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6588 bytes, as of the version from 2026-11-02 11:21 (current). Recorded here, not run.
# Argument selection for T675
Policy evidence: `agents/runs/20261102T104732Z-build-skills/manifest.json`.
Both archived canonical versions were read and matched SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attachment `skill-evidence.json` records both versions. This is the
builder's rationale, not the independent critique required by the campaign.
The named quantity is the unique slope-zero eigenvalue of geometric Frobenius
on the rank-four hypergeometric Calabi–Yau summand, in the same classical
coordinate and untwisted weight-three normalization as T674. The parameters
identify the motive template, rational fibre and reduction prime. None selects
a second quantity. The table holds the eigenvalue u, not the zero 1/u of the
constant-term-one Euler polynomial.
## template: discrete_index
Include all fourteen cyclotomic numerator templates of degree four with lower
multiset (0,0,0,0), weight three and Hodge vector (1,1,1,1). The keys and their
order are exactly T674's: 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. Each cyclotomic index d contributes all reduced a/d
with 1 <= a < d. Multiplicity is retained. These are a classified template
family; the upper parameters are not varied independently. Long–Tu–Yui–Zudilin,
Table 1 and Section 5, was checked for this class and its mirror geometry.
All fourteen receive coverage before increasing prime depth.
## z: arithmetic_structure
The domain remains Q minus {0,1}. Current coverage is every rational solution
of z,1-z in Z[1/2]^units: z=1/2,-1,2. This is an arithmetic slice of rational
fibres with controlled bad reduction, not a numerical mesh. The choice of the
allowed extra-prime set S={2} is noncanonical, but independent of numeral
notation. The prime 2 is a chosen ramification bound, not a base for digits.
If 0<z<1, both z and 1-z are powers of 2 with negative exponent; after clearing
denominators the smaller numerator must be 1 and the sum a power of 2, forcing
both to be 1/2. If z<0 or z>1, the difference of two positive powers of 2 is 1;
clearing denominators and considering parity gives powers 1 and 2. This yields
-1 and 2 and proves exhaustion of the chosen S-unit equation.
For a reduced fibre z=u/v, good primes satisfy p not dividing d*u*v*(u-v),
where d is the template denominator. Each chosen fibre therefore adds no
potentially bad prime outside 2 beyond the wild primes of the template. This
family-specific reduction criterion is documented in Roberts–Rodriguez
Villegas Section 10 and the PARI HGM manual. The three global fibres may have
the same reduction at a particular p; their separate global identities remain.
There is no claim that every fibre is CM or modular. More general S-unit
families and analytic period values need separate coverage decisions.
## p: discrete_index, restricted by intrinsic existence
Take every good rational prime p<=31 for the three chosen fibres, then retain
exactly those with a_p not congruent to 0 modulo p. The prime cutoff is a
practical finite fingerprint choice inherited from T674: nine or ten good
reductions per fibre include different splitting classes and already have
independently checked exact factors. It is not part of the domain definition.
There is no split-prime filter, no omission of coincident roots, and no choice
based on the apparent complexity of the digits.
The existence restriction is intrinsic. Newton-over-Hodge gives p dividing
b_p for this Hodge vector (Roberts–Rodriguez Villegas Section 11). Consequently
Q(X)=X^4-a_p X^3+b_p X^2-p^3*a_p X+p^6 reduces to X^3(X-a_p).
For a_p nonzero modulo p, its single nonzero residue root is simple, with
Q'(a_p)=a_p^3 modulo p. Hensel's lemma gives exactly one unit eigenvalue in Q_p.
For a_p=0 modulo p the reduction is X^4, so no unit eigenvalue exists, even in
an algebraic extension. This is not a claim that the other three Newton slopes
must be 1,2,3; a unit root alone is weaker than full ordinarity.
There are 390 checked source factors, 32 with no unit eigenvalue, leaving 358
entries. Counts by template are 20,25,27,25,23,25,26,28,25,27,26,29,26,26 in the
order listed. Every template and each of the 42 global fibres contributes.
The 32 excluded keys are reconstructible from frobenius-input.json by a_p%p=0.
The generator's default enumeration and complete-note describe exactly this
selection. No reference_sample subgroup or multidimensional exploratory grid
is present. The discrete prime index does not justify unrelated continuous
inputs; z has the separate arithmetic argument given here.
## Precision and size
For each retained p, N_p is the least integer with p^N_p >= 10^100. This is
100 decimal digits' worth of residue information, not real significant
figures. Every value retains its O(p^N_p). The measured N_p range is 68..210;
all residues have valuation zero. The longest printed entry is 113 characters
and the estimated block including entrywise links is about 105 KiB.
Precision and size do not determine the argument selection.
## Checked sources and dependency boundary
- https://arxiv.org/html/1705.01663v4 : Table 1, Section 5 and Appendix A;
fourteen templates, hypergeometric summand, and unit-root interpretation.
- https://arxiv.org/html/2109.00027 : Sections 10–11 and (11.1); reduction,
Frobenius convention, weight-three duality and Newton-over-Hodge inequality.
- https://doc.sagemath.org/html/en/reference/modfrm/sage/modular/hypergeometric_motive.html :
opening coordinate warning and Euler-factor routines.
- https://pari.math.u-bordeaux.fr/dochtml/html/Hypergeometric_Motives.html :
good-prime criterion, reciprocal coordinate and published factor controls.
- https://numberdb.org/T674 : authenticated API document and its attached
generator, checks and numerical evidence. It is a populated unpublished
companion. This run rechecks all its factors with PARI and all a_p modulo p
with hypergeometric truncations; the earlier explicit Gauss-sum certificates
remain on T674. Nothing in T674 is edited by this run.
Selection-relevant attachments to bind during independent critique:
`generate.py`, `frobenius-input.json` and `argument-selection.md`.
The checker also defines alternative enumeration checks and should be hashed
as `checks.py`; `check.py` is the shared validation helper. The table document,
including all entry identities and comments, is bound by the selection digest.
A fresh independent critique is required if these sources or the selection
change. This file contains no independent acceptance attestation.