cft-cmi

Exact relative-entropy identity on unitary orbits (bounded generators; affiliated generators with Omega in D(A))

Status: proved — a complete structured proof.

Area: Kubo-Mori (relative entropy) coefficient s2 · Card type: theorem · Knowledge-base id: km-exact-identity

Statement

(a) Bounded generators (thm:exact, rigor/kubo_mori_second_variation.tex:215): for A = A* in M (bounded), u_s = e^{isA}, omega_s = omega o Ad u_s: D(omega || omega_s) = || h(Delta)^{1/2} (u_s - 1) Omega ||^2 in [0, infinity], h(lambda) = lambda - 1 - log lambda, with the sum rule int (lambda - 1) d nu_s = 0 along the orbit; liminf s^{-2} D >= g_KM/2 with no further hypothesis (thm:lower), g_KM = -2 <A Omega, log Delta A Omega> (lem:kms); D = (s^2/2) g_KM + o(s^2) under (U_eps) (thm:main), which holds in particular when A is modularly analytic in a strip, (A_eps) => (U_eps) (lem:AtoU). (b) Affiliated generators (prop:unbounded, l. 774-778): for A = A* affiliated with M and Omega in D(A), the exact identity, the sum rule, lem:kms and the lower bound hold verbatim, and the second-order expansion holds under (U_eps); for unbounded A the modular-analyticity route (A_eps) => (U_eps) is not available (lem:AtoU is for bounded A). (c) Not applicable to the recovery curve of Note 7, whose generator T(g_tot) is affiliated with the global algebra but not with the interval algebra, so e^{isA} is not in M (Gap QK2, l. 822).

How to verify

rigor/kubo_mori_second_variation.tex: setting l. 102 (A bounded, in M), thm:exact (l. 215), lem:kms (l. 264), thm:lower (l. 310), thm:upper (l. 592), thm:main (l. 616), lem:AtoU (l. 524), prop:unbounded (l. 774-800, Omega in D(A) used in <1>2 and <1>3), Gap QK2 (l. 822), summary table (l. 851-853). The overview table l. 66-67 carries the split since 2026-09-24 (Correction C1, Sec. 11, l. 901-925): l. 66, A = A* affiliated with M and Omega in D(A), proved by prop:unbounded, with the upper bound and the expansion under (U_eps) and (A_eps) => (U_eps) only for bounded A; l. 67, A affiliated only with a larger algebra, open (Gap QK2). Until then both cases stood in one row marked open (DOCUMENT ERROR note).

Evidence

Related claims

None recorded.

Review

Review record: passed 2026-09-24 by REF-CARD-CFT.

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.

Documentation map · All claims by area · What is open · Symbols · Definitions