Proof of The Wick-Ordered $Phi^4$ Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every $L^p$, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff

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

Throughout, every $V_{N}$ is continuous, hence Borel, by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, $V_{N}$ being an admissible cylindrical potential 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 §admissible; and, by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity again, $\partial_{j}V_{N}$ is the $j$-th partial derivative of $V_{N}$ in the sense of Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial, $V_{N}$ being differentiable on $X$. The dyadic index is written $i\in\mathbb{N}$, and we use $i\le2^{i}$ (induction on $i$). Put $r=2^{-1/8}$; then $0<r<1$ and $(2^{i})^{-1/8}=r^{i}$ by 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, and $(r^{i})_{i}$ converges to $0$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric. For $N\le M$, 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 §increment and claims 1 and 5 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 $\lVert V_{M}-V_{N}\rVert_{2}\le C^{1/2}N^{-1/8}$, where $C^{1/2}$ is the square root of $C\ge0$.

We use one elementary fact (F): if a real sequence $(a_{i})$ converges to $a$, then $a\le\sup_{k}\inf_{i\ge k}a_{i}$. Indeed, for $\varepsilon>0$ there is $k$ with $a_{i}>a-\varepsilon$ for $i\ge k$, so $\inf_{i\ge k}a_{i}\ge a-\varepsilon$; as $\varepsilon$ is arbitrary the claim follows.

Step 1 (claim 1). Let $L$ be the set of $x\in X$ at which $(V_{2^{i}}(x))_{i}$ converges, equivalently (by The Real Numbers: Standing Notation and Background §sequences) is Cauchy. Thus

$$L=\bigcap_{q\in\mathbb{N}}\ \bigcup_{I\in\mathbb{N}}\ \bigcap_{i,i'\ge I}\{x:|V_{2^{i}}(x)-V_{2^{i'}}(x)|\le1/q\},$$

and each set on the right is Borel since $|V_{2^{i}}-V_{2^{i'}}|$ is Borel by claims 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; so $L\in\mathcal{B}(X)$. The functions $\mathbf{1}_{L}V_{2^{i}}$ are Borel by claims 1 and 3 of that lemma, and at every $x\in X$ they converge to $V(x)$: to $\lim_{i}V_{2^{i}}(x)$ on $L$ and to $0$ off $L$, matching The Wick-Ordered Phi^4 Potential of the Free Field on the Torus of Dimension at Most Two §potential. By claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $V$ is Borel.

Step 2 (almost-everywhere convergence). Put $D_{i}=V_{2^{i+1}}-V_{2^{i}}$. By Hoelder's Inequality, for Two and for Finitely Many Factors §holder with the conjugate exponents $2,2$ and the constant $1$ (with $\int_{X}1\,d\gamma_{c}=1$), $\int_{X}|D_{i}|\,d\gamma_{c}\le\lVert D_{i}\rVert_{2}\le C^{1/2}(2^{i})^{-1/8}=C^{1/2}r^{i}$. Let $G_{I}=\sum_{i=1}^{I}|D_{i}|$ and $G=\sup_{I}G_{I}$, with values in $[0,\infty]$. By Monotone Convergence Theorem and additivity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\int_{X}G\,d\gamma_{c}=\sup_{I}\sum_{i=1}^{I}\int_{X}|D_{i}|\,d\gamma_{c}\le C^{1/2}\sum_{i=1}^{\infty}r^{i}=\frac{C^{1/2}r}{1-r}<\infty,$$

using Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §finite the set $\Omega_{0}=\{x:G(x)<\infty\}$ is Borel with null complement. For $x\in\Omega_{0}$ the series $\sum_{i}|D_{i}(x)|$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion; since $|\sum_{i=I_{1}+1}^{I_{2}}D_{i}(x)|\le\sum_{i=I_{1}+1}^{I_{2}}|D_{i}(x)|$ for $I_{1}<I_{2}$, the Cauchy criterion Elementary Properties of Series of Real Numbers §cauchy, applied twice, shows that $\sum_{i}D_{i}(x)$ converges, and then $(V_{2^{i}}(x))_{i}$ converges by Elementary Properties of Series of Real Numbers §telescoping. Hence $\Omega_{0}\subseteq L$ and $V(x)=\lim_{i}V_{2^{i}}(x)$ for every $x\in\Omega_{0}$, which is the first assertion of claim 2.

