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 := − min_θ 2 Σ_{m≥1} λ_m p^{−m/2} cos(mθ), so K_p^(d) ≥ 0 ⟺ c_p ≤ d.
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
so c_p ≤ d is exactly the local bound |α_j| ≤ √p — a decision procedure, not a heuristic. 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. - Precision response. ζ's threshold matches its closed form
2/(√p+1)to 12 digits with truncation bounded below 3.4e-14; 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 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 scope() says so in the module rather than only here.
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.