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# Argument selection for T634
This build uses the coordinated skill manifest at
`agents/runs/20261031T091932Z-build-skills/manifest.json`.
Both archived versions have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The local canonical source also matches. This is the builder's justification;
the controller must obtain an independent critique and its argument sidecar.
## One quantity and two parameters
Every row is the normalized monstrous McKay–Thompson value
T_g(i sqrt(r)), with zero constant term and q=exp(2 pi i tau).
The class g identifies the Monster group element up to conjugacy in the
single class-indexed family of graded traces; it does not switch quantities.
The number-header applies to every row. No Fourier coefficients, Faber
polynomials or polynomial zeros are stored as values in this table.
## g: discrete_index
The selected classes in order are 2A, 2B, 3A, 3B, 5A, 5B, 7A, 7B, 13A, 13B.
They include both Fricke and non-Fricke classes for every prime p for which
the ordinary modular curve X_0(p) has genus zero. Thus the level selection is
a complete structural subfamily, rather than five primes chosen by a cutoff.
Matsusaka, Introduction, the Hauptmodul formulas immediately before Theorem
1.1, explicitly identifies these classes and eta quotients:
https://doi.org/10.1007/s40993-017-0090-x .
The zero-constant convention is fixed by that formula, including 13A.
The identity class, composite levels and other Fricke primes are outside the
initial coverage; none is inferred to be uninteresting. This follows the
family boundary in proposal #254 without extending this task to another table.
## r: arithmetic_structure
The positive integer r identifies the primitive principal binary quadratic
form [1,0,r], its upper-half-plane root i sqrt(r), and the rectangular lattice
Z+Z i sqrt(r). Its endomorphism order is Z[sqrt(-r)], of discriminant -4r:
a multiplier preserving the lattice belongs to the lattice by applying it to
1, and every a+b i sqrt(r) with integer a,b preserves it. Squared factors
therefore specify distinct orders or lattices and are retained. This is a
family-specific arithmetic reason for integer inputs, not a real input mesh
relabeled as discrete. The cyclic level structure is fixed by the sublattice
Z+Z p i sqrt(r), and is not independently varied.
The finite extent is all 1 <= r <= 30, or 4 <= |D| <= 120 with D=-4r.
This is our noncanonical practical cutoff for a reference covering small
principal rectangular CM orders. It gives breadth across fields and conductors
before deeper coverage in one field, keeps the smallest cases first, and
retains all ten classes at each chosen order. It is not a claim of documented
use for each individual point. The interactions of g and r have one meaning:
the same class-indexed invariant on each fixed CM lattice and its specified
level structure. This is not a Cartesian product of independently chosen
continuous reference meshes. General CM ideal classes, nonrectangular
representatives and other level structures would require separate selection.
Matsusaka, Introduction, defines CM arguments using positive definite binary
quadratic forms. Tomiyama, Generalized modular equations and singular values
of Hauptmoduln, Corollary 3.11, proves algebraic integrality of these CM values:
https://arxiv.org/html/2505.05135v2 . Neither source selects the bound 120;
that reference extent is explicitly ours. DLMF 23.17.8 fixes the eta product
and 23.18.5 fixes the transformation convention. DLMF uses exp(pi i tau) as
its nome, whereas this table consistently uses exp(2 pi i tau).
## Breadth and consistency
The 300 rows cover all ten selected classes and all thirty orders, rather
than only Fricke fixed points or individually famous CM values. Expanding
r instead of the genus-zero class breadth would omit a structural part of the
proposal. Extending to all CM classes or other levels would change its scope.
The measured storage and precision results are in numerical-evidence.md.
The generator's default enumeration and completeness note use exactly this
selection; the parameter declarations retain the full domains of g and r.
No group of exploratory continuous reference inputs is hidden in this
arithmetic family, and no quantiles are included.
## Corpus and tag review
The current corpus was searched by the prescribed already_here and
already_asked helpers, by McKay, moonshine and Hauptmodul, and through
api/lookup at expected values. It has eta ingredients (T506), q-coefficients
(T79, T80), modular equations (T309), rational singular moduli (T83), Hilbert
class polynomials (T129), Weber invariants (T297), and sporadic character
degrees (T255), but no existing table of this quantity. T96, T215 and T217
concern different polynomial objects. Full documents and searches are archived.
Use existing number theory, modular function, groups, special values and
algebraic tags. The live tag index lacks moonshine and complex multiplication.
Propose moonshine for this table, the two planned Faber companions, T79 and
T255; propose complex multiplication for this table, T83, T129, T297 and T506.
No published table is retagged and no unbuilt companion receives a guessed link.