Phi(zeta) and D(zeta) are nondecreasing in zeta (check_note5_theory.tex Prop. 8.1)
Status: proved — a complete structured proof.
Area: Theorem A: exact recovery error of the free fermion · Card type: result · Knowledge-base id: phi-monotone-in-zeta
Statement
For the free chiral fermion the zero-collar recovery error Phi(zeta) = -log F(omega, omega o beta-bar_s) and the relative entropy D(zeta) of the recovered state are nondecreasing functions of zeta = a s/(a+L) on (0, infinity) (rigor/check_note5_theory.tex prop:mono). Proof: at fixed (L, s) the glued map k_s does not depend on a, so for a_1 < a_2 the normal map on F_r(I_2), I_i = (-a_i, L), restricts to the one on F_r(I_1) (uniqueness of the normal extension, card petz-zero-collar-channel); root fidelity and Araki relative entropy are monotone under restriction to a subalgebra, so Phi(zeta(a_1, L, s)) <= Phi(zeta(a_2, L, s)); every pair zeta_1 < zeta_2 is realised by s > zeta_2, L = 1, a_i = zeta_i/(s - zeta_i); and Phi depends on (a, L, s) only through zeta (Note 5 Lemma 1.1, card mobius-reduction-cross-ratio). Note 5 itself left the monotonicity numerical (proof of cor:collapse).
How to verify
rigor/check_note5_theory.tex Prop. 8.1 (prop:mono, l. 1233-1255), steps <1>1-<1>6; the verdict box after it notes that Note 5's own reason was not an argument.
Evidence
- rigor/check_note5_theory.tex — document, at
prop:mono - fidelity_recovered_quasifree.tex — document, at
cor:collapse
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 - Weak link: Phi(zeta) <= 0.0085 zeta^2 on (0, 1] (rigorous on (0, 0.8868], numerical on [0.8868, 1]) —
numerical phi-derivative-nondecreasing—refuted, no page of its own
Review
Review record: passed 2026-09-23 by REF-DEP-CFT-B4.
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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