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 computeU (theta): e1 = exp (1j * theta) e2 = exp (-2j * theta) U = matrix ([e1, 0, 0, 0, e1, 0, 0, 0, e2]) return Udef consensusᵥiolation (theta): U = computeU (theta) I = matrix ([1, 0, 0, 0, 1, 0, 0, 0, 1]) diff = U - I J = sum (abs (diffi, j) **2 for i in range (3) for j in range (3) ) return J**0. 5theta₀ = findroot (lambda t: consensusᵥiolation (t), mpf ('6') ) piₑxtracted = theta₀ / 2print (f"pi (extracted) = piₑxtracted") print (f"pi (mpmath) = mp. pi") print (f"difference = abs (piₑxtracted - mp. pi) ") print (f"J (theta₀) = consensusᵥiolation (theta₀) ")