Proof of 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

Each result cited is universally quantified over the data in its own statement. Real order and arithmetic, including squares, absolute values, maxima and minima, finite sums of real numbers and sums over finite index sets, are used through the results placed in force by The Real Numbers: Standing Notation and Background §background, which is in force here by The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian and A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background; such a result is named, with its claim, wherever a step rests on more than routine arithmetic.

Step 0 (Notation and elementary facts). We write $d=d_{N}$, $\chi=\chi_{N}$, $J=J_{N}$, $C=C_{N}$, $f=f_{N}$, $v=v_{N}$, $\varphi[u]=\varphi_{N}[u]$ ($u\in\mathbb{R}^{d}$) and $\Phi[x]=\Phi_{N}[x]$ ($x\in X$). For $j\in\mathbb{N}$ the weight $a_{j}$ is positive, $a$ being 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 as recorded in The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis, and the variance $c_{j}$ is positive, as recorded in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads. The number $a_{j}^{1/2}$ is the positive square root of $a_{j}$ and $a_{j}^{-1/2}$ its multiplicative inverse, as fixed in Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space (cited by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads); the nonnegative square root written $a_{j}^{1/2}$ in 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 is the same number, by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. Hence

$$a_{j}^{-1/2}a_{j}^{1/2}=1,\qquad\bigl(a_{j}^{-1/2}\bigr)^{2}=a_{j}^{-1},\qquad a_{j}\,a_{j}^{-1/2}=a_{j}^{1/2},\qquad a_{j}\bigl(a_{j}^{-1/2}\bigr)^{2}=1,$$

the second because $(a_{j}^{-1/2})^{2}(a_{j}^{1/2})^{2}=1$ and $(a_{j}^{1/2})^{2}=a_{j}$, the third because $a_{j}=a_{j}^{1/2}a_{j}^{1/2}$. For $k\in\mathbb{Z}^{n}$, $1\le\mu_{k}$ by Summability of the Negative Powers of the Fourier Weights of the Torus §product, so $\mu_{k}$ and $\mu_{k}^{-1}$ are positive. For $j\in[d]$, $\chi(j)\in\{0,1\}$, so $\chi(j)^{2}=\chi(j)$ and $0\le\chi(j)\le1$, and $\chi(j)=1$ exactly when $j\in J$, by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension. For $k\in\mathbb{Z}^{n}$, $\langle E_{k},E_{k}\rangle_{L^{2}}=1$ by The Trigonometric System on the Torus is Orthonormal §orthonormal, so $\lVert E_{k}\rVert_{L^{2}}=1$, the norm of $L^{2}(\mathbb{T}^{n})$ being by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product the nonnegative square root of $\langle E_{k},E_{k}\rangle_{L^{2}}$.

For $j\in[d]$ let $w_{j}:\mathbb{R}^{n}\to\mathbb{R}$ be the map $w_{j}(z)=\chi(j)\,a_{j}^{-1/2}\,\psi_{\kappa(j)}(z)$. It belongs to $C_{\mathrm{per}}$, by The Trigonometric System on the Torus is Orthonormal §classes and Elementary Properties of Lattice-Periodic Functions §algebra, and it vanishes identically when $j\notin J$. By The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §field, for $u\in\mathbb{R}^{d}$ and $z\in\mathbb{R}^{n}$,

$$\varphi[u](z)=\sum_{j=1}^{d}u_{j}\,w_{j}(z),$$

so by claims 2 and 3 of Properties of Finite Sums, $\varphi[u+u'](z)=\varphi[u](z)+\varphi[u'](z)$ and $\varphi[tu](z)=t\,\varphi[u](z)$ for all $u,u'\in\mathbb{R}^{d}$, $t\in\mathbb{R}$ and $z\in\mathbb{R}^{n}$, where $u+u'$ and $tu$ are the sum and scalar multiple of $\mathbb{R}^{d}$.

Let $0_{n}\in\mathbb{Z}^{n}$ be the point all of whose components are $0$. By The Trigonometric System on the Torus §system and The Trigonometric System on the Torus §one-dimensional, $\psi_{0_{n}}(z)=\prod_{i=1}^{n}\phi_{0}(z_{i})$ is a finite product all of whose factors equal $1$, so $\psi_{0_{n}}(z)=1$ for every $z\in\mathbb{R}^{n}$ by claim 3 of Properties of Finite Products (used with $i=1$). Thus the constant map $\mathbb{R}^{n}\to\mathbb{R}$ with value $t\in\mathbb{R}$ is $t\,\psi_{0_{n}}$, and it belongs to $C_{\mathrm{per}}$ by The Trigonometric System on the Torus is Orthonormal §classes and Elementary Properties of Lattice-Periodic Functions §algebra.

Put

$$\sigma=\mathfrak{m}-3\varkappa C,\qquad\gamma_{0}=\frac{3\varkappa}{4}C^{2}-\frac{\mathfrak{m}}{2}C,\qquad B=\frac{K_{N}^{2}}{2\varkappa}+|\gamma_{0}| ,$$

so that $K_{N}=\max\{-\sigma,0\}$; hence $0\le K_{N}$ and $-\sigma\le K_{N}$ by claim 1 of Elementary Properties of the Maximum of Two Elements, and $0\le B$. Expanding The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §polynomial,

$$f(s)=\frac{\varkappa}{4}s^{4}+\frac{\sigma}{2}s^{2}+\gamma_{0}\qquad(s\in\mathbb{R}).$$

Let $f',f'':\mathbb{R}\to\mathbb{R}$ be the maps $f'(s)=\varkappa s^{3}+\sigma s$ and $f''(s)=3\varkappa s^{2}+\sigma$; thus $f'(s)=\varkappa(s^{3}-3Cs)+\mathfrak{m}s$ and $f''(s)=3\varkappa s^{2}+\mathfrak{m}-3\varkappa C$. These are names only; that they are derivatives is shown in Step 8.

Step 1 (Sums over the active indices). The set $J$ is nonempty, and for every map $g:\Gamma_{N}\to\mathbb{R}$

$$\sum_{j=1}^{d}\chi(j)\,g(\kappa(j))=\sum_{k\in\Gamma_{N}}g(k).$$

Indeed, $0_{n}\in\Gamma_{N}$ by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite. Since $\kappa$ is a bijection, for every $k\in\Gamma_{N}$ there is $j\in\mathbb{N}$ with $\kappa(j)=k$; then $j\le d$, because $d$ is the greatest element of the set of those $j'\in\mathbb{N}$ with $\kappa(j')\in\Gamma_{N}$ by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension, so $j\in J$. Taking $k=0_{n}$ shows that $J$ is nonempty, and the same argument shows that the image under $\kappa$ of $J$ is $\Gamma_{N}$ (the inclusion of the image in $\Gamma_{N}$ being the definition of $J$); so the restriction of $\kappa$ to $J$ is a bijection from $J$ onto $\Gamma_{N}$ by claim 3 of Injectivity, Composition, and Restriction of Bijections. The map $j\mapsto\chi(j)g(\kappa(j))$ on $[d]$ vanishes outside $J$ and equals $g(\kappa(j))$ at $j\in J$; hence

$$\sum_{j=1}^{d}\chi(j)g(\kappa(j))=\sum_{j\in[d]}\chi(j)g(\kappa(j))=\sum_{j\in J}g(\kappa(j))=\sum_{k\in\Gamma_{N}}g(k),$$

by claim 1 of Properties of a Sum over a Finite Index Set, claim 4 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, and claim 2 of Properties of a Sum over a Finite Index Set (reindexing along the bijection $J\to\Gamma_{N}$). The argument uses nothing particular to the cutoff $N$, so it holds at every cutoff $N'\in\mathbb{N}$, with the head dimension, indicator and active set of $N'$.

