Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $q,N,p\in\mathbb{N}$. Block maps $\mathfrak{p}_{k}$, configurations, product maps $h^{\oplus}$ and diagonal points $a^{\oplus}$ are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks and the clauses just cited, formed in dimension $q$, and in dimension $p$ for configurations in $\mathbb{R}^{pN}$. Continuity and Lipschitz bounds refer to the Euclidean distances.

1. (Linearity)¶ Each $\mathfrak{p}_{k}$ is linear and Borel. Every $x\in\mathbb{R}^{qN}$ satisfies $x=[\mathfrak{p}_{1}(x),\dots,\mathfrak{p}_{N}(x)]$, and $[y_{1}+ty'_{1},\dots,y_{N}+ty'_{N}]=[y_{1},\dots,y_{N}]+t\,[y'_{1},\dots,y'_{N}]$ for $y_{k},y'_{k}\in\mathbb{R}^{q}$ and $t\in\mathbb{R}$.

2. (Inner products)¶ For $x,x'\in\mathbb{R}^{qN}$,

and in particular $\lVert\mathfrak{p}_{k}(x)\rVert\le\lVert x\rVert$ for every $k\in[N]$.

3. (Product maps)¶ Let $h:\mathbb{R}^{q}\to\mathbb{R}^{p}$. Then $\mathfrak{p}_{k}\circ h^{\oplus}=h\circ\mathfrak{p}_{k}$ for every $k\in[N]$, the block map on the left being that of $\mathbb{R}^{pN}$. If $h$ is Borel, so is $h^{\oplus}$; if $h$ is continuous, so is $h^{\oplus}$; and if $h$ is Lipschitz with constant $L$, so is $h^{\oplus}$.

4. (Diagonal shifts)¶ For $a\in\mathbb{R}^{q}$ and $x\in\mathbb{R}^{qN}$: $\mathfrak{p}_{k}(x+a^{\oplus})=\mathfrak{p}_{k}(x)+a$ for every $k\in[N]$, $a^{\oplus}\cdot x=a\cdot\sum_{k=1}^{N}\mathfrak{p}_{k}(x)$, and $\lVert a^{\oplus}\rVert^{2}=N\lVert a\rVert^{2}$.

5. (Adding a particle)¶ Let $n\in\mathbb{N}$, write $\mathfrak{p}^{(n)}_{k}$ for the block maps of $\mathbb{R}^{qn}$ (the case $N=n$) and $\mathfrak{p}^{(n+1)}_{k}$ for those of $\mathbb{R}^{q(n+1)}=\mathbb{R}^{qn+q}$, and let $\iota^{qn,q}$ be the concatenation. For $u\in\mathbb{R}^{qn}$ and $v\in\mathbb{R}^{q}$,