cft-cmi

The first-order gain constant theta = 0.384 +- 0.003 (S10's 0.300 is a taper-suppressed lower bound)

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: constant · Knowledge-base id: theta-value

Statement

theta = sup_G cos^2(u,v_G) = 0.384 +- 0.003, hence 1-theta = 0.616 +- 0.003. The two published values are reconciled: they are the SAME number at different resolutions, not different limits. Five independent discretisations of F1's geometry-free Wiener-Hopf problem (Prop 6.2) extrapolate to 0.3835-0.3847: T0 uniform cells in the modular coordinate y=-log(1-x) with y_max=10 (theta = 0.31367, 0.34236, 0.35934, 0.36945, 0.37544 at h = 0.24, 0.12, 0.06, 0.03, 0.015) and with y_max=20 (0.31581, 0.34395, 0.36036, 0.37009, 0.37380 at h = 0.24, 0.12, 0.06, 0.03, 0.02); the S11 modular Galerkin frame at a=1, L=2 with Lambda=12 (0.29797 ... 0.36644 at h = 0.32 ... 0.04) and Lambda=8 (0.36608, 0.36962, 0.37346 at h = 0.04, 0.03, 0.02); and the same Galerkin functional on the circle model's own mesh (0.34058, 0.35830, 0.36904, 0.37534, 0.37893 at L = 128...2048). The measured convergence order is h^0.77 +- 0.02 (successive-increment ratio rho = 0.577...0.592 per family -- A: 0.581-0.591, B: 0.581, 0.591, C: 0.577, 0.588, C': 0.592, D: 0.570-0.606 -- stable over four octaves down to h=0.015; fractional because the defect kernel -(i/2pi) sinh(y1/2) sinh(y2/2)/sinh^2((y1-y2)/2) is homogeneous of degree 0 at the corner y=0, confirmed by corner-graded meshes which saturate). The modular-window systematic obeys theta(y_max) = theta_inf - c(h)/y_max with c ~ h^0.6 (residual 1.5e-5), worth +0.0009 to +0.0032. theta is geometry independent: the spread is 5.1e-4 over the four Galerkin rows (a,L) = (1,2), (2,1), (1,4) and s = 0.1, 0.3 at fixed h=0.1, Lambda=12 (0.34843, 0.34894, 0.34854, 0.34863), and 2.0e-3 over the whole table once the T0 (inf,1) row is included (0.34695 at h=0.1, Y=Ymax=8) -- that extra spread is the frame change (T0's smaller modular window, worth about -1.5e-3 by the y_max law above), not the geometry, as Moebius covariance requires. S10 Sec 7(c)'s 0.300 +- 0.005 is a LOWER BOUND: its ultraviolet taper removes the modes carrying ~20% of theta (see finding circle-model-taper-suppresses-theta), and its 1/L Richardson uses the wrong variable (theta_circ rises linearly in log L: 0.28958 -> 0.30173 from L=256 to 1024 with no saturation). Closed forms: 3/pi^2, 1/3, 1-2/pi, (pi^2-6)/12, 2-12/pi^2, 3/8, pi/8 and 16 further candidates are EXCLUDED; 1-pi^2/16 = 0.38315, 2 ln2 - 1 = 0.38629 and 2/pi - 1/4 = 0.38662 lie inside the window; with 26 candidates tested, a window of width 0.006 and 0.28 hits expected by chance, no selection is made (f3_candidates.py / f3_candidates.out). This card SUPERSEDES the window 0.30-0.40 of theta-modular-wiener-hopf / exact_optimum_tangent_problem.tex Sec. 6 (both rewritten 2026-09-15 17:27-17:29 by F1).

How to verify

cd numerics/optimality_all; $V/bin/python f3_smoke.py reproduces theta_galerkin.py (0.29926) and s10_diag.py (0.28958). f3_t0.py smoke|Y|ymax|ymax2|hladder|hladder2|final|grade is the T0 Wiener-Hopf frame; f3_gal.py uni|ladder8|lamsmall|misc the Galerkin frame; f3_mimic.py main|big|deep|geom the Galerkin functional on the circle mesh; f3_circ2.py Lsweep|taper the circle model; f3_close.py prints the y_max law, the measured order and the Richardson closure; f3_candidates.py (output f3_candidates.out) is the 26-candidate closed-form test. Tables and interpretation in numerics/optimality_all/F3_RESULTS.md.

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.

Documentation map · All claims by area · What is open · Symbols · Definitions