Manuscript Oai227-A@1 · Oai227-A.tex · lemma · 7f45f607

Lemma: The exact spectral trace is a dynamical trace

Stage 1: finite identities and the five-stage audit › Locally proved identities that an audit may use · lines 450–473 · proof included or linked

Oai227-A@1#lem:spectral, Oai227-A@1#eq:dynamical-trace

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Let $(λ_{α}, ψ_{α})$ be an orthonormal eigensystem of $- L_{J}$ on $L^{2} (π)$ , with $λ_{0} = 0$ and $ψ_{0} = 1$ . Then

A_{n, J} (t) = n^{- 2/3} α \geq 1 \sum e^{- λ_{α} T_{n} t} i = 1 \sum^{n} ⟨ f_{i}, ψ_{α} ⟩_{π}^{2} .

The weights are nonnegative. The eigenvalues here are those of the heat-bath generator, not those of the $n$ -by- $n$ interaction matrix.

Proof. Expand each $f_{i}$ in the reversible eigenbasis. Its constant coefficient vanishes by Lemma lem:dirichlet. Apply $P_{T_{n} t}^{J}$ , take its inner product with $f_{i}$ , and sum.

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Oai227-A@1#eq:dynamical-trace · line 455
$A_{n, J} (t) = n^{- 2/3} α \geq 1 \sum e^{- λ_{α} T_{n} t} i = 1 \sum^{n} ⟨ f_{i}, ψ_{α} ⟩_{π}^{2} .$

Lemma: The exact spectral trace is a dynamical trace

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