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4139 bytes, as of the version from 2026-10-31 10:48 (current). Recorded here, not run.
# Argument selection for T635
This is the build's justification, not the independent critique required by the
campaign. The archived skill manifest is
`agents/runs/20261031T094636Z-build-skills/manifest.json`, also attached as
`skill-manifest.json`. Both archived skill copies have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
## Quantity and conventions
The one stored quantity is the exact monic Hecke-type Faber polynomial
P_(n,g)(X), determined by P_(n,g)(T_g(tau)) = q^(-n) + O(q). The nome is
q=exp(2*pi*i*tau), the constant coefficient of T_g is zero, and T_1A=j-744.
The indexing order is (g,n); X is the polynomial variable, not a free selected
argument. There is no division of P by n. These conventions agree with family
issue #254 and the live CM-value companion T634, read before calculation.
## g: discrete_index
The selected classes, in display order, are 1A, 2A, 2B, 3A, 3B, 5A, 5B, 7A,
7B, 13A, 13B. The ten nonidentity classes include both the Fricke and ordinary
Hauptmodul for every prime p for which X_0(p) has genus zero. Thus all five
primes 2,3,5,7,13 are represented before increasing any one class's degree
range. The identity class supplies the classical j-744 anchor. Class g names
which Monster element supplies the graded trace, not a second quantity.
This is a structurally delimited discrete family. It is not a continuous input
mesh or a selection of only famous low-order examples. Other prime Fricke
classes have no ordinary genus-zero X_0(p) partner and are outside this initial
slice; composite levels and other Monster classes also remain uncomputed.
The parameter domain in the table remains Monster conjugacy classes; the
finite selection is recorded in the completeness note.
Checked source: Matsusaka, *The Fourier coefficients of the McKay–Thompson
series and the traces of CM values*, Research in Number Theory 3 (2017),
article 23, §1, the eta formulas preceding Theorem 1.1,
https://doi.org/10.1007/s40993-017-0090-x . This identifies all five primes,
the labels pA and pB, and both eta expressions. Bannai–Kojima–Miezaki,
*On the zeros of Hecke-type Faber polynomials*, Kyushu J. Math. 62 (2008),
15–61, pp.16–17 and Definitions 2.2–2.3, fixes the ATLAS convention and
zero constant coefficient; https://doi.org/10.2206/kyushujm.62.15 .
## n: discrete_index
The selected degrees are every integer 2 through 20 for each selected class.
Degree is the pole order q^(-n) as well as the polynomial's degree. This is a
complete initial segment of the nontrivial indexed family for each g. Degree
one is omitted because P_(1,g)=X for every class and is fully stated in prose.
Degree zero is only an auxiliary recurrence initial condition, not an entry.
The degree-20 cutoff is a noncanonical practical reference extent, retained
from the proposal after measurement rather than assumed from it. The 209
polynomials occupy approximately 45.0 KiB in the standard dry-run estimate;
the longest is P_(20,1A), 748 characters. The largest polynomial remains
readable and the block leaves substantial extension room. This covers all
eleven useful structural class slices before extending depth, supplies an
initial reference for replication and polynomial-zero computations, and does
not attempt an exhaustive Monster catalogue. The planned zero table is a
separate quantity and has not been built in this run.
Checked source: Bannai–Kojima–Miezaki, §2.2 (pp.21–22), defines the initial
polynomial family and its cancellation and recurrence calculations. The
source studies degrees through 50; our smaller cutoff is a readability choice,
not a claim that 20 has intrinsic mathematical significance.
There are no continuous, reference_sample, or mixed-role parameters. The
product of eleven class indices and nineteen degrees is an initial segment
of one discrete polynomial family, not a Cartesian mesh of exploratory real
inputs. Enumeration, completeness note and the Programs example use the same
normalization. A separate critique must independently assess both parameters
and fingerprint generate.py plus any reviewed source dependencies.