cft-cmi

Tangent SDP over (G, N1, Y1) in the circle model: orbit-only optimum matches (1-theta) to 1.1e-7, full optimum not resolvable (measured Gram cond 2.4e22)

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: tangent-sdp-noise-directions

Statement

The fixed-geometry tangent problem, min over G = iH anti-Hermitian on D, N1 >= 0, 0 <= Y1 <= N1 of g_Q(delta_c + [G,Q] + noise(N1,Y1)) with noise_DD = (1/2){N1,Q_DD} - Y1, noise_DA = (1/2) N1 Q_DA, noise_AA = 0, was posed as an SDP in the S10 spectral circle model (cvxpy, CLARABEL/SCS). SOLID OUTPUTS. (a) The ORBIT-ONLY restriction (N1 = Y1 = 0 forced) reproduces (1 - theta) g_Q(delta_c) to 1.1e-7 relative at L = 64 (cap 1e10) and 4.0e-7 (cap 1e6) -- this is agreement to 1.1e-7, not "exactly", and it is a statement about the orbit-only problem, i.e. a validation of the SDP formulation against the S10 CG, not about the full tangent optimum. (b) Homogeneity: delta_c -> 2 delta_c scales the ORBIT-ONLY optimum by exactly 4.00000000 in both the direct and the Gram form (L = 64, cap 1e6). (c) The rank-one gains along the KKT-violating directions, computed exactly without a solver, are 6.3e-9, 1.8e-12, 3.5e-11, 1.2e-10, 1.1e-10 in units of g_Q(delta_c) at L = 64, 96, 128, 192, 256 (cap 1e10), and the isometric rotations cannot absorb those directions (g_red/g_Q = 1 - 5e-4); no statement about kappa_opt is made here (that is the business of card kkt-noise-criterion-numerics, in a frame where rigidity is exact). (d) Subspace lower bounds on the gain (f2_sub_gain_fixed.out; each returned (N1,Y1) is checked for cone feasibility, and l and g_red are recomputed exactly outside the solver, so the numbers are valid lower bounds independently of the solver's own objective): at L = 64, subspace eigMtau k=4 gives gain/g_Q(delta_c) = 2.843e-07 (status optimal, cone feasibility -2.4e-12), svdQDA k=4 gives 9.204e-09 (optimal_inaccurate, -4.4e-14), mix k=8 gives 2.106e-04 (optimal_inaccurate, cone feasibility -3.4e-17, l recomputed -1.0000, g_red recomputed 3.2049e+05 by CG; the solver's own objective 6.970e+05 is 2.17x higher, i.e. the returned point is better than the solver thought, and a non-converged CG would only weaken the bound); at L = 128, eigMtau k=4 gives 5.586e-10 (optimal_inaccurate, -7.8e-16). Hence kappa_full <= 1 - theta - 2.106e-4 = 0.64883481 at L = 64 (1 - theta = 0.64904541) and kappa_full <= 0.70882131 at L = 128 (1 - theta = 0.70882131 to 9 digits). The remaining subspaces return no point, with honest solver statuses: infeasible_inaccurate (eigM and rand at L = 64, eigM and mix at L = 128) and infeasible (svdQDA at L = 128). CORRECTION TO THE FIRST VERSION OF THIS CARD. The earlier statement "at L = 128 and 256 every subspace solve failed" was wrong in its cause: the script crashed with "ValueError: matmul: Input operand 1 does not have enough dimensions" whenever the solver returned no point, because of an unguarded basis @ Nr.value with Nr.value = None. That is a code defect, not conditioning. It has been fixed (status caught, SCS fallback, cone feasibility measured) and the run redone; the table above is the corrected result. WHAT IS NOT RESOLVED: the FULL tangent optimum kappa_full in the circle model. The quadratic form is numerically singular there: measured Gram condition numbers (f2_gram_cond.out, 3|D|^2 real coordinates) are 2.50e+18 (L = 64, cap 1e4), 2.65e+19 (L = 64, cap 1e6), 2.37e+22 (L = 64, cap 1e10) and 2.51e+26 (L = 96, cap 1e10), with the smallest positive eigenvalue at the double-precision floor -- a direct consequence of the circle model's O(1/L) violation of Moebius rigidity, since the individual blocks of delta_c carry g_Q of order 1e5 x g_Q(delta_c). Consequences observed: CLARABEL fails with SolverError on the direct form at L = 64 for caps 1e4 and 1e10 and on the scale-free cone form for all caps; SCS does not converge in 1e5 iterations; the Gram-reduced form appears to succeed but its answer is a rank-cutoff artefact (it drops 161 of 675 directions below tol*max and then reports kappa_full = 0.5990 against 1 - theta = 0.6490 at L = 64, a spurious 5 percent gain obtained by treating small-curvature directions as free); and the UNCONSTRAINED relaxation (arbitrary Hermitian N1, Y1, no cone) is genuinely ill-posed -- CG on Ghat x = l does not converge and the gain estimate grows monotonically with the iteration count (8.7e-4 -> 6.8e-3 of g_Q(delta_c) between iterations 75 and 200 at L = 64), because with Y1 free the DD block of the defect can be set arbitrarily. The cone constraints are therefore essential and the correct scale-free formulation is q* = min{g_Q([G,Q] + noise(v)) : l(v) = -1, v in the cone} with gain = 1/(4 q*), l(v) = Tr(N1 M_*) - Tr(Y1 tau_*) in the half-SLD normalisation of t_*. This card reports BOUNDS in the circle model, not a value, and in a frame whose criterion verdict is shown to be artefact-dominated (card kkt-noise-criterion-numerics).

How to verify

cd numerics/optimality_all; $V/bin/python f2_tangent_sdp.py 64 (orbit-only check and homogeneity); f2_sub_gain.py 64,128 1e10 4 -> f2_sub_gain_fixed.out (each subspace prints its solver status, the cone feasibility of the returned point, and the exactly recomputed l and g_red); the Gram condition numbers are the one-liner quoted in F2_RESULTS.md T1 -> f2_gram_cond.out; f2_gram_verify.py 64 1e6 1e-14,1e-10,1e-6 (rank-cutoff artefact); f2_gain.py 64 1e10,1e4 200 (non-convergent CG ladder of the unconstrained relaxation)

Evidence

Related claims

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A2.

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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