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4141 bytes, as of the version from 2026-11-02 08:29 (current). Recorded here, not run.
# T661 presentation repair, 20261101T225849Z
The live draft was fetched through the authenticated API and viewed in Chromium before any write. All three ranked findings in `agents/critiques/T661-recovery-20261101T225130Z.md` still applied. The raw formula notation caused MathJax startup to request a missing `boldsymbol.js` extension (HTTP 404); no expressions were typeset. The cube-base paragraph used an ambiguous antecedent. The data-properties renderer displayed `digits: 100 (Unknown key)`.
The multipartition symbol is now consistently `\Lambda`. The cube-base comment now explicitly assigns the character-degree sum to `H_m\wr S_2`. The unsupported table-level digits key was removed; the generator and precision explanation still specify 100 significant digits. Six Similar tables links received readable captions, with destinations preserved. All 154 stored values, ordered arguments, parameter declarations, completeness, reader program, and existing generator sources were preserved.
The checked primary source is Cohn–Kleinberg–Szegedy–Umans, https://arxiv.org/pdf/math/0511460, Section 2, printed page 381 (PDF page 3). Its group is `H^2 semidirect C_2`, with the factors exchanged, for `H=(C_m)^3`. This is `H wreath S_2`, not H itself.
For any finite abelian H of order q, the q diagonal pairs of characters of H give two linear extensions each to H wreath S_2. Each of the q(q-1)/2 unequal unordered pairs gives an induced irreducible representation of degree two. Their squared degrees sum to `2q + 4*q(q-1)/2 = 2q^2`, the group order, exhausting the irreducibles. Thus the degree power sum is `2q + q(q-1)/2 * 2^s`, independently of the isomorphism type of H. H itself has q linear characters and constant sum q. This proves the corrected distinction for the whole stated family.
Run the attached repair checks with Sage, using the already attached unmodified generator:
sage -python repair-checks-20261101T225849Z.py repair-records-20261101T225849Z.json generate.py
The script recomputed all 154 values: 77 cubic sums agree exactly, and 77 noninteger sums have finite balls strictly inside the stored decimal intervals, with at least 590 relative accuracy bits at the verification precision. This recomputation shares the generator and is not an independent check. Separately, all 78 cube-slice entries passed the orbit formula; integer fourth-root isolation supplied rigorous rational bounds for every noninteger entry. All inputs are exact integers or rationals, with no rounded arguments. GAP controls verified C_2, S_3, the abelian cube C_2^3, and its wreath product with S_2. Their last two degree histograms are respectively {1:8} and {1:16,2:28}. No values were added or changed.
An initial attempt to import embedded libgap stopped before checking anything. The successful script uses Sage's external GAP interface; the failed log is retained among the local run artifacts. The completed checks ran under Sage 10.9.
The governing archived manifest is `agents/runs/20261101T225849Z-repair-skills/manifest.json`. Both archived local and live skills were read and verified byte-identical to the source snapshot. Both SHA-256 digests are `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`. The checks JSON records the same manifest and both digests.
This is a repair, not an independent critique. The original critique and its argument sidecar are untouched. The notation and comment changes alter fingerprinted mathematical context, so a fresh independent critique is required on this same T661 draft. The generator and argument-selection attachment hashes are unchanged. No offer, publication, or review action was taken.
After the repair, Chromium 156.0.8078.4 with Playwright 1.64.1 typeset 81 mathematical expressions on both desktop (1440 pixels) and mobile (390 pixels). There were no page errors, failed HTTP responses, MathJax error elements, unknown-key warnings, or document-wide horizontal overflow. Screenshots were inspected. The authenticated audit returned `clean: true` and an empty findings list. The live document readback exactly matched the proposed repair.