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5057 bytes, as of the version from 2026-10-29 07:39 (current). Recorded here, not run.
# Build handoff: T603
Built and filled **Dirichlet spectral determinants of Laplacians on hemispheres**,
https://numberdb.org/T603, as an unpublished draft. No offer, review or publication
endpoint was called. The separate controller critique and repair stages remain.
The draft was created in this run and filled with 40 values once.
The family issue #240 was read first. The live corpus screening found no existing
hemisphere determinant table; it found T595 for full spheres and T539/T540 for
homogeneous oscillators. `already_here`, `already_asked`, number lookups, tag
checks and actual related-table documents are retained in this work directory.
An initial lookup used the wrong query key and is explicitly an error in
screening.json; lookup.json contains the successful `text=` requests. Their
matches belong to other families, not a duplicate hemisphere table.
Policy: both archived local and live skills match. The authoritative manifest is
`agents/runs/20261029T070407Z-build-skills/manifest.json`. Each version has SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The manifest and the numerical evidence are attached to T603.
Selection: every integer dimension 1–20, determinant then natural logarithm.
The radius is one, the operator is the nonnegative unshifted scalar Laplacian,
and Dirichlet conditions hold at the equator (both endpoints for dimension 1).
There is no zero mode. Dimension is a discrete geometric index; 20 is a practical,
noncanonical cutoff providing ten dimensions of each parity. `convention` uses
exactly the invertible determinant/logarithm exception. The attached
argument-selection.md contains the breadth review. There are no exploratory
input grids or excluded quantiles. No mathematical convention was guessed.
Numerics: 100 significant digits, type R, rigour proven. The finite formulas of
Cunha–Freitas Theorem 3.2 use exact combinatorial coefficients and Arb enclosures.
At 150 working digits, all values retain at least 500 relative-accuracy bits;
200-digit recomputations agree. A separate spectral-sum continuation, coded
without generator imports or coefficients, uses 240 working digits, explicit
Laurent pole cancellation and an absolute tail bound below 1e-170. It agrees
in all dimensions. Both methods use Arb special-function primitives; this is
independence of analytic formulation and implementation, not of the Arb library.
Controls include log(2*pi), all negative-even derivative identities needed,
and changed tail split points. Independent Neumann sums reproduce the full-sphere
splitting in all 20 dimensions, and every value agrees with the six-decimal
source table. The original source files and their hashes are preserved locally.
The reported source inconsistency was resolved: Dowker's printed page-7 d=3
Neumann-minus-Dirichlet zeta-derivative difference uses +1/2 where its Barnes
formula implies -1, a discrepancy of 3/2. The latter agrees with Theorem 3.2 and
the direct spectral continuation. See source-check-notes.md and relations.py.
Stored-data verification: `verify(sample=None)` reports 40/40 matched, no
missing/differing/extra entries. Each stored decimal interval contains both the
primary 200-digit and independent spectral enclosures. Conversions and sphere
relations were checked again on stored values. Enumeration and parameter order
match. Longest stored number: 104 characters. Full logs are attached.
Audit: the long-definition finding was repaired by moving geometry to Comments.
One advisory remains and is intentionally reported, not suppressed: the symbolic
`convention` parameter structurally resembles two quantities. The definition
explicitly states that positive determinant and natural logarithm are one number
in two invertible conventions, as permitted by the current skill and family.
`audit.json` has `clean: false` only for that advisory. The critic should assess
this exception independently.
Tags: existing `special values`, `analysis`, `functional analysis` are used.
No spectral-theory tag exists in the checked tag index. A future shared tag
`spectral theory` would fit T603, T595, T539 and T540; none was created here.
Attachments include the self-contained generate.py, independent.py, the reusable
campaign check.py, checks.py, relations.py, readback.py, source input snapshots
for the checks, argument justification, policy manifest and numerical logs.
Only generate.py controls this table's selected parameters and defaults; it has
no local source dependencies. selection-fingerprint.json records a build-time
fingerprint. This is not an independent review sidecar. The critique actor must
read the live draft, check the mathematics and selection, and write its own
fingerprint-bound sidecar using its own run identifier. No acceptance evidence
was manufactured by the builder.
The API refusal encountered when updating an empty draft is documented in
environment-findings.md. Final prose was uploaded preserving the filled entries.
No source, prompt, skill, Git state or published table was edited.