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3175 bytes, as of the version from 2026-11-09 10:44 (current). Recorded here, not run.
# Argument selection for T687
This build uses the coordinated skill archived in
`agents/runs/20261109T100832Z-build-skills/manifest.json`.
The local and live versions both have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
The attached `skill-manifest.json` is an exact copy. This document is the
builder's justification; it does not replace the separate argument-selection
critique or its fingerprinted sidecar.
The single named quantity is the weighted Fricke trace t_m^(N*)(d).
N indexes its modular curve, m its Faber transform, and -d its CM discriminant.
No parameter selects a second quantity. All have integer type and stable
keys, in order (N,m,d). No quantiles or continuous reference inputs occur.
* N is intrinsic. Include all seven square-free N>1 with genus-zero X_0(N):
2,3,5,6,7,10,13. Matsusaka (2017), section 1 and sections 3–4, supplies this
family, including both composite levels and all Atkin–Lehner involutions.
This breadth is retained before increasing depth at a familiar prime level.
* m is a discrete index. The initial orders 1,2 give the Hauptmodul trace and
its second Faber trace. Order two has the additional Fourier-coefficient
identities of Matsusaka, Theorem 1.1. There is no division by m. The cutoff
is practical and noncanonical; it does not assert that higher orders lack
mathematical interest. The family definition still permits every m>0.
* d has arithmetic structure. Include every 1<=d<=200 for which -d is a
square modulo 4N. These are complete initial discriminant-height segments,
not meshes of function arguments. They span imaginary quadratic fields,
nonmaximal orders, level-ramified cases, and all imprimitive contributions.
The definition follows Matsusaka, section 1, without importing Zagier's
differing ramified example weights. The bound is noncanonical, independent
of decimal notation, and chosen to give full orbit sums across all levels
while allowing unstarred coefficient identities through exponent 50.
It controls orbit cost and trace growth. Inadmissible d are omitted; actual
zero traces at admissible indices are retained.
| N | Admissible d | Entries for m=1,2 |
|---|---:|---:|
| 2 | 75 | 150 |
| 3 | 66 | 132 |
| 5 | 60 | 120 |
| 6 | 49 | 98 |
| 7 | 57 | 114 |
| 10 | 45 | 90 |
| 13 | 53 | 106 |
| Total | 405 | 810 |
Enumeration orders N increasingly, then m increasingly, then d increasingly.
The congruence condition couples N and d, so this is not an unrestricted
Cartesian product. Each selected row is the entire stabilizer-weighted orbit
sum, not a principal CM point or a primitive-only subsum. The longest value
has 21 characters; dry_run measured the entries with their identities at
40.6 KB. The finite selection, generator default and completeness note agree.
Checked sources: Matsusaka, DOI 10.1007/s40993-017-0090-x, section 1,
Theorems 3.2, 3.4, Corollary 4.1 and Lemma 4.2; Zagier, Traces of Singular
Moduli, section 8; Amir, arXiv:1905.11323v5, Part II, Proposition 13 and the
Jacobi-basis construction. These sources support the family and conventions;
the two numerical cutoffs are the present reference choice.