# Argument selection for T665

This is a build justification, not an independent critique or an acceptance.
The governing archived manifest is
`agents/runs/20261102T013956Z-build-skills/manifest.json`; the attached
`skill-manifest.json` preserves its contents. Both local and live skill bytes
have SHA-256 `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The source snapshot has the same digest. Family issue 294 was read before
selection; its coverage map distinguishes these marked periods from beta
values, real covolumes, Jacobi sums, and local zeta polynomials.

## Every free parameter

* `N` is a discrete family index: the degree of the smooth projective curve
  X^N+Y^N=Z^N. Its domain is all integers N>=3. The selection includes every
  N from 3 through 10. This initial segment spans the elliptic cubic and
  higher-genus curves of prime, even and composite degree, including both
  primitive characters and characters pulled back from lower degree. The
  stopping point is a noncanonical practical choice. The largest curve has
  genus 36, so its complete holomorphic basis remains easy to inspect.
* `a` is a discrete character index, with 1<=a<=N-2. For each selected N,
  every such a is included, with every compatible b. It identifies the
  alpha-eigencharacter of the algebraic differential, not an independently
  sampled beta argument.
* `b` is a discrete character index, with 1<=b<N-a. Every compatible b is
  included. The triangular coupling a+b<N is exactly the holomorphic
  condition; no rectangular parameter mesh is introduced. The ordered pairs
  (a,b) and (b,a) identify different basis vectors when a!=b even though their
  scalar integrals on this particular cycle agree.

There are 120 entries, ordered lexicographically by (N,a,b), with counts
1,3,6,10,15,21,28,36 for N=3,...,10. The count is a consequence of complete
basis coverage, not a size target. The breadth review retains every basis
vector before considering deeper coverage in N, and includes imprimitive as
well as primitive characters. It excludes translated cycles as separate rows
because their integrals follow by multiplying by roots of unity, as recorded
in conventions.md. It does not add normalized periods, covolumes, regulators,
or the family's separate zeta-polynomial quantity.

All parameters have type Z and the role discrete_index. There are no
reference_sample groups, continuous choices, twists, free moduli, or hidden
probability arguments. The beta inputs a/N,b/N are forced by the algebraic
curve and differential. The source establishes the family and conventions;
it does not prescribe the cutoff 10. The completeness note and generator
must continue to state precisely this initial segment and full basis.

## Sources actually checked

* Otsubo, *On the Abel-Jacobi maps of Fermat Jacobians*,
  https://arxiv.org/html/1003.0357v1, Section 4.1: the real path, integral
  cycle, unnormalized differential omega_0, holomorphic index set, and (4.1).
  That section starts at N>=4 for its Ceresa-cycle application. The path and
  differential calculation also applies to N=3; the independent derivation
  in conventions.md and the next source cover the elliptic case.
* Otsubo, *On special values of Jacobi-sum Hecke L-functions*,
  https://arxiv.org/html/1404.7476v1, Sections 2.1, 2.2 and 5.1, especially
  (2.3), the holomorphic residue condition, and (5.2). These cover primitive
  reduction, character orbits and the integral cycle for general degree.
* https://dlmf.nist.gov/5.12.E1 gives Euler's beta integral and gamma quotient.
* https://dlmf.nist.gov/19.8 gives the independent AGM elliptic-period control.

Live companions T177, T9, T25, T143, T213, T214 and T216 were inspected.
The new values retain the algebraic differential's scale and the cycle's
complex phase, unlike the real beta ingredients. T213 holds a covolume,
not a marked scalar period. No existing family duplication was found;
three complex-number lookups also returned no matches. The draft title
was claimed before the full computation. A future zeta companion has no
invented identifier or link here.