(\kappa(j))=a_{j}^{-1/2}x_{j}$ for $j\in[d_{L}]$, so by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §field

$$\Phi_{L}[x](z)=\sum_{j\in J_{L}}a_{j}^{-1/2}x_{j}\psi_{\kappa(j)}(z),\qquad \Phi_{M}[x](z)=\Phi_{N}[x](z)+\Psi[x](z),\quad \Psi[x](z)=\sum_{j\in J_{N,M}}a_{j}^{-1/2}x_{j}\psi_{\kappa(j)}(z)\tag{0.1}$$

($\Psi=0$ if $J_{N,M}=\emptyset$). Call $F:X\times(\mathbb{R}^{n})^{q}\to\mathbb{R}$ ($q\in\{1,2\}$) elementary if $F(x,y)=\sum_{t=1}^{\nu}g_{t}(y)m_{t}(x)$ for some $\nu\in\mathbb{N}$, each $g_{t}$ a finite product of maps $y\mapsto\psi_{k}(y_{i})$ and each $m_{t}$ a real multiple of a finite product of coordinate maps $x\mapsto x_{j}$ (empty products being $1$). By (0.1) and expansion of finite products, every polynomial with constant coefficients in finitely many of the quantities $\Phi_{L}[x](z)$, $\Psi[x](z)$, $\Phi_{L}[x](w)$, $\psi_{k}(z)$, $\psi_{k}(w)$ is elementary. For elementary $F$: each $m_{t}$ is a finite linear combination of functions $H_{\beta}$ by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §monomials, each Borel and in $L^{p}(\gamma_{c})$ for all $p\ge1$ by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical; so $m_{t}$ and $\mathbf{1}_{A}m_{t}$ ($A$ Borel) are Borel and $\gamma_{c}$-integrable. Each finite product of maps $\psi_{k}$ lies in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra, so its restriction to $Q$ is $\lambda_{Q}$-integrable by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member. For $q=1$, $\int_{Q}F(x,z)\,dz=\sum_{t}(\int_{Q}g_{t})m_{t}(x)$, and linearity of both integrals gives, for Borel $A\subseteq X$,

$$\mathbb{E}\Bigl[\mathbf{1}_{A}\int_{Q}F(\cdot,z)\,dz\Bigr]=\sum_{t}\Bigl(\int_{Q}g_{t}\Bigr)\mathbb{E}[\mathbf{1}_{A}m_{t}]=\int_{Q}\mathbb{E}[\mathbf{1}_{A}F(\cdot,z)]\,dz.\tag{0.2}$$

For $q=2$ each $g_{t}(z,w)=\alpha_{t}(z)\beta_{t}(w)$ with $\alpha_{t},\beta_{t}$ finite products of maps $\psi_{k}$, and the same computation gives

$$\mathbb{E}\Bigl[\int_{Q}\Bigl(\int_{Q}F(\cdot,z,w)\,dw\Bigr)dz\Bigr]=\int_{Q}\Bigl(\int_{Q}\mathbb{E}[F(\cdot,z,w)]\,dw\Bigr)dz.\tag{0.3}$$

Step 1 (Wick-power form). For $z\in\mathbb{R}^{n}$ put

$$h^{L}_{z}=\sum_{j\in J_{L}}c_{j}a_{j}^{-1/2}\psi_{\kappa(j)}(z)\,e_{j},\qquad g_{z}=\sum_{j\in J_{N,M}}c_{j}a_{j}^{-1/2}\psi_{\kappa(j)}(z)\,e_{j}.$$

