cft-cmi

Two-chirality law for the massless Dirac fermion: -log F^(lambda) = Phi(z(1+e^{-pi lambda})) + Phi(z(1+e^{pi lambda})), even in lambda

Status: proved — a complete structured proof.

Area: Theorem A: exact recovery error of the free fermion · Card type: result · Knowledge-base id: two-chirality-law

Statement

For the 1+1-dimensional massless Dirac fermion (c = 1: one complex chiral component per chirality), the rotated Petz map of F(O_B) subset F(O_BC) with modular parameter t (lambda = 2t in the convention of Vardhan-Wei-Zou) factorises into the two chiral halves, which act as the geometric maps gamma_{s_t^+-} with s_t^+ = lambda_0 (1 + e^{-2 pi t}) and s_t^- = lambda_0 (1 + e^{+2 pi t}), lambda_0 = c/b the compression parameter of Note 4 (c, b the lengths of C, B); hence -log F^(lambda) = Phi(z(1+e^{-pi lambda})) + Phi(z(1+e^{pi lambda})), with Phi the single-chirality zero-collar recovery error and z = eta_V/(1 - eta_V) (rigor/check_note5_theory.tex prop:chirality <1>1-<1>7). This is even in lambda (interchange of the two arguments; for real lattice states evenness also holds identically by the antiunitary argument, l. 1422-1428), and at lambda = 0 it equals 2 Phi(2z). For a single chirality the rotated family collapses onto one function (Note 5 cor:collapse): -log F(omega, omega-tilde_t) = Phi(z(1 + e^{-2 pi t})), which is nonincreasing in t because Phi is nondecreasing (card phi-monotone-in-zeta); the ordinary Petz map (t = 0) sits at Phi(2z), and Phi(z) <= -log F(omega, omega-tilde_t) for every t: the geometric compression of ABC onto AB (s = lambda_0), the t -> +infinity endpoint, which is a limit point and not a member of the family, is at least as good as every member. Strict monotonicity and continuity of Phi are not proved. NOT claimed here: (i) that the Dirac error is minimal at lambda = 0 for every z (card two-chirality-minimum-at-lambda-zero, conjectural: its only proof step, prop:chirality <1>8, used the false card phi-derivative-nondecreasing); (ii) the law for a general 2D CFT (card two-chirality-law-general-cft, conjectural: it needs a chiral factorisation that is not proved).

How to verify

rigor/check_note5_theory.tex Prop. 8.2 (prop:chirality, l. 1369-1420), steps <1>1-<1>7 (step <1>8, the minimum, is not claimed); evenness also from the paragraph l. 1422-1428; single chirality: fidelity_recovered_quasifree.tex Cor. 5.1 (cor:collapse, l. 409-420) with check_note5_theory.tex Prop. 8.1 (prop:mono).

Evidence

Related claims

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B6.

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