Proof of The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant

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

For $\nu\in\mathcal{D}_{\Sigma}$ and $k\in\mathbb{N}$ write $m_{k}(\nu)=\int_{X}x_{k}^{2}\,\nu(dx)$, a real number as noted in the statement. The variances $c_{k}$ are positive by Variance Sequences and Their Truncations §variances, $c$ being a variance sequence by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian. Let $\varpi$ be the sequence with $\varpi_{k}=w_{k}c_{k}/(\beta a_{k})$ of The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space §corrector, which is admissible by that clause.

Clause 1. Let $\nu\in\mathcal{D}_{\Sigma}$. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain and The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain, $\nu\in\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$, and $\nu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, with noise score field $Z^{a}_{\nu}$. By The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space §cost and The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space §corrector (which gives $\nabla\Pi_{w}(\nu)=V_{\varpi}(\nu)$ for $\nu\in\mathcal{D}$),

$$G_{w}(\nu)=\beta\,\langle Z^{a}_{\nu},V_{\varpi}(\nu)\rangle_{\nu}.$$

By The Gaussian Relative Score Paired with the Gradient Field of a Diagonal Quadratic Profile §pairing applied with $b=\varpi$, the series

$$\sum_{k=1}^{\infty}a_{k}\,\varpi_{k}\Bigl(\frac{m_{k}(\nu)}{c_{k}}-1\Bigr)$$

converges with sum $\langle Z^{a}_{\nu},V_{\varpi}(\nu)\rangle_{\nu}$. Since $a_{k}\varpi_{k}=w_{k}c_{k}/\beta$ and $c_{k}\ne0$, its $k$-th term is

$$\frac{w_{k}c_{k}}{\beta}\cdot\frac{m_{k}(\nu)-c_{k}}{c_{k}}=\beta^{-1}\,w_{k}\bigl(m_{k}(\nu)-c_{k}\bigr).$$

Multiplying by $\beta$, Elementary Properties of Series of Real Numbers §linearity shows that $\sum_{k=1}^{\infty}w_{k}(m_{k}(\nu)-c_{k})$ converges, with sum $\beta\,\langle Z^{a}_{\nu},V_{\varpi}(\nu)\rangle_{\nu}=G_{w}(\nu)$.

Clause 2. For $\nu\in\mathcal{D}_{\Sigma}$ and $N\in\mathbb{N}$, by the arithmetic of finite sums,

$$G_{N}(\nu)-e_{N}=\sum_{k=1}^{N}w_{k}m_{k}(\nu)-C_{N}-\sum_{k=1}^{N}w_{k}c_{k}+C_{N}=\sum_{k=1}^{N}w_{k}\bigl(m_{k}(\nu)-c_{k}\bigr),$$

which is the $N$-th partial sum of the series of clause 1. By clause 1 and Series of Real Numbers §convergent, $\lim_{N\to\infty}(G_{N}(\nu)-e_{N})=G_{w}(\nu)$.

For the equivalences: (ii) implies (i) because $\mathcal{D}_{\Sigma}$ is nonempty, as recorded in the statement. (i) implies (iii): if $(G_{N}(\nu_{0}))_{N\in\mathbb{N}}$ converges for some $\nu_{0}\in\mathcal{D}_{\Sigma}$, then $e_{N}=G_{N}(\nu_{0})-(G_{N}(\nu_{0})-e_{N})$ is the difference of two convergent sequences and converges by the limit law for differences. (iii) implies (ii), together with the final formula: if $e_{N}\to e_{\infty}$, then for every $\nu\in\mathcal{D}_{\Sigma}$, $G_{N}(\nu)=(G_{N}(\nu)-e_{N})+e_{N}$ converges to $G_{w}(\nu)+e_{\infty}$ by Arithmetic of Limits of Real Sequences §sums.

Clause 3. Let $\nu\in\mathcal{D}_{\Sigma}$ and $M\in\mathbb{R}$. For $N\in\mathbb{N}$ put

$$s_{N}=\sum_{k=1}^{N}w_{k}\bigl(m_{k}(\nu)-c_{k}\bigr),\qquad t_{N}=\sum_{k=1}^{N}w_{k}c_{k},\qquad\text{so that}\qquad\sum_{k=1}^{N}w_{k}m_{k}(\nu)=s_{N}+t_{N}.$$

By clause 1 the sequence $(s_{N})_{N\in\mathbb{N}}$ converges, so it is bounded by claim 2 of Uniqueness of Limits and Boundedness of Convergent Real Sequences: there is $B>0$ with $|s_{N}|\le B$, hence $s_{N}\ge-B$, for every $N$. Each $w_{k}c_{k}$ is nonnegative, as $w_{k}\ge0$ and $c_{k}>0$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion, $t_{N}\le t_{N+1}$ for every $N$, and, since $\sum_{k=1}^{\infty}w_{k}c_{k}$ does not converge, the set $\{t_{N}:N\in\mathbb{N}\}$ is not bounded above. Hence $M+B$ is not an upper bound of it: there is $N_{0}\in\mathbb{N}$ with $t_{N_{0}}>M+B$. By induction from $t_{N}\le t_{N+1}$, $t_{N}\ge t_{N_{0}}$ for every $N\ge N_{0}$, and therefore

$$\sum_{k=1}^{N}w_{k}\int_{X}x_{k}^{2}\,\nu(dx)=s_{N}+t_{N}>-B+M+B=M\qquad\text{for every }N\ge N_{0}.$$