# Build handoff: T457

- Table: https://numberdb.org/T457
- Title: Quantiles of the studentized range distribution
- Family: numberdb-data#197
- State: filled, unpublished, not offered for review. The controller must run
  the separate critique and repair stages before offering it.
- Entries: 392; 100 significant digits; rigour: proven.
- Parameter order: nu, k, p. The parameters are exact integer/rational strings.
- Stored rectangle: k=2,3; nu=1,...,10,20,30,60,120; the family's fourteen
  lower-tail probabilities.

## Deliberate scope and convention choices

The definition is the standard single studentized range of independent
standard normal variables, divided by an independent chi-squared estimate
of standard deviation. It agrees with R's nranges=1 convention. The range
already includes both directions of pairwise differences. For k=2 the
conversion to Student quantiles is sqrt(2)*t(nu,(1+p)/2), and the two-sided
level-alpha formula uses t(nu,1-alpha/2).

The proposal allowed lower precision for the general distribution. This
build instead supplies certified 100-digit values for two and three means.
Three means is the first nontrivial simultaneous comparison. Its planar
Gaussian geometry reduces the CDF to a one-dimensional angular integral,
allowing a power series with a rigorous remainder bound. Larger numbers of
means and the infinite-degree-of-freedom limit are not included. The full
parameter domain remains k>=2 and positive integer nu; the completeness
note states the finite selection explicitly. The degrees of freedom are a
subset of rows actually tabulated in the cited University of Washington
reference, and the probabilities follow the family's conventions.

## Numerical certification and independent evidence

- mpmath at 135 decimal digits proposes roots, but does not certify them.
- Arb at 525 bits certifies strict CDF signs at exact rational endpoints;
  the final bracket has width 4e-115.
- For k=2 the certified CDF is a regularized incomplete beta value.
- For k=3 the coefficient recurrence has a Cauchy bound on its infinite
  remainder. The derivation and exact bound are in math-notes.md.
- The smallest returned relative accuracy was 374 bits before formatting.
- The standard dry_run.py completed all 392 entries, with no exactness,
  prose or disguised-exact-value findings. The formatted values have at
  most 104 characters; the measured flat block is about 55.0 KB.
- A separate 2048-bit checker uses finite elementary integration formulas,
  sharing neither the generator's beta implementation nor its coefficient
  recurrence. All 392 stored decimal intervals have strict opposite CDF
  signs at their endpoints; all 392 shifted intervals are rejected.
- Four controls precede the checks. All 42 applicable closed-form rows,
  upper-tail conversions, ordering in p, and stochastic ordering in k pass.
- SciPy's general CDF agrees on all 392 rows; the largest probability
  discrepancy is 1.8174350913113813e-13.
- 55 of 56 shared printed reference values agree within half a displayed
  unit. The remaining value is an explicitly documented source discrepancy.
- The two Python Programs snippets were executed. The mpmath snippet agrees
  beyond 55 decimal places. The R snippet follows the official R manual;
  it was not executed in this environment.
- Generator.verify(sample=None), after filling: 392/392 matched, zero
  differing, missing or extra. The independent checks were rerun on the
  database values and passed.
- Site audit: clean, with zero findings. A public unauthenticated API read
  is refused, confirming the table remains private.

Software: SageMath 10.9, SciPy 1.17.1, mpmath 1.4.1.

## Findings for the critique and controller

1. The family checklist says “answers #121”, but queue.py show 121 returns
   “Lagrange interpolation polynomials”. DO NOT close #121 for this table.
   No queue.py built call, request comment, or issue closure was made.
2. The statistics tag is absent (zero tables). The draft uses the existing
   probability theory tag. A statistics tag would serve T452, T453, T454,
   T455, T456 and T457; this is a batch-level taxonomy proposal.
3. At nu=1,k=2,p=99/100, the printed source gives 90.03. The exact Cauchy
   formula sqrt(2)*cot(pi/200) gives 90.02422668900665..., rounding to 90.02.
   Both certified calculations agree. The entry comment, checker and
   printed-discrepancies.json preserve this finding.
4. T454, T455 and T456 are currently drafts and absent from title search.
   Their documents were read through the authenticated API, and the draft
   uses stable T-number HREFs for them. Published neighbours were resolved
   through their returned search URLs and their documents checked again.
   A later publication check should ensure the linked draft tables are
   available to readers.

## Artifacts

The authoritative entries and document are in the website database. Attached
there are generate.py, checks.py, math-notes.md, reference-values.json,
check-report.json, measurement.json and stored-verification.json. The local
run directory also preserves values.json, document.json, stored-document.json,
the screening results, numerical logs, the source HTML, audit results,
request-121.txt, and the proposed lessons.

The generator's first line names T457. Its default action verifies, while
--fill-draft writes entries only. It contains no publication or offer action.
The build changed no repository source, did not commit or push, and did not
create a batch-exhausted marker. No published table was edited.