WHY π/2? THE STRUCTURAL ORIGIN OF THE CRITICAL CATENOID VOLUME COEFFICIENT

The companion paper The Volume of the Critical Catenoid (doi:10.5281/zenodo.18808912) shows that the solid of revolution bounded by the critical catenoid spanning two coaxial circles of radius R separated by distance h has volume exactly (π/2)R²h, and that this identity holds if and only if the shape parameter satisfies coth(x) = x. This note explains why π/2 is forced by structure rather than numerology. The coefficient decomposes as π/2 = π · (1/2): π comes from rotation, while the factor 1/2 arises because the cylinder-bisection condition factors through the boundary value of the Jacobi field of the catenoid family. The same Jacobi boundary value is independently the fold (criticality) condition. The hyperbolic Pythagorean identity cosh² − sinh² = 1 is the algebraic mechanism that produces this shared factor. The paper gives two independent proofs: (1) an explicit factorization of the bisection functional via the Pythagorean identity, showing that bisection and criticality are algebraically equivalent; and (2) a short proof from criticality to bisection that avoids evaluating the volume integral entirely, using instead a minimal-surface first integral relating volume to lateral surface area. This paper is a companion to the five-paper series on the Skwedge genus and Phylum Apicalia taxonomy; see The Skwedge Disparity (doi:10.5281/zenodo.18809247) for the series overview.