Theorem B: f2 = c/(12 pi^2) for every diffeomorphism-covariant net

Status: refereed — a complete structured proof that an independent referee pass has accepted. Stated under hypotheses H1, H2, H3, quoted below.

Area: Theorem B: universality for diffeomorphism-covariant nets · Card type: theorem · Knowledge-base id: thm-b-universality

Statement

For every diffeomorphism-covariant net on S^1 (separable, CDIT axioms, Bisognano-Wichmann, Haag duality, Reeh-Schlieder) with (H1) an implementation U_s of the C^{1,1} map, (H2) U_s Omega = Omega + i s T(g_tot) Omega + o(s) with T(g_tot) Omega in the stress-tensor subspace, (H3) first-order differentiability on a core: -log F = (s^2/8) g_B + o(s^2), g_B = 4 dist(T(g_tot) Omega, cl A(I)'_sa Omega)^2 = 2c/(3 pi^2). The stress-tensor subspace reduces Delta and J, c-linearity is structural, the constant comes from the fermion.

Hypotheses

The statement is made under named hypotheses. The sentences are quoted from the statement above; read the status badge only together with them.

Hypothesis labels used: H1, H2, H3.

For every diffeomorphism-covariant net on S^1 (separable, CDIT axioms, Bisognano-Wichmann, Haag duality, Reeh-Schlieder) with (H1) an implementation U_s of the C^{1,1} map, (H2) U_s Omega = Omega + i s T(g_tot) Omega + o(s) with T(g_tot) Omega in the stress-tensor subspace, (H3) first-order differentiability on a core: -log F = (s^2/8) g_B + o(s^2), g_B = 4 dist(T(g_tot) Omega, cl A(I)'_sa Omega)^2 = 2c/(3 pi^2).

How to verify

rigor/universality_theorem_B.tex (25 pp, rigor/universality_theorem_B.log of 2026-09-24, with its section 'Corrections of 2026-09-24', l. 1128-1177; 24 pp before) and its referee report; inverted factor V^-1 a = (c/48 pi^2)^{1/2}[g_tot] was corrected.

Evidence

• rigor/universality_theorem_B.tex — document, at thm:B
• rigor/referee_universality_theorem_B.tex — document, at r:edits
• universality_normalization.tex — document, at sec:setting

Related claims

• (H1)-(H3) hold for all diffeomorphism-covariant nets — refereed
• Free-fermion one-particle metrics of the compression direction: g_B = 2r/(3 pi^2), g_KM = r/6 (c = r) — proved

Review

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

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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