Proof of Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes

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

For $\mu\in\mathcal{P}(\mathbb{T}^{d})$ the class of a square-integrable Borel map $G:\mathbb{R}^{d}\to\mathbb{R}^{d}$ in $L^{2}(\mu;\mathbb{R}^{d})$ is written $[G]$; by The Space of Square-Integrable Random Vectors §classes (on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$, see Optimal Transport on the Flat Torus: Standing Notation §fields), $[G]=[H]$ exactly when $\mu(\{y:G(y)=H(y)\})=1$, and $a[G]+b[H]=[aG+bH]$ for real $a,b$. A sequence $(\mu_{n})$ converges to $\mu$ in $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$ exactly when the real sequence $(W_{\mathbb{T}}(\mu_{n},\mu))$ converges to $0$, by Convergent Sequence in a Metric Space.

Step 0 (a Lipschitz bound in the torus distance). Let $u\in C^{1}_{\mathrm{per}}$. We show that there is a real $L_{u}\ge0$ with $|u(y)-u(x)|\le L_{u}\,d_{\mathbb{T}}(x,y)$ for all $x,y\in\mathbb{R}^{d}$. For each $i\in[d]$ the function $\partial_{i}u$ belongs to $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §derivative, so there is a real $M_{i}\ge0$ with $|\partial_{i}u|\le M_{i}$ on $\mathbb{R}^{d}$ by Elementary Properties of Lattice-Periodic Functions §bounded; put $L_{u}=M_{1}+\dots+M_{d}$. Let $x,y\in\mathbb{R}^{d}$, put $v=\varpi(y-x)$ and $z=x+v$. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, $y-z=(y-x)-\varpi(y-x)\in\mathbb{Z}^{d}$, so $u(y)=u(z)$ by the periodicity of $u$; and $\lVert v\rVert=d_{\mathbb{T}}(x,y)$ by The Wrapped Displacement and the Flat Torus Distance §distance. If $v=0$, then $u(y)=u(x)$. Otherwise let $F:[0,1]\to\mathbb{R}$, $F(\tau)=u(x+\tau v)$. Since $u$ is of class $C^{1}$ on the open set $\mathbb{R}^{d}$ (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), Derivatives Along a Segment for C^1 Functions on a Euclidean Open Set (claims 2 and 3, with $J=[0,1]$) shows that $F$ is continuous on $[0,1]$ and differentiable at every $\tau\in(0,1)$ with $F'(\tau)=\sum_{i=1}^{d}\partial_{i}u(x+\tau v)\,v_{i}$. By the mean value theorem there is $c\in(0,1)$ with $u(z)-u(x)=F'(c)$, whence $|u(y)-u(x)|=|F'(c)|\le\sum_{i}M_{i}|v_{i}|\le L_{u}\lVert v\rVert=L_{u}\,d_{\mathbb{T}}(x,y)$.

Consequently, if $G:\mathbb{R}^{d}\to\mathbb{R}^{d}$ has components $G_{1},\dots,G_{d}\in C^{1}_{\mathrm{per}}$, then with $L_{G}=L_{G_{1}}+\dots+L_{G_{d}}$,

$$\lVert G(y)-G(x)\rVert\le\sum_{i=1}^{d}|G_{i}(y)-G_{i}(x)|\le L_{G}\,d_{\mathbb{T}}(x,y)\qquad(x,y\in\mathbb{R}^{d}).$$

Claim 1. Let $g$ and $h$ be gradient fields of $\varphi$.

