Free-fermion one-particle metrics of the compression direction: g_B = 2r/(3 pi^2), g_KM = r/6 (c = r)

Status: proved — a complete structured proof.

Area: Theorem A: exact recovery error of the free fermion · Card type: constant · Knowledge-base id: const-gb

Statement

For the r-component free chiral fermion (c = r), the one-particle Bures (Fisher) form of the zero-collar compression direction is g_B = 2r int |D^(1)(kappa, kappa')|^2 / N(kappa, kappa') dkappa dkappa' = (2r/3) int tanh(u/2) / (u (u^2 + 4 pi^2)) du = 2r/(3 pi^2) (rigor/quadratic_limit.tex prop:gB with the v-integral lem:V, proved from scratch; the first-order kernel D^(1) in the modular frame from the kernel identity, rigor/kernel_identity.tex thm:main), and the one-particle Kubo-Mori form is g_KM = (r/3) int du/(u^2 + 4 pi^2) = r/6 (Note 7 Theorem A, universality_normalization.tex thm:A; rigor/kubo_mori_second_variation.tex eq:ff-result). These are the one-particle sums eq:one-particle to which Note 5 Prop. 2.3 (card quasi-free-second-order-fidelity-expansion, finite-dimensional, here applied to the continuum kernel) reduces the many-body metrics. NOT part of this card: f2 = g_B/8, i.e. the identification of g_B/8 with lim Phi/zeta^2 (Theorem A, card thm-a-quadratic-law); g_B = 2c/(3 pi^2) for general nets (card const-gb-general-nets, from Theorem B); g_KM = c/6 for general nets (card const-gkm-general-nets, conjectural: only the analytically continued value exists, and the gauge infimum is (11 + 5 sqrt 5) c/12 = 11.09 x c/6). Only the kernel-identity computation is rigorous, and only at c = r: the modular-frame computation (Note 7 eq:result-general) and the x-space Mellin reassembly (card w-hat-mellin-formula) evaluate the same analytic-continuation prescription, so they are checks, not independent routes. Caution: g_KM is the metric of the direction, not the coefficient s2 of the relative entropy along the recovery curve (see conj-s2).

How to verify

rigor/quadratic_limit.tex def:gB (l. 398), prop:gB (l. 492) and lem:V (l. 503); rigor/kernel_identity.tex thm:main (l. 106) and its referee report rigor/referee_kernel_identity.tex; universality_normalization.tex thm:A (l. 365) with eq:one-particle (l. 324, Note 5 Prop. 2.3) and eq:V (verified only numerically there; proved as lem:V); g_KM: rigor/kubo_mori_second_variation.tex eq:Iu, eq:ff-result (l. 749-760).

Evidence

• rigor/kernel_identity.tex — document, at thm:main
• universality_normalization.tex — document, at thm:A
• rigor/quadratic_limit.tex — document, at prop:gB
• rigor/quadratic_limit.tex — document, at lem:V
• rigor/kubo_mori_second_variation.tex — document, at eq:ff-result

Related claims

• g_B = 2c/(3 pi^2) for the compression direction of every diffeomorphism-covariant net (corollary of Theorem B) — proved
• g_KM = c/6 for the compression direction of a general conformal net (conjecture; only the analytically continued value exists) — conjectural
• Theorem A: lim Phi/zeta^2 = c/(12 pi^2) — refereed
• s2 = c/12 for general nets — conjectural

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B3.

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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