Proof of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound

The setting adopted by the statement is in force: that of The Flat Torus: Standing Notation, used with $n=d$, and that of Probability Measures on Euclidean Space and Random Vectors: Standing Notation; $\varpi$ is the wrapped displacement and $d_{\mathbb{T}}(x,y)=\lVert\varpi(y-x)\rVert$ is the flat torus distance.

Conventions. $\tfrac12$ is the inverse of $2=1+1$; by claim 8 of Elementary Order Arithmetic in an Ordered Field (with $\varepsilon=1$) we have $0<\tfrac12<1$ and $\tfrac12+\tfrac12=1$. A point of $\mathbb{R}^{d}$ is determined by its coordinates and may be prescribed through them (claims 1 and 2 of Euclidean Points as Tuples of Real Numbers); coordinates of sums, differences and negatives of points are the corresponding sums, differences and negatives of coordinates. By the definition of the lattice a point lies in $\mathbb{Z}^{d}$ exactly when all its coordinates are integers, so by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers, applied coordinatewise, $0\in\mathbb{Z}^{d}$ and $-k$, $k+m$, $k-m$ lie in $\mathbb{Z}^{d}$ for $k,m\in\mathbb{Z}^{d}$. Weak inequalities are added by axiom 1 of Ordered Field and transitivity, strict and weak ones by claim 3 of Elementary Order Arithmetic in an Ordered Field; termwise inequalities between finite sums of reals are summed by claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers. For a real $s$ we have $s^{2}=|s|^{2}$, because $|s|$ equals $s$ or $-s$ (claim 1 of Properties of the Absolute Value in an Ordered Field). As in the definition of the canonical map, the real number $d$ is $\sum_{i=1}^{d}1$, so by claim 3 of Properties of Finite Sums a sum of $d$ terms all equal to a real $c$ is $d\,c$.

For $z\in\mathbb{R}^{d}$ and $i\in[d]$ we write $n_{i}=\lfloor z_{i}+\tfrac12\rfloor\in\mathbb{Z}$ and $t_{i}=\varpi(z)_{i}=z_{i}-n_{i}$, the point $z$ being clear from the context.

Step 0 (Integer parts and measurability of $\varpi$).

(a) If $s\in\mathbb{R}$ and $p\in\mathbb{Z}$ satisfy $p\le s<p+1$, then $\lfloor s\rfloor=p$: this is the uniqueness assertion of Existence and Uniqueness of the Integer Part of a Real Number, which also gives $\lfloor s\rfloor\le s<\lfloor s\rfloor+1$ for every real $s$.

(b) If $m\in\mathbb{Z}$ and $m\ne0$, then $1\le|m|$. Indeed $|m|$ equals $m$ or $-m$, an integer by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers, and $0<|m|$ by claim 1 of Properties of the Absolute Value in an Ordered Field; so $1\le|m|$ by claim 3 (discreteness) of Arithmetic, Order, Discreteness and Intervals of the Integers.

(c) For $p\in\mathbb{Z}$ and $s\in\mathbb{R}$: $p\le\lfloor s\rfloor$ if and only if $p\le s$. If $p\le\lfloor s\rfloor$, then $p\le s$ because $\lfloor s\rfloor\le s$. Conversely let $p\le s$ and suppose $\lfloor s\rfloor<p$; then $\lfloor s\rfloor+1\le p$ by discreteness, so $s<\lfloor s\rfloor+1\le p$ and $s<p$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, contradicting $p\le s$ since $\le$ is a total order.

(d) For $a\in\mathbb{R}$ and $p\in\mathbb{Z}$: $a<p$ if and only if $\lfloor a\rfloor+1\le p$. If $a<p$, then $\lfloor a\rfloor\le a<p$ gives $\lfloor a\rfloor<p$ (claim 2 of Elementary Order Arithmetic in an Ordered Field) and so $\lfloor a\rfloor+1\le p$ by discreteness; conversely $a<\lfloor a\rfloor+1\le p$.

