r_c7f779: the Cohn–Elkies-style certificate route for T1

Run 872d7dce-5f49-4e7c-8fcf-f470fd066e6f, opened 2026-08-24. Parent: hunts/r_b9552d run 37fb06a9 (RESULTS-37fb06a9.md §5, loose thread 2), which recorded the route as assessed and not settled.

The question

A Cohn–Elkies-style certificate for T1 is a function G with

G(s) >= f(s) := Dam(2y, s) - Kpair(s) for every s != 0, Ghat(w) <= 0 for every w,

since then sum_{p,q} G(tau_pq) = int Ghat |That|^2 <= 0, so GAS <= -k G(0) for every configuration at once, and T1 follows for all k simultaneously whenever -G(0) < Shq(1/2) = 0.06750840786062762.

Parametrising G(s) = -int mu(w) cos(sw) dw with mu >= 0 makes Ghat <= 0 structural, and the route's value is the linear program

V := min int mu subject to G >= f on (0, infinity).

Two thresholds bracket the verdict, both inherited from run 37fb06a9:

• V >= L/2 = 0.05716501969327026 always, because the uniform 2*pi lattice is a dual-feasible configuration. A computed value below 0.05716 is a truncation artefact and carries no upper-bound content.
• V < Shq(1/2) = 0.06750840786062762 is what the route needs. A value above 0.06751 kills the route with a witness.

Run 37fb06a9 reached 0.05410 at horizon s_max = 200 and its solver failed at 400 and 600. The window between the two thresholds is 15% wide and is exactly 2*(c2(0) - A^2) = 0.010343....

This hunt's question: where in that window does V actually sit, or is it above the top of it?

Preregistered kill conditions (written before any search)

1. If a re-solved, well-conditioned LP whose feasibility is verified a posteriori on a dense grid returns a value above 0.06750841, the route is dead with a witness and this hunt says so.
2. If the LP value stalls below 0.05716502 as the horizon grows and the verified certificate class cannot be repaired, the discretisation is still biting and this hunt reports not settled, not a bound.
3. If a certificate is produced whose value clears 0.05716502 and stays under 0.06750841, it is only a result if a dense independent re-evaluation of G - f (different code path from the LP's constraint matrix) finds no violation, and the tail s > s_max is covered by an argument rather than by absence of grid points.
4. Any claimed certificate that a red-team arm breaks is withdrawn, not repaired-and-kept.

Scope

Writes only inside hunts/r_c7f779/ plus one case-log entry (Hunt #110) in hunts/README.md. No changes to zeta/, ontology/, harness/, lean/, meta/, hunts/frontier_math/, or any root markdown file. Nothing here bears on RH (docs/08). The reserved word is not used.

id: r_c7f779
question: Does the Cohn-Elkies-style certificate LP for T1 have value below Shq(1/2) = 0.06750841, above it, or is it unresolvable at this cost?
frontier: V >= 0.05716501969327026 (the 2*pi lattice is dual-feasible); route works iff V < 0.06750840786062762; run 37fb06a9 reached 0.0541015 at s_max = 200 with solver failure above
proposed_attack: replace the uniform s-grid and the piecewise-constant measure by a class whose G has an exact closed form and a 1/s^2 tail, solve by constraint generation with a posteriori dense verification, and cover the tail by the leading asymptotic comparison
dead_routes:
  - per-pair domination of the gas charge
  - separation-assuming arguments
  - Cauchy-Schwarz on the pair sum
  - band-limited Fejer certificates, short by 1.15554665 with the frequency-domain witness ghat(0.87493) = +0.0839055 (r_b9552d run 1)
  - piecewise-constant mu on a uniform s-grid, which cannot be feasible at large s because its G decays only like 1/s
required_oracles:
  - exact closed-form evaluation of G for the chosen measure class, checked against numerical quadrature
  - independent dense re-evaluation of G - f on a grid disjoint from the LP constraint grid
  - HiGHS simplex and interior-point cross-check on the same program
  - the achievability floor 0.05716501969327026 computed from the 2*pi lattice by an independent route
kill_conditions:
  - the verified LP value exceeds 0.06750841, killing the route with a witness
  - the LP value stalls below the achievability floor 0.05716502 and the class cannot be repaired
  - a produced certificate fails dense a posteriori verification of G >= f
  - the tail s > s_max is not covered by an argument
agents_may:
  - search
  - derive
  - code
  - attack
  - formalize
agents_may_not:
  - declare novelty
  - declare theorem status
  - promote their own claim
  - open or merge a pull request