Step 3 (mean-square bounds). Fix $N\in\mathbb{N}$ and let $u_{i}=(V_{2^{i}}-V_{N})^{2}$, Borel and nonnegative. For $x\in\Omega_{0}$, $u_{i}(x)\to(V(x)-V_{N}(x))^{2}$ by The Real Numbers: Standing Notation and Background §background, so $(V-V_{N})^{2}\le\liminf_{i}u_{i}$ on $\Omega_{0}$ by (F). For every $k$, choosing $i\ge\max\{k,N\}$ gives $N\le2^{i}$ and $\inf_{i'\ge k}\int_{X}u_{i'}\,d\gamma_{c}\le\int_{X}u_{i}\,d\gamma_{c}\le CN^{-1/4}$ by 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 §increment. Hence by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and Fatou's Lemma,

$$\int_{X}(V-V_{N})^{2}\,d\gamma_{c}\le\int_{X}\liminf_{i}u_{i}\,d\gamma_{c}\le\liminf_{i}\int_{X}u_{i}\,d\gamma_{c}\le CN^{-1/4}.$$

So $V-V_{N}\in L^{2}(\gamma_{c})$ with $\lVert V-V_{N}\rVert_{2}\le C^{1/2}N^{-1/8}$; as $V_{1}$ is square-integrable by 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 §chaos, $V=(V-V_{1})+V_{1}$ is square-integrable by Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions §minkowski. Taking $N=2^{i}$: $\lVert V-V_{2^{i}}\rVert_{2}\le C^{1/2}r^{i}\to0$.

Step 4 (chaos and moments; claim 2). $\mathcal{H}_{\le4}$ and $\mathcal{H}_{\le3}$ are closed linear subspaces of $L^{2}(\gamma_{c})$ by The Wiener Chaos Decomposition: Cylindrical Hermite Polynomials Are Dense and the Gaussian Lebesgue Space Is the Orthogonal Sum of the Chaoses §subspaces; so (by Sequential Characterization of the Closure in a Metric Space and claim 4 of The Closure is the Smallest Closed Superset) each contains the $L^{2}$-limit of every convergent sequence of its elements. For fixed $N$, the classes $V_{2^{i}}-V_{N}$ lie in $\mathcal{H}_{\le4}$ by 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 §chaos and converge in $L^{2}$ to $V-V_{N}$ by Step 3, since $\lVert(V_{2^{i}}-V_{N})-(V-V_{N})\rVert_{2}=\lVert V_{2^{i}}-V\rVert_{2}$. Hence $V-V_{N}\in\mathcal{H}_{\le4}$, and $V=(V-V_{N})+V_{N}\in\mathcal{H}_{\le4}$. For real $p\ge2$, Moment and Tail Bounds on the Wiener Chaos up to a Given Order §moments with $m=4$ gives $V\in L^{p}(\gamma_{c})$ and, as the real power $(p-1)^{4/2}$ is the natural power $(p-1)^{2}$ by claim 1 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 ($p-1\ge1>0$),

$$\lVert V-V_{N}\rVert_{p}\le(p-1)^{2}\lVert V-V_{N}\rVert_{2}\le(p-1)^{2}C^{1/2}N^{-1/8}.$$

Step 5 (claim 3). Fix $N$; for $i\ge N$ we have $N\le2^{i}$, so $V_{N}$ is a conditional expectation of $V_{2^{i}}$ given $\mathcal{F}_{N}$ by 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 §conditional. In particular $V_{N}$ satisfies conditions (i) and (ii) of Conditional Expectation of a Square-Integrable Random Variable, and $V$ is a square-integrable random variable on $(X,\mathcal{B}(X),\gamma_{c})$ by Steps 1 and 3. For $A\in\mathcal{F}_{N}$ and $i\ge N$, condition (iii) for $V_{2^{i}}$ and linearity in Linearity and Monotonicity of the Lebesgue Integral §integrable give

$$\int_{X}V\mathbf{1}_{A}\,d\gamma_{c}-\int_{X}V_{N}\mathbf{1}_{A}\,d\gamma_{c}=\int_{X}(V-V_{2^{i}})\mathbf{1}_{A}\,d\gamma_{c},$$

whose absolute value is at most $\lVert V-V_{2^{i}}\rVert_{2}\lVert\mathbf{1}_{A}\rVert_{2}\le\lVert V-V_{2^{i}}\rVert_{2}$ by Linearity and Monotonicity of the Lebesgue Integral §integrable and Hoelder's Inequality, for Two and for Finitely Many Factors §holder. The left side does not depend on $i$ and the right side tends to $0$ (Step 3), so it vanishes: condition (iii) holds for $V$ and $V_{N}$, and $V_{N}$ is a conditional expectation of $V$ given $\mathcal{F}_{N}$.

