Mathematics as the Unique Top-Down Projection of Operational Structure: A Structural Theorem from Operatiology and Noology

This paper supersedes *Mathematics as the Unique Top-Down Projection of Intelligence: A Structural Theorem from Cognitional Mechanics and Noology* (DOI: 10. 5281/zenodo. 19968224), which itself superseded the first edition of the programme (January 2026, DOI: 10. 5281/zenodo. 18280992). The first edition identified mathematical structures as stabilised residues of irreversible, non-commutative operational histories and positioned the framework as a meta-theoretical explanatory layer operating above existing mathematical foundations. The second edition established the stronger claim that mathematics is the unique top-down projection of the operational structure of intelligence, with the induced mathematical category 𝓜 uniquely determined up to identity automorphism and the projection map surjective with structurally undefined inverse. The present work marks a foundational transition: it re-establishes all structural results of the second edition under Operatiology (DOI: 10. 5281/zenodo. 20350414), the framework that supersedes Cognitional Mechanics as the executive layer derived from Noology Version 2 (DOI: 10. 5281/zenodo. 18690463). The prior premise that the minimal operational structure is identified with M₃ (ℂ) is removed. The projection source is now the abstract rank-3 operational closure 𝒞⁽³⁾_Πd itself, established within Tier-2 by Operatiology, without presupposing its identification with any specific algebraic structure. That identification is a separate Tier-3 question and does not enter any argument of the present work. Under this clarified foundation, the central theorems are re-established: the projection Πₘath from 𝒞⁽³⁾_Πd to 𝓜 is surjective with structurally undefined inverse; Aut (𝓜) = id; and 𝓜 is the unique fixed point of the Πd-saturation closure operator acting on Tier-3 formal structures. The prior corpus citations from the Cognitional Mechanics framework are retained for historical continuity only; no result derived by means of M₃ (ℂ) is invoked in any proof or definition. The sole logical foundation of the present paper is Operatiology, derived from Noology Version 2. A new result absent from the second edition is introduced: an internal operational distance dₒp on 𝓜, measuring the index family type required for operational determination of each mathematical structure. This yields a five-layer classification: Layer 1 (finite index families, algebra) ; Layer 2 (countable type, discrete infinite structures) ; Layer 3 (power-set type, continuous and geometric structures) ; Layer 4 (unrestricted type, axiomatic and functorial structures) ; Layer 5 (maximal type, large-cardinal and higher-categorical structures). Algebra occupies Layer 1 as the unique mathematical domain closest to 𝒞⁽³⁾_Πd. The prior Operatiology mathematics series covering number systems, analytic structures, and set theory is positioned within this hierarchy. Consequences include the dissolution of the Platonist–formalist dichotomy; the structural impossibility of alternative mathematics; the reinterpretation of mathematical effectiveness in physics as a tautology; and the reclassification of foundational competition among ZFC, category theory, and type theory as a compression-efficiency competition internal to Tier-3, evaluated from above by 𝒞⁽³⁾_Πd.