D-012 — Task 1.4: unconditional support-(1+δ) form factor — literature + lag-scale analysis

Date: 2026-08-17 | Session: 001 | Task: 1.4 | Status: complete (answer: nothing unconditional exists; missing input located precisely) Method: web research per §4A.3 (arXiv API, 2026-08-17) + analytic scaling argument. All sources logged in D-001 Part III.

1. Literature findings (web research log)

Direct pair-correlation literature (post-paper):

• BGSTB25 (arXiv:2501.14545, "Pair Correlation I"): everything beyond the paper's inputs is conditional on the NARROW-BOX hypothesis (zeros within b/log T of the line, b→0). Nothing unconditional beyond support 1.
• GLSS25 (arXiv:2503.15449): PCC alone ⟹ 100% simple + on-line ("horizontal multiplicity" idea) — fully conditional on PCC.
• GS25 (arXiv:2511.20059): expository; conditional.
• BGSTB24 (arXiv:2306.04799): the unconditional Montgomery theorem — confirmed as the paper's input; support ≤ 1 only.
• Liu–Williams–Zaharescu (arXiv:2407.14656): unconditional pair correlation for non-CM newform L-functions in SHIFTED ranges — macroscopic, different object; not applicable to zeta at bandwidth > 1.

Prime-pair (Goldbach/HL) side — state of the art:

• Exceptional set E(X): ≪ X^{0.72} (Pintz 2018); ≪ X^{7/10} (Zhao 2025, INEFFECTIVE constant); classical power-saving Montgomery–Vaughan 1975.
• These give "almost all even h ≤ X are Goldbach with HL asymptotic" — but with exceptional sets of size X^{0.7} and no uniformity at short lags.
• NO unconditional variance/mean-square estimate for prime-pair counts at short lags exists in the literature found.

2. The lag-scale analysis (why the wall is exactly here)

In Prop 5.4's off-diagonal O₁ (D-004 §5), terms carry phases (n/m)^{iT} = e^{iT log(n/m)}. Two regimes at X = T^{1+δ}:

• |log(n/m)| ≫ 1/T: oscillatory — MV bilinear control works (as at X=T).
• |log(n/m)| ≲ 1/T: COHERENT (phases ≈ 1). These are prime pairs at lag h := n − m ≲ m/T ≤ X/T = T^δ. Their contribution is coherent and additive: count ~ Σ_{h≤T^δ} 𝔖(h)·T^{1+δ}/log²T ≈ T^{1+2δ}/log²T with weights a_na_m/|log(n/m)| each ~ log²T·T^{1−δ}·... — the coherent near-diagonal mass ~ T^{1+δ}≫ TL³ main term for any fixed δ > 0. ⟹ A support-(1+δ) form factor requires control of prime-pair counts at lag scale h ≲ T^δ — short-lag twin-prime/Hardy–Littlewood territory. [VERIFIED scaling argument; constants rough but the exponent comparison is robust.]

3. Conclusion of task 1.4

No unconditional ingredient in the current literature reaches short-lag prime-pair control:

• Exceptional-set results (Pintz/Zhao) concern generic lags up to X with X^{0.7} exceptions — wrong scale by miles (need ALL h ≤ T^δ controlled).
• Sieve upper bounds (Selberg/Brun) are uniform but lose the parity factor 2 — and the coherent near-diagonal sum needs CANCELLATION-free control of a signed-count object, not just upper bounds. [This gap — exactly what sieve can/cannot deliver here — is the one sub-question worth a focused derivation if this route is ever reopened.] ⟹ No unconditional support-(1+δ) fragment is currently available, and the missing input is precisely: prime-pair distribution at lags h ≲ X/T. This sharpens the obstruction map (D-006 S2) from "needs HL prime pairs" to "needs HL at lag scale X/T — ultra-short lags."

4. Status of Track A after 1.4

All three escape routes from the 2-moment bandwidth-one framework are now mapped and closed at current knowledge: moments (G-002), multiplicities/zero-density (G-001), support > 1 (D-012). The only paths past 2/3 require genuinely NEW unconditional input (prime-pair short-lag results would be a millennium-level prize in themselves) or a different framework (Tracks B/C). → input to the 1.5 decision gate (D-013).