Step 1 (agreement at heat-smoothed measures). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, let $0<s\le\tfrac12$, let $S_{s}$ be the heat semigroup at time $s$, and put $\nu=S_{s}\mu$; then $\nu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})\subseteq\mathcal{P}(\mathbb{T}^{d})$ by Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §score. By Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §represents, applied to $g$ and to $h$, $[g(\nu,\cdot)]=\nabla\varphi(\nu)=[h(\nu,\cdot)]$, so the set $D=\{y\in\mathbb{R}^{d}:g(\nu,y)\ne h(\nu,y)\}$ satisfies $\nu(D)=0$. For each $i$ the functions $g_{i}(\nu,\cdot)$ and $h_{i}(\nu,\cdot)$ lie in $C^{1}_{\mathrm{per}}$ by Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §regular, hence are continuous (C^k Maps on a Euclidean Open Set, clause 1) and periodic; so $D$, the union over $i\in[d]$ of the sets where $g_{i}(\nu,\cdot)-h_{i}(\nu,\cdot)\ne0$, is open (hence Borel), and $D+k=D$ for $k\in\mathbb{Z}^{d}$.

Suppose that $D$ has a point $y_{0}$. By The Half-Open Unit Cell Tiles Euclidean Space §wrap, $z_{0}=\pi(y_{0})\in Q$ and $y_{0}-z_{0}\in\mathbb{Z}^{d}$, so $z_{0}\in D$, and there is $r>0$ with the open ball $B(z_{0},r)\subseteq D$. Choose a real $\delta$ with $0<\delta<1-(z_{0})_{i}$ for every $i\in[d]$ and $\delta\sqrt{d}<r$, let $c\in\mathbb{R}^{d}$ have coordinates $c_{i}=(z_{0})_{i}+\delta/2$, and let $B'=B(c,\delta/2)$. For $x\in B'$ and $i\in[d]$ one has $|x_{i}-c_{i}|\le\lVert x-c\rVert<\delta/2$, so $0\le(z_{0})_{i}<x_{i}<(z_{0})_{i}+\delta<1$, whence $x\in Q$; and $\lVert x-z_{0}\rVert\le\lVert x-c\rVert+\lVert c-z_{0}\rVert<\delta/2+\delta\sqrt{d}/2\le\delta\sqrt{d}<r$, whence $x\in D$. Thus $B'\subseteq D\cap Q$, and $\lambda_{d}(B')>0$ by Balls Have Positive Lebesgue Measure and Bounded Sets Have Finite Lebesgue Measure (claim 1). Let $p$ be the positive periodic density of Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §density, so that $\nu(B')=\int\mathbf{1}_{B'}\mathbf{1}_{Q}\,p\,d\lambda_{d}=\int\mathbf{1}_{B'}\,p\,d\lambda_{d}$. If this integral were $0$, then $\mathbf{1}_{B'}p=0$ almost everywhere by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing, hence $\mathbf{1}_{B'}=0$ almost everywhere since $p>0$ everywhere, hence $\lambda_{d}(B')=\int\mathbf{1}_{B'}\,d\lambda_{d}=0$ by the same clause, a contradiction. So $0<\nu(B')\le\nu(D)=0$, which is absurd. Therefore $D$ is empty: $g(S_{s}\mu,\cdot)=h(S_{s}\mu,\cdot)$ on $\mathbb{R}^{d}$.

Step 2 (general measures). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $x\in\mathbb{R}^{d}$. For $n\in\mathbb{N}$ put $s_{n}=1/(2(n+1))\in(0,\tfrac12]$ and $\mu_{n}=S_{s_{n}}\mu$. By Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §distance, $W_{\mathbb{T}}(\mu_{n},\mu)^{2}\le d\,s_{n}$, which tends to $0$; so $W_{\mathbb{T}}(\mu_{n},\mu)^{2}\to0$ by claim 3 of Order Properties of Limits of Real Sequences (with $L=0$), and hence $W_{\mathbb{T}}(\mu_{n},\mu)=\bigl(W_{\mathbb{T}}(\mu_{n},\mu)^{2}\bigr)^{1/2}\to0^{1/2}=0$ by Properties of Real Powers of Nonnegative Real Numbers §continuity (the first equality by Properties of Real Powers of Nonnegative Real Numbers §agreement and Properties of Real Powers of Nonnegative Real Numbers §exponents, the last by Properties of Real Powers of Nonnegative Real Numbers §values); so $(\mu_{n})$ converges to $\mu$. By Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §joint-continuity, applied to $g$ and to $h$ with the constant sequence $x_{n}=x$, for each $i\in[d]$ the sequences $(g_{i}(\mu_{n},x))_{n}$ and $(h_{i}(\mu_{n},x))_{n}$ converge to $g_{i}(\mu,x)$ and $h_{i}(\mu,x)$ respectively. By Step 1 the two sequences coincide term by term, and a real sequence has only one limit (claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences), so $g_{i}(\mu,x)=h_{i}(\mu,x)$. Hence $g=h$, proving claim 1.

