Proof of The Renormalised Operator of the Wick-Square Problem: the Penalised Form on the Whole State Space, the Wick Domain without the Penalty, and the Forced Counterterm

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

Conventions. Sums over a cube $\Gamma_{N}$ are sums over a finite index set in the sense of Sum over a Finite Index Set, the cubes being nonempty and finite by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite; for a family $a:\mathbb{Z}^{n}\to\mathbb{R}$ we write $S_{N}(a)=\sum_{k\in\Gamma_{N}}a(k)$ for its $N$th cube sum, as in Cube Sums of Families on the Integer Lattice §cube-sums. Natural numbers are read in $\mathbb{R}$ through the canonical map $\iota$, as in The Real Numbers: Standing Notation and Background §numbers. The vector operations of $H^{-1}$ are the pointwise ones, because $H^{-1}$ is a linear subspace of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ by The Wick-Square Problem on the Torus: Standing Notation §state-space; so for $u,v\in H^{-1}$ the vector $u-v$ is the family $k\mapsto u(k)-v(k)$, and by Real Inner Product Space §distance, applied to the inner product of The Negative-Order Sobolev Spaces of the Torus §inner-product, $d_{H^{-1}}(u,v)=|u-v|_{H^{-1}}$, where $0\le|u-v|_{H^{-1}}$ by Real Inner Product Space §norm. By Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §sobolev with $m=1$ (as recorded in The Wick-Square Problem on the Torus: Standing Notation §state-space), a family $v$ lies in $H^{-1}$ exactly when $k\mapsto v(k)^{2}/\mu_{k}$ is cube-summable, and then $|v|_{H^{-1}}^{2}=\sum_{k\in\mathbb{Z}^{n}}v(k)^{2}/\mu_{k}$.

Preliminary facts.

(C) Constant sequences converge. For $a\in\mathbb{R}$, every $\varepsilon>0$ and every $N\in\mathbb{N}$ we have $|a-a|=|0|=0<\varepsilon$ by claim 1 of Properties of the Absolute Value in an Ordered Field; so the constant sequence with value $a$ converges to $a$ in the sense of Limit of a Sequence of Real Numbers.

(Nn) Products of nonnegative numbers. If $0\le a$ and $0\le b$, then $a\cdot0\le ab$ by claim 5 of Elementary Arithmetic in an Ordered Field, and $a\cdot0=0$ by claim 1 of Zero Products and Elementary Identities in a Field; so $0\le ab$.

(Sq) Squares are nonnegative. Let $r\in\mathbb{R}$. If $0\le r$, then $0\le r^{2}$ by (Nn). Otherwise $r\le0$, the order being total, so $0\le-r$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, and $r^{2}=(-r)(-r)$ by claim 2 of Zero Products and Elementary Identities in a Field, which is nonnegative by (Nn).

(Tw) A bound for the square of a difference. Let $p,q\in\mathbb{R}$. By claim 5 of Zero Products and Elementary Identities in a Field, $(p+q)^{2}=p^{2}+(pq+pq)+q^{2}$, and, applied to $p$ and $-q$ together with claim 2 of that lemma, $(p-q)^{2}=p^{2}-(pq+pq)+q^{2}$. Adding, $(p-q)^{2}+(p+q)^{2}=2(p^{2}+q^{2})$, so $2(p^{2}+q^{2})-(p-q)^{2}=(p+q)^{2}$, which is nonnegative by (Sq); hence $(p-q)^{2}\le2(p^{2}+q^{2})$ by claim 3 of Elementary Arithmetic in an Ordered Field.

