# Argument selection for T621

This is the Neumann member of numberdb-data family #247. Its single quantity is
the zeta-regularized determinant of the nonnegative scalar Euclidean Laplacian
on the regular polygon of circumradius one, with homogeneous Neumann boundary
conditions and the constant zero mode omitted. The determinant and its natural
logarithm are two invertible representations of this same quantity.

The governing skill evidence is
`../20261030T080603Z-build-skills/manifest.json`, archived by this run.
Both `skill-local.md` and `skill-live.md` have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The canonical source file has the same hash. `skill-evidence.json` records these
reads. This build is not an independent critique or an argument-review attestation.

## Every free parameter

* `n` has type `Z` and domain n >= 3. Its category is `discrete_index`:
  it counts the sides of a regular polygon and fixes its cyclic symmetry and
  corner angle pi*(1-2/n). The selected extent is every integer 3 through 30.
  This contiguous prefix covers small and moderate symmetry orders, including
  nonconstructible polygons, rather than retaining only the triangle, square
  and hexagon. Thirty is a noncanonical practical cutoff: deeper prefixes
  increasingly concentrate on shapes near the disk, without adding another
  geometric family. Nothing asserts that 30 has intrinsic significance, or
  that each individual polygon has a separately documented application.
* `convention` has type `Symbolic` and exactly the values `det`, `logdet`.
  Its category is `intrinsic` to the stated representation choice. Both rows
  occur for every side count; exp and real log recover one from the other.
  Their written lengths and numerical costs are comparable, so both fit the
  one-object-in-invertible-conventions exception. This parameter does not
  select different boundary problems or different spectral quantities.

There are no other free inputs. The physical circumradius is fixed at one.
The area, interior angle, conformal-map scale and rational Gamma arguments
are derived from n. No reference input mesh, root coupling, decimal grid,
Cartesian product of sampled continuous inputs, or statistical quantile is used.

## Breadth, boundary and evidence

The selection covers the full proposed discrete family through moderate n.
The Dirichlet problem belongs to companion T620, not to extra rows here.
T485's square and triangle eigenvalues use side length one; we explicitly
convert to side lengths sqrt(2) and sqrt(3) in our independent spectral checks.
Non-square rectangles and other triangles require another geometric argument;
Robin or massive operators introduce additional continuous arguments. The disk
is a limit, not a polygon with an integer side count. None is silently absorbed.

Checked mathematical sources are Aurell and Salomonson, Proposition 6 and
(18), (20), (23)-(24), for the regular-polygon formula and corner integral;
Kalvin 1910.00104v2, (3.15) and Lemma A.1, for finite rational-corner evaluation;
and Kalvin 2610.35383v1, Section 1 and Theorem 1.1, for the Neumann zero-mode
convention and determinant ratio. Its hypotheses apply because our polygons
are topological disks, flat in the interior, with straight sides and positive
corner angles. Mårdby and Rowlett, Corollaries 2.2 and 3.3 and the square and
triangle spectral discussions, supply independent integrable-domain checks.
These sources establish the family and conventions, not the editorial endpoint.

The database searches, family document, four numerical lookup probes, resolved
companion addresses and existing tags were checked before computation. T620
stores the distinct Dirichlet quantity. The generator enumerates increasing n,
then `det`, then `logdet`, for exactly 56 entries. The completeness note states
this same extent and rationale. Measurement and precision results are preserved
with the numerical evidence; the extent is not chosen to meet a count target.

A separate critic must assess both parameters, verify numerical claims, and use
`agents/argument_review.py fingerprint T621` with every reviewed source file.
No independent acceptance sidecar is supplied by this build.