Brief: theory worker (out-of-band envelopes)

Read first: hunts/oob_envelope/MISSION.md, then AGENTS.md (certainty ladder, reserved words, original vs novel), then arXiv:2608.24827v2 (Zhu) §1–4 and §14–16. Your worktree: <worktree> (branch teal-sea/oob-cert-theory). Write only in hunts/oob_envelope/theory/.

Tasks, in order

1. The lemma, proved. State and prove: for f ∈ L^2, supp f ⊆ [-L, L], and a real almost-periodic H(t) = Σ_w b_w cos(λ_w t) with Σ|b_w| < ∞ and every |λ_w| ≥ 2L, ∫ |F(t)|^2 H(t) dt = 0. Handle the boundary case |λ_w| = 2L explicitly (it is a measure-zero issue for L^2 f, but say why). Then restate Zhu's Theorem 1.1 with A_L replaced by S_L(H) = sup_t (P_L − H)(t) and check line by line that his proof (§4, eq. (13) and the tail/coupling bounds) goes through unchanged. Flag every place where it does not.
2. Optimal constant. Prove weak duality S_L(H) ≥ λ_max(P_L on L^2[-L,L]) for every admissible H. Then decide whether equality (strong duality) holds, for instance via Perron–Frobenius on the positive-coefficient comb, characters of the Bohr compactification, or a finite-dimensional Carathéodory–Fejér argument after truncating to finitely many primes. A proof, a counterexample, or an honest "unresolved" are all acceptable.
3. Asymptotics. Upper and lower bounds on S*_L = inf_H S_L(H) as L → ∞. The measured data suggests S*_L ≍ e^L with constant near 1.1 at L = 2 (Zhu's A_L ~ 4 e^L). Derive the constant if you can (hint: a positive test function on [-L, L] against the comb gives lower bounds; the per-prime Toeplitz construction gives upper bounds). This decides whether the doubly-exponential barrier keeps its shape (expected: yes).
4. Prior art. Has anyone used out-of-band freedom in Weil positivity or in the explicit formula? Check at least: Yoshida 1992, Bombieri 2000 (Rend. Lincei, and "Remarks on Weil's quadratic functional"), Burnol, Connes–Consani arXiv:2006.13771, Connes–Consani–Moscovici arXiv:2511.22755, Connes–van Suijlekom arXiv:2511.23257, Suzuki arXiv:2606.09096, Liu (alphaXiv, 2026-09-14, "Source-exact block-Schur and tail-compensation bounds"), and the Beurling–Selberg majorant literature (Carneiro and coauthors, which uses band-limited majorants on the zero side). Record exact citations with arXiv IDs and section numbers. Also pin down the true current record for unconditional window positivity, in one normalization (half-width L of supp f).
5. Liu's obstruction. Liu proves a tail-dropping localization has a negative high-frequency limit past half-width log 8 / 2 and that no fixed compact correction repairs it. Does that obstruction touch the out-of-band route? Answer from the paper's actual statement.

Output

hunts/oob_envelope/theory/RESULTS.md, opening with a 5-line graded summary (one line per task finding, each with its ladder grade), then the statements and proofs. PROGRESS.md updated at each step. Small commits to your branch; do not push. No heavy computation: anything over a minute of CPU belongs to the numerics worker (write the request in PROGRESS.md).

When blocked, write QUESTION: ... at the top of PROGRESS.md and end your turn. The supervisor answers there and in your terminal.