Pulling Back Intrinsic Test Functions along Tensor Powers

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Intrinsic test functions, their gradients along couplings and their translation Hessians are those of the setting, read at the particle dimension on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and at the configuration level on $\mathcal{P}_{2}(\mathbb{R}^{dN})$. Tensor powers of measures in $\mathcal{P}_{2}(\mathbb{R}^{d})$ lie in $\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments, and $(\mu^{\otimes N})^{[1]}=\mu$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor; for $Q\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $\Pi_{Q}:L^{2}(Q;\mathbb{R}^{dN})\to T_{Q^{[1]}}$ is the one-particle projection, and $a^{\oplus}\in\mathbb{R}^{dN}$ is the diagonal point of $a\in\mathbb{R}^{d}$.

Let $\mathcal{Q}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\mathcal{Q}_{N}\subseteq\mathcal{P}_{2}(\mathbb{R}^{dN})$ satisfy $\mu^{\otimes N}\in\mathcal{Q}_{N}$ for every $\mu\in\mathcal{Q}$, let $\Phi$ be an intrinsic test function on $\mathcal{Q}_{N}$ at the configuration level, and let $\varphi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ have the value $\varphi(\mu)=\Phi(\mu^{\otimes N})$.

1. (Test function)¶ $\varphi$ is an intrinsic test function on $\mathcal{Q}$, and for every $\mu\in\mathcal{Q}$ its gradient along couplings is

2. (Translation Hessian)¶ For every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and every $a\in\mathbb{R}^{d}$,