# T498 prose repair, 2026-10-25

The fresh authenticated document still contained all three critique findings.
The repair distinguishes the family domains from the stored selection, explains
similar representatives, relocates the normalization identity, and gives the
three existing companion links their titles. The 114 entries are unchanged.

Run `sage -python repair_checks.py` alongside the attached `generate.py` for
exact basis checks and the normalization identity. To check the entries too,
pass a JSON document downloaded from `/api/table?id=T498` as the first argument.
In this run `agents/sage.sh repair_checks.py generate.py T498-before.json`
executed the checks in Sage 10.9. The result is `repair-checks-output.json`.
All 114 written intervals or exact fractions contain the generator's exact
positive radical forms. Independently transcribed formulas in Conway and
Sloane (1982), equations (32)-(36), additionally match the 72 cubic, A, D and E
entries. This repair does not replace the original independent geometric checks.

For A1 and A2, exact Gram identities establish dual similarity. For D4, an
explicit scaled orthogonal map sends a root basis to a unimodular basis of the
dual. The integral E8 Gram matrix has determinant one, proving self-duality.
D3 has the A3 Gram matrix after a basis permutation. The generator's exact nth
powers agree for A1/A1*, A2/A2* and D4/D4*.

The low-dimensional laminated identifications are given by the named links in
the Catalogue of Lattices:
https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/
They identify Lambda1,...,Lambda8 with A1,A2,D3,D4,D5,E6,E7,E8 up to scale.
The E8 basis used in the check is the catalogue's basis:
https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html
The convention for I and its conversion to G is on page 212 of:
https://neilsloane.com/doc/Me82.pdf
The general identity follows directly from v^(1+2/n)=v*v^(2/n) for v>0.
The script also verifies it exactly for n=1,...,24 with v=t^n.

The private rendered page returns HTTP 404 without an owner browser session.
The site's read-only `/preview` renderer was therefore used for the changed
sections, preserving the whole changed paragraphs. These previews do not test
the complete draft page or its pagination. All three companion pages were
fetched again and returned HTTP 200 with the titles used as captions.