G-002 — Third-moment / moment-ladder route past 2/3 (incl. oversampling hatch)
Date: 2026-08-17 | Session: 001 | Track: A Tag: [FAILED-ATTEMPT(obstruction below)] + [NUMERICAL-EVIDENCE] (E-005) Opened: task 1.3 (D-011). Closed: same day by E-005.
Attempt
Use tr G̃³ (or higher moments) to power a CMS/Christoffel certificate past 2/3. Unconditional availability (Rudnick–Sarnak Xᵏ ≤ T^{2−ε}) forces bandwidth λ ≤ 2/k. The oversampling hatch (D-011 §4): denser Gabor grids (q ≥ 2) keep Poisson–Gabor exact and lift the d = λN dimension cap — tested numerically on the sine-process model.
E-005 verdict (sine-process model, 30 GUE samples; calibrated:
reproduces paper's μ = (1,4/3,2,13/4), CMS2 = 3/4, CMS4 = 31/36 at (λ,q) = (1,1) within model tolerances)
| λ | q | cap qλ | CMS2·(d/N) | MARK3·(d/N) | CMS4·(d/N) |
|---|---|---|---|---|---|
| 0.5 | 1 | 0.50 | 0.469 | 0.480 | 0.502 |
| 0.5 | 2 | 1.00 | 0.467 | 0.478 | 0.502 |
| 2/3 | 1 | 0.67 | 0.590 | 0.607 | 0.655 |
| 2/3 | 2 | 1.33 | 0.589 | 0.605 | 0.655 |
| 0.8 | 1 | 0.80 | 0.671 | 0.691 | 0.761 |
| 0.8 | 2 | 1.60 | 0.669 | 0.689 | 0.761 |
| 1.0 | 1 | 1.00 | 0.764 | 0.785 | 0.876 |
| 1.0 | 2 | 2.00 | 0.762 | 0.784 | 0.876 |
| (values = one-sided bound × d/N; raw numbers in | |||||
| data/E-005-oversampled-third-moment.csv) |
Obstruction (three independent walls, each sufficient)
- Availability/usefulness coincidence: at λ = 2/3 (the largest bandwidth with an unconditional 3rd moment), even the SHARP 3-moment bound gives 0.607N < 2/3 N — and conversion to simple zeros loses a further factor ≈ 0.84–0.89 (paper's λ=1 calibration). Even the sharp 4-moment bound (not available there) gives only 0.655N < 2/3.
- Oversampling conservation (D-011 Q1 answered): n₊(G̃) is q-invariant (0.607N vs 0.605N at λ=2/3) — denser grids pad the spectrum with spectrally empty directions; no new positive index. The dimension cap is real information-theoretically, not an artifact of grid choice.
- Ladder trend is adverse: k=4 needs λ ≤ 1/2, where even the sharp 4-moment cap is 0.502N. Higher moments are available only at bandwidths where their caps fall further below 2/3.
Lesson
The moment ladder within this framework is closed at every rung: availability (λ ≤ 2/k) always bites before usefulness (cap vs 2/3). The paper's "higher moments add nothing" is now mechanistically confirmed from BOTH sides (analytics D-011 + model numerics E-005). Only TWO escape directions remain consistent with all evidence: support > 1 form-factor input (task 1.4 — prime pairs), or a wholly different data class (Tracks B/C).
Salvage
- E-005 is a calibrated CMS testbed (reproduces 3/4 and 31/36 from first principles) — reusable for any future certificate idea.
- The q-invariance of n₊ (Poisson no-aliasing) is a clean lemma-grade observation for the paper's methods section.
- The (λ, q) cap table itself is exhibit material for the obstruction map (success criterion 4).