Mass–Energy Equivalence as a Constraint Surface and Its Dimensional Embeddings

Abstract

Einstein's mass–energy relation E=mc² is, as physics, a constraint among three positive scalars. This preprint develops it as a theory of embeddings: the constraint surface S ⊂ ℝ³ is a smooth saddle; a stated filling operator Φ places S uniquely into a 15-dimensional bookkeeping space M₁₅=ℝ¹⁵; dimensional folding is ordinary projection of that image. The 15-embedding is proved as a smooth proper injective immersion with a global left inverse (dictionary dimension 15; manifold dimension 2 at rest, 3 with a collinear boost). The scaling rule α(n)=2n/3 is a postulate about renamed slots, not a replacement of SI exponents; E=m c^{2n/3} fails as physics for n≠3. Laboratory checks (four independent runs, N=4000) confirm I=E/(mc²)=1 on S, negative Gaussian curvature, unique structured filling versus a random pad, nested PCA composition, and existence witnesses (exact left inverse, Jacobian rank 2/3, zero collisions, thirteen image equations). Manuscript 2.2.0 (theory edition). Zenodo deposit version 2.0.0. PDF only. Theory to share and explore; laboratory numbers are corroborating instances under stated regimes, not a claim that spacetime has fifteen axes.