By 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) their coordinates vanish beyond $d_{M}$, so the series of The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §space has finitely many nonzero terms and $h^{L}_{z},g_{z}\in H_{c}$; the partial sums of The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §partial-sums are constant from $d_{M}$ on, so by The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §functional and (0.1), $\ell_{h^{L}_{z}}(x)=\Phi_{L}[x](z)$ and $\ell_{g_{z}}(x)=\Psi[x](z)$. Since $c_{j}(a_{j}^{-1/2})^{2}=c_{j}/a_{j}=\mu_{\kappa(j)}^{-1}$ 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, The Cameron-Martin Inner Product of Two Cameron-Martin Vectors §inner-product, The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §square, $J_{N}\subseteq J_{M}$ and Step 0 give

$$\langle h^{L}_{z},h^{L'}_{w}\rangle_{c}=\mathcal{K}_{L\wedge L'}(z,w),\qquad \mathcal{K}_{L}(z,w):=\sum_{j\in J_{L}}\mu_{\kappa(j)}^{-1}\psi_{\kappa(j)}(z)\psi_{\kappa(j)}(w)=\sum_{k\in\Gamma_{L}}\mu_{k}^{-1}\psi_{k}(z)\psi_{k}(w),$$

and, by The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §wick, $|h^{L}_{z}|_{c}^{2}=\mathcal{K}_{L}(z,z)=C_{L}$ and $|g_{z}|_{c}^{2}=\mathcal{K}_{M}(z,z)-\mathcal{K}_{N}(z,z)=C_{M}-C_{N}=:\vartheta\ge0$. Hence, by The Wick Powers of a Paley-Wiener Functional §wick,

$$f_{L}(\Phi_{L}[x](z))=b_{4}{:}\ell_{h^{L}_{z}}^{4}{:}(x)+b_{2}{:}\ell_{h^{L}_{z}}^{2}{:}(x),\quad f^{\sharp}_{L}(\Phi_{L}[x](z))=\varkappa{:}\ell_{h^{L}_{z}}^{3}{:}(x)+\mathfrak{m}{:}\ell_{h^{L}_{z}}^{1}{:}(x),\quad H^{\vartheta}_{r}(\Psi[x](z))={:}\ell_{g_{z}}^{r}{:}(x).\tag{1.1}$$

Step 2 (The derivatives). For $k\in\mathbb{Z}^{n}$ put $D^{L}_{k}(x)=\int_{Q}f^{\sharp}_{L}(\Phi_{L}[x](z))\psi_{k}(z)\,dz$. We claim $\partial_{j}V_{L}=a_{j}^{-1/2}D^{L}_{\kappa(j)}$ for $j\in J_{L}$ and $\partial_{j}V_{L}=0$ for $j\in\mathbb{N}\setminus J_{L}$. By Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, $\partial_{j}V_{L}=0$ for $j>d_{L}$, $\partial_{j}V_{L}(x)=\partial_{j}v_{L}(p_{d_{L}}(x))$ for $j\le d_{L}$, which vanishes for $j\in[d_{L}]\setminus J_{L}$ by The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §hessian, and $\nabla_{a}V_{L}(x)=\sum_{j=1}^{d_{L}}a_{j}\partial_{j}V_{L}(x)e_{j}$. By The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §gradient and $\hat{E}_{\kappa(j)}=a_{j}^{1/2}e_{j}$ (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-vectors), $\nabla_{a}V_{L}(x)=\sum_{j\in J_{L}}a_{j}^{1/2}D^{L}_{\kappa(j)}(x)e_{j}$. Pairing with $e_{j}$ ($j\in J_{L}$) gives $a_{j}\partial_{j}V_{L}(x)=a_{j}^{1/2}D^{L}_{\kappa(j)}(x)$, and $a_{j}>0$.

