Physics and Logic: Generative Foundations of Dynamical Isomorphism

Abstract Physics and logic are two distinctly separate domains in the classical framework, but they share an unquestioned root: are the evolution rules of physics and the inference rules of logic emergent signatures of the same dynamical mechanism in different semantic spaces? Within the framework of generative mathematics, this paper reveals that the structure operator Û and the inference rule operator share the same dynamical mechanism of the evolution equation of Axiom 4—dynamical isomorphism. This is not a pre-existing algebraic correspondence, but a dual instantiation of the same mother equation in physical state space and logical proposition space. There exists a fourfold dynamical parallel between the algebraic characteristics of the structure operator Û (scalar direction, phase rotation, mixed direction, multi-dimensional flow) and the dynamical characteristics of the inference rule operator (deterministic pulling, statistical superposition, reverse search, rule search). This paper further diagnoses the root of why physics and logic have not converged for two hundred years—the two kingdoms share the same epistemological blind spot: the beauty contest paradigm (presupposing that the system "knows" the whole and selects the optimal path or the truest conclusion from it). The walking paradigm of generativism subverts this presupposition: the system does not know where the endpoint is; it only knows the slope of the ground beneath its feet at this very moment, and walks downhill. Dynamical isomorphism can only be revealed from the perspective of the walking paradigm. On this basis, this paper establishes the ontology of physical computation: computation is not symbol replacement on a Turing machine, but the convergence process of the coupling dynamics of Axiom 4 in state space. Verification and solution share the same topological charge locking mechanism—not two operations, but different observational stages of the same process. Physical computation is not "simulating logic with physics," but logic itself is the signature of physical evolution. Classical computers are the degenerate limit of physical computers under specific parameters—when evolution is frozen into execution, dynamics degenerates into a program. As an application of the theory, P vs NP is repositioned as a "physical realizability" problem, and the design principles of physical AI—inference capability is not a rule externally programmed and implanted, but a physical property that emerges under specific configurations of the structure operator—provide a dynamical legitimacy foundation for the L4 Holomorphic Architecture. Keywords: dynamical isomorphism; structure operator; inference rule operator; Axiom 4; walking paradigm; physical computation; physical AI; P vs NP; paradigm shift; generative unification