This paper presents the Consensus-Pi Engine, a numerical method that extracts the fundamental constant π by identifying the Consensus Recovery Point of M₃(ℂ) algebraic operations. Unlike traditional series-based approximations (Tier-3), this approach operates on the SU(3) manifold using a minimal symmetric generator H = diag(1,1,-2), enforcing unitary constraints at every computational step through Consensus Enforcement routines: SVD-based unitary projection, determinant correction, and traceless projection. The engine demonstrates that algebraic structure from M₃(ℂ) geometry acts as Logical Back-pressure, suppressing numerical drift beyond floating-point precision limits. Results show 100-digit precision with final consensus violation J ≈ 10⁻²⁰², outperforming classical Machin-like series in stability. This work provides empirical validation of Tier-2 operational structure in Cognitional Mechanics, establishing that π emerges as a finite algebraic invariant whose infinite decimal expansion is a representational artifact. The method bridges theoretical algebraic constraints with practical numerical implementation, with implications for computational mathematics, fundamental constant derivation, and quantum simulation. Aligned with Noology and IASER frameworks.
T.O. (Thu,) studied this question.