# Argument selection for T680

This build follows family issue numberdb-data#301 and the current archived skill:
`agents/runs/20261102T145807Z-build-skills/manifest.json`. Both the local and live
archives have SHA-256 `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The independent critique must assess the selection itself; this build does not
supply an acceptance sidecar or offer the draft.

## Template: discrete_index

Use all fourteen cyclotomic numerator multisets in Long–Tu–Yui–Zudilin,
Table 1 and Section 5 (arXiv:1705.01663v4). The generator parses the reduced
fractions of each cyclotomic factor, with multiplicity. Independent checks use
an explicit list of four upper fractions for each template. These four
fractions are not freely sampled parameters. The template order is
2.2.2.2, 2.2.3, 2.2.4, 2.2.6, 3.3, 3.4, 3.6, 4.4, 4.6, 6.6, 5, 8, 10, 12.
It agrees with T679, T674 and T675. This is complete classification coverage
before any single system is extended more deeply.

## Rational fiber z: mixed, with two disjoint exhaustive groups

* **intrinsic**, exact identity `1`: the conifold is the other finite singular
  point of the equation normalized to have singularities 0, 1, infinity. The
  finite limit uses continuation along 0<z<1. The local exponents are 0,1,1,2;
  logarithmic terms must be retained during matching even though their values
  vanish in the limit. Krattenthaler–Rivoal, Section 4, Lemma 4.2, and
  Bönisch–Klemm–Scheidegger–Zagier, Section 2, support this boundary convention.
* **arithmetic_structure**, exact identities `1/2`, `-1`, `2`: all rational
  solutions of z and 1-z both being units of Z[1/2]. This constrains the
  primes at which the global fiber can have bad reduction. Roberts–Rodriguez
  Villegas, Section 10, states the criterion in terms of numerator and
  denominator of z and numerator of z-1, in addition to template primes.
  T674 and T675 cover these same smooth fibers; their common indexing is a
  useful analytic-to-arithmetic comparison, not the sole selection rationale.

The choice of the allowed extra-prime set {2} is a noncanonical reference
coverage choice, independent of numeral notation. It is the smallest nonempty
prime set and supplies negative, interior positive and continued complex
regimes with only three smooth fibers. It makes no claim that these fibers
are all CM, modular, or individually famous. The group is the entire finite
S-unit solution family once this arithmetic restriction is selected, not a
sample of a dyadic mesh or a hidden reference_sample group.

For completeness of that arithmetic group, write the units as signed powers
of 2. When 0<z<1, the two positive powers of 2 sum to 1, so after factoring out
the smaller power the odd factor must be 1; the powers must coincide and both
are 1/2. When z is outside [0,1], two positive powers of 2 differ by 1.
Writing the difference as 2^m(2^k-1)=1 forces m=0 and k=1, giving z=-1 or 2.
Endpoints 0 and 1 are not units; 1 is included only in the separate intrinsic
group. There are no additional chosen rational arguments.

Enumeration is 1, 1/2, -1, 2 at every template. The conifold plus all three
S-unit fibers account for every one of the 56 underlying pairs. The origin is
only a normalization control: duplicating q(0)=0 under all templates would
not extend identification coverage. Larger S, inverses, other ranks and
integral period vectors are separate future selections or named quantities.

## Normalization: discrete_index

Use `unit` for Q_A(z)=z exp(G_A/F_A) and `integral` for q_A(z)=Q_A(z)/C_A,
in that order at each fiber. This is one number in two invertible conventions;
the conversion uses a known integer C_A and neither loses information nor
changes the fiber parameter. Krattenthaler–Rivoal, Section 1, equations
(1.1)–(1.4), gives the scaling. Their variable is x=z/C_A and their canonical
coordinate at x is our q_A(z). In x it begins x and has integral coefficients.
The parameter does not select F, G, a derivative, or the compositional inverse.

The result has 112 entries. The finite scope comes from the classification and
arithmetic restriction, not a size target. Every real component and each
nonzero complex component aims at 100 significant digits. The complete-note
states this exact coverage and why it was chosen, while the parameter domain
remains all nonzero rational fibers with the declared boundary prescription.