On the Structural Impossibility of Hash Collision Finding\\ via Direct SAT Duplication Why Encoding $H(x_1) = H(x_2) \wedge x_1 \neq x_2$ as a Duplicated SAT Circuit is Inherently Unsatisfiable

We present a fundamental structural observation regarding the encoding of hash function collision search as a Boolean Satisfiability (SAT) problem. The standard approach to finding collisions $H(x_1) = H(x_2)$ with $x_1 \neq x_2$ requires instantiating two copies of the hash circuit and constraining their outputs to be equal. We demonstrate that this ``duplication approach'' produces formulas that are inherently unsatisfiable due to deep structural reasons rooted in the interaction between circuit determinism, the pigeonhole principle at the clause level, and the symmetry-breaking effect of the inequality constraint. We argue that this unsatisfiability is not merely a consequence of hash function strength but rather an intrinsic property of the encoding itself, rendering the SAT-based collision search approach fundamentally misconceived. Even differential cryptanalysis paths, when fully encoded, collapse into the same structural trap. This observation has significant implications for understanding why decades of SAT-based cryptanalysis have produced limited results on collision finding for standard hash functions.