(Wt) The weights and the variances. For every mode $k$: $1\le\mu_{k}$ by The Wick-Square Problem on the Torus: Standing Notation §modes and $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, so $0<\mu_{k}$ by claim 2 of that lemma and $0<\mu_{k}^{-1}$ by its claim 7. Also $0<c_{k}$ and $c_{k}=\nu(2\mu_{k})^{-1}$ by The Free-Field Variances of the Fourier Modes §variances. Since in a field the inverse of a product of nonzero elements is the product of the inverses, and $\mu_{k}^{2}=\mu_{k}\mu_{k}$ (The Real Numbers: Standing Notation and Background §numbers),

$$c_{k}=\frac{\nu}{2}\,\mu_{k}^{-1},\qquad \frac{c_{k}}{\mu_{k}}=\frac{\nu}{2}\cdot\frac{1}{\mu_{k}^{2}}.$$

Since $n\le3<4$ by The Wick-Square Problem on the Torus: Standing Notation §dimension, the family $k\mapsto1/\mu_{k}^{2}$ is cube-summable by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §convergent with $s=2$, and therefore so is $k\mapsto c_{k}/\mu_{k}$, by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear with $\lambda=\nu/2$.

Clause 1 (penalised). Let $\psi=\varphi-P$, the function $u\mapsto\varphi(u)-P(u)$ on $H^{-1}$; it is regular by hypothesis, so by Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3 it has a curvature family and a cylindrical part, which we fix, and Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations applies to $\psi$. For every $u\in H^{-1}$ we have $1\cdot P(u)+\psi(u)=\varphi(u)$, so $\varphi=\theta P+\psi$ with $\theta=1$. By Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations §gradient with $\theta=1$, $\varphi$ is twice differentiable along the modes and $|D\varphi(x)|^{2}$ is defined for every $x\in H^{-1}$; that is, by The Gradient Energy with a Mode Cutoff §limit, the sequence $(|D_{N}\varphi(x)|^{2})_{N\in\mathbb{N}}$ converges to $|D\varphi(x)|^{2}$.

Fix $x\in H^{-1}$. By Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations §generator, $L\psi=L(\varphi-P)$ is defined at $x$, that is, by The Free-Field Generator with a Mode Cutoff §limit, the sequence $(L_{N}(\varphi-P)(x))_{N\in\mathbb{N}}$ converges to $L(\varphi-P)(x)$. Since $\varphi$ is twice differentiable along the modes, The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §penalised gives, for every $N\in\mathbb{N}$,

$$F_{N}[\varphi](x)=\gamma\,\varphi(x)-L_{N}(\varphi-P)(x)+\tfrac12\,|D_{N}\varphi(x)|^{2}-g(x).$$

The constant sequences with values $\gamma\varphi(x)$ and $g(x)$ converge by (C). By claims 1 and 3 of Arithmetic of Limits of Real Sequences, the right-hand side converges, as $N\to\infty$, to $\gamma\varphi(x)-L(\varphi-P)(x)+\tfrac12|D\varphi(x)|^{2}-g(x)$. Hence $(F_{N}[\varphi](x))_{N\in\mathbb{N}}$ converges, so $(\varphi,x)\in\mathcal{D}$ by The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §domain, and by The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §operator together with claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, $F[\varphi](x)$ equals this limit, which is the asserted formula.

Clause 2 (wick-only). Let $\psi=\varphi-\theta P$, regular by hypothesis; fix a curvature family and a cylindrical part of it as in Clause 1. For every $u\in H^{-1}$, $\varphi(u)=\theta P(u)+\psi(u)$, so Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations §gradient, applied with this $\theta$, shows that $\varphi$ is twice differentiable along the modes and that $(|D_{N}\varphi(x)|^{2})_{N\in\mathbb{N}}$ converges for every $x\in H^{-1}$ (The Gradient Energy with a Mode Cutoff §limit).

