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# Argument selection for T652
This table holds one named quantity: the signed coefficient c_g(n) of the
normalized simple-pole Monster McKay–Thompson series. The class g identifies
which element acts on the graded space; it does not select a second quantity.
The exponent n selects the canonical shifted grading. The coefficient of q^-1
is 1 and the constant term is zero. They are omitted. The identity-class
positive coefficients already occur in T79 and are also omitted.
Both parameters are discrete_index selections. These classifications are the
builder's justifications; independent acceptance belongs to the later critic.
## Monster class g
The domain is the ATLAS conjugacy classes of the Monster. The selection is
all 32 nonidentity classes of element order at most 10:
2A, 2B; 3A, 3B, 3C; 4A, 4B, 4C, 4D; 5A, 5B;
6A, 6B, 6C, 6D, 6E, 6F; 7A, 7B;
8A, 8B, 8C, 8D, 8E, 8F; 9A, 9B; 10A, 10B, 10C, 10D, 10E.
The bound is a noncanonical practical reference extent of a genuine finite
discrete family. Element order is independent of numeral notation. Including
every class in this segment supplies breadth across prime and composite
orders, Fricke and non-Fricke behavior, and sparse and dense series. With 1A
included internally this segment is closed under powers, supporting exact
replication tests without importing unselected high-order series. It extends
beyond the prime-class examples in T634–T636, rather than only deepening them.
No arbitrary continuous coupling is attached to the class index.
The class list, orders and power maps are from GAP CTblLib 1.3.11, the Monster
character table in ATLAS order. Labels are converted to uppercase. The
archived class-data.json records the full class list and orders and the
selected power maps. The first 33 class labels and prime-power identifications
agree with Conway–Norton Table 2 (printed p. 327); Duncan–Griffin–Ono's appendix
provides modern modular-group conventions. Do not infer a power class solely
from element order, or treat matching individual coefficients as aliases.
## Positive Fourier exponent n
The domain is all positive integers, with q=exp(2*pi*i*tau). We keep every
1 <= n <= 30 for each selected class, including zeros and negative traces.
The shared initial graded segment is appropriate because every selected
class acts on the same canonical spaces V_n^natural. It supports recognition
of coefficients computed as modular expansions or symmetry-inserted graded
traces. The bound 30 is noncanonical: it extends past Conway–Norton's ten-term
controls, while keeping each integer readily readable and verification modest.
It is not an attempt to fill a rectangular mesh of unrelated numerical inputs.
Higher orders and deeper exponents can extend this same table after validation.
The final measured sizes and check counts are recorded in numerical-evidence.md.
## Sources and normalization
- Duncan, Griffin and Ono, Moonshine, §3, equations (3.1)–(3.4),
https://arxiv.org/html/1411.6571#S3: the shifted module grading, trace
definition and zero-constant normalization; Appendix A: Monster classes.
- Conway and Norton, Monstrous Moonshine (1979), §10 and Tables 1a, 2, 4,
https://gwern.net/doc/math/1979-conway.pdf: head characters, labels,
prime powers and initial coefficients. The H_0 row in Table 4 is a
Rademacher constant, not the Fourier constant term (printed p. 323).
- Borcherds, Monstrous moonshine and monstrous Lie superalgebras (1992),
§9, equation (9.1), https://math.berkeley.edu/~reb/papers/monster/monster.pdf:
exact recurrences and the determination by the n=1,2,3,5 seed coefficients.
- GAP Character Table Library 1.3.11,
https://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/:
independently recorded class data and low-degree character values.
There are no continuous reference samples or quantiles. The cutoffs are
reference choices within two discrete indices, not intrinsic maximum indices.
The ordered enumeration is element order, class letter, then exponent.
The generator, completeness note and entry count must remain consistent.
The skill copies were archived at
agents/runs/20261101T140515Z-build-skills/manifest.json in the run source snapshot.
Both local and live copies have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.