# Argument selection for T671

This assessment uses the coordinated skill archived at
`agents/runs/20261102T064212Z-build-skills/manifest.json`.
Both archived versions have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
It is the builder's rationale; the controller requires a separate independent
critique and fingerprint-bound argument review before offering this draft.

The quantity is the genus-character sum of J_m at CM classes of discriminant
-Dd, with stabilizer weights and imprimitive forms included. The integer
coefficient normalization divides by the positive sqrt(D). It is invertible
on the same indices, has similar displayed lengths, and costs one square root.

* **m: discrete_index.** The positive integer is the pole/Hecke order of the
  canonical modular function J_m = q^(-m) + O(q), with q = exp(2 pi i tau).
  Orders 1 through 4 cover the original function, both first prime operators,
  and the first composite operator. The cutoff is a practical initial segment,
  not a claim of canonical completeness. Increasing this index grows the
  values exponentially, so breadth in discriminants comes before larger m.
* **D: arithmetic_structure.** Positive fundamental discriminants D>1 select
  the nontrivial real quadratic genus characters, not a numerical input grid.
  Every admissible D in the coupled bound Dd<=300 is included. This covers
  every positive fundamental discriminant through 100, rather than only the
  familiar character of discriminant 5.
* **d: arithmetic_structure.** The negative order discriminant -d is allowed
  to be fundamental or nonfundamental and to share prime factors with D.
  Every d>=3 congruent to 0 or 3 modulo 4 with Dd<=300 is included. The
  product bounds the actual CM discriminant, controlling the number of reduced
  forms and the exponential height of the CM values. It is not a Cartesian
  product of unrelated simple inputs. The finite bound 300 is noncanonical
  and independent of numeral notation; it keeps full orbits inexpensive and
  readable while traversing fields, conductors and all character cases.
* **normalisation: intrinsic.** Exactly two fixed conventions, trace and
  coefficient. For known D, t_m=sqrt(D) A_m and A_m=t_m/sqrt(D), with the
  positive square root. Neither selector changes the mathematical quantity
  or the chosen range. Signs and exact zeros are preserved.

The 139 discriminant pairs are assessed in four arithmetic subgroups:
84 with fundamental -d and gcd(D,d)=1; 32 with fundamental -d and shared
prime factors; 13 with nonfundamental -d and coprime discriminants; and 10
with nonfundamental -d and shared prime factors. The first samples maximal
order CM orbits; the second covers ramified/shared-prime character behavior;
the last two include nonmaximal orders and the imprimitive contributions.
The generator checks no output-dependent threshold. All four groups are
arithmetic_structure, so no exploratory reference_sample group is hidden
inside an intrinsic claim. Four orders and two normalizations give 1112 rows.

Sources read: Zagier, *Traces of Singular Moduli*, sections 6–7, equations
(16), (20), (21), and the remarks following Theorem 6; Jenkins, *Traces of
Singular Moduli, Modular Forms, and Maass Forms*, section 5.1, equation (5.1)
and Theorem 5.1; Andersen–Duke, *Asymptotic Distribution of Traces of Singular
Moduli*, section 2, equations (2.1)–(2.7). Jenkins explicitly covers general
negative order discriminants, avoiding extrapolation from the coprime
fundamental statement alone. Andersen–Duke specifies the zero character and
represented-integer rule. The sources establish the family and conventions;
the finite product and order cutoffs are our reference-coverage choices.

The family map and all eleven listed live companion documents were read.
T643 uses the same J_m, weights and imprimitive convention without the genus
character. T635 supplies the identity-class Faber polynomials, T79 the j
expansion, and T129 the primitive CM orbit polynomials. Their existing
coverage does not provide genus labels or the signed twisted sums.