Algebraic Collapse of Fermat's Equation via Non-commutative Homological Friction: The ROA Framework

For over three centuries, Fermat’s Last Theorem (FLT) has been exclusively investigated within the confines of commutative scalar arithmetic and flat geometries, culminating in monumental yet profoundly circuitous proofs. In this paper, we decisively shatter this commutative illusion. By elevating the classical integer solutions to eigen-projection operators within a non-commutative rough Hilbert space H_R governed by Rough Operator Algebra (ROA) and Seonggil Matrix Theory (SMT), we reframe FLT as a structural boundary problem of geometric friction. We demonstrate that for exponential degrees n ≥ 3, the nested star-productcommutators generate a Non-linear Torsion Cascade, forcing an inevitable dimensional expansion of the topological residue. The rigid, 1-dimensional integer spectrum cannot endure this high-dimensional geometric rupture. Through the extraction of the topological refractive index f(n) via non-commutative Taylor expansion, we rigorously prove that the homological friction strictly exceeds the finite information capacity bound (C_SMT) of the super-manifold, triggering a catastrophic topological trace collapse.Consequently, the underlying physical and algebraic structure of the rough lattice dynamically rejects the existence of integer eigen-states. This framework not only provides a definitive, algebraic collapse proof of FLT within three pages, but also establishes a revolutionary paradigm bridging non-commutative geometry, information dynamics, and prime number theory.