cft-cmi

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

Status: numerical — measured, with an error bar and a convergence order; not a theorem. Stated under hypotheses H3, quoted below.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: result · Knowledge-base id: kkt-noise-criterion-numerics

Statement

Numerical evaluation of the F1 Thm 4.4 criterion for the tangent problem (orbit = global optimum iff (a) M_* >= 0 and M_* - tau_* >= 0, and (e) the endpoint functional E(zeta) >= 0 for every admissible zeta with A(zeta,0,0) in the form domain F (thm:kktmain(e), exact_optimum_tangent_problem.tex:796-798), which on F1's model cone reads E(zeta_0) >= 0 and E(zeta_inf) >= 0; t_* is the SLD normalised by g_Q(delta_*) = (1/4) Tr(t_* delta_*), tau_* = E_D t_* E_D, M_* = Herm((t_* Q)_DD)). READ THE SIGN CLAIM AS AN EXTRAPOLATION: at every finite discretisation the run prints "CRITERION (M>=0 and M-tau>=0): False" (f2_t4_h024.out l.4, f2_t4_hladder.out l.4 and l.12, f2_t4_h003.out l.4, every Ymax row of f2_t4_ymax12.out); what the numerics establish is that the violation scales to zero in the double limit, with clean power laws, while the endpoint functional is positive at every finite discretisation (the regularised junction test, rem:H3space(2), tex :1223-1227); under hyp:H3, (e) holds outright from the proved E(zeta_inf) >= 0 (rem:H3space). (1) PRIMARY FRAME, parameter-free modular Wiener-Hopf (F1 Prop 6.2 as built by F3 in f3_t0.py; I = (-inf,1), y = -log(1-x), A = {y<0}, D = {y>0}, Q exact in the cell basis, ddot in closed form, Moebius rigidity EXACT: ||ddot_DD|| = 0 identically). Structural checks all pass: SLD identity (1/4)Tr(t_* delta_*)/g_Q(delta_*) = 1.0000000000; ||AntiHerm((t_* Q)_DD)||/||M_*|| = 2.8e-9, 8.5e-8, 1.5e-7, 3.5e-7 at h = 0.24, 0.12, 0.06, 0.03; F1's derivative formula d g_Q = (1/2)[Tr(N1 M_*) - Tr(Y1 tau_*)] verified by central finite differences to <= 3e-8 (measured relative errors 3.94e-10, 1.08e-08, 7.56e-09, 2.63e-08 at h = 0.24, 0.12, 0.06, 0.03: f2_t4_h024.out l.5, f2_t4_hladder.out l.5 and l.13, f2_t4_h003.out l.5); theta reproduces F3's ladder (0.31367, 0.34236, 0.35934, 0.36945 at h = 0.24, 0.12, 0.06, 0.03, Y = Ymax = 10). (a) SCALING OF THE VIOLATION. h-ladder at Y = Ymax = 10: lam_min(M_*)/||M_*|| = -4.076e-3, -1.859e-3, -8.228e-4, -3.667e-4 at h = 0.24, 0.12, 0.06, 0.03 (ratios 2.19, 2.26, 2.24 per halving, i.e. ~ h^1.15), and lam_min(M_* - tau_*) equals lam_min(M_*) to 5 digits (different eigenvectors with the same localisation and modulus profile and opposite <p,tau_*p>: at h = 0.24, <p,M_*p> = -5.4316e-4 and +2.7654e-3, <p,tau_*p> = -3.3086e-3 and +3.3086e-3, f2_t4_h024.out l.6-7). Outer-cutoff scan at h = 0.12, Y = 6: lam_min(M_*)/||M_*|| = -3.014e-3, -2.286e-3, -1.865e-3, -1.339e-3, -9.459e-4, -6.360e-4 for Ymax = 6, 8, 10, 14, 20, 30, i.e. lam_min/||M_*|| * Ymax = -0.0181, -0.0183, -0.0186, -0.0187, -0.0189, -0.0191 (~ 1/Ymax), and the rank-one gain Delta^2/g_Q(eta) in units of g_Q(ddot) = 3.325e-4, 1.874e-4, 1.230e-4, 6.236e-5, 3.080e-5, 1.380e-5 with gain * Ymax^2 = 0.0120 ... 