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5427 bytes, as of the version from 2026-11-02 14:37 (current). Recorded here, not run.
# T679 repair evidence
The live unpublished draft was repaired through the authenticated NumberDB API
as zeta3. All three ranked findings still applied when checked. The changes
remove the ambiguous qualifier “ordinary”, replace six URL-identifier captions
with the actual companion titles, and link the first Dedekind eta mentions in
Formulas and Numbers. The generator changes only the same entry-comment string.
The displayed modular identity and its Rogers–Wan–Zucker citation are retained.
## Governing instructions
The coordinated archive is
`agents/runs/20261102T143152Z-repair-skills/manifest.json`, SHA-256
`2ed18a5ba549629b2149341ece7834c3c10445d5345622bccc42c66178a837ba`.
Both archived versions were read and their bytes and manifest digests checked:
| Version | SHA-256 |
| --- | --- |
| Canonical local `skill-local.md` | `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278` |
| Archived live `skill-live.md` | `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278` |
The canonical skill in this snapshot has the same digest. The build prompt,
campaign gate and argument-review implementation were also read. The artifact
policy controls this repair: no publication, offer, human review, source edits,
Git actions, or modification of independent critique evidence.
## Exact mathematical check
The live Frobenius-polynomial companion stores
`P(T) = 1 + 2*T - 18*T^2 + 54*T^3 + 729*T^4` at (2.2.4, 1/2, 3).
The eigenvalue polynomial is `Q(T) = T^4*P(1/T)`.
Sage 10.9 computes its root valuations as 0, 3/2, 3/2, 3.
An independent exact lower-hull calculation on the coefficient valuations of
P gives vertices (0,0), (1,0), (3,3), (4,6), with the same multiplicities.
The known-root control `(T-1)*(T-3)*(T-9)*(T-27)` gives 0, 1, 2, 3.
The live unit-root companion's residue r also satisfies `Q(r) = 0 mod 3^210`,
`r = 1 mod 3`, and `Q'(r) != 0 mod 3`. A unit root therefore exists even
though the Newton polygon differs from the Hodge slopes 0, 1, 2, 3 stated for
this rank-four family. The repair removes the ambiguity without making a new
general ordinarity assertion. The companion itself explicitly distinguishes
unit-root existence from full ordinarity.
The exact check has no rounded input or unpropagated argument uncertainty.
No period value was recomputed or changed in this repair, and the existing
100-significant-digit heuristic precision claim remains unchanged. The
earlier independent critique's numerical checks were not counted as this
repair's computations.
Run the attached check with SageMath and the attached inputs in one directory:
```text
sage -python repair-checks-20261102.py
```
In this campaign it was run through the required wrapper:
```text
agents/sage.sh "$TMPDIR/repair_checks.py" "$TMPDIR/repair-inputs-20261102.json"
```
`repair-inputs-20261102.json` contains the two live entries, their API URLs and
the full companion-snapshot SHA-256 values. `repair-results-20261102.json`
records the successful exact results. Local scratch retains the complete
snapshots and both completed check runs. The second run verifies the compact
input artifact used by the attached reproducible check.
The Sage root-valuation convention was checked against its official
[polynomial documentation](https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_element_generic.html).
The eta target's convention agrees with the standard product recorded by
[LMFDB](https://www.lmfdb.org/knowledge/show/mf.gl2.history.infinite).
All seven target definitions and titles were freshly fetched from NumberDB;
their full snapshots are retained locally.
## Preservation and rendering
The proposed diff was printed and read before submission. Readback equals the
entire proposed document. All 56 numeric strings, ordered argument identities,
parameter declarations, completeness notes, mathematical definitions, program
snippet and precision metadata are unchanged. An AST comparison confirms that
the generator differs only in the intended eta-link string. Its arithmetic and
default enumeration are unchanged. Earlier checks and generator revisions are
retained.
Chromium 156.0.8078.4, driven by Playwright 1.64.1 from the shared installation,
rendered the live page before and after repair. The final DOM has six readable
Similar tables captions, two eta links, no page exceptions and no MathJax
errors. `links-after.png` was inspected. The authenticated final audit returned
`clean: true` and `findings: []`.
## Independent critique boundary
The selection fingerprint changed from
`971ba48323231acbd6fe4ca8c8b3c0ead035a982991f0f0e91a6a2828df2a2dd`
to `04f765a24441e5138e9f21790c6127c5d0df151f17c3a8196fa59fc1406d59cf`.
This is caused by the formula and entry annotation links, not different
arguments. The attached generator changed from
`93d8d529350c21abca359b7809ac5cc8a4ffcd6ffef813e965ea3a163a16a477`
to `6405c5827cf3d8446bf7e0e2da880397d604c3ff97c0938d0a577f40b7e23334`.
`argument-selection.md` remains
`3c0c855d90748f206a0198034d2217554d2b8bab21f26ad124ecccf4e8585465`.
The independent `agents/critiques/T679-arguments.json` was neither edited nor
accepted by this repair. Its preserved file SHA-256 is
`3ae049c0edb7a598a542a262e0f082b558a907f5afce5e1f486a2efda980f5f9`.
The controller must recover this same T679 draft for a fresh independent
critique. No offer, publication or human-review action was performed.