Multiplication-Free Matrix Determinants: Bitplane Semantics and Structured Overlays

We present a structured framework for multiplication-light and multiplication-free determinant computation. Classical algorithms—Gaussian/LU elimination, Bareiss fraction-free elimination, Schur complements, and modular (CRT) methods—are dominated by GEMM-shaped trailing updates [1–3]. We distinguish two execution models: (i) scalar arithmetic, where we count and suppress multiplications tile by tile using overlays (peel, rank-1 updates, annihilations); and (ii) bit-sliced Boolean GEMM, where every integer is decomposed into binary planes and updates are carried out entirely by bitwise AND/XOR, population count, shifts, and additions. In the scalar model, we reduce multiplication counts significantly (e.g. 14 → 5 multiplies in a worked 4 × 4 example). In the bit-sliced model, the scalar multiplication count vanishes completely for arbitrary integer ranges. Both approaches preserve exactness and pivoting semantics, and extend across LU, Bareiss, Schur, and CRT pipelines.