Algebraic Reduction in Vision Operators via Canonical Re-parameterization and Quotient Structure

We present a framework for algebraic reduction in computer vision operators based on canonical reparameterization, kernel quotienting, and representation-theoretic factorization, governed within a Functor Model Architecture. We show that Cantor-style bijections from N^k -> N provide insertion-stable canonical indexing for sparse and dynamically expanding domains, enabling uniform operator formulation without reindexing. While such mappings do not reduce computational complexity directly, they expose latent structure that admits reduction. Specifically, structured operators arising in vision, including Toeplitz, Kronecker, and circulant forms, enable factorization via Fourier diagonalization and permutation chains, reducing dense computation to O(n log(n)) or better. Further, when nontrivial kernels are present, computation can be restricted to quotient spaces V/ker(A), yielding true parametric reduction. Representation-theoretic decomposition provides additional block-diagonal structure, reducing operator complexity over irreducible components. The Functor Model Architecture supplies the categorical framework required to validate and safely replace canonical specifications with reduced implementations. The result is a principled shift from parameter optimization to algebraic reduction, enabling efficient, auditable, and structure-aware computation in vision systems.