Moebius reduction: the fidelity and relative entropy of the geometric recovery depend on (a, L, s) only through zeta = a s/(a+L), including the half-line chart a = infinity (zeta = s) (Note 5 Lemma 1.1; check_note5_theory prop:closed)
Status: proved — a complete structured proof.
Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: result · Knowledge-base id: mobius-reduction-cross-ratio
Statement
For the chiral free fermion, F(omega, omega o gamma_s) and the relative entropy of the geometrically recovered state depend on (a, L, s) only through zeta = a s/(a+L), where 1 + zeta is the cross ratio of the four points -a < 0 < L/(1+s) < L; the proof uses only Moebius covariance of the vacuum (the half-density action commutes with the Hardy projection and carries F(X) onto F(g(X))). Note 5's proof (lem:mobius) assumes g(I) bounded (fidelity_recovered_quasifree.tex:208; check_note5_theory.tex:180-185), so it covers finite configurations only. The reduction to the half-line chart a = infinity, L = 1 (zeta = s), used by uhlmann_upper_bound.tex:69-71, :923-926 (the Moebius map with its pole at -a), follows from the closed-form proof of check_note5_theory.tex prop:closed (Prop. 6.1). In the xi-representation the defect kernel is zeta S_p S_p' / (sinh M (sinh M + 2 zeta S_p S_p')), with zeta = s beta_0/L = a s/(a+L), a function of zeta alone, and that proof holds unchanged at a = infinity.
How to verify
No verification recipe recorded on this card.
Evidence
- fidelity_recovered_quasifree.tex — document, at
lem:mobius - rigor/check_note5_theory.tex — document, at
prop:closed
Related claims
None recorded.
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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