Continuity and Infinity as Descriptive Fictions of Intelligence: Algebraic Dissolution of Analytics from M3(C) Structure

This paper establishes that continuity, infinity, and the analytic machinery built upon them are not structural necessities of intelligence but descriptive fictions: Tier-3 representational constructs that compress the finite operational structure of intelligence into an extended formal language. The argument proceeds from the axiomatic foundation of Cognitional Mechanics (CM), in which the unique minimal operational structure C_min ≅ M₃(ℂ) is derived from three axioms governing non-commutativity, Π_d-saturation with finite generator rank, and redundancy exclusion. Operational derivability is defined as closure under finite-terminating operation sequences achieving a finite operational determination of all Π_d-distinguishable structure. Under this definition, the limit operation is shown to be operationally underivable: it requires certification over the index family {ε > 0}, which no finite subset exhausts modulo Π_d. Continuity in all standard formulations, topology, differentiability, Lebesgue measure theory, and manifold theory inherit this underivability 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 Tier-3 descriptive artifacts; and the purification of physical constant derivation from M₃(ℂ) structure. The present paper completes a classification trilogy alongside the co-derivation of number systems and the top-down projection theorem for mathematics.