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# Argument selection for T604
This build follows the archived canonical skill manifest
`agents/runs/20261029T075206Z-build-skills/manifest.json`. Both local and live
copies have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
This is the builder's justification, not an independent acceptance sidecar.
## Dimension d: discrete_index
The domain is every positive integer dimension. The radius-one round
hemisphere with its unshifted scalar Neumann Laplacian is an established
geometric family. Dimension changes the manifold and spectral multiplicities,
not a continuous coupling disguised as an integer. Dowker, Section 2,
equations (1)–(3), identifies this spectrum and its Neumann zero-mode
omission. Cunha and Freitas, Section 2 and Section 3.1, supply finite formulas
for the full sphere and the complementary Dirichlet problem.
We include every d from 1 through 20. This noncanonical practical prefix
contains the interval control, the low-dimensional hemispheres, and ten
dimensions of each parity. It retains breadth across the two parity formulas
without extending to the source's hundred-dimensional listing. Degree-19
multiplicity polynomials remain inexpensive to check with a separate direct
spectral continuation. The measured generator and independent calculation
retain far more than the 100 significant digits stored; numerical evidence
records the sizes and accuracy. No claim is made that each dimension has a
separate physical application. The integer cutoff is independent of numeral
base and was selected for useful geometric coverage and verification cost,
not a target entry count.
The breadth review considered both canonical boundary conditions. Dirichlet
hemisphere determinants belong to their own table, T603, since they are a
different boundary value problem. The same dimension prefix allows their
spectral relation to be inspected, but agreement with another table's range
is not the reason for the prefix. Variable radius, a mass or curvature shift,
and a variable cap angle would introduce additional free inputs and separate
selection questions; they are fixed by this table's definition.
## Convention: intrinsic, exhaustive
The parameter has exactly the two identities `det` and `logdet`. The positive
determinant and its real natural logarithm determine each other by exp and
log. They refer to the same operator, have the same dimensional domain and
finite range, and cost essentially the same to store. Conversion is elementary.
They therefore satisfy the one-number-in-two-invertible-conventions exception.
This does not combine different boundary conditions or unrelated quantities.
The positivity follows from exponentiating the real spectral zeta derivative;
there is no logarithmic branch choice.
## Enumeration and completeness
`HemisphereNeumann.enumerate()` yields ascending d, then `det`, then `logdet`.
It yields 40 entries and no omitted dimension in 1–20. Both parameter
declarations describe the full domain; the finite extent belongs in the
completeness note. The Programs loop has the same range.
There are no continuous sampled inputs, reference meshes, mixed-role argument
groups, or excluded quantile families. `reference_sample` is consequently not
the category for either parameter. All selection-relevant source code is in
`generate.py` and its dependency `finite_formulas.py`; the separate critic
must fingerprint both. Also include `independent.py` and `checks.py` when
binding all attached numerical source dependencies. This file does not attest
to that later review.
Sources checked: https://arxiv.org/pdf/hep-th/9312080, Section 2, equations
(1)–(3); https://arxiv.org/html/2404.12114v1, Theorems 2.7, 2.8 and 3.2,
Appendix A. Exact downloaded source hashes are in `source-evidence.json`.