A Continuous Generalization of the Cohn–Umans Framework via Quotient Topology and Fiber Morphisms

This preprint introduces a continuous, topological analogue of the Cohn–Umans group-theoretic framework for fast matrix multiplication.  By treating the computational domain as a topological vector space and applying a quotient topology that collapses redundant computations, the paper defines a global reduction in which bilinear maps descend to continuous morphisms on equivalence classes.  Each class carries a smooth internal fiber structure permitting local refinement through morphic maps that preserve the quotient projection. The resulting Morphic Complexity Reduction Theorem establishes a rigorous two-level structure—global quotient reduction and local fiber-wise refinement that generalizes algebraic tensor-rank decomposition to continuous settings. The work includes complete proofs, a topological convolution theorem, and discussion of applications to matrix multiplication, convolution, and attention mechanisms.  This version is posted as a working preprint for open peer feedback and collaboration.