Let $h=\varphi-P$. For every $u\in H^{-1}$, $h(u)=(\theta-1)P(u)+\psi(u)$. Mode derivatives of $h$. Fix $x\in H^{-1}$ and a mode $k$, and let $\sigma^{h},\sigma^{P},\sigma^{\psi}:\mathbb{R}\to\mathbb{R}$ be the sections of $h$, $P$ and $\psi$ at $x$ along $k$ (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes); then $\sigma^{h}(s)=(\theta-1)\sigma^{P}(s)+\sigma^{\psi}(s)$ for every $s\in\mathbb{R}$. The function $P$ is twice differentiable along the modes by The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §derivatives, and $\psi$ by Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations §derivatives; so, by Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice, $\sigma^{P}$ and $\sigma^{\psi}$ are differentiable at every point of $\mathbb{R}$ and their derivatives are differentiable at $0$. By claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, applied on the interval $\mathbb{R}$, all of whose points are interior points, first to $(\theta-1)\sigma^{P}$ and then to the sum, $\sigma^{h}$ is differentiable at every $s\in\mathbb{R}$ with $(\sigma^{h})'(s)=(\theta-1)(\sigma^{P})'(s)+(\sigma^{\psi})'(s)$; thus $(\sigma^{h})'=(\theta-1)(\sigma^{P})'+(\sigma^{\psi})'$ as functions on $\mathbb{R}$, and the same claim applied at $0$ gives $((\sigma^{h})')'(0)=(\theta-1)((\sigma^{P})')'(0)+((\sigma^{\psi})')'(0)$. By Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives,

$$\partial_{k}h(x)=(\theta-1)\,\partial_{k}P(x)+\partial_{k}\psi(x),\qquad\partial_{k}^{2}h(x)=(\theta-1)\,\partial_{k}^{2}P(x)+\partial_{k}^{2}\psi(x).$$

The generator of $h$. Fix $x\in H^{-1}$ and $N\in\mathbb{N}$. By the displayed identities and the field axioms, for every $k\in\Gamma_{N}$ the summand $\tfrac{\nu}{2}\partial_{k}^{2}h(x)-\mu_{k}x(k)\partial_{k}h(x)$ of $L_{N}h(x)$ (The Free-Field Generator with a Mode Cutoff §generator) equals $(\theta-1)\bigl(\tfrac{\nu}{2}\partial_{k}^{2}P(x)-\mu_{k}x(k)\partial_{k}P(x)\bigr)+\bigl(\tfrac{\nu}{2}\partial_{k}^{2}\psi(x)-\mu_{k}x(k)\partial_{k}\psi(x)\bigr)$. Summing over $\Gamma_{N}$ with claims 3 and 4 of Properties of a Sum over a Finite Index Set, and then using The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §corrector,

$$L_{N}(\varphi-P)(x)=(\theta-1)\,L_{N}P(x)+L_{N}\psi(x)=-(\theta-1)\beta\,{:}x^{2}{:}_{N}+L_{N}\psi(x).$$

The cutoff operators. Put $\lambda=(\theta-1)\beta$ and $A_{N}=\gamma\varphi(x)-L_{N}\psi(x)+\tfrac12|D_{N}\varphi(x)|^{2}-g(x)$. By The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §penalised (applicable since $\varphi$ is twice differentiable along the modes) and the last display, for every $N\in\mathbb{N}$

$$F_{N}[\varphi](x)=A_{N}+\lambda\,{:}x^{2}{:}_{N}.$$

The sequence $(A_{N})_{N\in\mathbb{N}}$ converges: $(L_{N}\psi(x))_{N}$ converges by Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations §generator and The Free-Field Generator with a Mode Cutoff §limit, $(|D_{N}\varphi(x)|^{2})_{N}$ converges as shown above, the constant sequences converge by (C), and claims 1 and 3 of Arithmetic of Limits of Real Sequences apply. Moreover $\lambda\ne0$: $\theta-1\ne0$ because $\theta\ne1$, $\beta\ne0$ because $0<\beta$ (The Wick-Square Problem on the Torus: Standing Notation §parameters), and claim 3 of Zero Products and Elementary Identities in a Field applies.

