`hunts/` — exploratory studies

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

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.

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 #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 #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:

• The DH height-85.7 pair's image in J^{d,0} tracks the PDE flow trajectory under that dictionary at 7.8e-8 → 4e-4 relative over half the flow; the landing degree d\* = 20785.13 reads t\_J = 0.0441690 against hunt #4's t\ = 0.0441263 — 0.097%, while the isolated-pair value is 7.2% away. Pair 2 (γ ≈ 114): d\* = 146365, clock agreement 0.011%. Flowing before damping, the two clocks add: t_land(d) + |x₀|/(8d) = t\* to 0.03–0.06%.
• The corollary: a blind hyperbolicity scan at d ≤ 32, n ≤ 250 sees nothing, and the dictionary says that is structural — degree d ≲ 2·10⁴ has already flowed this violation past its landing before the polynomial is inspected. Textbook-degree Jensen scans are the wrong side of the landing, not weak detectors.
• Null control, the explanation: a pair planted into ζ's γ-table at exactly x₀ lands at 0.0436212; the arithmetic-free N-body dynamics run on the plant's actual configuration lands at 0.0435805 — 0.094% — while DH's clock sits 1.1% away and the isolated formula 8.3% away. The clock reads zero configuration, not arithmetic (hunt #4's null, now on the coefficient side). The mission's own quantitative guess for this control was wrong (it forgot the ordinate doubling in the x-plane) and is recorded as a failed prediction in jensen_clock/RESULTS.md.
• Controls: ζ specificity (winding 1.3e-63, Newton relative Im 1.2e-200); instrument checked against zeta.li's twice-derived γ-table to 1.3e-51 and hunt #4's polished pair to 40 digits; d\* precision-flat to the printed digits across dps 130 → 160 and two quadrature geometries; truncation margin 36 orders at the deciding contour.

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.

The wide search (wide_search/)

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.

• The sharp constant for that method is 0.86864150052976706411... (simple and on the critical line; 0.93432075... distinct), attained at lambda = 1. Full account, with the functional and its provenance, in wide_search/RESULTS-xiprime.md.
• It does not reach Wu's 0.86957, falling short by 9.285e-4. No admissible window closes the gap, so the comparison the paper leaves open is settled negatively. The paper's own quartic was already within 1.5e-6 of sharp.
• Checks: the instrument reproduces the paper's Theorem D to 10 digits before being pointed at anything unknown; the functional reproduces all four published constants of Remark 7.3 to every printed digit; three independent derivations of it agree and five independent computations of the optimum agree to 14 digits; the coefficient formula is checked exactly by Dirichlet convolution; and Bian's 2008 thesis reproduces the same k = 1 coefficients exactly, from sixteen years earlier.
• Higher derivatives are blocked, and RESULTS-higher-derivatives.md records why: F_k for k >= 2 already exists in an unpublished 2008 Rochester thesis, it has no closed form, its author states he cannot bound the tail, and the tabulated 11-term truncations are measured here to diverge at the bandwidth the method needs (F_4(1) ~ 2476 against F_1(1) ~ 2.78, giving proportions of 1.198 and −2.64).

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:

• The scalar-moment formulation collapses, exactly. Retaining only (tr, ||.||_F^2) per window makes the joint feasible set the intersection of the single-window half-lines, hence sup_v H(v) = 0.6725007037.... That route cannot move the number, and is closed rather than unfinished.
• The published ceiling data reproduces. wide_search/pair_ceiling.py recomputes, from the enclosures in the public LawN256.lean alone and with exact rationals, the 2^140 scale, 256 rows, the worst interior error 1.83670992316e-40, D(1) = 0.8239531607128352, the stability coefficient 2.5431315104166665e-6 and the simple fraction 0.6818286874638315. All agree with the repository. The extremal law itself needs a certificate file the authors state is available on request.
• Remark 1.1's ceiling is stated more uniformly than its finite instance supports. The N = 256 law gives the 0.68185 sentence for certificates with abs(r'(1)) + integral abs(r'') <= 8.38043022204.... The formalisation carries the error terms and does not make that elision.

