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5279 bytes, as of the version from 2026-10-30 22:06 (current). Recorded here, not run.
# Argument selection for T633
This is the build's justification, not the independent critique. The policy
is the matched local/live skill archived in
`../20261030T215605Z-build-skills/manifest.json`. Each copy has SHA-256
`10c1d8bd62d6ddfa2c016375ad82879151f0ab7a9516c2f371432108a85be009`.
The source paths and digests are also preserved in `skill-evidence.json`.
The named quantity is the full one-form Hodge Laplacian determinant on the
unit-radius round three-dimensional lens space with a flat complex line.
Its exact and coexact spectra both contribute, and each complex dimension
is counted once. A determinant and its real natural logarithm are two
invertible conventions of that same positive quantity. Neither component
operator is separately stored here; they have companion tables T632 and T631.
The quotient generator is g(z1,z2)=(zeta_p*z1,zeta_p^q*z2), and the character
satisfies chi_u(g)=zeta_p^u. Lifted forms obey g*omega=zeta_p^u*omega.
All four parameters have the following justification, reassessed from the
current skill and primary sources rather than accepted as a queued grid.
| Parameter | Role | Family and finite extent |
| --- | --- | --- |
| p | discrete_index | Fundamental-group order and covering degree. Include every 2 <= p <= 20, a noncanonical practical complexity prefix. It covers prime, prime-power and mixed-composite orders while keeping exact finite character sums and numerical checks modest. The domain stays p >= 2. |
| q | discrete_index | A unit residue modulo p specifies a round quotient. Select the smallest positive member in each orbit under negation and inversion. These are all unoriented types within the bound, including inhomogeneous ones. The smallest-residue label is noncanonical and independent of numeral base. |
| u | discrete_index | Include every character up to conjugation, 0 <= u <= floor(p/2), on each selected presentation. This includes trivial, faithful and nonfaithful characters of the finite fundamental group. These are discrete flat-holonomy sectors, not sampled continuous angles. The domain stays 0 <= u < p. |
| convention | intrinsic | Both `det` and `logdet`, related by real logarithm and exponential, with exactly the same geometric parameters and range. No second named operator is selected by this parameter. |
This dependent selection has 46 presentations, 332 space-and-character pairs
and 664 entries. The pairs consist of 46 trivial and 286 nontrivial characters.
Enumeration increases p, then q, then u, with det before logdet. The default
enumeration, Programs example, and completeness note use these definitions.
Residual automorphisms may identify bundles or give repeated numbers; no claim
of unique values or of a minimal set of marked bundles is made.
The extent prioritizes breadth over more homogeneous spaces L(p,1): all types
and all character sectors in the bound are retained. The cutoff is a practical
reference extent, not an intrinsic threshold or an entry-count target. It is
not necessary for every finite-group character to have an individually famous
application. This is a discrete geometric family, with no continuous
reference-sample group and no arbitrary Cartesian product of simple inputs.
Keeping the same geometry as T631 and T632 makes the Hodge decomposition
comparable; matching their count alone would not justify the selection.
Under coordinate exchange q becomes r=q^(-1) mod p and u becomes ur mod p.
Negating q keeps u, and conjugating the character sends u to -u. These
identifications are tested on the complete exact multiplicity quasipolynomial
for both spectra, including all unreduced labels through p=20. The inverse
character transform is essential; treating q inversion as leaving u fixed
would in general select a different marked bundle.
Sources checked for the family and conventions:
* Dowker–Chang, arXiv:0904.0744, Section 3, equations (4), (6)–(9), and the
total/coexact zeta relation immediately before equation (3): spectra,
character projection, and Hodge decomposition. Complex dimensions are
used once, without their optional factor for realifying a complex line.
* Sağlam, Münster J. Math. 15 (2022), Theorem 1.3, p.392: unoriented
classification by q modulo negation and inversion. Coordinate exchange and
conjugation of the second coordinate are round isometries.
* Nash–O'Connor, arXiv:hep-th/9212022, Sections 2–4 and equation (1.1):
determinant use in flat gauge sectors and two nonzero determinant controls.
Their torsion sign/power convention is not imported.
* Kitaoka, SIGMA 18 (2022), 091, equation (4.3) and Proposition 5.1: the
trivial and nontrivial character torsions. Use g_std at unit radius in
(4.3), not 4*g_std from other parts of that article.
The longest returned ball and serialized block are measured in the dry run;
the evidence separately measures the intended 100-significant-digit strings.
There is no evidence of a readability or precision reason to prune sound
entries. Extending the order merely because more entries would fit is also
unnecessary. A separate critique must assess the choices and all source files,
including generate.py, independent.py and checks.py, before the controller
may offer the draft. This build creates no argument-review acceptance.