Step 2 (The Wick constant as a cube sum). Step 1 with $g(k)=\mu_{k}^{-1}$ and The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §wick-constant give

$$C=\sum_{j=1}^{d}\chi(j)\mu_{\kappa(j)}^{-1}=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1},$$

and likewise $C_{N'}=\sum_{k\in\Gamma_{N'}}\mu_{k}^{-1}$ for every $N'\in\mathbb{N}$. 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 $c_{j}/a_{j}=c_{j}\,\mu_{\kappa(j)}^{-1}c_{j}^{-1}=\mu_{\kappa(j)}^{-1}$, $c_{j}$ being positive; hence $C=\sum_{j=1}^{d}\chi(j)\,c_{j}/a_{j}=\sum_{j\in J}c_{j}/a_{j}$ in the notation of the statement. Finally the summands $\chi(j)\mu_{\kappa(j)}^{-1}$ are nonnegative, and for the index $j_{0}\in J$ with $\kappa(j_{0})=0_{n}$ (Step 1) the summand is $\mu_{0_{n}}^{-1}$; so $0<\mu_{0_{n}}^{-1}\le C$ by claim 6 of Properties of Finite Sums.

Step 3 (The sign-orbit identity). We show that for every $z\in\mathbb{R}^{n}$

$$\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}\psi_{k}(z)^{2}=C,\qquad\text{hence}\qquad\sum_{j=1}^{d}\chi(j)\,\mu_{\kappa(j)}^{-1}\psi_{\kappa(j)}(z)^{2}=C,$$

the second by Step 1 with $g(k)=\mu_{k}^{-1}\psi_{k}(z)^{2}$; it is the displayed identity of claim 1.

(a) For $m\in\mathbb{Z}$ and $t\in\mathbb{R}$, $\phi_{m}(t)^{2}+\phi_{-m}(t)^{2}=2$, with $\phi_{m}$ the maps of The Trigonometric System on the Torus §one-dimensional, where exactly one of $m=0$, $0<m$, $m<0$ holds. If $m=0$ then $-m=0$ and the left side is $1+1=2$. If $0<m$ then $-m<0$ and $-(-m)=m$, so $\phi_{m}(t)=\sqrt{2}\,C_{m}(t)$ and $\phi_{-m}(t)=\sqrt{2}\,S_{m}(t)$, and the left side is $(\sqrt{2})^{2}\bigl(\cos(2\pi mt)^{2}+\sin(2\pi mt)^{2}\bigr)=2$ by The Pythagorean Identity for Sine and Cosine §identity, since $(\sqrt{2})^{2}=2$ by the choice of $\sqrt{2}$ in The Trigonometric System on the Torus. If $m<0$ then $0<-m$, $\phi_{m}(t)=\sqrt{2}\,S_{-m}(t)$ and $\phi_{-m}(t)=\sqrt{2}\,C_{-m}(t)$, and the same computation applies.

(b) Sign flips. Let $[2]^{n}$ be the set of maps $\epsilon:[n]\to[2]$, nonempty and finite as recorded in Generalized Distributivity: Expanding a Product of Finite Sums. For $\epsilon\in[2]^{n}$ and $k\in\mathbb{Z}^{n}$ let $\epsilon\cdot k\in\mathbb{Z}^{n}$ be the point with components $(\epsilon\cdot k)_{i}=k_{i}$ if $\epsilon(i)=1$ and $(\epsilon\cdot k)_{i}=-k_{i}$ if $\epsilon(i)=2$, for $i\in[n]$; here $-k_{i}\in\mathbb{Z}$ by Arithmetic, Order, Discreteness and Intervals of the Integers. Since $-(-k_{i})=k_{i}$, $\epsilon\cdot(\epsilon\cdot k)=k$; and since $-N\le k_{i}\le N$ holds exactly when $-N\le-k_{i}\le N$, $\epsilon\cdot k\in\Gamma_{N}$ exactly when $k\in\Gamma_{N}$. Hence $k\mapsto\epsilon\cdot k$ maps $\Gamma_{N}$ into itself, is injective (if $\epsilon\cdot k=\epsilon\cdot k'$ then $k=\epsilon\cdot(\epsilon\cdot k)=\epsilon\cdot(\epsilon\cdot k')=k'$) and maps $\Gamma_{N}$ onto itself ($k=\epsilon\cdot(\epsilon\cdot k)$), so it is a bijection $\Gamma_{N}\to\Gamma_{N}$. Moreover $\lVert\epsilon\cdot k\rVert^{2}=\sum_{i=1}^{n}(\epsilon\cdot k)_{i}^{2}=\sum_{i=1}^{n}k_{i}^{2}=\lVert k\rVert^{2}$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, since $(-k_{i})^{2}=k_{i}^{2}$; so $\mu_{\epsilon\cdot k}=\mu_{k}$ by the formula $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$ of Summability of the Negative Powers of the Fourier Weights of the Torus. Finally $\epsilon\cdot0_{n}=0_{n}$, since $-0=0$.

(c) Let $P=\sum_{\epsilon\in[2]^{n}}1$. We show that $\sum_{\epsilon\in[2]^{n}}\psi_{\epsilon\cdot k}(z)^{2}=P$ for all $k\in\mathbb{Z}^{n}$ and $z\in\mathbb{R}^{n}$, and that $1\le P$. Fix $k$ and $z$, and let $c$ be the $n$-tuple of $2$-tuples with components $c_{i1}=\phi_{k_{i}}(z_{i})^{2}$ and $c_{i2}=\phi_{-k_{i}}(z_{i})^{2}$ ($i\in[n]$). For $\epsilon\in[2]^{n}$, by The Trigonometric System on the Torus §system and claim 2 of Properties of Finite Products (used with $b=a$),

$$\psi_{\epsilon\cdot k}(z)^{2}=\Bigl(\prod_{i=1}^{n}\phi_{(\epsilon\cdot k)_{i}}(z_{i})\Bigr)^{2}=\prod_{i=1}^{n}\phi_{(\epsilon\cdot k)_{i}}(z_{i})^{2}=\prod_{i=1}^{n}c_{i\,\epsilon(i)} .$$

Hence, by Generalized Distributivity: Expanding a Product of Finite Sums (with $m=2$ there), claim 1 of Properties of Finite Sums and part (a),

$$\sum_{\epsilon\in[2]^{n}}\psi_{\epsilon\cdot k}(z)^{2}=\prod_{i=1}^{n}\bigl(c_{i1}+c_{i2}\bigr)=\prod_{i=1}^{n}2 .$$

For $k=0_{n}$ the left side is $\sum_{\epsilon\in[2]^{n}}\psi_{0_{n}}(z)^{2}=\sum_{\epsilon\in[2]^{n}}1=P$, by part (b) and Step 0; hence $\prod_{i=1}^{n}2=P$, and the assertion follows for every $k$. Every summand of $P$ is $1$, which is nonnegative, so $1\le P$ by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §term.