Step 6 (Cameron-Martin partial derivatives in coordinates). In this step and the next, $\delta_{k}F$ denotes the Cameron-Martin partial derivative of $F\in\mathbb{D}^{1,2}_{c}$ of Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights §partial, and $\partial_{k}$ always means a classical partial derivative. The items of that definition are read with $c$ in place of $a$, and The Gaussian Sobolev Space: Linear Structure, Closedness, Hermite Polynomials and the Number-Operator Form of the Dirichlet Form, and the Log-Sobolev and Poincare Inequalities and The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis are likewise applied with the weight sequence $c$ in place of $a$, $c$ being a weight sequence by Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights (the rates $\theta_{k}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §rates then all equal $c_{k}/c_{k}=1$). So $f_{k}=c_{k}^{1/2}e_{k}$ is an orthonormal basis of $X^{c}$ with $\langle y,f_{k}\rangle_{c}=c_{k}^{-1/2}y_{k}$ for $y\in X^{c}$, by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis, where $y_{k}=\langle y,e_{k}\rangle$ is the $k$-th coordinate of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. For a representative $g$ of $\nabla_{c}F$, therefore, $\langle g(x),e_{k}\rangle$ is the $k$-th coordinate $g(x)_{k}$ and $c_{k}^{-1}\langle g(x),e_{k}\rangle=c_{k}^{-1}g(x)_{k}=c_{k}^{-1/2}\langle g(x),f_{k}\rangle_{c}$, that is,

$$\delta_{k}F=c_{k}^{-1/2}(\nabla_{c}F)_{k},$$

with the coordinates of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates (the space $L^{2}(\gamma_{c};X^{c})$ being that of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields). Consequently: (a) $\delta_{k}$ is linear on $\mathbb{D}^{1,2}_{c}$, by The Gaussian Sobolev Space: Linear Structure, Closedness, Hermite Polynomials and the Number-Operator Form of the Dirichlet Form, and the Log-Sobolev and Poincare Inequalities §linear and the linearity of coordinates; (b) $\lVert\nabla_{c}F\rVert_{\gamma_{c}}^{2}=\sum_{k=1}^{\infty}c_{k}\lVert\delta_{k}F\rVert_{2}^{2}$ by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, so $\lVert\delta_{k}F\rVert_{2}^{2}\le c_{k}^{-1}\lVert\nabla_{c}F\rVert_{\gamma_{c}}^{2}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates; (c) for every multi-index $\beta$, $H_{\beta}\in\mathbb{D}^{1,2}_{c}$ and $(\nabla_{c}H_{\beta})_{k}$, the coordinate of $\nabla_{c}H_{\beta}$ along $f_{k}$, is $c_{k}^{1/2}\beta_{k}H_{\beta-\varepsilon_{k}}$ if $\beta_{k}\ge1$ and $0$ if $\beta_{k}=0$, by The Gaussian Sobolev Space: Linear Structure, Closedness, Hermite Polynomials and the Number-Operator Form of the Dirichlet Form, and the Log-Sobolev and Poincare Inequalities §hermite; so, by the display and Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §derivative, $\delta_{k}H_{\beta}=c_{k}^{-1/2}c_{k}^{1/2}\beta_{k}H_{\beta-\varepsilon_{k}}=\beta_{k}H_{\beta-\varepsilon_{k}}=\partial_{k}H_{\beta}$ if $\beta_{k}\ge1$, and $\delta_{k}H_{\beta}=0=\partial_{k}H_{\beta}$ if $\beta_{k}=0$, as classes; that is, $\delta_{k}H_{\beta}$ is the class of $\partial_{k}H_{\beta}$.

Step 7 (claim 4). Hermite form of $V_{N}$. Fix $N$. 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, $\Phi_{N}[x](z)=\sum_{j=1}^{d_{N}}x_{j}b_{j}(z)$ with $b_{j}=\chi_{N}(j)a_{j}^{-1/2}\psi_{\kappa(j)}\in C_{\mathrm{per}}$, and $f_{N}(s)=q_{0}+\sum_{h=1}^{4}q_{h}s^{h}$ with real $q_{0},\dots,q_{4}$, by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §polynomial. Expanding the powers of the finite sum,

$$f_{N}(\Phi_{N}[x](z))=q_{0}+\sum_{h=1}^{4}q_{h}\sum_{\iota\in[d_{N}]^{h}}x_{\iota_{1}}\cdots x_{\iota_{h}}\,b_{\iota_{1}}(z)\cdots b_{\iota_{h}}(z).$$

