cft-cmi

Full primal SDP over F in the modular Galerkin frame: min_F g_Q = 8.41 and 7.13 times g_Q(delta_c) at h = 0.4, 0.3 -- the S11 representability obstruction, not a statement about non-isometric channels

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: galerkin-full-f-uninformative

Statement

In the block-exact modular Galerkin frame of kkt_continuum.py (a = 1, L = 2, Lambda = 12, s = 0.1, kind = 'first'), the full primal SDP over F = {X : h_B -> h_D, Z = Z* : X Q_BB X* <= Z <= 1 - X(1-Q_BB)X*} was solved at two rungs (f2_galerkin_a.out; 9425 s at h = 0.4, 24923 s at h = 0.3): h = 0.4, N = 69, (nA,nB,nC) = (30,9,30): g_Q(delta_c) = 8.1495e-06 = 0.8687 f2 zeta^2, theta = 0.28578, min_F g_Q = 6.8546e-05, min_F/g_Q(delta_c) = 8.411034, ||1 - X X*||_F = 5.872, tr(1 - X X*) = 35.16 of m = 39, ||Z - X Q_BB X*||_F = 3.556, sv(X) in [0.0061, 0.9999]; h = 0.3, N = 91, (nA,nB,nC) = (40,11,40): g_Q(delta_c) = 8.4021e-06 = 0.8956 f2 zeta^2, theta = 0.29926, min_F g_Q = 5.9870e-05, min_F/g_Q(delta_c) = 7.125692, ||1 - X X*||_F = 6.780, tr(1 - X X*) = 46.71 of m = 51, sv(X) in [0.0002, 0.9999]. Both rungs land a factor 7-8 ABOVE the compression defect, not below the isometric-orbit value, and the optimal X is a strong contraction, nothing like a co-isometry. WHAT IS BEING COMPARED (do not over-read the ratio): min_F g_Q is an exact finite-s minimum over the discretised F, whereas theta, 1 - theta and g_Q(delta_c) on the same rung are FIRST-ORDER (tangent) reference numbers produced by orbit_theta() (the CG of theta_galerkin.py, which it reproduces exactly: 1 - theta = 0.71422 at h = 0.4 and 0.70074 at h = 0.3 against the published 0.714 and 0.701). The two are not values of one and the same programme -- they are computed in one script and share Q and delta_c, but not the same optimisation -- so the ratio must not be read as "the orbit beaten or not beaten by that factor". WHAT THE RUNGS DO SHOW is the S11 representability obstruction (rigor/sdp_certificate_numerics.tex l.484-497): in a finite Galerkin cell frame the Moebius compression is not representable at all (its V maps cells to non-cells), so the discretised F does not contain the compression point, and min_F g_Q is dominated by that representation error rather than by the physics of non-isometric channels. H-TREND: the two rungs move in the right direction at fixed Lambda and collar, 8.411034 (h = 0.4) -> 7.125692 (h = 0.3). Two points are a trend, not a verdict, and no extrapolated "factor still to be closed" is claimed. The earlier runs kc_h04.out and kc_h05.out (8.2635 and 10.0467) are NOT comparable rungs of the same ladder: they differ in LamC (6 and 4 vs 12), n_B (9 and 7 vs 9 and 11) and n_C (15 and 8 vs 30 and 40); S11 identifies n_B/n_C and the collar s, not h alone, as the knobs, and kc_h04.out's Delta/(2 g_Q) = -10.325 is S11's n_B < n_C corner. CONCLUSION: this test cannot decide whether non-isometric channels beat the isometric orbit; the tangent (first-order) problem is the right place to ask, and the card kkt-noise-criterion-numerics answers it only as far as the numerics can: condition (a) as an extrapolation, condition (e) under H3 (F2_RESULTS.md:250, corrected 2026-09-24, F2 Correction C8).

How to verify

cd numerics/optimality_all; $V/bin/python f2_galerkin_full.py 0.4,0.3 (prints, per rung, the orbit CG value and min_F g_Q) -> f2_galerkin_a.out l.1-6; compare kc_h04.out, kc_h05.out for the non-comparable earlier runs; the representability obstruction is rigor/sdp_certificate_numerics.tex l.484-497

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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