0.0124 (~ 1/Ymax^2); on the h-ladder the same gain is 3.85e-4, 1.23e-4, 3.51e-5, 9.69e-6 (~ h^1.85). So lam_min(M_*)/||M_*|| ~ -c(h)/Ymax with c(0.24) = 0.041, c(0.12) = 0.0187 (c ~ h^1.1), and the gain ~ c'(h)/Ymax^2 with c'(0.24) = 0.039, c'(0.12) = 0.0123: both -> 0 in the double limit h -> 0, Ymax -> infinity, i.e. lam_min -> 0^- (a marginal, not a strict, sign). The violating eigenvector p supports this reading: in the Ymax scans at fixed h it is a delocalised mode at y ~ (0.37-0.6) Ymax (0.37 Ymax at Ymax = 30), tracking the artificial outer end of the y window (the moving endpoint x = 1); the A-side cutoff is irrelevant (Y = 6, 10, 14 give lam_min/||M_*|| = -1.8645e-3, -1.8586e-3, -1.8584e-3 at h = 0.12, Ymax = 10: the Y-scan block of f2_t5_endpoint.out, l.102, l.112, l.122; the h = 0.24 triple -4.0904e-3, -4.0756e-3, -4.0751e-3 is reported in F2_RESULTS.md:306, and of it only the Y = 10 value -4.0756e-3 is in an output file, f2_t4_h024.out l.2). At fixed Ymax = 10 the mode moves inward as h falls (y ~ 5.5, 5.1, 4.3, 1.9) while its eigenvalue keeps shrinking; the two vanishing scalings, not the location, carry the conclusion. (e) ENDPOINT CONDITION (T5, f2_t5_endpoint.out). E(zeta) = 2 b(delta_*, A(zeta,0,0)) = (1/2)Tr(t_* A(zeta,0,0)) with A from F1 eq:Ablocks. Convention check, exact to machine precision at all four h: E(zeta_inf)/g_Q(ddot) = (1/2)Tr(t_* ddot)/g_Q(ddot) = 1.37265519, 1.31527744, 1.28132522, 1.26109117 (f2_t5_endpoint.out l.3, 13, 23, 33) against 2(1-theta_h) = same to 2e-16..1e-15; the identity E(zeta_inf) = 2 lambda (1-theta) g_Q(ddot) is proved (F1 Prop 4.10 <1>4), its positivity needs theta < 1, which is numerical (theta_h = 0.314-0.369 here). For zeta_0 = -E_D d/dy E_D discretised on the uniform D-mesh, only the backward (upwind) difference is the admissible shift semigroup (sigma = Herm(zeta_0) <= 0); it gives a TOTAL E(zeta_0)/g_Q(ddot) = +34.122, +66.873, +131.298, +257.998 at h = 0.24, 0.12, 0.06, 0.03, but that total is the scheme-DEPENDENT quantity: the discrete boundary form decomposes as E(zeta_0) = (1/2)[M_*(0,0) + h Tr((-Laplacian) M_*) + M_*(Ymax,Ymax)] and the total (~ h^-0.97) is dominated by the O(h) upwind-viscosity middle term, an artefact of the difference scheme. The scheme-invariant piece, the one that represents the continuum E(zeta_0) (tex :1169, :1588), is the CORNER row (1/2)(M_*)_00/h = +10.304, +16.096, +24.672, +37.210 (~ h^-0.60); it diverges, and it is the value quoted below for the regularised (finite-h) junction directions. The centred difference is exactly anti-Hermitian (sigma = 0, lies in L-bar) and gives E = 0 to 2e-5; the forward difference has sigma >= 0 (decompression, inadmissible) and gives the negative of the backward value. Both quantities saturate in Ymax (E_bwd = +66.90, +67.07, +67.23, +67.36, +67.45 and corner = +16.46, +16.15, +16.03, +15.95, +15.89 for Ymax = 6, 10, 14, 20, 30 at h = 0.12 and Y = 6, the T5b scan) and are independent of Y (+67.07, +66.87, +66.87 for Y = 6, 10, 14). The corner term does NOT saturate in h (it grows like h^-0.6), i.e. M_*(y,y) diverges as y -> 0+ and E(zeta_0) = +infinity in the sense of the second case of eq:Efun (A(zeta_0,0,0) not in F): a convention, not a margin (tex :1033-1037). Cross-check (ii): the first-cell diagonal of M_*/h is +0.160, +0.261, +0.409, +0.623 (same sign as E(zeta_0)) and the last-cell diagonal is +3.2e-3, +3.9e-3, +4.4e-3, +4.8e-3 (the last at h = 0.03, f2_t5_endpoint.out l.40; same sign as E(zeta_inf) > 0). VERDICT ON (e), QUALIFIED: the document's reading applies. The corner row diverges, so A(zeta_0,0,0) is not in F and E(zeta_0) = +infinity: (e) is VACUOUS at zeta_0 rather than satisfied with a margin (exact_optimum_tangent_problem.tex:1171-1176, :1591-1593). In the model-cone form E(zeta_0) > 0, the corner row is +10.3, +16.1, +24.7, +37.2 g_Q(ddot) at h = 0.24, 0.12, 0.06, 0.03, with no sign change at any h, Y, Ymax, and E(zeta_inf) = 2(1-theta_h) g_Q(ddot) (the proved identity) > 0 because theta_h < 1 numerically. At finite h the absorption identity E(zeta_0) = (1/2)Tr((-2 sigma) M_*) holds to <= 7.5e-8 (f2_t5_endpoint.out l.7, 17, 27, 