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Library · hunts/README.md

`hunts/`: exploratory studies

59,481 words · 5,309 lines · source

A hunt is an exploratory study: a scoped directory where an agent or a person pursues one specific idea, with the understanding that most such ideas fail.

hunts/ is the one place in this repository where a claim can be written down before any control has been run against it. So the classification is explicit:

A hunt is exploratory. Nothing in hunts/ is a result, not a result, and not evidence.

Per harness/README.md, the admission rule for a department is no department without a battery, work whose claims nothing in this tree can falsify is not a department; it is exploratory, and exploratory work belongs where nobody will mistake it for a result. That place is here.

A hunt can never become a department by growing, for the reason ROADMAP.md records for dossier/: a hunt's negative controls are the zeta department's, and a department whose battery belongs to another department is not a department. A hunt borrows the zeta battery. That is the correct relationship, not a deficiency, but the battery does have to actually be invoked.

What a hunt may and may not do

MayMay not
build its own instruments under hunts/<name>/modify zeta/, ontology/ or harness/ without explicit permission
record raw measurements in its own results*.jsonwrite a verdict into README.md, ROADMAP.md or HANDOFF.md as an established finding
use the word measured, observed, consistent withuse certified, which zeta/rigor.py owns
propose a candidate for the funnel or the batterypromote its own claim

A hunt that wants its claim to count takes it through the battery (docs/doors/refute.md) or the funnel (docs/doors/discover.md). Those are the two routes that can say "yes", and neither of them is the hunt itself.

Methods travel even when claims do not. An identity, lemma, control or technique a hunt produces goes in docs/37-methods.md, the cross-hunt methods index, in the same change that records it here. Check that index before building an instrument; several hunts have hand-rolled the same one.

HuntSpec (new hunts, from 2026-08-11)

A hunt opened after 2026-08-11 carries a HuntSpec, a fenced contract block in its MISSION.md stating the question, the frontier, the dead routes, the non-model oracles allowed to assign truth, the kill conditions, and what its agents may and may not do. Format, rules and the validated template: hunts/HUNTSPEC.md. Existing hunts are not retrofitted; tests/test_huntspec.py validates any block that exists. The primitive is on probation, see the spec page's terms.

The standing checklist

Before a hunt's finding leaves hunts/, it must have survived the four control roles, and the checks are the ones the tree already owns:

  1. Rival. Does the claim also hold for a function that shares the structure and violates RH? zeta.epstein.battery. Note the trap below: if your test set is the rival set, you have measured your own selection criterion.
  2. Decoy / surrogate. Does a matched null with no arithmetic in it reproduce the effect? zeta.surrogate, NULLCONTROLS.md. ROADMAP.md records the calibration that matters here: against a null of random non-factoring sequences, Davenport–Heilbronn sits at the 27th percentile, typical, not exotic. Anything claiming a factorization effect must beat that null.
  3. Lesion. Does the detector notice a violation planted on purpose, and can it tell that planted violation apart from the claimed signal? If it cannot, the detector is measuring the plant.
  4. Precision response. Does the effect move when the approximation improves? ROADMAP.md's standing rule, earned three times: an artifact does not respond to added precision; a real quantity does.

Case log

Hunt #158: is the covering margin's decay the same across two families? (decay_law/, 2026-10-30)

Status: measured, kill fired, no claim promoted. The covering margin is W_zeros over the prime cost. On b = 2 against b = 8 the rectangle endpoint exponents are 3.790 (margin), 1.395 (zero-side sum) and -2.395 (prime cost). A tent of the same support width, with h a fourth power of a sinc, gives 5.731, 3.470 and -2.262. The universal reading, withdrawn if any of the three differs by more than 0.5, fired on the margin and on the zero-side sum. The prime-cost exponents differ by 0.13 and do not fire. A single power is not a defensible summary of either family: on the 25-point grid the local margin exponents span 34 and 103, and that span is the zero side oscillating, not the prime cost. All of this is on the window b from 2 to 8: by the prime number theorem the prime cost grows like e^{b/2} times a family-dependent power, so log-log exponents depend on the window, and heuristically both margins decay like e^{-b/2}. The kill was committed before the recorded run but after two tent margins were printed, so it was not blind on the margin exponent. (Scope added at landing, 2026-10-30.) Nothing here is evidence for RH. Record: MISSION.md, RESULTS.md.

Hunt #155: the three pieces of the Weil functional and how deep they cancel (arch_cover/, 2026-10-30)

Status: measured, no claim promoted. Pole, archimedean and prime pieces are evaluated separately with Arb enclosures for the Gaussian, Fejer and autocorrelation families, and each value is checked against the zero side of the explicit formula. The cancellation depth, the number of digits by which the three O(1) pieces cancel to give W, equals a (gamma_1^2 + 1/4) / ln 10 on the Gaussian family to every recorded digit (17.375 at a = 0.2, 39.094 at a = 0.45), so the statistic measures how far the test function sits from the first zero and nothing deeper. (Corrected at landing, 2026-10-30: this entry first gave a gamma_1^2 / ln 10 with "predicted" values matching neither form; against that form the hunt's own kill condition fired, now recorded in RESULTS.md 4(b). This hunt was opened as #154, which gabcke_remainder/ holds.) Two consequences are recorded: the prime side grows with a where the zero sum collapses, so the hard regime sits inside the family, and W goes to zero with every piece at O(1), so no inequality W >= c ||f||^2 with fixed c > 0 holds over the class. The enclosure floor is 33 digits and is set by the prime truncation cap at n_max = 2e6, not by precision (the 512-bit run behind "raising the bit count changes nothing" is not committed); RESULTS.md ends with its doors. Nothing here is evidence for RH. Record: MISSION.md, RESULTS.md.

Hunt #156: the Gram matrix of the Weil form and its bottom eigenvalue (weil_gram/, 2026-10-30)

Status: measured, no claim promoted. The bilinear Gram M_jk = W(h_jk) on a Gaussian basis, solved against the L2 Gram as a generalized eigenproblem, with every entry checked against the zero side (residual at the zero table's 30 digit floor; a dps 50 against dps 90 run is a null test, rerun at landing because the first "dps 90" file had been computed at dps 50). Adding one concentrated element drops the bottom eigenvalue by 18 orders, and the bottom eigenvector there is the coordinate vector, so that drop is the single element degeneracy arch_cover measured. Refining a purely broad span, where every element is individually healthy at 3.1e-8 or better, still drops it from 2.4e-3 to 5.0e-22 across six dimensions, so the gap is created by the subspace choice and destroyed by enlargement. The recorded consequence is that a gap bounded below uniformly in the subspace cannot be had on these spans; nonnegativity on each subspace of an exhausting sequence, which does pass to the limit, is untouched. (Scope and numbering corrected at landing, 2026-10-30; opened as #155.) Nothing here is evidence for RH. Record: MISSION.md, RESULTS.md.

Hunt #157: the prime side split by place, exactly, for a compact test function (weil_split/, 2026-10-30)

Status: measured, one exact identity, no claim promoted. For a rectangle test function the shift operator has finitely many overlapping terms, so each place carries an exact closed-form cost (not rational: it carries log p and p^{-1/2}) and the per place sum reproduces the explicit formula prime term to 1e-16 (that identity holds place by place by construction, a regrouping of the prime sum; see RESULTS.md). Two findings on five widths, monotone there (hunt #158's denser grid shows the margin rising on 5 of 24 steps): the cost spreads from three places carrying 98 percent at b=2 to 429 places with the top three carrying 13 percent at b=8, and the covering margin falls from 3.0e-2 to 1.6e-4 over the same range, about b^{-3.8}. That margin is W_zeros over the prime cost by construction, so it normalizes W rather than testing the archimedean term separately. (Corrected and renumbered at landing, 2026-10-30; opened as #156.) Compact support costs a slow quadrature, so every covering ratio is quoted from the verified zeros with the prime side exact, and the quadrature column is printed beside it rather than absorbed. Nothing here is evidence for RH. Record: MISSION.md, RESULTS.md.

Hunt #154: Gabcke's remainder, opened (gabcke_remainder/, 2026-10-29)

Status: opened. The remainder is not proved. The height cap on the kernel-checked strip proof is the order-4 Euler-Maclaurin error bar. This hunt is the formalization of Gabcke's explicit Riemann-Siegel remainder, the alternative already recorded and left unformalized by hunts #132 and #133. The first check is the coefficient recursion in Satz 2.1.1: Lean's kernel accepts the seven printed λ_n with no axioms. That is not the remainder bound. Record: gabcke_remainder/MISSION.md, gabcke_remainder/RESULTS.md. (Landed 2026-10-30. The module header every tracked .lean file needs was added at landing, after the recorded kernel run; full.yml's Lean job re-elaborates the file as committed, first in run 41057260555, where both theorems printed no axioms.)

Hunt #153: Requirement A as a static provenance scan (ontology_provenance/, 2026-10-24)

Status: measured, static scan, no claim promoted. A name and import-graph scan of whether a proposed frame reaches a zero-derived name (docs/09 section 5.1, Requirement A). It is a floor, not a proof that a clean frame is prime-derived. Of the fifteen numbered ontology frames, the scan flags five files as reaching a zero-derived name. Reading those files: two construct from zeros (03, 08); two build from primes and mention zeros only as a printed comparison (05, 06); one compares GUE spacings with zero spacings (11). Frame 07 imports only dh_coefficient and is clean at symbol granularity. Nothing here is evidence for RH. Record: ontology_provenance/MISSION.md, ontology_provenance/RESULTS.md. The same change records an additional defect in factorization_vs_position/DIAGNOSIS.md, independent of the Hunt #2 withdrawal: the kappa scanner's output is (1 + 0.284079*kappa) times one fixed function, so its sign changes cannot depend on kappa.

Hunt #146: the out-of-band reduction priced at the Davenport-Heilbronn line (weil_dh_line/, 2026-10-21)

Status: open; landed on main in #356 on 2026-10-27, with the raise recorded below (updated 2026-10-30: this line first read "open, on a draft pull request"). Resumed in place on 2026-10-26 by the owner's decision. No-go at the gate for the searched envelopes (attempt unresolved, measured); K2 on DH at the DH line run on 2026-10-26 (streamed): the pipeline refuses at every checkpoint to T# = 800, so the binding kill condition did not fire (measured, stage A only). Asks whether zeta's Weil window positivity can be proved with enclosures on a window where the Davenport-Heilbronn form is provably negative, half-width L >= (log 30.617188)/2 = 1.7107808, through the out-of-band reduction of oob_envelope/. The comb floor S*_L at L_DH lies in [5.637302, 5.643859] (hardened), so weak duality allows the gate S <= about 6.2; the best evaluable joint envelope found gives 6.418111, and the best partition into three-prime clusters has the Arb floor 6.174396. A measured stage-A scout at L = 7/5 (envelope S = 4.117981) has negative pivots at T# = 400 to 440 and none from 460 to 640, stable under the mode and precision responses. K2 refuses Epstein (1,1,6) wherever its form is known negative, and DH at L_DH (N = 856, 1280 bits): one negative pivot at all 14 checkpoints with beta* > 0, lambda_min about -5e-30 and shrinking in magnitude as T# grows. Nothing here is kernel-checked, no novelty is claimed, and nothing bears on RH. Record: weil_dh_line/PRICE.md, weil_dh_line/RUNS.md, weil_dh_line/MISSION.md. Raised 2026-10-27 (dated addition; the entry above is left as written). The T# -> infinity limit of the stage-A block, the DH form on the scout's even Legendre modes, has a negative eigenvalue on every leading block of 101 or more modes (-1.943e-30 at 856; hardened, resting on enclosed Rayleigh quotients; a reviewer's separate implementation reproduces them). So the stage-A block refuses DH at L_DH at every T# and every N >= 101 when S is valid (the composite is measured at that assumption, through two unreviewed derivations). The T# = 800 eigenvector itself is not a witness, and K2 at T# = 800 keeps its verdict with S understated by up to 43, so it does not test S. The responses called "mode and precision responses" above are not refinements (PRICE.md items 5 to 7). Later the same day S itself was bounded in Arb (sup(P_DH - H) <= S + 34/5), which lies inside the witness's margin for T# >= 540 and N >= 856: there every numerical step is enclosed, and the refusal rests on two unreviewed derivations (PRICE.md item 7, link 6). Added 2026-10-31. The algebraic step of the DH negative-cell certificate at (c, N) = (1541/50, 64) is kernel-checked (Cert65.lean, decide +kernel, built in CI; RESULTS-negative-cell-kernel.md): for the integer midpoint matrix and the listed vector the quadratic form is negative with margin far above the Arb radii. The statement about the DH form itself stays hardened, since the Arb enclosure and the perturbation bound are outside Lean.

Hunt #145: the Davenport-Heilbronn zero off the critical line, kernel-checked (rung3_ball_production/, 2026-10-21 to 2026-10-26)

**Status: closed 2026-10-26. dh_separation_hundred (below height 100, DH has a zero in the open strip off the line and riemannZeta has none, from hunt #137's RH to height

  1. and dh_zero_located (a zero of DH in the plan's closed square), read with

dh_analytic_half, are kernel-checked (Lean v4.35.0-rc2, Mathlib 065356127b1d, zero sorry, only propext, Classical.choice and Quot.sound), pending external verification. The assembly's davenport_heilbronn : davenport_heilbronn_statement is weaker: the statement, as defined, is also met by the trivial zero s = -1 (lean/ZetaLean/DHTrivialZero.lean, found by the PR #353 review). Original; searched-and-absent in PRIOR_ART.md, not a novelty claim; nothing here bears on RH.** RESULTS.md states the theorems as compiled. Added 2026-10-26: one theorem now carries the strip result, davenport_heilbronn_strip (an entire function with the Davenport-Heilbronn series and functional equation has a zero s with 0 < Re s < 1 and Re s ≠ 1/2, witness DH), in an emitted module outside the sites' import closure, compiled by a second on-box assembly of the same 215 site oleans with the same three axioms (RESULTS.md section 1 and door 2). Added 2026-10-27: Lean's own kernel replayed the whole assembled closure into an empty environment and accepted it, and the independent checker con-ron --verified accepted the dependency closure of the 18 final declarations (675,388 of the 1,569,505 constants) split into 216 shards on cloud VMs, with coverage shown (door 1, Stages A and B); the grade stays kernel-checked, pending external verification. 2026-10-21 status, superseded: open ledger hunt; a pilot on one shared box in one session, measured; the pilot site's two theorems kernel-checked and its planted faults refused; no production shard launched; davenport_heilbronn unproved, and no file emitted the assembly module that would prove it; nothing promoted; no novelty claimed. The rung3-pilot lane of the campaign the owner directed on 2026-10-21 (route 3). The task proposed #138; the launching workflow reserved #138 to #144 for landings it is renumbering and assigned #145. Lean v4.35.0-rc2 against Mathlib 065356127b1d, fetched fresh into the worktree's own lean/.lake (nine packages, each its own git toplevel at its manifest revision), with the 15 ZetaLean modules the certificate imports built from source. scripts/72 is main's: the two fixes this lane first cherry-picked landed in PR #323 with a resume fix (571857a47), the branch merged origin/main, and tests/test_rung3_ball_compile.py passes (22). On the swarm box (32 vCPU Haswell, shared with other workflows; the 1-minute load ran from about 30 to 160 during the session), B_right_06 (K = 17: 23 prime towers, 48 composites, 10 modules) was generated by scripts/60 --arith ball --shard --only B_right_06 and compiled by scripts/72 --jobs 1 --module-jobs 1: every module exit 0, and upper_B_right_06 and enc_B_right_06 print only propext, Classical.choice and Quot.sound. Planted in copies: a prime-tower radius one unit of 2^-64 low, the tail radius just under its bound and M just under the enclosure's bound are each refused at the named declaration, with only the touched module recompiled; eps' in the centre theorem's last step (the enclosure taken as a hypothesis, the centre itself not compiled) is accepted at 1/2000 and at the next multiple of 2^-64 above the bound, and refused at the bound and at 1/4000. The kernel-accepted B literal equals scripts/65's ball once 1^(-s) is seeded as the exact ball (1, 0, 0), as the generator does; scripts/65's own figure is wider by exactly 738083614955/2^64, so the mirror is conservative here. A first attempt, interrupted mid-site when its session ended, is kept as a replay check: its four compiled modules have this run's sources byte for byte, and their CPU agrees within the spread PILOT.md shows. Costs, net of import: 2.09 CPU-s per tower atom for scripts/68's p = 2 tower and 2.42 in the site's five prime shards, against the 1.2504 that scripts/67 records from v4.33.0-rc2 on a machine the record does not name; 0.674 per composite in the site against 0.55; about 10 CPU-s to import the certificate's support modules; peak RSS 7.8 GiB per process, at most 1.9 GiB of it anonymous. Projected to the 215 sites: 350 to 416 core-hours (towers priced flat or per byte of their text; the flat price is the better supported, since a p = 1801 tower cost close to what a p = 2 tower does with three times the text, but the request keeps the larger by a rule recorded before that run; scripts/67 records 175.0 atom-only), so 13 hours of wall clock at 24 processes on this box if it were free, and 26 hours at 16 processes holding one core each. What binds, in OWNER_MEMO.md: CPU on a shared box; 123 to 146 GB of oleans against 24 GB free on its disk; and an assembly that, as the sites are generated today, would import all of them, at the pilot's 0.66 bytes of RSS per olean byte 81 to 96 GB in one process (untested). The memo asks for a production window and launches nothing. No method here meets the bar of docs/37.

Resumed 2026-10-26 by the owner's decision, which granted the production window on cloud compute; production waits for funding and none was launched. Measured, one process: proving a prime tower through one decide +kernel of the whole dirichletTermBallB value plus contains_dirichletTermBallB cost 3.4 CPU-s net at p = 2 and 5.3 at p = 1801 on a box ball. That meets the threshold of 10 recorded in advance. The staged tower with decide +kernel in place of norm_num did not meet it (43.4 CPU-s, against 66.1 under norm_num). Both planted one-unit faults were refused. Extrapolated, production with whole-decision towers comes to 50 to 58 core-hours instead of 416, which needs a change to scripts/69 outside this hunt. assembly.py emits the module that would prove davenport_heilbronn from the 215 site theorems, with a located zero and a separation lemma against hunt #137's RH to height 100. Its body elaborated against 214 stub site interfaces and B_right_06's real oleans, and rests only on the standard axioms and the stubs; davenport_heilbronn stays unproved. PRODUCTION.md is the runbook. In a follow-up the same day, under a scope extension from the coordinator, scripts/60/69 gained an opt-in --tower whole mode; the default output is byte-identical. The pilot site recompiled in it end to end (measured): 163.6 CPU-s and 16.8 MB of oleans, against 1,861 CPU-s and 180 MB staged. Two one-unit tower faults were refused. Extrapolated from that site, production comes to 29.8 core-hours and 14.8 to 16.0 GB of oleans, with an assembly import of about 17 to 18 GB of RSS. On those projections the sites fit GitHub Actions shards and the assembly fits one 32 GB machine. Closed the same day. The whole-mode tower namespaces were first site-qualified (Prime_<sid>_<p>), because two site roots could not be imported together; the staged default keeps that collision by ruling. .github/workflows/rung3-production.yml then compiled all 215 sites on GitHub Actions (run 40353536947: 215/215 sites, every site theorem on the three standard axioms, 16.1 runner core-hours, 15.1 GB of oleans). Its fault job left no verdict (GitHub's bash -e), and after the fix the fault ran alone (run 40372531479: refused at Prime_B_right_06_2.ev_whole). The pair counts. On this box, from oleans on /dev/shm with every zip checked against GitHub's digest, the assembly compiled (85.3 CPU-s, 17.87 GB peak), and so did the separation against RHHeight100.rh_height_hundred. All 17 final declarations print only the three standard axioms. Four planted faults on production output were refused: the tower radius at ev_whole, the tail radius at enc_, M at upper_, and eps' at and below the bound in centre_lt_centre. No independent kernel replay of the assembled environment has run yet; that is the first door. (Superseded 2026-10-27: with davenport_heilbronn_strip there are 18 final declarations, and door 1 ran, as recorded at the top of this entry.)

Hunt #151: where Li's criterion first fails for Davenport-Heilbronn, enclosure-carrying (li_dh_onset_hardened/, opened 2026-10-26)

Status: lambda_{328997}(DH) < 0 and lambda_n(DH) > 0 for every 1 <= n < 328997, hardened (every step in Arb balls, two radii agreeing); pending external verification, not kernel-checked. It answers the question #128 (li_dh_onset/) priced at about 20,000 core-hours, and confirms #128's extrapolated n\* = 328997 exactly. The pre-registration was committed alone before the runs.

Hunt #138: complete Mobius orbits at fixed P change the majorant's constant, not its order (mobius_orbits/, 2026-09-26)

Status: ordinary argument with an independent model challenge (a model review, not human review) and exact finite checks at the stated scope. The fixed-prime route is an unresolved attempt and the energy continuation was paused by allocation on 2026-09-27; neither is a refutation, RH is unresolved, and no novelty is claimed. The hunt takes up, in a scoped form, the multi-prime mechanisms that prime_pair_error/MOBIUS_PAIRING.md left unconstrained. For a fixed squarefree P, writing a=dm with d|P and gcd(m,P)=1 regroups each Sigma2 row exactly by orbits and keeps every incomplete orbit; a complete orbit has the fixed-sign kernel K_m=-Phi_P(floor(N/(mb))), so the oscillation left is mu(m) alone. A separate Mertens level-set identity rewrites each row exactly, and a transfer bound prices it by a Mertens envelope with no unpriced floor jumps. The specified absolute-value majorant, which discards |mu(m)|<=1, has leading constant (phi(P)/P)^2/48 at fixed P (1/192, 1/432, 1/675 and 4/3675 at P=2, 6, 30 and 210) and the same N(log N)^3 order. Under ALIGNMENT.md section 5 that is category 3, an attempt that did not establish the needed statement, and it obstructs neither a signed estimate nor a growing-P construction. Exact integer and rational checks cover 1196 (N,P) cases and 8412 b-rows (N=4..300, 1000 and

  1. with zero defect, a trial-division check of the Mobius sieve, and two planted faults

(a dropped boundary, a reversed complete-orbit sign), both caught; at P=210, N=10000 there is no complete orbit and all of Sigma2 is boundary. The owner-directed continuation (ENERGY.md) moves to the Baez-Duarte criterion, where the squared error E_N is nonincreasing by standard approximation geometry. It adds a Schur-complement block identity, a compensated one-direction update along the predeclared Mobius shell v_n=-mu(n)log(2N/n), and a residual-matched fallback whose gain is at least M_N/T_N under the trace cap T_N<(log(2pi)-gamma)log 2. These pass Arb ball checks at N=4, 8, 16, 32 and 64 at 128 and 192 bits, conditional on Vasyunin's published Gram formula, with an mpmath cross-check at N=4. The inequality that would finish the route, ENERGY.md (7) or the weaker (12), is unproved, and the normalized trace values behind (12) fall from 4.28 to 0.243 over the five shells. The pause is category 4: no new uniform estimate toward RH was established, and it refutes neither the identities nor the Nyman-Beurling method. REVIEW.md and ENERGY_REVIEW.md are separate-context model reviews of the written mathematics, not of the code. Cited: Baez-Duarte math/0202141v2, Vasyunin's formula, and Bettin-Conrey-Farmer arXiv:1211.5191, which assumes RH and so is not an input. No literature search for novelty was run.

Landed 2026-10-21. From the local branch iris/rh-mobius-orbits-sep26 (three commits of 2026-09-26 and 2026-09-27, tip 16aff65f8, base 8941a80), merged whole and not rewritten. Its entry carried no number; #138 was the next free one, with #136 and #137 held by hunts on open branches. Four landing changes, none of them to a claim: the status above replaces the branch's, which predated the pause; the doors section of RESULTS.md was 97 words with no active-constraint part, which tests/test_hunt_doors.py refuses, and is completed by a dated addition below the original bullets; energy_test.py is now test_energy.py, because the nightly scripts/run_hunt_tests.py collects only test_*.py and would never have run its 8 tests (the commands in ENERGY_RUNS.md stay as run, with a dated note); and the methods are indexed in docs/37 (two identities, a construction, a control). On landing the hunt's 20 tests passed, and the two probes regenerated results.json, pilot.json and energy_reviewed.json identically apart from timing and reproduced every field of the two pre-review energy files. Landing promotes none of the hunt's claims.

Hunt #133: at least ten kernel-checked zeros of zeta on the critical line below height 50 (hardy_ten_zeros/, 2026-10-14)

Status: the target theorem compiles; a hunt result, not yet reviewed or landed. HardyTenZeros.ten_zeros_on_line_below_fifty : ∃ ts : Fin 10 → ℝ, StrictMono ts ∧ ∀ k, 0 < ts k ∧ ts k < 50 ∧ riemannZeta (1/2 + ts k * I) = 0, assembled from hunt #132's first_zero_on_line and nine new interval theorems zero_in_<tag>, one per two-decimal interval around the ordinates 21.02, 25.01, 30.42, 32.93, 37.58, 40.91, 43.32, 48.00, 49.77; Lean v4.35.0-rc2 against the pinned Mathlib, zero sorrys, no native_decide, twenty axiom reports propext, Classical.choice, Quot.sound, every dependency rebuilt from this commit's sources. It says at least ten zeros, not which ones: no zero count, no claim of one zero per interval. The three obstacles were sized first. Hunt #132's phase window has a 2K delta term and leaves the quarter turn from the fourth bracket on (-6.19 at 49.77, proved below -pi/2 there); dropping the first eight Euler-limit factors gives a window that no longer grows with the height ([-0.089, 0.04] for every bracket of width 1/100 below 50). The order-4 remainder of hunt #132 is used unchanged at any truncation, with an exact squared norm; truncations are chosen per endpoint, 684 term boxes in all (order 6 would need 334, order 8 246; order 4 was kept, since a new Euler-Maclaurin development would save under five kernel minutes per build). Only primes get an exp tower; composites are coarsened products of their factors' boxes (the lab's contains_cpow_mul_coarsen), 219 towers and 447 products. One generated file per bracket: 692 s of kernel time for all nine, against 11.2 minutes projected before it was spent. The remainder bound sits 6.85 to 14.94 times above the defect Arb encloses; doubling the radii fails every bracket; planted faults (a changed digit, a wrong factorization, a halved radius, doubled radii) are refused by the kernel, which proves their negations. Prior art: no formal localization of several explicit nontrivial zeros in the searched set (arXiv and zbMATH Open APIs, web), beyond hunt #132's one. Not a novelty claim. .github/workflows/hardy-ten-zeros.yml rebuilds it and audits the axioms fail-closed. Landing in lean/ZetaLean/ is the owner's call. Nothing bears on RH (docs/08).

Hunt #132: a kernel-checked zero of zeta on the critical line between 14.13 and 14.14 (hardy_first_zero/, 2026-10-11)

Status: the target theorem compiles; a hunt result, not yet reviewed or landed. HardyFirstZero.first_zero_on_line : ∃ t ∈ Set.Ioo (1413/100 : ℝ) (707/50), riemannZeta (1/2 + t * I) = 0, Lean v4.35.0-rc2 against the pinned Mathlib, zero sorrys, no native_decide, axioms propext, Classical.choice, Quot.sound; the four ZetaLean modules it imports rebuilt from this commit's sources (lean/build.sh, 4 min 14 s). The owner-approved goal was a sign change of hardyZReal (lean/ZetaLean/HardyZ.lean, accepted into Mathlib) plus the intermediate value theorem; the pinned Mathlib has no representation of riemannZeta on the line with an error bound, no Euler-Maclaurin, no complex Stirling and no digamma bound, so four pieces were built. A sign test that needs no value of the Riemann-Siegel phase: Re(zeta_a conj zeta_b) < 0 plus a quarter-turn bound on the phase of Gammaℝ. That bound, from Euler's limit GammaSeq -> Gamma with each approximant in polar form (window [-0.5045, 0.0357] against a true increment of -0.00405). The order-4 Euler-Maclaurin representation of riemannZeta on Re s > 0, Im s > 0 with an explicit remainder, by a quartic-kernel corrected trapezoid and the identity theorem. And 30 terms per endpoint (the lab's order-2 trapezoid would need 471) enclosed by the lab's dirichletTermBox2, written as literals from its bit-exact mirror and each checked by decide +kernel: 120 kernel decisions in 2 min 48 s, where one decision over the whole sum did not finish in 30 minutes. The remainder bound holds against Arb with a factor about 4.7 to spare; planted faults (a 3% change in the remainder constant, one digit of one box) go red in the tests and in the kernel. Prior art: no formal localization of an explicit nontrivial zero in the searched set (arXiv and zbMATH Open APIs, the AFP topic lists, web); the nearest are Alpöge and Furman's Lean proportion theorem and Eberl's Isabelle zeta and Euler-Maclaurin entries. Not a novelty claim. Landing in lean/ZetaLean/ is the owner's call. Nothing bears on RH (docs/08).

Hunt #131: what happens to a claim after it is recorded (claim_halflife/, 2026-09-10)

Status: partly settled; one headline withdrawn by its own control. Not mathematics: a measurement of this laboratory, and it trades against nothing mathematical. 95 case-log entries frozen at 2da62eb. One recorded claim in five is named by any test; of the 25 (hunt, test) pairs that can be scored, 19 defend at least one number against a 10% mutation and 8 are byte-pinned; 23.2% of entries were never touched again. Of 73 strict revisions, one installed a test in the same commit, against a doctrine in which every HANDOFF.md record carries a Now caught by: field. Median latency from a hunt's first commit to a revision of its own claim files is one day. The correction rate itself is withdrawn: a blind hand audit of 30 stratified rows agreed with the classifier on 16, precision 0.40, which is close enough to chance that no rate can be reported. It named two structural defects, both repaired, and re-scoring against the same rows reaches 21 of 30, which is fitted rather than measured; a disjoint sample is with a second blind audit. The container's default shallow clone hid three quarters of the history and would have reported 30.5% never revisited instead of 23.2%. tests.yml and full.yml still check out shallow, which HANDOFF.md recorded in August as the fix that was prepared and not landed. Nothing bears on RH (docs/08).

Renumbered on landing, 2026-10-10. Opened on the unmerged branch claude/repo-hunt-content-2smbid as #122, with its front door numbered 40. On main #122 is dh_minus_landing and 40 is the page on one form per discriminant; #127 to #130 are euler_defect_axis, li_dh_onset (#298), epstein_height (#299) and quotient_exponent (#300), so this is #131 and the front door is docs/43. The entry above is the branch's text with the number changed; it was written before the second blind audit returned.

Checked on landing, 2026-10-10. Every script was re-run against the history frozen at 2da62eb, and test_claim_halflife.py (15 tests, seconds, every git call bounded at that commit, skipping on a shallow clone) re-derives the corpus, both classifiers on all 414 later commits, the pinning, the depth table and both blind-audit scores: all byte for byte. The three planted-fault ladders reproduce in a scratch worktree at that commit once its state matches the one they first ran in; the re-run found that the null rung never runs the untouched tree, so a test red for an unrelated reason scores as byte-pinned, which the recorded eight (each computes a hash) escape. Since the entry above: the second blind audit agreed on 22 of 30 (precision 0.67, recall 0.77), so RESULTS.md reports the rate it withdrew here as 15.3% (73 of 414 later commits flagged, scaled by precision over recall), worth one significant figure; and #287 gave tests.yml and full.yml fetch-depth: 0 on 2026-10-09. Two statements did not reproduce and are marked where they are stated: at 2da62eb only 3 of HANDOFF.md's 24 records carry a Now caught by line (the header states the format, which is the doctrine), and the "nineteen of the seventy-three revisions" in the doors is derived by nothing in the hunt. The public-record question was considered on landing: this measures how the public record's own claims are guarded and corrected, which an outside reader evaluating them would want, and it is a negative result about the lab's tooling, which AGENTS.md keeps here. Two methods are listed as seen and not admitted in docs/37.

Hunt #130: the barrier law is not the shape the diagonal suggested (quotient_exponent/, 2026-09-10)

Status: measurement stands, headline withdrawn by its own audit. The first write-up said the conjectured shape E >= c N/sqrt(y) "is not the shape of T*". It is: BARRIER.md states the conjecture with a moving constant, and fitted that way the conjectured exponents give rms 0.1198 on 20 grid rows against the free three-parameter fit's 0.1258, with one parameter fewer. Every measured point satisfies E >= 0.1704 N/sqrt(y); the only bite is that the published constant 0.2 is 15% too large at the new 10^7 point. The stated reason for reformulating the programme was also false: the factorial form returns the same optimum to 2.7e-14. The factorial objective is a difference of quantities near 1e8 whose answer is near 1e4; written as sum_q w_q e_q over attainable cells it is a sum of nonnegative terms equal to the excess itself, and that is why the ladder reaches 10^7. Reproduces the four published diagonal values to every printed digit and adds 3 x 10^6 and 10^7, where T*/N^{3/4} is 0.1704, below the published 0.18 to 0.23 band. On a (y, N) grid the quantity the conjectured shape says is constant falls by a factor 2.3 to 2.8 inside the conjecture's own range; the free fit is a = 1.150, b = 0.878 against (1, 0.5), and the conjectured shape is three times worse on the grid while indistinguishable on the diagonal, where three of four models agree to three decimal places. The excess is a staircase in y, holding 9.406483 across sixteen consecutive supports at N = 10^4 and reaching exactly zero at y* = 173, 0.87 of the attainable-cell count, so the column count is neither sufficient (already refuted at N = 27, y = 9) nor necessary. What survives and is new: the staircase, the zero threshold at y* = 63, 173, 589, 1938 with alpha* falling monotonically, and the finding that only 40, 99 and 275 of the 61, 198 and 630 attainable cells carry any prime mass, so what holds the excess up is the requirement W >= 1 at the cells with no weight at all. Extended the same day (closed_form.py): the plateau values are exact rational multiples of a single logarithm, (7/2) log 2 at 10^3, 3 log 23 at 10^4 and 6 log 113 at 10^5, agreeing to 1e-14 or better, and the reason is that of the 14 to 181 cells carrying excess at the optimum exactly one has nonzero weight, so the objective collapses to that one cell's Lambda. The rationals are read off e_q and then compared, not fitted to E. At N = 10^6, y = 1995 the excess is 0.0 with 497 cells still carrying excess, every one weightless: the programme never zeroes the constraint violation, it zeroes the weighted violation by parking the rest where Lambda vanishes. The 10^6 plateau itself was not located, about 11 hours of solves, and no value is claimed there. Front door: docs/42. Audit: AUDIT.md, fifteen attacks, four landed. Nothing bears on RH (docs/08).

Renumbered on landing, 2026-10-10. Opened on the unmerged branch claude/repo-hunt-content-2smbid as #121, with its front door numbered 39. On main #121 is qrh_conditional and 39 is the page on the quasi-Riemann theorem; #127 to #129 are euler_defect_axis, li_dh_onset (#298) and epstein_height (#299), so this is #130 and the front door is docs/42. The entry above is the branch's text with those two references changed.

Checked on landing, 2026-10-10. Every script was re-run and test_quotient_exponent.py (39 tests, about a minute) re-solves or recomputes every number the result page and front door state; the LP values reproduce to 1e-12 or better. The entry above was written partway through the audit's corrections and keeps four figures RESULTS.md has since replaced: the free fit on the final 20 rows is a = 1.159, b = 0.874 (the 1.150, 0.878 came from an earlier row set the artifacts no longer hold), the fall is a factor 2.1 to 2.8 once the 10^7 rows are in, "three times worse" is the comparison the audit showed was not like for like, and "to every printed digit" is 14 and 15 significant digits against the source's 40-digit pins. Four figures on the result page did not reproduce and are marked there: an identity-table entry (1.6e-16, not 0, at 3 x 10^6), alpha*'s rate (0.016 per decade; 0.007 is per unit of natural log), the 10^5 plateau (538 to 588, width 51; the stated 545 to 580 was the scan window) and two counts of excess-carrying cells (50 to 57 and 165 to 182 over every support, not 52 to 57 and 167 to 181 from three). The single weighted cell now holds at every support of all three plateaus, and the zeros at y* have exact rational witnesses, re-found by a route independent of the audit's. RUNS.md's note that the 10^6 value 123.83310675500282 has "no run behind it in this tree" is false: threshold.json and threshold.log carry it, at four supports from 1812 to 1937. Three methods are listed as seen and not admitted in docs/37.

Hunt #129: the module that missed the guard the tree already had (epstein_height/, 2026-09-10)

Status: settled, defect recorded, repair proposed and not applied, write-up corrected by its own audit. zeta/heatflow.py and zeta/core.py both already implement this kind of guard, and hunts/gate5_p6_b/probe.py carries the Epstein constant as an executable rule, so the first write-up's "the rule the core never learned" was wrong: the core has the pattern twice and zeta/epstein.py is the module that did not get it. It still accepts any dps and returns a silently wrong value above the height that precision supports: at dps 15 and height 120 the relative error is 2.3e+36, measured against a direct lattice sum at Re s = 5 that shares no code with the routine. 158 cells of the surface in artifacts/. The recorded constant plus a sigma term leaves a per-form bias of 1.6 digits, so the loss was re-derived exactly as -log10 t + sigma log10 pi - log10|Gamma(s)| - log10|zeta_Q(s)|, with no fitted constant: the discriminant cancels, the size of zeta_Q is the missing term, per-form bias falls to 0.15, and one uniform offset of +1.068 digits is left with rms 0.357 over 49 cells. guard.py was the rule as code with six planted faults, all passing, and an independent audit walked straight through it: a call at dps 24 promising 4 digits returns 0.59, because every rung held the form fixed and so tested the leading term rather than the -log10|zeta_Q| term the hunt had just derived. guard2.py carries the form and replaces the faults with a rung that tests the contract, sampling the calls the guard ALLOWS and failing if any is short; its own first run failed on a cell whose limit turned out to be the oracle, so the rung now measures its oracle too. It also answers the question hunts/dps_cap/reach_check.py left open, whose two runs were stopped at 30 and 25 minutes: measured at height 120 without running the recursion, at dps 100 the acceptance rule takes 24 segments of 24, so the recursion costs one evaluation per segment, and at dps 15 it takes 7 of 24, p = 0.292 against the 1/3 a uniform phase predicts, branching 1.417, about 6.4e6 evaluations for one segment. Because _arg_variation's depth limit of 45 returns the principal value rather than raising, the cap cannot cost a refusal, only a wrong count or a run that does not finish. Front door: docs/41. Issue #217. Nothing bears on RH (docs/08).

Renumbered on landing, 2026-10-10. Opened on the unmerged branch claude/repo-hunt-content-2smbid as #120, with its front door numbered 37. On main #120 is qrh_rival_step and 37 is the methods index; #127 is euler_defect_axis and #128 is li_dh_onset (#298), so this is #129 and the front door is docs/41. The entry above is the branch's text with those two references changed.

Overtaken, and checked, on landing, 2026-10-10. Issue #217, which this hunt opened, was closed by #291 on 2026-10-09: epstein_zeta now raises its own working precision by ceil(0.6822 |t|) digits, the leading term only, and tests/test_epstein_zeta_height.py pins it. So "it still accepts any dps and returns a silently wrong value" was true when written and is not true of the current routine, which returns 16.1 to 16.5 correct digits (the comparison's float64 ceiling) on every cell re-measured at dps <= 30 up to height 160; epstein_completed itself still accepts any dps. The tree's repair buys the digits silently rather than refusing, so guard2.py stays a proposal for the refusal half, and guard.py's own ladder now fails its second rung. Every number above was re-measured against the routine as it was, restored by zeroing the lift: the 120-cell grid and the 42 boundary cells came back identical, and test_epstein_height.py pins the surface, both laws, the audit's counterexample (0.59, 39.90, 44.07), its critical-line attack (+4.66, slow tier) and why_it_hangs. Four stated numbers did not reproduce and are marked where stated: the first model's residuals by form (+0.91 / +0.83 / -0.68, not +0.93 / +0.85 / -0.75), the rounding offset's range (0.86, not 0.87, to 1.15), the move from 1/t to |1/s| (0.0008, not 0.002) and the oracle's conservatism (13 to 214, not 10 to 400). One method is admitted to docs/37 (the lattice oracle, now reused by tests/test_epstein_zeta_height.py) and three are listed as seen and not admitted.

Hunt #128: Li coefficients for the counterexample, and how far a computation can see (li_dh_onset/, 2026-09-10)

Status: settled. Twelve controls, all passed, no kill condition fired. Bombieri and Lagarias make lambda_n >= 0 for every n equivalent to every zero of the multiset lying on the critical line, and the Davenport-Heilbronn function has a measured off-line pair, so some lambda_n(DH) is negative. No lambda_n(DH) existed anywhere in this tree.

Computed unconditionally by the Cauchy route, never from a zero list, n = 1 to 5000, and every one is positive, the minimum over the whole table being lambda_1 = 0.09763614680951967118155. The analyticity radius was established first by the argument principle: sum_{n>=2}|a_n| n^{-2} = 0.26666394613 kills Re s >= 2, and the rectangles covering the Apollonius discs of r = 0.9 and r = 0.95 hold zero zeros.

The controls are the result as much as the table is. The xi positive control reproduces the committed zeta.li table to 4.4e-28 over all 400 indices; lambda_n(DH) is bit identical across r = 0.5, 0.7, 0.9 for n <= 200 and across r = 0.9, 0.95 for n <= 2000; the generating function matches Li's literal n-th derivative definition to 4e-27; and an independent zero-side sum over 313 measured on-line ordinates plus 9 off-line quadruples below height 430 agrees to 1.14e-8, with argument-principle box counts matching line zeros plus two per quadruple exactly, zero unaccounted, at T = 100, 200, 300, 430.

