π as the Finite Number: Algebraic Derivation of the Circumference Ratio Version 2.0

This paper is Version 2. 0 of "π as the Finite Number: Algebraic Derivation of the Circumference Ratio" (V1: DOI: 10. 5281/zenodo. 18782315). The structural content—definitions, theorems, and proofs—is unchanged. Version 2. 0 introduces three revisions. First, and primarily, the transcendence of π established by Lindemann (1882) is explicitly addressed. Within the CM Tier framework, transcendence is a Tier-3 classification: it characterizes π as an element of ℝ relative to ℚ-polynomial equations. The Tier-1 specification of π—eigenvalue pattern (1, 1, −2), closure condition U (Ω) =I, division by 2—is a group-theoretic condition, not a ℚ-polynomial equation. That π is transcendental at Tier-3 and structurally finite at Tier-1 are not contradictory; they describe different layers. The same principle applies to the transcendence of e established by Hermite (1873). Second, the separation between definitional structure and computational evaluation is formalized. The proof that Ω=2πclassical employs πclassical as a Tier-3 computational tool, not as a definitional assumption. This separation is now made explicit in a dedicated Remark with a reference table distinguishing the definition level (no π), the computation level (πclassical as tool), and the resulting characterization π: =Ω/2. Third, the companion numerical implementation—Consensus-Pi Engine (DOI: 10. 5281/zenodo. 18726805) —is explicitly cited as independent validation, extracting π as the Consensus Recovery Point of U (θ) =I with 100-digit precision and final consensus violation J≈10⁻²⁰². The V1 description below remains accurate for the structural content of this paper. --- This paper establishes that the fundamental constant π, traditionally viewed as an infinite transcendental number, is a structurally finite algebraic invariant emerging from the minimal operational algebra M₃ (ℂ). Within the framework of Cognitional Mechanics (CM), we resolve the Physical Reference Paradox—how finite physical systems can reference constants with infinite decimal expansions—by distinguishing between three hierarchical levels of reality: Tier-1 (Structural): π is defined by the finite eigenvalue pattern (1, 1, −2), the unique minimal traceless integer partition allowed by M₃ (ℂ) necessity. Tier-2 (Executive): π emerges as the half-period of the minimal closure parameter Ω within the compact group SU (3). Tier-3 (Projective): The infinite decimal expansion (3. 14159. . . ) is identified merely as a representational artifact of 10-base projection onto the real number line. We prove that n=3 is the minimal and unique dimension required for the unambiguous algebraic specification of π. By deriving π from group-theoretic compactness rather than geometric measurement or infinite series, this work demonstrates that classical π is a computational result of underlying algebraic structure, not a definitional assumption. This shift moves π from the realm of discovered empirical values to necessary structural invariants, providing a rigorous foundation for a finite-resource physics where the universe knows π through its algebraic embodiment rather than infinite calculation.