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/.

• Six defects in recorded claims, each reproduced by the director before being written down. The most serious: zeta/rigor.py's abscissa conversion parsed unrecognised numeric types from their printed decimal, so proven_sign returned a wrong nonzero sign on a numpy.float32 input — identically on both backends, because the fault sits upstream of the split.
• A strategic conclusion corrected. The claim that a coefficient functional is "blind to the position of the critical line by construction" is false as written and mis-cited (docs/18 §6 says ordinate statistics). The true statement is a threshold — c_p ≤ d for ζ(s−δ) exactly when δ ≤ ½ — and it is the Selberg-class axiom's θ < ½, i.e. known. The ROADMAP.md call that hunt #5 asked for is answered: no, this does not become a standing constraint on the coefficient-side programme.
• Two positives that held under attack. A blind replicator, given the statement only and forbidden the package, reproduced the flagship Weil enclosure to 43 digits from scratch; and the Lean arm rebuilt on a cold machine, 8715 jobs, zero sorrys.
• The formal arm's blocker was mis-diagnosed and is now measured. Rung 3's centre could not have passed at any Taylor order — its budget counts the tail radius once where the norm counts it twice — and the width floor is the κ enclosure, not the exponential Taylor order. A configuration with ≥1.1× margins exists at 1.63× the term count and the same literal sizes.

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.

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:

• The prime side factors place by place into a manifest norm. With Φ_p f = Σ_{m≥0} p^{−m/2} f(· − m log p) it decomposes as −Σ_p log p (Q_p(f) − ‖f‖²), Q_p = (1−1/p)‖Φ_p f‖² — a norm at every place, from coefficients only, with no zeros entering any definition. Reconstruction agrees with zeta.weil.explicit_formula_sides to 22 digits.
• And the local norms do not assemble, which is the result. Q_p − ‖f‖² is not of definite sign: 52 of the first 60 places positive, 8 negative. Local positivity is therefore compatible with either sign of W. Files under docs/09 §5.1's taxonomy item #5, finite approximants.
• Gate #3, with a crisp answer. The place kernel has the closed form K_p^(d)(θ) = Σ_j (1−|α_j|²/p)/|1−α_j p^{−1/2}e^{iθ}|², so c_p ≤ d is exactly the local bound |α_j| ≤ √p, with d read off each object's own gamma factors and never chosen. ζ and L(χ) quadratic mod 5 measure c_p = 0.8284 and PASS (matching 2/(√p+1) to 12 digits); Davenport–Heilbronn measures 1.8361 and FAILS at p = 2, 3; both disc −23 Epstein forms FAIL, at 5.995 and 6.462. "Where exactly does DH fail to embed?" — at p = 2, excess 0.836.
• Decoy / surrogate. Swapped coefficients move the verdict by 15 orders of magnitude — the control whose absence made the Imposter Gauntlet vacuous (docs/15). Against 300 random period-5 sequences 100% fail, median excess +5.88, and DH sits at the 6th percentile: a mild failure, not an exotic near-miss, echoing ROADMAP.md's 27th-percentile calibration from an unrelated statistic.
• Lesion. Interpolating ζ → DH, blindness sets in at ε* = 0.184, so a PASS means "no violation above ~18% of the way from ζ to DH at the tested places", and nothing stronger. The PASS side is not vacuous either: across 60 Satake angles the genuine degree-2 family keeps margin ≥ 0.343.
• The honest boundary, stated so nobody overclaims it. The gate is not a test for "has an Euler product". A genuine degree-2 product with α = 2.3, 1/α — legitimate in the Selberg class, violating Ramanujan — is rejected at p = 5 with c_p = 65.24. It tests the local Selberg bound, and localpos.scope() says so inside the module rather than only in prose.

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 exactly δ = ½ (c_p = 2x/(1+x), x = p^{δ−½}). 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.