Step 3 (Claim 1). Take $L=N$. Expanding the polynomial $f_{N}$ of degree $4$ in (0.1), $f_{N}(\Phi_{N}[x](z))=\sum_{t}g_{t}(z)m_{t}(x)$ is elementary with each $m_{t}$ a multiple of $\prod_{s=1}^{r}x_{j_{s}}=\prod_{k=1}^{d_{N}}x_{k}^{\alpha_{k}}$, $r\le4$, $j_{s}\in J_{N}$, $|\alpha|=r$. So $V_{N}=\sum_{t}(\int_{Q}g_{t})m_{t}$ (The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §potential) is, by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §monomials, a finite linear combination of functions $H_{\beta}$ with $|\beta|\le4$; these are Borel and square-integrable by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical, so $V_{N}$ is square-integrable and its class lies in the span of $\{H_{\beta}:|\beta|\le4\}$, contained in $\mathcal{H}_{\le4}$ (The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space §chaos-up-to). For $j\in J_{N}$, Step 2 and the same argument with $f^{\sharp}_{N}$ of degree $3$ give the assertion for $\partial_{j}V_{N}$ with $\mathcal{H}_{\le3}$; for $j\notin J_{N}$, $\partial_{j}V_{N}=0$, whose class is the zero vector of $\mathcal{H}_{\le3}$. As $N$ was arbitrary, claim 1 holds.

Step 4 (Independence of coordinate blocks). Let $d\in\mathbb{N}$, let $I_{1},I_{2}\subseteq[d]$ be disjoint and nonempty, and let $\mathcal{G}_{i}$ be the $\sigma$-algebra generated by the maps $x\mapsto x_{j}$, $j\in I_{i}$. We show $\gamma_{c}(A_{1}\cap A_{2})=\gamma_{c}(A_{1})\gamma_{c}(A_{2})$ for $A_{i}\in\mathcal{G}_{i}$, so that $\mathcal{G}_{1},\mathcal{G}_{2}$ are independent (Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras). Write $I_{i}=\{j_{1}<\dots<j_{p}\}$ ($p=p_{i}$), $\pi_{i}(x)=(x_{j_{1}},\dots,x_{j_{p}})$, and let $\mathcal{B}_{p}$ be the product Borel $\sigma$-algebra of Joint Distribution, Expectations, and Block Independence for Independent Random Variables. The sets $\pi_{i}^{-1}(D)$, $D\in\mathcal{B}_{p}$, form a $\sigma$-algebra (preimages commute with complements and countable unions) containing each $x_{j_{l}}^{-1}(B)$, $B$ Borel, which is $\pi_{i}^{-1}$ of the preimage of $B$ under the $l$-th coordinate projection, jointly Borel by claim 4 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables. Hence $A_{i}=\pi_{i}^{-1}(D_{i})$ with $D_{i}\in\mathcal{B}_{p_{i}}$. By Existence of Independent and Identically Distributed Sequences there is a probability space $(\Omega,\mathcal{F},P)$ with an independent sequence $(\xi_{k})$ of standard normal random variables; with $Y$ and $G$ as in The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections, claim The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections §unique and Diagonal Gaussian Measures on a Hilbert Space §measure give $\gamma_{c}=P_{Y}$, The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections §convergence gives $P(G)=1$, and The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections §random-element gives $\langle Y,e_{k}\rangle=\sqrt{c_{k}}\xi_{k}=:\zeta_{k}$ on $G$. The $\zeta_{k}$ ($k\in[d]$) are independent (Independence of Events and of Random Variables), since $\{\zeta_{k}\in B\}=\{\xi_{k}\in B'\}$ with $B'$ the preimage of $B$ under the continuous map $t\mapsto\sqrt{c_{k}}t$. With $\zeta^{(i)}=(\zeta_{j})_{j\in I_{i}}$, claim 3 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables, applied to the indicators $\mathbf{1}_{D_{i}}$ (jointly Borel by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), shows that $\mathbf{1}_{D_{1}}(\zeta^{(1)})$ and $\mathbf{1}_{D_{2}}(\zeta^{(2)})$ are independent, whence $P(\zeta^{(1)}\in D_{1},\zeta^{(2)}\in D_{2})=P(\zeta^{(1)}\in D_{1})P(\zeta^{(2)}\in D_{2})$. Since $\pi_{i}(Y)=\zeta^{(i)}$ on $G$ and $P(\Omega\setminus G)=0$, $\gamma_{c}(A_{1}\cap A_{2})=P(\pi_{1}(Y)\in D_{1},\pi_{2}(Y)\in D_{2})$ equals the left side and $\gamma_{c}(A_{i})=P(\zeta^{(i)}\in D_{i})$. Consequently, if $U$ is $\mathcal{G}_{1}$-measurable and $W$ is $\mathcal{G}_{2}$-measurable, both integrable, then $U,W$ are independent random variables on $(X,\mathcal{B}(X),\gamma_{c})$ (their events $\{U\in B\}$, $\{W\in B''\}$ lie in $\mathcal{G}_{1}$, $\mathcal{G}_{2}$), and by Expectation of a Product of Independent Random Variables

