Is the compression beaten in EXACT fidelity?

Status: numerical — measured, with an error bar and a convergence order; not a theorem.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: question · Knowledge-base id: exact-fidelity-optimality-open

Statement

ANSWERED 2026-09-15 (Phase 4): yes, modulo one numerical input. Proved (rigor/exact_fidelity_upper_half.tex, refereed 'stands with repairs', repairs applied): exterior directions are dense in the invisible space (Thm 5.3), so the optimised Uhlmann bound equals the second-order functional for every isometric quasi-free channel (Thm 4.3, Cor 4.4, Thm 7.1 in g_Q-free form) and E_rec <= (1 - theta_fr) f2 z^2 (1+o(1)) per chirality (Thm 7.3), with theta_fr > 0 iff E_D u E_D is not self-adjoint. Numerical: theta_fr >= 0.38, hence kappa_opt <= 0.616 +- 0.003 (Phase 5, F3, card theta-value: theta = 0.384 +- 0.003 from the modular Galerkin and modular Wiener-Hopf frames, agreeing to 3e-4; the spectral model's 0.300(5) is a taper-suppressed LOWER bound). Lower half: E_rec >= the unconditional dual form of optimality_all_channels.tex eq. (10); the sharper statement E_rec >= D_z (1-o(1)) is conditional on an admissible test family (Cor 7.5 of the E1 document, referee repair R5). Earlier state (2026-09-10 to 14): open, with the finite spectral model's exact fidelity UV-polluted and S11's windowed ratios inconclusive; superseded by the exterior-extension route.

How to verify

rigor/exact_fidelity_upper_half.tex Thms 5.3, 7.1, 7.3 and Cor 7.5; rigor/referee_exact_fidelity_upper_half.tex; numerics/optimality_all/e2_dist_ub.out (UB ratios).

Evidence

• rigor/exact_fidelity_upper_half.tex — document, at thm:main
• rigor/exact_fidelity_upper_half.tex — document, at thm:UHall
• rigor/referee_exact_fidelity_upper_half.tex — document, at sec:verdict
• rigor/exterior_extension_numerics.tex — document, at sec:ub
• rigor/optimality_all_channels.tex — document, at eq:liminf

Related claims

• The compression is not the minimiser of the second-order functional — numerical
• Exact single-observable fidelity bound (Alberti-Uhlmann) and its reduction to the recovered symbol — proved
• Exterior directions exhaust the invisible space: closure(Ext) = N^perp (question (Q) settled, YES) — proved
• (UH) proved: optimised Uhlmann bound = g_Q of the defect, for every isometric quasi-free channel — proved
• E_rec <= (1-theta) f2 z^2 (1+o(1)): the zero-collar compression is NOT optimal in exact fidelity — numerical

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A1.

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