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1740 bytes, as of the version from 2026-11-10 01:43 (current). Recorded here, not run.
## A torsion relation can silently use an extra lattice symmetry
What happened: Samart arXiv:1309.7730v2 Theorem 2.11(ii), equation (2.13),
prints the 3-torsion combination 9 L(tau/3)+9 L(1/3)+18 L((tau+1)/3)+4 L(0)
as zero without restricting tau to rectangular lattices. The proof invokes
an equality of values at (tau+1)/3 and (tau-1)/3. For the marked rhombic
CM lattices this equality fails. At Delta=-3 the printed combination is
4.7480008262162248837842548395005858984..., with a rigorous nonzero enclosure.
The defining q-average, independently summed Fourier expansion and direct
lattice check agree. The full E[3] distribution identity passes for every
selected discriminant and explains the discrepancy: the two diagonal
3-torsion directions cannot be combined without the extra symmetry.
What the skill says now: verify identities before writing them, and trace
numerical disagreements to a cause. This rule worked as intended.
What it should say: optionally add this as a concrete example of checking the
hypotheses of a symmetry used in a source's proof. A formula tested only on
rectangular lattices does not establish its extension to rhombic markings.
No change to the governing definition or a weaker numerical standard is
needed. This is proposed evidence for a person, not a skill modification.
Evidence: T702; checks.py; numerical-evidence.json, samart_2_13_diagnostic.
The full relation has 9 times the sum of all four inversion representatives
of nonzero 3-torsion, plus 4 L(0). The restriction is stated in the draft's
comment-three-torsion. Archived skill manifest:
agents/runs/20261110T013256Z-build-skills/manifest.json, both versions SHA-256
886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b.