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6432 bytes, as of the version from 2026-11-02 02:40 (current). Recorded here, not run.
# T666 build evidence
Draft: https://numberdb.org/T666 (Zeta polynomials of Fermat curves).
This build has not offered, published, or reviewed the draft. Independent
critique and any repair remain the controller's work.
## Instruction provenance
Canonical manifest:
`agents/runs/20261102T021426Z-build-skills/manifest.json`.
Both archived versions and the snapshot canonical skill were read and match
SHA-256 `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attached `skill-manifest.json` records both versions. The campaign gate
and argument-review implementation were read. No critique acceptance sidecar
was created by this builder.
## Coverage and conventions
There are 48 exact integer polynomials at every pair with 3 <= N <= 8,
p <= 101 prime, and p = 1 modulo N. The model is X^N+Y^N=Z^N; the variable is
T and the constant coefficient is 1. All ordered cohomological characters
occur with multiplicity. Otsubo's negative Jacobi sum j is related to the
positive-sum convention of T143 by j = -J; a numerator factor is 1+J*T.
No sign twist of the even-degree equation, additional quantity, or nonsplit
local factor was introduced. Exact values need no decimal precision claim.
The argument-selection attachment explains both free parameters, their
interaction, the noncanonical cutoffs, and the readability decision to retain
two entries of lengths 1332 and 1335 characters. The dry-run size warning is
advisory, not an exactness failure or a website size-limit exception. The
entries block estimate is 15.97 KiB. Nonsplit good primes remain uncomputed;
bad primes are outside the table's domain.
## Executed numerical checks
* `agents/table-build/dry_run.py` computed every entry and passed exactness,
ring imports, and prose checks. Its only advisory was the maximum length.
* `checks.py` recomputed every complete polynomial independently via additive
Gauss sums in Q(zeta_(Np)), selecting a different primitive root and using
explicit coefficient convolution. All 48 polynomials agreed exactly.
* `context_checks.py` reproduced all 48 complete polynomials with Sage's
DirichletGroup/Jacobi-sum implementation. The exact code shown in Programs
was executed separately against all 48 generator outputs and agreed.
* Direct projective counts over F_p and F_(p^2) agreed for every row. These
counts test the first two power sums, not every coefficient. The Gaussian
computation provides the full-coefficient comparison. Controls first gave
q+1 for lines and conics over F_7 and F_49, 9 and 63 for the Fermat cubic,
and the known cubic and quartic polynomials at (3,7) and (4,5).
* All cubic F_p counts agreed with v^2=u^3-432. The birational map on x+y != 0
is u=12/(x+y), v=36(x-y)/(x+y); substituting x^3+y^3=1 gives v^2=u^3-432.
* All rows passed degree, constant coefficient, leading coefficient,
reciprocity, Jacobi conjugate-norm and coordinate-exchange tests. Every
rational Galois-character-orbit factor was integral, and their product
reproduced the full polynomial. Every divisibility relation for represented
degrees d|N was also checked. No simplicity conclusion follows from this.
* After filling, every row was read back through the API in enumeration order,
compared with the independent Gaussian polynomial, and checked again for
projective counts and structural identities. `verify(sample=None)` passed
all 48 entries. See `numerical-evidence.json` for every coefficient and count.
The computations used SageMath 10.9, Python 3.12.3 and numberdb 0.1.10. Lazy
Sage initialization warnings occurred but every computation completed.
## Corpus, references and audit
`already_here` found neighboring curves and the Jacobi-sum table, with no
existing table of these local Fermat polynomials. `already_asked` returned the
selected family #294. Public lookup returned no matches for the cubic
numerators at (3,7) and (3,13). The live documents T143, T214, T216, T213 and
T665 were read. T143, T214 and T216 addresses came from live search results;
T665 is linked by its existing T-number, which the site's HREF resolver accepts.
T665 is API-readable with this account but is absent from public search and
returns 404 anonymously. No published companion was edited.
Otsubo, arXiv:1404.7476v1, Sections 2.1 and 2.2, was read for the model,
character set, orbit decomposition and local-eigenvalue sign. The reviewed
LMFDB `av.fq.weil_polynomial` knowl was read for the reversal convention.
The `av.fq.l-polynomial` page returned a browser challenge and was not used.
The table's links are to the checked sources and specific existing companions.
The website audit reports exactly one finding:
`Tags: "curves" reaches only this table; a tag that leads nowhere else leads nowhere`.
The tag API actually reports the published T107, Equations satisfied by Igusa
invariants of split Jacobians, with table_count=1. The audit uses cached public
counts and does not count T665 and T666 as draft members. The requested family
tag is retained: T107, T665 and T666 give three appropriate tables, counting
companions as the canonical skill permits. This is an explained audit warning,
not a claim that the audit is clean. No other audit finding was returned.
The absent, more specific tag `Fermat curves` remains a proposal for T143,
T665 and T666, as in the family issue. It was not created by this build.
## Reproduction and review dependencies
Download `generate.py`, `checks.py`, `context_checks.py`, and `check.py` into
one directory. With Sage and numberdb installed, run `sage -python checks.py`
and `sage -python context_checks.py`. An authorized client can run
`sage -python checks.py --stored`; configure the key without placing it on a
command line. `sage -python generate.py` runs the client verification.
`character-orbits.json` records each orbit's factor and multiplicity context.
For the independent argument review, fingerprint at least `generate.py`,
`checks.py`, `context_checks.py`, `check.py` and `argument-selection.md` from
the live attachments. The generator has no imported numerical data or local
source dependencies beyond the installed Sage and numberdb packages. The
checks import the attached generator and the unchanged campaign `check.py`.
The character-orbit JSON and numerical-evidence JSON are outputs, not generator
inputs. The builder's fingerprint is only a reproducibility record and is not
an independent acceptance.