HANDOFF — `wide_search`

Written for an agent starting cold. Read MISSION.md for scope, RESULTS-xiprime.md for the finished piece, RESULTS-higher-derivatives.md for the blocked one. This file is the operating state.

Where things are

• Everything is landed. main carries the merge of PR #10 (hunt/wide-search-xiprime, the sharp constant) and of PR #12 (hunt/pair-ceiling-reproduction, 6bc5869, the ceiling reproduction). Nothing this hunt owns is sitting unpushed.
• Everything this hunt owns is under hunts/wide_search/:
• xiprime.py — the instrument. Takes a pair-correlation form factor, returns 1/c, H, Hd, and the optimal window. Its zeta control reproduces the source paper's Theorem D to 10 digits.
• pair_ceiling.py — reads the public LawN256.lean and recomputes the ceiling data from the enclosures with exact rationals. Takes the path to that file as its one argument; reports the scale, the worst interior row, D(1), the stability coefficient, the simple fraction, and whether the underlying certificate artifact was supplied.
• probe.py — the four standing controls (precision response, rival battery, null band). precision_response is calibrated in both directions: zeta(2) settles to 41 digits, a random value to 0.
• RESULTS-xiprime.md, RESULTS-pair-ceiling.md, RESULTS-higher-derivatives.md, MISSION.md.

The source paper — get it first, it is not in the repo

More than two thirds of the zeros of the Riemann zeta function lie on the critical line, author "Claude", dated 10 August 2026. Public PDF:

https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf

curl -sL <that url> -o /tmp/cz.pdf && pdftotext -layout /tmp/cz.pdf /tmp/cz.txt

2260 lines. The sections that matter: §1.4 (the idea), §3 (linear algebra), §4 (zero side), §5 (prime side), §7.1 (the window optimisation for zeta), Remark 7.3 (zeros of xi', near line 1521), §7.5 (limits of the method), and Remark 1.1 (near line 180), which states the 0.68185 ceiling.

Note the OCR renders 2/L^3 in eq. (7.1) as L23. The reading that makes c_lambda(1) = F(lambda) = lambda/(1+lambda^2/3) come out right is 2/L^3; this was checked independently and is not negotiable.

What is established

The paper's machinery turns a pair-correlation form factor into an unconditional proportion. With v >= 0 even on [-1/2,1/2]:

H = 2 - 1/c, Hd = (1+H)/2, 1/c_lambda(v) = [ int v^2 + lambda * iint F(lambda(s-s')) v(s)v(s') ] / ( lambda (int v)^2 )

• F(x) = |x| (Montgomery) is zeta. Optimum cos(sqrt(2) s), H = 0.6725007.
• F_1(x) = |x| - 4x^2 + sum_{k>=1} ((k-1)!/(2k)!)(2|x|)^{2k+1} (Farmer-Gonek, arXiv:0803.0425 Thm 1.1) is xi'.

The finished result. The paper's Remark 7.3 never solves the variational problem for xi'. Solved here:

H* = 0.86864150052976706411... (simple and on the critical line) Hd* = 0.93432075026488353205... (distinct)

at lambda = 1, maximiser strictly positive (v*(+-1/2)/v*(0) = 0.671042). Consequences: the paper's quartic was within 1.5e-6 of sharp, and no admissible window reaches Wu's unconditional 0.86957 (short by 9.285e-4), which settles the comparison the paper leaves open.

Reproduce in one line from the repo root:

.venv/bin/python -c "import sys; sys.path.insert(0,'hunts/wide_search'); \ from xiprime import optimise; print(optimise(kernel='xiprime')['H'])"

THREAD 1 (partly closed, still worth the most) — can zeta's 0.6725 move?

Read RESULTS-pair-ceiling.md before spending anything here. The cheap half of this thread is done and the answer was the one predicted below: the scalar-moment formulation collapses. What survives is narrower and stated at the end of this section.