(e) Let $g:\mathbb{R}\to\mathbb{R}$, $g(s)=\lfloor s+\tfrac12\rfloor$. Fix a real $a$ and put $c=\lfloor a\rfloor+\tfrac12$. For $s\in\mathbb{R}$, by (d) with $p=g(s)$, then by (c) with the integer $p=\lfloor a\rfloor+1$, and then by adding $-\tfrac12$ (axiom 1 of Ordered Field, in both directions),

$$g(s)>a\iff\lfloor a\rfloor+1\le\lfloor s+\tfrac12\rfloor\iff\lfloor a\rfloor+1\le s+\tfrac12\iff c\le s .$$

Since $\le$ is total, $c\le s$ fails exactly when $s<c$, that is (claim 4 of Elementary Order Arithmetic in an Ordered Field) when $-c<-s$. The identity map of $\mathbb{R}$ is measurable with respect to $\mathcal{B}(\mathbb{R})$ and $\mathcal{B}(\mathbb{R})$ (the preimage of a set is the set itself), so $s\mapsto-s$ is measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and the set $\{s:-s>-c\}$ belongs to $\mathcal{B}(\mathbb{R})$ by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line. Hence $\{s:g(s)>a\}=\mathbb{R}\setminus\{s:-s>-c\}$ belongs to the $\sigma$-algebra $\mathcal{B}(\mathbb{R})$. As $a$ was arbitrary, $g$ is measurable by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line. For $i\in[d]$ the projection $p_{i}:z\mapsto z_{i}$ is Borel 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; the composite $g\circ p_{i}$ is Borel since $(g\circ p_{i})^{-1}(B)=p_{i}^{-1}(g^{-1}(B))$; so $z\mapsto\varpi(z)_{i}=p_{i}(z)-g(p_{i}(z))$ is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By the componentwise criterion, claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets (with claim 5 there), $\varpi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is Borel. This proves the last assertion of claim 7 concerning $\varpi$.

Claim 1 (Range). Let $z\in\mathbb{R}^{d}$, $i\in[d]$ and $k\in\mathbb{Z}^{d}$. By (a), $n_{i}\le z_{i}+\tfrac12<n_{i}+1$; adding $-n_{i}-\tfrac12$ (axiom 1 of Ordered Field and claim 1 of Elementary Order Arithmetic in an Ordered Field) and using $1-\tfrac12=\tfrac12$ gives $-\tfrac12\le t_{i}<\tfrac12$. The point $z-\varpi(z)$ has coordinates $z_{i}-t_{i}=n_{i}\in\mathbb{Z}$, so it lies in $\mathbb{Z}^{d}$. Next, $n_{i}+k_{i}$ is an integer and, adding $k_{i}$, $n_{i}+k_{i}\le(z_{i}+k_{i})+\tfrac12<(n_{i}+k_{i})+1$; by (a), $\lfloor(z+k)_{i}+\tfrac12\rfloor=n_{i}+k_{i}$, hence $\varpi(z+k)_{i}=z_{i}+k_{i}-n_{i}-k_{i}=t_{i}$ for every $i$, and $\varpi(z+k)=\varpi(z)$. Finally $-\tfrac12\le t_{i}\le\tfrac12$ gives $|t_{i}|\le\tfrac12$ by claim 6 of Properties of the Absolute Value in an Ordered Field, so $t_{i}^{2}=|t_{i}|^{2}\le\tfrac14$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and summation, $\lVert\varpi(z)\rVert^{2}=\sum_{i=1}^{d}t_{i}^{2}\le d\cdot\tfrac14=d/4$.

Claim 2 (Minimality). Let $z\in\mathbb{R}^{d}$, $k\in\mathbb{Z}^{d}$ and $i\in[d]$. We show

$$|t_{i}|\le|z_{i}-k_{i}|,\qquad\text{and}\qquad |t_{i}|<|z_{i}-k_{i}|\ \text{ if }k_{i}\ne n_{i}\text{ and }|t_{i}|<\tfrac12 .$$

