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5025 bytes, as of the version from 2026-10-30 22:25 (current). Recorded here, not run.
# Argument selection for T631
This build follows the matched archived local and live canonical skill in
`../20261030T204529Z-build-skills/manifest.json`. Both copies have SHA-256
`10c1d8bd62d6ddfa2c016375ad82879151f0ab7a9516c2f371432108a85be009`.
`skill-evidence.json` preserves that manifest's source paths and digests.
This is the build's justification, not an independent critique or approval.
The named object is the coexact one-form Hodge Laplacian determinant on a
round three-dimensional lens space with a flat complex line. Both helicities
and complex eigenspace dimensions are counted once. The metric is induced
from the sphere of radius one, and the quotient generator and character are
`g(z1,z2)=(zeta_p*z1,zeta_p^q*z2)` and `chi_u(g)=zeta_p^u`.
The selection is a bounded discrete geometric and representation-theoretic
family. It has no continuous exploratory argument, reference mesh or
unjustified multidimensional Cartesian product. No scalar or full one-form
determinant is stored in this table. Independent scalar computations in the
checks are evidence, not additional entries.
| Parameter | Role | Justification and finite extent |
| --- | --- | --- |
| p | discrete_index | The order of the fundamental group and covering degree. Every integer 2 through 20 is included. This noncanonical practical prefix covers primes, prime powers and mixed composite orders, with small exact character calculations. The mathematical domain remains p >= 2. |
| q | discrete_index | All unit residues modulo p, reduced by negation and inversion. The smallest positive member of each orbit {q,-q,q^(-1),-q^(-1)} labels its representative. This is a noncanonical labeling convention independent of numeral base. It covers all unoriented types within the chosen order bound, including inhomogeneous spaces. |
| u | discrete_index | All flat line characters up to conjugation: 0 <= u <= floor(p/2), for every selected presentation. This includes the trivial character, all faithful sectors, and all nonfaithful sectors for composite p. These are the complete characters of the finite fundamental group, not sampled continuous angles. Residual bundle automorphisms can give duplicates; no claim of distinct numerical values or distinct marked bundles is made. |
| convention | intrinsic | `det` and `logdet` denote one positive number and its real natural logarithm. The conversion is invertible and elementary; the parameter domain and coverage are identical for both. |
The default enumeration has 46 presentations and 332 space-and-character
pairs, hence 664 stored entries. It orders p, q and u increasingly, then det
before logdet. The 46 trivial and 286 nontrivial cases have the same metric
and character conventions as the family's torsion table T599. Character
conjugation, negation of q, and coordinate exchange are verified on the
entire exact multiplicity quasipolynomial. In particular, replacing q by its
inverse r replaces u by ur modulo p; keeping the old character would change
the marked bundle in general.
Breadth takes priority over a deeper prefix of homogeneous L(p,1): the full
set of unoriented types and characters is retained. The p=20 cutoff is a
cost/coverage choice, not an intrinsic threshold or a target entry count.
Every written value has about 100 significant digits, so even the largest
entries remain readable. The dry run and numerical evidence record actual
block size and longest-entry length. There is no need to prune to famous
spaces or to extend to higher orders merely because more would fit.
Sources actually checked:
* Dowker and Chang, *Analytic torsion on spherical factors and tessellations*,
arXiv:0904.0744, Section 3, equations (6)-(9): unit-radius scalar and coexact
spectra and the character projector. Their extra factor for real dimensions
of complex forms is not applied: this family counts complex dimensions.
* Sağlam, *Holomorphic curves in the symplectizations of lens spaces*,
Münster J. Math. 15 (2022), Theorem 1.3, page 392: the unoriented
classification. Coordinate exchange and conjugation are round isometries,
giving the representative identifications without changing metric scale.
* Nash and O'Connor, arXiv:hep-th/9212022, Sections 2-4 and equation (1.1):
gauge-theoretic use of flat holonomy sectors and a nontrivial-character
determinant control. The paper's torsion convention is not imported.
* Kitaoka, SIGMA 18 (2022), 091, equation (4.3) and Proposition 5.1: spectral
torsion for the trivial and nontrivial characters. The unit-radius metric
in (4.3), not the scaled metric elsewhere, is used for the trivial case.
The live corpus was rechecked with `already_here`, `already_asked`, targeted
text searches and four value lookups. T599, T622, T623, T595, T600, T14, T228
and T94 were read. The table's tags already exist. Actual resolved slugs were
read before composing companion links; unbuilt siblings receive no invented
identifiers. The controller must obtain an independent selection review after
this build.