The paper's Remark 1.1 states that no certificate reading bandwidth-one data "configuration by configuration" can exceed 0.68185, while its Theorem D attains 0.6725. So the best such certificate lies in [0.6725, 0.68185], and closing any of that moves a headline number.

Framing, worked out but barely begun:

• For a single window v the prime side pins both tr G = N (configuration independent) and ||G||_F^2 = (1/c(v)) N. So every window yields the same shape of constraint, s_1 >= (2 - 1/c(v))N, and the max over windows is exactly 0.6725. That is why one window cannot do better.
• The live question: the configuration must satisfy the constraints of every admissible window simultaneously, and different windows are different compressions of the same Weil form. Does the joint constraint set beat the best single member? Formulate "minimise s_1/N over configurations consistent with the known (tr, ||.||_F^2) for all windows" and solve it — it looks like an SDP/LP.
• A clean proof that the joint problem collapses to the single-window one is an equally good answer, and is what I would bet on.
• Strong check available: reproduce 0.68185 itself. If you cannot reconstruct the extremal configuration attaining it, you have not understood the ceiling.
• §7.5(b) says Prop 4.4 is sharp given only tr, ||.||_F^2, the block structure and tr P_1 <= s_1; §7.5(d)-(e) close off higher moments unconditionally (Rudnick-Sarnak range k*lambda < 2 allows only k=3, and odd moments do not help). So the extra juice, if any, is not more moments.

An agent was launched on this and stopped almost immediately; nothing from it was kept.

What is now closed, and what is left

Measured in RESULTS-pair-ceiling.md, landed as PR #12:

• The scalar-moment LP collapses, exactly. If the datum retained per window is only (tr, ||.||_F^2), the joint feasible set is the intersection of the half-lines s_1/N >= H(v), which is s_1/N >= sup_v H(v). So that formulation cannot move 0.6725007037..., and the bet recorded above was right. Any non-collapsing formulation has to keep cross-window information, or act on the whole bandwidth-one form-factor measure before it is reduced to one Rayleigh quotient.
• The 0.68185 check was run as far as the public data allows. pair_ceiling.py reproduces, from the published enclosures alone, the 2^140 scale, 256 rows, the worst interior error 1.83670992316e-40 at j = 1 against the advertised 3e-40, D(1) = 0.8239531607128352, the stability coefficient 2.5431315104166665e-6, and the simple fraction 0.6818286874638315. All agree. Reconstructing the extremal law itself needs cert_N256_blk_b128m.json, which is not in the public repository; the authors state it is available on request. Ask them for it before treating "cannot reconstruct the extremal configuration" as understanding failure.
• One gap fell out that is worth more than the collapse. Remark 1.1 states a bare uniform ceiling of 0.68185, but the N = 256 law delivers it only for certificates with abs(r'(1)) + integral abs(r'') <= 8.38043022204.... The Lean statement carries the error terms and makes no such elision, so this is visible only by putting the paper and the artifact side by side. Nothing here says the sentence is false.

What is left of this thread: the full-data LP over marked periodic configurations, which does not reduce to the single-window bounds and remains a legitimate route to something in (0.6725007, 0.68185). That is now the whole thread. Do not re-run the scalar-moment version.

Worked 2026-08-11 by hunts/frontier_math/ (see its RESULTS-frontier-math.md): at the pair-measure level the LP reduces exactly to 2 − sup D and descends to the single-window optimum — the measure relaxation collapses, so what the interval measures is configuration realizability. The attempted constructive residue was a Cheer–Goldston gap-rigidity transplant, now withdrawn: it used conjugate-transpose zero-side terms where the pinned upstream argument uses transpose terms, and the resulting on/off interaction can be negative. See ../frontier_math/CLEAN-KILL-REPORT.md. This thread's remaining open half is the configuration-level LP proper, for which the paper's N = 256 extremal law artifact (cert_N256_blk_b128m.json, available on request per RESULTS-pair-ceiling.md) is the natural starting point.

