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7144 bytes, as of the version from 2026-11-02 10:32. Recorded here, not run.
# T674 build evidence
Policy: agents/runs/20261102T102032Z-build-skills/manifest.json. Both archived
canonical skill versions were read and matched SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attachment skill-evidence.json records both source names and digests.
Environment: SageMath 10.9, PARI 2.17.1, numberdb client 0.1.10. Every Sage
computation in this build was run through agents/sage.sh. The artifact files
and logs are in agents/runs/20261102T102032Z-build-work.
## Selection and exactness
390 integer polynomials for all fourteen templates, z=1/2,-1,2, and all good
primes <=31. There are 42 global fibres and 9 or 10 primes per fibre. The
arithmetic selection and noncanonical prime cutoff are justified parameter by
parameter in argument-selection.md. Exact integers replace any decimal
precision target: there are no approximate polynomial coefficients.
The canonical dry_run.py passed: 390 entries; all coefficients exact; no
exact-in-disguise or prose findings. Longest entry 52 characters at
(template,z,p)=(3.4,1/2,29). The helper's estimated serialized block was
37.3 KiB when parameter names are included in its single synthetic key
(35.38 KiB with tuple keys in checks.py); neither estimate is a size target.
## Independent numerical checks
The generator uses Sage's Gross–Koblitz trace calculation at f=1,2 with an
explicit integer modulus exceeding twice the Weil bound. Its integer outputs
are compared with three further p-adic digits and standard Sage Euler factors.
Neither comparison is presented as independent of the underlying Sage routine.
Every full polynomial agrees with PARI's hgmeulerfactor evaluated at 1/z.
PARI is a distinct implementation but also uses p-adic trace methods.
The stronger independent computation in checks.py directly implements
Beukers–Cohen–Mellit, Theorem 1.3, arXiv:1505.02900v2. It derives gamma vectors
from cyclotomic indices by Möbius inversion without using Sage's HGM object,
constructs the fields F_p and F_(p²), and evaluates all multiplicative and
additive characters explicitly in 384-bit complex balls. It checks the
trivial Gauss sum -1 and the nontrivial Gauss norm q first. All 780 trace
balls are finite; their real and imaginary radii are less than 10^-75;
the imaginary part contains zero, and coercion to ZZ isolates the unique
integer contained in the real ball. The theorem's integrality makes this an
exact trace identification, not rounding a numerical answer.
Published controls: template 5, z=-1, p=3 has coefficients
[1,5,45,135,729], in the PARI manual. Template 2.2.2.2, z=-1, p=5 has
[1,4,-130,500,15625], in the Sage manual. The asymmetric template 5 control
at p=7 has [1,-35,805,-12005,117649] at z=1/2 and
[1,10,420,3430,117649] at z=2. Sage(z) agrees with PARI(1/z), while
failing to invert would exchange these distinct polynomials.
For all 390 factors, checks.py verifies degree, endpoints, weight-three
reciprocity, Newton's trace identity, divisibility of b_p by p, coefficient
bounds, and the exact real-root criterion for the quadratic in U=X+p³/X.
The last check proves that the four eigenvalue roots have absolute value
p^(3/2), including repeated-root cases, using integer inequalities only.
For the half-parameter template, eighteen additional checks at p=3,5,7,
f=1,2 and all three fibres compute a direct quadratic-character convolution.
Its normalizing Jacobi factor is J(phi,phi)^3=-phi(-1). Omitting that factor
first caused a sign discrepancy at p=7 in point-probe.py; the factor follows
from the quadratic Jacobi sum and the finite Euler convolution, and accounts
for every discrepancy. No choice between disagreeing answers was made.
The same fields are used for literal point counts of Legendre elliptic curves,
compared with BCM Theorem 1.2 and the rank-two PARI factors. The nonzero control
at p=5,z=2 has exactly eight projective points. The elementary character-sum
control sum_x phi(x²-1)=-1 is also enforced in each field.
Offline result: PASS 390 polynomials; 780 explicit Gauss-sum trace enclosures;
18 quadratic convolution and elliptic point-count controls. See
checks-offline.log and values.json in the run artifacts.
## Sources actually checked
- Long–Tu–Yui–Zudilin, https://arxiv.org/html/1705.01663v4 : Introduction,
Table 1, Section 5; template list, geometry, and distinction from z=1.
- Roberts–Rodriguez Villegas, https://arxiv.org/html/2109.00027 : Sections
10–11, equation (11.1); good primes, traces, odd-weight duality.
- Beukers–Cohen–Mellit, https://arxiv.org/html/1505.02900 : Definition 1.1,
Theorems 1.2 and 1.3; independent Gauss formula and elliptic normalization.
- https://doc.sagemath.org/html/en/reference/modfrm/sage/modular/hypergeometric_motive.html :
coordinate convention, H_value, padic_H_value and euler_factor examples.
- https://pari.math.u-bordeaux.fr/dochtml/html/Hypergeometric_Motives.html :
good-prime criterion, reciprocal convention, and known polynomial controls.
The source pages were read through the web tool. A direct archive download
attempt of the Sage documentation returned HTTP 403; that is not treated as
an absent mathematical source or replaced by cached agent transcripts.
Stored-value verification and site audit are recorded in the build addendum.
This is builder evidence, not an independent critique or approval.
## Build addendum: stored data and final audit
`verify(sample=None)` passed on T674: 390/390 matched, 0 differing,
0 missing and 0 extra. The complete independent check was repeated on the
stored values, including explicit containment of each recovered integer in
its trace ball. All 390 PARI polynomial comparisons, 780 Gauss-sum traces,
exact identities and 18 quadratic controls passed. The final run is
`readback-final.log`. The numerical generator was unchanged after the dry
run; its docstring gained an explicit installation command, and the checker
gained the explicit integer-containment assertion.
The site audit returned `findings: []` and `clean: true`. Its earlier
findings were resolved by naming the Gauss-sum computation instead of saying
“the latter”, and adding `sage -pip install numberdb` to the generator
docstring. All four linked companion URLs returned HTTP 200.
Run: 20261102T102032Z
Evidence SHA-256 values:
- `generate.py`: `84dd5f3ec93f183fad9594b6f36d114aa33d8052bc22d871156157a03e8b4296`
- `checks.py`: `48f9b022b52707423c9b8015bcc227f5db8689f3aab89f617f1e3e6774c08610`
- `check.py`: `1b69965212b74b8e9fa1d38b507f5629f9f4319707aefcf0bcb0dc94a6fb611b`
- `values.json`: `8ac53d6b87b610de775ed02084eee1d5aa26458943669ca4858ab9424a05256d`
- `dry-run.log`: `79cd4253d270acd88a8dc75a59090572fa290d1cca351172bb91cc51a92e7015`
- `checks-offline.log`: `9af27618efe2597f4d78018712f5309500fd0b3ee027dbd5eed5127e7215d6c6`
- `readback.log`: `320ef385d8a1b3dfc25895ceda93c982adc418e26f821ebe02fe21624738c0cd`
- `readback-final.log`: `320ef385d8a1b3dfc25895ceda93c982adc418e26f821ebe02fe21624738c0cd`
This draft has not been offered, reviewed or published. Independent
critique and its argument-selection sidecar remain controller stages.