back to table · edit · history · where entries came from · files · download
4209 bytes, as of the version from 2026-11-02 11:34 (current). Recorded here, not run.
# Argument selection for T674
The selected quantity is one Frobenius polynomial of one hypergeometric motive
at one good prime. Parameters are ordered (template, z, p); none chooses a
second quantity. The family domain has arbitrary rational z other than 0 or 1
and arbitrary good primes, while this computation is a finite part of it.
## Template: discrete_index
All fourteen rank-four, weight-three rational Calabi–Yau templates are included.
They are the ten unordered pairs from {1/2,1/3,1/4,1/6} and the four pairs
(1/5,2/5), (1/8,3/8), (1/10,3/10), (1/12,5/12), with upper multiset
(r,1-r,s,1-s). The denominator is Phi_1^4. Stable keys list numerator
cyclotomic indices in increasing order. Long–Tu–Yui–Zudilin, Table 1 and
Section 5, establish this class. It has fourteen distinct monodromy templates;
we do not take products of independently chosen upper hypergeometric parameters.
Covering the entire class takes priority over deepening one familiar template.
## Rational fibre z: arithmetic_structure
In the classical hypergeometric coordinate, take every rational solution of
z,1-z in Z[1/2]^×. These are precisely -1, 1/2 and 2. To see exhaustiveness,
write each unit as a signed integral power of 2. When both terms are positive,
clear the least power of 2: their sum is 1 only when both terms are 1/2.
When one is negative, the difference of two positive powers of 2 is 1;
factoring out the lesser power forces it to be 1 and the larger power to be 2.
This gives z=-1 or 2. The argument permits negative exponents and proves the
whole S-unit equation, not just a denominator-limited search.
The good-prime criterion makes this arithmetic restriction relevant: these
fibres introduce no potentially bad primes outside {2} beyond the template's
wild primes. The allowed extra-prime set {2} is a noncanonical, practical
coverage choice. Its mathematical definition is independent of decimal or
binary notation. These are global fibres of controlled ramification suitable
for arithmetic recognition; no CM or modularity claim is made for each fibre.
Keeping all three avoids selecting only the self-reciprocal point -1, which
cannot detect the Sage/PARI coordinate inversion. Equal reductions at small
primes retain distinct global-fibre identities. Order is 1/2, -1, 2 (increasing
absolute value), not a long negative-first grid.
The sources for the criterion are PARI's Hypergeometric Motives introduction
and Roberts–Rodriguez Villegas, Section 10. The elementary S-unit solution
argument is supplied here rather than attributed to a source using each fibre.
## Prime p: discrete_index
Include every good rational prime p <= 31, with no splitting-class restriction.
For this selected z-family, good means p does not divide 2*d. This gives ten
primes for templates with denominators supported only at 2, and nine when 3
or 5 is also wild. The cutoff is computational, not a claim of mathematical
completeness. It supplies several local factors and splitting classes per
object, while rank four needs trace calculations only over F_p and F_(p²).
Increasing prime depth before covering all templates would lose breadth.
Bad-prime factors, conifold factors, unit roots, slope lists and separate trace
values are not additional quantities in this table.
The construction gives 390 entries: 42 global fibres with 9 or 10 good primes.
Every polynomial is retained, including reducible cases and zero traces.
The generator, table completeness note and this selection must agree.
An independent critique is still required; this author justification is not
an acceptance sidecar.
## Sources and policy
- https://arxiv.org/html/1705.01663v4 , Introduction, Table 1, Section 5.
- https://arxiv.org/html/2109.00027 , Sections 10–11.
- https://doc.sagemath.org/html/en/reference/modfrm/sage/modular/hypergeometric_motive.html , coordinate warning and Euler factors.
- https://pari.math.u-bordeaux.fr/dochtml/html/Hypergeometric_Motives.html , Templates, Motives and hgmeulerfactor.
Policy manifest: agents/runs/20261102T102032Z-build-skills/manifest.json.
Both archived skill versions were read and matched SHA-256
a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278.