Current task: —

Updated 08-20-2026

D-043 — Task 2.13 analysis: λ-flow response, the discretization floor, and the locking correction (D-029 erratum)

Date: 2026-08-19 | Session: 008 | Task: 2.13 (E-025) Status: complete. E-025 checks: V2/V4 PASS; V1/V3 FAIL — the failures ARE the findings (machinery limit, below). Post-hoc: correct-direction locking analysis (E-022 data) + h-refinement runs.

A. The λ_a discretization floor (machinery limit, confirmed)

The Galerkin (Q,M) pencil bottom at h = 0.01: +5.1e-11 (a=4), −2.9e-2 (a=8), −6.8e-2 (a=12). h-refinement at a=8: −2.87e-2 (h=0.01) → −1.57e-2 (0.005) → −8.3e-3 (0.0025): O(h) discretization artifact → 0 as h → 0 (rough kernel g: cusp + prime kinks slow the conforming rate). At small a the floor is positive but refinement-chasing (a=2: 7.4e-11 → 3.3e-12): the true λ_a is below the floor at every a ≳ 2. Consequences:

B. λ-flow response (qualitative confirmation of the channel)

Approaching the discrete bottom (a=4: clean, bottom +5.1e-11; a=8/12: toward the artifact mode): | a | Δ(λ=−1) → Δ(near bottom) | residual | z₁·a | |---|---|---|---| | 4 | 2.16 → 6.99 | 48.3 → 10.7 | 4.50 → 6.98 | | 8 | 2.74 → 5.03 | 58.7 → 12.7 | 4.50 (stable) | | 12 | 2.94 → 4.34 | 57.6 → 15.4 | 4.50 (stable) | The channel response is REAL and as predicted (near-singular T ⟹ deeper valleys, shrinking residual), but at a ≥ 6 the approached mode is the artifact mode: sister-conjecture test gives corr(v̂_bottom, Ξ) ≈ 0.00 (a=8/12), −0.30 (a=4) — the bottom eigenvector is NOT ξ-like at any measured a. Nothing learned about (1.2) yet; the true ground state is unresolved by this machinery. (λ=0 rungs at a=8/12 excluded: T=Q is past the artifact bottom, indefinite → wild Δ; K-resid 2e-4 flags it.)

C. MAJOR CORRECTION — positional locking is real (D-029 erratum)

E-019's B2 "γ-locking" metric was wrong-direction: median over ALL ~10³ zeros (spanning z ≤ 300) of distance to the FIRST 30 γ's (z ≤ 77) — median ≈ 49 trivially, hence "chance-level" (D-029). Correct direction (each γ to its nearest W-zero), E-022 data: | a | mean min-dist | chance (π/4a) | normalized | |---|---|---|---| | 4 | 0.0395 | 0.196 | 0.201 | | 6 | 0.0244 | 0.131 | 0.186 | | 8 | 0.0114 | 0.098 | 0.116 | | 10 | 0.0058 | 0.079 | 0.074 | | 12 | 0.0042 | 0.065 | 0.064 | Fit: min-dist ≈ 0.65/a² — quadratic positional locking, monotone improving. ERRATA: D-029's "γ-proximity chance-level" is retracted (see D-029 erratum appended); D-040 F4's "weights not positions" framing is softened — positions track γ's AND weights concentrate. The stiff-lattice findings (maximal count, gap CV ↓, interlacing) are unaffected and remain true: the lattice is maximal AND locked — the excess zeros are the issue, not position capture.

D. Revised consolidation picture (supersedes parts of D-040/D-042)

  1. Locking: positions approach γ's as ~0.65/a² (measured, E-022 data).
  2. Weights: valley atoms carry ≈ m_γ = 1 already (E-022b).
  3. Residual: → Z/6 ≠ 0 (linear-K law, D-042) — the obstruction. [ERRATUM 2026-08-19 (session 010, D-060): superseded by D-045 §2 — the plateau law is K/(2a) → 1 (free PW diagonal), so the residual asymptotic is Z/(2π) (≈ 44.6 at Z = 280; measured 48–58, the gap being the finite-a prefactor and floor drift). The qualitative conclusion — nonzero residual equilibrium = the obstruction — is unchanged.]
  4. Outflow: none at a ≤ 12 (count stays maximal).
  5. λ-channel: response confirmed (B) but machinery-limited (A). So the (★)-route open content narrows to: the excess-zero outflow and the residual mass, both governed by the linear-K equilibrium. The sharpest next analysis target: derive the linear-K law K_plateau ≈ (6/π)a and the Z/6 residual analytically from T_a's spectral bulk (Szegő-type); identify what breaks the equilibrium.

E. Errata applied