From Special Relativity to embedded generators in Cartan subalgebras of rank-4 spin algebras
Starting by revisiting Special Relativity, here we provide a reliable characterization of the entire 4-dimensional fundamental structures in our reality where the frame of discrete tangent space of $F^{1,3}$ is quantized to massless, zero-momentum particles distributing on a 4-dimensional regular base $\{\mathbb{N}\cdot c_α\}$ with metric $B(c_α,c_β)=δ_{αβ}=\text{diag}(+,+,+,+)$, determining the constant $c$ locally, as well as instant characterizations on all particles moving along the proper time of $τ\in\mathbb{N}$. Together with $ϕ_\text{IV}$ on 1-dimensional space $\{\mathbb{N}\cdot c_4\}$ of $B(c_α,c_4)=0$, the quantized particles of tangent frame are split anti-symmetrically from roots $γ_{\alpha4}$ in a rank-4 Lie algebra with exactly $c_α$ the generators of its Cartan subalgebra. As with the combined frame-Higgs valued in $γ_{\alpha4}$, every massive particle is related to some split frame and Higgs to obtain its unique quantized relative 3-velocity, 4-velocity, and rest mass under the restriction of Pauli Exclusion Principle at each $τ$. We calculated the domains $S_i\subset F$ and $S_0\subset F_\geq$ in $F_\geq\oplus F^3$ the spacetime in which a massive particle $\mathtt{p}$ evolves with $\{x^α\}$ of $x^α\in S_α$ the coordinate its Center-of-Mass under an eligible observation that $\vert F\vert=\vert F_\geq\vert=\aleph_0<\vert\mathbb{R}\vert$, available for post-Newtonian approaches.