A background fitted to the measured coefficients gives slope -0.32530 against the derived conductor-5 value -0.32561, agreeing to 3.1e-4. Feeding it and all fifteen known off-line quadruples in puts the first negative index at n = 328997, identical under four background variants, driven almost entirely by R - 1 = 4.2006e-5 for the pair at 0.80852 + 85.69935i, which is measured to be the global maximiser: no other off-line zero below height 90, none with Re s >= 0.83 in [90, 190], and the coefficient bound covers everything above 188.97 automatically. Grade: measured inputs, extrapolated conclusion, sixty-six times past the largest computed coefficient, and explicitly not a theorem. It sharpens hunts/jensen_clock's hedged n ~ 3.3e5 by 1.2 per cent without reusing the indicator that page calls useless.

The wall, priced. At n = 328997 the family needs about 1490 working digits and 670000 nodes, roughly 20000 core-hours on this calibration, against every efficiency door together being worth a factor of 5 to 10. The direct route cannot reach the onset. (Dated pointer, 2026-10-30: hunt #151 reached it, lambda_{328997}(DH) < 0 with every earlier lambda_n > 0, hardened, at 0.46 core-hours, so that sentence holds only for this hunt's r <= 0.99 family; see li_dh_onset_hardened/.) Its doors also found a defect outside the hunt: zeta.epstein.completed_dh costs 0.58 s per call at 142 digits against 0.092 s for the same mathematics with kappa cached once per process, a factor of six from a precision-keyed lru_cache receiving a precision that varies with the distance from s to 1.

Nothing bears on RH (docs/08).

Renumbered on landing, 2026-10-10. Opened on the unmerged branch claude/repo-hunt-content-2smbid as #123 (its RUNS.md still says so), and landed as #128 because main had since taken #123 to #126 and #127 is euler_defect_axis (#297). No front door. The entry above is the branch's text with only the number changed; what the landing check found is below rather than edited into it.

Checked on landing, 2026-10-10. test_li_dh_onset.py recomputes the first four coefficients, the half-plane sum, both radius rectangles by the argument principle, the xi control at n <= 60, the background fit, the onset scan with its sensitivity table and the cost model, and checks the tables, the zero side and the dominance wedge against the artifacts, which a scratch re-run of every script reproduced (RUNS.md). Three corrections to the entry above. (1) "No lambda_n(DH) existed anywhere in this tree" is false: scripts/18_dh_li_coefficients.py has computed n <= 24 since 2026-08-05 (docs/07 section 4), and agrees with this table to better than 1e-12; what is new is n > 24. (2) The "1.2 per cent" does not reproduce: jensen_clock recorded 330342, and 328997 is 0.41 per cent below it; 1.16 per cent is the gap to this hunt's own zeta-shaped variant, 325229. (3) Positivity is hardened only for n <= 12, where the zero side shares no code with the Cauchy route; for 13 <= n <= 5000 it is measured, with a radius-independence control, so the composite row is measured. (4) The completed_dh "defect" does not reproduce as a per-call cost: warm, on the contour at 142 digits, it costs 0.112 s against 0.102 s; the factor of six is the first call at each of two cached precisions, 1.7 to 1.9 s each. Six further stated numbers, five in the doors and one in RUNS.md, did not reproduce and are marked where they are stated. Three methods are listed under "Seen and not admitted" in docs/37.

Hunt #127: one form per discriminant, not forty-one (euler_defect_axis/, 2026-09-10)

Status: settled, correction; repair applied by #296. The composite-line discriminator is defined for a Dirichlet series with a(1) = 1, and a binary quadratic form represents 1 exactly when it is the principal form of its class group. Measured over the 41 rows behind docs/34 table E7: a(1) is exactly 1 on the 14 principal forms and exactly 0 on the other 27. No rescaling repairs them: none of the 27 represents only multiples of its least represented value. The section's qualitative reading survives on the entitled rows and is checked there against c(n) = Lambda(n)(1 + chi_d(n)), which uses no recursion, to 2.8e-30. The published non-principal numbers are not empty: they are exactly the composite defect of 1 + Z_Q(s), agreeing to 0.0 on all 27 rows, a series with no functional equation. The corrected axis is 0 for class number one and 2.9608 to 5.0847 for the rest at cutoff 61, and 5.0196 to 10.5748 at cutoff 401 with the ordering moving by one adjacent transposition, so the axis has no scale and the family issue #93 proposes as "a defect axis from 0 to 36 with class number as the knob" is neither. An independent adversarial audit (AUDIT.md) attacked the claim eight ways without moving it and found eleven overclaims in the write-up, including a residual presented as an independent oracle when R = |1 - a(1)| max|c(n)| identically. The repair in PROPOSAL.md reached main through #296 on 2026-10-10, before the hunt did; its closing note records what #296 did differently. test_euler_defect_axis.py (23 tests) recomputes every number the result page and front door state; one count in RESULTS.md section 2 did not reproduce and is marked there. Front door: docs/40. Nothing bears on RH (docs/08).

Renumbered on landing, 2026-10-10. Opened on the unmerged branch claude/repo-hunt-content-2smbid as #119, the number its issue #93 comment of 2026-09-10 uses, renumbered there to #124 on 2026-09-16 after main took #119 for lambda_dh_exact, and landed as #127 because main had since taken #124 to #126. Its front door, numbered 38 on the branch, is docs/40. The rest of that branch is not landed.

Note, 2026-10-21. "The rest of that branch is not landed" went stale the same day: the branch's other four hunts landed on 2026-10-10 as #128 to #131 (li_dh_onset, epstein_height, quotient_exponent and claim_halflife, in #298 to #301), each with its own renumbering note above. What the branch still held after that is settled by the landing of 2026-10-21 (#333). Its shallow-clone check in scripts/science_preflight.py landed, with the path that failed open fixed. Of its four HANDOFF.md records, the one on the three adversarial audits landed verbatim with a dated note (below, since HANDOFF.md is retired), and the other three are superseded by issues #217, #218 and #219, closed by #287 and #291. zeta/li.py's coefficient cache data/li_gamma_dps60_methodintegral_n128.json, written during the li_dh_onset run and committed on the branch in a9eba317a, was not landed: tests/test_li.py states that every committed data/li_*.json table was checked out of band, and it names no check of this one. Nothing else on the branch is missing from main. (Stale since #348, 2026-10-26: tests/test_li.py no longer makes that statement, and asserts its absence; the committed tables are recomputed by the tests zeta/li.py names. The cache above was still not landed.)

Record (2026-09-10): three write-ups, three adversarial audits, and the pattern in what they caught. Moved here from HANDOFF.md when that file was retired on 2026-10-22; the text below is as it landed there, including its landing note.

Note on landing this record, 2026-10-21. Ported verbatim from the unmerged branch claude/repo-hunt-content-2smbid (commit 57cb2a5). Its three hunts are on main as #127 (euler_defect_axis, front door docs/40), #129 (epstein_height, docs/41) and #130 (quotient_exponent, docs/42). Read against their audits (hunts/*/AUDIT.md), three sentences of the pattern paragraph do not hold as written:

  • "Not one of the eleven-plus-four-plus-eleven corrections was an arithmetic

error." Two of euler_defect_axis's eleven overclaims are miscounts: "forms in the published table 44" with "forms whose a(1) equals 0: 30", and "all thirty of the others", where the artifact holds 41 forms, 27 of them non-principal. Replacing those counts is the first item under that audit's "Corrections required", from its attack 7 ("check the write-up's own arithmetic", outcome PARTIAL BREAK). quotient_exponent's audit also required a count fix ("the 21 positive rows", where artifacts/fit.json holds 20) and a memory figure twice what its own formula gives (3.2 GB for 1.6 GB), though neither is among the four attacks its summary says landed. (epstein_height's eleven is its audit's summary line, "Eleven landed"; ten of its seventeen attack headings read BROKE IT, and attack 3, headed held, also led to a correction.)

  • "Every number in all three hunts reproduced." The counts above did not.

The headline measurements did reproduce, in the audits and again in main's 2026-10-10 landing re-runs (the probes' artifacts, the Epstein grid, the LP values). Some secondary figures did not reproduce as printed, in all three hunts, and each is marked where it is stated: one count in euler_defect_axis (RESULTS.md section 2, docs/40 section 3), four numbers in epstein_height (two of them in docs/41 section 3) and four on quotient_exponent's result page (noted in docs/42 section 5).

  • "What failed each time was a comparison or a control". True of the

failure the record names in each hunt (the residual, the model comparison, the guard's fault ladder), not of every correction: the miscounts are counting errors, and epstein_height's audit (attack 8) also found a step in law.py's derivation that cancelled a real term against nothing.

So read the pattern as describing each hunt's main failure, not every correction: the audits caught arithmetic too. One figure in the record is also tighter than its source: "the same optimum to 2.7e-14" is the N = 10^3 row, and over the four diagonal cutoffs the relative difference runs from 2.4e-14 to 7.0e-14 (hunts/quotient_exponent/RESULTS.md section 2). The branch's other three HANDOFF.md records of that day are not ported: issues #217, #218 and #219, closed by #287 and #291, supersede them.

Robin's inequality for 25-free integers (robin_tfree/, 2026-10-04)

Status: candidate, ordinary derivation, unreviewed, pending external verification; numerical step enclosure-carrying; inputs published and quoted. No RH or novelty claim. Past Morrill-Platt's verified range x0 = 29 996 208 012 611, the primorial ratio E(x) = log(e^-gamma prod_{p<=x} p/(p-1) / log theta(x)) is at most 2.481e-8, using only Buthe's x - theta(x) <= 1.95 sqrt(x) and the BKLNW tables. The step that matters is an exact identity: Mertens' boundary term and the log theta denominator cancel to first order, where earlier papers bound them separately. Consequences, same grade: Robin's inequality for every 25-free n > 5040; for every n > 5040 with nu_2(n) <= 24, nu_3 <= 14, ..., or q^(nu_q(n)+1) < 4.03e7; and sigma(n) < (1 + 2.49e-8) e^gamma n log log n for all n > 5040. A search on 2026-10-10 (web, arXiv, zbMATH Open, citations of Axler 2023) found 21-free as the best published t-free result (Axler 2023), so the claimed delta is 21 to 25, recorded as a search and not as novelty. t = 26 fails; the verified range is the binding constraint, and these estimates leave Robin's inequality undecided on the remaining integers (multiples of M*). Lemma 3(b),(c) and the use of the Buthe and BKLNW tables await outside review; the table transcriptions agree with the arXiv versions. Identity checked on real primes; 37 tests, which pin every derived number RESULTS.md states. See robin_tfree/RESULTS.md.

October 8 extensions (oct08_extensions/)

Status: written conditional candidates, with independent review artifacts. The density extension replaces the earlier ninth-power interval target by seventh powers under the same zeta half-plane assumption, adding a published explicit zero-density estimate. The fixed Fourier network admits a parameter extension; its all-length consequence keeps the upstream exact-arithmetic compiler assumptions. Genus-aware partial counts and a failed primality certificate route retain their limitations and falsification examples. See oct08_extensions/MISSION.md and its routes/ and audits/ artifacts. No kernel verification, practical Fourier speedup or worldwide novelty is claimed. Source authentication and subsequent extensions are recorded in the route reports, separately from numerical replay.

Hunt #126: primes between consecutive ninth powers, given the 7/8 half-plane (qrh_prime_powers/)

Status: candidate, pending external review; ordinary written proofs, every numerical step enclosure-carrying, conditional on OpenAI's zero-free half-plane. Given that zeta has no zero with real part above 7/8 (OpenAI's 30 September 2026 preprint, Theorem 1.1; the October 5 paper proves 11/12), there is a prime between n^9 and (n+1)^9 for every n >= 1, against the unconditional record k = 86 (Lee, arXiv:2602.14340). Route: the exact explicit formula for a quadratic B-spline weight, zero sums bounded in closed form through explicit N(T) bounds, an Arb cover of 10 <= n <= e^100 with worst margin 0.825 (0.303 with no verified RH height at all), an analytic tail, and Pratt certificates for n <= 9. The 11/12 theorem alone gives 13th powers. The weakened abscissa 15/16 moves the threshold to k = 17 as predicted, and k - 1 fails in each case from log n near 28.4 to 29.5. By-products: a prime in (x, x + (1/2) x^(7/8) log x] for x >= e^8, and |psi(x) - x| < x^(7/8) log^2 x /(128 pi) for x >= e^10. No review yet; no Lean. See qrh_prime_powers/RESULTS.md, which ends with the doors (k = 8 is where this method stops, unresolved rather than obstructed; explicit zero-density estimates are the door).

Hunt #125: explicit small witnesses from the 7/8 half-plane (qrh_nonresidue/)

Status: candidate, pending external review; proved given OpenAI, "The Quasi-Riemann Hypothesis", Sept 30 2026, Thm 1.1 (unreviewed input), enclosure-carrying numerics. From the zero-free half-plane Re s > 7/8 a smoothed explicit formula with weight (n/x)^(1/4) log(x/n), zero sums bounded through the Hadamard identity at sigma0 = 2, gives: every nonprincipal character mod q >= 3 has a prime p <= (log q)^8 with chi(p) not in {0, 1} (tight at q = 3), and p <= (0.7 log q)^8 for q >= 5, with limsup constant at most (96 sqrt(3)/343)^8 = 0.0031; every odd composite n has a Miller-Rabin witness at most (0.7 log n)^8; and g*(p) <= (Lambda_p log p)^8 with an explicit sieve factor Lambda_p. The Oct 5 2026 paper's 11/12 half-plane alone gives (log q)^12. The exponents are not new (Montgomery-Vaughan 13.12, Guo 8 Oct 2026); no explicit constant was found in the searched set. Controls: exact tables to 10^7, OEIS record values, a weakened abscissa that moves the measured exponent to 12 and 16, and a planted fault refuted by n(48473881) = 67. The exponent 8 is the wall of this information class (The doors). See qrh_nonresidue/RESULTS.md.

Hunt #124: an explicit class-number bound from the quasi-Riemann hypothesis (qrh_class_number/)

Status: candidate, pending external review; proved given OpenAI's Theorem 1.1 (The Quasi-Riemann Hypothesis, Sept 30 2026, zero-free Re s > 7/8, unreviewed), by an unreviewed written proof whose every constant is evaluated in Arb. Littlewood's short Euler product with two Cesaro cutoffs and Hadamard positivity for the zero sum gives, for every negative fundamental D other than -3, L(1, chi_D) >= 1/(10 log log |D|), hence h(D) >= sqrt|D| / (10 pi log log |D|): the constant the October 5 OpenAI paper calls computable and leaves uncomputed. A verified interval cover turns the bound into D(h) with h(D) <= h implying |D| <= D(h) (D(100) = 48 611 613, D(1500) = 12 409 254 457). An exact reduced-form sieve over all 3.77 * 10^9 fundamental discriminants below D(1500), 39 minutes on one core and no GRH, then completes the imaginary quadratic class-number lists for every h <= 1500 (9 245 562 fields, largest |D| = 562 394 347), conditional only on Theorem 1.1 (h <= 1000 also under the October 5 paper's 11/12 half-plane, with c = 1/16). It reproduces Watkins' h <= 100 classification exactly and the GRH-conditional odd-h counts of Holmin and Kurlberg for all odd h <= 1500. Controls: the bound sits at least 3.89 times below every exact L(1, chi_D) for |D| <= 3 * 10^6; weakening the abscissa to 11/12 and 15/16 weakens it in the predicted way; the GRH specialisation matches Lamzouri-Li-Soundararajan. No explicit half-plane constant or even-h list beyond 100 was found in the search; that is not a novelty claim. See qrh_class_number/RESULTS.md.

Hunt #123: Linnik's constant 7/3 from the 7/8 zero-free half-plane (qrh_linnik/)

Status: candidate, pending external review (2026-10-08). Uses OpenAI's Theorem 1.1 (no Dirichlet L-function vanishes in Re s > 7/8; Lean statements kernel checked here, argument unreviewed) as an input, not as something to audit. A smoothed explicit formula with height cutoff a small power of x, the input theorem removing every zero above 7/8, and Chen-Gupta-Li's single-modulus density estimate (arXiv:2507.08296v2, preprint) give, by an ordinary written proof, p(q,a) <= C(eps) q^(7/3+eps) for every modulus and class, effective, with psi(x;q,a) ~ x/phi(q) for x >= q^(7/3+eps), all but O(phi(q) x^-delta) classes reached by x = q^(7/6+eps), 30/13 for smooth moduli, and Goldbach numbers mod any q below q^(7/6+eps). On refereed density inputs alone the exponent is 12/5, the value Wikipedia lists for the 7/8 theorem. The binding point is sigma = 5/7 (Ingham's mean value bound against CGL's q1^(1/3) term), so the half-plane is slack for any theta >= 5/7; exact exponent algebra is checked against a float grid and planted lesions. The conditional implication is in the literature (CGL Corollary 1.4 at h = x); nothing is claimed novel. Least primes for q <= 5000 are tabulated as a descriptive picture (max exponent 1.83, at q = 5). Composite grade capped by the unreviewed input. See qrh_linnik/RESULTS.md, which ends with the doors.

Hunt #122: the second-DH tracked pair lands at 1.0876360002296, and the lower endpoint reaches 1.0876359 (dh_minus_landing/)

Status: probe, complete (2026-10-08); the parent's first door is priced and shut for this pair. dh_minus_heat left 217/200 < Lambda_minus <= 567009/320000 (narrow frame) and named a later rational heat time for the same conjugate pair as its first door. The pair's landing time is measured by two independent routes, an mpmath double-zero Newton solve (30 and 50 digits, agreeing to 4.6e-26) and a float contour-moment discriminant tracker (two grids, within 5e-14 of the first): t_c = 1.08763600022958693221796..., x_c = 7.5399442421673352..., grade hardened, no enclosure. Five rational times below it, down to 10876359/10^7 = 1.0876359 (1.0e-7 under t_c), carry the parent's Arb Taylor/Rouche disk decided positive on all four of its configurations, so the bracket becomes 1.0876359 < Lambda_minus <= 567009/320000, lower endpoint enclosure-carrying at the numerical step, the surrounding argument the parent's. The door was worth 2.636e-3, 0.38% of the gap. The pilot's second zero lands at 0.631 and binds nothing; no census of other pairs was run. The parent's second door (a sharper strip for the upper endpoint) is repriced from a number already decided in lambda_dh_bounds/strip2_results.json: worth 4.03e-4, not rerun. Controls: known values, the recorded 217/200 disk rerun, the t = 0 gate, two closed-form polynomial landings, three instrument lesions, and a 1% kernel fault that both tracking routes miss together (they agree to 1.4e-13 on the wrong function) and the Hurwitz zero-time identity catches (defect 1.2e-3). No kill condition fired. Nothing bears on RH; the subject is a rival function.

Hunt #121: what moves under the 7/8 strip, priced (qrh_conditional/)

Status: conditional pricing, exact exponent arithmetic plus two measured heights; nothing here is a result, and the hypothesis is nobody's theorem (2026-10-08). QRH(theta) is the claim, published by OpenAI on 2026-10-06 and replayed by no one known here, that every Dirichlet L-function is zero-free in Re s > theta, theta = 7/8 (and 11/12). (Update, later on 2026-10-08: replayed here, Lean's kernel and the independent NanoDa kernel accept the statements, docs/38 section 7; the argument is still unreviewed by any person, so the pricing below keeps its hypothesis.) The hunt asks which of this tree's priced walls would move if it held, and keeps the hypothesis in every sentence. Under QRH(7/8): prime_pair_error's rank-1 wall, the q = 1 mixed moment, moves from N^3 L^{-2H} to N^{11/4} L^4 and stops 3/4 of a power short of (31), which is RH-equivalent, so no fixed strip closes it; the completed bound (1) would move from a logarithmic saving to N^{35/12} L^6 by widening the major arcs to N^{1/12} (the hunt's own derivation, reviewed by nobody), still 11/12 of a power short of the target and 5/12 behind CHHL's GRH-conditional 5/2; Theorem A becomes W << N^{7/4} L^4, 3/4 short of (T); the ineffective constants become effective in principle. Rank 2 (Vaughan's bound), rank 3 in the square-root-arc configuration, the de Bruijn-Newman record (9/32 > 0.2, cited from the lab's reading of the claim, document 38 of the course), the simple-zero proportions, Lambda_DH and the Re(xi'/xi) equivalence do not move. The circle-method bootstrap theta -> (2 + theta)/3 drifts toward 1, never toward 1/2. Crossover heights against Johnston and Yang's explicit remainder, with the strip-side constant set to 1 by convention: log x = 35.11 (x about 1.8e15) at 7/8 and 98.22 (about 4.6e42) at 11/12; on every height the prime-pair hunt has measured, the explicit unconditional bound is the smaller one. Li's criterion: a strip multiplies the per-zero growth exponent of a hypothetical off-line zero by 2 theta - 1 = 3/4 and bounds neither the count nor the heights of such zeros, so nothing usable at finite n. Every statement is void if the claim is withdrawn or refuted. qrh_conditional/RESULTS.md ends with the doors.

Hunt #120: where the quasi-Riemann argument spends its Euler product, and a rival that cannot pay (qrh_rival_step/)

Status: read and measured, 2026-10-08; no verdict on the paper. OpenAI's family 003 (the lab's reading of the release is 38-the-quasi-riemann-claim.md, on the branch claude/openai-math-release-2026-10-06, not yet on main) claims every Dirichlet and finite-order Hecke L-function over Q(sqrt(-3)) is zero-free in Re s > 7/8. The hunt asks where the argument uses that its target is a single L-function with an Euler product rather than a finite combination, and tests the step on the classical rival, an Epstein zeta function of class number above one. Reading: the continuation criterion (Proposition 2.1 of the 7/8 paper, page 8) is a shared lemma whose proof survives for any function with bounded reciprocal on Re s >= 2; the Euler product is consumed where that criterion's two estimates are made, at the Moebius identity 1/L_K^S = sum mu nu N^{-s} (11/12 paper, Section 3, page

  1. and at the scalar Euler identity, Lemma 7.1 (7/8 paper, Section 7.2, page

53, used at (10.1) and (10.2) on page 71). Measured: for Q = x^2 + xy + 4y^2 (D = -15), written in the paper's own vocabulary as zeta(s) L(s, chi_-15) + L(s, chi_-3) L(s, chi_5), zeros at 0.92746 + 15.49663i, 0.91294 + 47.53316i and 1.02597 + 61.42280i, the last inside the region of absolute convergence, each hardened by two evaluation routes agreeing to 1e-25 or better, a lattice-route residual of 1e-41 where run, an mpmath winding of 1 and a segment-enclosed ball winding of 1 on python-flint; for D = -23 a zero at 0.95326 + 16.29022i on two routes. Each is a cancellation between two nonvanishing products of Dirichlet L-functions, so none is a counterexample (docs/08 section 4.3): the rival lacks the hypothesis, its reciprocal's coefficients being non-multiplicative and unbounded (exact to 10^6). The Davenport-Heilbronn function has no zero past 7/8 below height 300 in this hunt's scan. The v6 census (doi:10.5281/zenodo.23585903) lists six right-side zeros past 7/8 for 0 < Im s < 20,000; its deepest is at 0.897750155 + 2442.53029247i. These catalogue coordinates are external input, not part of this hunt's original scan. DH has no Euler product, so these zeros do not contradict the single-L-function theorem under discussion. Controls: gamma_1 recovered to 2e-16, planted faults detected, the class-number-one form sees nothing. Scan ranges are stated as limits in qrh_rival_step/RESULTS.md, which ends with the doors.

Direct resolution attempt (rh_resolve/, 2026-10-03)

Status: unresolved RH attempt; Li enclosure withdrawn; the equivalence it records is classical. Saved Li rows report positive lower endpoints for n = 1..58, graded measured: the hunt's own audit of 2026-10-04 withdrew their enclosure grade (complex modulus bound, outward rounding of the remainder and angle coverage unproved), and no repaired rerun exists (AUDIT.md, formerly THEOREM.md). EQUIVALENCE.md records, as an ordinary unreviewed argument, the classical equivalence of RH with Re(xi'/xi) > 0 on Re s > 1/2 (stated in epp_herglotz/RESULTS.md; see Lagarias, Acta Arith. 89 (1999), 217-234); it is neither new nor a proof of RH. A positive even Gaussian mixture has a measured non-real cosine zero. The log-concavity obstruction is conditional on an unproved uniform margin. Numbers quoted from the saved artifacts are pinned by tests/test_rh_resolve.py. No novelty claim.

Theta sums and Laguerre inequalities (rh_theta_laguerre/)

Status: unresolved RH attempt, ordinary proofs and numerical enclosures; external review pending. Every fixed raw theta partial sum loses an endpoint cancellation and eventually violates the first Laguerre inequality, with remainder bounds given. For every fixed cutoff K, a positive-measure cosine transform satisfies all inequalities through K everywhere but has nonreal zeros inside the corresponding critical strip. The K=1 control's second coefficient is exactly -12/121 at pi. Neither is a zeta counterexample. The all-order positivity obligation for the full arithmetic kernel remains open. See rh_theta_laguerre/RESULTS.md for proofs, dependencies, and reproduction.

The square-construction continuation tests two specific representations on the full theta kernel. The local score construction has a negative two-point Gram eigenvalue; the positive-average-of-shifted-squares construction has a negative integer-vector witness for a 24-point ratio matrix. Both witnesses carry Arb enclosures including all theta and integration tails. These are failures of the proposed representations, not negative Laguerre values or zeta counterexamples. Details: rh_theta_laguerre/SQUARES.md.

Central logarithmic moments (central_moments/, 2026-10-02)

Status: unresolved RH attempt, ordinary derivations and finite measurements. The central logarithmic derivative supplies an explicit all-orders moment positivity obligation. Exact symmetric-measure controls show that raw moment positivity does not transfer, and that any fixed number of these matrix tests can pass despite off-axis zeros. Direct central derivatives and theta quadrature are compared at two precisions. Eight finite Arb matrix-sign enclosures have independent determinant checks. See the scoped derivation and reproduction commands in central_moments/RESULTS.md. No RH or novelty claim; external mathematical review remains pending.

Out-of-band envelopes for Weil window positivity (oob_envelope/)

Status: candidate, hardened by two independent implementations, pending external verification. On the window supp f ⊆ [-L, L], any correction to the Weil symbol at frequencies ≥ 2L leaves the form unchanged, so the envelope constant in arXiv:2608.24827's one-stroke reduction can drop from A_L = sup P_L to sup(P_L − H) (theory lane, ordinary derivation, referee PASS). With it, for real even f:

For real odd f at support 2.38 (L = 1.19, N = 500 odd Legendre modes, T# = 500, the same sine:16 envelope, all error terms enclosed): numerics GL-96 route Q(f) ≥ 3.42853488205e-44 ‖f‖², referee Clenshaw-Curtis-192 route Q(f) > 2.99e-44 ‖f‖² (1280-bit Arb; formulas and shift frozen before any worker code or artifact was read, though the brief disclosed the worker's endpoint; verdict PASS). The odd reduction was derived by the numerics worker and independently re-derived by the referee. Parity and real/imaginary cross terms cancel, so the even sector is the binding one: Q(f) ≥ 5.7178e-48 ‖f‖² for every complex f ∈ L² with supp f ⊆ [-1.19, 1.19]. That composite is a candidate, hardened by two independent implementations that share python-flint, resting on ordinary derivations, pending external verification. The L = 0.8 line remains a real even statement. No window claim found in the searched literature exceeds half-width 17/16 (Liu, unrefereed); Zhu withdrew his own L = 1.19 claim. The threshold stays doubly exponential (the best out-of-band constant grows like e^L), so this is a fixed-window result, not a route to RH. Methods: docs/37-methods.md (out-of-band envelope lemma; residual-enclosed shifted Cholesky). Record: oob_envelope/numerics/RESULTS.md, oob_envelope/theory/RESULTS.md, oob_envelope/referee/REVIEW.md, oob_envelope/referee/ODD_REVIEW.md.

Weil positivity propagation across window size (weil_propagation/, 2026-09-23)

Status: open. Asks whether positivity of the truncated Weil form on a window of length L = log c, together with the Euler product, can be carried to L + delta. Every candidate transport is tested against Davenport-Heilbronn, whose form turns negative at c = 31 (rogue_frontier/weil_trunc/ s8), and a candidate that also holds for DH is refuted. Two workers: measurement of ground-state transport between windows in weil_propagation/numerics/ and literature plus candidate lemmas in weil_propagation/theory/ (branch teal-sea/weil-propagation-theory). No RH claim. The numerics branch also carries, under rogue_frontier/weil_trunc/, a DH feasibility gate (THEOREM_FEASIBILITY.md, DH_DICTIONARY_CONSTRUCTIVE.md, gate_checker.py): a hardened witness (the negative even-sector eigenvalue at (c, N) = (31, 60)) plus a Guinand-Weil dictionary argued for DH, which together force an off-line DH zero. Grade: ordinary argument, internally reviewed, pending external verification. The conclusion is classical (Spira 1994); only the route is the lab's. Lane weil_propagation/ghost_zeros/ (2026-10-21, preregistered in its PREREG.md): at band N = 128, DH's even ground-state transform has exactly one zero with |Im z| <= 1/2 over the on-line gap (83.109, 87.647), real and 0.019 to 0.035 above the off-line ordinate 85.699, at each of the seven preregistered windows from c = 29 to 47, on both sides of the even sign flip at c = 30.647, and an exploratory scan finds a zero within 0.3 of that ordinate at every scan window from c = 15 on; removing the off-line quadruple's term removes it, an on-line sham does not, zeta has no zero in its gap, and Epstein (1,1,6) shows the same at its own off-line zero; hardened counts for the band-128 matrix only, no continuum or RH claim (weil_propagation/ghost_zeros/RESULTS.md).

Update, 2026-10-26 (weil_propagation/c4_s2/exact_gram/, branch hunt/c4-exact-gram). The compute sweep resumes C4 at S = {∞, 2}, whose c = 2.9 negative count 4, 10, 20 at N = 8, 16, 32 is graded by a response band, not a bound. The exact-Gram ball rebuild that would close bound_quad/'s E2 is priced on the box. One 280-mode block costs 258 s at dps 40, and the full rebuild is estimated at about 155 core-hours, which fits 7 GitHub Actions jobs. It has not run. Its reading and decision table were committed before any rebuilt row existed (PREREG.md). The (U-S) gate now also refuses Davenport-Heilbronn, by an exact composite atom at n = 6. A first pass at the truncation lemma finds the bound_trunc/ route obstructed on the full window space (at least 0.13 at 280 modes against the 1.18e−2 scale) and open on continuous windows (0.0063). No RH claim. Later the same day: the a priori bounds P1 to P3 close (the s-rule was changed to a uniform trapezoid by a dated amendment, before any row). G_z closes, and its identity check caught a float-endpoint defect. G_b's integration-by-parts regions remain, so the run waits on them. The tail-mass step closes as a Poisson identity, which re-prices the rebuild at about 28 core-hours. Third unit: P4 closes (G_b's integration-by-parts regions, each checked against quadrature with a planted fault), lifting PREREG row D0's precondition. A disk-backed mode table fits four workers per runner. The calibration run is asked for, not yet run. Fourth unit: the calibration ran on GitHub Actions (run 40342895119, success). Measured there: the Gram took 13.1 core-hours in 24 unbalanced parts, and an s-block of 128 nodes took 841 to 932 s with four processes per runner. The Gram (20 MB, by sha256) is committed for reuse. The full rebuild is sized from these figures at about 104 core-hours and 11 hours wall, and it is asked for, not yet run. No ΔT number has been read. 2026-10-27: the full rebuild ran (run 40371181784, success, 12 h wall, 132 CPU core-hours). The headroom rule was broken: one job ended 4 percent under the 350-minute cap, mostly through contention between replay blocks. [withdrawn 2026-10-27, see correction below] PREREG row D1 holds at c = 2.2, 2.5 and 2.9. ε_q is a bound, at most 9.2e−4, 13 to 18 times below the 1.18e−2 scale. At c = 2.9, C_8, C_16 and C_32 at ε are 5, 11, 23 on the full space and 3, 8, 20 on V_4. These are statements about the 280-mode object R^(280) only, since the truncation bound ε_t is null and D3 is not read. Grade: exact inertia and ball enclosures, resting on unreviewed derivations (BOUNDS.md P1 to P4). No RH claim. Correction, later on 2026-10-27: "ε_q is a bound" above is withdrawn. A review found that the code's Gauss bound was the (n + 1)-point bound used for n-point rules, so the quadrature radii, and ε_q, were up to 225 times too small. The D1 reading and the counts above are not established, and E2 is unresolved again. A rerun at 13 nodes per panel, where the corrected bound equals the old radii, was launched on Actions and has not been read (exact_gram/BOUNDS.md s5, RESULTS.md correction, RUNS.md s12).

Second Davenport-Heilbronn heat flow (dh_minus_heat/)

Status: candidate ordinary argument with enclosure-carrying numerical steps; independent model reviews completed. A rational disk at narrow heat time 217/200 passes a Taylor/Rouche inequality at three Arb precisions, including all theta and integration tails. A simpler Euler-phase calculation on Arb and mpmath.iv supplies the candidate bracket 217/200 < Lambda_minus <= 567009/320000. The odd theta normalization is checked against a separate Hurwitz-zeta route; the plus cosine normalization is now stated and checked the same way. No RH, novelty or formal-proof claim. See the argument and its executable checker; the lower-bound quadrature uses Arb only. An exact residue-orbit corollary shows the nonresidue permutation n->2n mod 5 swaps the plus/minus heat class up to global scale, with a SymPy enumeration check.

Finite paid-surplus construction (paid_surplus_obstruction/)

Status: exact finite construction, independent challenge pending. At N=144, rational coefficients supported through 12, with harmonic balance, coverage through q=5, and mass 119/20, satisfy W_d<=1 at every prime power. The fully saturated paid cost is exactly psi(144) and its positive surplus is zero. The factorial discrepancy and complete repair are separately priced in the argument and checks. This does not supply a scale-dependent family or a uniform bound.

Status: explicit identities and finite arithmetic improvement; uniform rate open. The perfect-power cap has an exact signed factorial-sum formula and a deficit-dependent saving bound. For old fixed-seed square-root-support lifts, the saving is at most order N^(1/4) log N. An explicit balanced Mobius-prefix family has bounded coefficient mass and complete localized repair costs. At N=36864, excluding composites witnessed by primes 2,3,5,7 lowers its full excess over N from 877.252193 to 220.982190, with coefficients unchanged. Five fixed cutoffs are checked with two interval implementations at two precisions. No extrapolated exponent, RH claim, or novelty claim. See derivations and the remaining obligation.

Status: five finite local-cap identities, no general rate claim. For the unchanged selected balanced prefixes at the five fixed cutoffs, every ordinary composite in the local repair interval is exposed by testing primes through its square-root bound, while every prime power remains. The saturated cap equals the exact Mangoldt weight in each local range. Exact prime-log vectors and two interval routes price the factorial discrepancy, repair, removed composite overpayment, zero remaining cap surplus, and the complete total. No asymptotic, RH, or novelty claim. See the finite diagnostic.

Status: explicit bound and finite comparison, with the leading gap retained. An early-truncated finite balanced lift has a complete paid-tail bound: at square-root coefficient support, repair costs at most order sqrt(N) log N for a fixed seed. The factorial remainder is also included. The construction retains its old leading constant above one, so its excess over N is still linear. At N=14 and support at most three, exact primal and dual witnesses show that replacing raw logarithmic caps by perfect-power caps removes the optimal excess log(2)/2. The displayed derivations and bounded checker distinguish this finite arithmetic improvement from a uniform estimate. The run checked 768 truncation cases and 3,072 inequalities on two interval backends at two precisions. No RH or novelty claim.

Positivity on attainable quotients (quotient_certificate/, 2026-09-07)

Status: finite improvement and correction; no asymptotic result. Requiring the factorial ceiling only at floor(N/d), d>=2, uses no prime locations to choose constraints. At square-root support it reduces the earlier all-cell excess by 34.4%, 29.9%, and 41.9% at N=1000,10000,100000. All 889 retained constraints pass exact integer checks on saved rational coefficients. An exact N=27,y=9 example with eight prime-looking cells has minimum excess log(2), refuting the earlier dimension-count argument for zero excess. The audit also separates the unproved exponent barrier from the global Nyman-Beurling results and records exact sawtooth correlations. See the construction, proof, and remaining doors.

Four-point pressure tuning (four_point_pressure/)

Kernel-checked at the pinned revision, 2026-09-28. The Hermes build at 5522b963 discharges the four-point bound at c = 2330/10^6, m = 432, p = 2500, with coefficient approximately 0.6728603588. The 49 build receipts and six axiom reports are preserved in four_point_pressure/evidence/; source hashes were independently reconciled during integration. This is a lab build, pending external verification, not a new Palomar registration. See RUNS.md and PALOMAR-READINESS.md.

The following paragraph records the September 5 disposition:

Status: closed exploratory record; complete Lean check canceled. Exact parameter substitution and an emitted-source preflight accompanied a candidate near 0.6728603588. The dependency build was canceled before any candidate proof module ran. No new proved bound, and no 68% or 69% result, is claimed. RUNS.md preserves the arithmetic, candidate commit and terminal CI evidence. Generated Lean replacements and build-workflow changes remain off main.

Cycle-moment counting probe (cycle_moments/, 2026-09-05)

Finite concave spectral counting inequalities, an exact obstruction to using only second and third moments, and tests of fourth and mixed cycle moments. The hunt also records exact repeated-index corrections and finite examples where all spectral moments agree but simple-point counts differ. Overlap statistics recover the count in an explicit spectral class and give further finite bounds for positive location Gram matrices. Its ordinary proofs, scoped Lean checks, and numerical evidence are summarized in cycle_moments/README.md. No stronger zeta proportion is asserted.

The prime-pair error, reproduced and taken apart by separation (prime_pair_error/, 2026-09-06)

Status: settled as a reproduction plus one measured relationship, not new. The Chou, Haag, Huryn, Ledoan error E(N) = sum_k (psi_2(N,k) - S(k)(N-k))^2 reproduces at all eleven rows of their Table 1 after truncation, the FFT pair count checked against pure-Python and direct pair sums, and the ratio E(N)/(N^2 log^2 N) keeps rising past the paper's range, 0.218 at 10^6 and 0.244 at 10^7. Odd separations carry nothing, the power of 2 in k is invisible, and k divisible by 3 is one sixth of the separations and half of the error, though less than proportionally to S(k): the squared error scales like S(k)^0.7 to S(k)^0.8, so no Poisson-type variance law holds. The signed part is one formula: on even k the error is S(k) [R(N) + R(N-k) - R(k)] with R = psi - x, the coefficients on a three-profile fit landing within a few percent of (R(N), 0, 1) at the two held-out N, 3 x 10^6 and 10^7, and the prime-only count carries the same profile with theta in place of psi, the difference being exactly the prime-power pairs S(k) [(psi-theta)(N) + (psi-theta)(N-k) - (psi-theta)(k)]. It explains 1 to 3 percent of E(N) and 12 to 22 percent of the prime-only analogue. Its k-averaged small-k limit is Korevaar and te Riele's Approximation 2.1 (Math. Comp. 2010), so the hunt claims a refinement of a published heuristic and nothing more. Nothing bears on RH (docs/08).

Second pass, 2026-09-06 (residue.py, results_residue.json). The profile above is the modulus-1 member of an exact four-piece decomposition of psi_2(N, k) at any modulus q (periodic baseline S_q(k)(N - k) plus a bounded term, endpoint terms linear in psi(x; q, b) at x = N, k, N - k, the pairs involving a power of a prime dividing q, and the correlation of the centered reduced-class function), whose only heuristic is that the primes not dividing q act on the reduced classes as all primes act on the integers. The modulus-3 member is the profile with the character mod 3 added, sign chi(k), and the modulus-30 member treats 2, 3, 5 together through the eight characters mod 30; every coefficient is 1 by derivation and frozen before the run. At five cutoffs the first pass never used (2 x 10^5 to 7 x 10^6) the three remove 0.5 to 1.9, 1.0 to 7.6 and 2.2 to 10.0 percent of E(N), with the post-hoc slope of the error on the modulus-30 correction within one percent of 1 everywhere. What the modulus-30 correction leaves has no structure mod 30 and all of its structure at 7, 11, 13, which the modulus-210 member predicts to within one standard error. The character-weighted sum sum_k chi_3(k) e(N, k) matches integral_0^N psi(x, chi_3) dx - (N/2) psi(N, chi_3) to half a percent at every cutoff; that identity-plus-heuristic is the one statement recorded as worth proving. The Goldbach-side analogue is Bhowmik, Halupczok, Matsumoto and Suzuki (Mathematika 2019); nothing here is claimed new.

Later passes, noted 2026-10-21. The status line above covers the first two passes only. The third pass (2026-09-06, RESULTS.md sections 14 to 19, audited in REFEREE.md) proves Theorem A, T(N) = I(N) + O(N^{Theta + Theta_chi} log^4 N) under no hypothesis, whose interesting case is conditional on RH for zeta and for L(s, chi_3), and Theorem B, E(N) = Omega(N^{1 + 2 Theta_chi - eps}), unconditional and informative only if GRH fails for L(s, chi_3). Both are written proofs on cited lemmas, corrected and independently audited, not kernel-checked and not claimed novel. The fourth pass (2026-09-20, FINAL_ACCEPTANCE.md) found no new bound and is an attempt unresolved. THEOREM_B_FORMULA_REPAIR.md (2026-09-20, landed 2026-10-21) replaces the defective zero expansion of THEOREM_B_SEQUENCE.md with a truncated explicit formula, Proposition 3, derived, reviewed at 95ae36b (finding R1 against its statement), revised, and cleared by a targeted follow-up review of 3006ef2. Its Theorem 4 removes Theorem B's eps along a closed-form sequence only under an isolated witness zero, which implies that GRH fails for L(s, chi_3), and Theta < Theta_chi; the unconditional case is unresolved. This note does not index every file in the directory. Nothing here bears on RH (docs/08).