$$\mathbb{E}[UW]=\mathbb{E}[U]\,\mathbb{E}[W].\tag{4.1}$$

Step 5 (Claim 2). We check (i)-(iii) of Conditional Expectation of a Square-Integrable Random Variable; $V_{M}$ and $V_{N}$ are square-integrable by Step 3, and $\mathcal{F}_{N}\subseteq\mathcal{B}(X)$ since the coordinate maps are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity. (i): by Step 3, $V_{N}$ is a linear combination of products of maps $x\mapsto x_{j}$ with $j\in J_{N}$, each $\mathcal{F}_{N}$-measurable, so $V_{N}$ is $\mathcal{F}_{N}$-measurable by claims 1-3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. (iii): by (0.1), Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §addition with variances $C_{N}$ and $\vartheta$ (so $C_{N}+\vartheta=C_{M}$), and $H^{\vartheta}_{0}=1$,

$$f_{M}(\Phi_{M}[x](z))=f_{N}(\Phi_{N}[x](z))+R(x,z),\qquad R(x,z)=\sum_{r\in\{2,4\}}b_{r}\sum_{i=0}^{r-1}\binom{r}{i}H^{C_{N}}_{i}(\Phi_{N}[x](z))\,H^{\vartheta}_{r-i}(\Psi[x](z)).$$

$R$ is elementary, and $V_{M}-V_{N}=\int_{Q}R(\cdot,z)\,dz$ by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §potential and linearity of the integral; so for $A\in\mathcal{F}_{N}$, (0.2) gives $\mathbb{E}[\mathbf{1}_{A}(V_{M}-V_{N})]=\int_{Q}\mathbb{E}[\mathbf{1}_{A}R(\cdot,z)]\,dz$. Fix $z$ and $0\le i<r$, and put $U=\mathbf{1}_{A}H^{C_{N}}_{i}(\Phi_{N}[\cdot](z))$ and $W=H^{\vartheta}_{r-i}(\Psi[\cdot](z))$, integrable by Step 0. If $J_{N,M}=\emptyset$ then $\Psi=0$, $\vartheta=0$ and $W=0^{r-i}=0$ by Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §low-orders. Otherwise $U$ is $\mathcal{F}_{N}$-measurable and $W$ is measurable for the $\sigma$-algebra generated by the coordinates with indices in $J_{N,M}$ (claims 1-3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), so Step 4 with $d=d_{M}$, $I_{1}=J_{N}$, $I_{2}=J_{N,M}$ gives $\mathbb{E}[UW]=\mathbb{E}[U]\mathbb{E}[W]$, and by (1.1) and Wick Powers of Paley-Wiener Functionals: Expansion in Cylindrical Hermite Polynomials, the Covariance Formula, Continuity in the Vector, and Membership in the Chaos §covariance (with ${:}\ell_{g_{z}}^{0}{:}=1$ and $r-i\neq0$), $\mathbb{E}[W]=\int_{X}{:}\ell_{g_{z}}^{r-i}{:}\,{:}\ell_{g_{z}}^{0}{:}\,d\gamma_{c}=0$. Thus $\mathbb{E}[\mathbf{1}_{A}R(\cdot,z)]=0$ for every $z$, and $\mathbb{E}[V_{M}\mathbf{1}_{A}]=\mathbb{E}[V_{N}\mathbf{1}_{A}]$. This proves claim 2.

