π as the Finite Number—an Algebraic Necessity of M3(C) Structure: A Structural Theorem from Operatiology and Noology

This paper is Version 3 of the π-as-finite-number programme. Version 1 (DOI: 10.5281/zenodo.18728748) established π as a structurally finite algebraic invariant of M₃(ℂ) within the Cognitional Mechanics framework, resolving the Physical Reference Paradox by distinguishing structural finiteness from projective-layer transcendence. Version 2.0 (DOI: 10.5281/zenodo.19014478) made the Tier-3 status of Lindemann transcendence explicit and introduced the Consensus-Pi Engine (DOI: 10.5281/zenodo.18726805) as independent numerical validation. The present version reframes the entire derivation within Operatiology, the successor framework to Cognitional Mechanics. The central advance is the replacement of the SU(3) half-period argument by a complete G1 volumetric grounding: π is established as an element of the finite set S of Operational Invariants (Japanese: Sousa Hensuu 操作遍数) via its appearance as an independent factor in the volume product Vol(U(1))·Vol(U(2))·Vol(U(3)) embedded in M₃(ℂ), and as A4-irreducible by Lindemann's theorem. The scope is extended from π¹ alone to all six independently grounded powers πᵏ (k=1,…,6); the upper bound k≤6 is an absolute consequence of the rank-3 constraint on M₃(ℂ). The derivation constitutes the complete proof of Lemma 3.8 (π∈S) of the general Operational Invariant solution (DOI: 10.5281/zenodo.20493369), and provides the retroactive structural justification for the use of π in all prior Cognitional Mechanics and Operatiology corpus papers. For those interested, the following code illustrates the algebraic extraction of π at 110-digit precision: starting from the integer initial value 6, it recovers π as θ₀/2 where θ₀ is the minimal positive solution of U(θ)=I, with no value of π appearing as input. The algebraic justification for the absence of circularity is given in Section 5.1 of the paper. from mpmath import mp, mpf, matrix, exp, findrootmp.dps = 110def compute_U(theta): e1 = exp(1j * theta) e2 = exp(-2j * theta) U = matrix([[e1, 0, 0],[0, e1, 0],[0, 0, e2]]) return Udef consensus_violation(theta): U = compute_U(theta) I = matrix([[1, 0, 0],[0, 1, 0],[0, 0, 1]]) diff = U - I J = sum(abs(diff[i, j])**2 for i in range(3) for j in range(3)) return J**0.5theta_0 = findroot(lambda t: consensus_violation(t), mpf('6'))pi_extracted = theta_0 / 2print(f"pi (extracted) = {pi_extracted}")print(f"pi (mpmath) = {mp.pi}")print(f"difference = {abs(pi_extracted - mp.pi)}")print(f"J(theta_0) = {consensus_violation(theta_0)}")