Two-chirality minimum: for the massless Dirac fermion -log F^(lambda) is minimal at lambda = 0 for every z (unproved)

Status: conjectural — believed, with partial evidence; not proved.

Area: Theorem A: exact recovery error of the free fermion · Card type: conjecture · Knowledge-base id: two-chirality-minimum-at-lambda-zero

Statement

Conjecture: for the 1+1-dimensional massless Dirac fermion, g(lambda) := Phi(z(1+e^{-pi lambda})) + Phi(z(1+e^{pi lambda})) (the two-chirality law, card two-chirality-law) satisfies g(lambda) >= g(0) = 2 Phi(2z) for every real lambda and every z > 0, i.e. the ordinary Petz map is the best member of the rotated family for two chiralities. No proof exists: the only argument in the documents, rigor/check_note5_theory.tex prop:chirality step <1>8 (l. 1413-1418), uses 'Phi' nondecreasing', which is false (card phi-derivative-nondecreasing, refuted); Note 5 (eq:collapse-vwz) and paper 1 (prop:twobranch) asserted the minimum with no valid argument until 2026-09-24; both now say that it is proved only at leading order (fidelity_recovered_quasifree.tex Correction C5; paper 1 fixed in place). Known: (a) g is even (card two-chirality-law); (b) for each fixed lambda != 0, g(lambda) > g(0) for all sufficiently small z, since Theorem A gives g(lambda) = f2 z^2 [2 + 4 cosh(pi lambda) + 2 cosh(2 pi lambda)] (1 + o(1)) and the bracket exceeds 8 for lambda != 0 (not uniform in lambda); (c) with t = lambda/2 and zeta_{+-} = z(1 + e^{+-2 pi t}), g'(t) = 2 pi [(zeta_+ - z) Phi'(zeta_+) - (zeta_- - z) Phi'(zeta_-)], so the minimum would follow from the weaker, UNPROVED property that zeta Phi'(zeta) is nondecreasing on (0, infinity) (then (zeta - z) Phi'(zeta) increases on zeta > z); the large-zeta values of card large-zeta-values (extrapolated estimates above rigorous windowed lower bounds, not certified; called the certified values here until 2026-09-24) are consistent with it (zeta Phi' = 5.7e-3, 1.27e-2, 2.11e-2, 3.01e-2, 4.42e-2 at zeta = 1.07, 2.13, 4.27, 8.53, 17.1) and the logarithmic bound that refutes 'Phi' nondecreasing' does not exclude it; (d) on the lattice the small-eta amplitude (-log F^(lambda))/eta^2 = 0.0722, 0.300, 4.49, 70.2 at lambda = 0, 0.5, 1, 1.5 (numerics/lattice/petz_lattice.out [7], eta = 0.0044, n = 64) increases with lambda, and VWZ observe the largest fidelity at lambda = 0.

How to verify

Broken step: rigor/check_note5_theory.tex l. 1413-1418 (<1>8). Sufficient condition and derivative formula: from <1>8's own expression for g'(t) with e^{+-2 pi t} = (zeta_{+-} - z)/z. Values: numerics/kb-checks-2026-09-23/phi_prime_check.out (last column); lattice: petz_lattice.out [7].

Evidence

• rigor/check_note5_theory.tex — document, at prop:chirality
• fidelity_recovered_quasifree.tex — document, at eq:collapse-vwz
• paper1/sec_2d_lattice.tex — document, at prop:twobranch
• numerics/lattice/petz_lattice.out — numerics
• numerics/kb-checks-2026-09-23/phi_prime_check.py — numerics
• numerics/kb-checks-2026-09-23/phi_prime_check.out — numerics

Related claims

• 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 — proved
• phi-derivative-nondecreasing — refuted, no page of its own
• Two-branch law confirmed on the lattice — numerical
• Theorem A: lim Phi/zeta^2 = c/(12 pi^2) — refereed

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.

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