37), but M_* >= 0 fails in every run and -2 sigma has full rank, so the positive sign is a numerical fact, not a forced one; in the continuum the identity is not available (prop:bdryfun(5), tex :1094-1096). Under hyp:H3 (tex :1191-1198), read in F, alpha = 0 is forced because A(zeta_0,0,0) is not in F: every admissible generator carrying a feasible first-order datum is, modulo L-bar, a non-negative multiple of zeta_inf, and (e) holds outright from E(zeta_inf) >= 0 alone (rem:H3space, :1209-1218). H3 is a hypothesis, not something these numerics test. So: numerically (a) in the limit; E(zeta_0) > 0 and E(zeta_inf) > 0; (e) for the whole cone under H3. CAVEAT ON THE PRIMARY FRAME. It needs no weight cap at the h computed, but the weights are not bounded uniformly in h: kappa_max = 5.38, 6.51, 7.66, 8.84 and max w = 1.09e2, 3.36e2, 1.07e3, 3.44e3 at h = 0.24, 0.12, 0.06, 0.03 (ratios ~3.1-3.2 per halving, i.e. max w ~ h^-1.65). So h -> 0 is exactly the limit in which this frame's weights diverge as well; the extrapolation above is over a window in which max w grows by a factor 32, and a much finer h would eventually re-enter the regime where the T2 artefact mechanism (below) operates. (2) SECONDARY FRAME, ARTEFACT-SUSPECT, S10 spectral circle model: the criterion FAILS at every L = 64, 96, 128, 192, 256, with lam_min(M_*)/||M_*|| = -0.206, -0.0075, -0.0352, -0.0579, -0.0498 and lam_min(M_* - tau_*) equal to it to 3 digits (at L = 96, 128, 192 only 3 digits agree, f2_kkt_f1_L.out l.43-45). That verdict must NOT be used, for three reasons, all in F1's normalisation as printed by f2_capscan.py (file f2_capscan_L128.out; the half-SLD convention of kkt_continuum.gQ_sld would halve lam_min and quarter the gain): (i) it is not cap-stable -- at L = 128, taper uv = 0.30, lam_min(M_*) = -3.211e-3, -3.134e-3, -7.887e-3, -3.477e-2, -1.050e-1 and lam_min/||M_*|| = -0.0122, -0.0067, -0.0105, -0.0352, -0.0753 for cap = 1e4, 1e6, 1e8, 1e10, 1e12, while theta is cap-stable to 4 digits (0.29117, 0.29119, 0.29116, 0.29118, 0.29119); (ii) the achievable gain collapses as the cap grows: rank-one gain/g_Q(dc) = 1.463e-4, 1.948e-6, 1.722e-10, 3.534e-11, 3.718e-12 on the same ladder (the violating directions have diverging curvature, g_Q(noise)/g_Q(dc) = 1.3e3 ... 5.5e13); the second taper uv = 0.20 repeats the pattern (lam_min/||M_*|| = -0.0125, -0.0070, -0.0088, -0.0258, -0.0287; gain = 5.808e-4 ... 6.444e-13) with theta shifted by 10% (0.2615); (iii) the mechanism: the circle model violates Moebius rigidity at O(1/L) (|dc_DD|/|dc| = 0.346, 0.214, 0.153, 0.101, 0.075 and |dc_AA|/|dc| = 0.048, 0.032, 0.024, 0.016, 0.012 at L = 64..256, both ~ 1/L, whereas the exact defect is purely A-D off-diagonal), and g_Q weights those spurious blocks by w up to the cap -- taken separately the blocks of delta_c carry g_Q of order 1e5 x g_Q(delta_c), so t_* and hence M_*, tau_* are contaminated exactly where the weights are largest. WHAT MAY AND MAY NOT BE CONCLUDED. The numerics are consistent with the isometric orbit carrying the tangent optimum and kappa_opt = 1 - theta, and they exclude any O(1) gain from non-isometric or boundary-moving directions (every measured rank-one gain is <= 1.05e-3 g_Q(ddot): the largest, 1.049e-3 at h = 0.24, Ymax = 6, is reported only in F2_RESULTS.md:302-303; in the runs with output files it is <= 3.85e-4, f2_t4_h024.out l.6; all fall like h^1.85/Ymax^2). They do NOT establish that the optimal N_1, Y_1 vanish: Thm 4.4(d) delivers a global minimiser, not uniqueness, and the marginal case lam_min -> 0^- found here is precisely where uniqueness is not delivered.

