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6351 bytes, as of the version from 2026-10-16 06:58 (current). Recorded here, not run.
# T455: Quantiles of the F-distribution
Read on 2026-10-16 against both the freshly fetched [live skill](https://numberdb.org/skill) and `.claude/skills/numberdb-table/SKILL.md`, using the stricter local rule where their argument-selection guidance differs. The table reads well overall. Two fixes are worth doing; one wording note is optional. No stored-value discrepancy was found. This is a report only: no table edits, publication, review actions, or commits were made.
**Rendering coverage has a limitation.** The specified SOCKS proxy was unavailable. Direct access to `/T455` returned HTTP 404; the authenticated API supplied the document and audit. I did not send the API key to website routes. To inspect actual server output, I submitted excerpts of the fetched document to the site's read-only `/preview` renderer, then loaded its returned HTML and live assets in Chromium. Five previews covered all prose sections, parameters, and representative entries. A combined preview exceeded the server's request-line limit. Thus the findings below include observed rendering, but this is not a full draft-page rendering or first-screen sign-off.
1. **Worth fixing: Data properties exposes two “Unknown key” warnings.**
The rendered section literally says `digits: 100 (Unknown key)` and ends the `parameter selection` paragraph with `(Unknown key)`. A reader sees validation debris beside the table's precision and coverage claims, although the API audit is clean. This is visible in both the server HTML and the browser screenshot.
Smallest fixes: preserve the precision metadata and have the site renderer recognise `digits` with a reader-facing label. For `parameter selection`, retain any additional source justification in the existing `complete-note` and remove the redundant, unrecognised field. The current completeness note already explains the selected degrees and probabilities. The `digits` warning is a renderer issue, not evidence that the numbers lack precision.
2. **Worth fixing: the NIST agreement claim omits the probability level it actually covers.**
The expanded rigour note says: “All 100 overlapping values in CITE{NISTCritical} agree within one unit of its last printed decimal place.” A reader cannot tell which 100 were checked. The cited page has **200** overlapping values: 100 each at upper-tail 5% and 1%, corresponding to `p=19/20` and `p=99/100`.
The claim holds for the 100 at `p=19/20`, including the explicitly discussed rounding discrepancy. It does not extend to the other block. For example, NIST prints `4999.52` for numerator degree 2, denominator degree 1, and upper-tail 1%; T455 correctly stores `9999/2 = 4999.5`, exactly two units of that printed last place away. The table's elementary formula independently gives this exact result. Seven values in the 1% block differ by more than one printed unit. [NIST critical-value tables](https://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm).
Smallest fix: begin the sentence “All 100 overlapping values at $p=19/20$ in CITE{NISTCritical} …”. This accurately scopes the existing assertion without changing any values or requiring an expanded source-error discussion.
3. **Optional wording improvement: name the normal variables and their sum.**
The error-function relation reads: “the error function gives the standard normal cumulative distribution function, whose independent squared variables define the chi-squared variables in the ratio”. Grammatically, “whose” refers to a CDF; a reader must supply both the random variables and the sum of their squares.
Smallest replacement: “the error function gives the standard normal cumulative distribution function; sums of squares of independent standard normal variables have the chi-squared distributions used in the defining ratio”. This is a minor clarity improvement, not a reason by itself to hold the table back.
The authenticated `GET /api/table/T455/audit` returned `{"findings": [], "clean": true}`. I agree that there is no family-splitting, parameter-range, or link-target defect apparent in the document. The clean result misses the rendered metadata warnings and the scope of the NIST claim, so it is not an overall reading pass.
I checked hardest the lower-tail versus upper-tail convention, the family/range separation, and the displayed labels. The definition identifies the central lower-tail quantile; the numerator and denominator degrees are distinguished explicitly. Parameters describe positive integer degrees and rational probabilities between zero and one, while the computed block belongs in the completeness note. The seven probabilities have conventional statistical meanings. This is one quantity with three arguments, and the value header names that quantity. Numeric parameter values remain numeric in the preview; no `param-latex` substitution, empty constraint parentheses, lost inequality text, or MathJax error appeared. The main related families are linked, and example exact values also exist as entries rather than being hidden only in comments.
Independent numerical checks used a new script, not the attached generator. At 140 decimal places, mpmath's regularized incomplete beta evaluation gave strict CDF signs at both ends of all **668** stored decimal intervals. Exact rational algebra verified the other **32** entries using reciprocal medians and the degree-two formulas. All 700 directed reciprocal comparisons, 157 applicable closed-form comparisons, and 100 probability-order checks passed. The stated 700-entry parameter block is complete. The displayed Python example ran and returned a value inside its stored interval. Both NIST blocks were compared, with the source discrepancies recorded separately. These fixed-precision checks provide independent numerical evidence; they do not re-certify the generator's ball enclosures, measured guard-bit claims, SciPy comparison, or Student-table comparison.
Reproducible scripts, fetched documents, audit output, HTML, screenshots, and numerical results are in `agents/runs/20261016T063815Z-critique-work/`, particularly `check_t455.py`, `T455-numerical-checks.json`, `T455-audit.json`, `T455-browser-checks.json`, and `T455-browser-properties.png`. The actual full draft page remains to be checked through an authorised website session.