Proof of The Structure Condition and Uniform Continuity on Bounded Sets Imply Their Forms at Bounded Positions

Each result cited is universally quantified over the data in its own statement. The natural number $d$ is read as the real number $\iota(d)$, with $\iota$ the canonical map; $1\le d$ by claim 2 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. For every real $a$, the constant finite sum $\sum_{j=1}^{d}a$ equals $da$: by induction on $d$, using the recursion of claim 1 of Properties of Finite Sums, claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field ($\iota(1)=1$ and $\iota(n+1)=\iota(n)+1$) and distributivity.

Step 1 (Positions at bounded laws have $L^{2}$ norm at most $dR$). Let $R>0$ be real, let $(H,M,\Omega)$ be a tracial W*-probability space and let $X$ be an $L^{2}$ $d$-tuple of it with $\mathrm{law}(X)=\kappa_{d}(\lambda)$ for some $\lambda\in\Sigma_{d,R}$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments and Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded,

$$\lVert X\rVert_{2}^{2}=\widehat{M}(\mathrm{law}(X))=\widehat{M}(\kappa_{d}(\lambda))=M(\lambda)=\sum_{j=1}^{d}\lambda(x_{j}x_{j}).$$

For $j\in[d]$, $x_{j}x_{j}=x_{w}$ for the word $w$ of length $2$ both of whose letters are $j$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials); $\lambda(x_{j}x_{j})=\lambda(x_{j}^{*}x_{j})$ is real and nonnegative by condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, as $x_{j}^{*}=x_{j}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint); and $\lambda(x_{j}x_{j})=|\lambda(x_{w})|\le R^{2}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound. Here $M(\lambda)$ is the second moment of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, a sum of the real numbers $\lambda(x_{j}x_{j})$, which is the real sum. Summing these $d$ inequalities, claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and the constant-sum identity above give $\lVert X\rVert_{2}^{2}\le\sum_{j=1}^{d}R^{2}=dR^{2}$, and $dR^{2}\le d^{2}R^{2}=(dR)^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, since $1\le d$ and $0\le dR^{2}$, and by commutativity of multiplication. As $\lVert X\rVert_{2}\ge0$ and $dR>0$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert X\rVert_{2}\le dR$. Moreover $R\le dR$ by claim 5 of Elementary Arithmetic in an Ordered Field.

Step 2 (Claim 1). Assume the structure condition, and let $R>0$ and $\eta>0$ be real. Apply it with the radius $dR>0$ and with $\eta$, obtaining a real $r>0$. Let $(H,M,\Omega)$, $X$, $Y$ and $\alpha$ be as in The Structure Condition at Bounded Positions for a Hamiltonian on Phase-Space Noncommutative Laws §structure for $R$ and this $r$. By Step 1, $\lVert X\rVert_{2}\le dR$ and $\lVert Y\rVert_{2}\le dR$, and $\alpha\lVert X-Y\rVert_{2}^{2}+\lVert X-Y\rVert_{2}<r$; so the structure condition gives $\mathcal{H}_{M}(Y,\alpha(X-Y))-\mathcal{H}_{M}(X,\alpha(X-Y))<\eta$. Hence $\mathcal{H}$ satisfies the structure condition at bounded positions.

Step 3 (Claim 2). Assume that $\mathcal{H}$ is uniformly continuous on bounded sets, and let $R>0$ and $\eta>0$ be real. Apply that definition with the radius $dR$ and with $\eta$, obtaining a real $r>0$. Let $(H,M,\Omega)$, $X$, $P$ and $P'$ be as in Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous in the Momentum at Bounded Positions §momentum for $R$ and this $r$. By Step 1, $\lVert X\rVert_{2}\le dR$, and $\lVert P\rVert_{2}\le R\le dR$, $\lVert P'\rVert_{2}\le R\le dR$. Take $X'=X$: then $X-X'$ is the zero $d$-tuple, whose $L^{2}$ norm is $0$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples), so $\lVert X-X'\rVert_{2}+\lVert P-P'\rVert_{2}=\lVert P-P'\rVert_{2}<r$. The uniform continuity gives $|\mathcal{H}_{M}(X,P)-\mathcal{H}_{M}(X,P')|<\eta$. Hence $\mathcal{H}$ is uniformly continuous in the momentum at bounded positions.