If $k_{i}=n_{i}$, then $z_{i}-k_{i}=t_{i}$. If $k_{i}\ne n_{i}$, put $m=n_{i}-k_{i}$, a nonzero integer, so $1\le|m|$ by (b); since $z_{i}-k_{i}=t_{i}+m$ and $m=(t_{i}+m)+(-t_{i})$, the triangle inequality and symmetry (claims 5 and 2 of Properties of the Absolute Value in an Ordered Field) give $|m|\le|z_{i}-k_{i}|+|t_{i}|$, hence

$$|z_{i}-k_{i}|\ge|m|-|t_{i}|\ge1-\tfrac12=\tfrac12\ge|t_{i}|,$$

using $|t_{i}|\le\tfrac12$ from claim 1; and if $|t_{i}|<\tfrac12$ the last step is strict, so $|t_{i}|<|z_{i}-k_{i}|$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. Squaring (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field) gives $t_{i}^{2}\le(z_{i}-k_{i})^{2}$ for every $i$, and summing, by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$,

$$\lVert\varpi(z)\rVert^{2}=\sum_{i=1}^{d}t_{i}^{2}\le\sum_{i=1}^{d}(z_{i}-k_{i})^{2}=\lVert z-k\rVert^{2};$$

as both norms are nonnegative, $\lVert\varpi(z)\rVert\le\lVert z-k\rVert$ by the same claim of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. For $x,y\in\mathbb{R}^{d}$ and $k\in\mathbb{Z}^{d}$ apply this to $z=y-x$: $d_{\mathbb{T}}(x,y)=\lVert\varpi(y-x)\rVert\le\lVert y-x-k\rVert$. The point $k_{0}=y-x-\varpi(y-x)$ lies in $\mathbb{Z}^{d}$ by claim 1 and $y-x-k_{0}=\varpi(y-x)$, which is the equality case. With $k=0$ we get $d_{\mathbb{T}}(x,y)\le\lVert y-x\rVert=\lVert x-y\rVert$, the last equality by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ with $\lambda=-1$.

Claim 3 (Regular points). Let $z\in\mathbb{R}^{d}$. We read the minimum over $[d]$ as the iterated minimum of two elements, $r(z)=\min\{\cdots\min\{\min\{h_{1},h_{2}\},h_{3}\}\cdots,h_{d}\}$ with $h_{i}=\tfrac12-|t_{i}|$ (and $r(z)=h_{1}$ if $d=1$); by induction on the number of terms, using claims 1 and 2 of Elementary Properties of the Minimum of Two Elements and transitivity, $r(z)\le h_{i}$ for every $i\in[d]$ and $r(z)=h_{i_{0}}$ for some $i_{0}\in[d]$.

(i) Since $|t_{i}|\le\tfrac12$ (claim 1), each $h_{i}\ge0$, so $r(z)=h_{i_{0}}\ge0$. Moreover $r(z)>0$ if and only if $h_{i}>0$ for every $i$ (for the forward direction use $r(z)\le h_{i}$ and claim 2 of Elementary Order Arithmetic in an Ordered Field; for the converse use $r(z)=h_{i_{0}}$), that is, if and only if $|t_{i}|<\tfrac12$ for every $i$.

(ii) For each $i$: $|t_{i}|=\tfrac12$ if and only if $t_{i}=-\tfrac12$, since $|t_{i}|$ equals $t_{i}$ or $-t_{i}$ while $t_{i}<\tfrac12$, and $|-\tfrac12|=\tfrac12$. Further $t_{i}=-\tfrac12$ if and only if $z_{i}-\tfrac12\in\mathbb{Z}$: if $t_{i}=-\tfrac12$ then $z_{i}-\tfrac12=n_{i}-1\in\mathbb{Z}$; conversely, if $z_{i}-\tfrac12=m\in\mathbb{Z}$, then $z_{i}+\tfrac12=m+1$ and $m+1\le m+1<(m+1)+1$, so $n_{i}=m+1$ by (a) and $t_{i}=(m+\tfrac12)-(m+1)=-\tfrac12$. Since $|t_{i}|\le\tfrac12$, $|t_{i}|<\tfrac12$ holds exactly when $|t_{i}|\ne\tfrac12$, that is, exactly when $z_{i}-\tfrac12\notin\mathbb{Z}$. With (i): $z$ is regular if and only if $|t_{i}|<\tfrac12$ for every $i$, if and only if $r(z)>0$.