Claim 2. Let $\varphi$ and $g$ be as in the hypotheses of claim 2 of the statement; for $\mu\in\mathcal{P}(\mathbb{T}^{d})$ we have $\nabla\varphi(\mu)=[g(\mu,\cdot)]$ by Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §represents. We verify the three clauses of Intrinsic Test Functions on the Torus Wasserstein Space §test with $\mathcal{K}=\mathcal{P}(\mathbb{T}^{d})$.

Step 3 (continuity). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $\varepsilon>0$, and put $\eta=\nabla\varphi(\mu)$. By Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §represents and Differentiability Along Couplings of a Function on the Torus Wasserstein Space, and Its Gradient §differentiable, applied with the value $1$ in place of $\varepsilon$, there is $\theta>0$ such that $|\varphi(\nu)-\varphi(\mu)-\mathcal{J}_{\mathbb{T}}(\eta,\gamma)|\le\sqrt{I_{\mathbb{T}}(\gamma)}$ whenever $\gamma\in\Pi(\mu,\nu)$ and $I_{\mathbb{T}}(\gamma)<\theta^{2}$. Let $\delta$ be the smaller of $\theta$ and $\varepsilon/(\lVert\eta\rVert_{\mu}+2)$, and let $\nu\in\mathcal{P}(\mathbb{T}^{d})$ with $W_{\mathbb{T}}(\mu,\nu)<\delta$. By The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §optimal there is an optimal $\gamma\in\Pi(\mu,\nu)$, so $I_{\mathbb{T}}(\gamma)=W_{\mathbb{T}}(\mu,\nu)^{2}<\theta^{2}$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §optimal. By the triangle inequality and The Torus Displacement Pairing: Linearity, the Cost Bound, and Vanishing of a Field with First-Order Small Pairings §bound,

$$|\varphi(\nu)-\varphi(\mu)|\le|\mathcal{J}_{\mathbb{T}}(\eta,\gamma)|+W_{\mathbb{T}}(\mu,\nu)\le(\lVert\eta\rVert_{\mu}+1)\,W_{\mathbb{T}}(\mu,\nu)<(\lVert\eta\rVert_{\mu}+1)\,\delta<\varepsilon .$$

So $\varphi$ is continuous at every $\mu$, which is Intrinsic Test Functions on the Torus Wasserstein Space §continuity.

Step 4 (differentiability). For every $\mu$, $\varphi$ is differentiable along couplings at $\mu$ by Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §represents, and $\nabla\varphi(\mu)=[g(\mu,\cdot)]\in T_{\mu}$ by hypothesis. This is Intrinsic Test Functions on the Torus Wasserstein Space §differentiability.

