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4550 bytes, as of the version from 2026-11-01 18:35 (current). Recorded here, not run.
# Argument selection for T657
This build uses the coordinated skill archived in
`agents/runs/20261101T181018Z-build-skills/manifest.json`. The local and live
versions both have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The family proposal is numberdb-data#289. This document is a builder's
justification, not the independent critique required by the campaign gate.
## The quantity and its parameters
The single quantity is the monic irreducible polynomial over Q of
T_g(i sqrt(r)), in the zero-constant McKay–Thompson normalization. The class
g indexes the graded Monster trace, and r indexes the CM lattice. The
polynomial variable is x. We preserve the numerical companion T634's parameter
order g,r, its positive square root, and its actual square factors of r.
* **g: discrete_index.** Include 2A,2B,3A,3B,5A,5B,7A,7B,13A,13B in this
order. This comprises both Fricke and non-Fricke Hauptmoduln at every prime
level p for which X_0(p) has genus zero. Matsusaka (2017), introduction,
explicitly identifies these classes, groups and eta formulas before
Theorem 1.1. These are ten indices of the same trace-defined object, not
a quantity selector. The identity class is already represented by the
classical j-invariant and Hilbert polynomial machinery; it is outside
this proposal, as are other levels and classes.
* **r: arithmetic_structure.** Include every integer from 1 to 10. The
principal rectangular lattices Z + Z i sqrt(r) have endomorphism order
Z[sqrt(-r)], of discriminant -4r. This samples fields and conductors within
one arithmetic family, retaining r=4 and r=9 as different orders rather
than discarding square factors. The level structure is the sublattice
Z + Z p i sqrt(r) inside Z + Z i sqrt(r). T129 defines the corresponding
CM constraint and T634 uses exactly this embedding and level structure.
## Extent, interactions and purpose
The purpose is exact recognition of algebraic modular values encountered in
CM elimination and class-field computations. A minimal polynomial is a
searchable exact object different from its numerical root. We do not claim
that every polynomial has its own documented application.
The r <= 10 endpoint is a noncanonical practical cutoff, not part of the
family's domain. The live T129 polynomials H_{-4r} in this prefix all have
degree at most two. Combining them with the canonical level-p equation gives
elimination degree at most 2(p+1) <= 28 before factorization. All five levels
and both class behaviors are retained before increasing depth to the numerical
companion's r <= 30. The build will measure polynomial lengths and preserve
those measurements with its numerical evidence. The enumeration has 100
entries, but that count is a consequence of the family, not a target.
The two parameters interact through a specified level structure on each CM
lattice. Completing this modest discrete slice supports comparison of ramified,
split and inert primes, and of Fricke and non-Fricke values. It is not a
Cartesian product of independently chosen real meshes. There are no continuous
reference samples, quantiles, decimal grids or famous-point-only pruning.
## Checked sources and companions
* Matsusaka, Research in Number Theory 3 (2017), article 23, introduction,
the displayed formulas preceding Theorem 1.1:
https://doi.org/10.1007/s40993-017-0090-x
* Tomiyama, arXiv:2505.05135v4, Proposition 3.7 and Theorem 3.9:
https://arxiv.org/html/2505.05135v4 . Our CM argument is an algebraic
integer, and the build separately checks monic integer coefficients.
* T634, https://numberdb.org/Values_of_monstrous_McKay_Thompson_series_at_rectangular_CM_points
supplies the same normalized scalar values at these arguments.
* T129, https://numberdb.org/Hilbert_class_polynomials gives H_{-4r}.
* T309, https://numberdb.org/Canonical_modular_polynomials specifies
m_p=p^{12/(p-1)}/R_p and its relation with j, rather than the inverse
eta ratio R_p used in the McKay–Thompson formulas.
This table stores the minimal polynomial of one specified value, not the
resultant or a class polynomial over all level structures. All irreducible
factors other than the one containing that value must be excluded by certified
bounds. T308's normalized Weber invariant and its discriminant restrictions
are different. T635's universal Faber polynomials are also different objects.
The independent critique must assess both parameters, their interaction,
correct-factor identification, and all attached source dependencies.