Theorem A: lim Phi/zeta^2 = c/(12 pi^2)

Status: refereed — a complete structured proof that an independent referee pass has accepted.

Area: Theorem A: exact recovery error of the free fermion · Card type: theorem · Knowledge-base id: thm-a-quadratic-law

Statement

For the zero-collar geometric compression of the free chiral fermion, Phi(zeta) := -log F(omega, omega^beta) satisfies lim_{zeta->0} Phi/zeta^2 = c/(12 pi^2) = f2 = g_B/8. Lower bound: Legendre form of the SLD metric plus data processing. Upper bound: Uhlmann with a Bogoliubov extension of the compression to a circle diffeomorphism, F >= det(1-V*V)^{1/2}, infimum over extensions = g_B/4 (three-way identity). No interchange of limits or tail bound needed.

How to verify

Compile rigor/quadratic_limit.pdf (lower) and rigor/uhlmann_upper_bound.pdf (upper); numerics/p1_bogoliubov.out reproduces the bound to 0.2% at zeta = 1/120.

Evidence

• rigor/quadratic_limit.tex — document, at thm:lower
• rigor/uhlmann_upper_bound.tex — document, at thm:main-upper
• rigor/referee_quadratic_limit.tex — document, at sec:thm42
• rigor/referee_uhlmann_upper_bound.tex — document, at sec:thm
• numerics/p1_bogoliubov.py — numerics

Related claims

• f2 = c/(12 pi^2) — proved
• Free-fermion one-particle metrics of the compression direction: g_B = 2r/(3 pi^2), g_KM = r/6 (c = r) — proved
• Remainder of Theorem A: global upper bound Phi <= -(1/2) log(1 - 2 f2 zeta^2); c3 <= 0 as a limsup; no positive log^-2 term — proved

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B8.

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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