Type-III second-variation lemma (fidelity half), under the vector hypothesis (V)

Status: proved — a complete structured proof.

Area: Foundations: CMI as a Markov defect (Notes 1-3) · Card type: theorem · Knowledge-base id: second-variation-lemma

Statement

Let M be a von Neumann algebra with a cyclic and separating vector Omega, omega = <Omega, . Omega>, and let omega_s = <xi_s, . xi_s>|M be vector states given by unit vectors xi_s with xi_0 = Omega that are norm-differentiable at s = 0 (hypothesis (V), rigor/lemma_second_variation.tex l. 119-121), with tangent representative a = i xi'_0; the model case is xi_s = e^{-isA} Omega with A = A* on H, Omega in D(A), a = A Omega. Then F(omega, omega_s) = 1 - (s^2/8) g_B + o(s^2) and -log F(omega, omega_s) = (s^2/8) g_B + o(s^2), with g_B = g_B(a) given by the three faces (thm:three; card bures-three-faces) (thm:main). Without (V) the conclusion is not available: norm-differentiability of s -> omega_s in M_* does not imply (V) (Powers-Stormer gives only Holder exponent 1/2), and then only F = 1 - O(s) follows (Gap 5, l. 1345-1355). The Kubo-Mori half is only available on unitary orbits of generators affiliated with M (card km-exact-identity).

How to verify

rigor/lemma_second_variation.tex hypothesis (V) (l. 119-121), thm:main (l. 962-979), thm:three (l. 418), Gap 5 (l. 1345-1355); referee notes inside.

Evidence

• rigor/lemma_second_variation.tex — document, at thm:main

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B4.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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