M-Maps: A Binary-Coded Visualization for Multivariable Boolean Functions, with an Algebraic Notation for Compound Atom Recognition

We introduce M-Maps, a binary-coded visualization techniquefor multivariable Boolean functions, together with analgebraic notation for compound atom recognition thatproduces factored exclusive-OR / AND / OR expressionsdirectly without an intermediate sum-of-productsrepresentation. M-Maps differ from the Karnaugh map inthree ways: (i) the layout is binary-coded rather thanGray-coded, allowing direct integer-to-cell addressing andtrivial geometric extension to N > 6 variables; (ii) thevisual representation preserves the diagonals of thefunction, which encode exclusive-OR structure invisible toGray-coded maps; and (iii) an associated atom-recognitionalgorithm identifies compound structural patterns -including AND-pair-XOR-literal atoms, paired complementedAND-pair atoms, and nested atoms - that producedramatically shorter algebraic output for functions withinternal exclusive-OR structure. On a hand-verified testcorpus of all 254 N=3 Boolean functions, an implementationmatches or improves upon hand-optimized factored expressionsin 100% of cases, with zero incorrect outputs. Applicationsinclude cryptographic circuit design, error-correctioncodes, arithmetic logic, and any digital synthesis problemwhere exclusive-OR structure dominates.