(d) Fix $z$ and let $S=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}\psi_{k}(z)^{2}$. For each $\epsilon\in[2]^{n}$, reindexing along the bijection of part (b) by claim 2 of Properties of a Sum over a Finite Index Set, and using $\mu_{\epsilon\cdot k}=\mu_{k}$,

$$S=\sum_{k\in\Gamma_{N}}\mu_{\epsilon\cdot k}^{-1}\psi_{\epsilon\cdot k}(z)^{2}=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}\psi_{\epsilon\cdot k}(z)^{2}.$$

Therefore, by claim 4 of Properties of a Sum over a Finite Index Set (homogeneity), claim 5 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set (interchange, for the map $(\epsilon,k)\mapsto\mu_{k}^{-1}\psi_{\epsilon\cdot k}(z)^{2}$ on $[2]^{n}\times\Gamma_{N}$), part (c) and Step 2,

$$P\,S=\sum_{\epsilon\in[2]^{n}}S=\sum_{\epsilon\in[2]^{n}}\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}\psi_{\epsilon\cdot k}(z)^{2}=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}\sum_{\epsilon\in[2]^{n}}\psi_{\epsilon\cdot k}(z)^{2}=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}P=P\,C .$$

Since $P\ne0$, $S=C$.

Step 4 (Coordinates; first assertion of claim 2). Let $x\in X$ and $j\in\mathbb{N}$. By Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, in force by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background, $x_{j}=\langle x,e_{j}\rangle_{H^{-m}}$; this is $\rho_{\kappa(j)}^{m}\,x(\kappa(j))$ 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 §basis, and $\rho_{\kappa(j)}^{m}=a_{j}^{1/2}$ 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 §basis-vectors. Multiplying $x_{j}=a_{j}^{1/2}x(\kappa(j))$ by $a_{j}^{-1/2}$ and using Step 0 gives $x(\kappa(j))=a_{j}^{-1/2}x_{j}$. Substituting this termwise into The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §field, and recalling $p_{d}(x)=(x_{1},\dots,x_{d})$,

$$\Phi[x](z)=\sum_{j=1}^{d}\chi(j)\,a_{j}^{-1/2}x_{j}\,\psi_{\kappa(j)}(z)=\varphi[p_{d}(x)](z)\qquad(z\in\mathbb{R}^{n}),$$

so $\Phi[x]=\varphi[p_{d}(x)]$, and then $V_{N}(x)=v(p_{d}(x))$ by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §potential; that is, $V_{N}=v\circ p_{d}$.

Step 5 (Gaussian moments of the field; rest of claim 1). Fix $z\in\mathbb{R}^{n}$ and put $b_{j}=w_{j}(z)$ for $j\in[d]$, so that $\Phi[x](z)=\sum_{j=1}^{d}b_{j}x_{j}$ for every $x\in X$ by Step 4 and Step 0. Integrals with respect to $\gamma_{c}$ are those of the measure space $(X,\mathcal{B}(X),\gamma_{c})$, as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures. By Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §coordinates, for $j,k\in\mathbb{N}$ the functions $x\mapsto x_{j}$ and $x\mapsto x_{j}x_{k}$ are integrable, $\int_{X}x_{j}\,\gamma_{c}(dx)=0$, and $\int_{X}x_{j}x_{k}\,\gamma_{c}(dx)$ equals $c_{j}$ if $j=k$ and $0$ if $j\ne k$. By Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, $x\mapsto\Phi[x](z)$ is integrable with integral $\sum_{j=1}^{d}b_{j}\cdot0=0$. Next, by claim 3 of Properties of Finite Sums used twice,

$$\Phi[x](z)^{2}=\sum_{j=1}^{d}b_{j}x_{j}\Bigl(\sum_{k=1}^{d}b_{k}x_{k}\Bigr)=\sum_{j=1}^{d}g_{j}(x),\qquad g_{j}(x)=\sum_{k=1}^{d}b_{j}b_{k}\,x_{j}x_{k}.$$

For each $j\in[d]$, Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear show that $g_{j}$ is integrable with $\int_{X}g_{j}\,d\gamma_{c}=\sum_{k=1}^{d}b_{j}b_{k}\int_{X}x_{j}x_{k}\,\gamma_{c}(dx)=b_{j}^{2}c_{j}$, the last equality by claim 7 of Properties of Finite Sums (only the summand $k=j$ can be nonzero). Applying the same two clauses once more, with all coefficients $1$, $x\mapsto\Phi[x](z)^{2}$ is integrable; and since $b_{j}^{2}=\chi(j)a_{j}^{-1}\psi_{\kappa(j)}(z)^{2}$ by Step 0 and $a_{j}^{-1}c_{j}=\mu_{\kappa(j)}^{-1}$ by Step 2,

$$\int_{X}\Phi[x](z)^{2}\,\gamma_{c}(dx)=\sum_{j=1}^{d}b_{j}^{2}c_{j}=\sum_{j=1}^{d}\chi(j)\,\mu_{\kappa(j)}^{-1}\psi_{\kappa(j)}(z)^{2}=C$$

by Step 3. Together with Steps 2 and 3 this proves claim 1.

Step 6 (Active coordinates and the noise norm; rest of claim 2). If $u,u'\in\mathbb{R}^{d}$ satisfy $u_{j}=u'_{j}$ for every $j\in J$, then $u_{j}w_{j}(z)=u'_{j}w_{j}(z)$ for all $j\in[d]$ and $z\in\mathbb{R}^{n}$ (for $j\notin J$ both sides are $0$), so $\varphi[u]=\varphi[u']$ by Step 0, and $v(u)=v(u')$ by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §potential.

