back to table · edit · history · where entries came from · files · download
5097 bytes, as of the version from 2026-11-02 11:01 (current). Recorded here, not run.
# T649 build numerical evidence
Quantity: ordinary logarithmic Mahler measure of x^3+y^3+1-cuberoot(t)*x*y, with the real cube root, natural logarithms and normalized torus measure. This is not the modified period.
Skill manifest: agents/runs/20261101T105238Z-build-skills/manifest.json. The local and live archived versions were both read and their hashes verified. Both SHA-256 values are a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278. See skill-evidence.json. These exact instructions govern the build.
Selection: 36 entries, 3 intrinsic controls (-1,0,27), 3 rational CM arguments and both conjugates of 15 quadratic source representatives. Exact comparisons identify 10 CM arguments in (-1,27). argument-selection.md justifies the two disjoint groups and the noncanonical degree-two extent. No independent argument-review acceptance is asserted by this build.
Exact checks: cm_checks.py verified all 33 CM j-invariants against Sage Hilbert class polynomials and the actual polynomials read from T129. checks.py symbolically checked the shared Weierstrass model's j-invariant formula and the torus monomial identity relating the cubic and quadratic polynomials.
Primary evaluation: adaptive tanh-sinh quadrature for the quadratic Jensen integral, at 140 and 190 decimal working digits. Minimum observed relative agreement: 136.0414200085515 digits. Each value is rounded to 100 significant digits. The completed run took 159.38 seconds. The initial incomplete runs are preserved; none of their rejected values was uploaded.
Independent numerical method: at 125 working digits, Cardano roots of the original cubic in y give a different Jensen integrand. Its integral over 0 <= theta <= pi/3 has factor 3/pi. All projected torus crossings are split, as are cubic discriminant crossings. At t=27, the explicit linear factors avoid cancellation at a multiple root. All 36 comparisons passed, with minimum relative agreement 119.8039682932619 digits. The check shares argument decoding with the generator, but no value-evaluation code.
Control: the direct cubic method first computed the nonzero N(0) control independently from 3*sqrt(3)*L(2,chi_-3)/(4*pi), with the Dirichlet value computed through two Hurwitz zeta values. The 125-digit strings coincide. The raw control log's 999 is the code's identical-string sentinel, not a claim of 999 accurate digits. The t=27 check agrees with three times the same control.
Additional check: 24 exterior arguments agree with the hypergeometric expression in Samart v3 equation (1.4), after substituting t=k^3. Minimum agreement: 120.00448088328191 relative digits. For abs(t)>27, the hypergeometric series is evaluated by an explicit scalar term recurrence with an absolute geometric tail estimate; elsewhere negative-real analytic continuation uses mpmath.hyper. The boundary t=27 uses its factorization control instead of a slowly converging unit-argument series.
Accuracy status: heuristic (agreement-checked). Neither floating quadrature nor its reported convergence estimates is an interval enclosure. The series tail estimate does not by itself certify rounding error. No ball is constructed around an uncertified midpoint. Stored decimals claim one unit in the last printed place under NumberDB's convention, supported by the measured agreement and independent checks rather than a proof.
Repository dry_run.py and check.py passed on all 36 entries. Longest numerical entry: 102 characters; measured annotated entry block: 6.7 KB. No exact-in-disguise or entry-prose findings. The helper's generic exactness success accepts decimal strings and does not upgrade this table's rigour.
Reproduce with SageMath and numberdb installed, keeping generate.py beside checks.py and values.json:
sage -python checks.py
sage -python cm_checks.py
sage -python generate.py
The last command needs the author's API key while T649 remains private. Campaign computations were run exclusively through agents/sage.sh. Exact dependency versions are recorded by fill.py in versions.json on the draft. The independent raw outputs are in independent-checks-complete.log; compact per-value outputs are in independent-values.json. Source downloads and hashes are in sources.json.
All values must be read back and compared after the single upload. The fill log records verify(sample=None), ordered identity checks, comparison of all stored decimal intervals against the independent values, the 24 hypergeometric comparisons, the intrinsic identities and actual T283/T145 companion digits. The separate handoff records the API audit result. Publication, offering and human review are controller/person actions and are not part of this build.
Source hashes at build preparation:
{
"generate.py": "bb666d38c350ff0e622548fcc60de1e587b13ada8559133a37548e1b92d4301f",
"checks.py": "16eb1a00c096458212b49fd9b537d363a2cb8c5599ed63b57702c30ee87ea4d7",
"cm_checks.py": "f64793ebce7ae91ddca5163f9d042d69ccd107d609f4072c63869a91a0081eb8",
"argument-selection.md": "ba54c13ab8906d7f82a678f82fe270f6ef19e890b7401c40b105dec5b7db86cf"
}