RESULTS — the frontier map

Status: a map, not a result. Every computed number below was produced by frontier.py (optimiser shared with wide_search), cross-checked against the paper's closed form where one exists, and run through the controls in probe.py. Every non-computed number carries a line citation to the 10 August 2026 paper (More than two thirds of the zeros of the Riemann zeta function lie on the critical line; URL in ../wide_search/HANDOFF.md) or to a wide_search RESULTS file, with that file's caveats inherited. Nothing here is evidence for or against RH; the paper itself states that nothing in its method distinguishes "two thirds" from "all" (§7.5(a)).

Reproduce everything:

.venv/bin/python hunts/frontier_map/frontier.py --json --figure .venv/bin/python hunts/frontier_map/probe.py

The picture is figures/frontier_map.png; the machine-readable map is frontier_map.json (regenerated, not hand-edited).

1. The λ-landscape (computed here)

The method has one dial: the bandwidth λ ≤ 1 of the pair-correlation data it consumes. For each λ the optimal window problem is a Rayleigh quotient (paper eq. 7.3, kernel-generalised as in ../wide_search/xiprime.py), solved in an even polynomial basis. Measured, on the grid λ = 0.05 … 1.00:

The certificate is empty for λ ≤ 1/2 (paper §7.5(a): "nothing at λ ≤ ½"); the measured onsets sit just above that line, and the ξ′ kernel switches on earlier and runs higher everywhere — consistent with F₁ ≤ F pointwise where it matters.

Cross-check (control 1). For ζ the paper solves the problem in closed form (eq. 7.4): c\*\_λ = √2·tanϑ/(1+ϑ·tanϑ), ϑ = λ/√2. Max deviation of the numeric curve from the closed form over the whole band: 9.5e-15. The closed form was not used in building the optimiser. A lesion — the same comparison against eq. (7.4) with its √2 mis-set to 1.5 — shows a minimum deviation of 1.5e-2, twelve orders larger, so the agreeing comparison is capable of disagreeing (control 3). Both λ = 1 constants are stable to < 6e-15 under a basis/quadrature ladder (control 4), and match the paper's Theorem D and RESULTS-xiprime.md's sharp constant respectively.

2. The fixed points (cited, not computed)

Prior art and ceilings, as pinned in frontier.py:REFERENCE_BARS with sources and caveats. Highlights:

• ζ, Levinson's line: 1/3 (Levinson 1974) → 2/5 (Conrey 1989) → 5/12 ≈ 0.4167 (PRZZ 2020, standing record before the paper). The paper's Theorem A lifts the unconditional on-line proportion to 2/3, and its Theorem D to 0.6725007 — by the other line (pair correlation), which had been RH-conditional since Montgomery 1973.
• ζ, the ceiling: no configuration-by-configuration certificate on bandwidth-one data can deliver more than 0.68185 (Remark 1.1). Standing caveat from ../wide_search/RESULTS-pair-ceiling.md: the published N=256 extremal law delivers this only for certificates with |r′(1)| + ∫|r″| ≤ 8.38043; the paper's bare sentence elides that hypothesis.
• ξ′: Wu 2015 has 0.86957 unconditionally for zeros merely on the line; the method's 0.8686415 carries simplicity and falls short of Wu's bar by 9.285e-4 with no admissible window able to close it (RESULTS-xiprime.md). The 0.79874 attributed to Conrey 1989 is carried as the paper's attribution only — wide_search could not locate it in that paper.

3. The lanes: where the frontier is open, closed, blocked, walled

Coordinates in frontier_map.json:open_lanes.

1. Open — ζ attained-to-ceiling, the interval (0.6725007, 0.68185). Width 9.3e-3. The scalar-moment joint-window LP collapses to the left endpoint exactly (PR #12); what remains is the full-data LP over marked periodic configurations. This is the one lane where bandwidth-one data might still yield something, and ../wide_search/HANDOFF.md THREAD 1 is its operating state.
2. Closed negatively — ξ′ vs Wu. The variational problem is solved sharp (H\* = 0.86864150052976706411); the gap of 9.285e-4 to Wu's bar is a property of the method, not of the window search.
3. Blocked — ξ^(κ), κ ≥ 2. Bian 2008 gives F_κ as a ~14-fold partition sum with no closed form and no tail bound; the 11-term truncations fail exactly at the λ = 1 optimum the map needs (impossible proportions 1.198, −2.64 for κ = 2, 3). Either a closed form or a real tail bound reopens the lane (RESULTS-higher-derivatives.md).
4. Walled — λ > 1. Remark 1.1: reaching 0.70 / 0.80 / 0.90 by this route needs pair-correlation input on Fourier support out to ≈ 1.04 / 1.26 / 1.70 — i.e. Hardy–Littlewood-strength information about prime pairs. The wall is drawn on the figure in the shaded region; the three crosses are the paper's own coordinates for what lies beyond it.

4. What the map says, in one paragraph

Both computed curves rise steeply from their onsets just above λ = 1/2 and hit the λ = 1 wall while still climbing — the method is input-limited, not optimisation-limited. For ζ the remaining slack within bandwidth-one data is the 9.3e-3 interval of lane 1 and no more; for ξ′ even the sharp optimum stops 9.3e-4 short of the strongest same-strength comparison bar. Every larger number on the chart is either RH-conditional (Montgomery–Taylor, CGdL) or on the far side of the shaded wall. A future session that wants to move anything on this map has exactly two doors that are not walled: lane 1's full-data LP, and lane 3's tail bound for Bian's F_κ.

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