cft-cmi

The all-channel second-order optimum equals the isometric-orbit value: kappa_opt = 1 - theta = 0.616 +- 0.003 (numerical; conditional on H1, H2, H3)

Status: numerical — measured, with an error bar and a convergence order; not a theorem. Conditional. The statement is not unconditional: it rests on H1, H2, H3, quoted below. Read the badge only together with them.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: result · Knowledge-base id: all-channel-optimum-value

Statement

Phase 5 answer to 'the exact value of the all-channel optimum (non-isometric channels)'. Analytic (rigor/exact_optimum_tangent_problem.tex, refereed, repairs applied): every feasible quasi-free channel is exactly (co-isometry V, isometry loss N = 1 - XX* >= 0, noise 0 <= Y = Z - X Q_BB X* <= N); the leading-order problem is a convex tangent programme, and the isometric-orbit optimum (value (1 - theta) g_Q(deltadot)) is its global optimum iff (a) M_* = Herm((t_* Q)_DD) >= 0 and M_* - tau_* >= 0 (tau_* = (t_*)_DD, as forms) and (e) the one-sided endpoint functional E(zeta) = 2 b(delta_*, A(zeta,0,0)) >= 0 for every admissible (dissipative) isometry generator zeta. For the two model boundary directions, E(zeta_inf) = 2 lambda (1 - theta) g_Q(ddelta) >= 0 is proved, and E(zeta_inf) > 0 is equivalent to theta < 1, which is numerical (1 - theta = 0.616 +- 0.003); E(zeta_0) > 0 is numerical; the reduction of the WHOLE cone of admissible generators to these two directions is proved only for vector fields (Prop 4.10) and is otherwise a declared hypothesis H3 (referee2_exact_optimum_tangent_problem.tex S1: multiplicity > 1 shift semigroups and unbounded skew-adjoint multiplication operators escape the argument). Numerical (F2, rigid parameter-free modular frame with exact Moebius rigidity): (a) holds in the limit (violation lambda_min/||M_*|| = -4.1e-3 ... -3.7e-4 at h = 0.24 ... 0.03, scaling as h^1.15 and 1/Ymax, achievable gain ~ h^1.85 -> 0); (e) on the model cone is equivalent to E(zeta_0) >= 0 (given the proved E(zeta_inf) >= 0; F1 Prop 4.10(4)-(5)), which holds numerically and vacuously: at zeta_0 the corner term (1/2) M_*(0,0)/h = +10.3, +16.1, +24.7, +37.2 at h = 0.24 ... 0.03 (~h^-0.60, the scheme-invariant continuum quantity; the discrete total +34 ... +258 is upwind-viscosity dominated and is not the margin) diverges, so A(zeta_0,0,0) is not in the form domain, E(zeta_0) = +infinity and (e) is VACUOUS at zeta_0 rather than satisfied with a margin (F1 Rem 4.11(ii), Prop 4.10(5)); no sign change in h, Y, Ymax. Hence, numerically and under H3, non-isometric quasi-free channels (isometry loss, noise, non-surjective isometries) do NOT lower the second-order recovery error: kappa_opt = 1 - theta with theta = 0.384 +- 0.003 (F3, two discretisations reconciled; S10's 0.300 was a taper-suppressed lower bound), i.e. E^(2) = (0.616 +- 0.003) f_2 z^2 per chirality. Conditional on H1 (differentiability along the scaled family), H2 (minimisers within the scaled-family form) and H3 (cone hypothesis) of the F1 document; the unconditional statements remain E_rec <= 0.616 f_2 z^2 (1+o(1)) (E1 upper half for isometric channels + theta) and E_rec >= dual form (optimality_all_channels.tex eq. (10)). Exact fidelity for NON-isometric channels was not needed: the optimum is attained on the isometric orbit, where E1 applies. Open: analytic proof of (a) and (e); H1, H2, H3; closed form of theta (Wiener-Hopf problem, prop:thetaframe); analytic proof that theta > 0 (E_D u E_D not self-adjoint).

Hypotheses

This result is conditional: at least part of the statement holds only where the hypotheses below hold. The sentences are quoted from the statement above; read the status badge only together with them.

Hypothesis labels used: H1, H2, H3.

The all-channel second-order optimum equals the isometric-orbit value: kappa_opt = 1 - theta = 0.616 +- 0.003 (numerical; conditional on H1, H2, H3).

E(zeta_0) > 0 is numerical; the reduction of the WHOLE cone of admissible generators to these two directions is proved only for vector fields (Prop 4.10) and is otherwise a declared hypothesis H3 (referee2_exact_optimum_tangent_problem.tex S1: multiplicity > 1 shift semigroups and unbounded skew-adjoint multiplication operators escape the argument).

Hence, numerically and under H3, non-isometric quasi-free channels (isometry loss, noise, non-surjective isometries) do NOT lower the second-order recovery error: kappa_opt = 1 - theta with theta = 0.384 +- 0.003 (F3, two discretisations reconciled;

Conditional on H1 (differentiability along the scaled family), H2 (minimisers within the scaled-family form) and H3 (cone hypothesis) of the F1 document; the unconditional statements remain E_rec <= 0.616 f_2 z^2 (1+o(1)) (E1 upper half for isometric channels + theta) and E_rec >= dual form (optimality_all_channels.tex eq. (10)).

H1, H2, H3; closed form of theta (Wiener-Hopf problem, prop:thetaframe); analytic proof that theta > 0 (E_D u E_D not self-adjoint).

How to verify

rigor/exact_optimum_tangent_problem.tex Thm 4.4 (conditions (a), (e)), Prop 4.10 (items (4)-(5) and the verdict box after Rem 4.7, :950-961, as corrected 2026-09-24, Corrections C2, C3), Sec. 5 and Sec. 7; rigor/referee_exact_optimum_tangent_problem.tex Secs. 11-12; numerics/optimality_all/F2_RESULTS.md Secs. T4, T5 with reproduce commands (f2_t4_wh.py, f2_t5_endpoint.py); numerics/optimality_all/F3_RESULTS.md Secs. 3-7 (theta).

Evidence

Related claims

Review

Review record: passed 2026-09-24 by REF-CARD-CFT.

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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