cft-cmi

E_rec <= (1-theta) f2 z^2 (1+o(1)): the zero-collar compression is NOT optimal 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: result · Knowledge-base id: compression-not-optimal-exact-fidelity

Statement

PROVED (no numerical input): per chirality E_rec <= (1 - theta_fr) f2 z^2 (1+o(1)) (Thm 7.3), where theta_fr is the isometric-orbit gain sup over FINITE-RANK anti-self-adjoint G = E_D G E_D with v_G != 0; and theta_fr > 0 iff R_a != 0, equivalently (S10 Cor. 11.2: E_D u E_D is purely anti-self-adjoint) iff E_D u E_D != 0. Proof: the g_Q-free (UH) (card uhlmann-upper-half-proved) minimised in lambda and maximised in G, plus ||u||_2^2 = g_B/4 (S10 Prop. 4.3(2)) so (1/2)||u||^2 = f2. NUMERICAL: theta_fr >= 0.38, hence kappa_opt <= 0.616 +- 0.003 < 1, i.e. the zero-collar compression is beaten in EXACT fidelity, not merely in the second-order functional. The bridge from the published theta to theta_fr is Remark 7.4: (i) theta_fr = theta_HS by S_2-density of finite-rank anti-Hermitian G supported in D (all quantities depend on G only through P^perp G P, S_2-continuously), only theta_HS = theta (bounded G with P^perp G P in S_2 but G not in S_2) is open and is irrelevant for a LOWER bound; (ii) the explicit admissible HS direction G = R_a gives theta_fr >= 0.0143; (iii) every discretisation behind the numerical theta is finite-dimensional, so its optimisers are finite rank -- and F3's extrapolations are themselves taken over finite meshes, so theta_fr >= 0.38 is as well supported as theta = 0.384 +- 0.003 (Phase 5, F3, card theta-value; the modular Galerkin frame of S12 Sec. 5.2 and the parameter-free modular Wiener-Hopf frame of rigor/exact_optimum_tangent_problem.tex Prop 6.2 agree to 3e-4 after extrapolation; S10 Sec. 7's 0.300 +- 0.005 is a taper-suppressed LOWER bound). Wording updated in rigor/exact_fidelity_upper_half.tex Remark 7.4 (rem:thetafr) and the dated addendum to its Sec. 10. Sandwich (Cor 7.5): sup over admissible test families of liminf Delta_z(t_z)^2/(8 V(t_z) z^2) <= liminf E_rec/z^2 <= limsup E_rec/z^2 <= (1 - theta_fr) f2; the identification of the left side with D_z = min_F g_Q is S12 Thm 4.1(2) and is CONDITIONAL on a test family with ||t_z||_1^2/V(t_z) bounded and Delta_z(t_z)||t_z||_1/V(t_z) -> 0. Also note (referee R7): the structural chain is duality-free but the constant f2 reaches back to PA Prop. 7.6, hence to twisted duality.

How to verify

rigor/exact_fidelity_upper_half.tex Sec. 7 (Thm 7.3, Remark 7.4, Cor 7.5), Verdict and Sec. 10; rigor/optimality_all_channels.tex Secs. 5.1-5.2 for theta.

Evidence

Related claims

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.

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