Metamorphosis Part III — The Three-Way Identity (Euclid, Emergence, and Calculus in Concordance)

This concluding part of the Sc-Rubs Metamorphosis Trilogy consolidates the volumetric and persistence behaviour of the scalar-field engine, with particular focus on the now-recurrent closure constant 2.915. Remarkably, this value arises from four completely independent derivations:
(1) Euclidean volumetrics using sphere–cube normalisation;
(2) the experimentally stable radial emergence gate at 0.7;
(3) global persistence closure of the variational field; and
(4) antipodal normalisation across the sphere, cube, and Sc-Rubs field.
That these four paths converge to the same constant (within <0.2%) strongly indicates the presence of a genuine geometric invariant governing the confinement–emergence–membrane structure of the field. This paper situates the constant 2.915 as a fundamental closure law for bounded emergence systems, completing the geometric arc that began in early Sc-Rubs experiments and now spans Euclidean, rectilinear, and variational domains.