(iii) Let $z$ be regular and $k\in\mathbb{Z}^{d}$ with $k\ne z-\varpi(z)$. The point $z-\varpi(z)$ has coordinates $n_{i}$ (claim 1), so $k_{i_{1}}\ne n_{i_{1}}$ for some $i_{1}\in[d]$. By Claim 2 and (ii), $t_{i}^{2}\le(z_{i}-k_{i})^{2}$ for every $i$ and $t_{i_{1}}^{2}<(z_{i_{1}}-k_{i_{1}})^{2}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). The numbers $e_{i}=(z_{i}-k_{i})^{2}-t_{i}^{2}$ are nonnegative and $e_{i_{1}}>0$, so by claims 2, 3 and 6 of Properties of Finite Sums,

$$\lVert z-k\rVert^{2}-\lVert\varpi(z)\rVert^{2}=\sum_{i=1}^{d}e_{i}\ge e_{i_{1}}>0,$$

and $\lVert\varpi(z)\rVert<\lVert z-k\rVert$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

(iv) Let $z$ be regular. By (ii) and claim 9 of Properties of the Absolute Value in an Ordered Field, $-\tfrac12<t_{i}<\tfrac12$, hence $0<\tfrac12-t_{i}<1$ (claims 1 and 4 of Elementary Order Arithmetic in an Ordered Field). As $(-z)_{i}+\tfrac12=-n_{i}+(\tfrac12-t_{i})$, we get $-n_{i}<(-z)_{i}+\tfrac12<-n_{i}+1$ with $-n_{i}\in\mathbb{Z}$, so $\lfloor(-z)_{i}+\tfrac12\rfloor=-n_{i}$ by (a) and $\varpi(-z)_{i}=-z_{i}+n_{i}=-t_{i}$. Thus $\varpi(-z)=-\varpi(z)$.

(v) Let $z$ be regular and $z'\in\mathbb{R}^{d}$ with $\lVert z'-z\rVert<r(z)$. For each $i$, claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 2 of Elementary Order Arithmetic in an Ordered Field give $|z'_{i}-z_{i}|\le\lVert z'-z\rVert<r(z)\le\tfrac12-|t_{i}|$, so by claim 9 of Properties of the Absolute Value in an Ordered Field, $-(\tfrac12-|t_{i}|)<z'_{i}-z_{i}<\tfrac12-|t_{i}|$. Put $s_{i}=z'_{i}+\tfrac12-n_{i}=(t_{i}+\tfrac12)+(z'_{i}-z_{i})$. Adding $t_{i}+\tfrac12$ (claim 1 of Elementary Order Arithmetic in an Ordered Field) gives

$$t_{i}+|t_{i}|<s_{i}<1+(t_{i}-|t_{i}|),$$

and $0\le t_{i}+|t_{i}|$, $t_{i}-|t_{i}|\le0$ by claim 3 of Properties of the Absolute Value in an Ordered Field; hence $0<s_{i}<1$, that is, $n_{i}<z'_{i}+\tfrac12<n_{i}+1$. By (a), $\lfloor z'_{i}+\tfrac12\rfloor=n_{i}$, so $\varpi(z')_{i}=z'_{i}-n_{i}=t_{i}+(z'_{i}-z_{i})$ for every $i$, which is $\varpi(z')=\varpi(z)+(z'-z)$. Moreover $\varpi(z')_{i}=s_{i}-\tfrac12$ with $0<s_{i}<1$ gives $-\tfrac12<\varpi(z')_{i}<\tfrac12$, so $|\varpi(z')_{i}|<\tfrac12$ for every $i$ (claim 9 of Properties of the Absolute Value in an Ordered Field), and $z'$ is regular by (ii) applied to $z'$.

(vi) Each $z\mapsto h_{i}(z)=\tfrac12-|\varpi(z)_{i}|$ is Borel: $z\mapsto\varpi(z)_{i}$ is Borel by Step 0(e), its absolute value by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and constants, sums and scalar multiples by claims 1 and 2 there. The minimum of two Borel functions is Borel by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; applying this $d-1$ times along the iterated minimum shows that $r$ is Borel.

