D-015 — E-006 positivity horizon: resolution-limited; ill-conditioning measured

Date: 2026-08-17 | Session: 002 | Task: 3.1 / Q-3.4b | Status: complete (measurement); follow-up designed Tag: [NUMERICAL-EVIDENCE] with instrument-limit caveat.

1. E-006 raw results (T = 2000, flat window ψ₀, critical-density grid)

(N′ = 2044 zeros in I′; at λ ≥ 1.5, d > N′ ⟹ dimensional rank deficiency — overcomplete basis, expected; λ_min+ = smallest eigenvalue above noise 1e-10·λ_max.)

2. Reading

1. Flat floor: λ_min+/mean ≈ 1e-9–1e-10 across λ — NO decay trend. A flat floor at the pipeline's accuracy scale = instrument resolution limit, not the mathematical floor. The true floor is below float64 + spline reach at these dimensions. (Compare Kim et al. 2026: Suzuki operator's lowest eigenvalue positive but superexponentially small; Groskin: spectral scale 1e-59 at c=100 would need T ~ 1e63 by brute cutoff — same phenomenon family.)
2. Genuine ill-conditioning at bandwidth 1: ~50 of 1834 eigenvalues below 1e-8 (relative) at λ=1 — the Weil form's Gabor compressions are nearly singular objects even where positivity is PROVED. This is a real structural fact (not noise: consistent across λ), quantifying why positivity certification is delicate.
3. Instrument verdict: float64 eigvalsh on raw G̃ cannot power a positivity certificate beyond trivial support. Certification must use exact/interval arithmetic (Groskin's cutoff-free LDLᵀ) or extended precision.

3. Consequences for the project

• Q-3.4b refined: the positivity horizon question splits into (i) empirical floor decay — needs extended-precision eigensolvers at SMALL dimension (mpmath, d ~ 100–300, exact ϕ̂ quadrature) → E-007 design; and (ii) certification theory — Groskin's B_T budget already answers the certification side; our n₊ input (Q-3.4a) remains the live cross-pollination.
• Obstruction-map entry (Track C flavor): positivity certification has a precision complexity wall — cost grows like 1/λ_min(support); if λ_min decays superexponentially (Kim et al. indicate it does for the Suzuki operator), the route is computationally closed beyond fixed support even though theoretically open. W-candidate for the map, pending E-007 quantification.
• No contradiction with Phase 1 results: E-003/E-005 conclusions used bulk spectrum statistics (tr, HS norm, CMS bounds) — robust to the tiny-eigenvalue floor; unaffected.

4. Options for next sub-stage (choose)

(a) E-007: extended-precision floor measurement, d ~ 150–300, mpmath eigensolver, λ ∈ {0.5, 1, 1.5, 2} — directly measures decay rate (slow but tiny; ~minutes). Answers the horizon question properly. (b) Q-3.4a analytic sub-stage: formulate precisely how the paper's unconditional n₊/rank lower bound interacts with Groskin's indeterminate zone [−B_T, 0) — pen-and-paper, no compute. (c) Move to task 3.2 (Li coefficients numerics to n ~ 10⁵ — needs the zero table; GPU-parallelizable; different flavor). Recommendation: (a) — finishes the horizon question while it's loaded; then (b) or (c).