Step 5 (uniform convergence of the fields). Let $(\mu_{n})$ converge to $\mu$ in $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$. We show: for every $\varepsilon>0$ there is $N\in\mathbb{N}$ with $\lVert g(\mu_{n},x)-g(\mu,x)\rVert<\varepsilon$ for all $n\ge N$ and all $x\in\mathbb{R}^{d}$. Suppose not. Then there is $\varepsilon>0$ such that for every $N$ there are $n\ge N$ and $x\in\mathbb{R}^{d}$ with $\lVert g(\mu_{n},x)-g(\mu,x)\rVert\ge\varepsilon$. Choosing successively $N=0$ and then $N=n_{k-1}+1$, we obtain a strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ and points $x_{k}$ with $\lVert g(\mu_{n_{k}},x_{k})-g(\mu,x_{k})\rVert\ge\varepsilon$. Put $y_{k}=\pi(x_{k})$; by The Half-Open Unit Cell Tiles Euclidean Space §wrap, $y_{k}\in Q$ and $x_{k}-y_{k}\in\mathbb{Z}^{d}$, so by periodicity of the components (Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §regular) $\lVert g(\mu_{n_{k}},y_{k})-g(\mu,y_{k})\rVert\ge\varepsilon$. Every point of $Q$ has coordinates in $[0,1)$, hence norm at most $\sqrt{d}$, so $Q$ is bounded; by Bolzano-Weierstrass Theorem in Euclidean Space there are $y^{*}\in\mathbb{R}^{d}$ and a strictly increasing sequence $(k_{j})_{j\in\mathbb{N}}$ with $y_{k_{j}}\to y^{*}$. The sequence $(n_{k_{j}})_{j}$ is strictly increasing, so $(\mu_{n_{k_{j}}})_{j}$ converges to $\mu$ by A Subsequence of a Convergent Sequence Has the Same Limit. By Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §joint-continuity, applied to the sequences $(\mu_{n_{k_{j}}})_{j}$ and $(y_{k_{j}})_{j}$, and again to the constant sequence $(\mu)_{j}$ and $(y_{k_{j}})_{j}$, for each $i\in[d]$ both $g_{i}(\mu_{n_{k_{j}}},y_{k_{j}})$ and $g_{i}(\mu,y_{k_{j}})$ converge to $g_{i}(\mu,y^{*})$ as $j\to\infty$. Hence $\lVert g(\mu_{n_{k_{j}}},y_{k_{j}})-g(\mu,y_{k_{j}})\rVert\le\sum_{i}|g_{i}(\mu_{n_{k_{j}}},y_{k_{j}})-g_{i}(\mu,y_{k_{j}})|$, which tends to $0$ by claims 1 and 3 of Arithmetic of Limits of Real Sequences and claim 4 of Order Properties of Limits of Real Sequences, contradicting the lower bound $\varepsilon$.

Step 6 (gradient continuity). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, let $(\mu_{n})$ be a sequence in $\mathcal{P}(\mathbb{T}^{d})$, and let $\gamma_{n}\in\Pi(\mu_{n},\mu)$ with $I_{\mathbb{T}}(\gamma_{n})\to0$. Since $W_{\mathbb{T}}(\mu_{n},\mu)^{2}\le I_{\mathbb{T}}(\gamma_{n})$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance, $(\mu_{n})$ converges to $\mu$. Use the representatives $g(\mu_{n},\cdot)$ and $g(\mu,\cdot)$, and let $A_{n}$ denote the $n$th term of the sequence in Intrinsic Test Functions on the Torus Wasserstein Space §gradient-continuity. By Step 0 there is $L\ge0$ with $\lVert g(\mu,y)-g(\mu,x)\rVert\le L\,d_{\mathbb{T}}(x,y)$ for all $x,y$. Let $\varepsilon>0$. First choose $N_{1}$ as in Step 5 for this $\varepsilon$; then, since $I_{\mathbb{T}}(\gamma_{n})\to0$, choose $N_{2}$ with $L^{2}I_{\mathbb{T}}(\gamma_{n})<\varepsilon^{2}$ for $n\ge N_{2}$. For $n$ at least the larger of $N_{1}$ and $N_{2}$ and $w\in\mathbb{R}^{d+d}$, with $x=\mathrm{pr}_{1}(w)$ and $y=\mathrm{pr}_{2}(w)$, the second inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (applied to $g(\mu_{n},x)-g(\mu,y)$ and $g(\mu_{n},x)-g(\mu,x)$) gives

$$\lVert g(\mu_{n},x)-g(\mu,y)\rVert^{2}\le2\lVert g(\mu_{n},x)-g(\mu,x)\rVert^{2}+2\lVert g(\mu,x)-g(\mu,y)\rVert^{2}\le2\varepsilon^{2}+2L^{2}d_{\mathbb{T}}(x,y)^{2}.$$

