From Relation to Quantum Dynamics, A Three-Level Construction of Spinorial States, Lorentzian Dynamics, and the Dirac Equation

We present a three-level organization of a bilocal relational theory inwhich structures usually associated with quantum mechanics arise atlogically distinct stages. The first level is purely geometric. Starting from the bilocalconfiguration space \(M× M\) and its canonical cotangent bundle,the relative sector admits a radial--angular decomposition. For \(N=3\),the angular phase space is \(T^*S^2\). Its non-singular sector isnaturally organized as\( R_+× SO(3)\) and therefore possesses the canonical doublecover\[ R_+× SU(2) C^2\{0\}.\] A classical two-component \(SU(2)\)-spinorial carrier thus appearsbefore any Lorentzian dynamics or quantum postulate; its identificationwith a Lorentz-covariant Weyl spinor is not assumed at this stage.Reduction at fixed internalCasimir yields the Hopf fibration with \(|c_1|=1\). For \(N=3\), defining\(p_μ=(p_t, P)\) and\(μᵢₙₜ=κ_1+iκ_2\)rewrites the full constraint as\[p_μ p^μ+|μᵢₙₜ|^2=0.\]Its Clifford factorization then admits an exact chiral first-order formwith a two-dimensional complex kernel. At this stage the constructionremains classical: the momenta are phase-space variables and the kernelspinor is not yet a quantum wave function.The third level promotes this geometric and dynamical structure toquantum spinor dynamics. Geometry supplies an integer topological classbut not a universal scale of action; the identification\(I=n/2\) therefore requires the quantum scale \(\).Together with the operator rule\(p_μ↦-i D_μ\), the classical chiral factorization becomesa quantum Dirac-type equation. If the internal complex contributionadmits a sector or limit in which it reduces to a real constant\(m\), the standard massive Dirac equation\[(iγ^μ D_μ-m)Ψ_D=0\]is recovered. The construction therefore separates three ingredients that arenormally introduced together: spinorial and topological statestructure, Lorentzian dynamics, and quantum operator dynamics. We donot derive quantum mechanics from classical geometry. Rather, we showthat the spinorial carrier and the algebraic architecture of the Diracequation can arise before quantization, while the quantum scale andoperator realization remain independent inputs. This identifiesprecisely which elements of quantum spinor dynamics are geometric,which depend on the Lorentzian dynamical postulate, and which remaingenuinely quantum assumptions.