Step 6 (Second moments). By (0.3) applied to $f_{L}(\Phi_{L}[x](z))f_{L'}(\Phi_{L'}[x](w))$, then (1.1), Wick Powers of Paley-Wiener Functionals: Expansion in Cylindrical Hermite Polynomials, the Covariance Formula, Continuity in the Vector, and Membership in the Chaos §covariance (mixed orders contribute $0$) and Step 1,

$$\mathbb{E}[V_{L}V_{L'}]=\int_{Q}\int_{Q}\bigl(24b_{4}^{2}\mathcal{K}^{4}+2b_{2}^{2}\mathcal{K}^{2}\bigr)\,dw\,dz,\qquad \mathbb{E}[D^{L}_{k}D^{L'}_{k}]=\int_{Q}\int_{Q}\bigl(6\varkappa^{2}\mathcal{K}^{3}+\mathfrak{m}^{2}\mathcal{K}\bigr)\psi_{k}(z)\psi_{k}(w)\,dw\,dz,$$

with $\mathcal{K}=\mathcal{K}_{L\wedge L'}(z,w)$; below, $\iint$ abbreviates this iterated integral $\int_{Q}\int_{Q}\cdots\,dw\,dz$. Expanding $\mathbb{E}[(V_{M}-V_{N})^{2}]=\mathbb{E}[V_{M}^{2}]-2\mathbb{E}[V_{M}V_{N}]+\mathbb{E}[V_{N}^{2}]$, and likewise for $D^{M}_{k}-D^{N}_{k}$, with $24b_{4}^{2}=\frac{3}{2}\varkappa^{2}$, $2b_{2}^{2}=\frac{1}{2}\mathfrak{m}^{2}$:

$$\mathbb{E}[(V_{M}-V_{N})^{2}]=\iint\Bigl(\tfrac{3}{2}\varkappa^{2}(\mathcal{K}_{M}^{4}-\mathcal{K}_{N}^{4})+\tfrac{1}{2}\mathfrak{m}^{2}(\mathcal{K}_{M}^{2}-\mathcal{K}_{N}^{2})\Bigr),\quad \mathbb{E}[(D^{M}_{k}-D^{N}_{k})^{2}]=\iint\bigl(6\varkappa^{2}(\mathcal{K}_{M}^{3}-\mathcal{K}_{N}^{3})+\mathfrak{m}^{2}(\mathcal{K}_{M}-\mathcal{K}_{N})\bigr)\psi_{k}(z)\psi_{k}(w).$$

For $k_{1},\dots,k_{r}\in\mathbb{Z}^{n}$ put $T(k_{1},\dots,k_{r})=\int_{Q}\prod_{i=1}^{r}\psi_{k_{i}}\,d\lambda_{Q}$. Expanding $\mathcal{K}_{L}^{r}$ by distributivity over $\Gamma_{L}^{r}$ (Step 1) and integrating term by term,

$$\iint\mathcal{K}_{L}^{r}=\sum_{(k_{i})\in\Gamma_{L}^{r}}\prod_{i}\mu_{k_{i}}^{-1}\,T(k_{1},\dots,k_{r})^{2},\qquad \iint\mathcal{K}_{L}^{r}\psi_{k}(z)\psi_{k}(w)=\sum_{(k_{i})\in\Gamma_{L}^{r}}\prod_{i}\mu_{k_{i}}^{-1}\,T(k_{1},\dots,k_{r},k)^{2}.$$

