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5316 bytes, as of the version from 2026-11-02 14:37 (current). Recorded here, not run.
# Argument selection for T679
This build uses the coordinated skill archived in
`agents/runs/20261102T134642Z-build-skills/manifest.json`. Both `skill-local.md`
and `skill-live.md` have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The family proposal is numberdb-data issue 301, read before the corpus checks.
This note is the builder's justification, not an independent acceptance.
## Quantity and conventions
The only quantity is F_A(z), the holomorphic solution normalized by F_A(0)=1
of theta^4 F - z product_j(theta+alpha_j) F=0. The classical lower parameters
are (1,1,1); the arithmetic lower multiset consists of four zero residues.
The conifold is z=1. For z>1 the table uses the upper-bank limit. No derivative,
logarithmic period, mirror coordinate, absolute value or integral-cycle period
is mixed into this table. Real fibers have imaginary component exactly zero.
## Template: discrete_index
All fourteen established rank-four hypergeometric Calabi–Yau systems are
included. A cyclotomic key selects its entire multiset, with multiplicity;
the four upper parameters are not independent free choices. The key 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.
The classification and the holomorphic series were checked in Long–Tu–Yui–Zudilin,
Introduction, Table 1 and Section 5:
https://arxiv.org/html/1705.01663v4 .
Bönisch–Klemm–Scheidegger–Zagier, Section 2.1, Table 1, supplies a second
parameter list and the local exponents. Their variable is mu times our z:
https://arxiv.org/html/2203.09426v1 .
The definition selects the rank-four differential system, without identifying
it with every geometric model's full middle cohomology.
## Rational fiber z: mixed, exactly two exhaustive disjoint groups
* `intrinsic`, identities ["1"]. This is the finite conifold limit. All
fourteen series have excess 3-sum(alpha)=1, so their coefficients are
asymptotic to a constant times n^-2 and the value exists. The conifold
location and local exponents 0,1,1,2 are intrinsic to this normalized system.
* `arithmetic_structure`, identities ["1/2", "-1", "2"]. These are precisely
the rational solutions of z and 1-z both being units in Z[1/2]. The choice
of permitted extra primes {2} is a noncanonical, notation-independent
reference-coverage choice; completeness of this finite arithmetic family
does not make the choice canonical. This is not a claim that every fiber
is CM, modular, or individually famous.
For 0<z<1 both units are negative powers of 2. Multiplying their sum by the
larger denominator shows that a sum of an odd number and a power of 2 is a
power of 2 only when both original terms equal 1/2. For z outside [0,1],
two positive powers of 2 differ by 1. Factoring their smaller power and using
2-adic valuations forces the smaller to be 1 and the larger 2. The solutions
are -1 and 2. Thus this is an exhaustive S-unit slice, not a dyadic grid.
The purpose of this slice is to compare analytic invariants of global fibers
with tightly controlled possible bad reduction. Roberts–Rodriguez Villegas,
Section 10, "Wild, tame, and good primes", states the dependence on the
numerator and denominator of z and z-1:
https://arxiv.org/html/2109.00027v1 .
The same three global fibers occur in the live T674 Frobenius-polynomial and
T675 unit-root drafts. Reading their definitions, coordinates and entries
confirmed compatibility. Links select actual local-factor and unit-root entries
of each smooth fiber; there is no local-factor link at the singular conifold.
## Extent and breadth
The order is template first, then z=1,1/2,-1,2, giving exactly 56 entries.
Every template has a negative real, interior positive real, conifold-limit and
continued complex value. Covering all these regimes is preferable to extending
one quintic example. The bounds are the complete 14-element classification and
the complete finite S-unit solution set for the selected prime set, together
with the conifold. No computational cutoff removes difficult cases. Expanding
the permitted prime set would change the coverage choice and require its own
review. Unrelated small rationals would require an explicit reference_sample
policy. No such hidden reference_sample group is included here.
The origin is a control, not 14 repeated exact-one rows. The natural domain is
nonzero rational z with the stated boundary prescription, so completeness is
"no". The completeness note states these exact selected families and why.
The pilot at 130 working digits covered both the all-half repeated exponents
and template 12 in every regime. The final run uses 140 and 190 digits and
independent checks, reporting heuristic agreement rather than a proof.
## Corpus and tags
The repeated screen.py already_here and already_asked checks found no duplicate.
Lookup of 1.037915197940 and 1.023094497870 found no matches. T25, T178, T348,
T665, T674 and T675 were read and their actual long URLs recorded. T323 contains
level-one L-values and does not cover the level-eight modular control.
Existing tags are number theory, special functions, special values and period.
Calabi-Yau is absent; propose it for T679, T674, T675 and the planned canonical
coordinate companion. No new tag or changes to companions were submitted.