If $x\in\mathcal{W}$, then $({:}x^{2}{:}_{N})_{N}$ converges by The Wick Square with a Mode Cutoff and the Wick Domain §domain, so $(F_{N}[\varphi](x))_{N}$ converges by claims 1 and 3 of Arithmetic of Limits of Real Sequences, and $(\varphi,x)\in\mathcal{D}$ by The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §domain. Conversely, if $(\varphi,x)\in\mathcal{D}$, then $(F_{N}[\varphi](x))_{N}$ converges, and ${:}x^{2}{:}_{N}=\lambda^{-1}\bigl(F_{N}[\varphi](x)-A_{N}\bigr)$ for every $N$, so $({:}x^{2}{:}_{N})_{N}$ converges by claim 3 of Arithmetic of Limits of Real Sequences; as $x\in H^{-1}$, this gives $x\in\mathcal{W}$ by The Wick Square with a Mode Cutoff and the Wick Domain §domain.

Clause 3 (dense). Let $y\in H^{-1}$ and $\varepsilon>0$ be given. The order of choice is: first $\eta$ (depending on $\varepsilon$), then $M$ (depending on $y$ and $\eta$), then $z$ (depending on $y$ and $M$).

The tail family. Let $a(k)=y(k)^{2}/\mu_{k}$ and $h(k)=a(k)+c_{k}/\mu_{k}$ for $k\in\mathbb{Z}^{n}$. The family $a$ is cube-summable since $y\in H^{-1}$ (Conventions), and $k\mapsto c_{k}/\mu_{k}$ is cube-summable by (Wt); so $h$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, with lattice sum $S=\sum_{k\in\mathbb{Z}^{n}}h(k)$. Also $0\le h(k)$ for every $k$, by (Sq), (Wt), (Nn) and claim 2 of Elementary Arithmetic in an Ordered Field.

Choice of $\eta$ and $M$. Since $0<\varepsilon$, $0<\varepsilon^{2}$ by claim 5 of Elementary Order Arithmetic in an Ordered Field; put $\varepsilon'=\varepsilon^{2}\cdot2^{-1}$ and $\eta=\varepsilon'\cdot2^{-1}$, so that by claim 8 of that lemma (twice) $0<\eta$, $\eta+\eta=\varepsilon'$ and $\varepsilon'<\varepsilon^{2}$. Since $(S_{N}(h))_{N}$ converges to $S$ (Cube Sums of Families on the Integer Lattice §lattice-sum), Limit of a Sequence of Real Numbers applied with $\eta$ provides $M\in\mathbb{N}$ with $|S_{N}(h)-S|<\eta$ for all $N\ge M$; in particular $|S_{M}(h)-S|<\eta$.

Definition of $z$. For $k\notin\Gamma_{M}$ let $r_{k}$ be the real number with $0\le r_{k}$ and $r_{k}^{2}=c_{k}$ given by Existence and Uniqueness of the Nonnegative Square Root (applicable since $0<c_{k}$ by (Wt)). Let $z(k)=y(k)$ for $k\in\Gamma_{M}$ and $z(k)=r_{k}$ for $k\notin\Gamma_{M}$, and let $v$ be the family $k\mapsto y(k)-z(k)$; so $v(k)=0$ for $k\in\Gamma_{M}$ and $v(k)=y(k)-r_{k}$ otherwise.

A dominating family. Let $h_{M}(k)=h(k)$ for $k\in\Gamma_{M}$ and $h_{M}(k)=0$ otherwise. Since $\Gamma_{M}$ is nonempty and finite, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support shows that $h_{M}$ is cube-summable with lattice sum $\sum_{k\in\Gamma_{M}}h_{M}(k)=\sum_{k\in\Gamma_{M}}h(k)=S_{M}(h)$ (the two maps agree on $\Gamma_{M}$). Let $e=2h-2h_{M}$; by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear it is cube-summable with $\sum_{k\in\mathbb{Z}^{n}}e(k)=2(S-S_{M}(h))$, and $e(k)=0$ for $k\in\Gamma_{M}$, $e(k)=2h(k)$ otherwise. We claim $0\le v(k)^{2}/\mu_{k}\le e(k)$ for every $k$. Nonnegativity follows from (Sq), (Wt) and (Nn). For $k\in\Gamma_{M}$, $v(k)^{2}/\mu_{k}=0=e(k)$. For $k\notin\Gamma_{M}$, (Tw) with $p=y(k)$, $q=r_{k}$ gives $v(k)^{2}\le2(y(k)^{2}+r_{k}^{2})=2(y(k)^{2}+c_{k})$, and multiplying by $\mu_{k}^{-1}$, which is nonnegative by (Wt), claim 5 of Elementary Arithmetic in an Ordered Field gives $v(k)^{2}/\mu_{k}\le2(a(k)+c_{k}/\mu_{k})=e(k)$. Since $|v(k)^{2}/\mu_{k}|=v(k)^{2}/\mu_{k}$ by Absolute Value in an Ordered Field, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison shows that $k\mapsto v(k)^{2}/\mu_{k}$ is cube-summable with $\sum_{k\in\mathbb{Z}^{n}}v(k)^{2}/\mu_{k}\le2(S-S_{M}(h))$.

