MISSION: out-of-band envelopes for Weil window positivity

Opened 2026-09-27 on branch teal-sea/oob-cert from origin/main (8941a80). Scope: hunts/oob_envelope/ only, plus this hunt's case-log entry in hunts/README.md. Everything else in the repo is a read-only input.

The gate

Every deliverable must be a statement strictly weaker than RH: a fixed-window positivity theorem, a lemma with a proof, or a measured cost law. A reformulation of RH is recorded as a reformulation, never as progress.

The observation this hunt tests

Weil's form on the window supp f ⊆ [-L, L] (arXiv:2608.24827, Zhu, eq. (2)):

Q(f) = 2 F(i/2)^2 + (1/2π) ∫ |F(t)|^2 Ψ_L(t) dt, Ψ_L(t) = Re ψ(1/4 + it/2) − log π − P_L(t), P_L(t) = Σ_{log n < 2L} (2Λ(n)/√n) cos(t log n).

|F|^2 is the Fourier transform of f ⋆ f~, which is supported in [-2L, 2L]. So for any bounded almost-periodic H whose frequencies all satisfy |λ| ≥ 2L, ∫ |F|^2 H dt = 0, and Ψ_L may be replaced by Ψ_L + H without changing Q on the window. Zhu's one-stroke reduction (Theorem 1.1) then runs with the envelope constant

S_L(H) = sup_t (P_L − H)(t) in place of A_L = sup_t P_L(t).

Zhu's barrier (Lemma 3.2, Theorem 1.4) is about bounding P_L itself: its sup is A_L, so T_1 = 2π e^{A_L} is optimal for pointwise envelopes of P_L. Remark 1.6 there says beating it needs Diophantine information about {log p}. The out-of-band freedom uses none: it is a band-limitation fact.

Weak duality (easy): S_L(H) ≥ λ_max of the windowed comb operator (Pφ)(x) = Σ_n (Λ(n)/√n)[φ(x − log n) + φ(x + log n)] on L^2[-L, L]. Per prime p, the best out-of-band correction of φ_p(θ) = Σ_{k log p < 2L} (2 log p / p^{k/2}) cos kθ has constant λ_max of the Toeplitz matrix with first row (0, t_1, ..., t_m), t_k = log p / p^{k/2} (Carathéodory–Toeplitz), realised to within a Fejér loss by a nonnegative kernel sum.

Seed numbers (Scholar's probes, float64, one route: measured)

Probes: probes/comb_operator.py, probes/separable.py. S_sep is the per-prime construction (Fejér degree 64), S_opt the comb-operator floor. N ≈ e·L·T#/2 is the matrix size with T# = 2π e^{S + 0.5}.

Calibration: N Zhu at L = 0.8 reproduces the N = 200 of Zhu's run. Zhu's valid-certificate estimate at L = 1.19 (support 2.38, the claim he retracted) was N ≈ 1.4–2 × 10^4; the separable correction brings it to the size of his retracted 950-mode run.

What this does not do: S_opt still grows like c·e^L (measured c ≈ 1.1 at L = 2), so the threshold stays doubly exponential. The gain is in the exponent's constant (roughly 4 → 1.1), not the shape. It cannot reach RH, and the Landau–Widom precision wall (Zhu §12) is untouched.

Records to beat (to be verified by the theory worker, one normalization)

• Yoshida 1992 / Connes–Consani (arXiv:2006.13771): support log 2.
• Zhu, arXiv:2608.24827v2: half-width 0.8 (support 1.6), computer-assisted.
• Liu, alphaXiv preprint dated 2026-09-14, unrefereed: half-width 1 and 17/16.
• Burnol 2000 and Bombieri 2000 are cited elsewhere with other ranges; pin down their normalization.

A valid, enclosure-carrying positivity bound at half-width 1.19 (support 2.38) would be a new record under any of the above.

Workers and ownership

• theory/, branch teal-sea/oob-cert-theory: the lemma, its proof, duality, asymptotics of S_opt, prior art. Brief: theory/BRIEF.md.
• numerics/, branch teal-sea/oob-cert: construction of H with enclosures, the modified reduction, the L = 0.8 replication, the L = 1.19 attempt. Brief: numerics/BRIEF.md.

• referee/, branch teal-sea/oob-cert-referee: adversarial review and an independent reimplementation, on a different model family. Started only once the lemma and the L = 0.8 replication exist. Brief: referee/BRIEF.md.

Team (chosen by the supervisor, 2026-09-27):

Method reference for all lanes (read, do not copy into this repo): <scholar-skills>/research/computational-math-research/SKILL.md.

Each worker writes only in its own subdirectory. Supervisor: Scholar (Hermes cron, every 5 minutes). Questions go at the top of your PROGRESS.md under a QUESTION: line; then end your turn and wait.

Kill-controls

• K1 (upper-bound consistency). Zhu's interval-arithmetic variational bound gives λ*(0.8) ≤ 2.27e-17. Any lower bound this hunt produces at L = 0.8 above that is a bug.
• K2 (unsound pipeline). Run the same reduction on a function whose Weil form is known negative on the window (Epstein (1,1,6) past c ≈ 28 to 29.5, or Davenport–Heilbronn past c ≈ 30.6; data in hunts/rogue_frontier/weil_trunc/ and the weil_propagation hunt). The pipeline must fail to produce a positive bound there. Note these have no Euler product, so use a valid joint H or H = 0, not the per-prime one.
• K3 (orthogonality). Check ∫|F|^2 H = 0 numerically on random window functions, and check that a deliberately in-band term breaks the identity.

Rules

• No claim about RH. The certainty ladder in AGENTS.md governs every statement; a composite claim takes the grade of its weakest step.
• The reserved word banned under hunts/ stays banned (tests/test_hunt_probe_discipline.py). Do not quote paper titles that contain it.
• Compute. This machine is Ghost: 8 GB, and it hosts the Hermes gateway. Local runs stay under 10 minutes and 2 GB. Heavy runs go to Modal (modal CLI, profile teal-sea), one unit per container, results written per unit. Never GitHub Actions: this overrides AGENTS.md compute rule 2 for this hunt. Write the estimate in numerics/RUNS.md and ask the supervisor before any Modal run.
• No push, no PR, no publication, no edits outside hunts/oob_envelope/ (and the one case-log entry). Commit small, often, to your own branch.