# Build handoff: T491

Built the draft **Values of the Kelvin function $\operatorname{bei}_\nu(x)$**, at https://numberdb.org/T491 (database address `Values_of_the_Kelvin_function_2`). It is unpublished and was not offered for review. This is a completed build, not a critique or a human review.

## Scope and convention

Family #205 was read first. The convention is DLMF 10.61.1: the imaginary part of the principal-branch Bessel value $J_\nu(xe^{3\pi i/4})$. Parameter order and keys are `nu`, `x`, both rational, with domains $\nu\geq0$ and $x>0$. The table holds the complete rectangle of orders 0, 1, 2 at the 120 positive reduced rationals $x\leq20$ of denominator at most 4, for 360 entries at 100 significant digits and `rigour: proven`.

This follows the family and companion T490. The small denominators select simple fractional arguments; the range covers the small-argument regime and four observed sign changes for each stored order. No unresolved mathematical convention remained. The origin values are stated exactly in a comment rather than added to the shared positive-argument range. The title, definition, and number-column header all identify bei, with its signs preserved.

The initial screening used `already_here`, `already_asked`, text searches for Kelvin, bei, Struve, and Bessel, and numeric `api/lookup` queries at (0,1), (1,1), (2,1), and (0,5). No existing bei table or value hit was found. Both requested tags already occur on T187, T190, T196, T199, and T490. The exact-title claim created T491 without a collision. Evidence is in `screen.json`, `family-205.md`, and `claim.json`.

## Completed numerical validation

- The repository's dry-run, exactness, ring-import, and size checks passed. Written values have at most 105 characters; the helper estimates the entry block at 51.73 KiB.
- At 397-bit precision, the largest generator ball radius was below $2.66\cdot10^{-113}$ and the largest relative radius was below $5.62\cdot10^{-115}$.
- Every stored decimal interval contains both the generator ball and a separately implemented 600-bit enclosure from the sine power series. The series uses 200 terms with exact rational coefficients and a rigorously bounded absolute tail below $1.62\cdot10^{-350}$ throughout the range.
- Every row also passed the rotated modified-Bessel identity, both integer-order reflection identities, mpmath at 160 decimal digits, and quadrature of Bessel's integral with elementary integrands at 125 decimal digits. The maximum quadrature relative discrepancy was about $1.082\cdot10^{-122}$. Quadrature was first tested against the known nonzero value $J_0(0)=1$ and the zeros $J_1(0)=J_2(0)=0$. Numerical quadrature agreement is a check; the error certificates come from ball arithmetic and the independently bounded series.
- The recurrence passed at all 120 arguments using the written intervals for bei orders 0, 1, 2 and the actual stored T490 ber values. All 360 companion values were also checked against independent series enclosures.
- Both displayed Programs snippets were executed and checked.
- `verify(sample=None)` returned **360/360 matched, 0 differing, 0 missing, 0 extra**. The entire independent suite was then repeated on the values read from the database.
- The site audit returned `{"findings": [], "clean": true}`.

See `stored-checks.txt`, `local-checks.txt`, `dry-run.txt`, `program-checks.txt`, and `audit.json`. Numerical runtime versions: Sage 10.9, mpmath 1.4.1, NumberDB 0.1.10.

## Website attachments and operational notes

The draft carries `generate.py`, `checks.py`, the unchanged repository helper `check.py`, `document.json`, and validation logs. Download the three Python files together to reproduce the independent checks. For the private draft, provide the account key through `NUMBERDB_API_KEY`; the campaign runners read it from stdin without putting it in an artifact.

The API refused a prose-only update to the empty draft, and the package's empty-upsert permission preflight failed before any numerical write. The recovery serialized the already checked generator output, rechecked each decimal against both certified enclosures, and submitted the complete document directly to the same empty T491. The numbers were filled once, with no numerical corrections or alternate draft. The ordinary generator verification and independent read-back checks succeeded afterwards. `environment-notes.md` and the retained failed-attempt logs record these deployment findings.

T490 is a **filled draft**, not publicly visible. T491 deliberately links to it with the supported stable target `HREF{T490}`; the four public Bessel slugs were read from actual search results. Any eventual publication must account for this companion dependency. No published table was edited.

No source, pipeline, skill, or Git history was changed. No publication, review, or offer action was attempted. The proposal remains for the controller's separate critique and repair stages; no claim of completed review is made here.