G-004 — λ → λ_a⁻ ground-state channel for (★)-consolidation

Date: 2026-08-19 | Session: 008 | Track: B (Suzuki line, task 2.13 follow-up)

Attempt

Drive T_a(λ) = A_a − λ toward singularity (λ ↑ λ_a, the Weil-form bottom) so the K-metric becomes ground-state-dominated: valley floors pinned (masses 1) while plateau weights collapse — the one mechanism that breaks the linear-K equilibrium (D-044 §4). Tested in E-025 (λ-ladder, a = 4/8/12) with sizing study D-046.

Obstruction

The two feasibility windows are disjoint by orders of magnitude:

• Resolving the true ground state needs the discretization floor below λ_a, which decays superexponentially (measured trustworthy points: 0.414 at a=0.25, 2.07e-4 at a=0.5, 1.44e-9 at a=1.0; < 3e-12 at a=2). Even at a=2 the floor (O(h) artifact from the rough kernel: prime-ramp kinks + cusp) out-ranks λ_a; the discrete bottom is an oscillatory artifact mode (frequency ~46, sign-indefinite) for a ≳ 5 at any achievable h.
• Resolving the zeros needs a ≳ 4–6 (lattice spacing π/a vs γ-gaps). Dense refinement infeasible (n > 10¹³); spectral methods with singularity panels reach maybe 1e-15 — covers only a ≲ 1.3. Measured partial response at the discrete bottom (a=4: Δ 2.16→6.99, residual 48→11) confirms the mechanism's direction but cannot be pushed through.

Lesson

The (★)-route's consolidation cannot be reached by Galerkin machinery on the screw-function operator — the ground state lives below any achievable discretization floor at any zero-resolving a. The viable direction is symbol-space: the comb-symbol picture (D-045: β → 1 + zero-comb) suggests formulating the ground-state regime directly on the symbol, no discretization.

Revival conditions

(i) theoretical lower bound on λ_a (no law known below a=1); (ii) symbol-space (discretization-free) formulation of the ground-state regime; (iii) singularity-adapted spectral Galerkin with demonstrated 1e-13+ floor at a ≥ 4.