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

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

Points of $\mathbb{R}^{m}$ are maps $[m]\to\mathbb{R}$ by Euclidean Points as Tuples of Real Numbers, as fixed in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces; equality of points is tested coordinatewise by claim 1 of Euclidean Points as Tuples of Real Numbers ("extensionality"). Unfolding Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks, for $x\in\mathbb{R}^{qN}$, $k\in[N]$ and $i\in[q]$,

where $b(k,i)\in[qN]$ by Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §range; the same holds in $\mathbb{R}^{pN}$ with $p$ in place of $q$, Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions being applied with $p$ in place of $q$. Every step below proved for $q$ holds likewise for $p$. Field identities in $\mathbb{R}$ are the numbered axioms of Field.

Step 1 (Blocks determine points). Let $x,x'\in\mathbb{R}^{qN}$ with $\mathfrak{p}_{k}(x)=\mathfrak{p}_{k}(x')$ for every $k\in[N]$. For $j\in[qN]$, Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection gives $k\in[N]$ and $i\in[q]$ with $b(k,i)=j$, and by (1) $x_{j}=(\mathfrak{p}_{k}(x))_{i}=(\mathfrak{p}_{k}(x'))_{i}=x'_{j}$. By extensionality $x=x'$.

Step 2 (Blocks of sums, multiples and differences). Let $x,x'\in\mathbb{R}^{qN}$, $t\in\mathbb{R}$ and $k\in[N]$. For $i\in[q]$, by (1) and Sum of Points of $\mathbb{R}^n$ (applied in $\mathbb{R}^{qN}$ and in $\mathbb{R}^{q}$), $(\mathfrak{p}_{k}(x+x'))_{i}=x_{b(k,i)}+x'_{b(k,i)}=(\mathfrak{p}_{k}(x)+\mathfrak{p}_{k}(x'))_{i}$; likewise by Scalar Multiple of a Point of $\mathbb{R}^n$, $(\mathfrak{p}_{k}(tx))_{i}=(t\,\mathfrak{p}_{k}(x))_{i}$, and by clause 1 of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, $(\mathfrak{p}_{k}(x-x'))_{i}=(\mathfrak{p}_{k}(x)-\mathfrak{p}_{k}(x'))_{i}$. By extensionality,

Step 3 (Claim 1). $\mathbb{R}^{qN}$ and $\mathbb{R}^{q}$ are real vector spaces by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and Step 2 gives conditions 1 and 2 of Linear Map, so $\mathfrak{p}_{k}$ is linear. For $i\in[q]$ the $i$-th component of $\mathfrak{p}_{k}$ is, by (1), the coordinate projection $\pi_{b(k,i)}:\mathbb{R}^{qN}\to\mathbb{R}$, measurable with respect to $\mathcal{B}_{qN}$ and $\mathcal{B}(\mathbb{R})$ by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets; by claim 2 there, $\mathfrak{p}_{k}$ is measurable with respect to $\mathcal{B}_{qN}$ and $\mathcal{B}_{q}$, which are $\mathcal{B}(\mathbb{R}^{qN})$ and $\mathcal{B}(\mathbb{R}^{q})$ by claim 5 there; so $\mathfrak{p}_{k}$ is Borel in the sense of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. Next, for $x\in\mathbb{R}^{qN}$, Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration gives $\mathfrak{p}_{k}([\mathfrak{p}_{1}(x),\dots,\mathfrak{p}_{N}(x)])=\mathfrak{p}_{k}(x)$ for every $k\in[N]$, so $x=[\mathfrak{p}_{1}(x),\dots,\mathfrak{p}_{N}(x)]$ by Step 1. Finally, for $y_{k},y'_{k}\in\mathbb{R}^{q}$ and $t\in\mathbb{R}$, write $Y=[y_{1},\dots,y_{N}]$ and $Y'=[y'_{1},\dots,y'_{N}]$. For every $k\in[N]$, Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration gives $\mathfrak{p}_{k}([y_{1}+ty'_{1},\dots,y_{N}+ty'_{N}])=y_{k}+ty'_{k}$, while Step 2 and the same clause give $\mathfrak{p}_{k}(Y+tY')=\mathfrak{p}_{k}(Y)+t\,\mathfrak{p}_{k}(Y')=y_{k}+ty'_{k}$. Step 1 gives $[y_{1}+ty'_{1},\dots,y_{N}+ty'_{N}]=Y+tY'$.