$z\in H^{-1}$ and the distance. By the Conventions, $v\in H^{-1}$ and $|v|_{H^{-1}}^{2}\le2(S-S_{M}(h))$. As $z(k)=y(k)-v(k)$ for every $k$ and $H^{-1}$ is a linear subspace, $z=y-v\in H^{-1}$, and $y-z=v$, so $d_{H^{-1}}(y,z)=|v|_{H^{-1}}$. Now $S-S_{M}(h)=-(S_{M}(h)-S)\le|S_{M}(h)-S|$ by claims 2 and 3 of Properties of the Absolute Value in an Ordered Field; multiplying by $2\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field), and then using $|S_{M}(h)-S|<\eta$ with $0<2$ and claim 10 of Elementary Order Arithmetic in an Ordered Field, we get $|v|_{H^{-1}}^{2}\le2(S-S_{M}(h))\le2|S_{M}(h)-S|<2\eta=\eta+\eta=\varepsilon'<\varepsilon^{2}$, the strict links combining by claim 2 of that lemma. Since $0\le|v|_{H^{-1}}$ and $0\le\varepsilon$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $d_{H^{-1}}(y,z)=|v|_{H^{-1}}<\varepsilon$.

$z\in\mathcal{W}$. Let $q(k)=z(k)^{2}-c_{k}$. For $k\notin\Gamma_{M}$, $q(k)=r_{k}^{2}-c_{k}=0$. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support with $E=\Gamma_{M}$, the family $q$ is cube-summable, i.e. $(S_{N}(q))_{N}$ converges. By The Wick Square with a Mode Cutoff and the Wick Domain §cutoff, $S_{N}(q)={:}z^{2}{:}_{N}$ for every $N$, so $({:}z^{2}{:}_{N})_{N}$ converges and, as $z\in H^{-1}$, $z\in\mathcal{W}$ by The Wick Square with a Mode Cutoff and the Wick Domain §domain.

Clause 4 (no-interior). Let $x\in H^{-1}$ and $\varepsilon>0$ be given. If $x\notin\mathcal{W}$, take $z=x$: then $d_{H^{-1}}(x,x)=0<\varepsilon$, because $d_{H^{-1}}$ is a metric (The Negative-Order Sobolev Spaces of the Torus §inner-product, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric) and condition 2 of Metric Space holds. Assume from now on that $x\in\mathcal{W}$. The order of choice is: the weight family $w$ (depending only on $n$) and the signs $s_{k}$ (depending on $x$); then the number $A$; then $\rho$ and $t$ (depending on $A$ and $\varepsilon$); then $z$.

The weight family. If $n=1$, let $w(k)=1$ for every $k$. If $2\le n$, let $w(k)$ be the real number with $0\le w(k)$ and $w(k)^{2}=\mu_{k}^{-1}$, given by Existence and Uniqueness of the Nonnegative Square Root since $0<\mu_{k}^{-1}$ by (Wt). In either case $0\le w(k)$ for every $k$ (claim 1 of Elementary Arithmetic in an Ordered Field when $n=1$), and $w$ has the following two properties.