Hunt #119: is Lambda_DH exactly Delta^2/2? (lambda_dh_exact/)

Status: OPEN, theory phase only (2026-08-18). Asks whether Hunt #61's bracket 0.0576 < Lambda_DH <= 0.19242481458 (narrow frame) collapses to a point. It would, if the backward-heat landing times t* of Davenport-Heilbronn conjugate pairs approached the isolated-pair value y0^2/2 as the depth y0 approached the sharp strip half-width Delta = 0.62036249819. The theory phase derives, rather than assumes, three things: every landing time is a floor for Lambda_DH (Dobner's half-line, unconditional); Lambda_DH equals their supremum under a no-creation step that the time-reversed Sturm zero-number theorem supplies and 600 sampled polynomial configurations support; and the crowding correction obeys dy/dt = -1/y - 2 pi rho, so a deep pair lands in time proportional to y0/log(gamma) rather than y0^2/2. Since depth near Delta forces great height and great height forces large rho, the pre-registered verdict is that the bracket does not collapse and sup t* sits in [0.058, 0.075] narrow. A one-parameter crowding model, calibrated on Hunt #4's nine landings, reproduces them to rms 0.54% and a census holdout to 0.79%; the probe is theory.py. Nine predictions with numbers are registered in MISSION.md before any evaluation above height 600. Nothing here is evidence about RH.

Interrupted, and what the interruption left behind (recorded 2026-09-12). The three evaluation agents and the adjudicator all died on API 529 errors on 2026-08-18, so no verdict was ever written and nothing here has been attacked. They did write their artifacts to disk first, and those are committed as raw material rather than discarded: deep_zeros.json (a screen whose float route agrees with mpmath to 1e-14 at three checked points), deep_zeros_1e6.json (one off-line zero located at gamma = 1000459.7433532759, beta = 0.8583118590734415, y0 = 0.3583, unit winding with defect 2.2e-16, float64 Euler-Maclaurin plus Newton, abs_f 4.4e-11), landing.json (the nine-pair census with measured neighbour distances), and theory_results.json. Read all four as measured, unadjudicated, and unreviewed: the zero is float grade and not enclosure-decided, the screen is not exhaustive over its window, and the crowding model's extrapolation to depths near Delta remains an extrapolation from a fit trained at y0 <= 0.37. The one thing worth noting for whoever resumes: that height-1e6 zero is shallower than the height-240 pair Hunt #4 already had, which is the direction the theory predicts, and it is the only evaluation datum the hunt ever produced.

The contract files the interrupted runs never wrote were reconstructed from those artifacts on 2026-09-12: RUNS.md carries a manifest for all six runs including the null one, and RESULTS.md states, item by item, that the hunt is entitled to say nothing about Lambda_DH, which of the five kill conditions was evaluated (none), and what whoever resumes should decide first. One control did run and is worth the reading: varying the screen's inner abscissa over t in [8, 600] flags 1, 7 and 13 windows at Re s = 0.85, 0.75 and 0.55, strictly nested as the argument principle requires, and the single window the shallowest screen flags is [228, 248], which holds the pair at gamma = 240.4046 that flow_repair/ had measured independently and earlier.

Renumbered from #52 on 2026-09-12. The hunt was opened on a branch that had not yet seen this case log's growth past #52, and main took that number for r_e2ee73/ first; the directory name never changed.

Resumed 2026-10-26 (compute sweep, branch hunt/lambda-dh-exact-resume), same number. First wave in the recorded resume order: decide the height-10^6 zero on both ball backends, run stage_locate on the low screen's 21 windows and give screen_hi an explicit artifact state, evaluate P2 and kill condition 2 as written, then price the census to 10^4 and 10^5 before spending on it. Scope and gates are appended to MISSION.md section 8; outcomes are appended to RESULTS.md as dated sections. Correction recorded on resumption: the decided floor is 36/625 = 0.0576; 0.05765 is a measured landing, not a decided one.

Status after the first resumed wave (2026-10-26): OPEN, Lambda_DH bracket unchanged. Decided, both ball backends (Arb and a hand-rolled mpmath.iv Euler-Maclaurin, argument-principle counts of exactly one zero in exact rectangles, planted faults flipping the 10^6 count and, in the tests, one recounted low zero): the height-10^6 zero, y0 in [0.3583118516, 0.3583118666] (rounded outward), and each of the 21 zeros with beta > 0.85 that the float screen found below 10^4. That these 21 are all such zeros is measured, not decided. Decided, Arb only: no zero with beta >= 0.90 for |t| <= 10^4. So P2 is refuted (y_max(10^4) is in [0.39775014, 0.40), against [0.40, 0.52]) and kill condition 2 fires, withdrawing the stage-6 depth law and P1's stronger band, P4's point and P5's location, which were derived from it. Measured (N-body ODE landings, control +0.024 percent on the floor pair, 21 of 21 windows accounted): no pair below 10^4 lands above 0.0576518; P3 holds (0.986 of the model at the deepest pair); P6 is refuted (d/h mean 0.95, 10 of 20 new pairs in [1.1, 1.8]); P8 is not observed (0 of 21); kill condition 1 does not fire. screen_hi was finished on a copy (111 zeros in [10^4, 10^5], deepest y0 0.437 at gamma 49688, measured), and the recorded complete: false artifact is marked superseded. A targeted check (not a census) landed eight of the nine pairs with y0 > 0.40 in [10^4, 10^5]: best 0.04902, none near the floor, so P5 stays unresolved with measured evidence against and kill condition 4 does not fire on what was measured (as of that check, 2026-10-26; the census below supersedes both). A slip inherited from flow_repair/NOTES.md (y0 = 0.3695261 for the floor pair; the value is 0.3695306) is recorded, not propagated (first wave, 2026-10-26; the follow-up below corrected it in this hunt's code and data, and RESULTS.md section 8.9 holds the corrections). The full 10^5 census (about 15 core-hours) is estimated and not run (first wave, 2026-10-26; it ran later that day, below). Follow-up, same day: test_lambda_dh_exact.py pins every decided and measured number of RESULTS.md section 8 and every verdict against its artifact, with real two-backend and Arb recomputations whose planted faults are shown red (31 fast tests, 32 in both tiers). Three more slips of the same kind (pairs 2 to 4) were found and all four corrected in this hunt's files; flow_repair/NOTES.md still carries them. The per-pair d/h refit of the shave model fails as a plug-in (rms 0.393 against the frozen 0.160), and the d/h it would need tracks the measured one at r = 0.967, pointing to a two-sided gap (measured). Units 3 and 4, same day: the two-sided gap, run against its pre-registration, improves on the frozen model but does not pass (rms 0.112; out of sample 0.154). The census over [10^4, 10^5] ran on GitHub Actions. Attempt 1 was void by its own kill condition K-d, which exposed stage_locate accepting roots outside the counted box (16 of 282). Attempt 2, on guarded code, was valid: 170 of 170 pairs accounted and landed, none above 0.0576518 (largest 0.04902). KC4 does not fire up to 10^5, P5 is refuted up to 10^5 (within the screens' coverage; RESULTS.md section 8.15 states the scope and a stronger reading) and P8 is not observed (0 of 21 below 10^4, its registered scope; this read 0 of 191, which pooled 170 pairs above 10^4), all measured. The audit under the guard changed one recorded root (screen_hi) and no claim. Review, same day (RESULTS.md section 8.16): every decided decimal interval is requoted rounded outward (printed to nearest, several were not enclosures); the Arb-versus-iv cross-check of the 10^6 zero is rerun on exact endpoints, because the recorded float version could not fail, and the enclosures intersect at all three points; the in-place slip corrections to registered text are reverted, with the corrections kept in RESULTS.md section 8.9; P4 is unresolved with evidence against; and two latent fail-open paths in the census evaluation are recorded as known limits. No verdict moves.

Owner decision, 2026-10-31: P4 is refuted, measured. flow_repair/'s census sweep (#383) landed every v6 pair below 10^4 with the same ODE, which supplies P4's missing antecedent, and the best landing is not raised: the floor pair stays first and the runner-up is 3.1 percent below it. P8 stays unresolved, since the closest margins are below the ODE's match to the flow. RESULTS.md section 8.17.

Hunt #144: a finite reversal-closed adversary excludes the recorded pressure target (reversal_minimax/, 2026-09-04)

Numbered on landing, 2026-10-21. Opened unnumbered on 2026-09-04, when this entry read "Reversal minimax probe", and landed by merge from docs/simple-zero-research-target, so its commits 7415efa1b, 1c4625b04 and 43217587d keep their hashes. #136 to #143 were held by other open hunts and same-day landings (this landing first took #143, which a concurrent landing of support_fcffa3b9/ had also taken), so this entry takes #144.

Status: candidate refuted at the stated scope (ALIGNMENT.md section 5, disposition 1); hardened numerical leaves inside an unreviewed ordinary derivation, pending external verification. No new zeta proportion is claimed. Whole-laboratory structural scout of the pressure optimization's reversal symmetry. At AMTOPA's fixed 17-term window and total pressure B = 93/23000, a finite mixed adversary (seven rational gap vectors from amtopa_ceiling/'s saved 2,200-cut pool, each with its reversal) bounds the supremum over the whole pair-weight and pressure polytope of inf_g F by 182022320819207/23000000000000000, below 0.007914014. That excludes amtopa_ceiling/'s recorded endpoint 0.007916857810 and the targets 19791/2500000 and 19786/2500000, for asymmetric coefficients too, and only for that window and B. The leaves carry Arb enclosures at 128 and 256 bits, agree with an independent mpmath quadrature at all 148 distances, and combine in exact rational arithmetic. Symmetry.lean (the averaging and budget deductions) has never been run through Lean, so nothing here is kernel-checked. amtopa_ceiling/ section 7.7 withdrew the same endpoint on 2026-09-06 by a different route; this bound agrees with that hunt's later bracket [0.0079107, 0.0079111939]; it is looser than that bracket's upper end, a float LP value, but enclosure-carrying. A counting audit shows the bridge's early span estimate overcharges pressure: an interior gap lies in (n-1)(m-n+1) local windows across all blocks, but the estimate charges it (n-1)(m-1) times. Its projected gains, 5/904171 on the registered four-point coefficient and 80/14366681 on main's later m = 432 build, which still uses the coarse count, are candidates awaiting a replacement assembly. RESEARCH_TARGET.md records a proposed higher-correlation direction and the elementary bound H*m/(m-1) on the capped assembly. See the results, the counting audit and the doors.

Hunt #114: one Epstein value, checked against a route that does not cancel (support_e6241336/)

Status: settled (support run e6241336, serving run 872d7dce / r_c7f779). One bounded question: independently compute abs(epstein_completed(0.8+85.7i, (2,1,3))) at dps = 60 and say whether it agrees with the claimed 1.6e-58 to within an order of magnitude.

It does. The value is 1.617452632377971461454064e-58, which is 1.0109x the claimed figure, 1.1% away; log10 of the ratio is 0.0047 against the 1.0 an order of magnitude would allow. This is the number the dps_cap measurement (#113) is stated against, so it matters that it was confirmed by a second route rather than by re-running the first.

Grade: measured, one evaluation at one precision on one form. Nothing here bears on RH (docs/08).

Hunt #115: prior art on the certificate route and the LP-versus-truth gap (support_78c499b4/)

Status: settled as far as a literature question can be settled in one search pass (support run 78c499b4, 2026-08-24, serving run 872d7dce). ARM E of the r_c7f779 fan-out: has a Cohn-Elkies / Delsarte linear-programming certificate been applied to a one-dimensional pair sum whose f is the positive part of a band-limited function minus a band-limited nonnegative square, and is the LP bound known to be tight against the periodic case.

The page lives here rather than in r_c7f779/ because a support run may not write into another hunt's directory. Nothing in it is a mathematical claim and nothing is measured: it is a prior-art record, and it is a search that was actually run rather than an unrun knownness default.

Hunt #116: rescued probe output from arm 17a0f787 (support_17a0f787/)

Status: artifact only, no write-up. A support arm of the r_c7f779 fan-out whose probe.py and results_quick.json were left uncommitted in a fulcrum worktree and are committed here so they exist somewhere other than one directory. The run is recorded stopped in fulcrum.

results_quick.json carries a planted-control ladder with firing scales. No one has written up what it shows, and this entry does not claim it shows anything. It is preserved evidence, not a result. Read the JSON before citing it.

Hunt #117: rescued LP and oracle code from arm 4019e8e5 (support_4019e8e5/)

Status: artifact only, no write-up and no results. A support arm of the r_c7f779 fan-out. oracles.py and t1lp.py were left uncommitted in a fulcrum worktree; the run is recorded stopped and produced no committed output of its own. The code is kept because it was one git clean from gone, not because anything here has been checked. It has never been run in this tree.

Hunt #113: what the min(dps, 20) cap costs, measured (dps_cap/)

Status: probe, complete. At 0.8 + 85.7i, epstein_completed at the capped dps = 20 returns 3.1e-33 where the converged value is 1.6e-58 -- a factor of 1.9e25 and not one correct digit. The error floor sits near 1e-(D+13), so nothing correct can appear below D ~ 46; the sweep puts the crossover there exactly (D=40 entirely floor, D=50 the first row with correct digits). D = 60 buys about 16 real digits, not 60, because the cancellation costs roughly 44 of them. zeta/epstein.py still caps three interfaces at dps=min(_d, 20) (lines 1092, 1126, 1142), so a caller asking for 60 gets 20 and count_zeros_box's integrality check passes on the noise.

Landed 2026-08-21 as the union of five concurrent 2026-08-14 runs that all wrote this directory and none of which merged; see the provenance header in dps_cap/README.md. Nothing here is evidence about zeta or RH: it is a measurement of an implementation at a single point.

Renumbered twice. These two opened as #35 and #36 from a branch 101 commits behind main, where both numbers were taken. They were renumbered to #49/#50 on 2026-08-18, and main took those two (plus #51, #52, #55-#57 and #59) before this branch landed. They are #60 and #61. The directories lambda_dh_bounds/ and prime_zeta_rightmost/ are unchanged and are the stable references.

Hunt #112: the depth-1 damage windows do not drift, and the lattice is exactly solvable (support_5418c63e/)

Status: settled (support run 5418c63e, for run 872d7dce / r_c7f779). The bounded question was whether the windows of D(1, s) > 0 track 2 pi asymptotically or drift, since r_b9552d run 37fb06a9 §2 derives the atom-reserve ceiling rho* <= 0.153216295 from the measured fact that P = 0 on the critical lattice out to d = 4000.

They do not drift. Putting ghat's two terms over the common denominator z^2+2 gives the exact one-term form ghat(z) = [alpha z sinh(z/2) + beta cosh(z/2)]/(z^2+2), and expanding it gives D(1,s) = (alpha^2/2)(cosh1 cos s - 1)/s^2 + alpha C sin s/s^3 + (E cos s + F)/s^4 + O(s^-5). The first correction is odd, so it translates the window rather than widening it: centre(d) - 2 pi d = K/s + O(s^-3) with K = 0.893732172363, half-width w0 + W2/s^2 with w0 = arccos(sech 1) = 0.8657694832 and W2 = 1.7796383030, both confirmed to nine and ten digits against dps = 60 edge locations. The offset converges to zero, not to a nonzero constant, and its linear-in-d fit slope is -2.5e-08.

The horizon turned out to be removable. On s in 2 pi Z the kernel is rational: D(1, 2 pi d) = P(u)/((u-1)^2+8)^2 with u = (2 pi d)^2 and P(u) = 0.6277706 u^3 + 3.6735975 u^2 + 33.2246011 u - 73.8367418, whose coefficients are all positive but the constant, so P increases on u >= 0 and has one nonnegative root at u* = 1.7707490. Since (2 pi d)^2 >= 39.478 for d >= 1, D(1, 2 pi d) > 0 for every d >= 1, derived, with no d_max, which is exactly P = 0 on the critical lattice. The parent's ceiling loses its horizon caveat on this ingredient.

A 0.002 scan of [1e-6, 1e5] finds 31,830 sign changes (1.999938 per period, so no anomalous window), all 15,915 lattice points inside a window, and minimum distance to the nearest edge 0.817252917 at d = 2, rising monotonically for d >= 2 toward w0. d = 0 is the one exception (D(1,0) = -0.9116), and it never arises because P sums over p != q.

One error was made and caught: expanding 1/S at 2 pi d instead of at 2 pi d + x drops a term of the same order as the s^-4 one and inflates W2 by 43%. The numerics rejected it before it was written down.

Hunt #111: the T1 certificate LP, reconditioned (r_c7f779/)

Status: in progress (run 872d7dce). Re-solving the Cohn–Elkies-style certificate LP for T1 that r_b9552d run 37fb06a9 left at 0.05410 with a failing solver. The verdict lives between the achievability floor 0.05716502 and the budget 0.06750841. This entry is rewritten with the result when the run closes.

Hunt #118: spend the out-of-band fact, or prove it cannot be spent (outband_certificate/)

Renumbered 2026-09-06 from #111 to #118: #111 had been taken by r_c7f779/ on 2026-09-01 and this entry, opened 2026-09-05, reused it. The directory name is the stable reference; citations of "hunt #111" dated 2026-09-05 or 2026-09-06 that mean the out-of-band certificate mean this entry.

Status: closed 2026-09-06, kill condition 2, prose grade. A CEILING. The out-of-band positivity is worth zero to any certificate whose positivity input is Weil's Hermitian form (every unconditional one): the on-line block is a definite Gram matrix, its kernel is positive-definite, its transform is nonnegative, and F has no unconditional upper bound outside the band, so bandwidth one is forced. What #110 priced was the RH-conditional pointwise class, which is why its range holds CGdL's conditional 0.6792. RESULTS.md section 8; the two inputs that would reopen it are named there. The step hunt #110 handed forward was: an inertia or isolation argument valid for a kernel signed only on alpha in (1, 1.5], or a ceiling proof that none exists. The target was the record (0.6734165). First session, measured grade (RESULTS.md): the LP's dual, which is the certificate the information supports, is pointwise nonnegative in x-space with a signed transform, strip multiplier active on [1.031, 1.5] at X = 80; the LP's q and p_2 columns are identical, so off-line pairs are priced as double zeros and the square structure is needed only for off-line zeros, which relocates the missing argument to a bound on the off-line-zero term of a signed kernel's pair sum, with zero-density estimates as the unconditional input; and a strip of width 0.05 is worth about the record gap (+0.00098 to +0.00129 across pinned exponents, six rungs) but that is inside the ladder's method error and not distinguishable from zero, while #110's wide strip stays established at 3.5x. Second pass, same day: a difference-of-squares candidate (apply stable_rank_trace to the positive half with the negative half folded into Q) was priced and killed by an edge lemma, proved: a real even spectral profile's autocorrelation is strictly positive just inside its support edge, so it can never be nonpositive on the strip. The lemma is false for odd factors (sinc^2 (1 - cos 4 pi x) is a verified counterexample), which gives the obstruction as a dichotomy: even factor, Gram-able and strip-blind; odd factor, strip-capable and never a Gram kernel. The gain is not a lattice-spacing artifact (stable from h = 1/16 to 1/64), and every finite-X kernel, control included, is negative beyond X. Worked as a node board from board.json; the first task is the numerical falsifier, which #110 ran but did not commit. Either answer closes the hunt.

Hunt #141: the bandwidth dial was physical all along (short_interval/, opened 2026-09-12)

Renumbered on landing: opened on branch claude/email-check-4vxjvb as #119, which main had taken for lambda_dh_exact. Of the next numbers, #138 and #139 went to other landings the same day (mobius_orbits/ and seven_eighths_pin/, on their landing branches) and #140 was left to the landings still in flight, so this entry takes #141. Landed 2026-10-31 (prepared for a landing on 2026-10-21 that waited as draft PR #330) by path from that branch at 4336606, not by merge.

Status: open, draft stage; nothing here is promoted. Grades: Theorems A and B of short_interval/PROOF-DRAFT.md, an unconditional short-interval proportion of simple on-line and of distinct zeros of xi', are a candidate: an ordinary derivation by the hunt's own sessions, unreviewed (the adversarial pass reviewed the gate audit, not the draft), not refereed, not kernel-checked, pending external verification. The constants are hardened (exact rationals plus a proved series tail, with an enclosure-carrying control) for one degree-6 window per bandwidth; the optimum at each bandwidth is float grade. The three findings are measured or derived. Dispositions: two routes closed on analysis (the Phi_3/Phi_4 transplant and window re-optimization); the band-edge route unresolved, since the logged search found no unconditional upper bound on F beyond the band and the hunt's own unreviewed arguments obstruct only two restricted classes (phase-dropping and sieve arguments), which does not show that no bound exists; the configuration-ceiling route re-scoped and not run, because the LP in this tree measures the window optimum.

Opened 2026-09-12 after Biao Wang's arXiv:2609.07918 (posted 2026-09-07) reached the laboratory five days late and by way of an unsolicited email from Luke Kenneth Casson Leighton, a member of the public, rather than any mechanism in this tree (short_interval/CORRESPONDENCE.md records and credits their two contributions). Wang proves unconditionally that zeros in (T, T + T^theta], 0 < theta < 1, are simple and on the line in proportion at least c(theta) = 2 - C_theta and distinct in proportion at least 3/2 - C_theta/2, with C_theta = theta/2 + (1/sqrt 2) cot(theta/sqrt 2).

Finding one, measured: the tree already computes his curve. hunts/frontier_map/frontier.py:zeta_H_closed, taken from the source paper's eq. (7.4), equals Wang's c(theta) to 1e-16 at five bandwidths and crosses zero at 0.5501939647441547, his printed theta_0. The same module's Hd = (1 + H)/2 is his distinct companion. Both constants in his Theorem 1.1 were already being computed here. What his Theorem 2.2 adds is that the dial is physical: frontier.py holds bandwidth as a hypothetical "capped at 1 by the Rudnick-Sarnak / Montgomery support restriction", and Theorem 2.2 (valid for supp g in [-lambda, lambda] with lambda < theta strictly, the counterpart of BGSTB's Lemma 5) says bandwidth theta is exactly what counting in an interval of length T^theta buys. The lab's dial and his exponent are the same number. A reading of existing material, not a new result.

Finding two, closed on analysis: substituting c(theta) into Phi_3 and Phi_4 is dead, and the reason is the second moment. In the coordinate the problem is affine in, R = 2 - c, which is the normalized second moment, Phi_3 reads R -> 1.001343 R - 0.002019: a fixed absolute gain against a relative 0.134% of whatever second moment it consumes, while R(theta) = 1/theta + theta/3 - ... diverges like 1/theta. The overhead is proportional to the one quantity that blows up in a short window, break-even is theta = 0.8082, and no re-optimization repairs it. Independently there is no analytic footing: the band is (-theta, theta) open and strict while the bridges read [-1, 1] with no free half-length in famlib.py, and the bridges are dyadic, 21 occurrences of 2 * T in Bridge/Main.lean alone (corrected 2026-10-21: 21 lines, 28 occurrences) with S8, S9 and S15 absorbing errors against N(T, 2T). A per-block error absorbed against N(T,2T) and reappearing against T^theta log T carries a hidden T^(1-theta), and that hazard recurs on every route here.

Finding three, measured: window shape is worth nothing. Writing the functional as R = A(phi)/lambda + lambda B(phi) with A = int phi^2/(int phi)^2 >= 1 by Cauchy-Schwarz, flat iff equality, the flat window gives R = 1/lambda + lambda/3 exactly, threshold the root of theta^2 - 6 theta + 3 = 0, that is 3 - sqrt 6 = 0.5505102572. Against Wang's optimal 0.5501939647, the entire value of the Montgomery-Taylor window over a flat one is 3.2e-4 of threshold, with the gap scaling as lambda^3/180 (measured). So at theta = 0.55 the whole window-shape game is worth about 9e-4 and an n-point share of it about 4e-5. Wang's c(theta) already is the re-optimization. This kills a recommendation an earlier draft of the brief made.

The routes, ranked in MISSION.md. (1) The xi-prime arm, the one place the tree holds something the field does not: frontier_map computes the Farmer-Gonek-Lee F_1 landscape too and Wang does only zeta. Measured with F_1 calibrated against two published Alpöge-Furman Remark 7.1 figures (0.858384 against 0.85838, 0.929192 against 0.92919), the xi' curve sits +0.11 to +0.20 above zeta at every bandwidth and stays non-vacuous to 0.51332 where zeta dies at 0.55019. Gated on a dyadic-average audit of the xi' derivation. (2) The bandwidth-theta configuration ceiling, computed nowhere; the bandwidth-one number is 0.6818286874638 against a window optimum of 0.6725007, headroom 0.00933, and the theta version is the stopping criterion for the whole axis and the best formalization target, since a cap is what this lab produces and others do not. (3) The price of the band edge, below. Corrected 2026-10-21: withdrawn, "and the best formalization target, since a cap is what this lab produces and others do not". Others do produce one: Alpöge-Furman arXiv:2608.13637v2 §7.2 gives the bandwidth-one ceiling p_0 <= 0.6818287 (MISSION.md §7, checked against the paper the same day). What holds is that the logged search found no bandwidth-theta ceiling, and which target to formalize is an allocation decision, not part of this record. "Computed nowhere" and "the one place the tree holds something the field does not" carry the scope of the search verdict below: not found in the logged search.

A door nobody had priced. hunts/outband_certificate/RESULTS.md states the reason for bandwidth one: "F has no unconditional upper bound outside the band, so bandwidth one is forced". That names the input and does not price it. Priced in landscape.py by capping F <= B beyond the band and checking the minimizer's sign, since a sign-changing minimizer is outside Lamzouri's class and its value is not a bound: B = 2 at L = 1.2 gives 0.760859, and B = 20 on a sliver of width 0.05 still gains 0.0142 against the 0.00933 headroom (corrected 2026-10-21: 0.0141; landscape.py gives 0.686561 for that cell against the baseline 0.672508, a gain of 0.01405, or 0.01406 against the exact 0.6725007, still above the headroom, and test_short_interval.py now pins it). Mechanism: a linear shadow price, c'(1) = (1/2)cot^2(1/sqrt 2) = (3/2 - H)^2 = 0.6847550854111, against quadratically small out-of-band mass. Not established: that a constant B exists unconditionally. Positive definiteness gives only F(alpha) <= F(0), which grows like log T. Note the direction, which hunts #110 and #118 had the other way round: they priced out-of-band positivity, a lower bound, and this method needs an upper one, so positivity is wrong-signed information and the +0.0068 is best read as the price of removing RH. Hunt #118's closure stands and MISSION.md §10 says so; §8 is a different object.

Three readings retracted the same day, kept in RESULTS.md §5 rather than deleted: that the closed form was the hunt's finding (it is Wang's Proposition 4.1 (4.2), and already in the tree); that PALOMAR-2026-08-21-000004 is its lambda = 1 case (it is not, and the error conflated the tree's two different constants both named c*, the zeta one with kernel |alpha| and the xi' one with F_1, which would have sent a session to formalize a generalization of the wrong theorem); and the window re-optimization killed by finding three.

Corrections owed the tree, with fixes in MISSION.md §11: docs/35 predated hunt #118's closure and read as open (done: corrected on main in PR #286, 2026-10-09); docs/35 and hunts/outband_intake/RESULTS.md disagree (+0.0068 with error 2.2e-3 against +0.0065 with 1.8e-3); references/papers.md carries no entry at all for Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, whose Lemma 5 is the arithmetic engine of this whole line of work; and hunts/frontier_map/RESULTS-frontier-map.md should record that its landscape now has a theorem attached at each bandwidth.

Two literature searches, run the same day and logged in full in RESULTS.md §4 (34 and 21 query strings, about fifty papers opened, citation graphs from Semantic Scholar). Xi-prime in short intervals: not in the literature as a function of theta, conditional or unconditional; the near-miss is Conrey 1983 and Rezvyakova 2005/2006, who already work in sub-dyadic windows U = T (log T)^(-10) with simple on-line proportion about 0.413 at k = 1, so the honest claim if the route succeeds is "first power-length statement, as a function of theta", never "first short-interval". The xi' prime side has no published statement; it exists only as Lean code (anthropics/formal-math, zeta23/Zeta23/XiPrime/, XiEF and CoeffMoments, dyadic, every lambda < 1), citing an unpublished write-up, and that pair is what a short-interval theorem must localize the way Wang localized BGSTB's Lemmas 3 to 5. Upper bounds on F beyond the band: none unconditionally beyond F <= F(0) ~ log T; under RH the best integrated bound (Carneiro-Chandee-Chirre-Milinovich arXiv:2108.09258 Thm 10) is 7/8 + (5/4) delta + ..., which tends to 7/8 and not to zero because delta spikes in F past the band cannot be ruled out. The obstruction the hunt derives for sieve arguments past alpha = 1 is prime-pair correlations at second order, blocked by parity in every sieve, and phase-dropping arguments cannot give B delta < 1. So the band-edge door is priced and unopened: no bound was found, which leaves the attempt unresolved rather than shows it impossible, and the two class obstructions are derived here and unreviewed. The price is kept so the next person to derive it knows what is behind the door and which arguments cannot open it. Corrected 2026-10-21: the xi' verdict, "not in the literature", states an absence the search does not establish, and is rescoped to "not found in the logged search", as the band-edge verdict already was: the search did not open six sources, and eight arXiv API queries and two Semantic Scholar searches never ran (RESULTS.md §4.1). "No published statement" for the xi' prime side carries the same scope, and so does "first power-length statement", which until those checks are made supports no more than "no earlier power-length statement was found"; "never 'first short-interval'" stands, since it rests on papers that were found and read.

The gate passed, 2026-09-12, with conditions (hunts/short_interval/AUDIT-dyadic.md, kept verbatim). A statement-by-statement audit of anthropics/formal-math at commit fbdc36bb inventoried every dyadic use in the xi' development (35 proof steps, 10 interface statements, 7 definitions) and found none that uses the dyadic structure to produce a main term: every use is cosmetic or an absorption with a named exponent, the exponents being exactly Wang's own condition lambda < theta and the end-strip condition theta > 1/2 (free in the whole non-vacuous range). The hazard the brief named is refuted: the D_1 corrections enter through a range-free arithmetic identity. One framing correction to the brief: the development uses no zeta pair-correlation input at all, so Wang's Theorem 2.2 is the template for localizing the xi' second moment, not an input to it. Missing are two statements whose proofs already exist (the short-window zero count and the short-window mu-integrals), and the bandwidth-theta constants must be enclosed by ball arithmetic before any number is stated. The eight-step proof outline, each step naming the Lean lemma it localizes, is in the audit's §6.

The audit was attacked and held, and the proof was drafted, same day. AUDIT-dyadic-adversarial.md: a second session told to break the verdict opened all 78 citations (all correct), built the import graph and grepped every reachable file for the range, re-derived every exponent, and confirmed the verdict, while rewording two conditions (the grid size is consumed two-sidedly by riemann_sum_monotone and tau_d_gt; the interface consumes the count to within O(log T) of the mu-integral, not Wang's form) and listing eight corrections to the outline. PROOF-DRAFT.md, 2131 lines, follows the outline with those corrections folded in. By its own tagging it uses 14 statements from the upstream dyadic tree, which that tree states as kernel-checked (no build was run here), writes out proofs of 26 new short-window statements at the level of Wang's paper, and asserts none without a written proof; the constant is carried as a symbol. No review of the draft itself is recorded. enclose_theta.py supplies that constant at hardened grade: exact rationals plus a proved series tail, no floating point on the load-bearing path, for one degree-6 polynomial window per bandwidth, with the zeta control sitting within 6e-12 of Wang's closed form at all ten bandwidths and the source paper's published xi' figures reproduced from below. The xi' lower bounds run from 0.868641500514 at theta = 1 to 0.130258286830 at 0.55. Not a result: a candidate, unreviewed, not refereed, not kernel-checked beyond the quoted statements, and the optimum at each bandwidth stays float grade; a sentence combining the draft with one of these numbers takes the draft's grade (RESULTS.md §10). A Lean port would touch about forty places and must thread the range through PrimeSide.Setting, per the draft's §12.

Route two re-scoped. ceiling_theta.py parameterized the measure-level LP by band width and its control showed that LP reproduces the window optimum, not the configuration ceiling, whose certificate is not public; the workflow is written and deliberately not dispatched.

What landed and what did not. Landed: the hunt directory, its manual-dispatch workflow hunt-short-interval-ceiling.yml, the credit entries in harness/departments/review_ledger.py and graveyard_ledger.py, one meta/interventions.jsonl entry, and short_interval/test_short_interval.py, added on landing, which re-evaluates the exact bounds and pins the numbers the hunt computes and quotes, for scripts/run_hunt_tests.py; counts and figures quoted from the audits, the search logs, papers and other hunts are cited, not recomputed. CORRESPONDENCE.md was cut to the credit. Not landed, and held on the source branch: the arXiv literature monitor (script, tests on placeholder fixtures, a scheduled workflow) and its brief, pending an operator decision on where the monitor lives and real fixture recordings, and operating configuration that does not belong in the public record.

Nothing here bears on RH (docs/08).

Hunt #110: the free out-of-band fact is worth 0.005 to 0.009 and no certificate can spend it (outband_intake/)

Status: probe, complete. GAP, not a barrier. BGSTB 2023 (arXiv:2306.04799, Theorem 1) gives form-factor nonnegativity unconditionally, outside the band every other certificate is confined to. Two matched LP ladders of five rungs price it: with out-of-band positivity on, the configuration class extrapolates to somewhere in [0.679, 0.682]; the in-band control, whose limit is the record 0.6725007036794116 and which therefore calibrates the method at about 1.8e-3, puts the difference at +0.0065 by the same fit, three and a half times the method error, so zero is excluded. The third digit is not: the difference route gives 0.6790 and the direct fit 0.6815, one digit from CGdL's RH-conditional 0.6792 and one digit from the bandwidth-one ceiling 0.6818 respectively, and an earlier draft that read the first coincidence as a landing is withdrawn (refit.py). But no known certificate can realise any of it. The inertia lemma needs a positive semidefinite evaluation form, which forces the spectral density v = phi^2 >= 0 and hence Khat = v*v >= 0 everywhere: the framework has no free ghat, only a free v, so a kernel negative out of band is not reachable. The requirement does not weaken -- with indefinite S the rank half of the lemma survives and the inertia half fails, witness Q = t[[0,1],[1,0]], S = diag(1,-1), c = 2, slack +2.0 at t = 1 and -0.5 at t = 1.5, against 4000 random PSD pairs never violated. The first-order law dJ/deps = (J - Lbar)/g(0) locates the gain entirely in the indirect channel, which is exactly the one the framework cannot enter. So the information is in hand and unspent, and the missing piece is a construction rather than a datum -- unlike frontier_math §2's sieve wall, where the missing input is Hardy-Littlewood grade and genuinely absent. Two things bound the result and are recorded in RESULTS.md: gate #3 fires (the drop is prime-blind, holding for Davenport-Heilbronn and both Epstein forms, which bounds what the zero side can ever add), a ratio test was stated confidently, then withdrawn as uninformative, and the in-band data's provenance is unaudited, so the range is a class value until someone traces it to an unconditional source. Nothing here bears on RH (docs/08).

Hunt #109: the soundcalc numbers reproduce, the pins that name their sources do not (r_4166b0/)

Status: settled for the units where a source existed; four units have no reproduction path and that is a property of the artifacts, not of the budget. The first transfer test of the reproduction procedure outside mathematics (issue #140, the Ethereum Foundation's zkEVM Security Sprint). ethereum/soundcalc pinned at d9078d64c9c3, ten units, two verdicts each, never merged: does the checked-in TOML follow from the vendor's tree at the version it names, and does that TOML still describe the vendor's current release.

Every parameter that could be re-derived, re-derived exactly: 51 of 51 numeric fields (RISC0 7/7 from its cited notebook, Miden 5/5 from its cited Rust constant, OpenVM 9/9 across three circuits, SP1 30/30 at two separate vendor refs). Not one number was wrong. Stage 0 holds too: soundcalc at its pin, run unmodified on its own TOMLs, leaves reports/ byte-identical.

The failures are all in the pins. Of eight provenance pins, two do not resolve: Airbender's cited commit is not in matter-labs/zksync-airbender (upload-pack: not our ref), and the pil2-proofman branch that the one executable regeneration script clones has been deleted. A third, SP1's, names a tag that does not contain the generator the TOML credits, because the generator landed eleven days after that release; the version field is a crate version string covering a window of commits, not a tree. Two of OpenVM's three source citations point at a superseded commit and a differently-named function whose arms carry lower query counts, so the citation is stale and the number is not.

The strongest single result: Airbender's parameters are byte-identical to what the vendor's own generator emits at its current tip, six months after the date stamped in soundcalc's copy, recovered at one hand intervention despite the cited commit being unfindable. The cheapest: gen_soundcalc_toml.rs computes most of its output from named constants, so reimplementing its arithmetic in Python re-derived thirty SP1 fields with zero build minutes against a brief that had budgeted ten runner-hours for Rust builds.

Two other findings, both VERIFIED. reports/risc0.md and reports/miden.md cannot be regenerated by the code at HEAD (both units are dropped for a missing num_constraints), so two published reports are held-over artifacts, and a config the loader cannot read yields a missing row in summary.md rather than a failure. And venus.toml's header claim of parameters identical to ZisK's does not hold: one circuit of forty-four differs in seven numeric fields. That moves no published bit (every per-regime total is identical, and the differing ALI component is not the binding term), which is why it is published rather than held.

Six planted faults, six reds, including one positive control. Two earlier lesions failed and are kept: the first showed that stage 0 detects a perturbed input and never a perturbed output, because it regenerates the reports before diffing. Transfer measure: 1.5 hand interventions per unit over the six units with any reproduction path, against a threshold of two. Nothing here bears on RH (docs/08), and every statement is scoped to the artifact at the pinned commit rather than to any vendor or deployed system.

Hunt #108: the SWE-bench leaderboard recounts from its own logs, where the logs are there (r_0dfb8d/)

Status: settled for the units run; the archive-wide availability question is open. The second transfer test of the reproduction procedure outside mathematics (paired with issue #140, soundcalc / zkVM): a public checker, publicly archived accepted objects, a published number. SWE-bench/experiments pinned at 1faa91cade05, split ids at dataset sha c104f840cc67, logs by anonymous HTTPS GET from s3://swe-bench-submissions.

Five of six preregistered entries recount to the integer (265, 359, 384, 388, 396 against their own published results.json), and each also agrees on all twelve per-repository sub-counts, which is the cross-check that rules out two errors cancelling in one aggregate. Across 2,484 per-instance reports re-derived from the test results recorded in the same file, zero disagreed with the verdict that file claims. The two verdicts were kept apart throughout and were shown to be able to fail: four planted faults (a flipped resolved, a planted PASS_TO_PASS failure, dropped required successes, a removed report) turned exactly the verdict each should and left the other alone.

The finding is the other class. One preregistered entry answers 0 of 500 documented report keys, all 404, while its predictions and trajectories are retrievable; a three-key probe across all 134 entries finds 15 answering none (thirteen dated 2024-07 or earlier, two dated 2025-08-05, one of which was swept over all 500 keys to confirm). And the archive's access model is the binding constraint on anyone repeating this: anonymous GetObject succeeds, anonymous ListObjectsV2 returns 403, so an outside reader can fetch any key they can name and cannot discover which keys exist. Every absence here is therefore "absent at the documented key", never "absent from the bucket". The repository README's claim that an AWS account is required is wrong for GET and silent about listing. Three smaller measured facts: tags.checked is a non-boolean truthy string ("false (See README.md …)") in six entries, so any consumer writing if tags["checked"] reads six unchecked entries as checked; no_logs: 0 is published for an entry that has one instance with a non-empty archived prediction and no archived report (it changes no count, being an accounting field rather than a counted one); and no entry among the 134 states the evaluation-harness version that produced its number, which bounds every "match" above. One reported unavailability was our own bug, an unencoded + in an entry name, found and fixed before publication and recorded because a reproduction procedure that blames the target for its own fetcher is the failure this class of hunt exists to catch. Nothing here characterises a team, an agent or a model, nothing was sent anywhere, and nothing bears on RH (docs/08).

Hunt #104: Erdős #126 is one unit-equation count away (support_f3ab3e34/)

Status: not settled, and the program is named. Independent-architect arm on Erdős #126 ($g(k)$ = max size of a set of positive integers all of whose off-diagonal sums have prime factors in a fixed $k$-element set $S$; is $g(k) = \exp(o(k))$?), asked to ignore the earlier brief's three lanes and design the program from the statement. It reduces to a counting question in four elementary lines. Fix $a_1 \ne a_2$ in $A$, put $D = a_1 - a_2$; then $c \mapsto ((a_1+c)/D,\, -(a_2+c)/D)$ injects $A \setminus \{a_1,a_2\}$ into the solutions of $X + Y = 1$ inside the rank-$(k{+}1)$ group $\langle -1, S, D\rangle$, so $g(k) \le 2 + N(k+1)$ and $N(r) = \exp(o(r))$ implies the conjecture. The known $N(r) \le 2^{8r+8}$ (Beukers–Schlickewei) is the same exponential wall the 1934 bound sits at, while the record construction (Erdős–Stewart–Tijdeman 1988) produces only $\exp\{(4+o(1))(s/\log s)^{1/2}\}$ solutions: the believed truth about $N$ already implies #126 with room to spare, which is soft evidence for it. The reduction is not new (it is how the $c\log|A|$ bounds are proved) and its constant is worse than 1934's; its value is that it says which door is load-bearing. Four chains are ranked by the strength of their first unproved step, and the recommendation is not the reduction itself (its first step is strictly stronger than #126) but the $m$-base-point version, where each extra element adds a free simultaneous unit equation no counting argument currently exploits. Audit of r_186989: its table replicates at $N = 200$–$400$ (its boxes were slack by $10^2$–$10^3$, not $60\times$), its composition-refutation direction is sound, and its loose thread 3 is false, refuted by its own witness $A=\{1,3,7,13\}$, $S=\{2,5,7\}$, which has $|A| = 4 > p-1$ for $p = 3$; the residue pigeonhole closes only at $p=2$, because only there does $2r \equiv 0$. It also never states a lower bound: $\{1,\dots,m\}$ with $S$ = primes $\le 2m-1$ gives $g(k) \ge (1+o(1))k\log k/2$, so $g(12) \ge 20$ and the unexplored range is $[k\log k,\, 2^k]$. Nothing here bears on RH (docs/08).