Hypotheses

The statement is made under named hypotheses. The sentences are quoted from the statement above; read the status badge only together with them.

Hypothesis labels used: H3.

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.

READ THE SIGN CLAIM AS AN EXTRAPOLATION: at every finite discretisation the run prints "CRITERION (M>=0 and M-tau>=0): False" (f2_t4_h024.out l.4, f2_t4_hladder.out l.4 and l.12, f2_t4_h003.out l.4, every Ymax row of f2_t4_ymax12.out); what the numerics establish is that the violation scales to zero in the double limit, with clean power laws, while the endpoint functional is positive at every finite discretisation (the regularised junction test, rem:H3space(2), tex :1223-1227); under hyp:H3, (e) holds outright from the proved E(zeta_inf) >= 0 (rem:H3space).

Under hyp:H3 (tex :1191-1198), read in F, alpha = 0 is forced because A(zeta_0,0,0) is not in F: every admissible generator carrying a feasible first-order datum is, modulo L-bar, a non-negative multiple of zeta_inf, and (e) holds outright from E(zeta_inf) >= 0 alone (rem:H3space, :1209-1218).

H3 is a hypothesis, not something these numerics test.

(e) for the whole cone under H3.

How to verify

cd numerics/optimality_all; $V/bin/python f2_t4_wh.py 0.24 10 10 > f2_t4_h024.out (smoke: theta=0.31367, |dc_DD|=0, SLD id=1.0000000000, lam_min(M_*)=-5.432e-4, CRITERION False); f2_t4_wh.py 0.12,0.06 10 10 > f2_t4_hladder.out; f2_t4_wh.py 0.03 10 10 > f2_t4_h003.out; the Ymax/Y scan one-liners quoted in F2_RESULTS.md ("Exact commands to reproduce") -> f2_t4_ymax12.out; endpoint test: f2_t5_endpoint.py 0.24 10 10 and the driver quoted in F2_RESULTS.md T5 -> f2_t5_endpoint.out; circle model: f2_kkt_f1.py 64,96,128,192,256 1e10 -> f2_kkt_f1_L.out and f2_capscan.py 128 1e4,1e6,1e8,1e10,1e12 0.30,0.20 -> f2_capscan_L128.out

Evidence

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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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