# T532 repair evidence, 20261026T234446Z

The P3 finding still applied: Definition's first mention of the derivative
zeros is unlinked, while Comments and Similar tables link the correct family.
The proposed HREF was verified, printed, rendered and saved through the API.
The mandatory post-edit audit then reported:

    Definition links to another table; a cross-reference belongs in Similar tables or a comment

This is a blanket HREF test at
numberdb_app/management/commands/audit_table.py:550. It conflicts with the
canonical skill's first-mention, once-per-section rule. The repair instructions
require fixing introduced audit findings, and source/audit-policy changes are
outside this run's authority. The link was therefore reverted through the API,
restoring the entire document exactly to its initial state. The single finding
is left for a person to resolve this policy conflict. No prose, entries,
parameter identities, selection, programs or generator sources changed in the
final document. Independent verification evidence was added as attachments.

The attempted link revision is
7ffeadbc9f4c2ac4f56730ba057774bc959b0156a98c012b7c2eea9f681811ce;
the restoration revision is
18e5e9591e9ad542dabdebf016e3923a4fe2a83009e65946388fa6bec6e0f59e.
Both were written as zeta3 through the authenticated API with reviewed=false.
The history was preserved. No offer, publication or review action occurred.

## Governing instructions

Read the canonical .claude/skills/numberdb-table/SKILL.md, both complete archived
versions in agents/runs/20261026T234446Z-repair-skills, and a fresh live
https://numberdb.org/skill response. All four byte streams are identical, SHA-256
b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.
The governing manifest is
agents/runs/20261026T234446Z-repair-skills/manifest.json, SHA-256
d03b756eaa1cd22ce39c8cee78fb3a44b052146786d6371925ac7bc9dec564f3.
The individual reads are recorded in skill-read-manifest.json in the repair work
directory and attached as repair-skill-read-manifest.json. Also read the build
prompt, campaign-gate.md, and argument_review.py. This evidence does not replace
or update the independent critic's verdict.

## Link claim and checks

The live target API document and rendered public page resolve the exact slug
Local_extrema_of_the_Airy_function_Ai. Its definition describes the negative
local extrema of Ai, indexed from n=1, with locations approximately -1.01879,
-3.24820 and -4.82010. Ai′ zeros are stationary points of Ai. By the Airy
ODE Ai″(x)=x Ai(x), a negative zero of Ai′ has nonzero curvature: simultaneous
vanishing of Ai and Ai′ would force the nonzero Airy solution to vanish
identically by uniqueness for the ODE. Thus these are extrema, as required
for this link. The root ordering agrees with DLMF 9.9(i).
Sources: https://dlmf.nist.gov/9.2#E1 and https://dlmf.nist.gov/9.9#i.

repair-checks.py provides a fresh Sage check. At 640 bits, it evaluated Ai′ at
both exact rational endpoints of the target's first three decimal intervals
(center ± one last-place unit). Opposite certified signs were obtained in all
three cases: (+,-), (-,+), (+,-). Argument rounding is enclosed by Arb; no
rounded midpoint is used as an exact root. This checks three target rows, not
the entire target table.

As a numerical preservation check, the same script independently recomputed
all 19 current T532 values. It builds Ai(z)/Ai′(0) from the linear Airy equation
and direct Arb evaluations at zero, divides formal power series to obtain
Ai/Ai′, and extracts Z_N(n)=(-1)^(n-1)[z^(n-2)]Ai/Ai′. This shares neither the
attached generator's nonlinear moment recurrence nor its Gamma initialization.
The finite formal coefficients have no series-truncation error. A nonzero
known control, (1+2z)/(1-z), returned coefficients 1,3,3,...,3 through order 20.
All 18 nonexact balls were finite, positive, had relative radii less than
2^-580, and lay strictly inside their stored decimal intervals. The n=3
coefficient and stored integer are exactly 1. All results agree with the
unchanged generator, whose default enumeration remains every n=2,...,20.
No values or precision claims were changed.

The successful live-read run is repair-checks-live.log; its complete JSON output
is repair-results.json (attached). Sage 10.9, numberdb 0.1.10. Earlier scratch
logs retain corrected checker API mistakes (ComplexBallField.real_field and
RealBall.rel_accuracy_bits are unavailable); these were not mathematical
failures. The final attached script passed using fresh API reads.

Reproduce with the attached generate.py and repair-checks.py in one directory:

    sage -pip install numberdb
    cat "$NUMBERDB_KEY_FILE" | NUMBERDB_KEY_FROM_STDIN=1 sage -python repair-checks.py

In this source snapshot, use agents/sage.sh instead of invoking Sage directly:

    cat "$NUMBERDB_KEY_FILE" | NUMBERDB_KEY_FROM_STDIN=1 agents/sage.sh \
      "$TMPDIR/repair-checks.py" "$TMPDIR/generate.py"

## Rendering and review boundary

GET https://numberdb.org/T532 returned 404 for this private draft; the full
private page was inaccessible without a browser login. A live read-only
/preview request containing the changed Definition, its declarations, and the
existing n=3 row rendered successfully. Chromium 156.0.8078.4 reported no page
or MathJax errors. The actual anchor resolves to
https://numberdb.org/Local_extrema_of_the_Airy_function_Ai with the expected
“zeros of Ai′” caption. The screenshot was inspected. This is a section
preview, not a claim to have inspected the complete private page.

The original and final selection fingerprint is
d693cc9b4e324ba5d8f67cda898389b0ab80d8d25ddd68c51d9b029e72096623.
The temporary link changed it to
f7b7a7093d7720692a3171574e0123c0102c8d1a901874ad344a2319b2d060d0,
which would have required a fresh independent critique. The restoration returns
the complete document and reviewed generator to their original bytes/content.
No selection, mathematical definition or reviewed generator change remains.
The controller decides the next step for the same T532 draft; this repair does
not accept or update the independent verdict. agents/critiques/T532-arguments.json
was not written or edited; its SHA-256 remains
9c20c15e7865d1e04a81e488d8fd7c856a0fd211bf4b50459d52a820437ba17b.
The original generate.py remains attached, unchanged, SHA-256
16eb9df4d4ced6515b06ea154580ac8b8c158c96dd80ddca0dd84a2f7d4aabbd.
The final audit and document/attachment verification are saved separately as
T532-audit-after.json and final-verification.json in the repair work directory.