M-Boolean Algebra and the M-Map System: A Coincidence-Primary Reformulation of Karnaugh Map Minimisation

We present M-Boolean Algebra, a self-contained algebraic Boolean system organised around three primary operators: the complement, the product, and the coincidence operator (equivalence, the dual of exclusive-or). We use it to define the M-Map, a Boolean map whose cells are numbered in natural binary order (0, 1, 2, ..., 2^n - 1) rather than in Gray code, and we show that this layout supports a complete minimisation procedure, including diagonal and non-adjacent cell groupings, when the coincidence operator is used as the primary combination operator. We further define a canonical enumeration of all 2^(2^n) Boolean functions of n variables (the M-List), a binary-coded identification scheme for each function (the M-Word), and a set of construction tables (M-Tables) that produce every function in the universal set from a small set of primary functions. The framework was originally developed and hand-typed in 1990; the seven-variable extension and a textbook-comparison worked example were prepared in 1991. A modern restatement, using the standard operator set {NOT, AND, OR, XOR}, was submitted as a United States provisional patent application on 21 January 2026 and is included here as a confirmation of equivalence. We argue that the two formulations, the 1990 coincidence-primary form and the 2026 classical form, are functionally equivalent restatements of the same underlying system, and we treat the 1990 form as primary on grounds of priority. We close with an account of the scaling wall (the doubly-exponential growth of 2^(2^n)), a connection of the recursive M-Map structure to Shannon expansion, and a separate connection of the author's spatial binary-tree construction to what is now called the Morton (Z-order) space-filling curve.