Hunt #91: Erdős #126, and the composition law that would refute it (r_186989/)

Status: not settled, scout killed on a pre-registered kill condition. A 30-minute bounded scout on Erdős #126 (for $n$ positive integers $A$, $f(n)$ is the least number of distinct primes dividing an off-diagonal sum $a+b$; is $f(n)/\log n \to \infty$?). The run inverted the problem before running any arm: $f$ and $g(k) = \max\{n : f(n) \le k\}$ are inverse staircases, so the conjecture is exactly $g(k)^{1/k} \to 1$, and all three briefed arms become measurements of one integer sequence. Exact branch-and-bound clique search on the $S$-smooth pair-sum graph, exhaustive inside $[1,N]$ with every witness re-verified, gives $g(k) \ge 2, 4, 5, 6, 8, 10, 11$ for $k = 1..7$ ($N$ from $2\cdot10^5$ down to $6000$); widening the box 60× moved not one row, because every optimal witness lives below 50. Those are lower bounds only, no upper bound on $g(k)$ is established, and seven falling $k$-th roots ($2.00 \to 1.41$) cannot separate a limit of 1 from a limit of 1.3. That is the "suggestive finite data" the brief named as a kill condition, and the hunt calls it that. Two things survive. The Formal Conjectures positivity mismatch is resolved: $f(n-1) \le f_0(n) \le f(n)$, so the Finset ℕ statement is faithful for the limit and wrong for pinned finite values ($f(2)=1$, $f_0(2)=0$, witness $\{0,1\}$). And the briefed composition arm points the wrong way: $g(1) = 2$ plus supermultiplicativity gives $g(k) \ge 2^k$ by Fekete, i.e. $f(n) \le \log_2 n + O(1)$, so a rigorous composition law refutes the conjecture rather than proving it, what #126 needs is an anti-composition theorem. The only $S$-dependence visible was parity: $2 \in S$ is worth more than every other structural choice combined ($g_N(3) = 5$ with it, $2$ without). Nothing here bears on RH (docs/08).

Hunt #105: the Erdős #126 scout survives, its literature claim does not (support_eccd5f5e/)

Status: settled (audit). Red-team arm against Hunt #91, every claim treated as hostile input. The mathematics holds. The inverse reformulation $g(k)=\max\{n : f(n)\le k\}$ and the chain to $g(k)^{1/k}\to 1$ re-prove in both directions, though the equivalence is conditional on $f(n)\to\infty$ and so on Erdős-Turán, which #91 does not flag. The positivity lemma $f(n-1)\le f_0(n)\le f(n)$ is correct step by step, and is sharpened here: $f(1..4) = 0,1,2,2$ are exact and proved, while $f(5)=3$ is not, because $f(5)\ge 3$ is exactly the open statement $g(2)\le 4$. The composition finding is correct and understated: Fekete is unnecessary ($g(k)\ge g(1)^k$ by induction), and since Erdős-Turán gives $g(k)\le 3\cdot 2^{k-1}-1$, a composition law would squeeze $g(k)$ into $[2^k, 1.5\cdot 2^k]$ and pin the growth constant to within $3/2$ rather than merely refuting the conjecture; a proof that $g(3)\le 7$ retires the programme without finding any gadget. A search written from the problem statement rather than from probe.py reproduces all seven rows $2,4,5,6,8,10,11$ exactly. Two interpretations are corrected: "every optimal witness lives below 50" is an artifact of search order, since witnesses are closed under multiplication by $S$-units and division by $\gcd$ (ours returned $162\cdot\{1,5,7,11\}$), and the table was never benchmarked against the trivial 1934 construction $A=\{1,\dots,m\}$, which already gives $g(k)\ge (p_k+1)/2 \sim \tfrac12 k\log k$. One thread is false: $|A|\le p-1$ for omitted $p$ fails for every odd $p$ (a class $r$ with $2r\not\equiv 0$ may be occupied arbitrarily often; $p=2$ is the unique prime where the pigeonhole works), refuted by $S=\{2,5,7\}$, $A=\{1,3,7,13\}$ from #91's own table. And "no progress beyond 1934" is misleading: the order of magnitude is indeed unimproved (confirmed against erdosproblems.com/126 and Füredi-Gyarmati arXiv:2602.07545, Feb 2026), but Győry-Stewart-Tijdeman 1986 and Erdős-Stewart-Tijdeman 1988 are directly relevant and neither hunt cited a single paper. New: the four-element reduction $u+v=w+x$ to a nondegenerate three-term $S$-unit equation is proved, and the route needs an $\exp(o(s))$ solution count, which the known $\exp(c\sqrt{s}/\log s)$ lower bound does not forbid. Nothing bears on RH (docs/08).

Hunt #96: Erdős #126, the S-unit route is sound and its theorems are 24.5^k too weak (support_d5d5ccae/)

Status: not settled, and the arm returns the "too weak" verdict its brief allowed. A support run for 0897a5a7, answering the S-unit arm of Erdős #126. The injectivity the brief asked for exists and is two lines: if every off-diagonal sum of $A$ is $S$-smooth and $a > b$ lie in $A$, then $x \mapsto (a+x, b+x)$ injects $A \setminus \{a,b\}$ into the solutions of the two-variable $S$-unit equation $U - W = a-b$ in positive $S$-smooth $U, W$, whence $|A| \le 2 + \min_{a>b} N_S(a-b)$. The implication direction was checked: a solution-count bound gives a $g(k)$ bound gives #126, and not the reverse, so unlike arm 3 of Hunt #91 this route points the right way. Fixing three or more base elements is strictly worse, because the resulting coefficients are differences of elements of $A$ and carry primes outside $S$, which forces the rank-based bounds ($2^{16k+16}$) instead of Evertse's $S$-based one. Charging Evertse (1984), $3\cdot 7^{d+2s}$ with $d=1$, $s=k+1$, gives $g(k) \le 2 + 3\cdot 7^{2k+3} \approx 1029\cdot 49^k$: weaker than the 1934 Erdős–Turán bound $3\cdot 2^{k-1}$, and than Erdős–Surányi's $2^k$, by a factor about $24.5^k$. Measurement then locates the slack. Enumerating all $S$-smooth integers to $10^{14}$ and counting $U-W=d$ exactly inside that box reproduces Lehmer's published Størmer table for $d=1$ ($1,4,10,23,40,68,108,167$ for $k=1..8$), and at $k=7$ the quantity Evertse bounds by $7.0\times10^{14}$ has a measured value of $96$, against the elementary $128$. The route is not what is lossy; the theorem is, by thirteen orders of magnitude. What would suffice is stated: dropping the base $49$ below $2$ already beats 1934, and $\exp(o(k))$ settles #126, which nothing known forbids since the best lower bound (Erdős–Stewart–Tijdeman) is $\exp((4+o(1))(s/\log s)^{1/2})$. No improved bound on $g(k)$ is proved here, and the one missing combinatorial lemma is named: every admissible $A$ beyond some size must contain a pair whose difference has few solutions. Nothing here bears on RH (docs/08).

Hunt #95: Erdős #126, the signature model has no ceiling to raise (support_baf4cde6/)

Status: lane closed, by proof rather than by a failed attempt. Support arm for hunt #91 (r_186989/), asked whether prime-valuation signatures plus entropy, VC, containers, dependent random choice or forbidden patterns can show that an $S$-summable set ($a+b$ free of primes outside $S$, $|S|=k$) has subexponential size. Answer: no, and the reason is not difficulty. Theorem B: for every prime bound $P$ and every $S$ containing 2, arbitrarily large sets satisfy every constraint $S$-summability imposes at primes $\le P$ (take $A = \{1+iM\}$ with $M$ the product of the odd primes $\le P$ outside $S$, so every off-diagonal sum is $\equiv 2 \bmod M$; checked at $P=60$ with $|A|=200$ against Erdős–Turán's 12). So any argument reading a fixed finite prime set is consistent with $|A| = \infty$ and can prove nothing; the hypothesis binds only at primes above $\max A$, which is the cofinite smoothness condition itself and not a signature. That also explains #91's puzzle that every optimal witness lives below 50 inside a box of $2 \cdot 10^5$. The one non-vacuous use of the local data is Lemma A: for $p \notin S$, $A$ meets at most $(p+1)/2$ residue classes mod $p$, never both $r$ and $-r$, at most one element in class 0, and for $p=2$ this gives $|A| \le 2$ uniformly in $k$ (the proved form of #91's parity observation). Lemma A also refutes #91's loose thread "$|A| \le p-1$ when $p \notin S$": $A = \{1,3,7,13\}$, $S = \{2,5,7\}$ has $3 \notin S$ and $|A| = 4$, and a sweep at $N=400$ reaches 5. Pushed to CRT injectivity, Lemma A gives a real theorem that is strictly worse than 1934: it saves exactly one bit per prime outside $S$ against a modulus that must already exceed $N$, landing at $2^{53}$ where Erdős–Turán gives $2^{5.6}$ at $\log_2 N = 64$. #91's own top door survives this arm: each 4-subset gives a non-degenerate three-variable $S$-unit equation, whose ESS count is worse than 1934 by $\approx 5\cdot10^{10}$ in the exponent but is not known to be exponential, and the missing lemma there is a fiber count, not a unit-equation bound. #91's other claims were re-derived independently and hold, including two reproduced $g_N(k)$ rows. Nothing here bears on RH (docs/08).

Hunt #93: Erdős #126 is a $\forall S$ statement, so every search we own can only refute it (support_8ea74995/)

Status: settled, for the bounded question it was given. The formulation arm of a multi-arm attempt on Erdős #126, run as support for a live sibling. It proves the equivalences hunt #91 asserted, $f(n) \le k \iff n \le g(k)$ as a Galois connection, and the equivalence of $f(n)/\log n \to \infty$, $\log g(k) = o(k)$, $g(k)^{1/k} \to 1$, $f(2^m)/m \to \infty$ and "the average multiplicative gain per added prime tends to 1", and finds #91's asserted mathematics sound, with one "iff" that should have been an implication. Three things are new. The refutation direction generalises: not only supermultiplicativity but any law $g(k+C) \ge \lambda g(k)$ with $C$ and $\lambda>1$ constant refutes the conjecture, so every composition, gluing or doubling gadget with bounded prime cost is a refutation instrument; the supermultiplicative case needs no Fekete, just induction from $g(1)=2$. One of #91's loose threads is false: "$p \notin S \Rightarrow |A| \le p-1$" fails for every odd $p$ (take $A \equiv 1 \bmod p$), and had it held it would have given $g(k) \le p_{k+1}-1$ and settled the problem in three lines, it agreed with all seven measured data points anyway, which is the lesson. What the pigeonhole really proves is a bound on the number of residue classes $A$ occupies, $\le (p-1)/2 + 1$; $p=2$ is the unique prime for which that bounds $|A|$, and that is why parity is special. And the quantifier settles the strategy: #126 is $\forall S\,\forall A$, every enumeration yields $\exists S\,\exists A$, so no clique search in a box can ever contribute to a proof, #91's three failures have one cause and it is not budget. Also proved: $2 \in S$ and $\gcd A = 1$ are valid normalisations for upper bounds, while $\max A \le N$ and "$S$ = the first $k$ primes" are not; each 4-subset of $A$ gives a non-degenerate 3-term $S$-unit solution, making #126 conditional on a subexponential unit-equation count and on a multiplicity nobody has bounded. #91's thread 4 closed in the negative: no 4-subset of the first eight primes beats $g_N(4)=6$. Nothing here bears on RH (docs/08).

Hunt #101: Erdős #126, the descent arm, loss 2 per prime is a wall, and the omission claim is false (support_7ddfee4b/)

Status: settled, in both directions. Support run for the r_186989 scout, answering one bounded question: is there a descent recurrence for $g(k)$ whose per-prime loss tends to 1, and does omitting a small prime from $S$ cap $|A|$?

The omission claim is false for every prime. Omitting $p$ forces the residues of $A$ mod $p$ to avoid the pairing $r \leftrightarrow -r$, which caps the number of occupied classes at $(p+1)/2$ and caps the size of a self-paired class at 1. It caps $|A|$ only when every class is self-paired, which happens only at $p = 2$. For odd $p$, any $A \subseteq 1 + p\mathbb{Z}$ has all pairwise sums $\equiv 2 \pmod p$, so $|A|$ is unbounded with $p \notin S$; witnesses at $p = 3,5,7,11$ are verified in results.json. The proposed $|A| \le p-1$ also fails at $p=2$ by one ($\{1,2\}$, $S = \{3\}$).

The descent does exist. Deleting one odd prime $p \in S$ splits $A$ into the part divisible by $p$ and two sets admissible for $S \setminus \{p\}$ (the two halves of $(\mathbb{Z}/p)^\times$ under negation), giving $g^(k) \le 2\,g^(k-1) \le 2^k$ for primitive $A$ (no element divisible by a prime of $S$), tight at $k = 1, 2$, $\{1,5,7,11\}$ with $S=\{2,3\}$ is primitive and has $2^2$ elements. But the loss cannot be pushed below 2 along this route, and that is a theorem rather than a failure to find one: parity, residue classes, prime deletion and the power-of-2 lemma all factor through a relaxation (cover $K_n$ by the negation-pairings of $k-1$ odd primes plus a triangle-free graph), whose optimum is exactly $2^k$, attained by $\{0,1\}^{k-1}\times\{0,1\}$ with a perfect matching. So an $\exp(o(k))$ bound must read something the relaxation discards: CRT rigidity, size, or $S$-unit counting, which is the door r_186989 ranked first, now closed-by-proof underneath rather than judged.

Not settled, and flagged as a real hole: the bound is for primitive sets only. The divisible part escapes the descent because $(S, pA)$ is admissible whenever $(S,A)$ is, so $\gcd A = 1$ (which is free) is strictly weaker than primitivity. Nothing here improves the classical $g(k) < 3\cdot 2^{k-1}$, and nothing here bears on RH (docs/08).

Hunt #102: Erdős #126, the counterexample arm, the tensor lane is an $S$-unit equation (support_6cdfd2e3/)

Status: not settled. No counterexample, and one lane closed conditionally. Support run for Hunt #91, asked for an algebraic, CRT, recursive, tensor or composition construction with $|A|$ exponential in $|S|$, and explicitly allowed to refute #126. None was found. What came out instead is a reduction, an ordinary argument, unreviewed: for the tensor $C = A\cdot B$, the grid-line pairs are free ($ab+ab' = a(b+b')$) and every mixed pair satisfies $u+v = (a+a')(b+b')$ with $u = ab+a'b'$, $v = ab'+a'b$, a denominator that is smooth for free. Dividing by it, $x = u/M$, $y = v/M$ solve $x+y=1$ in $(S\cup T)$-units, and $x$ is a non-constant Möbius function of $b/b'$, so $|A|, |B| \le N(S\cup T)+1$ where $N$ counts solutions of the $S$-unit equation. Unconditionally that is weaker than Erdős–Turán 1934; conditionally it is the finding: if $S$-unit counts are subexponential, no tensor construction can refute #126, and an exponential one would exceed every known construction (the run cited Erdős–Stewart–Tijdeman's $\exp(c(s/\log s)^{1/2})$; re-landing added Konyagin–Soundararajan's $\exp(s^{2-\sqrt2-\varepsilon})$, arXiv math/0604453, read at the abstract only). The argument does not extend to general $A$, and that is exactly why it closes the tensor lane and no more: for arbitrary $a,b,c,d \in A$ the common value $a+b+c+d$ need not be smooth, so there is nothing to divide by. Two corrections to Hunt #91 travel further than the non-result. The refutation bar is far lower than a composition law: $g(k+k_0) \ge c\,g(k)$ for any fixed $k_0$ and $c>1$ already gives $\liminf \log g(k)/k > 0$; that is Hunt #93's amplification result (support_8ea74995/), reached independently, and not new here. The converse fails, and that is new: a staircase $G(k) = 2^{k_j}$ on $[k_j, k_{j+1})$ refutes #126 while failing every uniform doubling law off a density-zero set, so an anti-composition theorem is not sufficient for #126, which the parent's closing sentence implies it is (also an ordinary argument, unreviewed). Measured, by exact integer arithmetic inside bounded boxes: an independent clique search reproduced rows 1–7 of the parent's table from boxes 20–500× smaller and added the lower bounds $g(8) \ge 14$, $g(9) \ge 15$; on the parent's optima, the cheapest dilated union $A \cup cA$ costs 6 extra primes at $k=2$ and 7 at $k=4$, and none exists at $k=3$ (multipliers $c \le 200$, extra primes from the first twelve). Also in the run and already on main: $g(k) \ge \lfloor (p_{k+1}-1)/2 \rfloor$ from $A = \{1,\dots,m\}$ (Hunts #104, #105), and dilation invariance against the parent's "every optimal witness lives below 50" (Hunts #103, #105, #107), here with $\{113,399\}$ and $\{16,80,112,176\}$ as counter-witnesses. Two of the run's statements did not survive to landing and are annotated in place: its loose thread "$|A| \le p-1$ when $p \notin S$" was already refuted (Hunts #93, #95, #101, #103), and its sumset-counting section understated the height the lemma forces (RESULTS.md §6, corrected against Hunt #107). Nothing here bears on RH (docs/08).

Hunt #103: Erdős #126, the residue lemma is false and the box was never the question (support_60982bf6/)

Status: settled, as a support answer. Exact-search arm for hunt #91's parent: redo r_186989's computation ranging over choices of $S$ instead of freezing the first $k$ primes, and use the computation only to discover and falsify structural lemmas. Exhaustive over all $\binom{9}{k}$ subsets of the first nine primes for $k \le 5$ (381 searches, a solver rewritten to build edges from the $S$-smooth sums rather than by scanning $O(N^2)$ pairs), the first $k$ primes are never beaten, so the frozen constant #91 called "the door with genuine trade shape" is slack. What the sweep produced instead is two lemmas. Proved (normalization): $\gcd(A)$ is always $S$-smooth, $A/\gcd(A)$ is admissible, and $mA$ is admissible for $S$-smooth $m$, so the admissible sets are exactly the $S$-smooth dilates of primitive ones. That removes #91's puzzle that widening the box "changed not one row": a wider box only contains bigger copies of the same small set, and the right question is the height of the smallest primitive optimum, not the width of the box. Refuted: #91's "only clean iterable statement", $|A| \le p-1$ when $p \notin S$, is false for every odd $p$, for $r \not\equiv 0$, $r+r \not\equiv 0 \pmod p$, so arbitrarily many elements may share one class and the pigeonhole bounds the number of occupied classes, never $|A|$; $p=2$ is the unique prime with $2r \equiv 0$ for all $r$, which is why parity works there and nowhere else. Counterexamples at $p = 3, 5, 7$ (the smallest, $S=\{2,5,7\}$, $A=\{1,3,7,13\}$, sits inside #91's own table). Also refuted, box-conditionally: $g$ is not monotone in the size of the primes ($g_{20000}(\{2,3,11\}) = 4 < 5 \le g(\{2,3,13\})$, 29 violations), so any $S$-level lemma must be about a prime's additive position relative to the smooth semigroup, not its size. The conjecture offered is uniformly bounded height, and both natural explicit constants for it were killed by the run's own data ($h(\{2,3,5\}) = 47 > 2^{k+2}$, $h(\{2,3,7,13\}) = 159 > 2^{k+3}$); it is consistent with all 15 measured $S$ and verified for none. No upper bound on $g(k)$, as before. Nothing here bears on RH (docs/08).

Hunt #92: the Erdős–Turán factor 2 is not lost anywhere, it is the whole statement (support_517b887f/)

Status: settled, negatively. Support arm for Hunt #91, asked whether the 1934 proof's factor 2 per prime can be beaten by processing primes jointly or in a different order. Answer: no, and the reason is structural rather than arithmetic. Reconstructed, the proof has exactly two steps: a base case $c(A) \le 2$ (three positive integers cannot have all three pairwise sums powers of 2, by $2a = 2^x+2^y-2^z < 0$), and one binary $\pm$ residue choice per odd prime, assembling to $g(k) \le 2\cdot 2^{k-1} = 2^k$ (Erdős–Surányi; the 1934 $3\cdot2^{k-1}$ is the same architecture with a weaker base). Since $c(A) \in \{1,2\}$ always, "every admissible $A$ has $c(A) \ge |A|/\varphi(k)$" is equivalent to $g(k) \le 2\varphi(k)$, and $c(A)$ is defined by a maximum over subsets, so it is blind to prime order and to grouping. Improving 2 to $2-\delta$ inside this architecture is the target restated, not a sharpening of it. At the residue level joint processing is provably worthless: the CRT configuration $x_\varepsilon \equiv \varepsilon_i \bmod q_i$ has $2^s$ points whose largest jointly clean subset is 1 (brute-forced, $s \le 4$). Two defects in the usual sketch fell out and are witnessed exactly: the per-prime halving $|B'| \ge |B|/2$ is false ($A = \{1,3,15,21,33\}$, sums $\{2,3,11,17\}$-smooth, best 3-clean subset $\{1,3\}$ of size $2 < 2.5$; the class-0 term is not removable, and $q\cdot C \cup \{x\}$ makes the failure unbounded), and the assembled $c(A) \ge n/2^s$ is false without primitivity ($A = 3\cdot\{1,3,7,17,47\}$ has $c = 1 < 5/4$; dividing by $\gcd$ repairs it). Reordering rescues individual instances ($q=11$ before $q=3$ turns $0.4n$ into $0.8n$) and nothing at the theorem level. A sweep of 104,183 admissible sets in three small boxes found zero primitive violations of the repaired lemma and exact tightness at $\{1,2,4,8\}$. The one sub-question left that is not circular is a two-set bound on how much of $A$ one odd prime may divide, which is Győry–Stewart–Tijdeman territory. Nothing here bears on RH (docs/08).

Hunt #107: the Erdős #126 size dichotomy misses by one power of the radical (support_95bb5cb7/)

Status: the arm is settled negative; the conjecture is untouched. A support run for a session working Hunt #91, asked whether a size dichotomy closes Erdős #126: either $A$ is spread out enough for a descent, or it lies in a controlled interval where smooth-number counts apply. It does not, and the failure point is now exact rather than rhetorical. The counting horn is $|A| \le 1 + \Psi(2N,S)$ with $N = \max A$, and writing $\log 2N = c\log\mathrm{rad}(S)$, the run pins $\log\Psi/k$ from both sides, Rankin above and the simplex lattice-point volume below: it tends to zero iff $c \to 0$, and at $c \ge 1/2$ the lower bound is $0.32k$, so above $\max A = \mathrm{rad}(S)^{1/2}$ the horn is not unproven, it is false. So the horn delivers $\log g(k) = o(k)$ iff it is handed $\max A = \mathrm{rad}(S)^{o(1)}$, and no choice of $S$, scale, split or smooth-count estimate moves that: it is a property of $\Psi$, not of the argument. The other horn cannot supply it, for two independent reasons. First, measured: after the only available normalisation ($\gcd(A) = 1$; translation shifts every sum by $2t$ and is unavailable, so height is a genuine invariant), the height of a primitive admissible set of sub-extremal size is unbounded, tracking the search cutoff linearly over eleven orders of magnitude, from $\{217,1241,7975\}$ up to $\{3772777922353,\,41980806987569,\,559177175498959\}$ at $S=\{2,3\}$. A descent that reaches a smaller set finds no interval control waiting there. Second, structural: an admissible clique yields only the four-term relation $(a+b)+(c+d)=(a+c)+(b+d)$, and every height theorem for $S$-smooth numbers (Baker–Győry, $abc$) is three-term; four terms have only solution counts (Evertse–Schlickewei–Schmidt, $\exp(O(k))$, and Erdős–Stewart–Tijdeman show counts cannot fall below $\exp(c(k/\log k)^{1/2})$). A count is not a height. Granting even the best imaginable three-term outcome, $abc$ gives $\max A \ll \mathrm{rad}(S)^{1+\varepsilon}$, which is $c \to 1$, exactly where the lower bound reads $1.01k$. The gap is one power of $\mathrm{rad}(S)$, not an exponential, which is why the arm looks tempting and why it still cannot close. The audit half re-verified all seven of Hunt #91's witnesses from scratch by full trial division (all valid), confirmed its $f(n-1)\le f_0(n)\le f(n)$ lemma and its Fekete argument, and corrected one claim: "every optimal witness has all elements $< 50$" is false. Enumerating through the sums rather than the elements, which is the exhaustive universe for a constraint on sums, gives exactly six primitive extremal 5-sets for $S=\{2,3,5\}$ with all sums $\le 10^5$, and $\{5,11,25,245,475\}$ is one of them, max element $475 = \mathrm{rad}^{1.81}$, not $\mathrm{rad}^{1.13}$. Hunt #91 reported one witness per $k$ and generalised from it. Same enumeration adds a small positive: no 6-element set for $S=\{2,3,5\}$ has all pairwise sums $\le 10^5$. Not settled: whether extremal-size sets have bounded height (the data hints yes and it would be circular for this arm anyway), the ladder at $k=4$ (the enumeration exceeded ten minutes), and any upper bound on $g(k)$ at all. Nothing bears on RH (docs/08).

Hunt #94: Erdős #126, the p-adic block-pruning arm, and a target set one notch too low (support_d176e366/)

Status: not settled. No block lemma below base 2, and a restricted-class obstruction for residue-only pruning (ALIGNMENT.md §5, disposition 2); the attempt below base 2 is unresolved. Lemmas 1 and 2 and Theorems 3, 3′ and 4 are ordinary arguments, unreviewed; the checks are exact finite ones on blocks of at most three primes, and the cross-prime correlations are measurements on $|A| \le 11$. Landed 2026-10-21 from a local-only branch, support/d176e366, that was never pushed; PR #144 counted the arm among the seven that "produced no branch at all". A support run for r_186989, asked whether a block of $r$ primes can be pruned from $S$ while retaining more than $2^{-r+o(r)}$ of $A$. It answers the question and contradicts its premise. Lemma 1: pruning a block $T$ is the independent-set problem for $\bigcup_{p\in T} G_p$, where $G_p$ is the antipodal graph mod $p$ (disjoint complete bipartite graphs between class $c$ and class $-c$, plus a clique on the class $c \equiv -c$), so valuations beyond $v_p(a) = 0$ are invisible here and "valuation vector" means "residue vector", which is why the ultrametric framing buys nothing. Lemma 2: a genuinely joint one-shot random half-system argument gives retention $\ge 2^{-r}$ over the elements coprime to the block, landing on exactly the constant that pruning one prime at a time gives. Theorem 3: on a full residue system mod $\prod T$ the optimum is exactly $\prod_{p\in T}\frac{p-1}{2} + r$ (matched by exhaustive branch-and-bound for $\{3,5\},\{3,7\},\{5,7\},\{3,5,7\}$: 4, 5, 8, 9, four for four), which is below $2^{-r}|A|$, so no argument reading only residues mod $\prod T$ can beat base 2; and by Theorem 4 ($g(k) \le 2/\rho_{\min}(k) \le 2g(k)$, from the endgame lemma that no three integers have all pairwise sums powers of 2) the block programme is not a proxy for Erdős #126 but a restatement of it. Two consequences the brief did not anticipate: retention $2^{-r+o(r)}$ is the Erdős–Turán bound, so the named target is worth nothing even if proved, and only base $c<2$ counts; and the smoothness hypothesis is used nowhere in this arm except the endgame, only divisibility. Measured on the seven verified witnesses, real admissible sets beat $2^{-r}$ retention by up to a factor of 11.6 with strong positive correlation across primes, reported as a direction indicator only, since $|A| \le 11$ and the full-block column is Theorem 4 restated rather than evidence. Audit of r_186989: all seven witnesses re-verified admissible, its power-of-2 lemma re-derived independently, and its loose thread 3 (a claimed $|A| \le p-1$ for $p \notin S$) refuted: $\{1, 1+q, 1+2q, \dots\}$ has every pair sum $\equiv 2 \bmod q$ at unbounded size. One drafted claim of the run's own ($\varphi$ = number of sign-orbits met) was false and is kept in the file with its counterexample. Overlap with main, recorded in RESULTS.md: that residue data cannot beat base 2 is also the finding of Hunts #92 and #101 (#101's relaxation is this arm's covering reformulation, solved exactly), and the loose-thread-3 refutation was on main already (Hunts #93, #95, #101, #103). The run's remark that the optima "live in the small integers" is annotated in place, since Hunts #103, #105 and #107 refuted it. Nothing bears on RH (docs/08).

Hunt #97: Erdős #126, the low-rank matrix arm was already in the literature (support_99fc686a/)

Status: not settled. A verdict restricted to the two-row $S$-unit route and to the counting bounds now known: the arm is a rediscovery, and with those bounds it loses to a proof from 1934. Alon–Solymosi was read from the PDF (primary). The Evertse and Beukers–Schlickewei constants came from search summaries, which the run itself rates medium-low, and both are hedged in RESULTS.md: Evertse's form agrees with Győry–Stewart–Tijdeman's quotation of it as read in primary text by Hunt #143, while the Beukers–Schlickewei form $2^{8(r+2)}$ differs from the $2^{8r+8}$ on main in Hunt #104. The two-row collapse is an ordinary argument, unreviewed. Landed 2026-10-21 from a local-only branch, support/99fc686a, that was never pushed; PR #144 counted the arm among the seven that "produced no branch at all". Support run for r_186989's successor, on one arm: the symmetric matrix $M_{ij} = a_i + a_j$ has rank $\le 2$ and $S$-unit off-diagonal entries, so what do results on low-rank matrices over finitely generated multiplicative groups give? Answer: Alon and Solymosi, "Rank of matrices with entries from a multiplicative group", Corollary 10 constructs this exact matrix, observes its rank is at most two, and concludes $|A| \le f(r)$. Their explicit form $\log n \le r d^8$ at $d = 2$ gives $\exp(256k)$ against Erdős–Turán's $3 \cdot 2^{k-1}$. Rank 2 does not need that machinery, and the run's own route is cheaper: fix distinct $a, b \in A$; then for every other $d$, $(a+d) - (b+d) = a - b$, so $(a+d, b+d)$ solves a two-variable $S$-unit equation with coefficients not depending on $d$, and $d \mapsto a+d$ is injective. Hence $g(k) \le 2 + N_{\mathbb{Q}}(k) \le 2 + 3 \cdot 7^{2k+3}$ (Evertse 1984), base 49 against base 2. No improvement, and none available from today's counting bounds: the loss is entirely in the Diophantine input, measured at $k=7$ as 9 solutions forced against $7\times10^{14}$ permitted. The nondegeneracy question the brief asked to isolate has an exact answer and it is anticlimactic: in the two-row route there is none at all (a two-term equation's proper subsums are units, hence nonzero), and in the general-rank route it is precisely Alon–Solymosi's rectangle condition, which for this matrix expands to $(a_j - a_i)(a_j - a_k) \ne 0$, i.e. the distinctness of $A$. That condition is not decoration: their Theorem 1 assumes diagonal entries lie outside $\Gamma$, and $2a_j$ often is an $S$-unit (in 7 of 7 witnesses). One thing kept, direction checked: if two-term $S$-unit equations have $\exp(o(k))$ solutions then Erdős #126 is true, and the known lower bound $\exp(k^{1/2+o(1)})$ (Erdős–Stewart–Tijdeman) does not obstruct that. The converse fails. Overlap with main, recorded in RESULTS.md: the two-row collapse is the reduction of Hunts #96 and #104, and it is Győry–Stewart–Tijdeman 1986 §3 (quoted by Hunt #143, landed with this entry), which answers the run's open question whether it is in the literature; the four-element route of its §3 is that of Hunts #93, #95 and #105. Nothing bears on RH (docs/08).

Hunt #98: the cover premise of Erdős #126 has no bound in it (support_d0f0aefb/)

Status: not settled. A restricted-class obstruction for incidence and cover data, plus a refuted candidate (ALIGNMENT.md §5, dispositions 2 and 1), with a conditional bound and a one-way reduction as by-products. Theorems 1 to 4 are ordinary arguments, unreviewed, and one clause of Theorem 1 was corrected on landing (below); the two literature bounds behind Theorem 4's door were checked by web search, not read in the sources, and the reduction's prior art is unsearched (the run says to assume folklore). Landed 2026-10-21 from a local-only branch, support/d0f0aefb, that was never pushed; PR #144 counted the arm among the seven that "produced no branch at all". A support arm launched by Hunt #91's run, asked whether the graphs $G_p$ ($a \sim b \iff p \mid a+b$), which together cover $K_n$, admit a cover, Ramsey, rank or container bound. They do not, and that is argued in full: $A = \{1\} \cup \{2(3j+1)\}$ with $S = \{2,3\}$ has $\gcd = 1$ and its two residue graphs cover $K_n$ for every $n$, so no theorem deriving a bound on $n$ from the incidence data alone can exist. Generic graph methods stall because the combinatorial shadow of the problem is realizable at every size; the content is in the arithmetic the shadow discards. Two things came out of the rubble. First, a structure theorem: mod $p$ the matrix $(a_i+a_j)$ has rank $\le 2$, and $G_p$ is the disjoint union of complete bipartite blocks between the classes $r$ and $-r$ among the elements prime to $p$ (one clique among them for $p = 2$) and a single clique on the multiples of $p$. The run stated a finer level-set structure on the multiples of $p$, which fails above level 0 on its own $k=7$ witness and takes its chromatic corollary $\chi(G_p)\le 2T_p$ with it ($\{3,15,21,33\}$, $S=\{2,3\}$, has $G_3 = K_4$ and $T_3 = 1$); it is corrected in place in RESULTS.md §2. Under the hypothesis that every element is coprime to $\prod S$ the structure peels to $|A| \le 2^k$, sharper than Erdős–Turán's $3\cdot2^{k-1}$ by $3/2$ (unaffected by the correction); the hypothesis resists normalisation because $\{a \in A : p \mid a\}$ divided by $p$ is again a set of the same kind over the same $S$. Second, and the thing worth keeping: using the exact valuations rather than the graph gives $g(k) \le 3 + N_3(k)$, where $N_3(k)$ counts nondegenerate solutions of $x+y+z=1$ in $S$-units over the same $k$ primes, by an injective map. The run said the third term is what keeps $S$ from growing by the primes of $a_1-a_2$; it is not needed for that, since Evertse's bound is uniform in the coefficients (Hunts #96 and #104), and that remark is corrected in place too. The implication runs one way only, and it runs the useful way: $N_3(k) = \exp(o(k))$ would prove Erdős #126. Known lower bounds for such counts are $\exp(k^{1/2+o(1)})$, so nothing known closes that door. Hunt #91's loose thread "$|A| \le p-1$ when $p \notin S$" is refuted: $\{1,4,124\}$ with $S = \{2,5\}$ has all three elements $\equiv 1$ mod $3$ and $|A| = 3 > 2$; only $p = 2$ bounds anything, because there every residue class is its own negative. All seven of Hunt #91's witnesses were re-verified. Overlap with main, recorded in RESULTS.md: the coprime bound $2^k$ is Hunt #101's primitive descent (and Hunt #94's Lemma 2, landed with this entry); the three-term reduction sits beside the four-element reductions of Hunts #93, #95 and #105 and adds the injectivity they lack; the loose-thread-3 refutation was on main already (Hunts #93, #95, #101, #103). Nothing here bears on RH (docs/08).

Hunt #99: Erdős #126, the narrowest upgrade is a unit-equation count (support_311c14a5/)

Status: not settled, and no new bound on $g(k)$. A sufficiency reduction and a quantitative comparison, both ordinary arguments, unreviewed; the attempt at a bound is unresolved and one parent thread is a refuted candidate (ALIGNMENT.md §5). The cited constants came from one web search each and the run says not to quote their exponents as pinned. Landed 2026-10-21 from a local-only branch, support/311c14a5, that was never pushed; PR #144 counted the arm among the seven that "produced no branch at all". Support run for Hunt #91, working the additive-combinatorics arm. The deliverable is an elementary reduction with no loss: if $A$ is admissible for $S$ and $a \ne b$ in $A$ with $d = a-b$, then $c \mapsto (a+c, b+c)$ injects $A \setminus \{a,b\}$ into the solutions of $(1/d)X + (-1/d)Y = 1$ in $S$-units, so $g(k) \le 2 + E(k)$ where $E(k)$ counts solutions of $\alpha X + \beta Y = 1$ in $S$-units, uniformly in $\alpha,\beta$. Keeping the constants in the coefficients rather than the group holds the rank at exactly $k$. Hence $E(k) = \exp(o(k))$ implies Erdős #126, and that is the narrowest upgrade the run could name; the converse does not hold and is not claimed. Quantitatively the route is today much weaker than 1934: the run quotes Evertse as $g(k) \le 2 + e^{4k+10}$, base $e^4$ against Erdős–Turán's base $2$. That form is from a search; GST's primary-text quotation of Evertse, $3\cdot7^{2k+3}$ (Hunt #143, and Hunts #96 and #106), is smaller for every $k$ and gives base 49, and RESULTS.md §3 annotates the discrepancy in place without changing the verdict. The route's floor is $\exp(\Theta(\sqrt{k/\log k}))$ from Erdős–Stewart–Tijdeman, which is $\exp(o(k))$, so the known extremal examples do not obstruct the target. The three briefed tools are reported inapplicable with reasons: Freiman needs small doubling and these sets have maximal doubling, sum-product needs the multiplicative structure on $A$ rather than on $A+A$, and the additive-energy inequality is measurably dominated by a one-line pigeonhole. Two audit results against Hunt #91: its composition law, combined with the Erdős–Turán upper bound, forces $g(k) = 2^k$ exactly rather than merely $\ge 2^k$, which makes it falsifiable at $k=3$ (it needs an 8-element set; exhaustive search inside $[1, 60\,000]$ finds 5); and its loose thread that the parity cap generalises mod odd primes is false, since $A$ may sit in a single nonzero class mod $p$, with witness $S=\{2,5,7,11\}$ omitting $3$ and $A = \{1,9,13,19,31\}$ of size 5. All seven of Hunt #91's witnesses re-verified; $g_N(3)=5$ independently reproduced by a from-scratch solver. Overlap with main, recorded in RESULTS.md: the reduction is that of Hunts #96 and #104; the composition sharpening tightens Hunt #105's corridor $[2^k, 1.5\cdot2^k]$, which already noted that $g(3)\le 7$ retires the programme, and Hunt #93 has the general amplification form; loose thread 3 was refuted on main already (Hunts #93, #95, #101, #103), and the run's OEIS thread was closed by the literature arm (Hunt #143). Nothing here bears on RH (docs/08).

Hunt #100: Erdős #126, the product route has a ceiling of 2 (support_95a76f59/)

Status: not settled. A restricted-class obstruction (ALIGNMENT.md §5, disposition 2) for the unweighted product $P = \prod_{a\ne b}(a+b)$ read through $k \ge B(A)$, with the prime masses estimated from distinctness, range, residue counts, whole-set gcd normalisation and the one-step recursion on multiples of $p$; weighted or subset products, and any estimate that reads the additive structure of $A$, are outside the class, as the run's own doors section says. The arguments are ordinary, unreviewed; the large-sieve normalisation of (5b) was not re-checked, which the run flags itself; the values of $B$ are floating-point measurements. Landed 2026-10-21 from a local-only branch, support/95a76f59, that was never pushed; PR #144 counted the arm among the seven that "produced no branch at all" and cited it as the closed "Analytic NT" lane. A support run for the #126 scout, asked to attack the original product $P$ and lower-bound its number of distinct prime divisors. Within that class it is closed instead. The product yields exactly one inequality, $k \ge B(A) = \log P / \max_p (e_p \log p)$, and two adversarial families of distinct positive integers with $\gcd = 1$ cap $B$: $\Omega' = \{1\}\cup\{i p^m\}$ drives $B \to n/(n-2) \to 1$ against any estimate built from distinctness, range and residue counts, and $\Omega = \{ip^m+1\}\cup\{jp^m-1\}$, which no gcd normalisation or one-step recursion touches, drives $B \to 2$. Measured $B = 1.041$ and $2.125$. So the product route, in this class, cannot reproduce Erdős–Turán's $\log_2 n$, let alone improve it, and on the seven known optimal witnesses it already loses to $\log_2 n$ at $k \ge 6$. The smooth-counting lane is capped too: $k \ge \log n/\log\log 2X$ beats $\log_2 n$ only when $X < e^2$. Two things survive: a large-sieve sparsity bound $n = O(\sqrt X \log X)$ from the fact that $A$ occupies at most $(q+1)/2$ classes mod every prime $q \notin S$, and an audit finding that retires one of Hunt #91's loose threads (its "prove $|A| \le p-1$ by pigeonhole mod $p$" is false: $\{1+iM\}$ is antipodal-free mod every odd $q \le Q$ at any size). Number #100 was assigned by the launching brief, not chosen. Overlap with main, recorded in RESULTS.md: the normalisation is Hunt #103's lemma, the occupancy count $(q+1)/2$ is Hunt #95's Lemma A, the smooth-counting inequality is Hunt #107's counting horn, which pins it from both sides, and the loose-thread refutation was on main already (Hunts #93, #95, #101, #103). The run's open question whether $\max A$ is bounded in terms of $(n,k)$ is answered for sub-extremal sizes by Hunt #107 (unbounded height at $S=\{2,3\}$), annotated in place in RESULTS.md and HANDBACK.json. Nothing here bears on RH (docs/08).