Claim 4 (Symmetry and periodicity). Let $x,y\in\mathbb{R}^{d}$, $k,m\in\mathbb{Z}^{d}$ and $v=y-x$. By claim 1, $k_{v}=v-\varpi(v)\in\mathbb{Z}^{d}$, so $-k_{v}\in\mathbb{Z}^{d}$, and Claim 2 at the point $-v$ gives

$$\lVert\varpi(-v)\rVert\le\lVert-v-(-k_{v})\rVert=\lVert-(v-k_{v})\rVert=\lVert\varpi(v)\rVert,$$

by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. Applying this with $-v$ in place of $v$ (note $-(-v)=v$) gives the reverse inequality, so $\lVert\varpi(-v)\rVert=\lVert\varpi(v)\rVert$, which is $d_{\mathbb{T}}(y,x)=d_{\mathbb{T}}(x,y)$ since $x-y=-v$. Next $(y+m)-(x+k)=v+(m-k)$ with $m-k\in\mathbb{Z}^{d}$, so $d_{\mathbb{T}}(x+k,y+m)=\lVert\varpi(v+(m-k))\rVert=\lVert\varpi(v)\rVert=d_{\mathbb{T}}(x,y)$ by claim 1. By The Half-Open Unit Cell Tiles Euclidean Space §wrap (with The Half-Open Unit Cell Tiles Euclidean Space §tiling), $\pi(x)=x+(-m_{x})$ and $\pi(y)=y+(-m_{y})$ for some $m_{x},m_{y}\in\mathbb{Z}^{d}$, so $d_{\mathbb{T}}(\pi(x),\pi(y))=d_{\mathbb{T}}(x,y)$ by what was just shown. Finally, if $v=k\in\mathbb{Z}^{d}$, then for each $i$ we have $k_{i}\le k_{i}+\tfrac12<k_{i}+1$, so $\lfloor k_{i}+\tfrac12\rfloor=k_{i}$ by (a), $\varpi(k)=0$ and $d_{\mathbb{T}}(x,y)=0$. Conversely, if $d_{\mathbb{T}}(x,y)=\lVert\varpi(v)\rVert=0$, then $\varpi(v)=0$ by claim 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and $v=v-\varpi(v)\in\mathbb{Z}^{d}$ by claim 1.

Claim 5 (Triangle inequality). Let $x,y,z\in\mathbb{R}^{d}$ and put $k=y-x-\varpi(y-x)$ and $k'=z-y-\varpi(z-y)$; both lie in $\mathbb{Z}^{d}$ by claim 1, hence so does $k+k'$. Since $z-x-(k+k')=\varpi(y-x)+\varpi(z-y)$, Claim 2 and the triangle inequality, claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, give

$$d_{\mathbb{T}}(x,z)\le\lVert\varpi(y-x)+\varpi(z-y)\rVert\le\lVert\varpi(y-x)\rVert+\lVert\varpi(z-y)\rVert=d_{\mathbb{T}}(x,y)+d_{\mathbb{T}}(y,z).$$

Claim 6 (Metric on the cell). We check the four conditions of Metric Space for the restriction of $d_{\mathbb{T}}$ to $Q\times Q$, a real-valued map. Nonnegativity holds since norms are nonnegative (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$); symmetry is Claim 4 and the triangle inequality is Claim 5. If $x=y$, then $y-x=0\in\mathbb{Z}^{d}$ and $d_{\mathbb{T}}(x,y)=0$ by Claim 4. Conversely let $x,y\in Q$ with $d_{\mathbb{T}}(x,y)=0$; by Claim 4, $y-x\in\mathbb{Z}^{d}$. For $i\in[d]$, the definition of $Q$ gives $0\le x_{i}<1$ and $0\le y_{i}<1$, hence $-1<-x_{i}\le0$ and, by claim 3 of Elementary Order Arithmetic in an Ordered Field, $-1<y_{i}-x_{i}<1$, so $|y_{i}-x_{i}|<1$ by claim 9 of Properties of the Absolute Value in an Ordered Field. The integer $y_{i}-x_{i}$ is therefore $0$, for otherwise (b) would give $1\le|y_{i}-x_{i}|$. Hence $x=y$.