(W2) The family $k\mapsto w(k)^{2}/\mu_{k}$ is cube-summable. If $n=1$, it is $k\mapsto1/\mu_{k}=1/\mu_{k}^{1}$, cube-summable by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §convergent with $s=1$, since $1<2$. If $2\le n$, it is $k\mapsto\mu_{k}^{-1}\mu_{k}^{-1}=1/\mu_{k}^{2}$, cube-summable by (Wt).

(W3) The set $\{S_{N}(w^{2}):N\in\mathbb{N}\}$, where $w^{2}$ is the family $k\mapsto w(k)^{2}$, is not bounded above. If $2\le n$, $S_{N}(w^{2})=\sum_{k\in\Gamma_{N}}1/\mu_{k}$ and this is Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §divergent. Let $n=1$; then $w(k)^{2}=1$ for all $k$, and a mode is a map on $[1]$, determined by its single component. For $N\in\mathbb{N}$ let $q_{N}$ be the mode with component $N$. Since $-N\le N\le N$ in the integers, $q_{N}\in\Gamma_{N}$; since $N<N+1$, $q_{N+1}\notin\Gamma_{N}$. Let $T$ be the set of $N\in\mathbb{N}$ with $\iota(N)\le S_{N}(w^{2})$. The constant family $1$ is nonnegative (claim 1 of Elementary Arithmetic in an Ordered Field). By the monotonicity claim Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §monotone with $E=\{q_{1}\}\subseteq\Gamma_{1}$, claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set and claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, $\iota(1)=1=\sum_{k\in\{q_{1}\}}1\le S_{1}(w^{2})$, so $1\in T$. Let $N\in T$. As $N\le N+1$, $\Gamma_{N}\subseteq\Gamma_{N+1}$ by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §nested, and $q_{N+1}\in\Gamma_{N+1}\setminus\Gamma_{N}$; so by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §monotone with $E=\Gamma_{N}\cup\{q_{N+1}\}$, and by claim 2 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, $S_{N+1}(w^{2})\ge\sum_{k\in\Gamma_{N}\cup\{q_{N+1}\}}1=S_{N}(w^{2})+1\ge\iota(N)+1=\iota(N+1)$, using $N\in T$ and claim 3 of Elementary Arithmetic in an Ordered Field for the second inequality and claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field for the equality. Thus $N+1\in T$, and $T=\mathbb{N}$ by Principle of Induction for the Natural Numbers (with $S(N)=N+1$, Natural Numbers). If some real $B$ satisfied $S_{N}(w^{2})\le B$ for all $N$, claim 1 of The Archimedean Property of the Real Numbers would give $N$ with $B<\iota(N)$, and then $\iota(N)\le S_{N}(w^{2})\le B<\iota(N)$, so $\iota(N)<\iota(N)$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, which is absurd.

Signs, $A$ and $t$. For each $k$ let $s_{k}=1$ if $0\le x(k)$ and $s_{k}=-1$ otherwise. Then $0\le s_{k}x(k)$ (in the second case $x(k)\le0$ and claim 4 of Elementary Order Arithmetic in an Ordered Field applies), and $s_{k}^{2}=1$ by claim 2 of Zero Products and Elementary Identities in a Field. Let $A=\sum_{k\in\mathbb{Z}^{n}}w(k)^{2}/\mu_{k}$, which exists by (W2) and satisfies $0\le A$ by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative, the family being nonnegative by (Sq), (Wt) and (Nn). Then $0<A+1$ by claim 3 of Elementary Order Arithmetic in an Ordered Field (from $0<1$ and $0\le A$), so $0<(A+1)^{-1}$ by claim 7 of that lemma; let $\rho$ be the real number with $0\le\rho$ and $\rho^{2}=(A+1)^{-1}$ (Existence and Uniqueness of the Nonnegative Square Root). Then $\rho\ne0$, since otherwise $\rho^{2}=0$ by claim 1 of Zero Products and Elementary Identities in a Field; so $0<\rho$, and $t=\varepsilon\rho$ satisfies $0<t$ by claim 5 of Elementary Order Arithmetic in an Ordered Field, and $t^{2}=\varepsilon^{2}(A+1)^{-1}$.

