Background. The derivation of physical dynamics from algebraic first principles remains a central open problem in foundational physics. This paper addresses the logical gap between Clifford-algebraic ontology and spacetime dynamics by constructing a unified framework that bridges noncommutative algebra, discrete spacetime structure, and the standard model.
Methods. Starting from the non-self-consistency of the commutative scalar complex field, we construct the \(\mathrm{Cl}(2,1)\subset\mathrm{Cl}(1,3)\) subalgebra and establish grade classification, dimensional upgrade (\(\hat{j}=\hbar j\)), and the \(3+1\to1+1\) dimensional reduction projection. The framework integrates Connes' spectral triple \((A,H,D)\), causal-set discreteness, the spectral action principle, and Wick rotation.
Results. The compact resolvent of D yields a discrete spectrum and UV cutoff, complementing the causal-set ontological discreteness. The core equation \(dx(v)=\frac{c^2}{v}(1-\frac{1}{\gamma})t_P\) is derived from causal-set counting under coarse-graining and shown to be unique within the physically motivated power-law family. The spectral action \(S=\mathrm{Tr}(f(D/\Lambda))\) automatically recovers the Einstein--Hilbert, Yang--Mills, and Higgs actions via heat-kernel expansion, with the gauge group \(U(1)\times SU(2)\times SU(3)\) derived from the noncommutativity of the internal algebra. Lorentz symmetry emerges as a low-energy asymptotic symmetry with deviations of order \((t_P/\epsilon)^2\), far below experimental constraints.
Conclusions. The framework provides a coherent algebraic-ontological foundation for spacetime dynamics. Seven logical gaps are assessed (G6 and G4 fully closed; G1, G2, G3, G5, and G2d basically closed with explicitly stated residual open directions), and the Dirac-sea positive-energy projection and CP-violation analysis establish the physical interpretability of the corrected basis.