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7250 bytes, as of the version from 2026-10-16 07:34 (current). Recorded here, not run.
# T456 — Quantiles of the Kolmogorov distribution
Reader critique, 2026-10-16. Two changes are worth making; one smaller wording improvement is optional. No numerical discrepancy was found. The table was not edited, and no publication or review action was taken.
I read both the freshly fetched [live skill](https://numberdb.org/skill) and the local `.claude/skills/numberdb-table/SKILL.md`. Where their argument-selection guidance differs, I used the stricter local convention. The 14 explicitly justified probability levels satisfy it.
**Access and rendering limitation.** The prescribed SOCKS proxy refused the connection. Direct HTTPS worked: authenticated `numberdb.table('T456')` and `GET /api/table/T456/audit` returned the document and audit. However, `GET https://numberdb.org/T456` returned HTTP 404 without a browser session. I kept the bot key confined to the API. The complete document exceeded the live preview endpoint's request-length limit, so I fetched non-saving previews of its sections separately, preserving their text, and inspected the returned HTML and Chromium/MathJax rendering. These establish the section-level observations below, but do **not** establish that the actual draft page renders correctly as a whole or that its first screen has the right balance. That check remains outstanding; the 404 is an access limitation, not a table-content finding.
1. **Medium priority; worth fixing: Data properties exposes two “Unknown key” messages.**
The live preview visibly prints `digits: 100 (Unknown key)` and appends `(Unknown key)` to the paragraph labelled `parameter selection`. A reader is being shown schema diagnostics where they expect information about accuracy and coverage. This also repeats the selection already explained in the completeness note. The clean audit does not catch either warning.
The smallest table-only fix is to remove those two unrecognized Data properties keys. The rigour details already state that the values are rounded to 100 digits, and `complete-note` already gives the probabilities and their rationale. Preserve those statements and the generator's precision setting. This needs no change to a value or to the website renderer. See the [live-preview screenshot](../runs/20261016T071512Z-critique-work/T456-browser-properties.png).
2. **Low priority; worth doing: the related-table list misses the direct mathematical connection.**
Four of the ten Similar tables entries explain how incomplete gamma, incomplete beta, and the error function produce **other distributions'** probabilities. Their relation text names chi-squared, Student, F, or normal critical values, leaving the reader to infer their relevance to Kolmogorov quantiles. Meanwhile, the corpus holds [T426, the Jacobi theta constant](https://numberdb.org/Values_of_the_Jacobi_theta_constant), which is the CDF used in this table's own mpmath example. I fetched T426 and checked its definition and actual address.
Replace those four indirect special-function entries with a single T426 link whose relation states: “The Kolmogorov cumulative distribution function is $F(x)=\theta_4(0,e^{-2x^2})$.” This follows by substituting the nome in T426's defining series. The existing normal, chi-squared, Student, and F quantile links already provide the useful statistical comparison. Removing the very broad entropy/divergence links is optional; their target documents contain no Kolmogorov entry, but their present descriptions do not assert otherwise.
3. **Minor; optional wording improvement: name the scaled statistic explicitly.**
Comment (3) says: “Its level-$\alpha$ critical value is $\kappa_{1-\alpha}$ after scaling by $\sqrt n$, with no division of $\alpha$ by two.” “Its” refers to $D_n$, so a reader has to decide whether the statistic or the critical value is being scaled. Formula (4) supplies the correct answer later, and Comment (2) already supplies the asymptotic caveat; this is ambiguity rather than a wrong formula.
Replace the sentence with: “The asymptotic level-$\alpha$ critical value for $\sqrt nD_n$ is $\kappa_{1-\alpha}$, with no division of $\alpha$ by two.” This matches the [SciPy definition of the limiting statistic](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kstwobign.html). No further explanation is needed.
**What reads well.** The definition specifies the CDF, its support, and the lower-tail inverse without digressing. The parameter describes the family, $0<p<1$, while the computed range belongs to the completeness note. This is one quantity indexed by a probability, with appropriate $p$ and $\kappa_p$ column headings, actual rational row labels, and no `param-latex` substitution or hidden comment-only values. The limiting-distribution and estimated-parameter caveats are useful and mathematical. The prose is otherwise complete English. The title and definition identify the object clearly, subject to the first-screen limitation above.
**Audit and checks actually performed.** The [saved audit](../runs/20261016T071512Z-critique-work/T456-audit.json) says `clean: true`, with no findings. I agree that there is no evident family-splitting or parameter-label problem; I do not take the clean result as an editorial pass, because it misses findings 1 and 2. All 11 explicit internal table links resolved through the API. The section previews preserved the inequalities, showed the lower-tail parameter constraint correctly, and produced no MathJax error nodes or “Math input error”. Both Programs snippets ran successfully; the mpmath example at $p=19/20$ differs from the stored midpoint by about 0.208 of its last-place unit, and SciPy agrees at machine precision.
As required by the artifact-run policy, I independently checked **all 14 stored decimal intervals**, using [Jacobi's infinite product, DLMF 20.5.4](https://dlmf.nist.gov/20.5.E4), rather than the generator's alternating-series inversion. With $q=e^{-2x^2}$, I evaluated $\prod_{k\geq1}(1-q^{2k})(1-q^{2k-1})^2$ in 600-bit MPFI arithmetic. After 600 factors the omitted product lies between $1-S$ and $1$, where $S=(q^{1202}+2q^{1201})/(1-q^2)$. Every computed enclosure had diameter below $10^{-170}$. At the exact endpoints implied by each stored decimal, $F(x)-p$ had strictly opposite signs. The checks also confirmed strict ordering, rejected intervals shifted by four last-place units, and used $0.72<F(1)<0.74$ as a control. The defining and transformed series agree with equations (3) and (16) of the [cited paper](https://arxiv.org/html/1803.00426v1).
The [document snapshot](../runs/20261016T071512Z-critique-work/T456-document.json), [independent check script](../runs/20261016T071512Z-critique-work/check_quantiles.py), [numerical results](../runs/20261016T071512Z-critique-work/numerical-checks.log), [program results](../runs/20261016T071512Z-critique-work/program-checks.log), and [browser results](../runs/20261016T071512Z-critique-work/browser-checks.json) are retained in this run's artifact directory. Reproduce the numerical check from the repository root with:
```sh
agents/sage.sh agents/runs/20261016T071512Z-critique-work/check_quantiles.py agents/runs/20261016T071512Z-critique-work/T456-document.json
```