Definition of $z$ and the distance. Let $u(k)=s_{k}w(k)$ and let $\delta$ be the family $k\mapsto t\,u(k)$. For every $k$, $\delta(k)^{2}/\mu_{k}=t^{2}s_{k}^{2}w(k)^{2}/\mu_{k}=t^{2}\,w(k)^{2}/\mu_{k}$, so by (W2) and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear the family $k\mapsto\delta(k)^{2}/\mu_{k}$ is cube-summable with lattice sum $t^{2}A$; by the Conventions $\delta\in H^{-1}$, and $z=x+\delta\in H^{-1}$ as $H^{-1}$ is a linear subspace. The vector $x-z$ is the family $k\mapsto-\delta(k)$, and $(-\delta(k))^{2}=\delta(k)^{2}$ by claim 2 of Zero Products and Elementary Identities in a Field; so by the Conventions $d_{H^{-1}}(x,z)^{2}=|x-z|_{H^{-1}}^{2}=t^{2}A=\varepsilon^{2}\cdot A(A+1)^{-1}$. From $A<A+1$ (claim 1 of Elementary Order Arithmetic in an Ordered Field, from $0<1$) and $0<(A+1)^{-1}$, claim 10 of that lemma gives $A(A+1)^{-1}<(A+1)(A+1)^{-1}=1$, and again, with $0<\varepsilon^{2}$ (claim 5), $\varepsilon^{2}A(A+1)^{-1}<\varepsilon^{2}$. As $0\le d_{H^{-1}}(x,z)$ and $0\le\varepsilon$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field yields $d_{H^{-1}}(x,z)<\varepsilon$.

$z\notin\mathcal{W}$. Fix $k$. By claim 5 of Zero Products and Elementary Identities in a Field, $z(k)^{2}=x(k)^{2}+\bigl(x(k)\delta(k)+x(k)\delta(k)\bigr)+\delta(k)^{2}$. Here $x(k)\delta(k)=t\,w(k)\,\bigl(s_{k}x(k)\bigr)$ is nonnegative by (Nn) applied twice (with $0\le t$, $0\le w(k)$, $0\le s_{k}x(k)$), so $0\le x(k)\delta(k)+x(k)\delta(k)$ by claim 2 of Elementary Arithmetic in an Ordered Field; and $\delta(k)^{2}=t^{2}w(k)^{2}$. Hence, by claim 3 of that lemma, $\bigl(x(k)^{2}-c_{k}\bigr)+t^{2}w(k)^{2}\le z(k)^{2}-c_{k}$. Summing over $\Gamma_{N}$ with Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison and claims 3 and 4 of Properties of a Sum over a Finite Index Set, and using The Wick Square with a Mode Cutoff and the Wick Domain §cutoff,

$${:}x^{2}{:}_{N}+t^{2}S_{N}(w^{2})\le{:}z^{2}{:}_{N}\qquad(N\in\mathbb{N}).$$

Suppose $z\in\mathcal{W}$. Then $({:}x^{2}{:}_{N})_{N}$ and $({:}z^{2}{:}_{N})_{N}$ both converge (The Wick Square with a Mode Cutoff and the Wick Domain §domain), hence are bounded by claim 2 of Uniqueness of Limits and Boundedness of Convergent Real Sequences: by Bounded Sequence of Real Numbers there are reals $R_{1},R_{2}>0$ with $|{:}x^{2}{:}_{N}|\le R_{1}$ and $|{:}z^{2}{:}_{N}|\le R_{2}$ for all $N$. By claims 2 and 3 of Properties of the Absolute Value in an Ordered Field and claim 4 of Elementary Order Arithmetic in an Ordered Field, ${:}z^{2}{:}_{N}\le R_{2}$ and $-{:}x^{2}{:}_{N}\le R_{1}$, so, adding (claims 2 and 3 of Elementary Arithmetic in an Ordered Field), $t^{2}S_{N}(w^{2})\le{:}z^{2}{:}_{N}-{:}x^{2}{:}_{N}\le R_{1}+R_{2}$. Since $0<t^{2}$ (claim 5 of Elementary Order Arithmetic in an Ordered Field), $0<(t^{2})^{-1}$ by claim 7 of that lemma, and claim 5 of Elementary Arithmetic in an Ordered Field gives $S_{N}(w^{2})\le(t^{2})^{-1}(R_{1}+R_{2})$ for every $N$, so $\{S_{N}(w^{2}):N\in\mathbb{N}\}$ is bounded above (Upper Bound and Least Upper Bound), contradicting (W3). Hence $z\notin\mathcal{W}$, and $z$ has the required properties.

