Computation as Proof: An Epistemological Revolution in Mathematics under the Hexad Framework

Abstract This paper proposes the "Computation-as-Proof" (CaP) paradigm, upgrading the Hexad framework Hsystem = ⟨M, A0, g, D, I, T⟩ from a computational engine to a proof engine. The core proposition: within the Hexad framework, the convergence of computation to a steady state is equivalent to the completion of an analytic proof—computation that runs through is proof that holds. This paper establishes the "Computation-Proof Isomorphism Theorem" (Theorem 2.1): each dynamic phenomenon of Hexad evolution (convergence, gradient vanishing, phase locking, etc.) corresponds to a logical component of an analytic proof (existence, uniqueness, optimality, etc.). When existing theorem libraries are insufficient, reverse distillation executes axiom tracing—tracing back to specific clauses of the information pixel axioms, generating new theorem proposals through rule composition § 2.2, rather than borrowing external physical analogies. On this basis, the paper establishes a theory of Hexad counterfactual reasoning: changing initial conditions and re-evolving to a new steady state, with different convergence paths corresponding to different counterfactual possible worlds—a natural extension of the isomorphism theorem. The Bootstrap Completeness Theorem (Theorem 2.4) guarantees a deterministic answer for each counterfactual hypothesis. Taking dynamic number theory as an instance, the paper demonstrates the complete reverse distillation pipeline from computational experiments to analytic proofs: primality testing (100% computational accuracy → periodic spectrum singularity theorem), the prime number theorem (computational density convergence → phase closure derivation), perfect number generation (computational structure emergence → isoperimetric optimality theorem), and the Riemann Hypothesis (computational self-dual fixed point → axiom tracing and rule composition proposal). The paper further proves that within the Hexad framework, truth conditions, computational convergence, and analytic proof are equivalent—truth is convergence, and convergence is provability. As a final corollary of the three core theorems, this paper proves the CaP Formalization Theorem (Theorem 6.1): there exists an algorithm A (Hexad evolution) such that encodable propositions necessarily generate analytic proofs, and non-convergent ones are judged undecidable or false. This paper argues for the revolutionary significance of the CaP paradigm over the classical "prove first, compute later" epistemology, and looks forward to its applications in the Millennium Prize Problems, AI for Mathematics, and automated theorem proving. Keywords: computation as proof; Hexad; computation-proof isomorphism; reverse distillation; axiom tracing; rule composition; truth-computation-proof unification; counterfactual reasoning; CaP Formalization Theorem