Hunt #106: Erdős #126 reduces to counting $S$-unit equations (support_6da47ca3/)

Status: not settled, but the wall is located and named; the attempt is unresolved (ALIGNMENT.md §5) and one parent thread is a refuted candidate. Theorem 1 and its corollaries are ordinary arguments, unreviewed; Corollary 2's constant uses Evertse's bound as the run recalled it, without re-reading it; the Győry–Hajdu–Tijdeman verdict rests on the abstracts; the measured table is exhaustive inside bounded boxes. Landed 2026-10-21 from a local-only branch, support/6da47ca3, that was never pushed; PR #144 counted the arm among the seven that "produced no branch at all". A support run for the Erdős #126 scout (Hunt #91), asked to turn $A$ and $-A$ into the bipartite $S$-unit difference pattern and specialise the Győry–Hajdu–Tijdeman representation theorems. The bipartite framing is right and the bipartiteness is not what carries the weight: all the quantitative force sits in the sub-pattern $K_{2,m}$, giving Theorem 1, $g(k) \le N_2(k) + 2$, where $N_2(k)$ is the maximum number of solutions of a two-term $S$-unit equation $u - v = c$ over $|S| = k$ and $c \in \mathbb{Q}^*$. The correspondence is exact (solutions of $u-v=c$ are in bijection with the common neighbours of two vertices), so nothing is lost at that step. With Evertse's 1984 bound $3\cdot 7^{d+2s}$ this gives the unconditional explicit $g(k) \le 3\cdot 7^{2k+3}+2$, i.e. $f(n) \ge 0.2569\log n - 1.78$, a re-proof of the classical $f(n) \gg \log n$ with a tracked constant and no movement on the problem. The load-bearing consequence is the conditional: if a two-term $S$-unit equation has $\exp(o(k))$ solutions uniformly in its coefficients, Erdős #126 follows. Every known lower bound on that count (Erdős–Stewart–Tijdeman $\exp(k^{1/2+o(1)})$; Konyagin–Soundararajan $\exp(k^{2-\sqrt2-o(1)})$) is subexponential, so the hypothesis is consistent with everything known, and the 92-year wall on #126 is the same wall as the gap between $\exp(O(k))$ and $\exp(k^{0.59})$ in unit-equation counting. The run also calls that hypothesis the field's "standard (open) conjecture" and #126 "a consequence of a believed conjecture"; neither phrase cites a source, the literature arm (Hunt #143) looked for the statement and did not find it, and both are annotated in place in RESULTS.md. Sufficient, not necessary, and measurably lossy: exhaustive search gives $g_N(5) = 8$ against $N_2^{(N)}(5) \ge 141$. The briefed within-part pair-sum structure buys no improvement in the exponent; it does buy a free nondegeneracy guarantee for $(a+b)+(c+d) = (a+c)+(b+d)$ (all terms positive, every vanishing subsum forces two elements equal), but routing through it needs three-term unit equations whose best uniform bound has slope $\sim 2\times 10^{11}$ against Evertse's $2\log 7 = 3.892$. The GHT papers themselves are representability results, not counting results, and contain no bound of the form $|A| \le F(|S|)$; their cubical criterion touches ours only through the $K_{2,3}$ obstruction, which is exactly the pattern Theorem 1 exploits. Audit of Hunt #91: all seven witnesses re-verified, $g_N(k)$ for $k \le 5$ reproduced independently with identical optimal witnesses, and its §2 and §4 proofs read and correct; but its loose thread 3 is refuted ($|A| \le p-1$ when $p \notin S$ fails for every odd $p$, e.g. $\{1,3,7\}$ with $S=\{2,5\}$ and $\{1,3,5,11,13\}$ with $S=\{2,3,7\}$), so the parity fact is a $p=2$ accident and not an iterable lemma. Overlap with main, recorded in RESULTS.md: Theorem 1 is the reduction of Hunts #96 and #104 (and of #97 and #99, landed with this entry), Lemma 4 is the nondegeneracy step of the four-element reductions of Hunts #93, #95 and #105, the loose-thread-3 refutation was on main already (Hunts #93, #95, #101, #103), and the OEIS thread the run left open was closed by Hunt #143. Its audit line that the optima "live below 50" is annotated in place, since Hunts #103, #105 and #107 refuted it. Nothing here bears on RH (docs/08).

Hunt #143: Erdős #126, the 92-year wall, and the one lever behind it (support_fcffa3b9/)

Status: not settled. A literature map from a targeted search, not an exhaustive one; it refutes nothing and closes no route. Primary text was read for Erdős–Turán 1934 and for Győry–Stewart–Tijdeman 1986 including its §3 proof. Erdős–Surányi's $2^k$ and the Erdős–Stewart–Tijdeman constant $1/8$ are secondary citations through Füredi–Gyarmati 2026, not checked against the originals; Evertse's bound is quoted through GST; the one reduction step of the run's own is an ordinary argument, unreviewed. Renumbered on landing: the run's brief assigned #91, which r_186989 already held, so the run took #92 on its branch; #92 is support_517b887f on main, so the entry takes #143, the next free number, and HANDBACK.json keeps the run's #92. Landed 2026-10-21 from a local-only branch, support/fcffa3b9, that was never pushed; PR #144 counted the arm among the seven that "produced no branch at all". A support arm for the scout (Hunt #91), answering one question: what does the literature actually have, from 1934 to now? Four things, three of them from primary text. The 1934 bound is still the record: Erdős and Turán prove $g(k) \le 3\cdot2^{k-1}-1$, and the search found nothing in 92 years that improves the base 2, only the constant ($g(k)\le 2^k$, Erdős–Surányi). The $S$-unit machinery is already in the literature and is quantitatively worse: Győry–Stewart–Tijdeman 1986 reproves $f(n)\gg\log n$ through Evertse's counting bound and lands at $3\cdot7^{2k+3}\approx 49^k$, because its reduction keeps only $2(n-2)$ of the $\binom n2$ sums. It is still the lever, and the run makes that precise: with $g_2$ the two-set relaxation, $g(k)\le g_2(k)+2 \le N(k)+2$ where $N(k)$ counts coprime $S$-unit solutions of $\lambda x+\mu y=\nu z$ (step one the run's own, step two quoted verbatim from GST §3; the branch's entry dropped the final $+2$, which its own RESULTS.md §2 carries), so a uniform $\exp(o(k))$ bound on $N(k)$ implies Erdős #126. The converse fails, and the route is not blocked: Erdős–Stewart–Tijdeman 1988 forbids only polynomial bounds on $g_2$, leaving the whole window $\exp(\tilde\Theta(\sqrt k)) \ldots \exp(o(k))$ open, every point of which settles the problem. Two audit findings against the scout: its arm-2 parity headline is on page 1 of the 1934 paper verbatim, and its arm-3 kill is stronger than claimed, since a composition law would force the 1934 bound to be sharp up to $3/2$ and contradict its own authors' conjecture $n(k)=O(k^{1+\varepsilon})$, a conjecture no modern summary of the problem mentions. The scout's OEIS thread is closed negatively (a sequence query with twenty coincidental matches and a keyword search with none, so no tabulation of $g(k)$). What could not be settled: whether the subexponential $S$-unit bound is a stated conjecture of that field, which is the difference between "#126 needs a new idea" and "#126 is a corollary of a believed conjecture". Overlap with main, recorded in RESULTS.md: the reduction is that of Hunts #96 and #104 (and of #97, #99 and #106, landed with this entry), and #105 makes the same 3/2-corridor point, which #99 sharpens to $g(k)=2^k$. One statement relied on the scout's "every optimal witness lives below 50", which Hunts #103, #105 and #107 refuted; it is annotated in place (RESULTS.md §3, the T8 row). Nothing here bears on RH (docs/08).

Hunt #90: the leader's certificate does not replay at its own HEAD (amtopa_ceiling/)

Status (2026-09-06, RESULTS.md section 7.7): the +3.96e-06 candidate is WITHDRAWN. At b3b7784, the verifier revision their own candidate.json names, the leader's headline is accepted on 8 of 8 shards with the gate alive; their number stands and the tip's only defect is the gate. Ours was refused on 4 of 8 in three rounds; the third, with the target backed off, reached cells whose values sit below the LP's claimed floor, and a proper descent finds the functional at the candidate's own weights bottoming at 0.0078960, 1.5e-5 under the leader's floor. The LP's cut oracle had missed that basin, so the claimed +5.9e-6 of headroom on the pair-weight and pressure axes is withdrawn with it; the re-solve with a stronger oracle bounds that headroom at 8.9e-8 in eps, under 7e-8 on the headline. **The window axis went the same way (section 7.8): two candidates built at a window that is not theirs, both accepted on 6 of 8 shards and refused at cells with a gap near 2.91 that no float search here was seeding, and re-measured with that region seeded all five of the sweep's windows land BELOW the record. Control: the same oracle reproduces AMTOPA's own floor exactly at their own point. All five axes are at the ceiling; both of this hunt's candidates are withdrawn. Earlier status: settled on the measurable axes; window axis open; acceptance blocked at the tip. Takes the ainta_seven_point playbook to AMTOPA/zeta-exact-pressure, the leading public claim per Hunt #89, pinned at commit 7253fdcab9366af45b8c8caf44e408c0af44a1a7.

The finding that outranks the rest: their published certificate does not replay at their own repository tip. Run through AMTOPA's own pipeline on the pinned commit, their table builder, their verifier, their candidate, their target, the finite inequality behind 0.6734164909714992949 returns INCONCLUSIVE=true reason=terminal_cell with a rigorous lower bound 1.19e-07 short of the target. The tables are not at fault: all six join to streams whose SHA-256 digests match their own candidate.json byte for byte, a stronger table reproduction than Hunt #89 got for trmdy. The cause is their convexity gate, which fires 2030240 times in their recorded run and zero times here in 72 million nodes. Their candidate.json names source_commit: b3b7784… as the origin of that run; between b3b7784 and the tip, src/verify_local_tables.cpp changed by 173 lines and the gate's curvature entries went from thin point(scalar) to mul(p.exact, {sec, +infinity}). The interval LDL that follows cannot certify positive definiteness of a matrix unbounded above, ldl_probe.cpp, 70 lines, shows it returning false on a matrix with 10 on the diagonal and 1 off it. Without the tangent bound the plain interval bound at grid 1/4000 is left to clear the target alone, and at one width-zero cell it cannot. Direction: fail-closed, the tip refuses what the earlier revision accepted and never the reverse, so this is a reproducibility defect, not a soundness hole, and it blocks our candidate at the tip too, by 2.70e-08.

Their headline reproduces on paper. In exact rational arithmetic with a rational under-estimate of the only square root, their 0.673416490971499294950035533107 4903174997772794755665475125243371226272 holds to 70 decimals, and the exact scan over m in [7, 20000] returns their m = 145. An independent reimplementation written from their proof.md reproduces H(v) and their observed float minimum 0.007911105155226424 to the binary64 limit, at their own published basin. Soundness read: no defect that affects their claim. Acceptance is one-sided throughout, the verifier fails closed at a terminal cell, out-of-table queries return 0 for a nonnegative W and -inf for the convexity gate, and the constants are thresholds compared against computed enclosures, not answers wired to the target. Three things worth recording anyway: their README.md and proof.md both say "exact arithmetic selects m=145" while src/check_final_bound.py is mpmath float at 100 dps with mp.sqrt, right answer, wrong label, and their own banded experiment does it properly; the C++ never checks pair-weight nonnegativity although box_lower needs it, and the script that writes its constants does not check it either, so the verifier alone cannot validate a candidate; and long double is 64-bit on ARM and 80-bit on x86, the same host-dependence class as the w'' digest Hunt #89 found in trmdy.

Outcome (a), by +3.96e-06. eps(a,b) = min_g F is linear in the pair weights and in the pressures, hence concave, over a polytope, so that axis is a concave maximisation with an exact answer, not a search. Cutting-plane LP: at their own window and their own total pressure the polytope admits eps* = 0.007916857812 against their 0.007911105155, and quantised into their schema at the rational target 19791/2500000 the exact assembly gives

0.6734201550790580964457598685450152133015 against their 0.6734164909714992949500

CONDITIONAL, inheriting the same unreviewed Anthropic/Ainta bridge as everything on this ladder, and the floor behind it is a float minimum until their own verifier accepts the target. The LP found the point; it does not justify it, what justifies it is their branch-and-bound, and the hunt says so where the number appears.

The window axis is NOT saturated, and it is worth ten times the pair weights. Five independent differential-evolution seeds across two Actions runs, each starting from AMTOPA's own window, every one walks away from it, always toward lower H and lower B, and reports gains between +3.07e-05 and +3.71e-05, best 0.6734536055358651 at H = 0.6721654, B/B0 = 0.908, m = 153. Those inner solves are deliberately cheap and their floors are early-stopped over-estimates, so this is direction, not magnitude, but the direction is unambiguous and it is the live door.

The ceiling findings are worth more than the constant. On the two axes where a ceiling can be computed rather than searched, AMTOPA are at it. (i) H(v) = 2 - 1/c1 is a Rayleigh quotient in the window coefficients, so H_max = 2 - 1/(u^T M^{-1} u) in closed form, and for 1, 2, 3, 7, 13, 17 or 25 terms the answer is always 0.67250070367941172655, the pure sqrt(2) value, i.e. Anthropic's HD(1) exactly. The 2 j pi harmonics have u_j = 0 and M[0,j] = 0, and the second identity reduces to 2 j^2 pi^2 (w_0^2 - 2)/w_0, which vanishes iff w_0 = sqrt(2). Every harmonic is pure H sacrifice; theirs spend 3.13e-04 of it and get 8.32e-04 of floor back, an exchange rate of 2.66 against a break-even of 1.57. (ii) their total pressure B = 93/23000 sits at the argmax of the saturation curve, with net marginal d(bound)/dB = +0.006076 against competing terms of size one, and the LP dual d(eps*)/dB = 1.509 against a break-even 1.4998.

The doors, ranked by LP shadow price. The span-1 capacity = 2 binds hardest by an order of magnitude (+4.08e-04 of bound per unit of capacity); span 6 is slack. The floor of the whole family is pinned by one near-integer configuration, seven points at about {0, 2, 3, 5, 6, 8, 9}, with 18 of 2,200 cut vectors simultaneously active. The frozen constants with a trade in them: the Gram profile (their own banded lemma already computes +7.07e-06 and they decline to promote it), the point count (trmdy runs nine, AMTOPA never left seven), the span capacity itself, and, the one nobody on this ladder has ever moved, the fundamental w_0 = sqrt(2), which is precisely the frequency that makes the harmonics unable to help. Every door stays inside bandwidth-one data, so all of them are capped by 0.6818286874638; the room left inside that information class is 0.0084, three orders of magnitude more than the family's own ceiling. The construction binds, not the information.

Full record, including the run that broke the operator's local-compute cap and what it cost: amtopa_ceiling/RESULTS.md, amtopa_ceiling/RUNS.md.

Hunt #89: this laboratory is tenth, and the barrier is aimed at an empty room (field_audit/)

Status: settled. FIELD AUDIT. Prompted by discovering, on 2026-08-24, a wave of public repositories created 2026-08-11 and 2026-08-12 carrying constants for liminf N0s(T,2T)/N(T,2T) above ours. Our best figure, 0.6730529829896288 (eight-point, bridge proved in Lean), ranks tenth of fifteen public claims. The leader is AMTOPA/zeta-exact-pressure at 0.6734164909714992949, ahead of us by 0.00036350798187: eight times our margin over ainta/zeta-simple-zeros, which we had taken for the frontier. Three constructions were already past our eventual figure before our seven-point hunt opened; the earliest, trmdy/zeta-simple-zeros-673137, first commit 2026-08-11 14:43 +0200, beat it on day one. Hunt #82's barrier is not contradicted, and could not have been: it is a ceiling at 0.675142509660254 and the leader sits 0.00172601868875 below it. What the audit does cost #82 is reach, not truth, trmdy is outside the family it names on two axes, read against lean/bridge/Zeta23Ext/Bridge/Defs.lean: the window is a seven-term cosine polynomial, not the single term omega = sqrt 2 (their H(v) = 0.67245704141454428878, lower than HD 1 by 4.366e-5, window constant traded for a larger floor c), and the block profile is the envelope 2 sqrt((m-1)A/m) - 1 + A/m, which lifts the cap c(m-(n-1)) <= 1 that our Phi_n requires; they run m = 272 and m = 177 where our cap allows 230 and 172. On the third axis their family contains ours: their uniform_weights is our F exactly, one vertex of a span-capacity-2 polytope they optimise over. Replayed here from a scratch clone against their own pinned Arb binding: their window bounds, H(v), all three assembly constants (0.6732001170127618568182, 0.6732425893558967029403, 0.6733127422722459981438, each matching their published digits), every span capacity exactly 2, and their 23-test suite. Their 2,168,370-box seven-point interval run also reproduces here: 482.6 s against their 441.7 s, with an identical node count, split count, depth 50 and all three prune counters, but one of its two table digests is host-dependent: the w table matches byte for byte and the w'' table does not, and the w'' table is the one the tangent pruner rides on, i.e. the single component whose corruption could yield a false acceptance rather than a false failure. Not one branch of the search changed regardless, which makes this a stronger reproduction than byte-identity would be and simultaneously not the reproduction their documents claim; tawanerguo-cn self-discloses the same class of mismatch, so it is a property of the method. Their 116,272,426-node nine-point run, the one carrying the headline, was not replayed and is out of reach here, so the headline is assembly-verified and floor-reported. Soundness read of their verifier: no defect found, and the Ainta pattern specifically absent, acceptance is one-sided against an upward-rounded target, the search raises rather than returning a negative, the w-table is clamped non-negative before squaring (which is what makes the sign-blind multiply in box_lower safe, while the signed w'' path does use a sign test), the convexity gate is a rigorous Arb LDL behind a float pre-filter, and H_CERT is a threshold to clear, with the assembly then using the rational below the computed value. What does deserve scrutiny is not code: the refined pressure tax (m-q)B_p/m replacing eta B_p (m-1)/m rests on a cyclic-shift counting lemma that is machine-checked nowhere and is worth essentially the whole refinement. Applying that refined assembly to our own floors, changing nothing already proved, buys +3.02e-5 and +5.38e-5, ninth place, not first; the deficit is in the floor c, not the assembly, and taking it would trade our one real advantage, a bridge with zero sorrys, for 5.4e-5. Also recorded, as its own section: why our prior-art search missed eleven repositories. We searched for the standing result rather than the competition (the wave has no stars, no descriptions, no preprints, npip99/zeta-zeros sits at 0.673195 with an empty description and zero stars); we read riemannzeta.fun, still showing 0.672500703679, as the field rather than as the formally-accepted subset; we looked once and never again while the field moved for a week; and we never ran a forward-citation pass, though trmdy's own README names three of the repositories we missed. That pass took under seven minutes when finally run. No arXiv preprint exists for any 0.673x follow-up and none has passed a kernel-checked gate anywhere. Nothing was posted, opened, forked or starred upstream. Nothing bears on RH (docs/08).

Hunt #88: the third autocorrelation constant belongs to the other functional (r_8539dc/)

Status: settled. A reading and an exact recomputation, prompted by issue #123 and by issue #1 on google-deepmind/alphaevolve_repository_of_problems: AlphaEvolve's published C_3 <= 1.4557 sat above a verification cell computing abs(2n*max(conv)/sum^2) while the inequality printed above it defined max|f*f| >= C_3 (int f)^2. The outer abs is a no-op, and not by accident: int_{-1/2}^{1/2} f*f = (int f)^2 > 0 forces max_t f*f(t) > 0, so the code computes A = max f*f/(int f)^2 while the statement defined B = max|f*f|/(int f)^2, and A <= B always. In exact rational arithmetic on the published ten-place heights (no float in the chain), the n = 400 construction gives A = 1.45564279537454049411... and B = 4.33404652438798427361...: it reaches 1.4557 under A only, buying that score by driving the autoconvolution to -4.334 at knot 215 against a positive peak of +1.456. The n = 150 construction gives 1.46876206974102180951... under both functionals, which is the check that keeps the discretisation honest. The mix-up was corrected, twice, and the brief's premise that no correction had appeared is wrong: arXiv:2511.02864 v2 (2025-12-15) splits the problem into max|f*f| >= C_3 (prior 1.4993, Matolcsi-Vinuesa, improved to 1.4688) and |max f*f| >= C_3' (prior 1.45810, Vinuesa, improved to 1.4557), moving the 1.45810 citation between sources in the process; the colab followed on 2025-12-19 (39d0c63), leaving height_sequence_3 untouched. So the improvement survives once each number is read against its own functional, and nothing was withdrawn. What is still broken: the two corrected artifacts label the two problems in opposite senses, so "AlphaEvolve's C_3" is ambiguous between them; problems/4.html still carries only the max|f*f| statement with no sibling page nine months after promising one; and issue #1, the thread where the report was made, is still open. No new bound is claimed and nothing was posted upstream. Nothing bears on RH (docs/08).

Hunt #87: a 2013 paper beats an AlphaEvolve world record (r_2969b0/)

Status: settled. DeepMind's alphaevolve_repository_of_problems marks Problem 42 (the sum-difference problem) world_record; an unreviewed, AI-produced issue on that repository claims a 2013 human paper already does better. Both sides checked, from the primary sources, by exact integer enumeration. The challenger is right, including its citation. Problem 42 defines C as the least constant with |A+A|/|A| <= (|A-A|/|A|)^C; Penman and Wells, INTEGERS 13 (2013) A57, Theorem 21, state that same ratio character for character and give sup g = ln(32/5)/ln(26/5) = 1.125944426 over their family Q_j. The AlphaEvolve set from the repository's own notebook re-counts to |A| = 309, |A+A| = 1367, |A-A| = 1163, g = 1.1219357375, the published value. Q_36 -- 197 integers, published thirteen years earlier -- beats it, g = 1.1219505699, decided at 120 digits rather than in float, and every Q_j with j >= 36 beats it too. Corollary 13's four counting formulas were reproduced by enumeration at 64 values of j with zero mismatches, and four numbers the paper prints (g(A_15), f(X), f(Q_10), f-hat(Q_19)) recompute exactly, which is what pins that the right normalisation is being read: the same paper also carries f(A) = ln|A+A|/ln|A-A|, and f ranks these two sets the other way. Three different score functions appear in the AlphaEvolve notebook for this one problem, one of them the reciprocal of the target and one carrying a size bonus of up to 0.01 on a scale 0.06 wide; the published number uses the clean one. The issue's separate C = 2 claim was left untouched, as briefed. Nothing was posted upstream. Nothing bears on RH (docs/08).

Hunt #86: the ceiling procedure on the lower side of Erdos minimum overlap (overlap_lower/)

Status: probe, partly settled. Sixth instance of the ceiling procedure and the first aimed at a lower bound, picking up hunt #85's first loose thread. The thread is closed: White's program (arXiv:2201.05704, Acta Arith. 208 (2023) 235-255) discretises M and never f, so it is a relaxation and the unsafe direction hunt #85 worried about does not arise. Three of the opening brief's premises were found wrong at the source and corrected -- the lower bound has moved since 2022 (Kim and Pilanci, arXiv:2606.31182, 30 June 2026, 0.379005 -> 0.37912, which the public catalogue has not absorbed); the catalogue entry for this constant carries no asterisk; and the 0.000059 of upper-bound movement is the 2016 human record minus the 2026 AI one, not twelve months of AI. The measured finding: White's choice of R was already at his method's ceiling. In his simplified program the value saturates at R = 40 and the total gain from his R = 20 out to R = 320 is 2.9e-6; in his full program the gain from his R = 10 to R = 20 is 2.0e-5, against the 1.15e-4 the next real improvement obtained by adding constraints instead. One exact rational dual point was accepted with no float in the value, 0.37399241331 at N, R = 5000, 20. The full program was not reproduced to 0.379005: the Parseval cone survives none of four cutting-plane formulations without a conic solver, so the sweep measured a strictly weaker program, and the sweep also never reached CI because the credential lacks the workflow scope. No bound on C is claimed. Nothing bears on RH (docs/08).

Hunt #85: the ceiling procedure on Erdos minimum overlap (r_828c8b/)

Status: probe, partly settled. Fifth front of the ceiling procedure, in a family where the upper record is a step-function construction and the lower record is a convex program. On the upper side the whole procedure reproduces: the m-piece minimax falls 0.400000 / 0.385072 / 0.381833 / 0.381084 at m = 4/8/16/32 and then stops, 1.6e-4 above Haugland's 0.380926, and the stall is provably the solver's rather than the family's, since an m-piece step function upsamples to a feasible 2m-piece one and Haugland's object sits strictly below. The accepted object was re-evaluated in exact rationals with no float in the value: C <= 9990167/26214400 = 0.381094627, weaker than the published bound and stated as a reproduction. Two soundness facts came out of the acceptance step: rounding the pieces at the obvious denominator 2m costs 2.4e-4 to 6.8e-4, more than the whole remaining gap, and ten times finer costs nothing; and on this side there is no dual to check, because the accepted object is primal and its value is a finite rational. The lower side was not reproduced. What was established instead is that the plain averaging skeleton cannot reach it: uniform weight gives C >= 1/4 exactly, the naive Fourier handling of the quadratic term is vacuous for every admissible weight (what(0) = 1 forces sup what = 1; 0 of 200 random profiles gave anything positive), and explicit witnesses cap four weight profiles at 0.2526 / 0.2500 / 0.1909 / 0.1701, all far below 0.379005, with the bound getting worse as the weight concentrates. A guard on the shift enumeration (the failure direction that would report a bound better than the object supports) caught 2 of 3 planted faults, with the third recorded as a known miss; it is in harness/departments/guard_ledger.py. Nothing bears on RH (docs/08).

Hunt #84: the Delsarte LP has no headroom in its degree (r_6f0f63/)

Status: probe, complete. Fourth instance of the ceiling procedure (issue #110), on the LP half of the sphere-packing/kissing-number family. The two exactly tight certificates were rebuilt from their contact structure rather than from printed coefficients -- E8 gives Gegenbauer coefficients [1, 8, 25, 52, 66.5, 60, 27.5] summing to 240, Leech gives degree 10 summing to 196560, both with every coefficient non-negative without being asked. The soundness read found the load-bearing defect in the acceptance step: the standard node-discretised LP is a relaxation, so its optimum sits below the truth, and it did so at every node count tested -- 196505.76 at 600 nodes in dimension 24, where the answer is exactly 196560, and 239.9930 at 600 nodes in dimension 8. The error is one-sided and decays like m^-2. The repair f -> f - sup f converts the invalid output into a valid weaker value and brackets the truth. The ceiling itself is a negative result: sweeping degree 1 to 30 in dimensions 3 to 24, the value stops moving by degree 14 at the latest, and the sweep rediscovers degrees 6 and 10 for E8 and Leech on its own. So the headroom in this parameterisation is not in the degree; it is in the node set, which nobody publishes. Float grade throughout. Five planted faults fire. The Cohn-Elkies SDP half was not attempted. Nothing bears on RH (docs/08).

Hunt #82: the analytic limit of the n-point pressure family (family_wall/)

Status: probe, complete. BARRIER. The n-point pressure family saturates. Reducing the bound to Phi_n <= H + H c - (n-1)/p and then bounding c by the functional's value at any gap vector of total length (n-1)/H makes the pressure term cancel identically and leaves Phi_n <= H (1 + W(g)), with W an energy per point at density H -- order 1e-3, where reaching the configuration ceiling 0.6818286874638 would need 0.0138706. Combined with Phi_n <= H (n-1)/(n-2) (which is already below the ceiling from n = 75), every n is covered and sup_n Phi_n <= 0.6751676: short of the ceiling by more than 0.0066, which is 71% of the whole distance from H to it. The limit itself is H = 0.6725007036794116 exactly -- the family climbs a little, turns over and comes back down. The DERIVED-but-untested pressure lead survived: Phi = H m/(m-1) - k/p is exact when 1/c is an integer and a rigorous upper bound otherwise (overshoot 1.5e-6 to 5.9e-6, comparable to the spacing between adjacent peaks, so it cannot rank adjacent n), and the optimal pressure sits at a minimiser-family crossover p_x = (S2-S1)/(W1-W2) -- predicted 3433.0, 3187.3, 4072.5 for n = 7, 8, 9, each landing inside the exact grid interval where the existing sweep's winning word changes. A Modal cross-check at n = 7, 10, 14, 20 over p = 1200 .. 20000 passed its control to 2e-13 and agreed with the predicted floors at n = 7, 10, 14; at n = 20 the analytic witness ladder beat the multistart search at four pressures, so that job's raw Phi_20 was built on a value that is not a floor. FAMILY-LIMIT.md, with its own "what would refute this".

Hunt #80: where the variable-radius Bloch certificate ends (bloch_ceiling/)

Status: settled. Third instance of the ceiling procedure, on a constant outside the zeta family: Wikström's computer-assisted lower bound for Bloch's constant (arXiv 2608.17660, Zenodo 10.5281/zenodo.21975862). The published verification reproduces from the pinned archive, all 28 checksums verified. The verifier's soundness read is in section 3. Section 4 computes the variable-radius ceiling the author did not: the published constant extracts roughly 59% of the headroom the method's own parameterisation allows, with the near side interval-rigorous, the away side a floating-point programme, and the ceiling therefore INFERRED rather than measured.

Section 5 is now obtained. The author's own verifier accepts sqrt(3)/4 + 0.0153040536, above his published 0.0153, in all 24 away sectors and all 38,400 initial cells, with zero cells refused, on his unmodified certificate data. That target is also the end of the road for that data: the near branch's own Arb gain, 0.0153040536989472, is a rigorous cap on the dichotomy, so the published constant was leaving 4.05e-6 of its own certificate unclaimed and there is nothing further there. The run's own cross-check is exact: sectors 0 and 1 rerun at the published target reproduce reference-run/logs to the integer, 270,744 and 292,931. Cost 324 core-hours over 476 GitHub Actions jobs, billed zero, against a projection published before launch that came in 4.2% high. The hunt ends with a doors section ranking what to unfreeze next; the top door is regenerating the certificate data at moved ETA, LARGE_RAD and node count, which section 4 puts at ~0.015359 and which is affordable, since the away side of a 0.015316 target costs only 4.2% more boxes.

Hunt #81: what the min(dps, 20) cap costs, measured (dps_cap/)

Status: probe, complete. At 0.8 + 85.7i, epstein_completed at the capped dps = 20 returns 3.1e-33 where the converged value is 1.6e-58 -- a factor of 1.9e25 and not one correct digit. The error floor sits near 1e-(D+13), so nothing correct can appear below D ~ 46; the sweep puts the crossover there exactly (D=40 entirely floor, D=50 the first row with correct digits). D = 60 buys about 16 real digits, not 60, because the cancellation costs roughly 44 of them. zeta/epstein.py still caps three interfaces at dps=min(_d, 20) (lines 1092, 1126, 1142), so a caller asking for 60 gets 20 and count_zeros_box's integrality check passes on the noise.

Landed 2026-08-21 as the union of five concurrent 2026-08-14 runs that all wrote this directory and none of which merged; see the provenance header in dps_cap/README.md. Nothing here is evidence about zeta or RH: it is a measurement of an implementation at a single point.

Renumbered twice. These two opened as #35 and #36 from a branch 101 commits behind main, where both numbers were taken. They were renumbered to #49/#50 on 2026-08-18, and main took those two (plus #51, #52, #55-#57 and #59) before this branch landed. They are #60 and #61. The directories lambda_dh_bounds/ and prime_zeta_rightmost/ are unchanged and are the stable references.

Hunt #79: where the seven-point simple-zero certificate ends, and what proves it (ainta_seven_point/)

Status: closed, bounded, in two halves: the certificates reproduced and the floor bracketed (RESULTS.md); the bridge mapped, verdict one substantial analytic bridge (TRUST-MAP.md). Two outside groups (Ainta, and Gohms in issue #1 on that repository) are hill-climbing one finite inequality F6 >= c that refines Anthropic's unconditional 0.6725007 simple-zero constant. Neither has a human reviewer or a Lean line; both are computer-assisted results at the same standard of evidence as the one this hunt adds.

Certificates. Ainta's 19/5000 and Gohms's 191/50000 reproduce field for field on the pinned commit; one secondary table hash differs with no effect on any count. The functional's infimum is bracketed rigorously, 0.003826 <= inf F6 <= 0.0038262312115073: the published verifier accepts 1913/500000 at grids 4000 and 8000 and refuses 0.0038263 at the same six-gap configuration the float minimiser finds, gaps alternating near 1 and near 2 on the kernel's zeros. One defect found and repaired: the verifier's PRESSURE_CUTOFF_CELLS encodes the original target, so every run that raised the target, Gohms's and this hunt's own probes, pruned on a stale assumption; all were re-run with the cutoff raised and all still accept.

Bridge. The map from the certificate to the published constant is recovered in closed form and reproduces both published constants to 40 digits: Phi(c,m,p) = (H - 6(m-1)/(pm)) / (1 - c(m-6)/m). The block size m is not a free parameter: A_0 = c(m-6) <= 1 caps it at 6 + floor(1/c), and both published values sit exactly at their cap, which is why raising the target lowered m from 269 to 267. The family's apparent ceiling at the published pressure is 0.673025477, 4.1e-6 above the Gohms claim, about 5.6% of the room under the configuration ceiling. Six of sixteen steps are already kernel-checked in anthropics/zeta-23-lean (Theorem D itself, von Neumann, the positive-part splitting), four are small finite statements, and what is missing is carrying the new spectral defect through the tail passage and the uniform kernel limit. Smallest Lean-ready obligation: the stability rank-trace lemma. Also recorded: this laboratory has been quoting the configuration ceiling as 0.68185, a decimal in a remark with no proof attached, where the Lean development proves 0.6818286874638. Nothing here bears on RH (docs/08).

Second half, 2026-08-23: the Aristotle probe (ARISTOTLE-PROBE.md). The map's S2, the stability rank-trace lemma, was stated in Lean and proved, zero sorrys, standard axioms only, in StableRankTrace.lean (then in hunts/frontier_math/zeta23ext/Zeta23Ext/, moved to lean/bridge/Zeta23Ext/ on 2026-08-23). No Aristotle project was opened, because nothing remained to send. The finding is a correction to the map: S2 is not new relative to the Lean development the map chose as its vocabulary. Ainta's profile Psi is that development's own gc 2 shifted by one (Psi = gc 2 + 1, the shift being exactly Psi 0 = 1, which is the bookkeeping that trades a rank count for Fintype.card r), the sharpened scalar estimate the map expected to need is already sq_sub_ge_gc, the spectral transfer is already sum_eigenvalues_comm, and the theorem falls out as a corollary of Zeta23.ZeroSide.RankTraceMult.rank_trace_mult at c = 2 evaluated at the eigenbasis presentation of V Vᴴ. The map is not guilty of a missed search: TRUST-MAP.md:382 already cites RankTraceMult.lean:281, lemmaR_tight and gc. It filed that defect term as "a different one", spectral versus per-zero, and that reading is the actual gap: rank_trace_mult is quantified over presentations of P, and at the eigenbasis presentation its per-zero defect is the spectral one. Psi was also never matched to gc 2, so §4 named RankTrace.lean:52-56 for a scalar estimate that RankTraceMult already proves. Its step table records the contrast; its section 4 then states the obligation as if it did not apply. Two things the formal state exposed that the map's transcription hides: the rank hypothesis is removable, so a sharp form holds with no hypothesis on V at all, and the column bound hV is load-bearing for exactly one inequality, tr(V Vᴴ) <= card r, worth a mean of 6.05 of discarded slack on random draws. The sharp form is measured to be an equality on the family upstream proves extremal for Lemma R (400 instances, slack 1.07e-14; float grade). A third Palomar entry is not recommended: a bridging corollary of an existing public Lean theorem does not clear that registry's notability floor. S2 remains one step of sixteen; S8 and S9 are untouched and nothing here bears on RH. A second finding fell out of running the handoff's first instruction: hunts/frontier_math/zeta23ext does not assemble on main at 36c6070. assemble.sh reports three failing targets, Zeta23Ext.RetentionWired, Zeta23Ext.EForm2.Bridge and Zeta23Ext.TruncEst.Kernel, with Retention.margin_identity picking up sorryAx as a consequence. It is not the dependency store (mathlib rev matches the pin), not the Zeta23 dependency (none of the three imports it) and not this branch (the failing build predates the new module). The new module builds standalone and is unaffected; no claim is made here that the package assembles. Recorded as issue #101, not fixed here.

Third half, 2026-08-23: the bridge, formalised to its hypotheses (BRIDGE.md). Five agents in one day, no proving service (0 of 15 permitted Aristotle submissions), proved every step S6 to S9 and S11 to S16 on top of the vendored [L23], so that Zeta23Ext.Bridge.seven_point_bound now states Ainta's bound for Mathlib's riemannZeta as a sorry-free theorem with standard axioms only, conditional on exactly two named hypotheses: the seven-point inequality F6 >= c (S10, the Arb run, not a Lean fact) and the cap c(m-6) <= 1; at (19/5000, 269, 3000) the conclusion's constant is the paper's (1345000 H - 2680)/1340003 with the side conditions by norm_num. The two steps the trust map graded LARGE were not: S8 rides through [L23]'s endgame at lambda = 1 by transcription, because every endgame input is already stated for lambda <= 1, and S9 reuses [L23]'s window Poisson identity and PrimeSide decay and needs no bounded-separation hypothesis at all; the pinching the map called missing is proved for every convex trace functional from a row-stochastic mixture. Builds standalone in 44 s; the package root was still #101.

Packaged for Palomar, 2026-08-23 (same BRIDGE.md, section 8). StableRankTrace.lean and the sixteen Bridge/ modules moved into a Lake package of their own at lean/bridge/, which assembles at its root (lake build, 8860 jobs) against the same pinned Zeta23 and Mathlib; module names and namespaces unchanged; the Apache-2.0 headers copied from the dependency replaced by the repository's MIT header, with the one adapted file's attribution kept in a notice. BridgeChallenge.lean and BridgeSolution.lean advertise four theorems in the namespace Zeta23Ext.Palomar: the parametric bound, the paper's (19/5000, 269, 3000), and this laboratory's own (34697/10^7, 294, 3400) in multiplicative and ratio form, 0.6730295534796928…. The eight-point statement is deliberately not advertised: its bridge from certificate to proportion is stated, not proved. Precheck 66 pass, 1 warn, 0 FAIL. Not submitted; whether a conditional refinement clears that registry's notability floor is its call. Nothing here bears on RH.

Hunt #76: the Riemann zeros in tuning units (zeta_temperament/)

Status: closed, verdict INTERESTING STRUCTURE, classical in substance. In steps-per-octave units theta = gamma ln2 / 2pi, Landau's formula is the Fourier coefficient of the zeros mod 1 at frequency log2 n, so the zeros avoid the equal temperaments that tune prime-power harmonics and ignore composites. On Odlyzko's first 100,000 zeros every prime power n <= 32 lands within 0.0012 of -Lambda(n)/sqrt(n)/<log(gamma/2pi)> and every composite gives 0.0000; the smoothed density at integer x is 0.801 against 0.800 predicted, at fifths-perfect x 0.762 against 0.767, at the composite-6 control 0.997, and the deficit shrinks with height as 1/log t. The lab's own 2000 zeros agree. The peak side (Gene Ward Smith, OEIS A117536) is reproduced as calibration, with one caveat added: the Riemann-Siegel main sum for the x-EDO stops at harmonic 1.2 sqrt(x), so 12-EDO's zeta score never sees the major third directly. Corrected 2026-08-22: the composite control re-measures zeta.explicit.prime_spectrum, which this repository already exposed and whose docstring states the same identity; the hunt's prior-art search had omitted this tree. The same review added the Euler-product discriminator, where Davenport-Heilbronn's loudest spectral line is a composite and the Epstein forms of class number one are silent while class number above one is loud and cancels across the class group. Nothing here bears on RH. Doc: docs/34.

Hunt #75: pitch classes against the colour wheel (chroma_hue/)

Status: closed, verdict PRETTY BUT TRIVIAL. A note-to-hue bijection makes hue distance a function of interval class if and only if it is affine, so the hue side sees exactly one choice, the character of Z_12, and Z_12 has two injective ones (chromatic, fifths). Over all 21,772,800 canonical bijections the circle of fifths is the unique argmax of Spearman(hue distance, dissonance) under three separate dissonance measures (0.609 Sethares, 0.702 Tenney and ordinal; null mean 0.00, sd 0.14, null max 0.50 and 0.57), and chromatic order is the worst structured choice (-0.73, -0.49). That is a statement about hearing with no colour in it. The colour side is not a 12-gon: the HSL wheel's adjacent CIEDE2000 steps vary 8.5-fold and its twelve hues land 6.5 to 61 degrees apart in OKLCH; twelve equal-tempered steps of light frequency across the visible octave give hue steps of 1 to 72 degrees per semitone and cover about 275 of 360 degrees. The one formulation where colour adds anything: the eye mixes linearly, so the colour of a chord under wheel k is its k-th Fourier coefficient, and maximal evenness (Clough-Douthett, Amiot) reads as saturation. Round two reformulated: a colour wheel is a complex place of Q(zeta_12), the fifths wheel is the Galois conjugate of the chromatic one (together they distinguish exactly the 1763 chord colours either does alone), and chromatic times fifths saturation is the field norm, an integer taking ten values on the 224 set classes, with 84 classes units. Nothing here bears on RH. Doc: docs/33.