For $1\le h\le4$ each product $b_{\iota_{1}}\cdots b_{\iota_{h}}$ 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, with integral $I_{\iota}$; the constant term $q_{0}$ has integral $q_{0}\lambda_{Q}(Q)=q_{0}$ over $Q$ (The Flat Torus: Standing Notation §measure). By linearity of the integral, $V_{N}(x)=q_{0}+\sum_{h=1}^{4}q_{h}\sum_{\iota\in[d_{N}]^{h}}I_{\iota}\,x_{\iota_{1}}\cdots x_{\iota_{h}}$. The constant function $1$ is the monomial $\prod_{k=1}^{d_{N}}x_{k}^{0}$ of order $0$, so by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §monomials it is a finite linear combination of functions $H_{\beta}$ with $|\beta|=0$. For $1\le h\le4$, each $x_{\iota_{1}}\cdots x_{\iota_{h}}$ is the monomial $\prod_{k=1}^{d_{N}}x_{k}^{\alpha_{k}}$ with $\alpha_{k}$ the number of $l$ with $\iota_{l}=k$, of order $h\le4$, so by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §monomials it is a finite linear combination of functions $H_{\beta}$ with $|\beta|\le4$. Hence $V_{N}=\sum_{\beta\in\mathcal{B}_{N}}t_{\beta}H_{\beta}$ on $X$ for a finite set $\mathcal{B}_{N}$ of multi-indices and reals $t_{\beta}$.

$V_{N}\in\mathbb{D}^{1,2}_{c}$. By Step 6(c) and The Gaussian Sobolev Space: Linear Structure, Closedness, Hermite Polynomials and the Number-Operator Form of the Dirichlet Form, and the Log-Sobolev and Poincare Inequalities §linear, the class of $V_{N}$ lies in $\mathbb{D}^{1,2}_{c}$, and by Step 6(a),(c), $\delta_{k}V_{N}$ is the class of $\sum_{\beta}t_{\beta}\partial_{k}H_{\beta}$. Each $H_{\beta}$ is differentiable on $X$ by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §derivative; a finite linear combination of functions differentiable on $X$ is differentiable on $X$, its gradient being the same linear combination of their gradients, directly from clause 1 of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (with $E=U=X$) and the uniqueness in its clause 2; and by Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial the $k$-th partial derivative is the inner product of the gradient with $e_{k}$, which is linear in the gradient. So $\sum_{\beta}t_{\beta}\partial_{k}H_{\beta}$ is the $k$-th partial derivative of $V_{N}=\sum_{\beta}t_{\beta}H_{\beta}$, which is $\partial_{k}V_{N}$ by the opening paragraph. Thus $\delta_{k}V_{N}$ is the class of $\partial_{k}V_{N}$ for every $k\in\mathbb{N}$.

Gradient increments. Let $N\le M$. Since $\kappa(d_{N})\in\Gamma_{N}\subseteq\Gamma_{M}$ by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §nested, $d_{N}\le d_{M}$, so $\partial_{k}V_{M}=\partial_{k}V_{N}=0$ for $k>d_{M}$. By Step 6(a),(b) and 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 §increment,

$$\lVert\nabla_{c}V_{M}-\nabla_{c}V_{N}\rVert_{\gamma_{c}}^{2}=\sum_{k=1}^{d_{M}}c_{k}\int_{X}(\partial_{k}V_{M}-\partial_{k}V_{N})^{2}\,d\gamma_{c}\le CN^{-1/4}.$$

Hence $(\nabla_{c}V_{2^{i}})_{i}$ is Cauchy, with $\lVert\nabla_{c}V_{2^{i'}}-\nabla_{c}V_{2^{i}}\rVert_{\gamma_{c}}\le C^{1/2}r^{i}$ for $i\le i'$, and converges to some $\mathcal{G}$ in the real Hilbert space $L^{2}(\gamma_{c};X^{c})$ (The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert). As $\lVert V_{2^{i}}-V\rVert_{2}\to0$ (Step 3), The Gaussian Sobolev Space: Linear Structure, Closedness, Hermite Polynomials and the Number-Operator Form of the Dirichlet Form, and the Log-Sobolev and Poincare Inequalities §closed gives $V\in\mathbb{D}^{1,2}_{c}$ and $\nabla_{c}V=\mathcal{G}$. For each $N$ and every $i\ge N$, $\lVert\nabla_{c}V-\nabla_{c}V_{N}\rVert_{\gamma_{c}}\le\lVert\mathcal{G}-\nabla_{c}V_{2^{i}}\rVert_{\gamma_{c}}+C^{1/2}N^{-1/8}$; letting $i\to\infty$, $\lVert\nabla_{c}V-\nabla_{c}V_{N}\rVert_{\gamma_{c}}\le C^{1/2}N^{-1/8}$.

