# T456 repair evidence

This repair used the live NumberDB API as zeta3. T456 remained a private draft; no offer, publication, review, new table, Git operation, or source-tree change was made. The run repaired prose and metadata only. All 14 values, parameter keys, entry order, precision, and the existing generator were preserved. This is a repair, not a new build, so no build table marker was created.

The artifact-run instruction takes precedence over the shared handoff path: `T456-repaired.md` is retained in this run directory and attached to the draft, alongside the original critique. The website remains authoritative.

## Finding 1: partial repair and site conflict

Fresh section previews reproduced both warnings. Removing `digits` and `parameter selection` eliminated them, but the post-edit audit then explicitly required `Data properties: parameter selection`. Its explanation was already present in the unchanged `complete-note`; the live audit did not accept that location. The original `parameter selection` text was restored. The final audit is clean; one “Unknown key” warning remains. The site must reconcile its audit and renderer before this finding can be fully resolved. No values were removed or altered to appease the audit.

The local audit source differs from the deployed behavior. The API results, not the isolated source snapshot, establish the conflict. See `T456-audit-after.json`, `T456-audit-final.json`, and `preview-evidence-final.json`.

## Mathematical checks

T426's live document defines theta4(0,q) = 1 + 2 sum((-1)^j q^(j^2)). Substituting q = exp(-2*x^2), with x > 0 so 0 < q < 1, gives T456's CDF term by term. Both series converge absolutely. The new relation explicitly states x > 0; the unrestricted identity would be false on the nonpositive support extension.

Sources read live:
- https://numberdb.org/api/table?id=T426
- https://numberdb.org/Values_of_the_Jacobi_theta_constant
- https://dlmf.nist.gov/20.2.E4
- https://dlmf.nist.gov/20.5.E4
- https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kstwobign.html

The attached generator was downloaded from the website and run unchanged with `agents/sage.sh`. It reports 14/14 matched, no missing, differing, or extra entries. `repair_checks.py` independently uses Jacobi's product with 600 factors and 600-bit MPFI intervals, bounding the omitted factors by their summed deficits. All 14 exact decimal endpoints have strictly opposite signs of F(x)-p; every computed enclosure has diameter below 1e-170. The checks include a gamma special-value control, a control at x=1, strict entry ordering, and rejection of intervals shifted four last-place units in either direction. The upper-tail conversion uses alpha=1-p, with no halving, on every row. mpmath theta and direct series also agree within 1e-140 at all 14 midpoints.

Both stored Programs snippets ran successfully. At p=19/20 the mpmath result differs from the stored midpoint by 0.207935997016 last-place units; the SciPy result differs by about 2.22e-16. Read-back proves that the final stored Numbers are identical to those independently checked.

Reproduce from the repository root (all Sage computations use the wrapper):

```sh
cat "$NUMBERDB_KEY_FILE" | NUMBERDB_KEY_FROM_STDIN=1 agents/sage.sh "$TMPDIR/generate.py"
agents/sage.sh "$TMPDIR/repair_checks.py" < "$TMPDIR/T456-final.json"
python3 "$TMPDIR/render_previews.py" final
```

## Rendering and remaining limitation

`https://numberdb.org/T456` returned HTTP 404 without a browser session. The key was sent only to NumberDB API endpoints. The API document and three non-saving preview sections were inspected freshly, preserving each changed section's text and using one actual row for preview validity. The final properties preview has one warning; the comments and related-table previews have none. These HTML inspections establish section content, not full-page browser layout or MathJax rendering. No full-page rendering claim is made.

The final audit reports `clean: true` and no findings. The original generator and build checks remain attached to T456; this repair adds its independent check, results, critique, and repair handoff for the controller and person responsible for the remaining site conflict.