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4390 bytes, as of the version from 2026-10-29 03:59 (current). Recorded here, not run.
# Argument selection for T599
This build follows the identical local and live skill copies recorded in
`agents/runs/20261029T032928Z-build-skills/manifest.json`. Each skill has SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The proposal was re-read from numberdb-data issue 236 and reassessed against
that skill. This is the builder's justification; an independent campaign
critique must assess it separately.
## The quantity and its arguments
This table holds one quantity, positive analytic torsion of a flat complex
line over a three-dimensional lens space. The ordered integer parameters
`p`, `q`, `u` specify the manifold presentation and its character. They do
not select different invariants. The quotient, generator, character and
spectral-zeta sign convention are pinned down in the table. The free
parameters have the following roles.
* **p: discrete_index.** The order of the finite fundamental group, p >= 2,
is a genuine topological size, not a discretized continuous variable.
Every order from 2 through 20 is included. This practical initial extent
spans primes and composite orders with several kinds of nonfaithful
characters, while retaining complete coverage at each included order.
Twenty is a noncanonical cost and readability cutoff, not a special
topological threshold. It is independent of numeral notation.
* **q: discrete_index.** Among units modulo p, the action exponent describes
the lens-space type. We include the least positive residue in every orbit
{q, -q, q^{-1}, -q^{-1}} modulo p. This is exactly the unoriented
diffeomorphism classification, not a grid of arbitrary angular inputs.
The representative is a bookkeeping choice, and it fixes the generator.
q is not bounded separately from p, and every type at the chosen p is
represented. In particular, the table is not restricted to L(p,1).
* **u: discrete_index.** All nontrivial complex characters of the chosen
cyclic fundamental group are present up to complex conjugation:
1 <= u <= floor(p/2). Nonfaithful characters are retained. Characters
are structural data of the torsion problem, not sample probabilities or
numerical evaluation angles. We do not claim that these rows are
inequivalent under every self-diffeomorphism, or that their values differ.
The general parameter domain remains p >= 2, 1 <= q,u < p, gcd(p,q)=1.
The finite selection belongs only in the completeness note and enumeration.
Coordinate exchange replaces (q,u) by (q^{-1}, u q^{-1} mod p). Character
labels must move with the chosen generator; only then may conjugation reduce
u to min(u,p-u).
## Breadth, extent, and cost
The initial segment gives 286 rows. It covers all lens-space types and
all character orders at each p, rather than pursuing only faithful characters
or extending the homogeneous spaces more deeply. The tuple constraints and
symmetries remove redundant presentations; this is a finite census of
manifolds with local systems, not a Cartesian mesh of independently chosen
continuous inputs. Lens-space torsion is a standard test case for an invariant
that distinguishes spaces beyond their fundamental and homology groups.
No claim is made of a separate published application for each row.
The numerical evidence records the number of types, exact rational cases,
precision, maximum string length and verification results. The generic dry
run measured 51.6 KiB of high-precision ball representations, comfortably
below the recommended 160 KiB allowance; stored 100-digit decimals are shorter.
The bound is chosen for coverage, not to reach a row-count or storage quota.
## Checked sources
* A. Kitaoka, *Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces*,
SIGMA 18 (2022), 091, pp. 2–3: cyclic actions and complex characters;
Section 4: the spectral convention; Proposition 5.1: the torsion product
with inverse residues. DOI: 10.3842/SIGMA.2022.091.
* M. Sağlam, *Holomorphic curves in the symplectizations of lens spaces:
An elementary approach*, Münster J. Math. 15 (2022), 389–440,
Theorem 1.3, p. 392: the unoriented diffeomorphism classification.
DOI: 10.17879/21089689454.
These sources support the family and conventions. The finite cutoff p <= 20
is our reference extent for this discrete family. There is no exploratory
continuous-input group to hide under an intrinsic or arithmetic label.