The Bound Theorem in Cognitional Mechanics: Proof via Spectral Flow Geometry

This work establishes the Bound Theorem in Cognitional Mechanics: the operational capacity of any formal reasoning system grounded in the algebra M₃(ℂ) is strictly bounded by C₃ = δ ⋅ K = 3/√2. Using spectral flow geometry, we show that the traceless constraint confines operator eigenvalues to a 2-dimensional plane, where the equilateral triangle configuration uniquely maximizes both discriminability and logical stress. The identification threshold δ = √(3/2) and Casimir scale K = √3—derived in prior CM works on optical theory and friction—are shown to combine multiplicatively as a geometric necessity, not an empirical fit. This result completes the axiomatic foundation of CM capacity theory, demonstrating that M₃(ℂ) is not merely convenient, but logically necessary as the minimal kernel for stable, non-commutative reasoning.