Let $h\in\mathbb{R}^{d}$ and put $\eta_{j}=\chi(j)a_{j}^{-1/2}h_{j}$ for $j\in[d]$, so that $\varphi[h]=\sum_{j=1}^{d}\eta_{j}\psi_{\kappa(j)}$ pointwise. By Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus §member and Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus §class (with $M=d$, $c_{j}=\eta_{j}$ and $g_{j}=\psi_{\kappa(j)}$) and The Trigonometric System on the Torus is Orthonormal §classes, $\varphi[h]|_{Q}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ and

$$U=[\,\varphi[h]|_{Q}\,]=\sum_{j=1}^{d}\eta_{j}E_{\kappa(j)} .$$

The map $U'\mapsto\hat{U}'$ from $L^{2}(\mathbb{T}^{n})$ to $X=H^{-m}(\mathbb{T}^{n})$ is linear by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §embedding; so by claim 4 of Properties of Finite Sums of Vectors, 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 and Step 0,

$$\hat{U}=\sum_{j=1}^{d}\eta_{j}\hat{E}_{\kappa(j)}=\sum_{j=1}^{d}\chi(j)a_{j}^{-1/2}h_{j}\,a_{j}^{1/2}e_{j}=\sum_{j=1}^{d}\chi(j)h_{j}e_{j}=\sum_{j\in J}h_{j}e_{j},$$

the last sum read with the convention of The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension. The $d$-tuple $(E_{\kappa(1)},\dots,E_{\kappa(d)})$ is orthonormal in $L^{2}(\mathbb{T}^{n})$: for $i\ne j$ in $[d]$, $\kappa(i)\ne\kappa(j)$ by claim 1 of Injectivity, Composition, and Restriction of Bijections, so $\langle E_{\kappa(i)},E_{\kappa(j)}\rangle_{L^{2}}=0$ by The Trigonometric System on the Torus is Orthonormal §orthonormal, and $\lVert E_{\kappa(i)}\rVert_{L^{2}}=1$ by Step 0. Hence, by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §norm and Step 0,

$$\int_{Q}\varphi[h](z)^{2}\,\lambda_{Q}(dz)=\langle U,U\rangle_{L^{2}}=\lVert U\rVert_{L^{2}}^{2}=\sum_{j=1}^{d}\eta_{j}^{2}=\sum_{j=1}^{d}\chi(j)\frac{h_{j}^{2}}{a_{j}}=\sum_{j\in J}\frac{h_{j}^{2}}{a_{j}} .$$

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 §noise-space, $\hat{U}\in X^{a}$ and $|\hat{U}|_{a}^{2}=\langle\hat{U},\hat{U}\rangle_{a}=\langle U,U\rangle_{L^{2}}=\lVert U\rVert_{L^{2}}^{2}$, the first equality because $|\hat{U}|_{a}$ is the nonnegative square root of $\langle\hat{U},\hat{U}\rangle_{a}$ by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product. Finally, for each $j\in[d]$, $\chi(j)h_{j}^{2}/a_{j}\le h_{j}^{2}/a_{j}$, since $0\le h_{j}^{2}/a_{j}$ and $\chi(j)\le1$; so $\sum_{j\in J}h_{j}^{2}/a_{j}\le\sum_{k=1}^{d}h_{k}^{2}/a_{k}$ by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison and claim 1 of Properties of a Sum over a Finite Index Set. This proves claim 2. For later use write $r(h)=\sum_{j\in J}h_{j}^{2}/a_{j}$ for $h\in\mathbb{R}^{d}$; thus $0\le r(h)=\int_{Q}\varphi[h](z)^{2}\,\lambda_{Q}(dz)$, the inequality by claim 5 of Properties of Finite Sums.

Step 7 (Pointwise bounds on the field). Let $\lVert\cdot\rVert$ be the Euclidean norm of $\mathbb{R}^{d}$ and put

$$D=C\sum_{j=1}^{d}\chi(j)\,\mu_{\kappa(j)},\qquad\alpha=\sum_{k=1}^{d}a_{k}^{-1},\qquad G=1+D\alpha .$$

$D$ and $\alpha$ are nonnegative by claim 5 of Properties of Finite Sums, so $1\le G$. We show that for all $h\in\mathbb{R}^{d}$, $z\in\mathbb{R}^{n}$ and $j\in[d]$

$$0\le\sum_{k=1}^{d}\chi(k)\psi_{\kappa(k)}(z)^{2}\le D,\qquad\varphi[h](z)^{2}\le D\,r(h),\qquad r(h)\le\alpha\lVert h\rVert^{2},\qquad|\varphi[h](z)|\le G\lVert h\rVert,\qquad|w_{j}(z)|\le G .$$

For $k\in[d]$, the summands of the second identity of Step 3 are nonnegative, so $\chi(k)\mu_{\kappa(k)}^{-1}\psi_{\kappa(k)}(z)^{2}\le C$ by claim 6 of Properties of Finite Sums; multiplying by $\chi(k)\mu_{\kappa(k)}\ge0$ and using $\chi(k)^{2}=\chi(k)$ gives $\chi(k)\psi_{\kappa(k)}(z)^{2}\le C\chi(k)\mu_{\kappa(k)}$. Summing over $k$ (claim 1 of Properties of a Sum over a Finite Index Set, Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison and claim 3 of Properties of Finite Sums) gives the first chain, the lower bound by claim 5 of Properties of Finite Sums.

Next let $\xi,\vartheta\in\mathbb{R}^{d}$ have components $\xi_{k}=\chi(k)a_{k}^{-1/2}h_{k}$ and $\vartheta_{k}=\chi(k)\psi_{\kappa(k)}(z)$. Their dot product is $\xi\cdot\vartheta=\sum_{k=1}^{d}\chi(k)^{2}a_{k}^{-1/2}h_{k}\psi_{\kappa(k)}(z)=\varphi[h](z)$, and by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and Step 0, $\lVert\xi\rVert^{2}=\sum_{k=1}^{d}\chi(k)h_{k}^{2}/a_{k}=r(h)$ and $\lVert\vartheta\rVert^{2}=\sum_{k=1}^{d}\chi(k)\psi_{\kappa(k)}(z)^{2}\le D$. By Cauchy-Schwarz Inequality for the Euclidean Dot Product, $|\xi\cdot\vartheta|\le\lVert\xi\rVert\lVert\vartheta\rVert$, both sides being nonnegative, so squaring by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\varphi[h](z)^{2}\le r(h)\lVert\vartheta\rVert^{2}\le D\,r(h)$.

Next, $0<a_{k}^{-1}\le\alpha$ for each $k\in[d]$ by claim 6 of Properties of Finite Sums, so $\chi(k)h_{k}^{2}/a_{k}\le\alpha h_{k}^{2}$, and summing as above, $r(h)\le\alpha\sum_{k=1}^{d}h_{k}^{2}=\alpha\lVert h\rVert^{2}$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square. Hence $\varphi[h](z)^{2}\le D\alpha\lVert h\rVert^{2}\le G^{2}\lVert h\rVert^{2}$, because $D\alpha\le G\le G^{2}$ (as $1\le G$); since $|\varphi[h](z)|^{2}=\varphi[h](z)^{2}$ and both $|\varphi[h](z)|$ and $G\lVert h\rVert$ are nonnegative (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square), claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|\varphi[h](z)|\le G\lVert h\rVert$. Finally, by Step 0, $w_{j}(z)^{2}=a_{j}^{-1}\chi(j)\psi_{\kappa(j)}(z)^{2}\le\alpha D\le G^{2}$, the middle inequality because $a_{j}^{-1}\le\alpha$ and $\chi(j)\psi_{\kappa(j)}(z)^{2}\le\sum_{k=1}^{d}\chi(k)\psi_{\kappa(k)}(z)^{2}\le D$ (claim 6 of Properties of Finite Sums); so $|w_{j}(z)|\le G$ in the same way.