Clause 5 (counterterm). Let $b$, $\varphi$ and $x$ be as stated. By Clause 1, $\varphi$ is twice differentiable along the modes and $(F_{N}[\varphi](x))_{N}$ converges. Subtracting the two identities of The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §penalised, for every $N\in\mathbb{N}$

$$F_{N}^{b}[\varphi](x)=F_{N}[\varphi](x)-\beta\,S_{N}(c-b),$$

where $c-b$ is the family $k\mapsto c_{k}-b(k)$. If $c-b$ is cube-summable, i.e. $(S_{N}(c-b))_{N}$ converges (Cube Sums of Families on the Integer Lattice §lattice-sum), then $(F_{N}^{b}[\varphi](x))_{N}$ converges by claims 1 and 3 of Arithmetic of Limits of Real Sequences. Conversely, if $(F_{N}^{b}[\varphi](x))_{N}$ converges, then, as $\beta\ne0$ (since $0<\beta$, The Wick-Square Problem on the Torus: Standing Notation §parameters), $S_{N}(c-b)=\beta^{-1}\bigl(F_{N}[\varphi](x)-F_{N}^{b}[\varphi](x)\bigr)$ converges by claim 3 of Arithmetic of Limits of Real Sequences, so $c-b$ is cube-summable.

Now let $2\le n$ and $b$ be the zero family, so that $F_{N}^{b}=F_{N}^{0}$ (The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm §bare) and $c-b=c$. By (Wt) and claim 4 of Properties of a Sum over a Finite Index Set, $\beta S_{N}(c)=\kappa\,T_{N}$ with $\kappa=\beta\nu\,2^{-1}$ and $T_{N}=\sum_{k\in\Gamma_{N}}1/\mu_{k}$; here $0<\kappa$ by claims 8 and 5 of Elementary Order Arithmetic in an Ordered Field, since $0<\beta$ and $0<\nu$. Suppose $(F_{N}^{0}[\varphi](x))_{N}$ were bounded below, i.e. (The Real Numbers: Standing Notation and Background §bounds, Lower Bound and Greatest Lower Bound in a Totally Ordered Set) there were a real $\ell$ with $\ell\le F_{N}^{0}[\varphi](x)$ for all $N$. Since $(F_{N}[\varphi](x))_{N}$ converges, it is bounded (claim 2 of Uniqueness of Limits and Boundedness of Convergent Real Sequences): there is $R>0$ with $|F_{N}[\varphi](x)|\le R$, hence $F_{N}[\varphi](x)\le R$ by claim 3 of Properties of the Absolute Value in an Ordered Field, for all $N$. With $-F_{N}^{0}[\varphi](x)\le-\ell$ (claim 4 of Elementary Order Arithmetic in an Ordered Field) and claims 2 and 3 of Elementary Arithmetic in an Ordered Field, $\kappa T_{N}=F_{N}[\varphi](x)-F_{N}^{0}[\varphi](x)\le R-\ell$, and, multiplying by $\kappa^{-1}$, which is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field gives $T_{N}\le\kappa^{-1}(R-\ell)$ for every $N$. So $\{T_{N}:N\in\mathbb{N}\}$ would be bounded above, contradicting Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §divergent. Hence $(F_{N}^{0}[\varphi](x))_{N}$ is not bounded below.