# Argument selection for T670

The only free parameter is the negative order discriminant Delta, of type Z,
with Delta congruent to 0 or 1 modulo 4. Its role is arithmetic_structure:
it identifies the endomorphism order of a CM elliptic curve. The unique
factorization Delta=f^2 D recovers the fundamental field discriminant and
positive conductor. Neither D, f, ideal classes, class numbers, nor Gamma
arguments are additional independently selected inputs.

We choose every order with |Delta|<=1000, in increasing absolute value,
beginning -3,-4. This covers 500 orders, including 305 maximal orders and
195 nonmaximal orders, with f<=18. The cutoff 1000 is a noncanonical practical
choice, not an intrinsic mathematical boundary or a count target. It keeps
each defining Gamma sum below 1000 terms and permits a separate complete
eta-class-product check for every row at 100 significant digits. The entries
have essentially constant written length, about 100 digits, so this adds
arithmetic breadth without long expressions or a multidimensional mesh.

Breadth was considered before depth. Restricting to class number one would
omit nonrational singular moduli such as discriminant -23. Restricting to
maximal orders would omit conductor effects. This complete discriminant
interval includes both discriminant congruence classes, class number greater
than one, split, inert and ramified conductor primes, prime-power conductors,
and conductors with several prime factors. D and f are coupled by the order
bound |D| f^2<=1000; taking independent lists of D and f would be a different
and less compact selection. The range extends beyond the rational curves in
T211, while T129 provides the same order index for its Hilbert class polynomials.

There is no exploratory continuous input or numerical reference mesh. The
arguments a/|D| of Gamma are forced by the quadratic character formula.
The role is uniform across maximal and nonmaximal orders, so no mixed or
reference_sample classification is proposed. No claim of independent demand
for each row is needed: orders organize a named arithmetic invariant.

Sources checked: Barquero-Sanchez, Cadwallader, Cannon, Genao and Masri,
Faltings heights of CM elliptic curves and special Gamma values (2017),
section 1 and Theorems 1.3 and 5.1, DOI 10.1007/s40993-017-0077-7;
Li, Long and Tu, Computing Special L-Values of Certain Modular Forms with
Complex Multiplication (2018), section 2.1 equation (2.1),
DOI 10.3842/SIGMA.2018.090. The finite cutoff is this table's choice.
The full coverage rationale is also in Data properties.complete-note.

Run instructions: agents/runs/20261102T053137Z-build-skills/manifest.json.
Both archived skill versions have SHA-256
 a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278.
This is the builder's justification. A separate critique must independently
assess it and fingerprint generate.py and this attachment against the live draft.