Step 4 (Claim 2). Let $x,x'\in\mathbb{R}^{qN}$. By clause 2 of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, $x\cdot x'=\sum_{j=1}^{qN}x_{j}x'_{j}$. Applying Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §sums to the family $a_{j}=x_{j}x'_{j}$, then (1) and clause 2 of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$ in $\mathbb{R}^{q}$,

Taking $x'=x$ and using $\lVert z\rVert^{2}=z\cdot z$ in $\mathbb{R}^{qN}$ and in $\mathbb{R}^{q}$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) gives $\lVert x\rVert^{2}=\sum_{k=1}^{N}\lVert\mathfrak{p}_{k}(x)\rVert^{2}$. Each summand is nonnegative by claim 2 of Nonnegativity of Squares in an Ordered Field, so claim 6 of Properties of Finite Sums gives $\lVert\mathfrak{p}_{k}(x)\rVert^{2}\le\lVert x\rVert^{2}$; as $0\le\lVert\mathfrak{p}_{k}(x)\rVert$ and $0\le\lVert x\rVert$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert\mathfrak{p}_{k}(x)\rVert\le\lVert x\rVert$. Combined with Step 2, for $x,x'\in\mathbb{R}^{qN}$ and $k\in[N]$,

Step 5 (Claim 3: blocks of $h^{\oplus}$). Let $h:\mathbb{R}^{q}\to\mathbb{R}^{p}$ and $x\in\mathbb{R}^{qN}$. By Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map, $h^{\oplus}(x)=[h(\mathfrak{p}_{1}(x)),\dots,h(\mathfrak{p}_{N}(x))]\in\mathbb{R}^{pN}$, so Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration, formed in dimension $p$, gives $\mathfrak{p}_{k}(h^{\oplus}(x))=h(\mathfrak{p}_{k}(x))$ for every $k\in[N]$. With (1) in dimension $p$,

For $j\in[pN]$ write $j=b(k,i)$ with $k\in[N]$, $i\in[p]$, which is possible by Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection in dimension $p$; by (3) the $j$-th component of $h^{\oplus}$ is $f_{j}=\pi_{i}\circ h\circ\mathfrak{p}_{k}$, where $\pi_{i}:\mathbb{R}^{p}\to\mathbb{R}$ is the $i$-th coordinate projection.

Step 6 (Claim 3: Borel). Let $h$ be Borel and $j\in[pN]$, $f_{j}=\pi_{i}\circ h\circ\mathfrak{p}_{k}$ as in Step 5. The map $\mathfrak{p}_{k}$ is Borel by Step 3, and $\pi_{i}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{p})$ and $\mathcal{B}(\mathbb{R})$ by claims 1 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. Since $\mathcal{B}(\mathbb{R}^{m})$ is the Borel $\sigma$-algebra of the metric space $(\mathbb{R}^{m},d_{E})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces, claim 4 of Borel Measurability and Bounded Integration on a Metric Space, applied first with $Y=\mathfrak{p}_{k}$ and $g=h$ and then with $Y=h\circ\mathfrak{p}_{k}$ and $g=\pi_{i}$, shows that $f_{j}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{qN})$ and $\mathcal{B}(\mathbb{R})$. By claims 2 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, $h^{\oplus}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{qN})$ and $\mathcal{B}(\mathbb{R}^{pN})$, that is, Borel.

Step 7 (Claim 3: continuity). Let $h$ be continuous on $\mathbb{R}^{q}$ in the sense of Continuous Map Between Metric Spaces, and fix $x\in\mathbb{R}^{qN}$. Recall that $d_{E}(z,z')=\lVert z-z'\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, that $d_{E}(z,z')=d_{E}(z',z)$ by condition 3 of Metric Space and Euclidean Distance is a Metric on $\mathbb{R}^n$, and that $\lVert z-z'\rVert^{2}=\sum_{l}(z_{l}-z'_{l})^{2}$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and clause 1 of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$.

