D-046 — Fork 2: spectral-Galerkin sizing study for the λ → λ_a⁻ channel

Date: 2026-08-19 | Session: 008 | Fork 2 of the D-044 follow-up fork Status: complete. Verdict: INFEASIBLE with current and refined machinery — the two feasibility windows are disjoint (→ G-004).

1. The requirement

The λ-channel (D-044 §4) needs discretization error floor ≪ λ_a(a) at an a where zeros are resolvable (a ≳ 2–4: lattice spacing π/a vs γ-gaps; valleys measurable).

2. The true λ_a law (from E-009b data, floor-corrected)

Trustworthy (above-floor) values only at a ≤ 1: a=0.25: 4.14e-1; a=0.5: 2.07e-4; a=1.0: 1.44e-9. Decay accelerates (ratios 2e3 per 0.25, then 1.4e5 per 0.5) — superexponential, consistent with Kim et al.'s report. All values at a ≥ 2 are discretization floor (refinement-chasing: a=2: 7.4e-11 → 3.3e-12 per halving; a=4: 5.6e-12 → 3.2e-13). The true λ_a(2) is < 3e-12; λ_a(4) < 3e-13 — plausibly vastly smaller.

3. The artifact mode (diagnosed)

At a=8, h=0.01 the discrete bottom (−2.87e-2) eigenvector is a frequency-46 oscillatory mode (1480 sign changes), NOT the smooth ground state; the negative eigenvalue decays O(h) (measured −2.87e-2 → −1.57e-2 → −8.3e-3 under halvings). So the discrete bottom is a quadrature-error mode of the rough kernel (cusp at 0 + prime-ramp kinks), and it out-ranks the true tiny λ_a for all a ≳ 2. At a=4 with maximal refinement (h=0.0025) the bottom eigenvector is still not ξ-like (sister corr −0.38, sign-unstable under refinement; crossings at chance level).

4. Sizing

• Dense P1 refinement: floor ~ C(a)·h with C(8) ≈ 2.9; resolving λ_a(4) < 3e-13 needs h < 1e-13/C — n > 10¹³ — infeasible.
• Spectral/exponential-order methods: kinked kernel (prime ramps (t−log n)₊) limits spectral order unless singularity-adapted panels at log n breakpoints are used; even granting a 1e-15 floor, coverage extends only to a ≈ 1.2 (λ_a(1) = 1.4e-9, extrapolated λ_a(1.3) ~ 1e-12) — where zero resolution is hopeless (need a ≫ log γ₁ ≈ 2.6).
• The feasibility windows are disjoint by orders of magnitude: ground-state resolution needs a ≲ 1.3; zero resolution needs a ≳ 4–6.

5. Verdict and revival conditions

The λ → λ_a⁻ consolidation channel is CLOSED as a numerics program (→ G-004). The partial response measured at the discrete bottom (a=4: Δ 2.16 → 6.99, residual 48 → 11) is consistent with the mechanism but cannot be pushed to the true ground state. Revival conditions (any one): (i) a theoretical lower bound on λ_a making the target explicit (currently no law below a = 1); (ii) a symbol-space formulation of the ground-state regime that avoids discretizing g (e.g., working directly with the zero-comb symbol of D-045); (iii) a singularity-adapted spectral discretization with a demonstrated 1e-13+ floor at a ≥ 4. Note (ii) connects to the one remaining theory opening from D-045: the symbol is 1 + zero-comb; the ground state is the direction that makes 1/β diverge — possibly expressible in closed form without any Galerkin machinery.

6. Feeds

• G-004 (graveyard): λ-channel closure with this file as evidence.
• D-044 §4's mechanism stands as the correct qualitative picture.
• D-045's symbol picture suggests the alternative: work with the comb-symbol directly (no discretization).