By The Trigonometric System on the Torus is Orthonormal §classes, The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and The Trigonometric System on the Torus is Orthonormal §orthonormal, $T(k,k')=\langle E_{k},E_{k'}\rangle_{L^{2}}$ is $1$ if $k=k'$ and $0$ otherwise. Put

$$\Sigma_{4}=\sum_{(k_{i})\in\Gamma_{M}^{4}\setminus\Gamma_{N}^{4}}\prod_{i=1}^{4}\mu_{k_{i}}^{-1}\,T(k_{1},k_{2},k_{3},k_{4})^{2},\qquad \Sigma_{2}=\sum_{k\in\Gamma_{M}\setminus\Gamma_{N}}\mu_{k}^{-2}$$

(empty sums being $0$). As $\Gamma_{N}^{r}\subseteq\Gamma_{M}^{r}$ and only diagonal pairs have $T(k,k')\ne0$,

$$\mathbb{E}[(V_{M}-V_{N})^{2}]=\tfrac{3}{2}\varkappa^{2}\Sigma_{4}+\tfrac{1}{2}\mathfrak{m}^{2}\Sigma_{2}.\tag{6.1}$$

For the derivatives, Step 2 gives for $j\in[d_{M}]$ with $k=\kappa(j)$: $\partial_{j}V_{M}-\partial_{j}V_{N}=a_{j}^{-1/2}(D^{M}_{k}-D^{N}_{k})$ if $j\in J_{N}$, $=a_{j}^{-1/2}D^{M}_{k}$ if $j\in J_{N,M}$, and $=0$ otherwise. With $c_{j}(a_{j}^{-1/2})^{2}=\mu_{k}^{-1}$ and the bijections of Step 0,

$$\sum_{j=1}^{d_{M}}c_{j}\,\mathbb{E}[(\partial_{j}V_{M}-\partial_{j}V_{N})^{2}]=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}\mathbb{E}[(D^{M}_{k}-D^{N}_{k})^{2}]+\sum_{k\in\Gamma_{M}\setminus\Gamma_{N}}\mu_{k}^{-1}\mathbb{E}[(D^{M}_{k})^{2}].$$

For $k\in\Gamma_{N}$, $\iint(\mathcal{K}_{M}-\mathcal{K}_{N})\psi_{k}(z)\psi_{k}(w)=\sum_{k'\in\Gamma_{M}\setminus\Gamma_{N}}\mu_{k'}^{-1}T(k',k)^{2}=0$, and the cubic part is $6\varkappa^{2}$ times the sum over $(k_{1},k_{2},k_{3})\in\Gamma_{M}^{3}\setminus\Gamma_{N}^{3}$; for $k\in\Gamma_{M}\setminus\Gamma_{N}$, $\mathbb{E}[(D^{M}_{k})^{2}]$ is $6\varkappa^{2}$ times the sum over $\Gamma_{M}^{3}$ plus $\mathfrak{m}^{2}\mu_{k}^{-1}$. A tuple $(k_{1},k_{2},k_{3},k)\in\Gamma_{M}^{4}$ lies outside $\Gamma_{N}^{4}$ exactly when $k\in\Gamma_{N}$ and $(k_{1},k_{2},k_{3})\notin\Gamma_{N}^{3}$, or $k\notin\Gamma_{N}$; hence

$$\sum_{j=1}^{d_{M}}c_{j}\,\mathbb{E}[(\partial_{j}V_{M}-\partial_{j}V_{N})^{2}]=6\varkappa^{2}\Sigma_{4}+\mathfrak{m}^{2}\Sigma_{2}.\tag{6.2}$$