Claim 7 (Lipschitz bound and measurability). Let $x,y,x',y'\in\mathbb{R}^{d}$. By Claims 5 and 4, $d_{\mathbb{T}}(x,y)\le d_{\mathbb{T}}(x,x')+d_{\mathbb{T}}(x',y)$ and $d_{\mathbb{T}}(x',y)\le d_{\mathbb{T}}(x',x)+d_{\mathbb{T}}(x,y)=d_{\mathbb{T}}(x,x')+d_{\mathbb{T}}(x,y)$; by claim 6 of Properties of the Absolute Value in an Ordered Field and Claim 2,

$$|d_{\mathbb{T}}(x,y)-d_{\mathbb{T}}(x',y)|\le d_{\mathbb{T}}(x,x')\le\lVert x-x'\rVert .$$

In the same way, $d_{\mathbb{T}}(x',y)\le d_{\mathbb{T}}(x',y')+d_{\mathbb{T}}(y',y)$ and $d_{\mathbb{T}}(x',y')\le d_{\mathbb{T}}(x',y)+d_{\mathbb{T}}(y,y')$, with $d_{\mathbb{T}}(y',y)=d_{\mathbb{T}}(y,y')\le\lVert y-y'\rVert$ by Claims 4 and 2, give $|d_{\mathbb{T}}(x',y)-d_{\mathbb{T}}(x',y')|\le\lVert y-y'\rVert$. The triangle inequality, claim 5 of Properties of the Absolute Value in an Ordered Field, applied to the sum of the two differences, yields the first displayed bound of claim 7.

For the second, put $a=d_{\mathbb{T}}(x,y)$ and $b=d_{\mathbb{T}}(x',y)$, both nonnegative. The real number $d=\sum_{i=1}^{d}1$ is nonnegative by claim 5 of Properties of Finite Sums, each summand being positive by claim 6 of Elementary Order Arithmetic in an Ordered Field. Let $\rho=\sqrt{d}$, the nonnegative square root; then $(\tfrac12\rho)^{2}=\tfrac14 d=d/4$, and $a^{2}\le d/4$, $b^{2}\le d/4$ by claim 1, so $a\le\tfrac12\rho$ and $b\le\tfrac12\rho$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, whence $0\le a+b\le\rho$. Since $a^{2}-b^{2}=(a-b)(a+b)$, claim 4 of Properties of the Absolute Value in an Ordered Field gives $|a^{2}-b^{2}|=|a-b|\,(a+b)$, and by claim 5 of Elementary Arithmetic in an Ordered Field, used twice with nonnegative factors,

$$|a^{2}-b^{2}|\le\lVert x-x'\rVert\,(a+b)\le\sqrt{d}\,\lVert x-x'\rVert .$$

Let $F:\mathbb{R}^{d+d}\to\mathbb{R}$, $F(w)=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, every $w$ equals $\iota(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$, and by the description of $\iota$ in the preamble of that lemma, $\mathrm{pr}_{1}(w)_{i}=w_{i}$ and $\mathrm{pr}_{2}(w)_{i}=w_{d+i}$ for $i\in[d]$. Hence for $w,w'\in\mathbb{R}^{d+d}$ the points $\mathrm{pr}_{j}(w)-\mathrm{pr}_{j}(w')$ and $\mathrm{pr}_{j}(w-w')$ have the same coordinates, and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections gives $\lVert\mathrm{pr}_{j}(w)-\mathrm{pr}_{j}(w')\rVert=\lVert\mathrm{pr}_{j}(w-w')\rVert\le\lVert w-w'\rVert$ for $j=1,2$. With the first bound of claim 7,

$$|F(w)-F(w')|\le2\lVert w-w'\rVert .$$

By claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\lVert w-w'\rVert$ is the Euclidean distance of $w$ and $w'$, and by The Euclidean Distance on the Real Line is the Absolute Value Metric the Euclidean distance of $\mathbb{R}$ is the absolute-value metric; so $F$ is Lipschitz with constant $2$, hence continuous by A Lipschitz Map is Uniformly Continuous, hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps (claim 3 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets). That $\varpi$ is Borel was shown in Step 0(e). $\blacksquare$