# Argument selection for T554

This table stores one quantity, the interior conformal radius r(D,0), for two discrete polygon families. The geometry parameter identifies its domain, rather than switching quantities. Parameters occur in the order geometry, n.

## Euclidean group

`geometry=euclidean` selects the open convex hull P_n of the nth roots of unity, centred at zero, with circumradius one and a vertex at one. The integer n>=3 is the number of sides and the order of rotational symmetry. This is a discrete index, not a sampling of continuous Gamma arguments. The conformal-radius family is explicitly studied in Barnard, Hadjicostas and Solynin (2005), Theorem 1.2, equation (1.5). Its area normalization is converted to circumradius one using A_n=(n/2)sin(2*pi/n), consistently with NumberDB T17 and T390.

## Ideal hyperbolic group

`geometry=ideal-hyperbolic` selects H_n in the Poincaré unit disk, with the same ideal vertices and centre. Its sides are orthogonal circular arcs, rather than Euclidean chords. The stored derivative is in the complex disk coordinate; it is not a hyperbolic distance. Equation (1.3) of the same paper identifies its conformal h-radius with the usual conformal radius at zero. Theorem 1.3, equation (1.6), supplies this discrete n>=3 family. There is no freely chosen angle, area, curvature normalization, or evaluation point hidden beside n.

## Breadth, extent and ordering

Both groups contain every integer 3<=n<=30, increasing within the Euclidean group and then within the ideal-hyperbolic group. The two geometries supply useful breadth before increasing depth. The contiguous range includes the small familiar symmetries as well as composite, prime, constructible and nonconstructible polygons. Thirty is a noncanonical editorial cutoff, not a mathematical boundary or an entry-count target: it extends beyond the elementary cases while avoiding deep coverage of polygons tending towards the unit disk. No point needs an individually documented application. The family-specific structure is the discrete symmetry, rather than the ability to evaluate Gamma at simple fractions. No continuous exploratory grid is selected; reference_sample and reference_policy are inapplicable.

Changing scale, centre, or the radius of nonideal hyperbolic vertices would introduce other choices, so they are not enumerated here. Scale conversion is stated as a formula for the Euclidean group. Extending n is straightforward; extending geometry requires a fresh assessment.

## Supporting sources and instruction record

Barnard, Hadjicostas and Solynin, The Poincaré metric and isoperimetric inequalities for hyperbolic polygons, Trans. Amer. Math. Soc. 357 (2005), 3905–3932, pp. 3905–3908, equations (1.3), (1.5), (1.6), DOI 10.1090/S0002-9947-05-03946-2. The full article was read through a public mirror after publisher and author PDF downloads failed: https://studylib.net/doc/10406440/transactions-of-the-american-mathematical-society-volume-...

DLMF 19.8(i), equation 19.8.5, fixes the AGM check's elliptic modulus convention. NumberDB T25 instead uses parameter m=k^2, so its lemniscatic argument is m=1/2.

This build follows the coordinated instruction archive `agents/runs/20261027T171124Z-build-skills/manifest.json`. Both `skill-local.md` and `skill-live.md` were read and their bytes SHA-256 checked: e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c. This is the builder's justification, not an independent critique acceptance. The separate critic must assess both groups and fingerprint the final document and selection-relevant attachments.