Step 7 (The quartic integrals). Let $k=(k_{1},\dots,k_{4})\in(\mathbb{Z}^{n})^{4}$, with $k_{i,l}$ the $l$-th component of $k_{i}$. We show: (a) $|T(k)|\le4^{n}$; (b) if $T(k)\ne0$, then for each $l\in[n]$ there are $\varepsilon_{1,l},\dots,\varepsilon_{4,l}\in\{1,-1\}$ with $\sum_{i}\varepsilon_{i,l}k_{i,l}=0$. Let $J^{0}=\{t:0\le t<1\}$ with the measure $\lambda_{J^{0}}$ (the set $J$ of The Integral over the Unit Cell of a Product of One-Variable Functions), let $\phi_{m}$ be the maps of The Trigonometric System on the Torus §one-dimensional, and $g_{l}(t)=\prod_{i=1}^{4}\phi_{k_{i,l}}(t)$. By The Trigonometric System on the Torus §system, $\prod_{i}\psi_{k_{i}}(z)=\prod_{l}g_{l}(z_{l})$; each $\phi_{m}$ is a constant multiple of $C_{|m|}$ or $S_{|m|}$ (with $C_{b},S_{b}$ as in Cell Integrals of the Trigonometric Monomials; see (b) below), which is continuous on $\mathbb{R}$ by Cell Integrals of the Trigonometric Monomials §calculus, so $g_{l}$, a product of such maps, is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, hence measurable on $J^{0}$, and $|g_{l}|\le4$ by The Trigonometric System on the Torus is Orthonormal §derivative. So The Integral over the Unit Cell of a Product of One-Variable Functions §integral gives $T(k)=\prod_{l}\tau_{l}$ with $\tau_{l}=\int_{J^{0}}g_{l}\,d\lambda_{J^{0}}$ and $|\tau_{l}|\le4\lambda_{J^{0}}(J^{0})=4$, proving (a). For (b), with $C_{b},S_{b}$ as in Cell Integrals of the Trigonometric Monomials, each $\phi_{m}$ is a constant multiple of $C_{|m|}$ or $S_{|m|}$ ($\phi_{0}=1=C_{0}$). A product of factors $C_{b_{i}}$ or $S_{b_{i}}$ ($i\le r$, $b_{i}\in\mathbb{Z}$) is a linear combination of maps $C_{b}$, $S_{b}$ with $b=\sum_{i}\varepsilon_{i}b_{i}$ for signs $\varepsilon_{i}$: induct on $r$, using (with $u=2\pi bt$, $v=2\pi b't$) $C_{b}C_{b'}=\frac12(C_{b-b'}+C_{b+b'})$, $S_{b}S_{b'}=\frac12(C_{b-b'}-C_{b+b'})$, $S_{b}C_{b'}=\frac12(S_{b+b'}+S_{b-b'})$ from Product-to-Sum Formulas for Sine and Cosine §cosine-cosine, Product-to-Sum Formulas for Sine and Cosine §sine-sine, Product-to-Sum Formulas for Sine and Cosine §sine-cosine, and commutativity. By Cell Integrals of the Trigonometric Monomials §single, $\int_{J^{0}}S_{b}=0$ and $\int_{J^{0}}C_{b}=0$ unless $b=0$. So $T(k)\neq0$ forces each $\tau_{l}\ne0$, hence $\sum_{i}\varepsilon'_{i}|k_{i,l}|=0$ for some signs, and $\varepsilon_{i,l}=\varepsilon'_{i}$ times the sign of $k_{i,l}$ ($1$ if $k_{i,l}=0$) works.

Step 8 (Lattice bounds and claim 3). Put $\eta=(4\pi^{2})^{-1/8}$. If $k\in\mathbb{Z}^{n}\setminus\Gamma_{N}$, some component has $|k_{l}|\ge N+1$, so $\mu_{k}\ge4\pi^{2}k_{l}^{2}\ge4\pi^{2}N^{2}$, and claims 1 and 2 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities give

$$\mu_{k}^{-1/8}\le\eta N^{-1/4}\qquad(k\notin\Gamma_{N}).\tag{8.1}$$

The sum $\Sigma_{2}$. Since $\mu_{k}^{-2}=\mu_{k}^{-15/8}\mu_{k}^{-1/8}$, (8.1) gives $\Sigma_{2}\le\eta N^{-1/4}\sum_{k\in\Gamma_{M}}\mu_{k}^{-15/8}\le\eta N^{-1/4}S_{15/8}$, where $S_{15/8}=\sum_{k\in\mathbb{Z}^{n}}\mu_{k}^{-15/8}$ exists by Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums §powers ($n\le2<15/4$) and bounds the finite sums by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative.

