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

Operatiology, derived from Noology through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽…Operatiology, derived from Noology through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽³⁾_Πd as the unique structure satisfying the executive axiom system {A1, A2, A4} together with the Operational-Geometric Coupling. The companion paper on the top-down projection of mathematics establishes that the induced mathematical category M is uniquely determined and that algebra occupies the minimum-distance layer M₁ within M, delegating the complete algebraic identification to the present work. This paper discharges that delegation. The argument proceeds in four stages. First, algebra is established as the M₁-layer: the unique minimum-distance projective structure from the operational substrate. Second, operational non-decomposability is derived from Axioms 1, 2, and 4: direct sum decomposition of the state space is excluded by Axiom 1, and non-trivial two-sided ideals are excluded via the maximality of the canonical irreducible generator set I_Πd, establishing simplicity. Third, finite dimensionality is derived from the finite generator rank and the Πd-saturation closure via an explicit saturation depth bound, and the Wedderburn-Artin theorem is applied to identify the representation as Mₙ(D). Fourth, a bridge lemma connecting Πd-separation to eigenvalue structure establishes that the Noological primitive Arbitrium forces D = ℂ, and the three-generator Πd-independence condition forces n = 3, yielding M₃(ℂ) as the unique minimal algebraic realisation of C⁽³⁾_Πd. Published on May 24, 2026 DOI: 10.5281/zenodo.20363937