Continuous Universal Analog Logic (LCUA): Smooth Functional Equivalents of Digital Gates Without Comparators

We present a framework to implement Boolean logic using only smooth functions (sums, products, and low-cost nonlinearities) without hard thresholds or explicit comparators, while exactly preserving truth tables on the Boolean vertices {0,1}. The core is a family of polynomial gate primitives (AND, OR, XOR, NAND, etc.) that are exact at the vertices and are interleaved with a smooth booleanizer Bn(x) = xn / (xn + (1−x)n) that stabilizes trajectories toward the attractors 0 and 1 without discretization. We prove (i) exactness on the Boolean vertices; (ii) existence of attractors with local contraction for Bn; (iii) a block-contraction condition that yields stability in deep cascades; and (iv) a phase/harmonic variant via the encoding s = 2b − 1∈ {−1, +1} (equivalently s = cos(π(1 − b))). We synthesize a continuous half-adder and a continuous SR latch (smooth ODEs), and we outline an analog-hardware route using standard op-amp adders/scalers, analog multipliers, and exponential (log-antilog) cells. We discuss sensitivity, cost, speed, and robustness under variation, and we identify research directions linking continuous logic to neuromorphic and AI-centric analog-digital co-design.