Linearity of SHA-256 and Chaotic Systems in Self-Defined Dimensions

This paper introduces Inverse Dimensionalization, a novel analytical framework for examining cryptographic hash functions through geometric coordinate transformations. The central thesis posits that apparent chaos in deterministic systems is fundamentally observer-dependent: by embedding hash outputs into self-referential dimensional spaces, hidden linear structures emerge that can guide efficient similarity searches. Applied to SHA-256, this framework constructs a three-dimensional space where message indices, hash values, and collision metrics form navigable coordinates. Experimental validation with N = 100, 000 samples demonstrates the identification of quasi-collision pairs achieving 174/256 matching bits (67. 97%) —a result requiring approximately 2^46 random comparisons under brute-force assumptions, yet obtained in under 4 seconds using geometric guidance. These findings suggest that the perceived "one-way" nature of cryptographic hashes may reflect representational limitations rather than fundamental mathematical barriers, opening new avenues for structural cryptanalysis.