The Operatorization of Classical Variables: Hyperkähler Gauge Rigidity and Clifford Commutation Laws

Abstract Classical complex analysis and analytical numbertheoryhave historically treatedthecomplex variable s = σ + it as a static, localized point within a flat, two-dimensional coordinatedomain. This geometric reductionism inevitably introduces artificial mathematical singularities and divergent non-physical infinities when computing highly non-linear operator kernels, such as the Riemann ζ function. Building upon the axiomatic foundation of ReflexiveSet Theory (RST) established in Paper I, this paper introduces the strict mathematical formalism for the Operatorization of Classical Variables. We lift the scalar variable s into anon-commutative, four-dimensional holistic operator acting on a 4D Hyperkähler manifold.ByemployingthestructuralholonomyofthecompactsymplecticgroupSp(1)andtheCliffordalgebra C 2,0, we prove that the global stability and uniform convergence of analytic continuation are governed entirely by Gauge Rigidity. This mathematical framework establishesa homogeneous co-moving commutation law [T,n] = 0, providing the essential dynamicalstabilization required for resolving global spectral conjectures