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

Lemma: Replica-coordinate identity and finite entrance density

Stage 1: finite identities and the five-stage audit › Locally proved identities that an audit may use · lines 475–507 · proof included or linked

Oai227-A@1#lem:replica, Oai227-A@1#eq:finite-entrance

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For any orthonormal basis $u_{1}, \dots, u_{n}$ of $R^{n}$ , set

x_{a} (σ) = n^{- 1/3} u_{a} \cdot σ .

Then

n^{- 2/3} σ \cdot τ = a = 1 \sum^{n} x_{a} (σ) x_{a} (τ) .

w_{s} = \frac{d ( U _{n} P _{s T_{n}}^{J} )}{d π},

then

A_{n, J} (t) B_{n, J} (s, t) = a \sum π [x_{a} P_{t T_{n}}^{J} x_{a}], = a \sum π [w_{s} x_{a} P_{t T_{n}}^{J} x_{a}] .

Moreover $w_{s + t} = P_{t T_{n}}^{J} w_{s}$ .

Proof. The first identity is orthogonality. Apply it to the two configurations in a two-time pair and condition on the first. Since $π$ has full support, the density exists. Reversibility identifies the evolution of a density relative to $π$ with $P_{u}^{J}$ , proving the last assertion.

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Oai227-A@1#eq:finite-entrance · line 492
$A_{n, J} (t) B_{n, J} (s, t) = a \sum π [x_{a} P_{t T_{n}}^{J} x_{a}], = a \sum π [w_{s} x_{a} P_{t T_{n}}^{J} x_{a}] .$
unlabeled · line 478
$x_{a} (σ) = n^{- 1/3} u_{a} \cdot σ .$
unlabeled · line 482
$n^{- 2/3} σ \cdot τ = a = 1 \sum^{n} x_{a} (σ) x_{a} (τ) .$
unlabeled · line 488
$w_{s} = \frac{d ( U _{n} P _{s T_{n}}^{J} )}{d π},$

Lemma: Replica-coordinate identity and finite entrance density

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