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2422 bytes, as of the version from 2026-10-27 17:40 (current). Recorded here, not run.
# Sources checked for T554
The family proposal was read with `python3 agents/queue.py show 222` before the build. Its centre, vertices, geometry keys, parameter order and discrete extent were retained after assessment. The coordinated skill manifest is `agents/runs/20261027T171124Z-build-skills/manifest.json`; both archived copies have SHA-256 e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
Barnard, Hadjicostas and Solynin, The Poincaré metric and isoperimetric inequalities for hyperbolic polygons, Trans. Amer. Math. Soc. 357 (2005), 3905–3932, DOI 10.1090/S0002-9947-05-03946-2, was read via the full-paper mirror https://studylib.net/doc/10406440/transactions-of-the-american-mathematical-society-volume-... . Browser retrieval exposed the article text, including definitions on pp. 3905–3908 and equations (1.3), (1.5), (1.6). Direct publisher PDF retrieval returned HTTP 403; the author's old PDF URL returned HTTP 404. The mirror's direct urllib download was also unavailable; no downloaded PDF is claimed.
The paper uses the disk metric |dz|/(1-|z|^2); its h-radius and ordinary conformal radius agree at zero by (1.3). The regular ideal n-gon has its vertices on the unit circle, and (1.6) gives Gamma(1-1/n) Gamma(1/2+1/n) / [Gamma(1+1/n) Gamma(1/2-1/n)]. Equation (1.5) gives the Euclidean radius squared divided by area as 2^(4/n)/pi times that ratio. Reflection and duplication yield the generator's circumradius-one Euclidean expression.
DLMF 5.5, https://dlmf.nist.gov/5.5, supplies Gamma recurrence, reflection and duplication. DLMF 5.12.1 and 5.12.3, https://dlmf.nist.gov/5.12, supply the beta integral and its integral over the positive half-line, from which the Gamma-free checks in checks.py are derived. DLMF 19.8.1 and 19.8.5, https://dlmf.nist.gov/19.8#i, supply the AGM iteration and its relation to the complete elliptic integral. DLMF uses modulus k; NumberDB T25 uses parameter m=k^2.
Corpus documents actually read include T17, T25, T390, T553, Gamma values, beta values and arithmetic-geometric means. The final table links only public destinations, using verified slugs. T553 is still private, so its cross-link is deferred. Sample lookups at approximately 0.56604668 and 0.45694658 were empty; the square's 0.76275976 matched hypergeometric entries in T348, a mathematical coincidence that does not duplicate this family. No repeats property is appropriate.