# Argument selection for T620

The archived canonical skill is recorded in
`../20261030T071029Z-build-skills/manifest.json`. Both archived versions have
SHA-256 `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The family conventions were read from numberdb-data issue 247 before any
computation. This is the builder's rationale, not an independent critique.

The number of sides n is a discrete index of regular Euclidean polygons.
It determines the cyclic symmetry, interior angle pi(1-2/n), and all rational
special-function arguments in the formula. The operator and its entire
boundary condition are fixed. The domain has circumradius one; the radius
is not sampled. A different side count is a different domain of the same
established spectral invariant. Aurell and Salomonson, Proposition 6,
hep-th/9304031, treats this family for general n. Kalvin, 2610.35383v1,
Section 7, supplies a proof for the polygon formula, and the integrable
triangle and square controls are treated by Mårdby and Rowlett (2025),
doi:10.1007/s00041-025-10202-6.

The complete finite prefix 3 <= n <= 30 includes every side count, covering
triangles and squares, tiling and nontiling polygons, constructible and
nonconstructible symmetry orders. The endpoint 30 is a noncanonical practical
cutoff. It gives a reference for low and moderate polygonal symmetries without
extending deeply into polygons that approximate a disk. Breadth within this
family is covered before depth: no gaps are removed merely because a polygon
is not individually famous. The range is justified by the domain family,
not by a size target or by the existence of similarly indexed tables. The disk
is not an integer-indexed polygon and is not a stored row. Other triangles,
rectangles, arbitrary moduli and boundary conditions are outside this proposal.

The parameter convention is intrinsic: det and logdet are the determinant
D_n^D and its natural logarithm L_n^D. All Dirichlet eigenvalues are positive,
so these real representations determine each other by exp and log. They have
identical finite ranges, cheap conversion, and similar written lengths. This
is one object in two invertible conventions, not two boundary problems.
Enumeration is increasing n, then det, then logdet: 28 polygons and 56 entries.

There is no freely selected continuous input and no exploratory reference
mesh. The Gamma and Barnes arguments are algebraic consequences of n. There
is no quantile, percentile or statistical critical value. The generator,
completeness note and stored parameter identities must agree with this
selection. The independent critique must bind its verdict to the live
selection and generate.py using agents/argument_review.py fingerprint.