Continuity and Infinity as Representational Artifacts of Operational Structure: A Structural Theorem from Operatiology and Noology

This paper establishes that continuity, infinity, and the analytic machinery built upon them are not structural necessities of any operational system but representational artifacts: formal constructs that encode the finite operational structure of the rank-3 minimal operational closure C⁽³⁾_Πd into an extended descriptive language. The argument proceeds from the axiomatic foundation of Operatiology, in which C⁽³⁾_Πd is derived from three axioms governing non-commutativity, Πd-saturation with finite generator rank, and redundancy exclusion. Operational necessity is defined as closure under finite-terminating operation sequences achieving a finite operational determination of all Πd-distinguishable structure; the criterion is not formal definability but operational exhaustibility. Under this definition, the limit operation lies outside operational necessity: it requires certification over the index family {ε > 0}, which no finite subset exhausts modulo Πd. Continuity in all standard formulations, topology, differentiability, Lebesgue measure theory, and manifold theory inherit this classification by a single uniform mechanism formalised as the Unbounded Index Obstruction lemma. Infinity is formally classified as the termination-failure equivalence class [ℐ/∼]: the structural label unifying the limit operation, actual infinity, arbitrary unions, σ-additivity, non-countable cardinality, and Dedekind completeness under one mechanism. Four consequences follow: the dissolution of the Platonist–formalist dispute; the reinterpretation of the unreasonable effectiveness of mathematics as compression efficiency; the classification of renormalization divergences as representational artifacts; and the purification of physical constant derivation from M₃(ℂ) structure. This paper supersedes the Cognitional Mechanics formulation (DOI: 10.5281/zenodo.20282144) and forms part of the Operatiology mathematics series alongside the co-derivation of number systems and the top-down projection theorem for mathematics.