D-013 — PHASE 1 DECISION GATE (task 1.5): is > 2/3 achievable in this framework?
Date: 2026-08-17 | Session: 001 | Task: 1.5 | Status: complete Inputs: D-004, D-005, D-006, D-009, D-010, D-011, D-012; E-001..E-005; G-001, G-002. Tags: [VERIFIED] / [NUMERICAL-EVIDENCE] as marked therein.
QUESTION (task 1.5)
Is > 2/3 achievable in the Weil-form-compression framework?
ANSWER: **No — not within the framework's current data class; and the
framework's walls are now mapped exhaustively.** (Formally: not by any certificate using only {tr G̃, ‖G̃‖²_HS, block structure} at bandwidth one, nor by moments k ≥ 3 at their unconditional availability bandwidths, nor by multiplicity/zero-density side inputs.)
THE WALL MAP (all verified this session)
| # | Wall | Precise mechanism | Evidence |
|---|---|---|---|
| W1 | Certificate sharpness | Lemma 3.2 tight at {1}_P/{2}_Q eigenvalues; all three counts sharp given the data | D-005, D-006(b) |
| W2 | 0.682 bandwidth-one ceiling | near-CUE 256-periodic law with 68.18% simple is indistinguishable by 2-moment data (LP-optimal, Lean + interval arith.) | D-006, D-009 |
| W3 | Moment ladder | availability λ ≤ 2/k bites before usefulness; caps 0.607N (k=3, λ=2/3), 0.502N (k=4, λ=1/2) < 2/3 N; n₊ q-invariant | D-011, E-005, G-002 |
| W4 | Multiplicity side-inputs | s₂/p substitution symmetry: depth→0 pairs mimic doubles in all 2-moment data | G-001 |
| W5 | Zero-density input | depth→0 off-line pairs invisible to N(σ,T) bounds | D-009 (A) |
| W6 | Support > 1 | needs prime-pair control at lag h ≲ X/T = T^δ; nothing unconditional exists at short lags (best: exceptional sets X^{0.7} at generic lags) | D-012 |
WHAT IS PROVABLY TRUE ABOUT ANY FIX (necessary conditions)
Any argument beating 2/3 (equivalently 0.6725, toward 0.682+) must:
- use data beyond {2 trace moments at bandwidth 1} — else W1/W2 bind;
- constrain OFF-LINE PAIRS at shallow depth — the unique invisible mass (G-001 mechanism) — i.e. input that sees |β−½| at scale o(1/log T) with multiplicative (not just additive) consequences;
- avoid D–H/Epstein blindness — the input must FAIL for those L-functions (sanity gate, NGUYEN.md §3);
- if via moments: break the RS range (Xᵏ ≤ T^{2−ε}) — needs new prime-side technology (short-lag correlations, W6);
- if via primes: control lags h ≲ X/T (W6) — currently prize-level open.
ROUTES STILL OPEN (ranked by evidence-based promise)
- Track C (positivity criteria) — Weil positivity on growing support, Li coefficients, de Bruijn–Newman. Different data class entirely; not constrained by W1–W6. The 2/3 density result might feed these (task 3.4: cross-pollination).
- Track B (spectral) — a self-adjoint realization would bypass the certificate framework wholesale. High risk, high ceiling.
- Track A with new arithmetic input — e.g., any future unconditional short-lag prime-pair theorem would immediately re-open 1.4 via D-012's exact insertion point (O₁ in Prop 5.4). Monitor literature.
- ξ′-analogue sharpening (thread T4) — the method is stronger on ξ′ (0.86864); understanding WHY the s₂/p symmetry breaks there might reveal portable structure. Cheap to investigate.
- GL(2) extension (fallback goal) — Theorem B → automorphic L-functions; new theorem territory without beating 2/3 (success criterion 4 / fallback 4).
RECOMMENDATION TO PHASE 2/3 PLANNING
Do not invest further in bandwidth-one certificate variants (proven exhausted: W1–W5 + E-005). Reorder upcoming phases: run 2.x/3.x in parallel with 4.x literature monitoring; keep 1.2 (SDP tooling) only as calibration infrastructure. Track A's residual value = the obstruction map itself (paper §7.2 companion material; success criterion 4).
GATE DECISION
[DOCUMENTED] > 2/3 is NOT achievable within the current framework's data class. Framework change required. Proceed to Phase 2/3 with Track A in monitoring mode.