Step 8 (The polynomial). (a) Derivatives. The real line $\mathbb{R}$ is an interval every point of which is interior, by Basic Facts about Intervals of the Real Line and Their Interior Points §whole-line. On it, $f$ is differentiable at every $s$ with derivative $f'(s)$, and $f'$ is differentiable at every $s$ with derivative $f''(s)$, in the sense of Derivative at an Interior Point: by claim 1 of Derivative of a Polynomial Function on the Real Line the maps $s\mapsto s^{4}$, $s\mapsto s^{3}$, $s\mapsto s^{2}$ and $s\mapsto s^{1}=s$ have derivatives $4s^{3}$, $3s^{2}$, $2s$ and $1$ at $s$, and by claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives constant maps have derivative $0$ and sums and constant multiples are differentiated termwise; so $f$ has derivative $\frac{\varkappa}{4}\cdot4s^{3}+\frac{\sigma}{2}\cdot2s=f'(s)$, and $f'$ has derivative $3\varkappa s^{2}+\sigma=f''(s)$.

(b) Local bounds. Let $R\in\mathbb{R}$ with $0\le R$ and let $s,t\in\mathbb{R}$ with $|s|\le R$ and $|t|\le R$. Using $s^{2}-t^{2}=(s-t)(s+t)$, $s^{3}-t^{3}=(s-t)(s^{2}+st+t^{2})$ and $s^{4}-t^{4}=(s-t)(s+t)(s^{2}+t^{2})$, with $|s+t|\le2R$, $|s^{2}+st+t^{2}|\le3R^{2}$ and $0\le s^{2}+t^{2}\le2R^{2}$, one obtains

$$|f(s)-f(t)|\le\Lambda_{0}(R)|s-t|,\qquad|f'(s)-f'(t)|\le\Lambda_{1}(R)|s-t|,\qquad|f''(s)-f''(t)|\le\Lambda_{2}(R)|s-t|,$$

where $\Lambda_{0}(R)=\varkappa R^{3}+|\sigma|R$, $\Lambda_{1}(R)=3\varkappa R^{2}+|\sigma|$ and $\Lambda_{2}(R)=6\varkappa R$, all nonnegative; and directly $|f'(s)|\le\Lambda_{0}(R)$ and $|f''(s)|\le\Lambda_{1}(R)$.

(c) Global bounds. For every $s\in\mathbb{R}$,

$$f(s)\ge\frac{\varkappa}{8}s^{4}-B,\qquad f'(s)^{2}\le2\varkappa^{2}s^{6}+2\sigma^{2}s^{2},\qquad f''(s)\le3\varkappa s^{2}+|\sigma| .$$

For the first: $\frac{\sigma}{2}s^{2}\ge-\frac{K_{N}}{2}s^{2}$, since $-\sigma\le K_{N}$ (Step 0) and $0\le s^{2}$; $\frac{\varkappa}{8}s^{4}-\frac{K_{N}}{2}s^{2}+\frac{K_{N}^{2}}{2\varkappa}=\frac{\varkappa}{8}\bigl(s^{2}-\frac{2K_{N}}{\varkappa}\bigr)^{2}\ge0$; and $\gamma_{0}\ge-|\gamma_{0}|$; so $f(s)\ge\frac{\varkappa}{4}s^{4}-\frac{K_{N}}{2}s^{2}+\gamma_{0}\ge\frac{\varkappa}{8}s^{4}-\frac{K_{N}^{2}}{2\varkappa}-|\gamma_{0}|$. The second is $(p+q)^{2}\le2p^{2}+2q^{2}$, which holds because $2p^{2}+2q^{2}-(p+q)^{2}=(p-q)^{2}\ge0$, with $p=\varkappa s^{3}$ and $q=\sigma s$. The third holds since $\sigma\le|\sigma|$.

Step 9 (Integrals over the cell). Let $P$ be one of $f$, $f'$, $f''$, let $u\in\mathbb{R}^{d}$ and let $w\in C_{\mathrm{per}}$. The map $z\mapsto P(\varphi[u](z))\,w(z)$ belongs to $C_{\mathrm{per}}$, being built from $\varphi[u]$, which is in $C_{\mathrm{per}}$ by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §field, from $w$ and from constant maps (Step 0) by sums, scalar multiples and products, by Elementary Properties of Lattice-Periodic Functions §algebra. Every member of $C_{\mathrm{per}}$, constant maps included, has a $\lambda_{Q}$-integrable restriction to $Q$, belonging to $\mathcal{L}^{1}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member with the exponent $1$. By Step 0, The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and The Trigonometric System on the Torus is Orthonormal §orthonormal,

$$\int_{Q}1\,\lambda_{Q}(dz)=\int_{Q}\psi_{0_{n}}(z)\,\psi_{0_{n}}(z)\,\lambda_{Q}(dz)=\langle E_{0_{n}},E_{0_{n}}\rangle_{L^{2}}=1,$$

so $\int_{Q}t\,\lambda_{Q}(dz)=t$ for every constant $t\in\mathbb{R}$, by claim 2 of Linearity and Monotonicity of the Lebesgue Integral; that claim also supplies the linearity, the monotonicity and the bound $|\int_{Q}g\,d\lambda_{Q}|\le\int_{Q}|g|\,d\lambda_{Q}$ used below for such integrable functions $g$. Finally, for $g\in C_{\mathrm{per}}$ and $k\in\mathbb{Z}^{n}$, $g|_{Q}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member, and by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, The Cauchy-Schwarz Inequality in a Real Inner Product Space, $\lVert E_{k}\rVert_{L^{2}}=1$ (Step 0) and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field,

$$\Bigl(\int_{Q}g(z)\psi_{k}(z)\,\lambda_{Q}(dz)\Bigr)^{2}=\bigl\langle[\,g|_{Q}\,],E_{k}\bigr\rangle_{L^{2}}^{2}\le\bigl\lVert[\,g|_{Q}\,]\bigr\rVert_{L^{2}}^{2}=\int_{Q}g(z)^{2}\,\lambda_{Q}(dz);$$

we refer to this as the inequality (CS).

