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3092 bytes, as of the version from 2026-10-27 16:59 (current). Recorded here, not run.
# Derivations and independent checks for T553
Put a=1/n and theta=pi/n. Theorem 1 of Solynin and Zalgaller gives
cap(P_n)^2 / Area(P_n) = n tan(theta) Gamma(1+a)^2 /
(pi 2^(4a) Gamma(1/2+a)^2). Their stated polygon has centre zero and a vertex
at one. Its area is n sin(2 theta)/2, so taking the positive square root gives
n sin(theta) Gamma(1+a)/(sqrt(pi) 2^(2a) Gamma(1/2+a)).
Gamma duplication gives
Gamma(1+a) Gamma(1/2+a) = 2^(-2a) sqrt(pi) Gamma(1+2a).
Reflection and recurrence give
Gamma(1-a) Gamma(1+a) = pi a / sin(pi a).
These transform the source expression into the generator's quotient
Gamma(1+a)/(Gamma(1-a) Gamma(1+2a)). All Gamma arguments are positive.
Independently, the exterior Schwarz–Christoffel derivative is
F'(w)=c(1-w^(-n))^(2/n), taking the branch tending to c at infinity.
Its integral along one unit-circle arc maps to a straight polygon side. Thus
2 sin(theta)=c*(4/n)*2^(2/n)*integral_0^(pi/2) sin(t)^(2/n) dt.
The tests evaluate this integral at 130 and 170 decimal digits, without Gamma
functions, after testing its exponents 0 and 1 against pi/2 and 1. This is an
agreement check, with no claim that mpmath quadrature is a certified enclosure.
Scaling F by s>0 scales its leading coefficient and all side lengths by s;
scaling to side one divides c by 2 sin(theta), and scaling to area one divides
c by sqrt(n sin(2 theta)/2). No extra normalization parameter is introduced.
The inscribed disk has radius cos(theta) and the circumscribed disk has radius
one. Capacity is monotone under inclusion and a disk's capacity is its radius,
giving cos(theta)<=c<=1 and the limit c->1 by squeezing. These are analytic
arguments; finite testing is not presented as a proof of an infinite limit or
the paper's minimization theorem.
For n=4, the Gamma quotient and the lemniscatic formula
K(1/2)=Gamma(1/4)^2/(4 sqrt(pi)) give c=sqrt(2)K(1/2)/pi, with K using parameter
m rather than modulus k. Gauss's AGM relation then gives
c=1/(sqrt(2) AGM(1,1/sqrt(2))). A separate 12-step AGM computation bounds its
true limit between the geometric and arithmetic iterates and checks the
(1,1) control. This enclosure is also inside the stored square interval.
No repeats declaration is appropriate for the square's coincident value in
T348: its hypergeometric entry describes a different quantity, linked by a
special-value identity, and is not a transcription of this construction.
The 100-digit serialized values denote centre +/- one unit in the last place.
Exact rational comparisons prove that these intervals contain the complete
461-bit computation balls and the independently evaluated 600-bit source
formula balls. All computed balls are finite, positive and narrow. The minimum
measured relative precision is 457 bits. No exact rational capacity was inferred
from a run of numerical digits.
The checked sources and their locators are recorded in sources.json. The
coordinated skill manifest is
agents/runs/20261027T162618Z-build-skills/manifest.json; both local and live
versions have SHA-256
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.