Curves over F_p — department #2, and the first second subject: the same referee architecture under a property that is decidable.
Declared in harness/departments/finitefield_department.py. Audited by tests/test_department_conformance.py.
What it studies
The Riemann hypothesis that is a theorem (Hasse, 1933; Weil, 1948): for a non-singular curve over F_p, every Frobenius eigenvalue sits on the circle |α| = √p, equivalently |a_p| ≤ 2√p. Point counts over the extension tower, the Lefschetz fixed-point formula, functional-equation self-duality αβ = p, Sato–Tate statistics — all implemented in zeta.finitefield with the Lefschetz predictions pinned against brute-force enumeration.
Module: zeta.finitefield.
First command:
.venv/bin/python scripts/11_finite_field_rh.pyWhat can refute a claim here
| Role | Members | What they are |
|---|---|---|
| Rivals (2) | counterfeit Lefschetz profiles, traces 70 and 200 at p = 1009 | integer positive counts with exact self-duality and both roots off the circle — the count-shape without the theorem |
| Decoys (2) | count jitter on the Hasse–Weil scale; tower permutation | the sequence without the eigenvalue pair; the numbers without the tower |
| Surrogates (2) | uniform-angle and Sato–Tate-angle trace draws | the Hasse bound satisfied by construction, from no curve at all |
| Lesions (3) | counterfeit profiles at traces 64, 80, 128 (magnitudes 0.129, 1.02, 2.76 off the circle) | planted violations for detector-power measurement |
Two facts worth knowing before bringing a claim:
- The payload is counts and nothing else. Every subject hands claims
{"p", "counts"}; no field says which subject it is. A claim distinguishes the target by mathematics or not at all. - Lesions are quantised. The smallest integer trace violating Hasse at p = 1009 is 64, so no lesion smaller than magnitude 0.129 can be planted without breaking integrality — which would be detectable for the wrong reason. Department #1 plants δ = 0.001; this department provably cannot. Detector power below the floor is unmeasurable here, and that is a fact about the subject, not a defect of the battery.
Why a counterfeit is a fair rival
Hasse's theorem is exactly the statement that no curve realises the counterfeit profiles: they satisfy every structural constraint the counting data wears on its sleeve (integrality, positivity, the Lefschetz recursion, the functional equation) and violate RH. A claim that fires for them is leaning on structure that provably does not force the property — the same modus tollens the Davenport–Heilbronn function supplies in department #1, made exact.