Integrating against the probability measure $\gamma_{n}$, by linearity and monotonicity of the integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) and Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost, $0\le A_{n}\le2\varepsilon^{2}+2L^{2}I_{\mathbb{T}}(\gamma_{n})<4\varepsilon^{2}$. As $\varepsilon>0$ was arbitrary, $A_{n}\to0$, which is Intrinsic Test Functions on the Torus Wasserstein Space §gradient-continuity.

Step 7 (conclusion of claim 2). By Steps 3, 4 and 6, $\varphi$ is an intrinsic test function on $\mathcal{P}(\mathbb{T}^{d})$, and $g$ is a gradient field of $\varphi$; so $\varphi$ is a Laplacian test function by Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §test.

Claim 3. Let $\chi=a\varphi+b\psi$, and put $c=|a|+|b|+1$.

Step 8 (continuity). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $\varepsilon>0$. By Intrinsic Test Functions on the Torus Wasserstein Space §continuity for $\varphi$ and $\psi$ there are $\delta_{1},\delta_{2}>0$ with $|\varphi(\nu)-\varphi(\mu)|<\varepsilon/c$ when $W_{\mathbb{T}}(\mu,\nu)<\delta_{1}$ and $|\psi(\nu)-\psi(\mu)|<\varepsilon/c$ when $W_{\mathbb{T}}(\mu,\nu)<\delta_{2}$. For $W_{\mathbb{T}}(\mu,\nu)$ below the smaller of $\delta_{1},\delta_{2}$, $|\chi(\nu)-\chi(\mu)|\le|a|\,|\varphi(\nu)-\varphi(\mu)|+|b|\,|\psi(\nu)-\psi(\mu)|\le(|a|+|b|)\varepsilon/c<\varepsilon$. So $\chi$ is continuous.

Step 9 (differentiability). Let $\mu\in\mathcal{K}$ and put $\eta=a\nabla\varphi(\mu)+b\nabla\psi(\mu)\in L^{2}(\mu;\mathbb{R}^{d})$. Let $\varepsilon>0$; by Intrinsic Test Functions on the Torus Wasserstein Space §differentiability and Differentiability Along Couplings of a Function on the Torus Wasserstein Space, and Its Gradient §differentiable, applied with $\varepsilon/c$, there are $\theta_{1},\theta_{2}>0$ for $\varphi$ and $\psi$; let $\theta$ be the smaller. For $\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma\in\Pi(\mu,\nu)$ with $I_{\mathbb{T}}(\gamma)<\theta^{2}$, The Torus Displacement Pairing: Linearity, the Cost Bound, and Vanishing of a Field with First-Order Small Pairings §linear gives

$$\chi(\nu)-\chi(\mu)-\mathcal{J}_{\mathbb{T}}(\eta,\gamma)=a\bigl(\varphi(\nu)-\varphi(\mu)-\mathcal{J}_{\mathbb{T}}(\nabla\varphi(\mu),\gamma)\bigr)+b\bigl(\psi(\nu)-\psi(\mu)-\mathcal{J}_{\mathbb{T}}(\nabla\psi(\mu),\gamma)\bigr),$$

whose absolute value is at most $(|a|+|b|)(\varepsilon/c)\sqrt{I_{\mathbb{T}}(\gamma)}\le\varepsilon\sqrt{I_{\mathbb{T}}(\gamma)}$. Hence $\chi$ is differentiable along couplings at $\mu$ with gradient $\eta$, so $\nabla\chi(\mu)=a\nabla\varphi(\mu)+b\nabla\psi(\mu)$ by the uniqueness in Differentiability Along Couplings of a Function on the Torus Wasserstein Space, and Its Gradient §gradient; and $\nabla\chi(\mu)\in T_{\mu}$ because $\nabla\varphi(\mu),\nabla\psi(\mu)\in T_{\mu}$ and $T_{\mu}$ is a linear subspace by The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients §subspace.

