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6972 bytes, as of the version from 2026-11-02 11:00. Recorded here, not run.
# T675 numerical build evidence
Archived policy manifest: `agents/runs/20261102T104732Z-build-skills/manifest.json`.
Both valid versions, local and live, were read and compared byte for byte.
Their SHA-256 is `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
`skill-evidence.json` preserves both manifest entries. All working artifacts
are under `agents/runs/20261102T104732Z-build-work`.
SageMath 10.9; PARI 2.17.1; numberdb 0.1.10. Every Sage computation used the
prescribed agents/sage.sh wrapper. No production computation, publication,
review or offer action is performed by this builder.
## Inputs and conventions
The authenticated API document for T674 was read, along with its family
coverage map in issue 299 and all eleven existing companion tables. The exact
input is frozen in frobenius-input.json; its metadata binds the downloaded
T674.json. The original generator and checks were read from T674's attachments,
not another agent's transcript. T674 is a populated draft, not a public table.
Its prior evidence reports 780 explicit Gauss-sum trace enclosures, PARI
comparisons and character-count controls. This run does not claim to have
repeated that entire Gauss-sum calculation.
This run did independently recompute all 390 complete factors with PARI at
1/z. It also computed the p-1 truncated classical hypergeometric series
modulo p from rational upper parameters and compared it with a_p for every
factor, including the 32 omitted ones. All comparisons passed. The generator
itself only reads the frozen exact factors and lifts roots; it shares no PARI
or hypergeometric-series code with these checks.
The Sage and PARI coordinate warnings were read. The asymmetric template 5
control at p=7 has P=[1,-35,805,-12005,117649] at z=1/2, and
P=[1,10,420,3430,117649] at z=2. The two differ, so this check detects failure
to invert the coordinate. The first has no unit root, the second does.
The PARI manual control is [1,5,45,135,729] at PARI t=1/2,p=3, hence classical
z=2; z=-1 has the same reduction at p=3 but is not the manual's stated input.
No unrelated quantity is stored: these are the eigenvalues satisfying Q(u)=0,
and not the reciprocal zeros satisfying P(u)=0. All constraints and symbols
follow T674. Calling the existence locus fully ordinary would be stronger
than justified, so the prose says only that a unique unit eigenvalue exists.
## Certificates and independent root calculation
All 390 input factors obey P=[1,-a,b,-p^3*a,p^6] and p divides b. Therefore
Q modulo p is X^3(X-a). Exactly 358 have a nonzero modulo p and hence a simple
unit root. The 32 remaining factors have no unit root. All fourteen templates
and all 42 rational fibres contribute retained entries.
For each retained factor, exact integer Newton iteration doubles its certified
exponent until N is the least integer with p^N>=10^100. The resulting residue
r is a unit, satisfies Q(r)=0 modulo p^N, and has Q'(r)=a^3 nonzero modulo p.
Hensel's lemma therefore proves the stated residue class contains exactly one
root. Its inverse satisfies P(r^-1)=0 at the same modulus. All 358 certificates
passed. Precision spans 68..210, with no conversion to real decimal error.
The Qp object has finite absolute precision N and valuation zero; capped
relative precision therefore has exactly the same exponent in this case.
Independently, integer Newton identities generate the power sums s_m from
P's coefficients, without polynomial root lifting. The ratio s_(N+1)/s_N,
computed by modular inversion, agrees with every stored-precision root.
Here is the bound making that a meaningful full-precision check. For any
non-unit root with 0<v_p(beta)<1, the term -a*beta^3 of Q(beta) would have
strictly smaller valuation than all other terms, an impossibility. Thus the
other roots have valuation at least 1. Consequently s_m-u^m has valuation at
least m, and s_m is a unit. This proves s_(m+1)/s_m=u modulo p^m. This proof
also explains the precision of the independent method rather than relying
on convergence observed in digits.
Before using either method, the explicit polynomial
(X-2)(X-5)(X-25)(X-125) was checked at p=5,N=40. Both methods returned 2
modulo 5^40, a known nonzero answer. A nonzero Hasse control at template
2.2.2.2,z=-1,p=5 gives 1 modulo 5. Exhaustive enumeration of all lifts through
p^4 finds exactly one lift at each stage for every retained factor and agrees
with the high-precision residue. This supplies a third implementation check.
## Dry run and size
The unchanged campaign dry_run.py and check.py passed on all 358 values.
Every value carries its own p-adic error bound; no exact-in-disguise or prose
finding occurred. Longest printed value: 113 characters. The helper estimates
106.9 KiB with its synthetic parameter keys, or 105.15625 KiB with tuple keys,
including the entrywise links. No size exception is needed. The Programs
example was executed and agrees with the generator at (5,2,7), including
its precision. See dry-run.log, checks-offline.log and program-check.log.
## Screening and source reads
The prescribed already_here and already_asked calls returned no duplicate
family or separate matching request. Three api/lookup controls with eight
p-adic digits at p=5,7,13 found no matching entries. Results are preserved in
screening.json and lookup-controls.json. Existing tags number theory,
L-function, p-adic and eigenvalue were found. Frobenius is absent; the proposed
shared tag would connect T675,T674,T214,T414,T666. No tag or companion was
edited in this run. The known companion URLs were read from API/search or
site links, never guessed from titles.
Sources actually read: Long–Tu–Yui–Zudilin Table 1, Section 5 and Appendix A;
Roberts–Rodriguez Villegas Sections 10–11; Sage HGM documentation; and PARI HGM
documentation. URLs and roles are in argument-selection.md. Direct downloads
of the two papers succeeded; the Sage documentation archive request returned
403, but its source was successfully read through the web tool. This is an
archive limitation, not an unresolved mathematical source.
## Issues resolved offline
The initial claim contained a parameter CITE whose Comments section had not
yet been included. The API returned HTTP 500, matching the known deployment
bug documented in docs/agent-environment.md. Retrying the same title alone
created T675; no second title or replacement draft was created.
The first offline computation attempted Qp(type='capped-abs'), which Sage
refuses. Qp(type='capped-rel') with a unit and explicit absolute precision N
has the required meaning. The failed log is retained as
checks-offline-initial.log; no value had been sent at that point. The corrected
full computation passed. This is an implementation correction, not a
numerical disagreement between the two methods.
Stored verification and the live audit are recorded in the final addendum.
This evidence is a builder report, not an independent critique or acceptance.