The sum $\Sigma_{4}$. Let $\mathcal{E}$ be the set of sign arrays $\varepsilon=(\varepsilon_{i,l})\in\{1,-1\}^{4\times n}$ ($16^{n}$ elements), and $\sigma_{\varepsilon}$ the involution of $(\mathbb{Z}^{n})^{4}$ multiplying the component $k_{i,l}$ by $\varepsilon_{i,l}$. It preserves each $\mu_{k_{i}}$ (a function of the squared components) and each $\Gamma_{L}^{4}$ (defined by $-L\le k_{i,l}\le L$), hence $\Delta=\Gamma_{M}^{4}\setminus\Gamma_{N}^{4}$. Let $Z_{0}=\{k:\sum_{i}k_{i}=0\}$. By Step 7(b), $T(k)\ne0$ implies $\sigma_{\varepsilon}k\in Z_{0}$ for some $\varepsilon$, and $T(k)^{2}\le16^{n}$ by Step 7(a). Substituting $k=\sigma_{\varepsilon}k'$,

$$\Sigma_{4}\le16^{n}\sum_{\varepsilon\in\mathcal{E}}\ \sum_{k\in\Delta,\ \sigma_{\varepsilon}k\in Z_{0}}\prod_{i}\mu_{k_{i}}^{-1}=256^{n}\sum_{k'\in\Delta\cap Z_{0}}\prod_{i}\mu_{k'_{i}}^{-1}.$$

Each $k'\in\Delta\cap Z_{0}$ lies in $\Gamma_{M}^{4}\cap Z_{0}$ with $k'_{i}\notin\Gamma_{N}$ for some $i\in\{1,2,3,4\}$, and there $\prod\mu_{k'_{i'}}^{-1}=\mu_{k'_{i}}^{-1/8}F_{i}(k'_{1},k'_{2},k'_{3})\le\eta N^{-1/4}F_{i}(k'_{1},k'_{2},k'_{3})$ by (8.1), where $F_{i}$ is the family of Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums §quartic with $\delta=1/8$ (as $k'_{4}=-(k'_{1}+k'_{2}+k'_{3})$). The map $k'\mapsto(k'_{1},k'_{2},k'_{3})$ is a bijection of $\Gamma_{M}^{4}\cap Z_{0}$ onto a nonempty finite subset of $\mathbb{Z}^{3n}$ (it contains $0$), so by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative, with $\Lambda_{i}=\sum_{\mathbb{Z}^{3n}}F_{i}$ the lattice sum given by Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums §quartic,

$$\Sigma_{4}\le256^{n}\eta N^{-1/4}(\Lambda_{1}+\Lambda_{2}+\Lambda_{3}+\Lambda_{4}).$$

Conclusion. Put $C=\eta\bigl(6\varkappa^{2}256^{n}(\Lambda_{1}+\Lambda_{2}+\Lambda_{3}+\Lambda_{4})+\mathfrak{m}^{2}S_{15/8}\bigr)\ge0$, which depends only on $n$, $\varkappa$ and $\mathfrak{m}$ ($S_{15/8}$ and $\Lambda_{i}$ depend on $n$ only). Since $\Sigma_{4},\Sigma_{2}\ge0$, (6.1) and (6.2) give that both quantities of claim 3 are at most $6\varkappa^{2}\Sigma_{4}+\mathfrak{m}^{2}\Sigma_{2}\le C\,N^{-1/4}$, for all $N\le M$. This proves claim 3.