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4174 bytes, as of the version from 2026-10-26 15:32 (current). Recorded here, not run.
# Build handoff: Slepian concentration eigenvalues, T519
This run built the draft at https://numberdb.org/T519. It must remain unpublished and unoffered pending the controller's separate critique and repair stages.
The family convention is numberdb-data #212: the continuous order-zero sinc operator on [-1,1], c=pi*S/2, decreasing mode index starting at zero. The parameters are (S,n), with S rational and positive, n a nonnegative integer. There are 48 rows: S=1,2,4,8 and n=0,...,11. Each value has 100 significant digits. These four Shannon numbers are the family's finite reference benchmarks for integral effective dimensions; no claim of a published application demanding all four is made.
The local and live skills were read. The live skill strengthens argument selection, so the benchmark rationale is explicit. Corpus screening used already_here and already_asked, live text searches, three api/lookup intervals, and both tag pages. No duplicate family was found. The title claim created empty draft T519. T518 is the distinct differential-eigenvalue table. Its actual slug was obtained through the client's no-op writable check (unchanged=true), and its document was read. The Legendre table's actual address was read from search and its document inspected. No published table was edited.
The generator uses the commuting Legendre differential matrix, Sturm counts to select the mode, and a backward coefficient recurrence. The finite-Fourier eigenvalue is evaluated at the origin (or from its derivative for odd parity), then converted to the concentration eigenvalue. The code compares (working decimal digits, terms per parity) = (160,80), (220,80), (220,120), (220,180). The minimum relative agreement across these comparisons is 125.992864 digits. At least 110 are required to retain 100.
The independent method is 120-node Gauss-Legendre Nystrom discretization at 170 decimal working digits. It uses the sinc kernel directly and shares no differential matrix or eigenvector code. All 48 entries agree with it to at least 148.802191 significant digits. Before comparison, the quadrature reproduced polynomial moments through degree 15, and a constant kernel reproduced the nonzero rank-one spectrum 2,0,...,0. A deliberately perturbed scalar failed the comparison control.
The complete discretized spectrum passes the trace S and squared-kernel norm identities at all four bandwidths. Maximum measured errors are 1.66e-170 and 2.49e-170 respectively. The stored 12-mode lists are not the full trace: their deficits range from 1.8704332832394e-27 at S=1 to 0.0000599438605607975 at S=8. The smallest stored eigenvalue is approximately 1.73880832652738e-24 at S=1,n=11; the leading S=8 value is approximately 0.999999999705392 and remains inexact.
Shared exactness, ring-import, equality-to-independently-rounded-values and measurement checks passed. The written values have maximum length 125 characters, with a 6.84 KB entries-block estimate. The program snippet computes only the largest eigenvalue in double precision and was tested at all four bandwidths, agreeing within 3e-15. The full package dry run, generator verify(), stored-value checks and site audit are recorded in the final handoff evidence.
Rigour is heuristic (agreement-checked). There is no certified infinite-dimensional truncation bound or quadrature error bound. Ball or interval certification is not claimed.
The proposed `signal processing` tag is still absent. Existing `special functions`, `eigenvalue`, and `analysis` were used. A future signal-processing tag would group T519, T518, T391 (Daubechies), T395 (Symlets), and T401 (Coiflets). No new tag was created.
No convention was guessed. Implementation choices beyond the proposal were the origin/derivative finite-Fourier ratios, the independent Nystrom formulation and its 120-node/170-digit settings, and a leading-eigenvalue-only double-precision program snippet. The generator and numerical checks are reproducible attachments. The numerical evidence and source snapshots are retained in the stage work directory. The controller should use the completion pointer only after the fill and read-back checks succeed.