The Dimensional Loss Theorem: Proof and Neural Network Validation

This paper presents the formalization and empirical validation of the Dimensional Loss Theorem, a universal principle governing the degradation of binary discrete patterns when embedded from 2D planes into 3D lattice volumes. Building upon prior empirical observations of an 86% scaling law, component-wise proofs are provided for the S (Connectivity), R (Volumetric), and D (Entropy) transformations. The connectivity tax is demonstrated to be a geometric invariant of Moore neighborhoods. Applying this framework to the final layers of GPT-2 and Gemma-2, numerical verification confirms exact component transformations (0.000% implementation error) while empirical validation demonstrates 84.39% ± 1.55% total information loss across N=60 patterns. Furthermore, the semantic invariance property is established, proving that topological information loss is content-independent.