Algebraic Derivation of the Proton-to-Electron Mass Ratio from M3(C) Structure

The proton-to-electron mass ratio μ = 1836.152 673 426(32) is one of the most precisely measured dimensionless constants in physics, yet the Standard Model offers no closed-form derivation of its value. Lattice QCD provides numerical estimates but requires non-perturbative inputs and external renormalization, leaving the algebraic origin of μ unexplained. Cognitional Mechanics (CM) is a framework that derives physical constants as structural projections of the unique minimal non-commutative algebra M₃(ℂ), selected by three axioms with no free parameters. The fine-structure constant α⁻¹ was previously derived with residual 6.82×10⁻¹⁶ against the Morel 2020 rubidium-recoil measurement. The present paper extends this framework to μ. The natural derivation variable is (μα)², not μ alone. The ratio μ = mₚ/mₑ presupposes distinct massive particles; α = e²/ℏc presupposes a charged particle. Both are Tier-3 quantities in the CM framework, where particles are spectral projections rather than primary entities. Their product (μα)² cancels all particle-ontological prerequisites, yielding a Tier-1 quantity derivable directly from the M₃(ℂ) axioms. The derivation produces a six-term closed-form expansion of (μα)² with all coefficients uniquely fixed by the spectral structure of the Cartan element H₂ = diag(1,1,−2) and two cyclotomic identities. The residual against CODATA 2022 is +1.73×10⁻¹³, a factor of approximately 100 below the current measurement uncertainty, with zero free parameters. The derivation further identifies ξ ≡ μα as the spectral coupling constant of M₃(ℂ).