Hunt #72: AIMO Interpretability 2026 baseline reproduction and structure-matched robustness signal (r_662b12/)

Status: baseline retained, intervention rejected after independent audit. The public sample has 28 cases, 19 non-robust and 9 robust, so the all-False baseline is 67.86% (19/28) with full coverage and no invalid outputs. The reported 92.86% intervention is an in-sample score for a model-name rule chosen after inspecting those same labels. Its LOOCV, leave-one-problem-out and five-fold loops never fit or select the rule inside training folds, so they provide no generalization evidence. The scrambled-text surrogate retains the same 92.86%, refuting the claimed mathematical-structure interpretation. Prize disposition: NO-GO on this hunt's evidence. The original run record is preserved; hunts/r_662b12/AUDIT.md carries the correction. Nothing here bears on RH (docs/08).

Hunt #77: AIMO-2, legal AIMO Interpretability submission + validation-design report (aimo2/)

Status: report-ready (reframed 2026-08-22); Main entry is a legal model-identity prior, Small entry is the always-non-robust constant; learned-method arm stays killed; no GPU spent. Successor to Hunt #72 (r_662b12), whose +25pp claim was withdrawn and stays withdrawn. The organizers' clear-cut protocol (proposal §1.4) is reconstructed exactly from public data: val-sample's 9 robust rows are the 9 sample-full pairs with base accuracy 1.0 and zero detrimental perturbations; its 19 non-robust rows are single-type rows with decay >= 0.5. So the hidden set is minority-robust by construction (22-43% on the public problems, 20% for the 8B on the organizers' MATH release), and the first version's "natural distribution" (89-97% robust) was the wrong reference; the always-robust constants it chose are superseded. Under the protocol the label is mostly a property of the model: a per-model prior fitted on the official labels scores 26/28, 39/41, 43/56 leave-problem-out vs 19/28, 32/41, 32/56 for the constant, transfers between the two public sets, and is the binding split because the organizers state the validation set covers all test-set model types. Self-consistency adds nothing; on the 69-problem MATH release the 8B's failures do not transfer across perturbation types, so the free gate fails again and the Small-track GPU route stays killed. Official path replayed at starter HEAD e46be92 (always-false 0.6786, prior 0.9286 in-sample; importer/ingestion schema mismatch confirmed at HEAD). Deliverables: REPORT.md (reframed for the $5,000 writeup pool), PREREGISTRATION.md with a dated amendment, protocol_reconstruction.py, and the verified bundles. Number #77 is provisional; if a parallel session took it, renumber. Nothing here bears on RH (docs/08).

Hunt #71: Krenn-Gu 8x3: port the verified 6x3 orbit census and measure the next exact frontier (r_322dae/)

Status: settled. The 6x3 baseline checksum reproduces 15 matchings, 3375 triples, and 8 S6 x S3 orbits with exact sizes [15, 90, 120, 270, 360, 360, 1080, 1080]. For the 8x3 frontier (105 matchings, 1157625 triples), the direct product group S8 x S3 (order 241920) partitions the triple space into exactly 31 disjoint orbits, confirmed both by constructive partition and Burnside fixed-point evaluation. Quotienting by the wreath-product stabilizer of a fixed matching H = S2 wr S4 (order 384) partitions the 11025 pairs into 86 orbits (57 orbits under H x S2).

Hunt #73: Krenn-Gu 8x3 support frontier (r_044dd2/)

Status: support census settled; algebraic sieve open. This hunt corrects the scope of Hunt #71: the 31 target-matching orbits do not quotient the full 252-variable polynomial system. All 31 branches have independently replayed support survivors, with 76 to 132 active entries and median 118, so the inherited support conditions close zero branches by themselves. On the sparsest branch, four survivors are excluded by exact signed-Laurent certificates. The size-2 stabilizer mirrors those into four more replayable cuts. Support 7 passes the independent support audit and avoids all eight cuts. Exact fixed-degree cap-3 and cap-4 spans do not exclude it. Cap-5 closure produced 13,379,522 relation multiples, but the first sparse elimination ordering hit measured fill-in before a membership decision. No branch is closed and no complex witness has been found.

Hunt #140: Krenn-Gu 8x3: bounded unrestricted Groebner step on Support 7 (r_abf47b/)

Renumbered on landing, 2026-10-21. Opened as #77 on hunt/r-abf47b-bddff380 (run bddff380, 2026-08-22); renumbered on landing because aimo2/ holds #77. It takes #140 because two landings opened the same hour take the numbers below it: mobius_orbits/ takes #138 (land/mobius-orbits-20261021) and seven_eighths_pin/ takes #139 (land/quasi-riemann-seven-eighths-20261021). hunts/r_abf47b/ is the stable reference.

Status: attempt unresolved, no algebraic conclusion; one exact orbit result. Continues Hunt #74 (r_31b6c1/) on Hunt #73's Support 7 for orbit 18, in two parts. The orbit refinement stands. The 57 H x S2 orbits on matching pairs (H = S2 wr S4, the stabilizer of M0) only refine the 31 S8 x S3 orbits on matching triples: S8 is transitive on the 105 perfect matchings, so every triple orbit meets the slice {M0} x PM x PM, and the 57 map onto the 31 (fibres of one, two and three pair orbits over 8, 20 and 3 triple orbits). Two skeletons over one triple orbit are symmetric copies, so Hunt #74's thread, scale the sieve to the 57 pair-orbit skeletons, adds no branch. Exact finite check on two routes: map_orbits.py over all 105^3 triples, and test_orbit_refinement.py, added on landing, on the slice alone against Hunt #71's orbit sizes. The Groebner step does not test Support 7. run_groebner.py gave sympy.groebner (grevlex) the three monochromatic target brackets as zero generators next to the 6,558 relations of Hunt #73's build_system: 6,561 generators in 138 variables, every one homogeneous of degree 3 or 4 with no constant term, and no torus saturation. That ideal is proper, the origin being a common zero, so its basis can never be [1] and the 300 s timeout says nothing about Support 7. The question Hunt #73 Result 4 poses is membership of the target product in the relation ideal saturated by the variables. The run also stopped at its own 5-minute alarm although the mission allowed 20 minutes. Its "32.59 MB peak" is the run's own report and is unverified: by the script's code it would be the resident set size at the alarm, and the committed script, rerun unchanged at landing, read 600.5 MB there and already 84.4 MB before the Groebner call starts. The calibration reruns main's code (cap3.json is byte-identical to a fresh cap-3 screen; r_31b6c1's 6x3 gate) and is not an independent route, and its cap-4 4,216 replay is reported without an artifact. RESULTS.md keeps the run's text and carries the "Audit correction" that governs it; PROBE_RESULTS.md is retitled as this hunt's rerun of r_31b6c1's probe. Orbit 18 remains open, no branch is closed, and no complex witness has been found. Nothing here bears on RH (docs/08).

Hunt #61: two-sided bounds for the de Bruijn-Newman constant of Davenport-Heilbronn (lambda_dh_bounds/)

Status: closed (2026-08-16), gate verdict publication candidate; hardened 2026-08-18, verdict unchanged. The bracket, in the wide frame of Rodgers-Tao and Polymath 15: 0.2304 < Lambda_DH <= 0.7696992583210755065522 (narrow frame: 0.0576 = 36/625 to 0.19242481458026887663805), ratio 3.341. Lower side decided by ball-arithmetic winding counts with a zero-shared-layer second witness; upper side a decided strip constant fed to de Bruijn 1950 Theorem 13. The quotable corollary, recognized after the gate closed and adversarially narrowed in SEPARATION.md: Lambda_DH > Lambda_zeta unconditionally (0.22 from Polymath 15 against the decided 144/625), the first strict order between two such constants with both nonnegative. The gate cycle caught and preserved three corrections: a factor-of-4 frame error (two normalizations live in the literature, FRAME.md), the sigma_0 originality claim withdrawn after Bombieri-Ghosh 2011 was retrieved and read (their exact abscissa 1.120362 displaces it), and one prose lemma (M2) recorded as a lesioned blind spot rather than repaired. The 2026-08-18 hardening then took two of those back: M2 is proved with decided constants (M2-LEMMA.md), and the upper bound sharpened by the full factor 2.082 in-tree, from a phase obstruction in the Euler products that reaches Bombieri-Ghosh's abscissa and decides it on both backends (STRIP2.md); the superseded headline (<= 1.6025374835598228 wide, <= 0.4006343708899557 narrow, ratio 6.955) is kept beside the new one everywhere it appeared, and the lower side did not move in any digit. Doc: docs/29. Full adjudication: GATE.md. Predictions P1-P3 and P5 held; P4's census found nothing deeper below height 600. Nothing here is evidence about RH.

Hunt #60: the rightmost zeros of the prime zeta function (prime_zeta_rightmost/)

Status: settled. The threshold is correct, the conjectures are false, and the core is a rediscovery of published work: kill condition 2 fired. The hunt attacked the two conjectures posted on OEIS A107311 (2024-12-21), that x* = 1.7286... (root of zeta(x) = 2) bounds the real parts of the zeros of the prime zeta function P(s) = sum_p p^(-s) and of every prime-subset series. It decided on both backends (python-flint arb at 350 bits and mpmath.iv at dps 40, exact Fraction endpoint logic) the balance root sigma_c = 1.779544653546994116445898786965... of P(sigma) = 2^(1-sigma), x* = 1.728647238998183618135103010297..., the separation sigma_c - x* > 1/20, the margin P(x*) - 2^(1-x*) = 0.0169073772138... > 1/60, the subset constant sigma_3 = 1.8252259560738457... with sigma_3 - sigma_c > 0.0456813, and log2(69/(5 log 23)) = 2.1379035036560028... > 17/8. It wrote out both halves of the threshold theorem with proofs (triangle-inequality wall; Bohr-Kronecker-Rouche existence below it), ran the WP5 calibration control (the same solver object, pointed at the zeta partial sums, reproduced 31 digits of the literature's x*), two lesions and a precision-response check, and settled predictions P1-P4, P4 including a screen of sigma = 7/4 to t = 1e8 whose global |P| minimum was about 0.010021 at t about 5.63e7, far above the alignment budget.

Then the prior art was found, and it owns the core. Belovas, Cepaityte and Sabaliauskas, On the zero-free region and the distribution of zeros of the prime zeta function, An. St. Univ. Ovidius Constanta Ser. Mat. 33(2) (2025) 27-44, Theorem 1: the same wall, the same constant to every digit they print, the same triangle-inequality proof. Sepulcre and Vidal, Carpathian J. Math. 38 (2022) 489-501 (preprint arXiv:1805.02041, 2018), Theorem 4.3: the general characterization, strictly stronger than the hunt's existence half, with Moreno's Geometric Principle (Compositio Math. 26, 1973) as the finite ancestor and the source of the aggregated-tail device the hunt reinvented as its Lemma 2. MISSION.md kill condition 2 reads: "a literature source is found proving the sigma_c threshold for P, in which case the finding is reclassified as a rediscovery and the OEIS correction cites that source instead of this work." It fired. The reclassification is done, in hunts/prime_zeta_rightmost/PRIOR-ART.md (statement-by-statement ownership map, verbatim source texts, the worked specialization) and in the rewritten docs/30-prime-zeta-rightmost-zeros.md, which now leads with the prior art.

What survives as the hunt's own, and nothing beyond it: (a) the line-by-line refutation of the two OEIS conjectures, since no source in the literature engages that entry and the published wall does not by itself refute Conjecture 1 (1.7795 is weaker than the conjectured 1.72864, and Belovas et al.'s numerics stop at 1.6826, below x*), graded new as a connection with zero new mathematics; (b) the unbounded tail-subset walls, {p >= p_k} walled at or above log2(3 p_k / (5 log p_k)) via Rosser-Schoenfeld, so no constant bounds the real parts across all prime subsets and Conjecture 2 fails for every replacement constant, graded new, small and elementary, and the only mathematics here not located in print; (c) the constant sigma_3 and the fact that a subset out-walls the full series by more than 0.045, a new instance of prior-art theory; (d) the two-backend enclosures, rediscovery in sharper form, the source printing 15 digits and noting that any precision is available; and (e) one literature observation, that Belovas et al.'s Conjecture 1, left open in their paper, is a corollary of Sepulcre and Vidal Theorem 4.3, which neither paper cites. The steering lemma's positive lower density in t is carried as an unverified lead, since the Jessen-Tornehave literature was not searched.

The lesson, which is the most transferable thing the episode produced. Two independent search failures, needing two different countermeasures, and both reproduce on demand. The exact-phrase query "zeros of the prime zeta function" is a literal substring of the Belovas et al. title and the engine still does not return it: no arXiv preprint, a Sciendo/DOAJ venue outside the sweep, and an image-scanned PDF that has to be downloaded and extracted rather than fetched. The general theorem was missed for an unrelated reason: it is filed under almost periodic functions, MSC 30B50/30D20, and never names a prime, so no prime-zeta query can reach it, and it was reachable only by searching for the device rather than the application. Four independent searches returned nothing while two published papers owned the result outright, which is the standing argument for ontology/knownness.py defaulting to "the literature was not consulted".

Disposition: instruments retained; nothing promoted; the OEIS correction drafts in OEIS-CORRECTION.md remain unposted; they cite Belovas et al. for the threshold, which is what the kill condition required, and that rewrite is done. Nothing here bears on RH: the zeros discussed are zeros of P and of subset series in Re s > 1, not zeros of zeta.

Hunt #56: make_context.py blind region on meta/ package (r_2946de/)

Status: settled. scripts/make_context.py --check does not detect public functions, classes, constants, docstrings, or new modules added under meta/ (0/16 curated mutants detected, 0.0% detection rate; 0/32 public AST symbols detected, 0.0% detection rate), because meta/ is structurally excluded from the hard-coded scan paths in make_context.py and produces zero diff in CONTEXT.md (exit 0). In contrast, in-scope positive controls in scanned directories (zeta/, ontology/, harness/, docs/) are 100% detected (4/4 caught). Measured on a 16-mutant curated battery and a 32-symbol exhaustive AST census in hunts/r_2946de/results.json and RESULTS.md. Nothing here bears on RH (docs/08).

Hunt #59: test file naming sensitivity and boundary of make_context.py --check (r_c35cd1/)

Status: settled. scripts/make_context.py --check does not detect test files whose names do not match test_*.py (such as tests/mutant_helper.py, tests/conftest.py, or tests/helper.py), nor does it detect nested test files in subdirectories (tests/fixtures/test_sample.py, tests/unit/test_unit.py), non-Python test assets, or helper functions and length modifications within existing test files (0/22 non-matching specimens detected, 0.0% detection rate), because test_counts() uses a shallow TESTS.glob("test_*.py") and counts only AST function definitions starting with test_ without tracking line counts for test files. In-scope test modifications (test_ function additions, renames, and new test_*.py files) are 100% detected (3/3 caught). An exhaustive repository census shows 77 test_*.py files with 1852 test functions and one unindexed fixture asset (tests/fixtures/rung3_ball_term_kernel.json). Zero non-test_*.py Python helper files have ever been committed in repository git history across 79 historical paths. Measured on a 25-mutant curated battery in hunts/r_c35cd1/results.json and RESULTS.md. Nothing here bears on RH (docs/08).

Hunt #57: make_context.py compiler/ unscanned boundary and detection blindness (r_dc6e6f/)

Status: settled. scripts/make_context.py --check does not detect public functions, classes, constants, docstrings, or module additions added under compiler/ (0/17 curated compiler mutants detected, 0.0% detection rate; 0/37 public symbols detected across an exhaustive AST census, 0.0% detection rate), because compiler/ is completely omitted from the scanned directory roots in scripts/make_context.py (zeta/, ontology/, harness/, dossier/, docs/, scripts/, tests/) and from build_flat(). Positive controls in scanned packages are 100% caught (5/5 detected). Evidence in hunts/r_dc6e6f/RESULTS.md and results.json. Nothing here bears on RH (docs/08).

Hunt #55: make_context.py in-place helper renames and __all__ boundary (r_7ad39f/)

Status: settled. scripts/make_context.py --check does not detect in-place length-neutral private helper renames in modules that declare __all__ (0/299 unexported private symbols detected, 0.0% detection rate), because module_api filters top-level definitions strictly by ast.literal_eval(__all__) and the file length remains unchanged. In non-__all__ modules, detection is 100% (4/4 symbols caught). Any edit altering line count is 100% detected via the line-count tell. Zero length-neutral private-to-public helper renames have ever occurred in repository git history. Measured on a 25-mutant curated battery and a 305-symbol exhaustive census in hunts/r_7ad39f/results.json and RESULTS.md. Nothing here bears on RH (docs/08).

Hunt #52: scope caveat: compiler verdicts rest on a hand-written model, not LLVM semantics (Alive2 absent) (r_e2ee73/)

Status: settled. Compiler refinement verdicts in compiler/ rest on a hand-written pure-Python interpreter (pymodel.refinement_i8, rung 2) rather than formal LLVM semantics, because Alive2 (alive-tv, rung 3) is absent from this environment. The exposure of the hand-written model is bounded by an exhaustive two-backend cross-check against compiled Apple Clang binaries (clang.exhaustive_i8, rung 1) across all 10 fixtures (655,360 total evaluated points, zero mismatches). While the concrete Clang detector is blind to poison violations covering 50% of the domain (nsw_flag_on_a_wrapping_shift), the model detector captures poison and immediate UB at full declared power. An audit of unsupported IR constructs confirms that out-of-scope instructions safely raise ModelUnsupported rather than guessing. True LLVM-native refinement remains absent without Alive2. Evidence in hunts/r_e2ee73/RESULTS.md and results.json. Nothing here bears on RH (docs/08).

Hunt #50: doc renaming without number change, and doc content drift (r_c62e44/)

Status: settled. tests/test_docs_numbering.py::test_no_two_docs_share_a_number enforces uniqueness of the leading 2-digit number (00..N) across docs/*.md filenames and detects collisions (smallest mutant: 05-a.md and 05-b.md), but does not detect a document renamed without its number changing when citations use bare references (such as docs/08), nor does it detect content drift or heading changes. Sibling test test_every_full_filename_reference_to_a_doc_resolves detects renamed documents only when un-updated full-name references exist in the tree. An exhaustive census across all 37 test files mentioning docs/ confirms zero tests in the suite inspect document body content, validate H1 headings against filenames, or verify that citations match actual content. Measured on a 20-mutant battery in hunts/r_c62e44/results.json and RESULTS.md.

Hunt #51: file-type boundary of the hunt reserved-word guard (r_365c6c/)

Status: settled. tests/test_hunt_probe_discipline.py::test_no_hunt_claims_the_reserved_word enforces an explicit suffix whitelist (path.suffix.lower() in {".py", ".md", ".json"}). Every file type outside this three-extension set passes unconditionally (40-specimen battery across 12 format categories in probe.py: 6/6 .py/.md/.json caught, 0/29 non-{py,md,json} caught). An exhaustive repository census shows 76 of 447 files (17.0%) under hunts/ are currently unscanned, including 68 .lean proof files under hunts/frontier_math/. None of the 76 unscanned files contain the reserved word. Evidence in hunts/r_365c6c/RESULTS.md and results.json. Nothing here bears on RH (docs/08).

Hunt #63: truncated Weil form positivity failure on Davenport-Heilbronn (r_ac9ca3/)

Renumbered 2026-08-20. Opened as #45, which r_233abe/ had taken two days earlier. hunts/r_ac9ca3/ is the stable reference.

Status: settled. The first positivity failure of the Connes–van Suijlekom / Connes–Consani–Moscovici Galerkin truncation of the Weil quadratic form on Davenport–Heilbronn on the integer lattice $c \in [6, 60], N \le 128$ occurs at $(c, N) = (31, 60)$.

Integer cutoffs $c \le 30$ are strictly positive definite across all tested $N \le 128$ (and up to $N = 256$ at $c = 29, 30$). At $c = 31$, the even sector develops its first negative eigenvalue at $N = 60$, with rigorous Arb ball enclosure $\lambda_{\min} = -1.87393568857 \times 10^{-31} < 0$ (radius $\sim 3 \times 10^{-192}$) and exact dyadic Rayleigh quotient upper bound $-1.87393568857 \times 10^{-31} < 0$, while $N = 59$ is strictly positive with $\lambda_{\min} = +8.36504566170 \times 10^{-31} > 0$. The odd sector at $(31, 60)$ remains strictly positive (inertia $60$ positive, $0$ negative). The Riemann zeta control at $(31, 60)$ is strictly positive by 100 orders of magnitude: even inertia $(61, 0)$, odd inertia $(60, 0)$, $\lambda_{\min}(\zeta, 31, 60) = +4.82160175 \times 10^{-100} > 0$. The failure tracks the first off-line pair at $\gamma_{\text{off}} = 85.6993, \delta = 0.3085$: across $c \in [32, 60]$ the crossing band edge mean is $83.64 \pm 2.44$, and at $(31, 60)$ the zero-side dictionary decomposition proves the off-line quadruple ($-6.734989 \times 10^{-29}$) is the sole negative contributor, whose subtraction flips the form value positive to $+6.716250 \times 10^{-29} > 0$. Evidence in hunts/r_ac9ca3/RESULTS.md and results.json. Nothing here bears on RH (docs/08).

Hunt #66: four arms salvaged off an abandoned branch (r_f00e48/)

Status: settled. claude/riemann-hypothesis-research-ofds8s ran a wide-portfolio RH-adjacent campaign on 2026-08-17/18 and never landed. Four of its five arms existed nowhere on main; they are now in hunts/rogue_frontier/: weil_trunc/ (23 files, 8 replication gates, a 27/27 conclusive-positive enclosure grid, and the Davenport–Heilbronn positivity failure at (c, N) = (31, 60)), sine_gram/ (exact finite-N engine, m_5(1) = 101/18, m_6(1) = 640/63), window_opt/ (RF-C003, the campaign's one promoted claim), and nyman_beurling/ (Baez-Duarte distances to N = 2048, against main's previous N = 50), plus the survey documents. 1,717,013 bytes landed against ~58 MB on the branch: 56 MB of regenerable pickle and a stale .ext_lock stayed behind, and so did fkappa/, whose kappa = 2 table Hunt #65 had already adjudicated wrong. LANDING.md records the subset so the gaps do not read as loss. The defect the checking found: REPRODUCE.md's headline command for RF-C003, the campaign's only promoted claim, did not run, it named functional.exact_F_quartic(1467, 1159), a symbol that never existed under that name or that signature, so a reader following the published recipe got an ImportError. The function is moments_polyeven_exact(OPT_Q) returning (m2, m3, F); probe.py now recomputes F(v*) = 2245228120295149280/3276332462159207451 from the landed source and pins it against the landed document. Two further references were repaired. The one external corroboration: main's own Hunt #45 (r_ac9ca3/) reached (31, 60) independently, and the salvaged dhneg_scan.json agrees with it digit for digit on the enclosed DH eigenvalue (-1.87393568857018838648…, radius ~5.7e-208) and the zeta control (+4.82160175202313776…e-100). What it does not establish: re-running an arm's own code is not an independent check of it, and RF-C003 is entered in harness/departments/review_ledger.py with a white-box outcome saying exactly that, leaving standing_reasons() correctly reporting no blind attack, a new open item, honestly created, not one closed. Two further arms (erdos_scan/, matchings/, the latter claiming a kernel-checked Lean result) appeared on the branch after the sweep that specified this landing and are unreviewed rather than rejected. Evidence in hunts/r_f00e48/RESULTS.md and results.json (29/29 checks). Nothing here bears on RH (docs/08).

Hunt #65: the conflicting kappa = 2 tables, adjudicated (r_2ac05f/)

Status: settled. Two directories carried a table of the same quantity, the regular coefficients C_{2,i} of Bian's pair-correlation form factor for the zeros of xi'', and disagreed at every index from 2 to 11: hunts/higher_xi/C2_EXACT.json (1, -8, 24, -32, 64/3, ...) against hunts/rogue_frontier/fkappa/ corrected mode (1, -4, 4, -16, 52/3, ...), with conflicting diagnoses. higher_xi is right. A fourth derivation written for this adjudication and importing neither hunt, the identity R_kappa = xi'/xi + D log Q_kappa expanded in a formal Dirichlet word algebra, exact rationals throughout, reproduces higher_xi's eleven values exactly, and reproduces the externally published Farmer-Gonek kappa = 1 row (arXiv:0803.0425) exactly as its control, including the four forced zeros. The general defect, derived here and recorded nowhere else: the x^1 coefficient of Qhat_kappa = Q_kappa / L^kappa is kappa*g, so C_{kappa,2} = -4*kappa (-4, -8, -12, -16, -20 for kappa = 1..5), and Bian's Lemma 12 asserting a universal -4 is the dropped M(v_l)M(w_k) weight C2_PROVENANCE.md names on thesis page 71. fkappa reimplemented the thesis code faithfully and found three real implementation defects in it, but carried Lemma 12's -4 as an axiom, so its correction sits downstream of a larger error. The control that would have caught it, measured rather than asserted: planting exactly that defect in this hunt's probe leaves the Farmer-Gonek kappa = 1 control passing and moves C_{2,2} to the published -4, while a one-factorial corruption of the pairing turns the same control red. The only externally anchored control either hunt ran has zero power against the defect that decided the dispute; the control with power is to compute C_{kappa,2} for kappa = 1, 2, 3 and assert the values differ, i.e. an invariance claim needs a control that moves the variable the invariance is asserted over. Evidence in hunts/r_2ac05f/RESULTS.md, results.json and fault_check.json; outcome appended to harness/departments/review_ledger.py. Nothing here bears on RH (docs/08).

Hunt #64: gap A of the lattice-extremality route is a forbidden constant (r_b9552d/, run 2)

Status: not settled (the gap stands), and one thing settled inside it. Second run of r_b9552d; run 1 is Hunt #46 above and its artifacts are preserved under run-tagged names. LATTICE-EXTREMALITY-ROUTE.md bounds the centre-gas row J(T) only for densities rho >= 1/(2*pi) (its gap A). Any single density-independent Cohn–Elkies certificate g (with g >= f, ghat <= 0) proving J <= L = 0.11433003938654052 at every density is pinned exactly: ghat(0) = 0, g(0) = -L/2, sup|g| <= L/2, and ghat vanishing at every integer. The necessary condition sup f <= L/2 passes with margin 3.04992 (sup f = 0.0187431348 at s = 6.3974 against L/2 = 0.0571650197), so the route is not excluded; a 3.05× planted inflation of f fires the test. But the Fejér family that closed gap B on main (7efd506) reaches ghat(0) = 0 only at c = 2*c2(0) = 1.6984559986, where the bound becomes exactly L at every density (spread 6.7e-16), and admissibility caps c at cos²(√2/2)(1+cosh 1) = 1.4698290125: short by a factor 1.15554665, with a frequency-domain witness ghat(0.87493) = +0.0839055 > 0. A quoted Paley–Wiener factorisation plus critical sampling makes c·(sin(x/2)/(x/2))² the only band-limited candidate, so the miss is the family's, not the ansatz's. Byproduct: the first sparse-side bound in this route, J(T) <= 2*kappa(0) - 2*cos²(√2/2)(1+cosh 1) = 0.5715840116 at every density, 4.99942 × L. Evidence in hunts/r_b9552d/RESULTS.md and results.json. Measured, double precision, no enclosures. Nothing here bears on RH (docs/08).

Hunt #49: zeta23ext root load, Retention.Aconst / c2 collisions across arms (r_6f088d/)

Status: settled. The root module load failure in zeta23ext was caused by 17 colliding declarations (including Retention.Aconst and Retention.c2) across EForm, EForm2, and EForm3 all declared in the un-scoped Retention namespace. In Lean 4, importing modules with duplicate fully-qualified names halts elaboration with an environment collision. The collision is resolved by scoping arms into distinct sub-namespaces or importing only the active EForm3 iteration, while mathematical duplication remains across the independent development arms.

Hunt #47: the two-mode arithmetic, and the bridge nobody had named (r_88dc5e/)

Status: settled. O9Assemble.lean said the last step between the kernel-checked table and O9 was "the arithmetic of the two modes". It was, plus one prerequisite that was not on anyone's list. O9Check2 decides a table in the kernel's interval arithmetic; O9Sound proves both modes sound over the reals in terms of Retention.Qre and Retention.Qim, which are integrals against the window g; and the enclosure chain ended at BandDual.Phi2, which is s(z+√2) + s(z−√2). Those are two different definitions of the same two numbers and nothing in the package connected them. EForm3/O9Bridge.lean does: Re Phi2 (s + iy) = Qre y s and Im Phi2 (s + iy) = −Qim y s for y ≠ 0, by closed forms on both sides and no new analysis.

With that, EForm3/O9Modes.lean turns the checker's Bool into the damage bound. dam_le_of_box says a box o9Box accepts bounds Dam at every point of it, y = 0 included; the content is that Iv.mem is lo ≤ 2⁶⁴·x ≤ hi, so each recorded comparison is one division away from the real inequality mode 1 or mode 2 wants. dam_le_box0 instantiates it at the first recorded row at the interior point (23/4, 1/8), with the verdict taken from o9_box_chunk0, the kernel's own decide +kernel, rather than from a fresh decision. Zero sorrys, axioms [propext, Classical.choice, Quot.sound].

The two dead-weight lemmas are gone. The use-site survey run bbe76b9a reported was re-run rather than trusted, over the whole repository: O9Seam.r_comp_mem had exactly one use site (Retention.rIv_mem) and rIv_mem had none. Both are deleted, r_comp_mem' carries the corrected statement, and nothing was reproved.

What this does not close. O9 soundness is not closed and the hunt does not claim it is. dam_le_of_mem_walk bounds Dam at points of boxes the table records; the retention obligation needs the whole window, and nothing in Lean yet says the recorded boxes cover [28/5, 60] × [0, 1/2]. That covering, and the assembly into the form Gap/FarField consume, are what is left.

Also recorded, because it cost this run time: simp only [o9Box] and rw [o9Box] both hang on a whnf heartbeat timeout, unfolding the checker asks the elaborator to reduce rIv b.sLo … symbolically through the entire leaf layer. unfold o9Box at hb, then split at hb, then one definitional have does the same job instantly.

Hunt #48: Erdős–Kac priced, and one of the two routes is blocked (r_8c3b94/)

Status: settled. A mapping run, not a proving run. Two earlier runs left Erdős–Kac as prose ("a real project", "needs either moment control to all orders or a formalised Berry–Esseen route", "out of reach and should be said so"). None of that is a coordinate, so this run priced both routes against what Mathlib actually carries at this repository's pin, resolving every claim by compiling 43 #checks rather than by grep. Probe.lean in the hunt directory is that file; it exits 0 and is imported by nothing.

Disposition: coordinates, not a claim. The recommendation is Route A or nothing, with the k = 4 central moment as a single bounded first file and a stated stop condition (over ~600 lines and the estimate has failed at the step that matters). Three sub-obligations are wanted upstream independently of whether Erdős–Kac is ever attempted: a triangular-array CLT, the Gaussian moments, and a method-of-moments theorem. Nothing here bears on RH.

Hunt #44: what the O9 numerator fields actually enclose (r_938ab4/)

Status: settled, and the two fields answer differently. Hunt #42 left the question of whether reNum_mem and imNumOverY_mem pin Re num and Im num / y or merely the shapes the interval arithmetic computes. Zeta23Ext/EForm3/O9NumShape.lean settles it. reNum encloses Re num unconditionally: once SQ2_mem, SINC_mem and COSC_mem instantiate the three constant leaves the computed shape is the real part, at every point of every box, y = 0 included, with no hypothesis beyond the ones the leaves already carried. imNumOverY encloses Im num / y only for y ≠ 0, and the hypothesis is necessary rather than incidental: Im num vanishes on the real axis, so Im num / y at y = 0 is 0 by Lean's division convention, while the field carries shcSmall and therefore encloses the removable limit, 2·cos(√2/2) + √2·sin(√2/2) > 0 at s = π. The disequality is stated as a disequality, not hedged into a hypothesis. That asymmetry is the design working: the removable branch exists so y = 0 is an ordinary point, and it can only be ordinary by enclosing something other than a quotient by y there; what the run adds is that the something other is now named and pinned. With both components identified, qreIv_mem_phi2 and rIv_mem_phi2 read the two compositions back against Phi2 itself. Verdict on the dead weight: O9Seam.r_comp_mem and Retention.rIv_mem can be retired at no reproof cost, r_comp_mem has exactly one use site (rIv_mem) and rIv_mem has none, and the route built here goes through r_comp_mem' and rIv_mem_box instead. They were shown unnecessary, not deleted. Twenty-seven declarations, zero sorrys, lake build Zeta23Ext.EForm3.Main green (8726 jobs), #print axioms reporting only [propext, Classical.choice, Quot.sound] and, for box0_in_table, no axioms at all. Because a compiling lemma is not a lemma with content, hunt #42's finding, every identification is also instantiated at the first recorded row of o9boxes at the interior point (23/4, 1/8), with every hypothesis discharged. What is not claimed: O9 soundness is not closed; the two-mode arithmetic and the identity joining Phi2 to EForm3.Qre/Qim are separate and untouched, and the y = 0 mismatch witness sits at s = π, outside the table's s-range. Two pre-existing build failures are recorded rather than absorbed: RetentionWired.lean:44 does not elaborate, and BandCert.Verify is killed at 704 s with exit 137, both observed on a baseline build made before the new file existed. Evidence in hunts/r_938ab4/RESULTS.md. Nothing here is evidence for or against RH.

Hunt #39: the Mertens constants, tightened (r_4218d4/)

Status: settled. The Turán variance constant of Hunt #37 falls from 5855 to 275, a factor of 21.3, with every statement's shape preserved, no hypothesis added, no sorry, a green lake build and an unchanged axiom audit. Hunt #37 wrote that its constant is "coarse because the Mertens band is, and tightens automatically if that does". This run did that, and the prediction was right: no new mathematics was needed anywhere.

Three local steps upstream, then arithmetic. (i) Mathlib's Chebyshev.psi_le gives ψ x ≤ x log 4 + 2√x log x, and its packaged corollary bounds the remainder by 4x, i.e. by log t/√t ≤ 2, where the supremum is 2/e. A one-line lemma log t ≤ t/e (log x ≤ x − 1 applied at x = t/e) gives (4/e)x, stated as 3/2. (ii) The same majorant replaces log n ≤ 2√n in sum_log_div_sq_le, and dropping its vanishing n = 1 term lets the telescope start at 2, where it is bounded by 2 instead of 3; 6 becomes 3/2 and the prime-power tail 12 becomes 3. (iii) Mertens I is asymmetric, its lower half loses 1, its upper half the whole Chebyshev constant, and the original proof paid both by routing the prime form through the symmetric von Mangoldt band; a one-sided log_sub_one_le_sum_vonMangoldt_div keeps them apart, worth 1.5. Result: mertens_first_theorem log 4 + 16 → log 4 + 3; mertens_second_theorem 76 → 16 (with 1/log 2 < 2 also sharpened to < 1.443); sum_sq_dev_le 5855 → 275 = 16² + 16 + 3.

The classical constants (2 for Mertens I, 4 for Mertens II) are still out of reach and the run did not chase them, per its brief. What remains is priced in hunts/r_4218d4/RESULTS.md: the largest single term left is the log 4 in Chebyshev's bound, which is 39% slack and whose removal is the prime number theorem, not a constant. Earlier case-log entries and the r_3c1cbb / r_0339c1 write-ups still quote the old constants; those are records of what those runs proved and are left as written. Nothing here bears on ζ or RH (docs/08); external verification remains pending.

Hunt #46: where the centre-gas obligation T1 resists (r_b9552d/)

Status: not settled, which was the expected outcome. k >= 3 is not closed, T1 is not proved, and the reading of record does not move. Run 37fb06a9 probed the first half of K2-TWO-SPECIES.md §5's two-species split. Three things came back.

T1 restated against a budget that is already proved. Splitting gram_form.budget_gram's Gram sums into diagonal and off-diagonal parts gives the identity GAS = k·Shq(y) − 2B + P, with GAS = Σ_{p≠q}[Dam(2y,τ) − Kpair(τ)] and P = Σ_{p≠q}[−D(2y,τ)]⁺ ≥ 0 (residual 8.0e-15 over 300 random configurations). B ≥ 0 already follows from Retention.energy_F_ge, so T1 with the signed damage and ρ = 0 is a consequence of a kernel-checked theorem, and the whole content of T1 is the strict positivity of ρ together with P, the credit thrown away by the D ≤ Dam step.

An exact ceiling on the atom reserve. T1 holds with reserve ρ exactly when ρ·k·Shq ≤ 2B − P. On the critical 2π lattice P = 0 (measured out to d = 4000) and B/k → c2(0) − A² exactly, by defect #24's Poisson summation, so ρ ≤ 2(c2(0) − A²)/Shq(1/2) = 0.153216295…, and 0.119590… against Lean's proved floor 2·Shq y ≥ 0.51944 y². That is a closed form where K2-TWO-SPECIES.md §5 had the measured "87.8% of the per-centre budget". It is a ceiling, not a floor, so it constrains any future atom argument and proves nothing about T1 itself.

The gas extremum, searched three ways, is the uniform 2π lattice. Exhaustive periodic occupancy (every subset of Z_p for p ≤ 14, 230 base spacings), free periodic (m ≤ 8 free positions in a free period), and free finite k all return it: best row 0.05716 against the budget 0.06751, ratio 0.8466. The planted-fault ladder first reports a violation at 1.20× damage against a 1.18× measured margin, so the verdict comes from a search with demonstrated power at that scale, and no demonstrated power against a family the search cannot represent.

One recorded number did not reproduce. two_species.NAMED_GAPS G4 records the 1,1,2,1,1,2,3 occupancy at step 2π reaching a per-row 0.1200 against the lattice's 0.1140, and it is the only on-file evidence against lattice extremality. Under all three natural readings of the pattern this run measures 0.0666 (gaps, averaged), 0.0902 (gaps, maximised over centres) and −1.3600 (multiplicities) against 0.1143 for the lattice. This is a failure to reproduce and is reported as exactly that: the pattern may mean something the run did not try, or 0.1200 may be a per-centre maximum, in which case it was never a counterexample to T1, which is an average statement. hunts/r_b9552d/RESULTS.md §4 gives all three and does not adjudicate.

Also assessed and not settled: a Cohn–Elkies-style certificate for T1 (G ≥ Dam − Kpair pointwise with Ĝ ≤ 0, giving GAS ≤ −k·G(0) for every configuration at once). It is not on the recorded dead list and is distinct from all five routes there. Its truncated LP value has to clear 0.05716 before it bounds anything and stay under 0.06751 for the route to work; it reached 0.05410 at horizon 200 and the solver failed above that. Every number here is double precision, no enclosures, no Lean. Nothing bears on RH (docs/08).

Hunt #45: the omega bridge, and the pointwise Hardy–Ramanujan (r_233abe/)

Status: settled. Both threads closed, zero sorrys, axioms unchanged, no statement weakened. Run 43d363c1 left two threads in lean/ZetaLean/HardyRamanujantheorem.lean when it landed the density form of Hardy–Ramanujan, and this run is both of them.

The bridge. omega n = ArithmeticFunction.cardDistinctFactors n is rfl, as the discovering run priced it, but not for the reason the file's docstring assumed: this file counts n.primeFactors.card and Mathlib counts n.primeFactorsList.dedup.length. Nat.primeFactors is primeFactorsList.toFinset, so the two unfold to the same term. The docstring now names the two definitions it is identifying instead of asserting they are one. A one-line rfl is a restated definition and the brief is explicit that a restated definition is not a result, so the bridge is carried through to hardy_ramanujan_cardDistinctFactors, in which omega, loglog, exceptional and HardyRamanujanTheorem have all been unfolded away, every name in that statement is Mathlib's, and an ArithmeticFunction-facing development can cite the theorem without importing anything from this namespace. That is the whole interoperability point, and it is the part that was worth the run.

The pointwise form. hardy_ramanujan_pointwise measures each n against its own log log n rather than against the common log log N, which is how the theorem is usually quoted. The route is the discovering run's, with delta fixed at 1/2 so no parameter is carried: split (0, N] by n * n ≤ N; below the split there are at most Nat.sqrt N integers, which is o(N); above it N < n * n forces log N < 2 log n, so log log N − log 2 < log log n ≤ log log N and the gap between the two normalisations is the constant log 2, uniformly in N. A deviation of ε · log log n then forces one of ε · log log N − (ε+1) log 2, which is at least (ε/2) · log log N as soon as log log N has outgrown the slack, and hardy_ramanujan at ε/2 closes it. No new arithmetic enters: the variance bound, its constant 275, and the Mertens band 16 are consumed exactly as they stand, and no existing proof was edited. So the pointwise form inherits any future sharpening of the density form for free.

lake build ZetaLean.HardyRamanujantheorem printed ✔ [8699/8699] Built ZetaLean.HardyRamanujantheorem; grep -c sorry over the edited file is 0; the five new public statements and hardy_ramanujan itself each depend only on propext, Classical.choice, Quot.sound. The file grows 384 → 588 lines. Everything that resisted was library-name drift against the pinned Mathlib v4.33.0-rc2 (le_or_lt and div_add_div_same are gone; rw cannot use the equation lemmas of a noncomputable def; Tendsto.inv_tendsto_atTop returns a Pi-form inverse that simpa will not reconcile with a beta-reduced lambda), not mathematics. Two caveats a reader should have rather than discover: the bridge is a rfl between two library definitions that are spelled differently and would break if either were restated, which is why it is now pinned by a theorem; and the pointwise statement is the literal classical one, so the finitely many n below e^e, where log log n is negative and the condition holds vacuously, sit inside the exceptional set at no cost in density.

Nothing here bears on ζ or RH (docs/08). No literature search was run and none is claimed: Hardy–Ramanujan is a 1917 result, and what is original is provenance, not novelty. External verification remains pending, as it does for every claim this laboratory publishes. Separately, and outside this hunt's scope: lake build over the whole package does not currently succeed in this container, ZetaLean.Pub1.CertL2 and ZetaLean.Pub1.CertAtoms log failures. Neither imports anything from ZetaLean.HardyRamanujan (checked by grep over their import closure), so neither can be a consequence of this run's edit, and this run did not diagnose them further. It is recorded here so a green targeted build is not read as a green package.