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:

• The nine known off-line quadruples (Spira 1994; Balanzario–Sánchez-Ortiz 2007, each re-polished in-tree) land on the real axis at measured times t\* between 0.00275 and 0.05765. Each is a lower bound for Λ_DH := inf{t : H_t^{DH} real-rooted}; the max comes from the height-240 pair (β ≈ 0.8695), not the famous height-85.7 zero, which places third. Λ_DH ≥ 0.0577 measured — inside [0, 0.2], the interval that historically bracketed Λ_ζ. In flow time, the counterexample fails RH by less than ζ's own uncertainty span: "Λ is small" distinguishes nothing (gate #3 on the flow axis).
• Null control, the headline: the arithmetic-free N-body zero dynamics ż = 2Σ1/(z−z') seeded with the measured t = 0 configuration reproduces every PDE repair time to ±0.04% (target was 1%). The repair clock reads zero geometry, not arithmetic — the flow-time distance to real-rootedness is a property of where the zeros sit, shared by any function with that layout. Position-sensitivity (docs/18) meets the counterexample gate.
• Method worth keeping: the pair is tracked through its collision by contour moments — Δ(t) = 2q₂ − q₁² is analytic through the landing, so t\* is a clean root even though the zeros themselves have a branch point. Zero-census accounting closed in all five checked windows (line + 2·pairs = strip count; e.g. 49 + 4 = 53 at pair 1).
• Lesions: a contour clipping one pair member is refused (N=1 ≠ 2); a contour through the zeros returns winding ~1.8e9 and is refused. Post- landing, the newborn real pair is invisible to the default mean-gap/20 sign scan at 3 of 5 grid phases until gap/step ≳ 1.4 — hunt #3's blind spot, measured on the other side of a collision.
• Precision response: t\*₁ = 0.0441263445516 identical to the last digit across dps 44/54/70 and 96/192 contour nodes; spread exactly 0.

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).

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:

• The near-miss bump between the pair was decided positive at 64 bits by both backends — proven_sign pattern −,+,− on exact rationals, midpoints agreeing on Z(7005.0819) = 0.003967335016595021 to all 16 digits — and a dense scan found exactly 2 sign changes with zero undecided samples.
• Lesion: the default grid policy of rigor.certified_sign_changes (mean_spacing/20 ≈ 0.0448) is wider than the Lehmer gap; sweeping the window phase, the default grid missed the pair entirely at 1 of 5 phases. Honest both times (a sign-change count is a lower bound), but blind.
• Precision response: enclosure widths shrink ~2^−prec with the midpoint pinned; at 32 bits flint straddles zero while mpmath.iv decides — the backends disagree about decidability at the boundary, never about value.
• Rival, the headline: the predicted "failed Lehmer bump" at Davenport–Heilbronn's off-line zero does not exist — Z_dh's closest approach on [85.2, 86.2] is −0.357, two orders of magnitude farther from zero than ζ's bump clears it, while hiding 2 strip zeros. So "|Z| gets small" flags nothing: it fires on ζ's healthiest close pair and stays silent at an actual RH violation. Magnitude heuristics die here; sign counting vs strip counting survives.

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:

• The completeness gate was never called. zeta/detector.py's own docstring states the load-bearing caveat: the residue measures "zeros unaccounted for by the supplied on-line list", so a missing on-line zero produces a residue indistinguishable from an off-line zero, and a scan whose completeness has not been checked reports nothing trustworthy. online_list_is_complete appears nowhere in hunts/. The on-line zeros were found by a step=0.05 sign-change scan, which skips close pairs.
• The lesion confirms the confound, measured. Give ζ — factorization defect 2.65e-32, a perfect Euler product — a zero list with one on-line zero removed, and the residue jumps from 0.0038 to 1.99. An O(1) position residue is therefore produced by an incomplete list at zero factorization defect, which is precisely the signal the hunt read as off-line zeros. The recorded residues (4.07–4.33) sit at about twice that lesion.
• The test set is the rival set. The discriminant −23 principal form (1,1,6) is a registered rival in zeta.epstein.battery, admitted precisely because it lacks a scalar Euler product while keeping the functional equation. Confirming that the battery's rivals lack an Euler product and have off-line zeros restates their admission criterion. Under gate #3 that distinguishes nothing.

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.