MISSION — `frontier_map`

Persona

A cartography exercise, not a search. wide_search mined one method — the 10 August 2026 paper More than two thirds of the zeros of the Riemann zeta function lie on the critical line — for a sharp constant and a ceiling reproduction. This hunt assembles what that work scattered across three RESULTS files into one instrument and one picture: the frontier map of unconditional critical-line proportions reachable by the paper's pair-correlation machinery, as a function of the one dial the method has (the bandwidth λ), against the ceilings and prior-art bars that box it in.

A map is not a discovery. Every number on it is either computed here by the laboratory's own optimiser, taken from the paper with a line citation, or inherited from wide_search with its caveats attached. The value of the map is that the open gaps become coordinates — intervals with two computed endpoints — instead of prose.

Scope

This hunt may write: hunts/frontier_map/ and figures/.

This hunt may not write: zeta/, ontology/, harness/, lean/ without explicit permission, and may not write a verdict into README.md, ROADMAP.md or HANDOFF.md as an established finding. It may not promote its own claim. It reads (and does not modify) hunts/wide_search/, whose xiprime.py optimiser is the shared instrument.

Objective

Three deliverables, all measurements:

1. The λ-landscape. For each kernel the paper's machinery accepts unconditionally — Montgomery's F (ζ) and Farmer–Gonek–Lee's F₁ (ξ′) — the optimal proportion H(λ) of simple on-line zeros over the admissible band λ ∈ (0, 1], with the distinct-zero companion Hd(λ) = (1+H)/2, the onset λ₀ where the certificate first becomes non-empty (H > 0), and the value at the wall λ = 1.
2. The cross-check. The paper's eq. (7.4) gives the ζ optimum in closed form; the numeric optimiser must match it along the whole curve, not only at λ = 1. A generator of a curve never judges it; the closed form was not used to build the optimiser.
3. The gaps, as intervals. Attained-versus-ceiling for ζ (Remark 1.1's 0.68185, with the hypothesis-scope caveat measured in wide_search/RESULTS-pair-ceiling.md), attained-versus-Wu for ξ′, the blocked κ ≥ 2 lane, and the paper's own structural wall: the bandwidths ~1.04 / 1.26 / 1.70 it states would be needed for 0.70 / 0.80 / 0.90.

The standing checklist, answered in advance

1. Rival. No structural claim about ζ is made here, so the battery has nothing to distinguish; the paper itself records (§7.5) that for Davenport–Heilbronn-type functions Proposition 5.6 fails and the certificate is empty, which is the correct rival behaviour for the method and is quoted on the map, not tested here.
2. Decoy / surrogate. Not applicable to a map of published constants; the computed curves get control 4 instead.
3. Lesion. probe.py runs the optimiser against a deliberately wrong closed form (a mis-set constant in eq. 7.4) and must see the discrepancy that the honest comparison does not show.
4. Precision response. Every computed point on the map is re-run up a convergence ladder (basis size, quadrature order) and the settled digits recorded before it is written down.

Vocabulary

This directory says measured, observed, consistent with. It does not say verified, confirmed, definitively, and it never uses the reserved word that zeta/rigor.py owns for enclosure-carrying quantities — a ban tests/test_hunt_probe_discipline.py enforces on the bytes of every file here. Nothing in this hunt is evidence for or against RH, in either direction; the map charts a method, and the method's own paper states that nothing in it distinguishes "two thirds" from "all" (§7.5(a)).