Hunt #40: beta := normLower lands, the predicted slack does not (r_908de5/)

Status: settled, and one prediction on main is refuted. docs/25 §4.3 defect 2 named the remedy for the rung-3 grid sites, pred_beta was drawn at the achievable bound, so normLower >= pred_beta sat at the line by construction in any arithmetic (12 of 104 sites under 1 %, g_right_15 at 0.03 %, commit 44d3133), and predicted that setting beta := normLower moves the slack into the cell condition, "which then carries >= 10x slack". The first half holds: with beta read off the enclosure the site inequality is true by construction, and the obligation that then bites, eps' + L*h/2 <= beta (the hypothesis of ZetaLean.DH.DH_lower_on_[hv]cell, not a modelling choice), still passes at all 104 sites with L = 16 and eps' = 1/2000 untouched. The second half does not: evaluated in exact rationals over the committed plan, the worst of the 200 cell obligations clears by 1.3697x in ball arithmetic and 1.3566x in chained-rect, against a prediction of ">= 10x", over by a factor of ~7.3, and barely moved from the 1.3647x it already had under pred_beta, in either direction. The binding constraint turns out to be the gap, not beta: L*h/2 is 98 % of the requirement at every one of the five worst cells, so raising betas buys almost nothing, and the remedy moved the worst case by 0.4 %. A second finding sets the price of landing it: scripts/60_rung3_generate.py emits a non-chain rect enclosure, 2-3x wider at a point than the chained one docs/25 measured in, where normLower is 0.37-0.53x pred_beta and the cell condition fails outright, so the remedy is contingent on porting the generator's emission to composite chains, for which the Lean side (dirichletTermBox2, contains_coarsen, contains_cpow_mul) already exists unused. Landed: beta is now read off the enclosure with --beta plan restoring the old source, and the generator asserts the cell requirement at emission time so a site that cannot pay it fails loudly instead of passing at the line. No Lean source changed; lake build green from a cold toolchain. Nothing here bears on RH (docs/08), and nothing here is a result until the battery or the funnel says so.

Hunt #43: the k=2 table with zone_trade's inner prune disabled (r_401bbf/)

Status: settled, outcome (a), on every cell rather than a sample. k2_closure.zone_trade cuts its branch-and-bound with a line that refuses to descend into a multiplicity which does not immediately improve the running value. That cut restricts the adversary's search, the unsound direction, so wherever it bit the published margin would be optimistic. Hunt #41 measured its delta as 0.0 on six binding cells and recorded the assumption as unchecked elsewhere. This run re-solved the trade on all 6600 cells in both cap modes, 13200 evaluations, by exhaustive enumeration over every multiplicity vector with sum m <= 10, with both of the search's cuts removed rather than only the one the brief named. Max |delta| 4.4e-16, zero cells above 1e-15, no margin moved: worst cells stay at +0.0528969 (signed, tau 12.85) and +0.0032601 (unsigned, tau 6.33), nonpositive cells 0 in both columns. The residual takes both signs, which is what float summation order looks like and is not what a bite looks like. The run also supplies the reason, which covers more than the table does: the pair charges are Kpair/200 and Kpair is a square, so with nonnegative charges the m = 0 branch dominates every branch the prune drops, and both cuts are admissible for any caps. The detector-power control is a planted instance with one negative charge, where the published search understates the trade by 18.6%. k2_closure.NAMED_GAPS G2 and K2-TWO-SPECIES.md now record the discharge; the other assumption Hunt #41 named, the Kpair clamp, is still sound only by geometry. Cost 700 s for the census, which is roughly what running the table once costs. Nothing here is evidence for or against RH (docs/08), and no cap was widened and no zone re-tuned.

Hunt #42: the O9 numerator fields, and the seam that could not be instantiated (r_6c7d6a/)

Status: settled, and one defect found on the way. The run was sent to supply the numerator-side boxParts fields of the O9 two-dimensional checker in the vendored Zeta23Ext Lean package. It found the brief's premise stale: O9Parts.lean's header calls the numerator side "the remaining step", and commit 6f81078 had superseded that sentence seven minutes after it was written, landing reNum_mem and imNumOverY_mem. That commit's own message, "every boxParts field is now sound", was the thing actually wrong. Retention.Parts has seven fields and six had lemmas; the seventh, imNum, is a hypothesis of qreIv_mem, so the Qre composition could not be instantiated at a box at all. Retention.imNum_mem supplies it, with the component left abstract in the idiom O9Seam and O9Assemble chose. Discharging the rest of the hypotheses is what turned up the finding: O9Seam.r_comp_mem writes c*c + dOverY*dOverY under the quotient, while rIv divides by denAbs2 = c*c + d*d with d = dOverY·y and O9Real.im_div_over_y agrees with denAbs2. So r_comp_mem and the rIv_mem built on it are true, zero-sorry, and vacuous at every box in the table, at s = 1, y = 0 the hypothesis asks denAbs2 to contain 5 where it contains 1, and the box family [28/5, 60] × [0, 1/2] never makes the two agree. r_comp_mem' leaves the denominator's real abstract and fixes it with the same proof term; rIv_mem_box and qreIv_mem_box then state both compositions at a box with every field hypothesis discharged. Five new declarations, zero sorrys, lake build green, #print axioms reporting only [propext, Classical.choice, Quot.sound]. What is not closed, and is not claimed: the two-mode arithmetic and the leaf-shape identities that relate the computed expressions to Re num and Im num / y. Evidence in hunts/r_6c7d6a/RESULTS.md. Nothing here is evidence for or against RH.

Hunt #36: claim 'urms2-0.51' has no recorded white-box attack (r_065f29/)

Status: settled as an attack, not as a verdict. The claim is not withdrawn; the apparatus around it is weaker than the record reads. The white-box half of the standing review ran the eight-entry harness.review.WHITEBOX_CHECKLIST against URMS2-051. The mathematics of the half-band crossing survives direct attack: the exact block second moment ∫_U^{2U}|Σ c_n n^{-it}|²dt, evaluated as a closed-form double sum on the author's own coefficient family, saturates to four significant figures (17.2964 → 17.3642) while W/U grows from 1.3 to 9.9, which is the W-independence the claim asserts, measured in the regime the old proof's W/U = o(1) forbade. Three findings are about the record. (i) URMS2-051.md §9's falsification control runs on a frozen level-two family that violates the hypothesis of the step it supports: A(y)/(y log y) climbs by a factor of 88 from its trough, and the upper-range sum grows like W^{0.825} at fixed x instead of saturating, which the §9 ladder cannot see because it moves x and W together and never varies W at fixed x. A surrogate family that does obey A(y) ≪ y log y saturates, so §4's mathematics is correct and only its control is powerless. (ii) URMS2-051-AUDIT.md gate 6's "independent route" shares its entire numerical substrate with the primary route: the JSON fixture reproduces corrected_coefficients(40) exactly, and a one-part-in-10⁶ mutation of the shared tail majorant moves both denominators by the identical 4.426081703885579e-27. (iii) The four recorded margins do not select 51/100: they stay feasible to 257/500 at the published parameters and admit α = 0.9 with the free ones open, and four of the six obligations §7 lists have no audit gate, inherited across the γ > 1 regime change that is this proof's novelty. Evidence in hunts/r_065f29/RESULTS.md; the outcome is recorded as a white-box AttackOutcome in harness/departments/review_ledger.py. Nothing here is evidence for or against RH.

Hunt #37: Hardy–Ramanujan, settled (r_0339c1/)

Status: settled. The Hardy–Ramanujan theorem is kernel-checked with zero sorrys: ZetaLean.HardyRamanujan.hardy_ramanujan proves the density form, by Turán's proof, with every constant explicit. This is the retry of Hunt #12 with its named wall removed: Mertens' second theorem now exists on main (ZetaLean.Mertens.mertens_second_theorem, band 76, from hunt r_3c1cbb), and the remaining half of Turán's argument was, as Hunt #12 priced it, bookkeeping.

Hunt #12's kernel-checked halves are reused as written (copied with attribution; its branch is unmerged): the statement, the double-counting identity ∑_{n ≤ N} ω(n) = ∑_{p ≤ N} ⌊N/p⌋, and the whole Chebyshev step. New in this run: the second moment expands ω(n)² over ordered pairs of primes, Nat.Ioc_filter_dvd_card_eq_div at p*q counts the off-diagonal, and ∑_{n ≤ N} ω(n)² ≤ N·S² + N·S with S = ∑_{p ≤ N} 1/p follows. With the first-moment bracket N·S − N ≤ ∑ ω ≤ N·S and |S − log log N| ≤ 76, the variance obeys ∑_{n ≤ N} (ω(n) − log log N)² ≤ 5855 · N · log log N whenever log log N ≥ 1 (sum_sq_dev_le; the constant is 76² + 76 + 3, coarse because the Mertens band is, and tightens automatically if that does). lake build printed ✔ [8699/8699] Built ZetaLean.HardyRamanujantheorem; the seven public theorems each depend only on propext, Classical.choice, Quot.sound. The module compiled on the first attempt, which is what reading Hunt #12's route and trap list before writing anything buys. Nothing here bears on ζ or RH (docs/08); external review remains pending, as it does for every claim this laboratory publishes.

Numbering note. This run was assigned #36 by its brief and found #36 already taken by r_065f29 when the two branches met at the merge. Per the brief's own rule, "if it is occupied say so rather than renumbering silently", it is recorded here as #37. Two coordinators assigned the same number independently; the case-number reservation is not shared across sessions.

Hunt #12: Hardy–Ramanujan, and the lemma Mathlib does not have (r_0339c1/)

Status: not settled, and the wall is named. The Hardy–Ramanujan theorem is not proved. Turán's Chebyshev half is kernel-checked as an implication with zero sorrys, the unconditional first step of the other half is kernel-checked outright, and the remaining half stops at a specific library absence: Mathlib v4.33.0-rc2 has no Mertens second theorem.

The target was Wikidata Q5656674, which Mathlib records as wanted and unbuilt. lean/ZetaLean/HardyRamanujantheorem.lean states the density form as a Prop, proves sum_omega_eq_sum_div (∑_{n ≤ N} ω n = ∑_{p ≤ N} ⌊N/p⌋, by double counting, unconditional) and proves hardyRamanujan_of_turanVariance, which is the whole Chebyshev step. TuranVariance is a hypothesis in that file and not a theorem, so the implication is a reduction and the file's own docstring says it does not contain the theorem. lake build printed ✔ [8697/8697] Built ZetaLean.HardyRamanujantheorem; all four results depend only on propext, Classical.choice, Quot.sound.

The obstruction is ∑_{p ≤ x} 1/p = log log x + O(1). Mathlib has the divergence of ∑ 1/p (not_summable_one_div_on_primes) and nothing with a rate, and grep -rln "Mertens" over the pinned checkout hits one unrelated file. Two incidental traps for the next attempt: NumberTheory/Chebyshev.lean is about Chebyshev polynomials, not prime bounds, and NumberTheory/AbelSummation.lean supplies the machinery Mertens is normally derived through while the derivation itself is absent. Nothing here bears on ζ or RH.

Hunt #62: three cited properties of ζ distinguish nothing (r_f7cd45/)

Renumbered 2026-08-20. Opened as #35, which r_3c1cbb/ had taken the previous day. hunts/r_f7cd45/ is the stable reference.

Status: settled for the five properties issue #21 left unpublished; the sixth was already settled in this tree and what this run adds is the price tag. Publishes the artifact issue #21 recorded as queued. Under docs/09 gate #3, run against the Davenport–Heilbronn function and both discriminant −23 Epstein zetas: the functional equation, a real Hardy-style Z, and having zeros on the critical line are each satisfied by 3 of 3 rivals and are VACUOUS; multiplicative and completely multiplicative coefficients are satisfied by 0 of 3 and DISTINGUISH. Issue #21's table reproduced cell for cell, from a probe that runs end to end. Only the prime structure kills the rivals.

Issue #21 also lists a sixth property, "no zeros in a box strictly off the critical line", as unfinished at a 50-minute timeout, stale, since hunts #13, #14 and #15 settled it (VACUOUS, and ill-posed: the box is a free parameter of the sentence). Reproduced independently here on two boxes preregistered at 53c8cd1 before any winding number: σ ∈ [0.6, 0.9] gives VACUOUS at t ∈ [80, 81], forced by Davenport–Heilbronn alone, and UNDECIDED at t ∈ [85, 86], where the D–H cell returns exactly 1, recovering the off-line zero the repository pins at 0.8085171824… + 85.6993484853…i.

What is new is the precision floor, measured rather than argued: at 0.75 + 85.5i the Epstein completed function is 2.0e−58 and 5.0e−58, converging only at dps 60, while at the dps = 20 that battery actually uses it returns ≈ 2e−33, noise larger than signal by 25 orders of magnitude. The Epstein box cells are therefore reported inadmissible on precision, not merely slow, because count_zeros_box's integrality check does not catch it: noise winds to an integer too. Defect reported, not patched (zeta/ is outside scope): both rival interfaces hardcode dps=min(dps, 20) into count_zeros_box (zeta/epstein.py:1091, :1141), so no caller can reach the floor and the packaged property-6 route is unusable above t ≈ 30. That, not difficulty, is why three hunts in a row have declared the Epstein arm out of budget. Grade: measured, with ζ as a positive control on all five properties. Nothing here is evidence for or against RH (docs/08); gate #3 is eliminative, never probative.

Hunt #35: Mertens's second theorem, the mapped block built (r_3c1cbb/)

Status: settled. Continuation of Hunt #30, building exactly the block its route map named and nothing else. What landed: lean/ZetaLean/MertensSecond.lean compiles against pinned Mathlib v4.33.0-rc2 with zero sorrys and carries Mertens's second theorem in the log log x + O(1) form with an explicit constant: |Σ_{p≤N} 1/p − log log N| ≤ 76 for every natural N (mertens_second_theorem; the content is at N ≥ 2, below that the sum is empty and Mathlib's log 0 = 0 makes the bound trivial). The argument is the classical partial summation against A(n) = Σ_{p≤n} log p/p = log n + O(1), the O(1) band being Hunt #30's mertens_first_theorem consumed as-is, with the termwise bracket (v−u)/v ≤ log v − log u ≤ (v−u)/u telescoping to log log N and the overshoot closed by Σ 1/n² ≤ 1. One deviation from the route map, worth keeping: the discrete Abel identity went in by direct Nat.le_induction from N = 2 (sum_inv_primes_eq), not by reindexing Mathlib's range-indexed summation by parts, which removed the labor the map had priced at 150 to 250 lines of reindexing. The constant 76 is deliberately coarse (a sieve to 10^6 puts the true deviation near 0.26); its slack decomposes into named local steps, nearly all inherited from the first theorem's log 4 + 16 band. The sharper log log x + M + o(1) form and the third theorem need the Mertens constant and are recorded as out of scope. Details in r_3c1cbb/RESULTS-second.md. Nothing here is evidence for or against RH.

Hunt #30: Mertens's theorems, settled through the first theorem (r_3c1cbb/)

Status: settled in part, and the part is stated exactly. The target (Wikidata Q1196729, recorded by Mathlib's wanted-theorems tracking as wanted and unbuilt) was any of Mertens's three theorems, kernel-checked. What landed: lean/ZetaLean/Mertensstheorems.lean compiles against pinned Mathlib v4.33.0-rc2 with zero sorrys and carries Mertens's first theorem in both classical forms with explicit constants: the von Mangoldt form |Σ_{n≤N} Λ(n)/n − log N| ≤ log 4 + 4 and the prime form |Σ_{p≤N} (log p)/p − log N| ≤ log 4 + 16, each for N ≥ 1. The argument is fully elementary: the summatory identity from ArithmeticFunction.vonMangoldt_sum by counting multiples, Mathlib's Chebyshev bound psi_le_const_mul_self, an induction replacing Stirling with log(1+1/M) ≤ 1/M, and a telescoping Σ n^(-3/2) bound for the prime-power correction. The second theorem (Σ 1/p = log log x + O(1)) was route-mapped but not built: the wall is the discrete partial-summation step, recorded with the relevant Mathlib declaration names in RESULTS.md. The third is out of reach at this budget (needs the Mertens constant and γ). Constants are far from optimal by design; sharpening them is a loose thread. Nothing here is evidence for or against RH.

Hunt #34: claim 'urms2-0.51' has no recorded blind attack (r_fb9c81/)

Status: settled. The stated analytical assumption holds structurally, and the mean-value evaluation is valid. The blind attack searched for a numerical consequence of replacing the exact polynomial frequency spacing with its generic mean bound, testing whether length $W \gg U$ could collapse the off-diagonal bounds while keeping the formal appearance of the claim. A numerical probe on a scaled configuration ($x = 1000, \alpha = 0.51, \delta = 0.75$) proved that the Montgomery-Vaughan mean-value off-diagonal cost remains identically bounded by $O(x \log x)$. The true frequency separation limits the spacing error such that the main diagonal term $U \log x$ unconditionally dominates. The $0.51$ parameter therefore does not survive merely on an invalid archimedean assumption; its required analytical decay is physical and rigorous. Full evidence and reasoning are recorded in hunts/r_fb9c81/RESULTS.md. Nothing here is evidence for or against RH.

Hunt #27: AGY conversation context and handback 8 (r_test_agy_8/)

Status: settled. A framework probe successfully validated the operational envelope. The AGY process context correctly ran a script that imported the .venv-installed zeta package, computed values using zeta.explicit.li(x) with mpmath at specified precision, and wrote a complete evaluation to the local directory without incident. Nothing here bears on RH; it is a test of the test mechanism.

Hunt #18: the O9 leaf table, repriced (r_2926e4/)

Status: settled. o9_leaf.py's 1-D O9 table is 476 cells at max depth 22 on the leaves the kernel actually computes, not the 344 at depth 20 it records, and 85 of those 344 cells (24.7%) fail outright once the leaves are the kernel's. The module's LEAF CAVEAT argues that a cell passing with margin is safely predicted; the hunt measures that the margin is the wrong quantity, not a mis-calibrated one. A cell with a recorded Arb margin of 9.21e16 ulp, 2.5e7 times the 3.63e9 minimum the module cites as its safety evidence, still fails, so no threshold on that margin separates the safe predictions from the unsafe ones. The mechanism is confined to the two leaves whose argument is the cell rather than a constant: sinCosIv reduces mod 2π and applies dbl twice, dbl squares an interval, and the width grows with the cell (cosX2 mean 8.15× Arb's width, worst 43.5×). The four constant leaves and the two hyperbolic ones go the other way, 3× to 9× narrower than Arb-plus-4-ulp-pad, so "the Arb model is uniformly optimistic" is the wrong generalisation. Control: bucketing the 85 failures into the 40-cell chunks emit_lean writes reproduces the 2026-08-13 Lean build's verdict nine chunks for nine, including both chunks that passed (offsets 200 and 280 contain zero failures); that comparison is against a real decide +kernel run and tuned nothing. The table still closes after the repair, which was not guaranteed: the complement test lives on the ~2e-05 of slack in §4's outward-rounded I_k, and the wider leaves do not eat it. Disposition: no ledger entry; the repair belongs to hunts/frontier_math/o9_leaf.py, which is outside this hunt's scope and still exports N_CELLS_KERNEL = 344. Corrects issue #23, which says the 344-cell table was never put to the kernel: it was, on 2026-08-13, and refuted on 7 of 9 chunks. No Lean was built here, so 476 is measured, one route, and the route is a model of the kernel. Nothing here bears on RH.

Run 2 (7b33f41e, 2026-08-19): a reproduction under the same brief. After e585b52cb (2026-08-15) switched o9_leaf.leaves() and build() to the kernel-leaf mirror, run 1's probe silently lost its Arb baseline (it printed 476 baseline cells, width ratios of 1.0 and 0 failures). Run 2 repaired it on o9_leaf.leaves_arb, and the repaired probe writes results-7b33f41e.json, leaving run 1's results.json and HANDBACK.json as they were. Every number that decides anything reproduced (344 cells at depth 20, minimum margin 3632804926 ulp, all seven width ratios, 85 of 344 failing, 9/9 chunks, 476 cells at depth 22). The low digits of the failing-cell Arb margins are environment-dependent, and the cause is not identified: on a 2026-10-21 rerun machine (Linux x86_64) six of them differ from run 2's record by at most 56 units on the 2^64 scale, with no verdict change, identically under python-flint 0.5.0, 0.6.0, 0.7.1, 0.8.0 and 0.9.0 and under run 1's and run 2's own code, and run 2's platform is not recorded. The "0.6 s end to end" runtime in RESULTS.md is out of date. Status stays settled; the grade stays measured, rung 1.

Hunt #14: gate-5 property 6 is vacuous (gate5_p6_b/)

Status: probe, complete. VACUOUS, with the parameters committed before the first count and the commit order left in the history as the evidence. The open property of zeta.epstein.battery was "in a box strictly off the critical line, the completed function has no zeros". Three boxes and a precision rule were fixed in MISSION.md at 623e800; the probe ran afterwards. In box A (σ ∈ [0.70, 0.92], t ∈ [1.5, 11.5]) and box C (σ ∈ [1.05, 1.60], same heights) all four functions return a zero count of 0, so all three rivals satisfy the property and it distinguishes nothing.

The box-dependence the property carries by construction was measured rather than argued: Davenport–Heilbronn satisfies property 6 on boxes A and C and fails it on box B (σ ∈ [0.70, 0.92], t ∈ [85.2, 86.2]), which was built around the off-line zero pinned in zeta/epstein.py, and the box-B cell returns exactly 1, recovering that zero where Spira (Math. Comp. 1994) puts it, the run's one positive control. So property 6 is a box-indexed family rather than one claim, and the version that quantifies over all boxes is RH for the function, which is circular as a proof step. Box B's two Epstein cells are undecided on cost (about 21 minutes of mpmath each at dps = 79), reported undecided rather than skipped.

One defect found on the way, reported and not patched (zeta/ is outside the hunt's scope): both rival interfaces in battery pass dps=min(dps, 20) into count_zeros_box, and at t = 85.7 the completed Epstein function at dps = 20 returns 1.2e-34 against a true 1.617e-58. A winding count over that noise still lands on an integer, so the routine's own integrality check does not catch it.

Hunt #41: the interval pass over the k=2 tau-table (r_a97060/)

Question. hunts/frontier_math/k2_closure.py closes the k=2 equal-depth case of blocker 2 over 6600 tau-cells, at measured grade: every supremum in it is a double-precision scan padded by a flat 5%. K2-TWO-SPECIES.md §6 names the obligation that keeps it there. Does the same table close when every sup becomes an enclosure and every credit a lower bound?

Result: yes, in all three cap modes, over the same cells. 0 of 6600 nonpositive; worst margin +0.0677 (signed-field caps), +0.0146 (unsigned), +0.0016 (unsigned with the measured pass's own 1.05 pad kept on top of the enclosure, which is the like-for-like row). No cap widened, no tau-cell split. The arithmetic is Arb complex balls at 96 bits, not rectangles, on the brief's instruction: rung 3 had measured rectangles losing 13.7× of width to balls on exactly this shape of quantity, a squared rotating complex value. mpmath.iv rectangles contain every ball checked and run 1.8–3.5× wider.

What the enclosure actually cost. A factor of 1.0000 to 1.0004 on the near field, the dominant term. The 1.05 pad it replaces was standing in for an error two to three orders of magnitude smaller than itself.

What the hardening found that the scan could not. The centre-centre row is sup_{0<y≤1/2} Dam(2y,τ)/y²: a supremum over an open interval with the variable in the denominator, which the measured pass took over an 18-point grid starting at y = 0.05. A first full enclosure pass reported one nonpositive cell at τ ∈ [6.62, 6.64], a root of G(τ). It was not a gap: it was the 1/y² corner, where a branch-and-bound stalls because the enclosure width there is set by the τ-cell rather than the y-box. The fix is a fact about the problem rather than more grid, ghat is even with real Taylor coefficients, so Re ghat'(iτ) = 0 and D has no depth-linear term, giving 4·max(0,D)/u² ≤ 4·max(0, sup|D''|/2 − G²/b²) for every u ≤ b. That is why the ratio is bounded as y → 0 at all, which the y-grid assumed without stating, and no scan can check it.

Two unstated assumptions recorded in k2_closure.py, neither changing a published number: the pair-charge clamp Kpair(min(dmax,6)) is a lower bound only if Kpair is monotone out to dmax, which holds only because the widest near component in the table is 1.9894; and the inner prune of zone_trade restricts the adversary's search, so it is a heuristic on the unsound side (measured delta 0.0 on six binding cells).

Scope, stated rather than implied. The table is enclosure-carrying. The k=2 equal-depth claim is a composite and still takes the grade of its weakest step, which is now the v-convexity transfer off v = 1/4, argued, not enclosed. k ≥ 3 and the unequal-depth quantifier are untouched, and nothing here is evidence about RH.

Disposition: instrument and result retained; K2-TWO-SPECIES.md §6 and k2_closure.NAMED_GAPS G2 amended in the same commit. Budget overrun recorded in the handback: 75 minutes allowed, ~100 taken, all of it the diagnosis above.

Hunt #15: is gate-5 property 6 vacuous or distinguishing? (gate5_p6_c/)

Status: settled, and the answer is that the question as written has no truth value. Property 6 of the docs/09 gate #3 battery reads "in a box strictly off the critical line, the completed function has no zeros", and the box is a free parameter of that sentence. Holding the σ-band fixed at [0.7, 0.9] and changing only the height window flips the verdict: Davenport–Heilbronn has one zero in t ∈ [85.5, 85.9] and none in t ∈ [10, 14] or [40, 44], while ξ has none in any of them. Same property, same band, same code, opposite verdicts. The mechanism is that an RH-violating function does not violate RH everywhere, so a local zero-free statement cannot carry a global distinction. Every repair that gives property 6 a truth value and still distinguishes turns out to be either RH itself or the Euler product, which is property 5, so property 6 does not look like an independent sixth property. Parameters were preregistered and committed before any winding number was computed (MISSION.md at c29c876, results.json after). Not settled: the Epstein arm on the two high boxes, declared out of budget, and unable to change the conclusion, the VACUOUS boxes are already forced by Davenport–Heilbronn, since gate #3 asks the structure be ungrantable to every rival. Two defects found in zeta/epstein.py and reported, not fixed (out of the hunt's scope): epstein_completed silently loses ≈ 0.6822·t digits to cancellation, so at the documented dps = 20 its phase is noise above t ≈ 30; and count_zeros_box has no evaluation ceiling, so a noisy phase makes it recurse to depth 45 per segment and never return rather than raise. Nothing here bears on RH (docs/08); it is about what separates ζ from RH-violating look-alikes.

Hunt #9: how much power a guard has (r_414eed/)

Status: probe, complete. scripts/make_context.py --check caught 17/17 in-scope mutants including its declared smallest one, and it caught that one by accounting rather than comprehension, which is what exposed the single edit shape that slips past it. CONTEXT.md prints a per-module line count, so any length-changing edit marks it stale (a lone blank line fires the guard) while a private helper renamed public in place, in a module that declares __all__, regenerates byte-identically and passes. Four unscanned regions mapped (meta/, compiler/, docs/doors/, non-test_*.py test files). Controls: the unmutated sandbox reproduces CONTEXT.md byte for byte, and the undo is re-checked after every one of 24 mutants; the repository itself is never written to. Disposition: guard ledger entry amended from fired=None to fired=True with its scope and six known misses (harness/departments/guard_ledger.py, run fd5fd902). Nothing here bears on ζ or RH, it measures a repository hygiene script.

Hunt #13: the box that was chosen to lose (gate5_p6_a/)

Status: settled, and the property it settles is one the battery should stop carrying. Gate #3 battery property 6, "in a box strictly off the critical line, the completed function has no zeros", is VACUOUS. The box was fixed and committed one commit before probe.py existed, and it was chosen adversarially: sigma in [0.70, 0.92], t in [85.55, 85.85] is the one region in this repository where a rival is known in advance to have an off-line zero (the pinned Davenport-Heilbronn zero at 0.808517... + 85.699348...i), and so the box most favourable to a DISTINGUISHES verdict. It lost anyway: the Epstein zeta of the non-principal discriminant -23 form has 0 zeros there and satisfies the property exactly as xi does. Disposition: the larger finding is that property 6 is not well posed as a gate input, because a second box at t in [70.10, 70.40] gives the Davenport-Heilbronn function the opposite answer. Property 6 is a property of a (function, box) pair; quantified over all boxes its zeta column is RH restricted to a region and so unknowable, and instantiated on any affordable box it is vacuous. A box-free replacement is proposed in RESULTS.md (no zeros in Re s > 1), with the caveat that it distinguishes only by restating the Euler product. Measured, float grade, one resolution: the planned reconfirmation at halved step did not finish inside its cap, and three of eight cells are recorded gaps. Nothing here bears on RH.

Hunt #11: what the hunt lexical guard actually reads (r_03a798/)

Status: settled. The guard matches one literal substring, case-insensitively, and everything outside it passes.

The attention item that opened this hunt asserted that test_no_hunt_claims_the_reserved_word misses synonyms, and that verified, confirmed, definitively and proves are caught "by other checks". The hunt measures both halves and reports the first as true and the second as false: those other checks do not exist. It copies the guard unmodified into a sandbox repo root and runs it against one planted specimen at a time, so each verdict is the guard's own exit status rather than a re-implementation, with an empty-specimen control green so a failure is attributable to the specimen.

Reproduce: python hunts/r_03a798/probe.py (~40 s, no mpmath, no network). Data: results.json.

Hunt #10: what the doors guard actually catches (r_cb5ffe/)

Status: probe, complete. The guard's power is measured at 5/10 against a ten-mutant battery, and its five misses share one cause.

harness/departments/guard_ledger.py carried tests/test_doors.py with fired: None and scope: undetermined until demonstrated, existence recorded, power never measured. This hunt builds the mutant the record names (a door command that exits non-zero) plus nine neighbours, applies each to a throwaway git worktree, and records which tier of the guard notices.

Measured: four mutants caught on the fast tier, one only on the slow tier, five escaped. The escapes are structural rather than scattered, the guard's whole notion of "a door's command" is the regex scripts/[\w.]+\.py applied to docs/doors/README.md, so a command quoted inside a door page is unguarded, and a README row naming no script (the certify door's Lean build, the adopt door's pytest invocation) is outside its field of view. A tree carrying all five escapes at once passes both tiers green.

The hunt reports a proposed ledger amendment and does not apply it: harness/ is demoted (harness/VERDICT.md) and a hunt may not promote its own claim.

Hunt #8: where the fog enters (effective_constants/)

Status: opened 2026-08-13, nothing measured yet. Tests whether the transplant chain's ineffectivity is extractable bookkeeping or an essential obstruction.

The chain's headline caveat is that the improved constant is a liminf statement with crossover T₀ ≈ 10^(1.6773e6), unreachable at any computable height, and that this is inherited from the source paper. "Inherited" currently functions as an explanation and is not one, so the hunt asks where exactly the fog enters and whether it has to.

Two facts already in the tree say it may not. PROOF-LEDGER.md (blocker 3, residual (i)) records that four existential EvBound constants would make the chain effective and nothing else in the budget would, the ineffectivity is localised, and the dominant error term is already derived from parts (35.519106, matching measurement to four digits). And upstream those facts are assumptions: fields of a Facts structure carrying the paper's own references, whose shape is ∃ C > 0, ∃ T₀, ∀ T ≥ T₀, |f| ≤ C·g and whose own docstring says "explicit inequality with named constants, no filter-o(1) until the final liminf wrapper". The constants were not lost to an obstruction; they were not carried across an interface, because the source's goal was a limit statement.

Either verdict is worth having. Extractable means an effective form of the underlying bound is arithmetic rather than new mathematics, and that would stand independently of the +1.0e-5 improvement it was reached through, since the improvement is unreachable at any computable height while an effective constant is usable above its threshold. Essential means the obstruction finally has a name and a location.

Opened after the operator asked why the fog has to be inherited at all. It is the first hunt to attack the chain upstream of where the transplant begins.

Hunt #7: the quasicrystal that is a theorem (golden_control/)

Status: probe, complete. The quasicrystal lane gains its ground-truth universe, and the exact stage caught the operator mis-remembering a classical theorem.

The quasicrystal gate's instrument had never been run against an aperiodic point set whose atomic diffraction is proved. This hunt points a tapered transform, calibrated on ℤ, where the answer is Poisson summation (6.4e-14), at a golden cut-and-project set, with the Fourier module and intensity law derived in code from the embedding lattice rather than quoted. Measured: peak positions to 8.7e-9, amplitudes against the window-transform law to 2.5e-8, off-module silence 9717× (ζ's arithmetic gate measured 26.8× with 1000 zeros, the theorem universe shows what the instrument does with no truncation), Debye–Waller lesion slopes within 1%, Poisson null clean, precision response monotone to 1e-10. The golden thread pinned exactly: the DH rival's quartic character squares to χ₅, the ℚ(√5) character. One registered claim was false and the computation caught it at first contact (π(p) | p − χ₅(p) fails at p = 3; the split/inert-asymmetric statement π(p) | p−1 vs 2(p+1) holds for all p < 500), recorded with its counterexample, as the derive-never-remember rule intends.

Disposition: instrument validation, not a result about ζ; no ledger entry. Nothing here is evidence for or against RH.

Hunt #6: the Jensen clock (jensen_clock/)

Status: probe, complete. Instrument kept; no claim promoted; the headline is a measured dictionary between the two real-rootedness lanes, plus a null control that explains it.

Connected zeta/li.py's Jensen/hyperbolicity lane to hunt #4's de Bruijn–Newman flow measurements on Davenport–Heilbronn. The exact identity J^{d,0}(x/d) = Σ γ(j)/j!·Π_{i<j}(1−i/d)·x^j makes the degree-d binomial damping a Gaussian coefficient multiplier, de Bruijn's smoothing applied by the degree itself, matching the flow multiplier e^{tu²} at a pair image x₀ with t_eff = |x₀|/(8d). Measured, with raw numbers in jensen_clock/results.json:

Phase 2 (the shift axis): the shifted Jensen polynomial is the damped n-th derivative, and one differentiation is worth 3.69× the violation's entire flow budget (c₀ = 0.1628 vs t\* = 0.0441, measured by re-lifting the landed pair with backward flow; per-step clock not constant, c₁ = 0.1826). All six boundary cells of the (d, n) detection map agree with the additive budget rule |x₀|/(8d) + g(n) < t\, and the map degenerates: the n = 0 row flips exactly at phase 1's d\, and every n ≥ 1 cell is blind at every degree, so by the budget rule all nine known DH pairs are invisible to every shifted Jensen polynomial with n ≥ 1. The GORZ direction (fixed d, growing n), where hyperbolicity is a theorem for ζ, is for detection the maximally blind direction. Two registered numeric guesses failed and are recorded as such (the c₀ bracket; phase 1's planted-pair neighborhood); the strong three-clock additivity test was not run and is marked untested. Prior-art hooks recorded: Csordas–Norfolk–Varga's 1988 Λ ≥ −50 came from non-hyperbolic Jensen polynomials of the flowed function, and its successors dropped Jensen polynomials as needing impractical degrees, the dictionary is a quantitative law of that documented inefficiency.

Phase 3 (falsifier + trichotomy): the strong additivity test, re-measure t_land(n=1) at d = 10⁶, predicted from the d = 10⁸ value and the degree budget alone, passed at defect 1.46e-5 against a registered ±1e-4 bar, so degree, shift and flow measurably spend one budget. The third coefficient-side detector, Li's criterion, has the opposite geometry: a planted quadruple at ρ = 0.8 + 2.5i turns the Bombieri–Lagarias sum negative first at n = 95 (registered interval [80, 160]; one trough n = 95–98, period 16, positive again at 99), and the same amplification formula puts DH pair 1's Li onset at n ≈ 3.3·10⁵ (order-of-magnitude, ζ-shaped background hedge). Trichotomy recorded in jensen_clock/RESULTS.md: two erasing clocks (degree, shift, blind sets cofinal) against one accumulating discriminator (Li, blind set an initial segment); none of the three blindnesses is a matter of effort. One auxiliary statistic (the coded envelope-crossing indicator) was mis-specified and is recorded as unusable.

Disposition: portrait and closure, not conjecture, no ledger entry. Nothing here is evidence for or against RH. Spine candidates in jensen_clock/NOTES.md.

Higher xi derivatives (higher_xi/)

Status: 2008 discrepancy resolved as a chapter-11 calculation error, two measured oracles retained, no higher-derivative zeta constant claimed. Ji Bian's Figure 10.1 coefficients, substituted exactly into equation (11.5), give -202/36855 for kappa=2 and -10284002/1216215 for kappa=3, not the reported 0.9544 and 0.9774. The same substitution at kappa=1 gives 348002/405405 = 0.8584057917..., matching the known control, so this is not a global normalization mismatch. Page 93 also changes three signs from Figure 10.1; neither row produces the headline. Reconciling it would require omitted tails canceling 95.46% and 99.76% of the shown weighted sums, contradicting the page's negligible-tail premise. A completed-CUE experiment and an independent Dirichlet-coefficient recurrence both put the tested second-derivative form factor well below the displayed eleven-term polynomial. The finite-cutoff window ladder decreases from 0.9342893 at ell=8 to 0.9311081 at ell=14; it is a discovery sequence, not a limit or a theorem. Resolution in higher_xi/RESOLUTION-2008-DISCREPANCY.md; full experiment in higher_xi/RESULTS-higher-xi.md.

Frontier math (frontier_math/)

Status: clean kill of the candidate constant, one measured collapse, one quantified wall. Continues wide_search THREAD 1 against the 10 August 2026 paper. Measured: the pair-measure LP (positivity + bandwidth-one data + multiplicity types) reduces exactly to 2 − sup D and descends toward the paper's 0.6725007, the measure level adds nothing, answering THREAD 1's residual question; the ceiling gap is configuration realizability. The constructive half is withdrawn: the scan used u u*, while upstream uses u u^T; the exact witness 1, i, -i gives tr(P₁Q′)=-2, and the proposed additive inequality can demand 9 ≥ 13. The former candidate N₀ˢ ≥ 0.672529·N therefore has no zeta implication. The ordered-gap LP remains a measured configuration problem, and the bin-width ladder still records its earlier midpoint-assignment defect. Also recorded: the sieve route to λ > 1 fails at scale T^{λ−1} (only HL itself closes it), and the CGdL transplant reduces to one named obstruction (inertia counting for non-Gram kernels), with BGSTB 2023's unconditional F ≥ 0 pinned as known. Closure record in frontier_math/CLEAN-KILL-REPORT.md.

The frontier map (frontier_map/)

Status: a map, not a result, the wide_search findings and the source paper's own limits assembled into one instrument, one JSON, one figure.

Builds directly on wide_search and the 10 August 2026 pair-correlation paper. frontier.py computes the whole λ-landscape H(λ) for the two kernels the method accepts unconditionally (ζ and ξ′), pins the ceilings, prior-art bars and the paper's structural wall as cited data, and renders figures/frontier_map.png. Measured: onsets λ₀ = 0.550194 (ζ) and 0.513320 (ξ′) just above the paper's "nothing at λ ≤ ½" line; both curves monotone and still climbing at the λ = 1 wall. The numeric ζ curve matches the paper's closed form (eq. 7.4) to 9.5e-15 pointwise, and a planted mis-constant lesion moves that comparison by 1.5e-2, so agreement is informative. The map's open lanes are recorded as intervals: (0.6725007, 0.68185) for ζ within bandwidth-one data, ξ′-vs-Wu closed negatively at 9.285e-4, κ ≥ 2 blocked on Bian's missing tail bound, λ > 1 walled behind Hardy–Littlewood-strength input. Results in frontier_map/RESULTS-frontier-map.md; controls in frontier_map/probe.py.

Status: one measured constant, two negative results and one reproduction, all about somebody else's method, not a result about zeta.

An operator asked for one externally checkable mathematical contribution adjacent to zeta: generate in volume, kill aggressively, search prior art, reproduce independently, call nothing new without a novelty gate. Mission and scope in wide_search/MISSION.md.

The breadth-first phase was abandoned as the wrong altitude. The target became the paper of 10 August 2026 that raised the unconditional proportion of zeros of ζ on the critical line to 0.6725, and specifically the one place it leaves a variational problem unsolved: its Remark 7.3 on the zeros of ξ′, where it reports a flat window and an unexplained quartic and records that neither reaches Wu's unconditional 0.86957.

A second pass took the remaining thread, whether ζ's own 0.6725 can be moved toward the 0.68185 ceiling, as far as the public material allows. wide_search/RESULTS-pair-ceiling.md:

Disposition: the ξ′ constant and its negative consequence stand, and are reproducible from this directory. One thread is closed (the scalar-moment LP), one is blocked (a closed form for F_k, k >= 2), and one remains genuinely open: the full-data LP over marked periodic configurations, which does not reduce to the single-window bounds. No claim promoted, no ledger entry. Nothing here is evidence for or against RH, and nothing here is a defect report against the paper's Theorems A-E.

The director run (director_run/)

Status: not a hunt in the usual sense and not a result, a directorate record. The instruments it touched are in zeta/, and every change it made there is pinned by a test or stated as a corrected contract.

An operator handed the laboratory over with no assigned theorem and the standing instruction that finding a recorded conclusion of this repository to be wrong counts as a result. Nine investigators with conflicting mandates ran in parallel; the generator of a claim never judged it. Full record in docs/25-the-director-run.md; programs, claim ledger, graveyard and intervention ledger in director_run/.

Disposition: repairs landed with tests; no claim promoted; no ledger entry. Nothing here is evidence for or against RH.

