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2609 bytes, as of the version from 2026-10-27 16:59 (current). Recorded here, not run.
# Logarithmic capacity of regular polygons (T553)
The sole parameter is the integer number of sides n >= 3. It indexes a
geometrically defined discrete family: the closed convex hull of the nth roots
of unity. Circumradius is one, the centre is zero, and a vertex is one. There is
no independently sampled continuous argument, shape parameter or Gamma input.
The rational Gamma arguments are consequences of the side count.
The default enumeration is every n = 3,...,30, in increasing order. This is a
`discrete_index` selection. It includes the familiar triangle, square, pentagon
and hexagon, and contiguous coverage of all subsequent small symmetry orders,
including nonconstructible polygons. Restricting to constructible, prime or
individually famous cases would remove members without a capacity-specific
reason. Thirty is a noncanonical editorial cutoff: the purpose is reference
coverage of small polygonal symmetries, without extending deeply into shapes
whose capacity approaches the unit disk's capacity. It is independent of the
numeral base and is not a precision, storage or entry-count target.
Breadth review: this table holds one invariant of one Euclidean family. Scaling
to side length one or area one is explained algebraically, rather than adding
normalization groups to the family proposal. Interior conformal radii and
uniformization radii are different invariants assigned to other proposals.
There are no exploratory continuous-input groups, arbitrary meshes or Cartesian
products. The complete-note and Programs enumeration must both say 3 <= n <= 30.
Supporting sources checked:
* Solynin and Zalgaller, Annals of Mathematics 159 (2004), Theorem 1 and equation
(1.2), https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n1-p05.pdf.
Their regular polygon is centred at zero with a vertex at one, exactly the
geometry here. The formula applies to every integer n >= 3; the cutoff 30 is
ours and is not attributed to the paper.
* DLMF 5.5.1, 5.5.3 and 5.5.5, https://dlmf.nist.gov/5.5, justify the Gamma
recurrence, reflection and duplication transformations used in the generator.
* The family conventions were read from numberdb-data issue 222 before work.
Existing T17's unit-R branch and the regular-polygon cases of T390 were read
from the database to check the shared scale.
Policy evidence: agents/runs/20261027T162618Z-build-skills/manifest.json.
The archived local and live skills both have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
This is a build justification, not an independent critique or an acceptance.