Relativity within the Framework of Spacetime Quantization: Derivation Chain and Theoretical Construction

Based on the first principles of ontology, this paper traces the core formulas of relativity back to the most primitive logical starting point and unfolds them step by step. The logical starting point of this paper is moved back from the postulates to special relativity: the Lorentz transformation and the Lorentz factor \(γ\) are first derived from Einstein's two basic assumptions, and a review of the Clifford algebra basis then shows that spacetime dynamics must be built on a noncommutative matrix algebra. A dimensional-upgrade interface and a dimensional-reduction projection are introduced next, projecting the complete 3+1-dimensional Clifford algebra onto 1+1 dimensions. On this basis, this paper introduces the time quantization assumption \(dt=dt'=t_P\) (explicitly flagged as an additional physical input, not derived from first principles) and inverts the differential form of the Lorentz transformation to obtain the core equation \(dx(v)=[1-√{1-v²/c²}]/v·√ G/c\), which is then rewritten by a formal transformation into \(dx(Γ)=l_P√(Γ-1)/(Γ+1)\), the form that connects with the system of postulates. Thereafter, starting from postulates A1--A9 and using the state count \(Ω∝(Γ²-1)3/2\), the entropy shape function \(S(Γ)\), the change of variable \(u≡Γ-1\), the redshift conversion, the effective entropy production rate, and the production--dilution equation E1, this paper rigorously solves, by the integrating-factor method, for the closed-form dark energy density \(ρDE(u)=N(u)/u³\), where \(N(u)=3/2u²-6u+12ln(u+2)+C\); derives, under the standard conservation law, the equation of state \(w(u)=-u³/[(u+2)N(u)]\); inversely calibrates, from \(w(uₘᵢₙ)=-1.65\), the integration constant \(C=-8.317960\); and further obtains the potential \(V(u)=[(u+2)N(u)+u³]/[2u³(u+2)]\) from the standard relations for a scalar field. All conclusions of this paper are confined to what has been proved, with no extrapolation beyond the evidence.