# Argument selection for T666

This build follows the coordinated skill archived in
`agents/runs/20261102T021426Z-build-skills/manifest.json`. Both `skill-local.md`
and `skill-live.md` were read and have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The source snapshot's canonical skill has the same bytes. The attached
`skill-manifest.json` preserves the runner's version records. This is the
builder's justification, not the independent critique required by the controller.

The object is the numerator P_(N,p)(T) of the zeta function of the smooth
projective curve X^N + Y^N = Z^N over F_p. It has constant coefficient 1;
its reversed characteristic polynomial is not a second quantity here. There
are exactly two free parameters, N and p. The summation indices a,b are
internal to a full cohomology product, not independently selected arguments.

## Degree N: discrete_index

The domain is every integer N >= 3. The current enumeration contains every
degree from 3 through 8. This initial segment includes genus 1 and higher
genus curves, odd and even equations, prime degrees 3,5,7, and composite
degrees 4,6,8 with imprimitive cohomological contributions. It is a geometric
family with a fixed equation, not evaluations of an arbitrary function at an
integer mesh. The global equation identifies which curve produced a local
polynomial, even when another curve or isogeny class has the same polynomial.

The cutoff 8 is a practical, noncanonical choice. Genus has reached 21 and
polynomial degree 42. The preflight measured 48 entries, a 15.97 KiB entries
block estimate, and a maximum expanded length of 1335 characters at (8,97).
The two entries at (8,73) and (8,97) are 1332 and 1335 characters respectively,
slightly beyond the dry-run advisory of about 1100 to 1300. They are retained
to finish the selected arithmetic slice; this is not a hard-limit exception.
Their character-orbit factorizations are attached as readable context.
Extending the curve degree would increase both polynomial degree and coefficient
length. The period companion reaches degree 10 because its scalar entries
have a different readability cost; this is the reason for the different extent.

## Rational prime p: arithmetic_structure

The domain is every rational prime p not dividing N. Current coverage takes
all primes p <= 101 with p congruent to 1 modulo N. Such primes split
completely in Q(zeta_N). Every character eigenspace can therefore be handled
over F_p itself, and the full local factor is an integer product of the
Jacobi-sum eigenvalues. The relation couples p to N. It is not a Cartesian
product of arbitrary small inputs and is independent of numeral notation.

The bound 101 is a noncanonical reference budget within this arithmetic
family. It gives several split reductions per curve while retaining enough
curve degrees to compare prime and composite degrees. The numbers of rows
for N=3,4,5,6,7,8 are respectively 11,12,6,11,3,5. All qualifying primes in
the slice are included; none is filtered by an interesting-looking value.
Nonsplit good primes require Frobenius orbits over extension fields and are
left for a later extension. Bad primes need different local data and are
outside the declared domain. Missing primes do not mean that their factor is 1.

## Sources, conventions and boundaries

Otsubo, *On special values of Jacobi-sum Hecke L-functions*, arXiv:1404.7476v1,
Sections 2.1 and 2.2, fixes x^N+y^N=1, the index set of all ordered nonzero
(a,b) with a+b nonzero modulo N, the Galois-orbit decomposition, and the local
Jacobi-sum eigenvalues. Section 2.2 uses j = -sum; T143 and this generator use
the positive sum J, so a numerator factor is 1+J*T. For even N no change to
X^N+Y^N+Z^N=0 is made. The full index set, including imprimitive pairs and both
coordinate orders, is retained with multiplicity. No simple-motive or
polarization-dependent scalar is inferred from a polynomial factorization.

The live documents T143, T214, T216 and the existing draft T665 were read.
T214 uses the reversed characteristic-polynomial convention and class labels;
T665 uses the same global model and holomorphic half of the character index
set. The LMFDB reviewed knowl `av.fq.weil_polynomial` confirms the reversal
convention. The family proposal #294 supplies shared conventions, not an
independent acceptance of this selection. No continuous reference_sample or
mixed-role parameter group is used, and no statistical quantile is involved.

The generator enumerates increasing N and then increasing p. Its defaults,
the 48 selected identities, and the completeness note agree. A separate
critique must independently review these choices and fingerprint the live
selection and attached source files before the controller offers this draft.