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4473 bytes, as of the version from 2026-11-02 08:35 (current). Recorded here, not run.
# Argument selection for O’Nan moonshine coefficients (T672)
Policy: the coordinated local/live skill recorded in
`agents/runs/20261102T075604Z-build-skills/manifest.json`, both SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attached skill-evidence.json records the archive paths. This is the build's
justification; the controller must obtain an independent critique and its
fingerprint-bound argument-review sidecar.
The single quantity is the signed trace a_[g](n) on W_n in Duncan–Mertens–Ono,
arXiv:1702.03516v4 (DMO), Theorems 3.1 and 4.1. The acting conjugacy class
and the positive Fourier grade are arguments of that trace, not selectors
between different quantities. The convention is q=exp(2*pi*i*tau), with
F_[g]=-q^(-4)+2+sum_(n>0) a_[g](n)q^n. Signs and structural zeros are kept.
The universal polar and constant terms are stated in the definition rather
than separately stored. Grades congruent to 1 or 2 modulo 4 vanish by the
Kohnen plus condition and are outside the declared parameter domain.
## g: discrete_index
The natural domain is all conjugacy classes of O’Nan, labelled as in DMO,
Appendix A. This first extent includes every class of element order at most
seven: 1A, 2A, 3A, 4A, 4B, 5A, 6A, 7A, 7B. There are no omitted classes
within that order bound. The selection covers identity, prime orders, the
first composite order with two classes, and mixed prime order six before
increasing any one's depth. All seven levels have genus-zero X_0(N); their
trace formulas in DMO, Table 5.1, Proposition 5.1 and Appendix D require no
cusp-form choices. Levels 8, 10 and 12 are other accessible extensions, but
the initial order-seven boundary keeps the independently checked class
prefix compact; it is a practical noncanonical extent, not a classification
of all accessible classes. Higher-order classes remain part of the natural
domain. The important cusp-form cases at 11, 14, 15, 19, 28 and 31, and the
nontrivial character at 16, require further construction before extension.
Separate 4A/4B and 7A/7B identities are preserved even though each pair has
an equal series. An acting class is a finite-group object independent of its
printed label; these choices do not depend on numeral notation. Class order
and letter determine row order, matching family issue #298.
## n: discrete_index
The natural domain is n>0 with n congruent to 0 or 3 modulo 4. This build
holds every such grade through 200 for every selected class: 100 grades per
class and 900 entries. Both residue classes, nonfundamental discriminants
and all imprimitive-form contributions are included. The cutoff is a
noncanonical practical reference bound: it covers successive low graded
representations, sign changes and structural zeros, gives full overlap
with both Hecke-order 1 and 2 of the independently computed Zagier trace
companion T643 for the identity class, and keeps the rapidly growing integer
entries readable. The longest value has 39 characters and the measured
entries block is 43.8 KB. That overlap supports verification; it is not the
reason the Fourier grades themselves are meaningful.
This is a complete initial segment of genuine discrete grades. It is not
continuous reference sampling, a decimal grid, or a Cartesian product of
arbitrary continuous parameters. The common grade prefix lets readers compare
class traces of the same W_n. No probability, quantile or critical-value family
is included. Broader class coverage is preferable to taking the identity
class to a much larger exponent. No individual application is claimed for
every grade.
## Sources and consistency
Checked sources: DMO Theorems 3.1 and 4.1 (module, normalization, support and
integrality); Table 5.1 and Proposition 5.1 (CM functions and trace weights);
Appendix D (the seven level-specific class-number corrections); Appendix A
(all classes and their order); Appendix B (independent small-grade
multiplicities). CTblLib 1.3.11 independently resolves a sign error in the
printed character table, documented in numerical-evidence.md. This does not
change the class labels, modular-form definition or selected arguments.
The generator default enumeration, parameter declarations, Programs example
and completeness note all use this same class list and n<=200. The generator
contains the entire selection and computation; checks.py, source-controls.json,
companion-controls.json and gap-ON-excerpt.txt are verification dependencies.