The Spectral Coupling Constant ξ = µα as Proof of Common Origin of Electromagnetic and Mass-Hierarchy Invariants from M3(C) Structure: A Structural Theorem from Operatiology and Noology

This paper establishes the structural foundation of the spectral coupling constant ξ ≡ μα, the product of the proton-to-electron mass ratio μ and the fine-structure constant α, within the Operatiology and Cognitional Mechanics framework. A two-stage proof is presented.

The first stage demonstrates that μ and α share a common algebraic origin in the matrix algebra M₃(ℂ): the product ξ = μα forces the cancellation of the common Casimir invariant Tr(H²) = 6 appearing in the leading terms of both independent derivations, making algebraically manifest the shared Cartan generator H = diag(1,1,−2) underlying both constants. This addresses a question that the Standard Model of particle physics has no framework to pose.

The second stage identifies the common origin algebraically: Axiom A1 of Operatiology forces the su(3)-type minimal non-commutative closure, whose tracelessness implies p₁ = Tr(Hλ) = 0. This constraint forces the invariant algebra to satisfy Inv(𝒪) ≃ ℝ[λ²], making λ² the unique minimal coordinate of the algebra's self-description. Since ξ ∝ λ, ξ lies outside the invariant algebra; since ξ² ∝ λ², ξ² is proportional to its minimal generator. The asymmetry between ξ and ξ² is therefore a necessary consequence of A1, not an external imposition.

Conditional on prior spectral completeness results, ξ² functions as the generator of all physical constants within the framework. The paper is the first of a two-part work; the companion paper derives ξ² independently with zero free parameters.