Version 2 Collapse of the Hypotenuse
Year: 2025
Type: article
Abstract: The central question is whether two genuinely positive legs can sum to a zero hypotenuse. The answer is yes, and version 4 organises the cases into a hierarchy of three levels. Level 1 (complex numbers) is a foil: i^2 + 1^2 = 0 looks like collapse but cheats, since i^2 = -1 is negative. Level 2 (Minkowski spacetime) is real geometry — null vectors, photon worldlines — but the indefinite metric assigns a negative squared norm to the timelike leg, so positivity is not unambiguous. Level 3 (Riemannian manifolds) is the strongest case: on the sphere, flat torus, and ellipsoid, both legs are measured by a positive-definite metric with no sign ambiguity anywhere, and collapse is forced entirely by curvature or global topology. The paper also extends collapse to unequal legs for any positive pair (m,n), introduces the geoplus notation to distinguish geometric summation from arithmetic, and shows via the 90° tilt construction that ellipsoidal surfaces — not just spheres — support the same collapse. The hierarchy is the structural contribution. Previous versions treated all mechanisms as equivalent. Version 4 makes clear that the Riemannian cases are the genuine article.
Author: John Eonas
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7082167133
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