Step 10 (Continuity of integrals depending on a point of $\mathbb{R}^{d}$). Let $(P,\Lambda)$ be one of $(f,\Lambda_{0})$, $(f',\Lambda_{1})$, $(f'',\Lambda_{2})$, let $w\in C_{\mathrm{per}}$ and $W\in\mathbb{R}$ with $0\le W$ and $|w(z)|\le W$ for every $z\in\mathbb{R}^{n}$, and let $F:\mathbb{R}^{d}\to\mathbb{R}$ be $F(u)=\int_{Q}P(\varphi[u](z))\,w(z)\,\lambda_{Q}(dz)$, defined by Step 9. Then $F$ is continuous at every point of $\mathbb{R}^{d}$. Fix $u\in\mathbb{R}^{d}$ and a positive $\varepsilon\in\mathbb{R}$; put $R=G(\lVert u\rVert+1)$, $\Theta=\Lambda(R)\,G\,W\ge0$, and let $\delta$ be the minimum of $1$ and $\varepsilon/(1+\Theta)$, which is positive. Let $u'\in\mathbb{R}^{d}$ satisfy $\sum_{i=1}^{d}(u'_{i}-u_{i})^{2}<\delta^{2}$, that is $\lVert u'-u\rVert^{2}<\delta^{2}$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square; then $\lVert u'-u\rVert<\delta\le1$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and $\lVert u'\rVert\le\lVert u\rVert+1$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §triangle. By Step 7 and the linearity of Step 0, for every $z$: $|\varphi[u](z)|\le R$, $|\varphi[u'](z)|\le R$ and $|\varphi[u'](z)-\varphi[u](z)|=|\varphi[u'-u](z)|\le G\lVert u'-u\rVert$; so by Step 8(b)

$$\bigl|P(\varphi[u'](z))\,w(z)-P(\varphi[u](z))\,w(z)\bigr|\le\Lambda(R)\,G\lVert u'-u\rVert\,W=\Theta\lVert u'-u\rVert .$$

Integrating with Step 9, $|F(u')-F(u)|\le\Theta\lVert u'-u\rVert\le\Theta\delta\le\Theta\varepsilon/(1+\Theta)<\varepsilon$, hence $(F(u')-F(u))^{2}<\varepsilon^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. This is continuity at $u$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces, used with $\mathbb{R}^{d}$ in place of its $\mathbb{R}^{n}$ and with $m=1$.

Step 11 (Partial derivatives under the integral). Let $(P,P^{*},\Lambda)$ be $(f,f',\Lambda_{0})$ or $(f',f'',\Lambda_{1})$, let $w$, $W$ and $F$ be as in Step 10 with this $P$, and let $u\in\mathbb{R}^{d}$ and $j\in[d]$. We show that the partial derivative of $F$ with respect to the $j$th variable exists at $u$ and

$$\partial_{j}F(u)=\int_{Q}P^{*}(\varphi[u](z))\,w_{j}(z)\,w(z)\,\lambda_{Q}(dz).$$

$\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Let $\varsigma>0$ be the number written $\rho$ in claim 1 of Slice Function and the Partial Derivative, for $F$, $u$ and $j$, and let $I$ be the set of those $s\in\mathbb{R}$ with $u_{j}-\varsigma<s<u_{j}+\varsigma$, the open interval with these endpoints, every point of which is an interior point of it by An Open Interval is an Interval All of Whose Points Are Interior. For $s\in I$ let $u[s]$ be the point of $\mathbb{R}^{d}$ whose $j$th coordinate is $s$ and whose other coordinates are those of $u$. Its coordinates differ from those of $u$ only at $j$, by $s-u_{j}$, so by Step 0 and claims 2, 3 and 7 of Properties of Finite Sums, $\varphi[u[s]](z)=\varphi[u](z)+(s-u_{j})w_{j}(z)$ for $z\in\mathbb{R}^{n}$. Let $\Upsilon:I\times Q\to\mathbb{R}$ be $\Upsilon(s,z)=P\bigl(\varphi[u](z)+(s-u_{j})w_{j}(z)\bigr)\,w(z)$. We verify the three hypotheses of Differentiation under the Integral Sign, on the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ of The Flat Torus: Standing Notation §measure and the open interval $I$, with endpoints $u_{j}-\varsigma<u_{j}+\varsigma$.

(i) For $s\in I$, $\Upsilon(s,\cdot)$ is the restriction to $Q$ of $z\mapsto P(\varphi[u[s]](z))\,w(z)$, which is integrable by Step 9.

(ii) Fix $z\in Q$. The map $\gamma:I\to\mathbb{R}$, $\gamma(s)=\bigl(\varphi[u](z)-u_{j}w_{j}(z)\bigr)+w_{j}(z)\,s$, is differentiable at every $s\in I$ with derivative $w_{j}(z)$, by claim 1 of Derivative of a Polynomial Function on the Real Line (the map $s\mapsto s^{1}=s$) and claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives; $P$ is differentiable at $\gamma(s)$, an interior point of the interval $\mathbb{R}$, with derivative $P^{*}(\gamma(s))$ by Step 8(a). By Chain Rule for One-Dimensional Derivatives and claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives (the constant multiple $w(z)$), $s\mapsto\Upsilon(s,z)$ is differentiable at every $s\in I$, with derivative $D_{1}\Upsilon(s,z)=P^{*}(\varphi[u[s]](z))\,w_{j}(z)\,w(z)$.

(iii) For $s\in I$ and $z\in Q$, by Step 7, $|\varphi[u[s]](z)|\le|\varphi[u](z)|+|s-u_{j}|\,|w_{j}(z)|\le G\lVert u\rVert+\varsigma G$; with $R=G\lVert u\rVert+\varsigma G$, Step 8(b) and Step 7 give $|D_{1}\Upsilon(s,z)|\le\Lambda(R)\,G\,W$, and the constant map on $Q$ with this value is integrable by Step 9.

By Differentiation under the Integral Sign, the map $s\mapsto\int_{Q}\Upsilon(s,z)\,\lambda_{Q}(dz)=F(u[s])$ on $I$, which is the slice function of $F$ at $u$ in the $j$th variable of claim 2 of Slice Function and the Partial Derivative, is differentiable at $u_{j}$ with derivative $\int_{Q}D_{1}\Upsilon(u_{j},z)\,\lambda_{Q}(dz)$. Since $u[u_{j}]=u$, claim 2 of Slice Function and the Partial Derivative gives the displayed formula.

