back to table · edit · history · where entries came from · files · download
5321 bytes, as of the version from 2026-10-28 22:35 (current). Recorded here, not run.
# T457 repair evidence, 2026-10-16
The live skill and local skill were read before editing. The local version's stricter argument-selection rules govern this run. The live API document and audit were fetched afresh. The authenticated API returned all 392 entries; anonymous API access hid T457 and its public HTML route returned 404, consistent with the draft-only state. No credential was sent outside NumberDB's API.
All three ranked critique findings were still present. The final repair preserves both `parameter selection` and `digits: 100`; changes the convention comment to refer to quantiles; names only the probabilities $p$ as rational; moves the upper-tail identity and its explicit definition into `formula-upper-tail`; and removes duplicated computational justification from the scope comment and completeness note. Numerical methods remain in `rigour details`. The parameters, all 392 stored values, precision, and rigour are unchanged.
The proposed deletion of `parameter selection` was tested and declined. Removing it produced a new live audit finding demanding an explanation specifically in that property, despite the explanation already present in `complete-note`. It is therefore consumed by the live auditor, although its renderer calls it an unknown key. The field was restored exactly, including its original order. The first write, its failed audit, the corrected document and the final clean audit are preserved as artifacts.
## Why the mathematical wording is valid
Let $G(x)=\Pr(Q\leq x)$. The studentized range has a continuous, strictly increasing CDF on its positive support: independent Gaussian samples have positive density, the range can fall in any positive interval, and division by an independent positive chi-squared scale preserves continuity and strictly positive probability on positive intervals. Thus $G(q_p)=p$. An upper-tail critical value at level $\alpha$ satisfies $\Pr(Q>Q^{\mathrm{upper}}(\alpha))=\alpha$, or $G(Q^{\mathrm{upper}}(\alpha))=1-\alpha$. Uniqueness gives $q_p=Q^{\mathrm{upper}}(1-p)$. The exact rational checks verify $1-19/20=1/20$ and $1-99/100=1/100$.
The [official R documentation](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Tukey.html) independently confirms the scale, independence and lower-tail convention. The [SciPy distribution documentation](https://docs.scipy.org/doc/scipy-1.14.1/tutorial/stats/continuous_studentized_range.html) gives the distribution's general continuous density. No new claim of numerical certification was added to the table.
## Checks actually run
`agents/sage.sh check_T457.py T457-before.json` completed successfully. The checker independently parsed all 392 stored values and verified their full 14-by-2-by-14 selection, including canonical exact rational probabilities. In 600-bit Arb arithmetic, all 42 applicable closed-form enclosures were finite and strictly contained in their stored decimal intervals. This containment itself establishes that the enclosures are narrow enough; no overlap-only test was used.
At 150 decimal digits, four known-answer controls passed, with maximum residual 7.63734090874901e-152. Opposite CDF signs were verified at both stored decimal endpoints for all 196 two-mean entries and 20 three-mean entries spanning degrees of freedom 1, 2, 10 and 120 and probabilities 1/200, 1/4, 1/2, 19/20 and 999/1000. These 216 mpmath checks are numerical cross-checks, not a fresh certification of every original Arb remainder bound. No numerical changes were needed.
The two-means checks use the regularized incomplete beta CDF. The three-means checks use direct angular quadrature. For the three-means, two-degree control, substituting $t=\tan\theta$ gives $G_{3,2}(\sqrt2)=6\arctan(1/3)/(\pi\sqrt3)$, an independent elementary evaluation of the quadrature.
Read-only live section previews confirm both initial metadata warnings and the revised comments and upper-tail formula. Removing the selection property initially suppressed its renderer warning but introduced the live audit finding described here. The final document retains both properties, so both `Unknown key` warnings remain. The snapshot's renderer omits both from its recognized property labels. The site must reconcile its renderer, live audit and skill; this was left for a person because the campaign forbids source changes and deployment. These HTML section previews do not establish full private-page layout or browser MathJax behavior.
## Reproduction
With the attached input snapshot and scripts downloaded, run:
```
python3 repair-T457-20261016T080819Z.py repair-T457-20261016T080819Z-input.json proposed.json
sage -python check-T457-20261016T080819Z.py repair-T457-20261016T080819Z-input.json
```
The first command constructs the document repair and prints its complete prose diff without sending it. The second checks the numerical evidence without writing to the table. The original table generator and existing attachments are not replaced by this prose-only repair.
## Stored result
The final API readback equals the proposed document, with all 392 values and the parameter order unchanged. Restoring the selection property returned the post-write audit to `clean: true` with no findings. The API write records `reviewed: false`; no offer, publication or review endpoint was called.