M20b Operational Logic: The Manifold as Ground of All Reasoning: Logic Hierarchies, Quantum Structure, Neural Computation, Cryptography, and the Self-Computing Universe

This monograph develops Operational Logic, a unified theory of propositional reasoning in which each rank of the Garycki Manifold defines a complete logical system. Classical Boolean logic, many‑valued logics, and quantum logic are recovered as specific rank restrictions, while higher and complex ranks yield genuinely new logical structures not present in the classical literature. At the foundational level, the work shows that Boolean logic arises from the restriction of the rank‑1 operation (multiplication) and the rank‑ (−1) operation (tropical max) to the binary set 0, 1. Extending the truth domain to 0, 1, the three fundamental fuzzy logics — Gödel, Łukasiewicz, and Product — are recovered precisely as the rank RC=−1, 0, 1 sector, reproducing Hájek’s classification without additional axioms. The Manifold is thus identified as the unique algebraic ground of all continuous propositional logics. Beyond the classical and fuzzy regime, the monograph derives quantum logic as a structural consequence of the non‑commutativity of the RC=i operation. The Birkhoff–von Neumann quantum logic appears naturally, with the classical Boolean algebra recovered in the commutative limit. Higher real ranks (RC≥2) generate Power Logic and Tetration (Toweration) Logic, extending logical operations beyond polynomial expressibility. In parallel, the star‑sector logic, defined over the unit circle, is introduced as a new many‑valued logic with phase‑based truth values and no classical analogue. From this logical foundation, four applied frontiers are developed. (i) Logic theory: a complete hierarchy of operational logics is established, including real, fractional, and complex ranks. (ii) Neural computation: standard activation functions are shown to be rank‑specific projections of the Manifold; Softmax arises as the gradient of the Gödel OR, ReLU as a tropical restriction, and new star‑rank activations are proposed for phase‑based and complex‑valued networks. (iii) Cryptographic logic: several hardness problems are formulated at the level of logical inversion and rank recovery, grounded in non‑associativity and cloning. (iv) Foundations: logic itself is reinterpreted as an emergent grammar of operations rather than an axiomatic primitive. The monograph does not propose new axioms of logic; instead, it derives logical systems as operational shadows of a single algebraic structure, unifying Boolean, fuzzy, quantum, and higher‑rank reasoning within one continuous framework.