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# Argument selection for T636
This table holds one quantity: a zero of a Hecke-type Faber polynomial associated to a Monster class. The parameters identify the class, the polynomial degree, and its distinct roots. The canonical local and live skills were read from ../20261031T100952Z-build-skills/manifest.json; both SHA-256 values are a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278. This is a build justification, not the independent critique required by the campaign.
- `g` (Symbolic; discrete_index): select 1A and both pA and pB for every prime p whose ordinary modular curve X_0(p) has genus zero: p=2,3,5,7,13. The identity series is the classical j-744 anchor; the two classes at each prime represent the Fricke and non-Fricke Hauptmoduln. These are class indices of the same Monster module's trace construction, not unrelated functions sampled on a common real mesh. Matsusaka, section 1 and the displayed eta formulas, supports the class assignments and the complete prime-level structural family. Other Monster classes and composite levels are outside this finite class selection.
- `n` (Z; discrete_index): polynomial degree is an intrinsic positive integer index. Include every 2<=n<=10 for every selected class. P_{1,g}=X has the same trivial zero for every class and is stated in the prose instead of repeated. Degree 10 is a noncanonical practical reference cutoff: the total root count grows quadratically with maximum degree, and this bound includes low-degree real and complex behavior across all structural classes before extending just one class's depth. It is not a decimal grid, a requirement of the function, or an assertion of an application for each degree. The companion polynomial table covers degrees through 20; roots and polynomials have different display costs.
- `k` (Z; discrete_index): enumerate every distinct zero for each selected (g,n), in lexicographic order of its exact algebraic real and imaginary parts, starting at one. The upper endpoint is determined by the polynomial, not a chosen coupling or sample size. Multiplicities are comments, not duplicated rows. No conjugate, sign, or nonreal root is discarded. Rational roots are exact; all remaining components are certified enclosures targeting 100 significant digits.
Bannai, Kojima and Miezaki, On the zeros of Hecke-type Faber polynomials, Kyushu J. Math. 62 (2008), 15–61, DOI 10.2206/kyushujm.62.15, studies precisely these X-plane roots; see pp. 16–17, equations (4)–(5), and sections 2 and 4–5. Matsusaka, Research in Number Theory 3 (2017), article 23, DOI 10.1007/s40993-017-0090-x, section 1, fixes the eta and class conventions. The sources justify the family, not the practical degree cutoff.
Breadth review: all five structural prime levels, both A/B cases, and the classical anchor are covered. Restricting to real roots or to 1A and 2A would omit the complex configurations that motivate this table. No continuous argument, exploratory reference_sample group, quantile, or mixed parameter role is concealed. Enumeration and completeness state the same class and degree selection. The independent critique must inspect all three parameters and fingerprint generate.py; it must not accept this document as an independent verdict.