Structure of the numerical optimum

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: optimal-channel-structure

Statement

The quasi-free optimum on lattices is a near-isometric squeeze (singular values 0.95-1.00) that does not reproduce the BC marginal (Petz's defining property is suboptimal); the all-channel optimum on tiny chains is Gaussian and gauge covariant. As of S8/S10, non-isometric channels (X X* < 1, genuine noise Y > 0) had not been analysed in the continuum (rigor/sdp_dual_certificate.tex verdict gaps :754-757 and (ii) :951-953). They were analysed afterwards in Phase 5: the exact (N, Y, V) chart and the tangent programme with N_1, Y_1 (tangent-problem-exact-parametrisation), the KKT criterion (kkt-noise-sign-criterion), its continuum-frame numerics (kkt-noise-criterion-numerics) and the circle-model SDP (tangent-sdp-noise-directions). The exact value of min_F g_Q is still not proved; numerically kappa_opt = 1 - theta = 0.616 +- 0.003, conditional on H1, H2, H3 (card all-channel-optimum-value).

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 exact value of min_F g_Q is still not proved; numerically kappa_opt = 1 - theta = 0.616 +- 0.003, conditional on H1, H2, H3 (card all-channel-optimum-value).

How to verify

S8 Table 1; numerics/optimality_all/analyze_opt.py output.

Evidence

• rigor/optimality_second_order.tex — document, at sec:numerics
• numerics/optimality_all/analyze_opt.py — numerics

Related claims

• 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) — numerical
• Exact (N,Y,V) chart of F and the tangent problem at the compression — proved
• KKT criterion numerics: every finite run FAILS (a); the violation extrapolates to 0^- as h -> 0, Ymax -> inf; under H3, (e) follows from the proved E(zeta_inf) >= 0 alone, and the finite-h junction values are positive — numerical

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A4.

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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