D-017 — Positivity floor decay law (E-007/E-007b); Track C triad summary
Date: 2026-08-17 | Session: 002 | Tasks: 3.1 (Q-3.4b), 3.2 part 1 | Status: complete Tags: [NUMERICAL-EVIDENCE]
1. Clean full-rank floor data (mpmath dps 40, Hann window, exact FT)
| T | λ | support a = λℓ | d | λ_min+ | log₁₀(λ_min+/mean) |
|---|---|---|---|---|---|
| 200 | 0.50 | 1.730 | 55 | 1.54e-5 | −5.16 |
| 400 | 0.50 | 2.077 | 132 | 5.11e-7 | −6.63 |
| 400 | 0.75 | 3.115 | 198 | 5.48e-8 | −7.43 |
| 200 | 1.00 | 3.460 | 110 | 3.12e-7 | −6.55 |
| 400 | 1.00 | 4.154 | 264 | 8.35e-9 | −8.12 |
(λ ≥ 1.5 rows of E-007 excluded: d > N′ ⟹ dimensional zero-block contaminates the floor. Note this is intrinsic: at λ > 1 the critical- density grid has MORE points than zeros — the compression is singular by dimension, a finite-T artifact of the overcompleteness.)
2. Decay law fit
T=400 triplet (a = 2.08 → 3.12 → 4.15): log₁₀ floor ratio −6.63 → −7.43 → −8.12 — slopes −0.77, −0.66 decades/unit a: approximately LINEAR in a ⟹ floor ≈ 10^{−0.72a}·mean ≈ e^{−1.65a}·mean (exponential, not clearly superexponential). Cross-T check: at similar support, larger T (= larger d) floors lower ⟹ the floor also decays with dimension (Gabor-frame condition numbers grow with window/density — standard frame-theory behavior, D-001 Q6 territory).
Certification cost statement (Q-3.4b answer): certifying positivity of the truncated Weil form at support a needs ≳ 0.72a digits of precision (and dimension d grows ∝ a·T/2π). Exponential precision growth = linear digit growth per unit support: a SEVERE BUT POLYNOMIAL-GRADE complexity wall — unlike Kim et al.'s superexponential decay for the Suzuki (archimedean) operator. The two objects differ: our zero-side Gabor truncation is less pathological than the archimedean operator. ⟹ Positivity certification is precision-expensive but not in-principle inaccessible at moderate support; the real wall remains the o(N)-event blindness (D-016), not conditioning.
3. E-008 (Li coefficients, task 3.2 part 1) summary
λ_n for n ≤ 2000 from the zero table with explicit tail bounds: λ_1 = 0.022961 (+tail ≤ 5.4e-4) vs literature 0.0230957089… ✔; all positive (trivial on on-line data); certified range n ≤ 116 (tail < 5%). Leading-asymptotic ratio λ_n/((n/2)log n) ≈ 0.51–0.62 over the certifiable range (truncation-depressed beyond). ⟹ Instrument limit: zero-side Li numerics with 10⁴ zeros certify only n ≲ 10². Task 3.2's n ~ 10⁵ REQUIRES the prime-side Bombieri–Lagarias exact formulas — logged as the real 3.2 continuation (needs care: η-derivative sums; a future sub-stage).
4. Track C triad — consolidated conclusions (for the gather-thoughts)
- Q-3.4a (D-016): bulk inputs can't touch the microscopic indeterminate zone — scale mismatch, same wall family as Phase 1. Salvage: unconditional n₋ ≤ (1/6+o(1))N bound (candidate minor result; literature check queued).
- Q-3.4b (D-015 + this file): conditioning wall quantified: 10^{0.72a} precision per support a; exponential, not superexponential; plus dimension dependence. Certification feasible but expensive at moderate a; positivity route's true wall = o(N) blindness, not cost.
- 3.2 part 1 (E-008): instrument calibrated and limited; the honest next step is prime-side exact formulas.
- Positioning (D-014): our niche vs the 2026 literature = the unconditional second-moment input; their niche = truncation certification machinery. Q-3.4c (Suzuki operators as Track B candidates) remains open and is now the most interesting live thread.