back to table · edit · history · where entries came from · files · download
5232 bytes, as of the version from 2026-10-30 22:26 (current). Recorded here, not run.
# Argument selection for T632
This build follows the two byte-identical skills archived in
`../20261030T212747Z-build-skills/manifest.json`. Each has SHA-256
`10c1d8bd62d6ddfa2c016375ad82879151f0ab7a9516c2f371432108a85be009`.
The four parameter choices in numberdb-data family #252 have been reassessed
against that skill. This is the build justification, not the independent
critique or its argument-review sidecar.
## Object and normalization
The quantity is the zeta-regularized minimal scalar determinant on a round
three-dimensional lens space with one complex flat line character. The metric
is induced from the sphere of radius one, without mass or curvature shift.
Complex dimensions are counted once. The constant scalar zero mode is omitted
for the trivial character. See Dowker–Chang, arXiv:0904.0744, Section 3,
equations (4), (6)–(9), taking the complex-character projector without their
additional realification factor. Both determinant and natural logarithm are
stored as invertible conventions of the same number. Neither the coexact nor
the full one-form determinant is an additional output of this table.
## Each free parameter
* **p: discrete_index.** The order of the fundamental group is the covering
degree, not a sampled continuous coupling. Include every integer 2 through
20, covering primes, prime powers and mixed composite orders. This is a
noncanonical practical prefix of a discrete geometric family. Increasing p
enlarges both the type list and the character list; complete breadth through
20 is preferred to a much longer list restricted to homogeneous spaces.
* **q: discrete_index.** Include every unit residue class modulo p up to
negation and inversion, taking the smallest positive member of each orbit
`{q,-q,q^(-1),-q^(-1)}`. This includes all unoriented lens-space types in the
bound (Sağlam, DOI 10.17879/21089689454, Theorem 1.3). These identifications
are isometries of the round quotient: coordinate exchange and conjugation
realize them explicitly. The minimum-residue representative is a labeling
choice, independent of decimal notation, not a preferred metric or a claim
that this presentation is canonical. All q types occur at each p; no
arbitrary slice or cross-product of continuous inputs has been introduced.
* **u: discrete_index.** Include every character exponent from 0 through
floor(p/2) on each selected presentation, covering all characters up to
complex conjugation. The trivial sector describes the ordinary scalar
operator. Nontrivial sectors describe flat holonomy; both faithful and
nonfaithful characters are included. All sectors have the same discrete
representation-theoretic role, so these are not mixed exploratory and
application selections. The characters and projector are specified in
Dowker–Chang, equations (2), (8); the marked generator and character
transformation agree with T599 and T631. Exchange sends (q,u) to
(q^(-1),u*q^(-1) mod p); negating q keeps u. Conjugation then reduces u.
* **convention: intrinsic.** `det` and `logdet` express a positive determinant
and its real natural logarithm. Exponentiation and logarithm are inexpensive
inverse conversions with the same domain and comparable storage length.
This is the one-number-in-two-conventions exception, not a quantity switch.
Nash–O'Connor, arXiv:hep-th/9212022, equation (1.1), explicitly uses the
logarithmic convention; Dowker–Chang, Section 3, defines the determinants
through spectral zeta derivatives.
There are 46 presentations, 332 space-and-character pairs (46 trivial and
286 nontrivial), and 664 entries. Enumeration is increasing p, then q, then u,
then det and logdet. The full parameter domains remain p>=2, unit q modulo p,
and 0<=u<p. The finite selection is recorded in the completeness note.
Residual automorphisms may identify marked characters within a selected
presentation; repeated numbers are retained and no uniqueness is claimed.
This is discrete geometric and representation-theoretic coverage. No
reference_sample group, arbitrary decimal grid, or statistical quantile occurs.
## Breadth, size and overlap
Full type and holonomy coverage includes homogeneous and inhomogeneous spaces,
ordinary and twisted scalars, and prime and composite group orders. The bound
20 is a computational/reference choice, not a theorem or a fame threshold.
The checks measure the actual decimal block and longest entry before filling.
The choice agrees with the companion extent because these are the same
geometric objects, not because equal entry counts are intrinsically desirable.
The (2,1,0) determinant and logarithm reproduce the radius-one RP3 member of
T600; the table declares this overlap. T595 gives the covering-sphere control.
T599 is torsion, T631 is coexact, and T622/T623 are odd-signature asymmetry,
so none duplicates the scalar family. They were read from the live database.
The selected proposal has no earlier request marked '(answers #N)'; no GitHub
request provenance is put in mathematical prose. The independent critique
must assess these choices and bind its own sidecar to the live document and
all generator dependencies. This build does not create or accept that sidecar.