VERIFICATION: every exact number by a second implementation, and planted faults

Hunt #148. Status: the cross-check run of 2026-10-26 (15 items) reproduced every exact number it recomputes; no row disagrees with the primary run (section 2 lists the rows that record source errata or findings, and the nine probe triples that rest on the shared empty-box argument rather than an independent sum). Files produced on branch hunt/ff-probe, landed on main 2026-10-30 (PR #355). Every count on this page is read from results_second.json (written by crosscheck.py) or measured by test_ffprobe.py, and pinned there. Grades as in MISSION.md.

1. The second implementation

ffprobe_pari.py writes the arithmetic again in PARI/GP, through cypari2, and imports nothing from ffprobe.py: residue symbols by resultant and by Frobenius at each prime factor, Gauss sums by enumeration of the residues, by the Davenport-Hasse closed form and by the prime-power reduction, cyclotomic numbers as polmods modulo polcyclo(N), Patterson's C and rho_0, the probe's weights and its direct sum, the local factor of eq. (local-P) by its definition, and the rival's L-polynomials, whose roots come from PARI's polroots. Python and numpy only add up integer exponent histograms (counts of roots of unity). The conventions are read again from the sources, not from ffprobe.py.

crosscheck.py runs it over the ranges of the primary run. Where the documents quote a value, PARI recomputes the value and the two are compared as exact elements of Q(zeta_N); where they quote a count of checks, PARI recomputes the count on its own. The identities the second implementation uses beyond the definitions (the Chinese remainder theorem for Gauss sums, Davenport-Hasse, the prime-power reduction, Gamma(c) = gamma_3(c) for squarefree c, which it uses at degree 5) are each checked inside PARI against enumeration at smaller degree (item pari_internal, and the Davenport-Hasse and C_p items).

2. What was reproduced

One row per comparison in results_second.json, rendered by render_verification.py (--check fails if this table is stale; test_ffprobe.py runs it). "short" marks a row whose held count is below its checked count, a false flag or a nonzero mismatch count; the prose after the table says, for each, whether it is a documented erratum or a failure.

<!-- results-table: begin (render_verification.py) -->

15 items, 193 rows, 37 marked short; 7,727.7 s of item time in total. <!-- results-table: end -->

No row is a failure to reproduce. The rows marked short are of three kinds, and each equals the primary run's own record (test_ffprobe.py, test_second_implementation_reproduces_the_primary_numbers):

• Source errata, reproduced exactly. Davenport-Hasse in the 2007 note's form, with the 2014 additive character: 45,129 mismatches at (7, 6) and 4,394 at (13, 4) (SOURCES.md row 2). The Hecke relations at infinity as printed: 40 of 60 hold in the 2007 reading and 20 of 40 in the 2014 one (rows 8 and 9). The corrected reading holds 60 of 60.
• A fit tried outside its range. s4.relations.hecke_infinity_corrected, 10 of 20, is the n = 3 sign rule applied at n = 4, recorded the same way in results_step3.json (relations_note). The n = 4 relation holds 20 of 20 in each additive character (s4.hecke_infinity_both).
• The finding itself. s4.three_body.*.all_equal_one is false for every admissible component choice. That is S4.md section 3's defect: the three-prime ratio is defined only at lambda = 5 and 11, and the defect values equal the primary run's.

Beyond the rows, PARI reproduces these values: the 66 off-circle rivals, 50 of them beyond sigma = 1; the rival's mismatched monomials, X^2 to X^3 W; the balanced probe values 1.529 and 0.609; and every probe triple's exact value. The nine triples with an empty frequency box are zero by the same box argument in both implementations, not by an independent sum. The crosscheck run of 2026-10-26 crashed once in item s4. The cause was a bug in crosscheck.norm2, which asked PARI for the degree of an exact zero; nothing disagreed. It was fixed and s4 was rerun alone (RUNS.md).

3. Planted faults

ffprobe.FAULTS switches on eight planted faults, four wrong Gauss-sum signs and four wrong residue symbols, each at a different place, and test_ffprobe.py checks which of six quick equality checks each one turns red: the two symbol routes, Davenport-Hasse against enumeration, the structural route against enumeration, Lemma 7.1 locally (definition against closed form at T + 1 for j = 0..5 and at T^2 + 1 for j = 0, 3), the probe's direct and Euler representations at (0,2,1), and the PARI implementation against ffprobe. Without a plant all six are green.

Every plant turns at least one check red. The Davenport-Hasse closed form of gauss_dh feeds nothing else (the structural route has its own prime formula), so its sign is caught by its own check alone. A fault in a component that both of ffprobe's own routes share is what the PARI comparison is for: it goes red for six of the eight plants.

4. Reproduce

.venv/bin/python hunts/ff_probe/crosscheck.py                 # results_second.json, checkpointed per item
.venv/bin/python -m pytest -q -n0 hunts/ff_probe/test_ffprobe.py -k "plant or lesion or second"