First, each component $f_{j}$ ($j\in[pN]$, $f_{j}=\pi_{i}\circ h\circ\mathfrak{p}_{k}$ as in Step 5), regarded as a map into $\mathbb{R}^{1}$, is continuous at $x$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces. Let $\varepsilon>0$. By continuity of $h$ at $\mathfrak{p}_{k}(x)$ choose, depending on $\varepsilon$, a real $\delta>0$ such that every $z\in\mathbb{R}^{q}$ with $d_{E}(\mathfrak{p}_{k}(x),z)<\delta$ satisfies $d_{E}(h(z),h(\mathfrak{p}_{k}(x)))<\varepsilon$. Let $y\in\mathbb{R}^{qN}$ with $\sum_{l=1}^{qN}(y_{l}-x_{l})^{2}<\delta^{2}$, that is, $\lVert y-x\rVert^{2}<\delta^{2}$; by the three facts just recalled, $\lVert x-y\rVert=d_{E}(x,y)=d_{E}(y,x)=\lVert y-x\rVert$, so $\lVert x-y\rVert^{2}<\delta^{2}$. Since $0\le\lVert x-y\rVert$ and $0<\delta$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert x-y\rVert<\delta$, and by (2) and claim 2 of Elementary Order Arithmetic in an Ordered Field, $d_{E}(\mathfrak{p}_{k}(x),\mathfrak{p}_{k}(y))=\lVert\mathfrak{p}_{k}(x)-\mathfrak{p}_{k}(y)\rVert<\delta$. Hence $w=h(\mathfrak{p}_{k}(y))-h(\mathfrak{p}_{k}(x))$ satisfies $\lVert w\rVert<\varepsilon$, so $\lVert w\rVert^{2}<\varepsilon^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Now $\lVert w\rVert^{2}=\sum_{i'=1}^{p}w_{i'}^{2}$ with nonnegative summands (claim 2 of Nonnegativity of Squares in an Ordered Field), and $w_{i}=f_{j}(y)-f_{j}(x)$, so claim 6 of Properties of Finite Sums and claim 2 of Elementary Order Arithmetic in an Ordered Field give $(f_{j}(y)-f_{j}(x))^{2}<\varepsilon^{2}$, which by claim 1 of Properties of Finite Sums is $\sum_{l=1}^{1}(f_{j}(y)-f_{j}(x))^{2}<\varepsilon^{2}$.

By Coordinatewise Criterion for Continuity of a Map Between Euclidean Spaces (with $n=qN$, $m=pN$, $E=\mathbb{R}^{qN}$, $f=h^{\oplus}$, $a=x$), $h^{\oplus}$ is continuous at $x$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces. Let $\varepsilon>0$ and choose, depending on $\varepsilon$, the $\delta>0$ provided there. If $y\in\mathbb{R}^{qN}$ and $d_{E}(x,y)<\delta$, then $\lVert y-x\rVert=d_{E}(y,x)=d_{E}(x,y)<\delta$, so $\sum_{l=1}^{qN}(y_{l}-x_{l})^{2}=\lVert y-x\rVert^{2}<\delta^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; hence $\lVert h^{\oplus}(y)-h^{\oplus}(x)\rVert^{2}=\sum_{j=1}^{pN}(f_{j}(y)-f_{j}(x))^{2}<\varepsilon^{2}$, and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $d_{E}(h^{\oplus}(y),h^{\oplus}(x))<\varepsilon$. Thus $h^{\oplus}$ is continuous at $x$ relative to $\mathbb{R}^{qN}$, and, $x$ being arbitrary, continuous on $\mathbb{R}^{qN}$.

Step 8 (Claim 3: Lipschitz). Let $h$ be Lipschitz with constant $L$, a nonnegative real, in the sense of Lipschitz Map Between Metric Spaces, and let $x,x'\in\mathbb{R}^{qN}$, $z=h^{\oplus}(x)-h^{\oplus}(x')$. By Steps 2 and 5 (in dimension $p$), $\mathfrak{p}_{k}(z)=h(\mathfrak{p}_{k}(x))-h(\mathfrak{p}_{k}(x'))$, so Step 4 in $\mathbb{R}^{pN}$ gives $\lVert z\rVert^{2}=\sum_{k=1}^{N}\lVert h(\mathfrak{p}_{k}(x))-h(\mathfrak{p}_{k}(x'))\rVert^{2}$. For each $k$, the Lipschitz bound, claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and Step 2 give

Both sides are nonnegative: the left by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, the right by claim 5 of Elementary Arithmetic in an Ordered Field applied to $0\le\lVert\mathfrak{p}_{k}(x-x')\rVert$ and $0\le L$, with $L\cdot0=0$ by claim 1 of Zero Products and Elementary Identities in a Field. So claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and axioms 5 and 8 of Field give $\lVert h(\mathfrak{p}_{k}(x))-h(\mathfrak{p}_{k}(x'))\rVert^{2}\le L^{2}\lVert\mathfrak{p}_{k}(x-x')\rVert^{2}$. By claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, claim 3 of Properties of Finite Sums and Step 4 in $\mathbb{R}^{qN}$,

As $0\le\lVert z\rVert$ and $0\le L\lVert x-x'\rVert$ (as above), claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert z\rVert\le L\lVert x-x'\rVert$, that is, $d_{E}(h^{\oplus}(x),h^{\oplus}(x'))\le L\,d_{E}(x,x')$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. So $h^{\oplus}$ is Lipschitz with constant $L$.