Step 10 (gradient continuity). Let $\mu$, $(\mu_{n})$ in $\mathcal{K}$ and $\gamma_{n}\in\Pi(\mu_{n},\mu)$ with $I_{\mathbb{T}}(\gamma_{n})\to0$. Choose Borel representatives $G_{n},G,H_{n},H$ of $\nabla\varphi(\mu_{n}),\nabla\varphi(\mu),\nabla\psi(\mu_{n}),\nabla\psi(\mu)$; then $aG_{n}+bH_{n}$ and $aG+bH$ represent $\nabla\chi(\mu_{n})$ and $\nabla\chi(\mu)$ by Step 9. For $w\in\mathbb{R}^{d+d}$, with $x=\mathrm{pr}_{1}(w)$, $y=\mathrm{pr}_{2}(w)$, the first inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (applied to $a(G_{n}(x)-G(y))$ and $-b(H_{n}(x)-H(y))$) gives

$$\lVert(aG_{n}+bH_{n})(x)-(aG+bH)(y)\rVert^{2}\le2a^{2}\lVert G_{n}(x)-G(y)\rVert^{2}+2b^{2}\lVert H_{n}(x)-H(y)\rVert^{2}.$$

Integrating against $\gamma_{n}$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), the $n$th term for $\chi$ in Intrinsic Test Functions on the Torus Wasserstein Space §gradient-continuity lies between $0$ and $2a^{2}$ times the $n$th term for $\varphi$ plus $2b^{2}$ times the $n$th term for $\psi$; the latter two sequences converge to $0$ by that clause for $\varphi$ and $\psi$, so, by claims 1 and 3 of Arithmetic of Limits of Real Sequences and the squeeze claim (claim 2) of Order Properties of Limits of Real Sequences, the former does too. With Steps 8 and 9, $\chi$ is an intrinsic test function on $\mathcal{K}$ with $\nabla\chi(\mu)=a\nabla\varphi(\mu)+b\nabla\psi(\mu)$.

Step 11 (the Laplacian case). Let $\varphi,\psi$ be Laplacian test functions with gradient fields $g,h$. By Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §test they are intrinsic test functions on $\mathcal{P}(\mathbb{T}^{d})$, so by Steps 8 to 10 with $\mathcal{K}=\mathcal{P}(\mathbb{T}^{d})$, $\chi$ is one as well. We check that $k=ag+bh$ is a gradient field of $\chi$. Clause (a): $k_{i}(\mu,\cdot)=a\,g_{i}(\mu,\cdot)+b\,h_{i}(\mu,\cdot)\in C^{1}_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §algebra. Clause (b): $\chi$ is differentiable along couplings at every $\mu$ with $\nabla\chi(\mu)=a[g(\mu,\cdot)]+b[h(\mu,\cdot)]=[k(\mu,\cdot)]$, by Step 9 and Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §represents for $g$ and $h$. Clause (c): by claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $\partial_{j}k_{i}(\mu,x)=a\,\partial_{j}g_{i}(\mu,x)+b\,\partial_{j}h_{i}(\mu,x)$, so if $\mu_{n}\to\mu$ and $x_{n}\to x$, then $k_{i}(\mu_{n},x_{n})$ and $\partial_{j}k_{i}(\mu_{n},x_{n})$ are the corresponding linear combinations of sequences that converge by Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §joint-continuity for $g$ and $h$, and so converge to $k_{i}(\mu,x)$ and $\partial_{j}k_{i}(\mu,x)$. Hence $\chi$ is a Laplacian test function with gradient field $ag+bh$ by Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §test. This proves claim 3.