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4613 bytes, as of the version from 2026-10-25 13:49 (current). Recorded here, not run.
# Build handoff: T488
Draft: https://numberdb.org/T488
Title: Eigenvalues of the clamped plate
Family: numberdb-data#204
Run: 20261025T132237Z-build
The draft holds the first 30 clamped-plate eigenvalues on the unit disk,
counted with multiplicity, at 100 significant digits and rigour `proven`.
The quantity is Lambda for Delta^2 u=Lambda u, with u=normal derivative=0.
The first value is 104.3631055588443069217226196733478504926323754398933759856840104698636921970114261209766249526384382.
## Scope decisions
The family permits a subset of T390's domains. This build includes only
`unit-disk`, using T390's display name and radius one. The first 30 modes end
after a full eigenspace and provide a low-mode reference. No values for the
square, other polygons, sectors or annulus have been established in this run.
Their absence is explicit in the completeness note. The angular/radial mode
map is in the check logs and rational certificates, rather than extra table
parameters: the table's n is the sorted full-spectrum index.
The existing `spectral geometry` tag has zero public tables. T485 currently
uses only `eigenvalue` and `analysis`, as does T390, so those established tags
are used. A coordinated application of `spectral geometry` to T390, T485 and
T488 is proposed for a person; no other table was edited. T485, T486 and T487
are accessible via the authenticated table API but absent from public search;
they are linked by their actual T identifiers. The Steklov family member
was not found in search and no nonexistent link was invented.
The general absence of additional clamped-disk multiplicities is not assumed.
It is proved for this range by disjoint interval enclosures. See the lesson
artifact and the source by Mangoubi and Rosenblatt.
## Checks completed
- Live /skill and local skill read; stricter local argument-selection rules
retained. Family #204 read before screening.
- already_here and already_asked called. Text searches for clamped,
biharmonic and related spectra found no existing table. api/lookup checked
104.3631, 452.0045 and 1216.407. Tag existence checked.
- Title claim succeeded as T488, unpublished and not offered.
- Official dry_run.py/check.py passed: 30 entries, 130-character maximum,
6.3 KB block; no float, nonfinite or disguised-exactness findings.
- Arb at 513 bits certified exact rational root brackets and derivative
signs. A certified J_m sign mesh, zero-spacing proof and interlacing
established each radial index. Monotonicity excluded omitted orders.
- Independent fourth-order radial ODE power series with geometric tail
bounds certified signs at the exact endpoints of every displayed decimal.
- Independent mpmath ODE roots at 140 and 180 digits agreed to 125 relative
places for all 17 distinct modes, covering all 30 rows.
- Nonzero controls: determinant at zero equals 2, and its first derivative
equals -1/[8(m+1)(m+2)(m+3)], for orders 0 through 9.
- Generator.verify(sample=None): 30/30 matched, no differences/missing/extra.
- All independent interval and precision checks repeated on the read-back
values; all passed. Minimum generator ball accuracy: 371 bits.
- Programs snippet executed and matched the fundamental value.
- Stored document lists n=1,...,30 in that order and labels Lambda_n(Omega).
- Site audit returned findings=[], clean=true.
## Evidence and reproducibility
The website attachments include generate.py, checks.py, certificates.md,
root-certificates.json, the official dry_run.py and check.py, numerical logs,
versions.json and this handoff. The local work directory also preserves the
family, screening results, the initial claim and fill responses, the full
submitted and read-back documents, and source downloads.
Sage 10.9; Python 3.12.3; numberdb 0.1.10; mpmath 1.4.1. All Sage runs used
agents/sage.sh. No computation remains running.
Sources inspected: Mangoubi and Rosenblatt arXiv:2411.07564v2; Baricz,
Ponnusamy and Singh arXiv:1507.01104 / DOI 10.1007/s12044-018-0398-z;
Garrett's open-access acoustics chapter, DOI 10.1007/978-3-030-44787-8_6.
The corroborating 104.363 value in Kawohl and Sweers, Remark 4.3, was visible
through the web-indexed source; its university PDF failed to download
directly. The proof and precision do not depend on that low-precision check.
No publication, review, offer action, Git operation or source edit was made.
The table was claimed first and filled atomically after offline checking.
The controller should perform its separate critique and repair stages before
it decides to offer the draft. This handoff is build evidence, not a critique.