Convergence of the derivatives. Fix $j\in\mathbb{N}$. By Step 6(a),(b), $\lVert\delta_{j}V-\partial_{j}V_{N}\rVert_{2}\le c_{j}^{-1/2}C^{1/2}N^{-1/8}$; here $N^{-1/8}\to0$ by claim 3(a) 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, so $c_{j}^{-1/2}C^{1/2}N^{-1/8}\to0$ by Arithmetic of Limits of Real Sequences §scalar, and $\lVert\delta_{j}V-\partial_{j}V_{N}\rVert_{2}\to0$ by claim 3(c) 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 with $L=0$. Each $\partial_{j}V_{N}$ has class in $\mathcal{H}_{\le3}$ (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 §chaos), which is closed (Step 4), so $\delta_{j}V\in\mathcal{H}_{\le3}$ and $\delta_{j}V-\partial_{j}V_{N}\in\mathcal{H}_{\le3}$. For real $p\ge2$, Moment and Tail Bounds on the Wiener Chaos up to a Given Order §moments with $m=3$ gives $\delta_{j}V\in L^{p}(\gamma_{c})$ and $\lVert\delta_{j}V-\partial_{j}V_{N}\rVert_{p}\le(p-1)^{3/2}\lVert\delta_{j}V-\partial_{j}V_{N}\rVert_{2}$, which tends to $0$ by Arithmetic of Limits of Real Sequences §scalar; so $\lVert\delta_{j}V-\partial_{j}V_{N}\rVert_{p}\to0$ by claim 3(c) 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.

Step 8 (claims 5 and 6). By The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §polynomial, with $A_{N}=(3\varkappa C_{N}-\mathfrak{m})/2$,

$$f_{N}(s)=\frac{\varkappa}{4}s^{4}-A_{N}s^{2}+\frac{3\varkappa C_{N}^{2}}{4}-\frac{\mathfrak{m}C_{N}}{2}=\frac{\varkappa}{4}\Bigl(s^{2}-\frac{2A_{N}}{\varkappa}\Bigr)^{2}-\frac{A_{N}^{2}}{\varkappa}+\frac{3\varkappa C_{N}^{2}}{4}-\frac{\mathfrak{m}C_{N}}{2}.$$

Since $0<C_{N}$ 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, $|A_{N}|\le(3\varkappa C_{N}+|\mathfrak{m}|)/2$, so $A_{N}^{2}/\varkappa\le(3\varkappa C_{N}+|\mathfrak{m}|)^{2}/(4\varkappa)$; also $3\varkappa C_{N}^{2}/4\ge0$ and $-\mathfrak{m}C_{N}/2\ge-|\mathfrak{m}|C_{N}/2$. Hence $f_{N}(s)\ge-B_{N}$ for every real $s$. Integrating over $Q$ by the monotonicity in Linearity and Monotonicity of the Lebesgue Integral §integrable, with $\lambda_{Q}(Q)=1$ (The Flat Torus: Standing Notation §measure), gives $V_{N}(x)\ge-B_{N}$ for every $x$. This is claim 5.

For claim 6, let $n=1$. The family $k\mapsto\mu_{k}^{-1}$ is cube-summable 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 with $s=1$ (as $1<2$), the real power $\mu_{k}^{-1}$ there being the multiplicative inverse of $\mu_{k}$ by claim 1 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 ($t^{-a}=1/t^{a}$ and $t^{1}=t$); so $C_{\infty}$ is defined. 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 and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative with $E=\Gamma_{N}$, $C_{N}=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}\le C_{\infty}$. As $0<3\varkappa C_{N}+|\mathfrak{m}|\le3\varkappa C_{\infty}+|\mathfrak{m}|$ and $|\mathfrak{m}|C_{N}\le|\mathfrak{m}|C_{\infty}$, we get $B_{N}\le B_{\infty}$, and $B_{\infty}\ge0$. If $x\in L$, then $V_{2^{i}}(x)\ge-B_{2^{i}}\ge-B_{\infty}$ for all $i$, so $V(x)=\lim_{i}V_{2^{i}}(x)\ge-B_{\infty}$ by The Real Numbers: Standing Notation and Background §background; if $x\notin L$, then $V(x)=0\ge-B_{\infty}$.