Agent Persona: The Hunter (ontology-attempt edition) Scope: hunts/local_positivity/ only. Nothing outside this directory is modified except the case-log entry in hunts/README.md and the record at docs/24-the-local-positivity-attempt.md.
Objective
docs/09 §5.1 states the positive target: not "prove the Weil form is positive" (that is RH restated), but factorization, construct, from prime data alone, a structure in which the Weil quadratic form is a norm square, so its sign becomes formal. Requirements A (arithmetic provenance), B (exact trace realization), C (structural positivity).
Every serious program is stuck at Gates 1 and 2. This hunt asks whether Requirement C can be reached one place at a time, where the local structure is small enough to write down:
Does the prime side of the explicit formula factor, place by place, into a manifest norm, and if it does, does local positivity buy anything global?
The answer is yes to the first and no to the second, and the second is the result. Recorded so the avenue is not reopened.
The construction (derived in the module, not recalled)
For f(s) = Σ aₙn^{-s} with a₁ = 1, let λ_m = b_{p^m}/log p be the local log-derivative coefficients at p. The place-p kernel is the Toeplitz form
K_p^(d)(θ) = d + 2 Σ_{m≥1} λ_m p^{−m/2} cos(mθ)
and the statistic is the normalization-free threshold c_p, the least diagonal for which every finite Toeplitz section with off-diagonals λ_m p^{−m/2} is positive semidefinite, so the form with diagonal d is PSD exactly when c_p ≤ d. On a summable sequence this is the cosine-series minimum
c_p = − min_θ 2 Σ_{m≥1} λ_m p^{−m/2} cos(mθ),
on a growing one it is +∞ (the module's "divergent" regime), and at |α_j| = √p it can still be finite (exactly d for ζ(s − ½), where the module reports INDETERMINATE). The module reads the regime off two weighted terms near the truncation edge, and for a conjugate pair of parameters, whose weights oscillate, that reading can be wrong on either side of the bound (CORRECTIONS.md, "Later correction").
Nothing is fitted: c_p is a minimum, and d is read off the object's own gamma factors. With Satake parameters α_j the series has the closed form
K_p^(d)(θ) = Σ_j (1 − |α_j|²/p) / |1 − α_j p^{−1/2} e^{iθ}|²
when all |α_j| < √p and the coefficients are real (parameters real or conjugate-paired). The forward direction, every |α_j| ≤ √p gives c_p ≤ d, is derived. The closed form does not establish the converse: at p = 2, α = (3√2, 0, 0, 0) the rational expression is 3 − 8/(10 − 6 cos θ) ≥ 1 although one parameter exceeds √p, and the defining series diverges (the gate itself reports "divergent" and FAIL there). The converse for c_p rests on a separate PSD-boundedness argument, recorded in CORRECTIONS.md ("Later correction") as an unreviewed derivation. Writing Φ_p f = Σ_{m≥0} p^{−m/2} f(· − m log p), the prime side decomposes as −Σ_p log p (Q_p(f) − ‖f‖²) with Q_p = (1−1/p)‖Φ_p f‖², a norm at every place.
Instruments
localpos.py, which writes results.json. Entry points: gate, reference_table, lesion_sweep, null_distribution, ramanujan_violator, satake_check, epstein_local_check, scope, provenance_report.
The laboratory is used for three things only: zeta.epstein.kappa for the DH coefficient, zeta.epstein.epstein_representation_count for the Epstein cross-check, and zeta.weil.explicit_formula_sides for the decomposition comparison. The gate itself imports nothing from zeta/.
The standing checklist, answered
- Rival. Davenport–Heilbronn and both disc −23 Epstein forms, the zeta department's own rivals. All three FAIL, at
p = 2, 3. The trap is noted: these are the rival set, so the gate is calibrated in the other direction too (ζ,L(χ), and a genuine degree-2 factor must PASS, and do). - Decoy / surrogate. Swapped coefficients move the verdict by 15 orders of magnitude, which is the control whose absence made the Imposter Gauntlet vacuous (
docs/15). Against 300 random period-5 sequences, 100% fail with median excess +5.88, and DH sits at the 6th percentile, an unusually mild failure, echoingROADMAP.md's 27th-percentile calibration with a different statistic. - Lesion. Interpolating ζ → DH, the blindness threshold is
ε* = 0.184. 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. Those are margins of the truncatedc; since the regime test of 2026-08-11 the gate's verdict on that family is FAIL at 22 of the 60 angles, a defect of that test recorded inCORRECTIONS.md("Later correction"). - Precision response. ζ's threshold matches its closed form
2/(√p+1)to 12 digits with truncation bounded below 3.4e-14; for a single parameter the closed form pins the Ramanujan failure point at√pto 8 digits. The statistic responds to precision the way a real quantity does.
Honest scope
Nothing here is evidence for RH. Every bound in this hunt is elementary floating point, carrying no enclosure at any step, so nothing here makes a claim in the ball-arithmetic regime zeta/rigor.py owns, and this file does not use that regime's reserved word, which tests/test_hunt_probe_discipline.py forbids anywhere under hunts/ except the case log. The gate is not a test for "has an Euler product": a genuine degree-2 Euler factor with α = 2.3, 1/α, violating Ramanujan, is rejected at p = 5. The place is in the "divergent" regime (2.3/√5 ≈ 1.0286), so c_p = +∞ by the module's convention; the 65.24 the module prints is the series truncated at depth 41 (22.46 at depth 23, 110.80 at depth 52), not the value. This file used to call that factor legitimate in the Selberg class; it is not, since docs/09 §3's axioms (4) and (5) both exclude |α| > √5. What the gate reads is the local bound |α_j| ≤ √p, forward direction derived, converse an unreviewed derivation. scope() says so.
Disposition
Instrument kept; no claim promoted. The construction reaches Requirement C locally and fails to globalise: Q_p − ‖f‖² is not of definite sign (52 of 60 places positive, 8 negative), so local positivity is compatible with either sign of W. Filed under docs/09 §5.1 taxonomy item #5, finite approximants.
A hunt, not a department: its rivals are the zeta department's rivals, and per harness/README.md a department whose battery is another department's battery is not a department. Full record in docs/24-the-local-positivity-attempt.md.