Step 9 (Claim 4). Let $a\in\mathbb{R}^{q}$ and $x\in\mathbb{R}^{qN}$. By Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §diagonal and Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration, $\mathfrak{p}_{k}(a^{\oplus})=a$ for every $k\in[N]$, so Step 2 gives $\mathfrak{p}_{k}(x+a^{\oplus})=\mathfrak{p}_{k}(x)+a$. By Step 4, $a^{\oplus}\cdot x=\sum_{k=1}^{N}a\cdot v_{k}$ with $v_{k}=\mathfrak{p}_{k}(x)$. The sum $\sum_{k=1}^{N}v_{k}$ is the finite sum of Finite Sum Notation in a Vector Space in the real vector space $\mathbb{R}^{q}$ of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space. Let $A$ be the set of $n\in\mathbb{N}$ such that $n\notin[N]$ or $a\cdot\sum_{k=1}^{n}v_{k}=\sum_{k=1}^{n}a\cdot v_{k}$. Then $1\in A$, both sides being $a\cdot v_{1}$ by claim 1 of Properties of Finite Sums of Vectors and claim 1 of Properties of Finite Sums. Let $n\in A$ with $S(n)\in[N]$; since $n<S(n)$ by claim 5 of Properties of the Order on the Natural Numbers, claim 1 there gives $n\in[N]$, and by claim 1 of Properties of Finite Sums of Vectors, claim 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, $n\in A$, and claim 1 of Properties of Finite Sums,

so $S(n)\in A$. By Principle of Induction for the Natural Numbers, $A=\mathbb{N}$; as $N\in[N]$ by claim 1 of Basic Properties of Initial Segments of the Natural Numbers, $a^{\oplus}\cdot x=a\cdot\sum_{k=1}^{N}\mathfrak{p}_{k}(x)$. Finally, by Step 4, axiom 6 of Field, claim 3 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field,

which is $N\lVert a\rVert^{2}$ by axiom 8 of Field and clause 3 of The Real Numbers and Standard Notation.

Step 10 (Claim 5). By identities 1 and 4 of Natural Numbers, $q(n+1)=q\,S(n)=qn+q$. Let $u\in\mathbb{R}^{qn}$, $v\in\mathbb{R}^{q}$ and $w=\iota^{qn,q}(u,v)$. By the definition of the concatenation map in Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (with $m=qn$ and $n=q$ there), fixed in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, $w_{l}=u_{l}$ for $l\in[qn]$ and $w_{qn+j}=v_{j}$ for $j\in[q]$. The number $b(k,i)=(k-1)q+i$ does not depend on the number of blocks. For $k\in[n]$ and $i\in[q]$, $b(k,i)\in[qn]$ by Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §range (with $n$ in place of $N$), so by (1) for $N=n+1$ and for $N=n$, $(\mathfrak{p}^{(n+1)}_{k}(w))_{i}=w_{b(k,i)}=u_{b(k,i)}=(\mathfrak{p}^{(n)}_{k}(u))_{i}$, and extensionality gives $\mathfrak{p}^{(n+1)}_{k}(w)=\mathfrak{p}^{(n)}_{k}(u)$. For $k=n+1$, which lies in $[n+1]$ by claim 1 of Basic Properties of Initial Segments of the Natural Numbers: with $\iota=\iota_{\mathbb{R}}$ as in clause 3 of The Real Numbers and Standard Notation, claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field and axioms 1, 3 and 2 of Field give $\iota(n+1)-1=\iota(n)$, so axiom 8 of Field and claims 5 and 4 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field give $b(n+1,i)=\iota(q)\iota(n)+\iota(i)=\iota(qn+i)$, and by claim 7 there $b(n+1,i)$ is the natural number $qn+i$. Hence $(\mathfrak{p}^{(n+1)}_{n+1}(w))_{i}=w_{qn+i}=v_{i}$ for every $i\in[q]$, and extensionality gives $\mathfrak{p}^{(n+1)}_{n+1}(w)=v$.