# T489 build handoff

Draft: https://numberdb.org/T489
Title: Steklov eigenvalues of the classical planar domains
Family: numberdb-data#204
Run: 20261025T135053Z-build

## State

Built and filled as a draft. Neither publication nor an offer for review was
requested. The controller owns the separate critique, repair and offer stages.
The generator and independent checks are attached to the draft. No source,
pipeline, skill, published table or Git checkout was modified.

The family checklist was marked built with queue.py. Its prescribed operation
answered and closed requests #133 and #137 and closed the completed family
#204. Request provenance is not included in the mathematical table prose.

## Coverage and choices

80 entries: n=1,...,20 for unit-square, rectangle-1x2, unit-disk and annulus-1-2.
These are the exact domain slugs, display names and physical sizes in T390.
The parameter order is domain,n. The zero eigenvalue has index 1; all
multiplicities count. The range stops at index 20 even if that eigenvalue's
multiplicity continues at index 21. This cutoff is stated explicitly.

This is the pure Steklov problem, with unit density on the entire boundary,
including both annulus components. The inner normal is -d/dr. Eigenvalues are
not multiplied by perimeter, rescaled to unit area, or square-rooted.

The family permits a subset. Sectors, triangles, the L-shape, GWW drums and
regular polygons were omitted to keep a complete low-mode rectangle with
certified values. The selected four domains admit a complete analytic
separation description. The first 20 modes provide a compact reference,
matching the low-mode range of the Neumann sibling.

24 values are exact integers or rationals. The other 56 are certified Arb
enclosures serialized to 100 significant digits. The smallest measured
relative enclosure precision is 118.280 decimal digits.

## Evidence

- `screening.json`: already_here('Steklov') found no table;
  already_asked found family #204. Sibling documents were read from the API.
- `cheap-checks.json` and `square-lookup.json`: number lookups did not find a
  Steklov table. The first annulus eigenvalue appears in T35, as expected;
  both order-one annulus roots now link to the actual T35 entry identities.
- `dry-run.log` / `final-local-checks.log`: all 80 entries pass exactness,
  ring naming and prose checks; longest value 148 characters; final measured
  entry block 13.6 KB.
- `stored-checks.log`: verify() matches 80/80, with no missing, differing or
  extra entries. Independent calculations at 140 and 180 digits check every
  actual stored last-place interval, rather than only its midpoint.
- Controls recover Rayleigh quotients 1 for disk u=x and 2 for the square's
  bilinear mode. Rectangle energy integrals agree on every positive mode.
- Annulus values are independently recomputed from the symmetric weighted
  Dirichlet-to-Neumann matrix, with boundary residuals checked on both circles.
- Every displayed rectangle mode also passes an independent dilation check.
  The displayed mpmath program reproduces the square's first positive value.
- All 18 nonpolynomial square entries have separation parameters matching
  the existing clamped-free/clamped-clamped beam roots of T486.
- Later prose edits were checked to leave Numbers, including annotations,
  exactly unchanged.

The generator proves branch enumeration and ordering: positive trigonometric
Robin quotients decrease and hyperbolic quotients increase. The exceptional
first sine/sinh branch is treated according to half-widths; the square's
bilinear limit is counted once. Strict lower bounds exclude all uncomputed
branches. For the annulus, C_m <= 5m/2 gives the lower-root bound m/5, certifying
that the omitted angular orders cannot enter the retained prefix.

## Audit disposition

The final API audit has one finding, so its raw `clean` value is false:
`domain` has four named values carrying the same index grid, which triggers
the structural several-quantities question.

This is an acknowledged structural warning, not a numerical failure. The
four names select the domain argument of the same operator and the same
quantity sigma_n(Omega), with one definition and one value-column symbol.
The family explicitly requires this parameterization, and it matches T390
and T485. A table comment explains the argument. The audit's implementation
does not read such explanations and intentionally flags small named domains.
The independent critique should assess this rationale; the build did not
alter the parameterization to silence the audit.

The initial definition-length finding was fixed by moving indexing and
normalization details into Comments. No other audit finding remains.

## Tags and cross-links

The existing eigenvalue and analysis tags are used. The vocabulary contains
spectral geometry, but its live table_count is zero. Adding it to this draft
alone would create a singleton. A coherent tag assignment could cover T390,
T485, T488 and T489; this build did not edit other tables. All four family
siblings are linked through verified T-numbers because their public pages
currently answer 404 while their authenticated documents are available.

## Remaining work

Separate critique and any resulting repair, then the controller may offer
the draft. No human review or publication action was taken in this session.
The source PDFs, numerical logs, full stored document and audit response are
preserved beside this handoff in the run artifact directory.