THREAD 2 (open, blocked) — closed form for F_k, k >= 2

Prior art that nearly went unnoticed: Ji Bian, The Pair Correlation of Zeros of xi^(kappa)(s), PhD thesis, Univ. of Rochester, 2008, advisor Gonek. Never published, not on arXiv, ~one citation. Rochester institutional repository, item 5500. It gives F_kappa under RH for 0 < |alpha| < 1.

What is still open, and why it is blocked:

• Bian has no closed form (a ~14-fold combinatorial sum over set partitions, his eq. 7.8, evaluated by Mathematica in his Appendix A).
• He states he cannot bound the tail for kappa >= 2; his 0.9544 and 0.9774 assume "the coefficients after 11 terms are negligible".
• Measured here: that assumption fails where it matters. The 11-term truncations give F_kappa(1) = 2.78, 31.9, 427.3, 2476.3 for kappa = 1..4, and proportions of 0.596, 1.198 (impossible) and -2.64. Usable only to alpha ~ 0.4; the optimum sits at lambda = 1.

So a closed form or a real tail bound is required, and either would repair a gap Bian flagged himself. The exact rational check grid (his Fig. 10.1, kappa 1..4, i 1..11) is tabulated in RESULTS-higher-derivatives.md, together with his proved stabilisation lemma C_{j,i} = C_{i-2,i} for j >= i-2. The kappa=1 row is reproduced exactly by the closed form above — that is what fixes the normalisation.

Do not fit coefficients to that grid and call it a derivation. Eleven targets per row makes fitting easy and worthless; three fitted two-parameter kernels already hit the published k=1 data points to machine precision and are all wrong.

Two derivation agents were stopped mid-flight. One had matched kappa=1 exactly, found its general-k formula failing kappa >= 2, and said it had located the cause without landing a fix. Neither produced a formula; nothing was kept.

Expected value warning before spending on this: Conrey's unconditional bar is 0.79874 (k=1), 0.93469 (k=2), 0.9673 (k=3). At the efficiency seen at k=1 the method lands ~1.6 points over Conrey at k=2 and under a point at k=3. Real, but much less striking than k=1.

Environment gotchas that cost time

• Python is always .venv/bin/python from the repo root. Never python3.
• Ball-arithmetic backend must be Arb: .venv/bin/python -c "from zeta import rigor; print(rigor.BACKEND, rigor.available_backends())" should say python-flint ['mpmath.iv', 'python-flint'].
• No Lean toolchain on this machine and the disk was at 90% (21 GB free). Installing elan + Mathlib would consume most of it. Deferred deliberately; do it only if something formalizable actually exists.
• gh is authenticated as tlince, which cannot merge in this repo. Use gh auth switch --user teal-sea, then switch back.
• The lexical ban under hunts/ is machine-enforced by tests/test_hunt_probe_discipline.py on the reserved word that zeta/rigor.py owns — which this file therefore cannot spell. It is a case-insensitive substring match, so it also catches the word with a negating prefix, and it applies inside a sentence disclaiming it. Writing this bullet the obvious way failed the suite; see CLAUDE.md for the term itself. verified / confirmed / definitively / proves are banned by documentation but not by a test. Say measured, observed, consistent with.
• Before handing back, run: tests/test_hunt_probe_discipline.py, tests/test_docs_numbering.py, tests/test_doors.py, and scripts/make_context.py --check.

Standing honesty constraints

• Nothing in this hunt is evidence for or against RH, in either direction.
• The finished result is an optimisation, not a new theorem: the functional is the paper's, the form factor is Farmer-Gonek's. Novelty of the sharp constant was gated at about 0.85 confidence by reading primary sources.
• It rests on a paper published 10 August 2026 that has not been peer reviewed.
• Two citation problems are recorded and should not be propagated: the "Conrey 1989 = 79.874%" figure was not located in that paper (its stated results concern zeta); and the alpha_j family appears to be Farmer's 1995 combination of Conrey 1989 + 1983-II rather than a display in either.
• A generator of a claim never judges it. Every number above that matters was reproduced by an independent route before being written down.