Step 12 (Regularity and the derivatives; first assertions of claim 3). By Step 0, $\psi_{0_{n}}$ is the constant map $1$, so $v$ is the function $F$ of Step 10 for $(P,w,W)=(f,\psi_{0_{n}},1)$, by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §potential. Step 11 for this choice gives, for $u\in\mathbb{R}^{d}$ and $k\in[d]$,

$$\partial_{k}v(u)=\int_{Q}f'(\varphi[u](z))\,w_{k}(z)\,\lambda_{Q}(dz);$$

so $\partial_{k}v$ is the function $F$ of Step 10 for $(P,w,W)=(f',w_{k},G)$, with $|w_{k}|\le G$ by Step 7, and Step 11 for this choice gives, for $j\in[d]$,

$$\partial_{j}\partial_{k}v(u)=\int_{Q}f''(\varphi[u](z))\,w_{j}(z)\,w_{k}(z)\,\lambda_{Q}(dz),$$

where $\partial_{j}\partial_{k}v$ is the partial derivative with respect to the $j$th variable of $\partial_{k}v$, as in clause 4 of C^k Maps on a Euclidean Open Set. This is the function $F$ of Step 10 for $(P,w,W)=(f'',w_{j}w_{k},G^{2})$, where $w_{j}w_{k}\in C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra and $|w_{j}w_{k}|\le G^{2}$ by Step 7. By Step 10 for the three choices, $v$, every $\partial_{k}v$ and every $\partial_{j}\partial_{k}v$ are continuous at every point of $\mathbb{R}^{d}$. By clause 1 of C^k Maps on a Euclidean Open Set, read through its scalar convention in clause 3, each $\partial_{k}v$ and $v$ itself are of class $C^{1}$ on $\mathbb{R}^{d}$; hence $v$ is of class $C^{2}$ on $\mathbb{R}^{d}$ by clause 2 there, used with $k=1$. If $j\in[d]$ and $\kappa(j)\notin\Gamma_{N}$, then $j\notin J$, $w_{j}$ vanishes identically (Step 0), and the formulas above give $\partial_{j}v(u)=0$, $\partial_{j}\partial_{k}v(u)=0$ and $\partial_{k}\partial_{j}v(u)=\int_{Q}f''(\varphi[u](z))\,w_{k}(z)\,w_{j}(z)\,\lambda_{Q}(dz)=0$ for all $u\in\mathbb{R}^{d}$ and $k\in[d]$, the integral of the zero map being $0$ by Step 9.

Step 13 (The Hessian; rest of claim 3). Let $u,h\in\mathbb{R}^{d}$. For each $j\in[d]$, by Step 12, Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear (coefficients $h_{j}h_{k}$; the integrands are restrictions of members of $C_{\mathrm{per}}$, integrable by Step 9), claim 3 of Properties of Finite Sums and Step 0,

$$\sum_{k=1}^{d}\partial_{j}\partial_{k}v(u)\,h_{j}h_{k}=\int_{Q}\sum_{k=1}^{d}h_{j}h_{k}\,f''(\varphi[u](z))\,w_{j}(z)\,w_{k}(z)\,\lambda_{Q}(dz)=\int_{Q}h_{j}\,f''(\varphi[u](z))\,w_{j}(z)\,\varphi[h](z)\,\lambda_{Q}(dz),$$

and summing over $j$ in the same way,

$$\sum_{j=1}^{d}\sum_{k=1}^{d}\partial_{j}\partial_{k}v(u)\,h_{j}h_{k}=\int_{Q}f''(\varphi[u](z))\,\varphi[h](z)^{2}\,\lambda_{Q}(dz)=\int_{Q}\bigl(3\varkappa\,\varphi[u](z)^{2}+\mathfrak{m}-3\varkappa C\bigr)\varphi[h](z)^{2}\,\lambda_{Q}(dz).$$

Pointwise, $3\varkappa\,\varphi[u](z)^{2}\varphi[h](z)^{2}\ge0$ and $\sigma\,\varphi[h](z)^{2}\ge-K_{N}\varphi[h](z)^{2}$ (Step 0), so by the monotonicity and linearity of Step 9 and by Step 6,

$$\int_{Q}f''(\varphi[u](z))\,\varphi[h](z)^{2}\,\lambda_{Q}(dz)\ge-K_{N}\int_{Q}\varphi[h](z)^{2}\,\lambda_{Q}(dz)=-K_{N}\sum_{j\in J}\frac{h_{j}^{2}}{a_{j}}\ge-K_{N}\sum_{k=1}^{d}\frac{h_{k}^{2}}{a_{k}},$$

the last inequality by claim 2 and $0\le K_{N}$. This proves claim 3.

Step 14 (Admissibility; claim 4). $0\le K_{N}$ by Step 0, $v$ is of class $C^{2}$ on $\mathbb{R}^{d}$ by Step 12, and $V_{N}=v\circ p_{d}$ by Step 4. The settings of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §admissible are in force here by The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian and The Flat Torus: Standing Notation §background; we verify its four conditions with head dimension $d$, profile $v$ and semiconvexity constant $K_{N}$. Let $u\in\mathbb{R}^{d}$ and write $r=r(u)$, so that $0\le r=\int_{Q}\varphi[u](z)^{2}\,\lambda_{Q}(dz)$ by Step 6.

A lower bound. By The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §potential, Step 8(c) and Step 9,

$$v(u)=\int_{Q}f(\varphi[u](z))\,\lambda_{Q}(dz)\ge\frac{\varkappa}{8}\int_{Q}\varphi[u](z)^{4}\,\lambda_{Q}(dz)-B .$$

By (CS) of Step 9 with $g=\varphi[u]^{2}$ (in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra) and $k=0_{n}$, using $\psi_{0_{n}}=1$ (Step 0), $r^{2}\le\int_{Q}\varphi[u](z)^{4}\,\lambda_{Q}(dz)$. Hence

$$v(u)\ge\frac{\varkappa}{8}\,r^{2}-B\ge-B .$$

(a) The last display is condition Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below with $b=B$.

(b) Condition Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §semiconvex with $K=K_{N}$ is the inequality of claim 3 (Step 13).

(c) From the lower bound, $r^{2}\le\frac{8}{\varkappa}(v(u)+B)\le\frac{8}{\varkappa}(|v(u)|+B)\le\frac{8}{\varkappa}(1+B)(1+|v(u)|)$, the last because $(1+B)(1+|v(u)|)=1+B+|v(u)|+B|v(u)|$ and $0\le B$. Both ends being nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $r^{4}\le\frac{64}{\varkappa^{2}}(1+B)^{2}(1+|v(u)|)^{2}$. Next, for $k\in[d]$, Step 12 and the linearity of Step 9 give $\partial_{k}v(u)=\chi(k)a_{k}^{-1/2}\int_{Q}f'(\varphi[u](z))\psi_{\kappa(k)}(z)\,\lambda_{Q}(dz)$; so by Step 0 and (CS) of Step 9 with $g=f'\circ\varphi[u]$ (in $C_{\mathrm{per}}$ by Step 9),

$$a_{k}\bigl(\partial_{k}v(u)\bigr)^{2}=\chi(k)\Bigl(\int_{Q}f'(\varphi[u](z))\psi_{\kappa(k)}(z)\,\lambda_{Q}(dz)\Bigr)^{2}\le\chi(k)\int_{Q}f'(\varphi[u](z))^{2}\,\lambda_{Q}(dz).$$