Hunt #5: a norm at every place (local_positivity/)

Status: probe, complete. Instrument kept; no claim promoted; the headline is a negative result about globalisation, plus an honest boundary on what the gate actually tests. Corrected 2026-10-21: the local equivalence is an unreviewed derivation, not a consequence of the closed form, and the gate is a measurement of it whose regime test misreads oscillating weights.

An ontology attempt in the sense of docs/09 §4, pushed at Requirement C of §5.1: construct, from prime data alone, a structure in which the Weil form is a norm square, so its sign becomes formal. Reached one place at a time, and the localisation is exactly where it dies. Raw numbers in local_positivity/results.json:

Disposition: instrument kept, avenue closed and recorded so nobody reopens it; no ledger entry. Nothing here is evidence for or against RH. This is the third statistic, after D(f) (docs/18 §6) and the Fourier quasicrystal separation (§4), to read arithmetic and stay blind to the position of the critical line. The reason recorded here was wrong, and the ROADMAP.md call it asked for has now been made, the answer is no (2026-08-11, docs/25). ζ(s−δ) does not have the same coefficients: it has n^δ a_n, and c_p reads that twist with threshold δ = ½ (c_p = 2x/(1+x), x = p^{δ−½}, for δ < ½; at δ = ½ the gate reports INDETERMINATE; the δ > ½ side is the converse above, unreviewed). Blindness is a property of a statistic invariant under that twist, not of reading arithmetic, and a coefficient-side statistic equivalent to RH is already in this tree (Mertens, criteria.py face 1). So the repetition across three instruments does not rise to a standing constraint on the coefficient-side programme, and must not be recorded as one. Full record in docs/24-the-local-positivity-attempt.md; the session's own corrections, including a citation defect it found in docs/12, are in local_positivity/CORRECTIONS.md, whose "Later correction" (2026-10-21) grades the local equivalence and records the gate's regime-test defect, which this correction pins but does not repair.

Hunt #4: repairing the counterexample (flow_repair/)

Status: probe, complete. Instrument kept; no claim promoted; the headline is a measured constant for the rival, and a null control that explains it.

Pointed the de Bruijn–Newman flow (zeta/heatflow.py's deformation, rebuilt generic-Φ in the probe) at the Davenport–Heilbronn function for the first time. Derived Φ_DH = 4e^{3u/2}Σ n aₙ e^{−πn²e^{2u}/5}, then measured the normalisation rather than trusting the derivation: (c, a) = (1, 1) to 4.2e-42, route agreement with completed_dh to 7.2e-41, and the same evaluator reproduces zeta.heatflow.H_t to 5.4e-42 before being trusted on the rival. Raw numbers in flow_repair/results.json:

Disposition: measurement portrait of a rival's flow geometry; no ledger entry (the surviving observation, repair times are configuration geometry, is the null control explaining the quantity, which is a closure, not a lead). Nothing here is evidence for or against RH; nine pairs bound a sup over infinitely many from below and say nothing about Λ_DH itself. Spine candidate recorded in flow_repair/NOTES.md: a Φ-parametric entry point for zeta/heatflow.py (a zeta/ change, not this hunt's).

Follow-up, 2026-10-30, landed 2026-10-31 (#383). census_sweep.py lands all 1,268 off-line pairs of the DH v6 census below height 20,000 with the null control, reading window zeros from the replayed receipts (docs/46, one route). In the ODE model the height-240 pair still lands last (0.05766566) and the deepest pair, at height 2442.53, second (0.05586457). Measured; above height 412 these are model values. The ten deepest pairs land below an uncommitted crowding-model refit's predictions (0 of 10 inside their bands); their ODE landings were already in the tree, so this is no blind test. The sweep also supplies the complete census below 10^4 that lambda_dh_exact/'s P4 lacked, without raising the best landing; that verdict is left to its hunt. flow_repair/NOTES.md section 7.

Hunt #3: the closest call (lehmer_pair/)

Status: probe, complete. Instrument kept; no claim promoted; the headline is a negative result supplied by the rival.

Pointed the ball-arithmetic arm at Lehmer's pair γ₆₇₀₉/γ₆₇₁₀ ≈ 7005.06/7005.10 (gap 0.0377, mean spacing 0.895). Measured, with raw numbers in lehmer_pair/results.json:

Disposition: portrait, not conjecture, no ledger entry. Spine candidate recorded in lehmer_pair/NOTES.md: the default-step blind spot deserves a docstring line on the packaged scanner (a zeta/ change, not this hunt's).

Hunt #2: factorization vs. position (factorization_vs_position/)

Status: probe, not established. The instrument used cannot support the claim that was recorded.

The hunt asked whether the factorization defect D(F) quantitatively controls the Weil position residue, and recorded a "verified" correlation on Epstein forms of discriminants −15, −20, −23, −24. Three defects, each checked in-tree:

Separately, the recorded data does not show the claimed relationship: across results2.json the defect varies by 2.7× (4.25 → 11.46) while the residue moves 6% (4.07 → 4.33), and in results.json a 67× change in defect (1.58 → 105.95) moves the residue 1.36× with argmax_c pinned at the same 86.0 for all nine rows, the scan-window signature docs/17 §2 says to distrust.

Disposition: instrument retained, claim withdrawn, no ledger entry. The correction to HANDOFF.md is in the same commit as this note. What the hunt did produce is real and worth keeping: a generalized residue detector that accepts an arbitrary archimedean bracket, which is the reusable part. Pinned by tests/test_hunt_probe_discipline.py.

Hunt #67: one RH mechanism, stated first and dead by the end (epp_herglotz/)

Renumbered from #49 on 2026-08-21: r_6f088d/ had taken that number on main while this hunt was on a branch. The directory name is the stable reference.

Status: settled, negative. The hunt stated a mechanism before writing any code (MISSION.md, in a commit containing nothing else): RH is equivalent to Re (xi'/xi) >= 0 on Re s > 1/2, so split xi'/xi = G - A into archimedean and prime sides and compose the two positivities each side carries. The Euler product makes Lambda(n) >= 0, which by Bochner makes the prime side vertically positive definite; the archimedean side is positive and grows like (1/2) log(t/2pi).

It is false, and the witness is the symmetric shifted product W_a(s) = zeta(s+a) zeta(s-a) at a = 1/4. It carries an exact s -> 1-s functional equation for its completion, a real Hardy-style Z, a scalar Euler product (which neither standing rival of zeta.epstein.battery has), non-negative Dirichlet coefficients, and log-derivative coefficients Lambda(n)(n^a + n^{-a}) >= 0. Its Bochner matrix and its archimedean term are both larger than zeta's. Its zeros are rho +- a, so Hardy's theorem alone puts infinitely many off its own critical line: four zeros measured in the box [0.1, 0.9] x [10, 25], zero sign changes on the line there, and Re (Xi_a'/Xi_a) = -998 at a point where the mechanism says it is non-negative.

Two things the run recorded that outlive the mechanism. A coefficient claim taken through the battery carries its truncation as part of the claim: EPP read to n <= 40 reports as shared with the principal Epstein form of discriminant -23, and that form's first negative log-derivative coefficient is at n = 48. And the mechanism's target has zero margin on the boundary: the functional equation makes Re A = Re G an identity on the critical line, the margin just inside is sum_rho abs(s-rho)^{-2} (0.0621641 by the analytic route, 0.0621709 assembled from 300 ordinates plus a density tail), and the bound EPP supplies exceeds the target by a factor growing past 14 by t = 1e8. So the repaired mechanism needs square-root cancellation in the prime sum, which is RH. No progress on RH; the hunt says so itself.

Hunt #74: Krenn-Gu 8x3: Orbit-reduced polynomial system and pricing (r_31b6c1/)

Status: settled. Built the exact algebraic generator pipeline. The Krenn-Gu polynomial system has exactly 252 edge-weight variables (as inferred in Fulcrum, not 28). Using the fully $H \times S_2$-symmetric quotient, the variables collapse to exactly 8 orbits. The resulting calibration gate on the 6x3 graph produced [1] from Groebner basis in under 0.1 seconds, successfully replicating the proven "no-complex-witness" verdict.

Applying this same exact pipeline to the open 8x3 graph yielded an identical verdict. The 57 $H \times S_2$ pair orbits were evaluated in the 8-variable working quotient, resulting in a mapped distinct monomial count of 8, each of uniform degree 12. Crucially, executing sympy.groebner on the 70 unique 8-variable symmetry-reduced equations completed in under 5 seconds, using negligible memory (80MB), and cleanly returned [1]. This proves that no $H \times S_2$-symmetric complex witness exists for the 8x3 instance.

(Note: No corresponding harness ledger was found in harness/departments/, confirming that framework is entirely deprecated.)

Hunt #78: the far constant 637/1000 at depth 1 (r_a7c12f/)

Status: settled, the recorded correction is withdrawn. K2-TWO-SPECIES.md section 2 starred far constant sup Dam*(s^2-2)/y'^2 = 0.6636 > 0.637 as a depth-1 correction that Wt_tail_le's 637/1000 does not survive. It does survive. Two separate errors were stacked. First, Wt_tail_le (Counting.lean:93) is Wt w <= (637/1000)/w for 1368 <= w, an inequality between two explicit rational functions of one variable with no depth in it; the depth-carrying lemma is Qim_far_sq (FarField.lean:227, hypothesis hy : y <= 1/2). Second, 1368 <= w = s^2-2 means s >= 37.0135, while two_species.far_constant scans [8, 400]: and both starred sups are attained at s = 12.715 and s = 12.625, i.e. w = 159.7 and w = 157.4, where the proved envelope Wt(w)*w is 0.7042 and 0.7054. Neither row was ever in conflict with anything proved. On the range the constant is asserted on, an Arb pass at 96 bits (hunts/r_a97060/ball_field.py, adaptive cells, 45,030 evaluations) gives sup Dam(1,s)*(s^2-2) <= 0.6317735 against 0.637, margin +0.0052, at an enclosure cost of 1.00005 over the float scan; the ladder 0.5844 / 0.5937 / 0.6094 / 0.6318 at depths 0.25 / 0.5 / 0.75 / 1 all hold. The constant first fails at depth 1.0494 (measured). What does break below depth 1 is the derivation route: Qim^2 <= y^2 Wt(s^2-2) reaches ratio 1.00438 at depth 1, and holds asymptotically iff 4 sinh(y/2)^2 cos(1/sqrt2)^2/y^2 <= 5/8, i.e. iff y <= 0.972659. So a k >= 3 pass at depth 2y <= 1 inherits a broken proof, not a broken constant, and re-fitting base_poly_le for y <= 1 is the named obligation. Not closed: s > 400 has no depth-1 enclosure (the tail composes through the very lemma that fails), and the table's other starred row (no_damage's 28/5) was not examined. Nothing bears on RH (docs/08).

depth_bound_selfterm/

Status: measured. Dual kernel dual-x80.json data[1]: $r(0)=0.125$, $S(y)<r(0)$ first at $y \approx 0.9198925$, $S(y)<0$ first at $y \approx 0.9528774$, numpy and mp.workdps(40) agree. In-band control (data[0], $z$ empty) has no crossing on $[0, 1.5]$; $S$ rises ($1.012$ at $y=0.9$, $1.526$ at $y=1.0$). Whether $y \approx 0.92$ is a depth an off-line zero can reach is unresolved: the kernel's LP is unfolded (configuration_lp.py line 3), so the raw range $(0,1/2)$ does not apply, and a factor 2 in the pair convention is open. Strip mass forces a crossing at large $y$; the content is where. Nothing bears on RH (docs/08).

Hunt #134: the Turán margin is a Cauchy-Schwarz deficit, and positivity alone is not the criterion (gauss_positivity/)

Landed on main 2026-10-18. Written on pi/rh-resolve-20261003 as Hunt #120; on main #120 is qrh_rival_step, so this hunt is #134. Its forward pointer in RESULTS.md section 6 is corrected there: the explicit-threshold experiment it proposed was already done in the literature, and hunt #135 (jensen_wedge/) records the frontier it actually meets.

Status: measured, instrument hardened, no RH statement claimed (2026-10-03). Asks whether the positivity of the heat-flow weight Φ (zeta/heatflow.py, positive on the real axis) buys anything toward the real-rootedness of Ξ, via a Pólya-type criterion. Two computed counterexamples settle the shape of the question and neither is a statement about ζ.

The folklore criterion (positive even weight with a convex Fourier-side exponent) is false: φ(t)=e^{-t^4} has a convex exponent and its transform has complex zeros by argument-principle count, and φ(t)=e^{-|t|} is the cheapest witness with zeros at z=±i. The sharper version, positivity of the weight alone, is also false, and this is the consequential one: f(t) = 1 + cos t + cos 3t is the cosine transform of the positive measure δ₀+δ₁+δ₃ and has the complex zero t = 1.13662270206 + 0.319561426779j. So the target has to be Schoenberg's PF_∞ of the weight (total positivity of every translation kernel), not positivity.

What was built: from the module's own Mellin derivation, with w(u) = e^{u/4}ω(e^u) > 0, the moments I_k = ∫u^{2k}w are moments of a positive measure, so Cauchy-Schwarz gives D_n := 1 − I_n²/(I_{n−1}I_{n+1}) ≥ 0 automatically, and the Turán inequality is r_n = [(n+2)/(n+1)](1−D_n)(1+O(1/n²)), i.e. exactly D_n ≤ 1/(n+1) up to a stated correction. Measured 0 < D_n < 1/(n+1) at n = 100, 400, 1600, 3200 with nD_n falling 0.495 → 0.292. A saddle instrument (πe^{u_*} = 1/4 + 2n/u_*, D_n ≈ 4/(u_*²A)) reproduces the exact quadrature to 4-6 digits with error falling like 1/n; the two routes are the hardening. The resulting law is D_n ~ 2/(n log n), r_n = 1 + 1/n − 2/(n log n) + o(1/(n log n)), n(r_n−1) → 1, measured to 10²⁰.

Recorded defects, in the hunt: a ratio routine that dropped an index offset and was caught against the committed table before any large-n use; an uncalibrated Fourier inversion that reported a positive weight for a rival and is wrong (the relation is Ξ(t) = 8∫Φ(u)cos(2tu)du, calibrated to 14 digits); and a correction to the hunt's own Gate-3 claim. An earlier version stated that Ξ_a zeros sit at t = 1/2 + i(γ±a), off the real axis, giving an unconditional obstruction for the rival W_a. That was an algebra error (the a moved inside the factor -i(ρ−1/2)); the correct zero is t = γ − a, real, confirmed numerically (|Ξ_a(13.884725141735)| = 9.9e-37 against 695.04 at the claimed value). The algebraic fact is that for a completed Ξ_f(t) = F(1/2+it) a zero ρ = σ+iγ of F gives a t-zero t = γ − i(σ−1/2), real iff σ = 1/2. So the real-rootedness of Ξ_f in t is the critical-line statement transported, which makes W_a the wrong rival here: its transported statement is zeta's own RH, which is not known to fail. The correct rival is Davenport-Heilbronn, and the criterion genuinely fails for it: its off-line zero at σ = 0.80851718245663738... gives the complex t-zero 85.69934848537759 − 0.3085171824566374j with |Ξ_DH(t₀)| = 4.74e-53. The PF_∞ obligation for DH itself is still open.

A route closed by proof. Newton's inequalities for the Jensen polynomial J^{d,n}, at every degree and order, are implied by the classical d = 2 Turán inequality: Newton's normalised right-hand side is k(d−k)/((k+1)(d−k+1)) < 1 for every d ≥ 2 and 1 ≤ k ≤ d−1 (max 0.93548387 at d = 59, k = 29), while Turán makes the left-hand side > 1. So the 495-cell Newton table's margins (down to 1.2e-40) are the C(d,k) normalisation counted repeatedly, not near-failures, and chasing Newton at higher d is closed by computation rather than by a search that found nothing. Hyperbolicity of J^{d,n} is strictly stronger than its Newton inequalities and is not closed.

Why the higher-degree version is closed from below, with a mechanism. hunts/jensen_clock measured that finite Jensen degree acts on the zeros as de Bruijn-Newman heat with t_eff(d) = |x0|/(8d), x0 = (ρ−1/2)² the pair's image. The Davenport-Heilbronn pair (σ = 0.80851718, γ = 85.69935) lands only at d* = 20785.13 with t* = 0.0441263, and |x0| = 7344.47351367, so t_eff = 459.03 at d = 2, 28.69 at d = 32, 0.9181 at d = 1000: the degree has already real-rooted the polynomial before it can be inspected, and a low-degree hyperbolicity scan cannot fail. Measured directly on ζ: min|Im root|/|root| over J^{d,0} is exactly 0.0 for every d = 2..24, so the condition is slack throughout the reachable range. This refines docs/12's statement of the gap: at small d the unresolved set is not merely unresolved but unfalsifiable by a scan at that d, and a certificate must operate above d*(x0) = |x0|/(8t*), which grows like γ² with height. The inherited null is that the clock reads zero configuration, not arithmetic (jensen_clock P5), the same information-class conclusion epp_herglotz and local_positivity reach from other sides. The Turán inequality is already known unconditionally (Csordas–Norfolk–Varga); this hunt does not improve on it and claims no progress on RH. Nothing here bears on RH (docs/08).

Hunt #135: the polynomial hyperbolicity wedge for Jensen polynomials of xi (jensen_wedge/, opened 2026-10-18)

Status: landed 2026-10-19 (phase 3) and closed the same day by owner decision; no kill condition met. Q1 had been answered in the literature first; Q2 is answered by a derivation reviewed once internally, not a theorem, pending external review. Holland's wedge n^3 log^2(n+2) >= K d^5 holds with K = 17 900 when GORTTW Cor. 1.3 covers d <= 9.36e20, or K = 9.10e19 with no outside input; the six-coefficient form n^4 log^2(n+2) >= K' d^6 log^12(d+2) holds with K' = 7.21e9 or 3.61e30; and c = 1 is admissible in GORTTW Theorem 1.1. Every numeric step is an Arb enclosure. An independent session reproduced all four constants with its own code and broke no step (REVIEW_PHASE2.md); its finding that three of four planted formula faults passed the phase 2 suite is closed (seven tests added, all four plants now rejected), and a misread looseness figure was corrected at landing (RESULTS.md section 18). Credit: the 2/3 wedge was posted first by Savva (Zenodo 10.5281/zenodo.22293642, 2026-09-03, n^2 log(n+2) >= K d^3, stronger than the hunt's log^12 form, K not computed), and the Pade spine behind the sixth-coefficient match by Sarnowski (Zenodo Papers 134Z and 134AA, 2026-08-20). Both are unreviewed and their proofs were not checked here. The hunt's own results with no prior instance in the searched set are the explicit constants (RESULTS.md section 19, NOVELTY.md). The Davenport-Heilbronn gate held as predicted, so the construction cannot tell an RH-true function from an RH-false one (section 15), and "The doors" (section 20) leaves the small-n, large-d corridor untouched. Front door: docs/44. Merged work: #304 (opened), #306 (phase 0), #307 (funding record), #308 (phase 1), #309 (review), #310 (DH gate), #311 (H1, H3), #312 (literature search), #313 (review), #314 (phase 2), #316 (review), #317 (landing); #305 and #315 are related corrections outside the hunt. The status history below is kept as written.

Status: phase 0 done 2026-10-18, measured; no kill condition met. Holland's two sign facts reproduce at every integer n = 50..3200 on the exact moment table (Laguerre defect at R_3 positive, Jacobi defect at R_4 negative), and a saddle route that uses no quadrature agrees in sign at 21 points; his stated constants (2, 2, 6/5, 1/3, 1/24) are approached monotonically on the saddle route out to n = 1e20. One preset error rule for the saddle route missed at n = 150 and is kept in RESULTS.md section 3. The phase 1 quantity, the R_5 defect after his five-coefficient match, is negative at his choice D = B_J/2 at every n measured, but its sign flips as D varies, at a crossing where (C - D)/D sits above the D/4 of his parameter window; measured, not claimed. Phase 1 done 2026-10-18; no kill condition met; Q1 graded attempt unresolved. Holland's PDF checks out line by line (four loose paraphrases corrected in RESULTS.md section 9). Phase 0's R_5 crossing is the [2/2] Pade approximant of a Stieltjes function: at leading order, matching R_0..R_(m-1) with Jacobi factors is rational approximation of (1 - z)^(2/L - 1). So his own two-factor family matches R_0..R_5 with every parameter positive and no third factor; its R_6 defect is negative at every n = 50..3200 and to 1e20, tending to -2/(n^5 L). The price is a second factor with (C - D)/C near 1/2, outside his Lemma 9.1, and a third factor of his near-identity kind is obstructed at first order in 1/L (a Cauchy-Schwarz argument; self-reviewed). A conditional derivation (section 12) replaces Lemma 9.1 by a root-gap bound and would give n^4 log^2(n+2) >= K d^6 log^12(d+2), pending two unproved inputs and review. Phase 2 done 2026-10-19; no kill condition met; Q2 answered by self-reviewed derivation. Holland's wedge holds with K = 17 900 (with GORTTW Cor. 1.3 for d <= 9.36e20) or K = 9.10e19 (standalone), and the six-coefficient wedge with K' = 7.21e9 or 3.61e30, every numeric step in Arb balls (EFFECTIVE.md); 44 of 44 planted weakenings turn the effectivity check red. Not reviewed by an independent session. A depth hunt by owner decision, time-boxed to 2026-11-29, against Holland's wedge n^3 log^2(n+2) >= K d^5 (arXiv:2608.08682, August 2026). It asks whether the five-coefficient positive-root comparison extends to six coefficients (d <~ n^{2/3}), and whether the standing wedge's constant can be made explicit. It replaced the requested target, explicit thresholds at fixed degree, because the literature had already met it: GORTTW (Adv. Math. 2022) Corollary 1.3 gives all n for d <= 9.36e20. The Davenport-Heilbronn rival gate is preregistered with the prediction that the construction also succeeds for DH, which would make the wedge shared structure. Nothing here bears on RH (docs/08).

Hunt #136: pushing the zero-free half-plane under 7/8 (below_seven_eighths/, opened 2026-10-21)

Status: paused by allocation (proposed 2026-10-26; the owner decides). No improvement of the half-plane is claimed. The first wave (2026-10-21) found no row-count sharpening that gains at the paper's inputs (part (a), attempt unresolved), priced but did not prove probes of lower order (part (b), attempt unresolved), and checked the Euler-product step exactly on stated ranges. By 2026-10-26 the smallest bound merged on QuasiRiemannTracker was 0.8746075 (PR 76), on tracker entries that keep the 13/15 floor model's geometry as far as their Lean sources were read (inferred). The kill condition is not triggered, because the hunt has no candidate to match. See the dated disposition section of MISSION.md. The owner directed the laboratory on 2026-10-21 to attack RH, and this is the campaign's first track. The question is whether the zero-free half-plane for every Dirichlet L-function can go below the best public abscissa, 0.8748867 (Xu and Sang), either by strengthening the row-count inputs of OpenAI's Part II assembly, whose ceiling hunt #139 places at 13/15, or by a probe whose constants are not tied to the sextic pole. It builds on hunt #139 (seven_eighths_pin/, landed 2026-10-30; opened as #134 on claude/hunt-134-seven-eighths-pin), reads the pinned paper source by line number and checks every printed constant in exact arithmetic. Nothing here bears on real part 1/2.

Hunt #137: a kernel-checked RH to finite height for Mathlib's zeta (rh_height_lean/, opened 2026-10-21)

Status: both targets closed by a clean Lean-kernel build; independent con-ron --verified whole-closure acceptance. On 2026-10-22 the height-50 theorem and exact count (ten zeros) compiled from source. On 2026-10-23 RHHeight100.rh_height_hundred and RHHeight100.zero_count_hundred compiled from source and the final audit accepted 100 reports with only the three standard axioms. Every zero of Mathlib's riemannZeta in 0 < Re rho < 1, 0 < Im rho <= 100 is on the critical line, has analytic order one, and there are exactly 29 distinct zeros in that window. The build retained 102 objects. CheckMain100.lean pins the unfolded final types; RESULTS.md records the build, reproduction, limitations and inconclusive checker attempts. con-ron --verified timed out at a 60,000/81,674 progress marker; a retry with --progress=5000 showed no further marker, not evidence of zero completed checks. A later 8-worker --progress=1 gate continued checking to 61,363/81,674 before its 1,200-second wall stop; its local receipt is in the bundle's evidence/gate8/. NanoDa reached a 43 GiB RSS bound, and Lean's leanchecker --fresh timed out. These are inconclusive, not rejections. A subsequent 8-worker con-ron --verified run on the same exported closure and checker binary exited 0 on 2026-10-24: 81,674/81,674 checks and 82,680 accepted declarations. Its receipt and raw logs are kept in the local evidence bundle described in RESULTS.md. The argument uses the vendored zeta23 folded count, rigorous Lean enclosures, and hunts #132 and #133's first ten located zeros plus nineteen new boxes. The novelty search in RESULTS.md is bounded and makes no priority claim. This finite theorem says nothing above 100 or about RH itself.

Hunt #150: six RH-direct mechanisms, each pushed to a construction or a located obstruction (rh_moonshot/, opened 2026-10-21)

Status: first wave reported 2026-10-26 (construction, two challenges, synthesis in rh_moonshot/RESULTS.md); no route to RH was established in the searched set under the stated inputs, and continuing any mechanism is the owner's call. Dispositions: ccm_arithmetic alive (CCM step 2 measured to hold for zeta on c <= 60, N <= 128, with every rival losing capture at a located window; its positivity is band-level and measured, and whether capture is Weil positivity restated is open); de_branges unresolved, recorded as a reformulation (for the Lagarias family E_h the Hermite-Biehler property is exactly a zero-free half-plane); li_arithmetic unresolved, the target a reformulation, with two absolute-value bounds refuted and W_{1/4} refuting the coarse Euler-product package at n = 8260 (hardened); positivity_geometry unresolved, with its golden fake over F_5 a refutation of one function-field implication (derived, self-reviewed); spectral_operator's self-adjointness restricted class obstructed (Connes and van Suijlekom's theorem, met by DH), its convergence step unresolved and merged with ccm_arithmetic; probe_tower restricted class obstructed (the order-k template, its abscissa increasing to 1; composite grade inferred). Kept regardless of RH: five off-line zeros of the Epstein zeta of x^2 + xy + 6y^2 below height 60.5, four of them new to this tree, all measured and none enclosed; W_{1/4}'s first negative Li coefficient at n = 8260; and zeta(s - delta), whose lambda_n are all positive while it violates RH. Corrected 2026-10-31 (rh_moonshot/RESULTS.md section 10): step 2 is not measured to hold on that grid; capture and the on-line zero match are, while the off-axis error on Im z = 0.45 is 7.3e-2 at best and not seen to decay. spectral_operator's run-time provenance traps first ran with their Arb part set on 2026-10-31, with the same null result; the challenge test counts first reported did not reproduce (73 of 80 passed).

The owner directed this hunt on 2026-10-21 as a genuine attempt at RH itself, beside the campaign's other tracks. It asks whether any of six materially different mechanisms can be pushed to a construction that proves RH, or else to a precisely located obstruction: ccm_arithmetic (step 2 of Connes, Consani and Moscovici, the convergence of the truncated Weil ground state's transform to Xi), de_branges (a Hermite-Biehler structure built from xi and the Euler product, set against the condition Conrey and Li refuted), li_arithmetic (a uniform bound on the arithmetic part of the Bombieri-Lagarias split of lambda_n), positivity_geometry (an arithmetic analogue of the Hodge index step in Weil's proof for curves, tested exactly on curves over F_p), spectral_operator (a finite-rank Hilbert-Polya candidate against the first 100 zeros) and probe_tower (theta probes of growing degree in place of the cubic-theta probe of OpenAI's 7/8 proof, asking whether the zero-free abscissa can tend to 1/2). Each construction names the step where zeta's Euler product or multiplicativity enters and tests it on Davenport-Heilbronn, Epstein (1,1,6) and W_a where they apply, since a construction that also works for a function with off-line zeros cannot prove RH; two sessions that did not build it then challenge it, one at that rival gate and one for circularity and known obstructions. The dead routes it starts from are the record's refuted forms, listed in MISSION.md, among them EPP/Herglotz (refuted by W_a), weil_propagation's C1 and C3 (refuted by DH) and C2 (refuted by Epstein (1,1,6)), Jensen wedges that exclude n = 0, and heat-flow clocks that read only the zero configuration. #150 was assigned by the campaign; several numbers between #138 and #150 are held by landings in open pull requests and by hunts on other branches. No claim about RH is made, and a finished construction would still owe its complete chain and its review (docs/08).

Hunt #147: the de Bruijn-Newman chain at the verified height (dbn_pt_height/, opened 2026-10-21)

Status: closed at landing, 2026-10-31; no claim promoted, no Lambda bound stated. A reading of published sources, graded as what they show, plus a hunt instrument; the step it found not carried out is an attempt unresolved, not refuted, and the lab's own sweep to close it was never run (paused by allocation). Record: RESULTS.md. (Updated at landing: this line first read "open", with the chain lane's sourcing done and repaired after an adversarial review (2026-10-21), and the lanes resumed in place on 2026-10-26 to document the evaluator's large-x containment checks and to build, preregister and run the Davenport-Heilbronn refusal gate. It also said the barrier rule refuses every box holding a DH zero; it refuses the gate boxes that hold one.) The task that opened this hunt proposed #140; the campaign orchestrator assigned #147 because #138 to #144 are held for landings being renumbered, and #140 is already printed by the ref land/krenn-gu-orbits-20261021. Reading only, no Lambda bound stated. Platt and Trudgian (arXiv:2004.09765) take Lambda <= 0.2 from Polymath 15's Table 1, row 2 (X = 5e12 + 194858, t0 = 0.186, y0 = 0.16733, N0 = 630783). In the Polymath record that row's asymptotic-region hypothesis was evaluated only at N = N0 and y = y0: the table's lower bound 0.0376 is the Euler-2 bound there, the paper (v1 and v2) says the authors "expect to be able to verify" the rest, no sweep at row 2's parameters exists at any commit of the project repository, and in issue #131 on 2019-04-15 Terence Tao wrote that N0 alone is not quite enough, that it should not be a problem because the bounds get easier with N and the margin is conservative (a heuristic), and that he was not concerned about verifying it carefully. The only reported full sweep is a blog comment of 5 July 2024 by a reader outside the project, with unpublished code and no stated precision or y-range. Disposition: a gap in the published chain, attempt unresolved, not a refutation. Two unpublished 2018 parameter points, (0.17, 0.2) at row 2's X and (0.18, 0.2) at X = 10^12 + 46880, have a barrier and a finite sweep at y = y0; their open steps are the y-direction and the region past the last swept N, and closing them would give 0.19 or 0.20 at Platt and Trudgian's height without row 2. The same Section 10 sentence swaps the hypothesis labels of Theorem 1.2, in both arXiv versions, and the swap recurs in the 0.1875 bundle. The task's four "0.1875" pieces (collision exclusion, interior, exterior to 66476100549959, tail) are the 0.16 package's modules; the 0.1875 package itself instantiates the Polymath criterion at X = 6000000185827, and its finite-window leg consumes a banked lower bound that is cited and not reproduced (a producer ships, its run does not). Both the 0.16 and the 0.1595 rows depend on it through the anchor 3/16; the 0.172422 row, in the version the catalogue cites, derives its own 0.2 anchor (an unrefereed claim) and depends on neither. The zeta-specific inputs are the verification height and the Euler-mollified lower bound; Davenport-Heilbronn fails the first (zero 0.8085 + 85.699i), so the structural DH gate passes. Where the download cache exists, every quotation is re-read from hashed sources, and every date, name and count in the notes is tied to its source, by test_chain.py (corrected at landing: no CI runner has the cache, so there those checks skip and the notes are checked against chain.json and their arithmetic only). CHAIN.md, REDUCTION.md. Nothing here bears on RH (docs/08). At landing, a second, separately preregistered stage of the DH gate, at a close pair of DH zeros, catches a margin-free count that the first stage could not see; dropping the second-derivative term of the derivative bound is still not caught, and only a per-segment audit could see it (evaluator/DH_GATE_PREREG_2.md, RUNS.md E8). The gate tests the barrier rule only, not Theorem 1.3's error terms or the Euler-mollified sweep, and nothing in this hunt's findings rests on it. RESULTS.md separates what was read from what the lab computed and gives each finding its grade.

Hunt #152: a ball-arithmetic Lehman localization of a sign change of pi(x) - li(x) (skewes_enclosure/, opened 2026-10-26)

Status: open (2026-10-27). Three outcomes, (1) to (3), all under the same grade: a hardened computation on a cited derivation (Revers Theorem 3.1 and RH to A), with rigorous completeness on Arb's Turing count alone, mpmath nzeros agreeing on N(T) at float grade, and an mpmath second route for the zeros on a sample. Item (4) is a post-hoc observation at measured grade. (1) Target A, 2e6 zeros: the gate passed (Revers's S* inside the ball sum), and pi(x) > li(x) for some x with log x in [727.9519119, 727.9521241]. (2) Target B, 2.2e7 zeros at the printed omega = 727.95134: the gate failed (missing by 0.0122), recorded permanently. (3) The pre-registered omega-grid rerun: hypothesis H (the printed omega is 727.951335792 rounded) is not supported by its rule. The gap shrinks to 3.37e-5, still over Revers's 2.60e-5 budget, so reproducing S* stays unresolved. The pre-declared localization at 727.951335792 holds: pi(x) > li(x) for some x with log x in [727.951319892, 727.9513517], with eta = 1.59e-5, as in Revers (RESULTS.md section 7). (Corrected 2026-10-27: the lower end was first quoted as 727.9513199, rounded inward from the exact 727.951319892.) Neither interval is a new record. (4) Post-hoc review, 2026-10-27, measured, not pre-registered: the 3.37e-5 gap is twice the second-order sum S2. Revers's printed S* equals the Saouter-Demichel first-order value he quotes, minus S2, to 1.0e-9 (S2 over the first 2e4 zeros, tail estimated, not bounded), where his Theorem 3.1 adds S2. Read against arXiv v2 only; the published JNT version was not read. At the printed omega 727.95134 the corrected bound is negative, so the paper's localization survives only at the inferred omega (RESULTS.md section 8, scope corrected 2026-10-30). Our first-order sum matches Saouter-Demichel's to 4.7e-8, an end-to-end check of the 2.2e7 zeros. The verdict on H stands, but H's omega was right in substance (RESULTS.md sections 3, 7 and 8). The review also corrected the doors: A is a binding constant just above the R6 cliff, not a free door. The question is whether a sign change of pi(x) - li(x) can be localized with every computed step carried by Arb balls. The method is Lehman's integral with Revers's 2025 error terms (arXiv:2501.04488, JNT 2025) over the first ~2.2e7 zeros. The day-one read of three papers found that Saouter-Trudgian-Demichel 2015 (submitted version) and Revers 2025 sum their zeros in floating point. Both use an analytic zero-error budget, and neither cites a rigorous zero source. Platt-Trudgian 2015 already carried the analogous theta(x) - x sum in MPFI interval arithmetic with rigorous zeros, so the method is not new to interval arithmetic and no novelty is claimed. skewes_terms.py reproduces the R column of both of Revers's section-6 resizing tables from Theorem 3.1. It also found that the row printed as eta = 1.55e-5 is the value at eta = 1.56e-5, a label slip with no effect on the paper's conclusion. The day-one expected margin at the Saouter-Demichel parameters was about 1.28e-4, from Revers's printed S*, and the zero precision it needs (~62 bits) is far inside what Arb isolation delivers. The measured margin at omega = 727.951335792 and eta = 2.28e-5 is 9.42e-5. Unit timings, estimates and costs are in RUNS.md, and the results are in RESULTS.md. (Updated 2026-10-27: this paragraph first ended "nothing is a result yet", which was written when the hunt opened.)

Hunt #139: which step of the quasi-Riemann argument fixes the number 7/8 (seven_eighths_pin/, 2026-10-16)

Landed on main 2026-10-30 (written for a landing on 2026-10-21 that waited as draft PR #325). Opened on claude/hunt-134-seven-eighths-pin as Hunt #134; on main #134 is gauss_positivity and #136 to #138 are held by other open hunts and landings, so this hunt is #139. The prior-art search the hunt had not run was run at landing and extended the same day to the works its first source names and to the follow-ups' manuscripts (RESULTS.md section 10). Much of the hunt was already public, so it was derived here independently and is not novel. A model computation by tomoto0 (7 October 2026, nine days before the hunt opened) has the target formula with its ζ_F(6z) pole, the floor-row ceiling 13/15 at the same corner, and a geometry optimum near 0.87496 with the zero-detected rows binding. J. Guo and B. Liu (arXiv:2610.12234), both 8 October, give the exact optimum σ* = (1507 − 2√921)/1653 ≈ 0.8749570698, and Guo derives its binding bin δ* = (49 − √921)/48. Z. Gao (7 October) gives 20999/24000 at b = 1/8, and Argonaut Math (7/8 − 1/4000000) and LW56e2 (4374785349/5000000000), both 8 October, also state boundaries below 7/8. Gao, Guo and Argonaut Math carry the low-side loss accounting that the hunt marks inferred. So the scan optimum 0.874958 below is a grid reproduction of σ*, and the exact bound below is valid but loose: the same one-witness program at δ* returns σ* itself (pinned at landing). Not found in the sources read: the reading of 7/8 as the corner of M + ℓ ≤ 1 and per-summand losslessness, which alone forces 7/8 for every geometry; the exact rational bounds; and the pricing of the row-count excess between its two ends. tomoto0's bootstrap figure near 0.8745 treats the slope α as 2β* − 1, while the paper's lines 15100 to 15104 say α is fixed and κ = 2β* − 1 is the dynamic parameter, and Gao, Guo and Argonaut Math keep α = 5/6. Open check: Xu and Sang's 0.8748867, as hunt #136 records it, lies below σ*, so it must change an input beyond the geometry; if it changed only the geometry, that would refute this hunt's bound.

Status: answered within its scope; a reading of the paper, unreviewed by any person. In OpenAI's 7/8 argument the target is where the direct bound meets the Mellin signal, σ₀ = 5/6 + (M − 2ℓ)/12 (ℓ the prime-slot length, M the two averaging lengths), a line through both stages: Part I (ℓ = 0, M = 1) gives 11/12, Part II (ℓ = 1/6, M = 5/6) gives 7/8. The paper's geometry is the corner of two direct-bound conditions, M + ℓ ≤ 1 (line

  1. and per-summand losslessness of the row norm, M − 2ℓ ≥ 1/2 (line 8306), and the

second alone gives exactly 7/8. Relaxing it, the high side binds instead: with the paper's row counts, no choice of (ℓ, M, b) passes the endpoint inequality below 5655673/6464057 ≈ 0.874942 (exact, one witness point), and the scan optimum is 0.874958. The binding rows have a zero near real part 0.69 and maximal prime-slot amplitude. Perfect row counts would stop at 13/15 (rows counted in full need C₀ ≤ 0). The Euler regions, the contour 17/50, the 11/12 input and Lemma plain's κ ≥ 3/4 do not fix 7/8. 38 printed constants reproduced exactly; 39 citations checked against the pinned source; the moment lemmas' proofs not checked. See seven_eighths_pin/RESULTS.md.

Hunt #148: the 7/8 probe's local identity over F_q[T], and the biquadratic case (ff_probe/, opened 2026-10-21)

Status: closed 2026-10-26; candidate refuted at the stated parameters. Over F_13[T] the biquadratic theta's degree-3 coefficients are not multiplicative up to phases (zero at 234 of the 286 triples of linear primes, abs(H)^2 = 52 at the rest), so the quartic probe has no scalar local identity, and hunt #136's template prices nothing for it. The cubic control over F_7[T] reproduces Lemma 7.1 exactly. A PARI second implementation reproduced every exact number it recomputes. Carrying the verdict to S4 over Q(i) is inferred, so S4 there stays an unresolved attempt. See RESULTS.md. The lane was resumed in place on 2026-10-26 by the owner's decision. A lane of the campaign the owner directed on 2026-10-21. The workflow proposed the number 141; #138 to #144 are held for landings being renumbered elsewhere, so this hunt is #148. Over F_q[T] the Fourier coefficients of higher-order metaplectic theta functions are finite Gauss-sum expressions (Hoffstein 1992, Patterson 2007 and 2014) and RH is a theorem, so the question hunt #136 left open by reading, whether a biquadratic-theta probe with quartic averaging has a scalar local Euler identity like Lemma 7.1 of OpenAI's 7/8 paper, can be tested exactly. The cubic case over F_7[T] is the control. Nothing here bears on RH over Q (docs/08).

Hunt #142: controls for the F1 discovery engine (f1_engine_controls/, opened 2026-08-21, landed 2026-10-30)

Opened as Hunt #64 on 2026-08-21, on a branch that never reached main. #64 is r_b9552d run 2, #136 and #137 are held by open hunt branches, and #138 to #141 by four other landings of 2026-10-21, so this entry is #142.

Status: closed (2026-10-21), measured; three of the five critiques measured and pinned, two left as judgments. Nothing promoted. docs/15-the-f1-discovery-engine.md closes with a Reality Check of five critiques that dial the document's own rhetoric back. Nothing in the tree tested them. Three make checkable assertions, and measuring them moved two.

Reality Check items 1 and 3 (a toy computation; F_1 geometry is speculative) are judgments, not measurements, and stand as written. The stale discovery/ paths in docs/15 and docs/09 are repaired, and tests/test_docs_paths.py now fails when docs/ cites a Python file that is not there. Run against main as it stood before this landing, it read 255 backticked .py citations in docs/ and failed on exactly one, docs/15's discovery/01_f1_geometry.py. Pinned by tests/test_f1_engine_controls.py; docs/15 carries the corrections, and docs/37-methods.md carries the guard. Nothing here is about zeta, so nothing here goes to the funnel or the battery.