By Step 8(c) and Step 7 (which gives $0\le\varphi[u](z)^{2}\le D\,r$, hence $(\varphi[u](z)^{2})^{3}\le D^{3}r^{3}$),

$$f'(\varphi[u](z))^{2}\le2\varkappa^{2}\bigl(\varphi[u](z)^{2}\bigr)^{3}+2\sigma^{2}\varphi[u](z)^{2}\le2\varkappa^{2}D^{3}r^{3}+2\sigma^{2}D\,r\le A\,(1+r^{4}),$$

where $A=2\varkappa^{2}D^{3}+2\sigma^{2}D\ge0$, because $r\le1+r^{4}$ and $r^{3}\le1+r^{4}$: if $r\le1$ then $r\le1$ and $r^{3}\le1$, and if $1<r$ then $r\le r^{3}\le r^{4}$. Integrating (Step 9) and summing over $k$ (Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison with claim 1 of Properties of a Sum over a Finite Index Set), with $\nu=\sum_{k=1}^{d}\chi(k)\ge0$,

$$\sum_{k=1}^{d}a_{k}\bigl(\partial_{k}v(u)\bigr)^{2}\le\nu A\,(1+r^{4})\le\nu A\Bigl(1+\frac{64}{\varkappa^{2}}(1+B)^{2}\Bigr)\bigl(1+|v(u)|\bigr)^{2},$$

using $1\le(1+|v(u)|)^{2}$. So condition Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope holds with the constant $\nu A\bigl(1+\frac{64}{\varkappa^{2}}(1+B)^{2}\bigr)$, which does not depend on $u$.

(d) Let $\varepsilon\in\mathbb{R}$ be positive. For $k\in[d]$, Step 12 and Step 0 give $a_{k}\,\partial_{k}\partial_{k}v(u)=\int_{Q}f''(\varphi[u](z))\,\chi(k)\psi_{\kappa(k)}(z)^{2}\,\lambda_{Q}(dz)$, since $a_{k}w_{k}(z)^{2}=a_{k}(a_{k}^{-1/2})^{2}\chi(k)\psi_{\kappa(k)}(z)^{2}=\chi(k)\psi_{\kappa(k)}(z)^{2}$. Summing with Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear,

$$\sum_{k=1}^{d}a_{k}\,\partial_{k}\partial_{k}v(u)=\int_{Q}f''(\varphi[u](z))\,Y(z)\,\lambda_{Q}(dz),\qquad Y(z)=\sum_{k=1}^{d}\chi(k)\psi_{\kappa(k)}(z)^{2},$$

and $0\le Y(z)\le D$ by Step 7. Pointwise, by Step 8(c) and Step 7, $f''(\varphi[u](z))\le3\varkappa\,\varphi[u](z)^{2}+|\sigma|\le3\varkappa Dr+|\sigma|$, whose right side is nonnegative; multiplying by $Y(z)\ge0$ and then using $Y(z)\le D$, $f''(\varphi[u](z))\,Y(z)\le(3\varkappa Dr+|\sigma|)\,D$. Integrating (Step 9) and using $r\le\alpha\lVert u\rVert^{2}=\alpha\sum_{k=1}^{d}u_{k}^{2}$ (Step 7 and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square),

$$\sum_{k=1}^{d}a_{k}\,\partial_{k}\partial_{k}v(u)\le D|\sigma|+3\varkappa D^{2}\alpha\sum_{k=1}^{d}u_{k}^{2}\le C_{\varepsilon}\Bigl(1+\sum_{k=1}^{d}u_{k}^{2}\Bigr)\le\varepsilon\sum_{k=1}^{d}a_{k}\bigl(\partial_{k}v(u)\bigr)^{2}+C_{\varepsilon}\Bigl(1+\sum_{k=1}^{d}u_{k}^{2}\Bigr),$$

with $C_{\varepsilon}=D|\sigma|+3\varkappa D^{2}\alpha$, which bounds both nonnegative numbers $D|\sigma|$ and $3\varkappa D^{2}\alpha$; the last inequality holds because $\varepsilon\sum_{k}a_{k}(\partial_{k}v(u))^{2}\ge0$, each $a_{k}$ being positive. This is condition Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §curvature, and claim 4 is proved.

Step 15 (The noise gradient; claim 5). Let $x\in X$. By Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, Step 12, Step 4 ($\varphi[p_{d}(x)]=\Phi[x]$) and Step 0, the coefficient of $e_{k}$ ($k\in[d]$) in $\nabla_{a}V_{N}(x)$ is

$$a_{k}\,\partial_{k}v(p_{d}(x))=a_{k}\,\chi(k)\,a_{k}^{-1/2}\int_{Q}f'(\Phi[x](z))\psi_{\kappa(k)}(z)\,\lambda_{Q}(dz)=\chi(k)\,a_{k}^{1/2}\int_{Q}f'(\Phi[x](z))\psi_{\kappa(k)}(z)\,\lambda_{Q}(dz),$$

and $a_{k}^{1/2}e_{k}=\hat{E}_{\kappa(k)}$ 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 §basis-vectors. Hence

$$\nabla_{a}V_{N}(x)=\sum_{k=1}^{d}\chi(k)\Bigl(\int_{Q}f'(\Phi[x](z))\psi_{\kappa(k)}(z)\,\lambda_{Q}(dz)\Bigr)\hat{E}_{\kappa(k)},$$

which, with $f'(s)=\varkappa(s^{3}-3Cs)+\mathfrak{m}s$ (Step 0) and the convention of The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension for sums over $J$, is the formula of claim 5.

Step 16 (The Wick constant diverges; claim 6). Suppose $2\le n$. Let $\vartheta:\mathbb{Z}^{n}\to\mathbb{R}$ be $\vartheta(k)=\mu_{k}^{-1}$, a positive family. By The Wick Square of the Free Field on the Torus as Diagonal Couplings: Summability in Dimension at Most Three, the Riccati Condition, and the Divergent Wick Constant §divergent, used with $w_{0}=1$, the series $\sum_{j=1}^{\infty}\mu_{\kappa(j)}^{-1}=\sum_{j=1}^{\infty}\vartheta(\kappa(j))$ does not converge. Since $\kappa$ is a bijection, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §enumeration shows that $\vartheta$ is not cube-summable, and then Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative shows that the set of its cube sums $S_{N'}(\vartheta)$, $N'\in\mathbb{N}$, is not bounded above. Here, by Cube Sums of Families on the Integer Lattice §cube-sums, $S_{N'}(\vartheta)=\sum_{k\in\Gamma_{N'}}\mu_{k}^{-1}$, the cube $\Gamma_{N'}$ being that of The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points (of which the version cited by that definition is an earlier version with the same cube), so $S_{N'}(\vartheta)=C_{N'}$ by Step 2. Let $M\in\mathbb{R}$. As $M$ is not an upper bound of this set, there is $N_{0}\in\mathbb{N}$ with $M<S_{N_{0}}(\vartheta)$, the order of $\mathbb{R}$ being total. For $N'\in\mathbb{N}$ with $N_{0}\le N'$, $S_{N_{0}}(\vartheta)\le S_{N'}(\vartheta)$ by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative, so $M<S_{N'}(\vartheta)=C_{N'}$. This proves claim 6.