Entropy Shape Function and Closed-Form Equation of State: A Comparative Test against DESI Survey Data

Based on the first principles of ontology, and starting from the spatial quantization postulate, this paper constructs a closed-form derivation chain for dynamical dark energy. The postulate specifies only that the minimum resolvable length dx=lP√(Γ-1)/(Γ+1) varies with the scalar field strength Γ, from which there follow, in turn: the decrease of the field strength φ=1/dx with expansion (space softens as it expands), the state count Ω∝(Γ²-1)3/2, the entropy shape function S(Γ)=32ln[(Γ²-1)/(Γₘᵢₙ²-1)], and the evolution equation for the dark energy density determined by the competition between entropy production and dilution by expansion. That equation is a first-order linear ordinary differential equation, and the integrating-factor method yields the exact closed-form solution ρDE= N(u)/u³, where N(u)=32u²-6u+12ln(u+2)+C, u=Γ-1, and the integration constant $C=-8.317960$ is uniquely calibrated by the present-day value of the equation of state. Substituting into the conservation law gives the equation of state w(u)=-u³/[(u+2) N(u)], which is a strictly increasing function: at early times $w<-1$ (phantom phase); at u^*=0.2473196823 it crosses $-1$, after which $w>-1$ (quintessence phase) and approaches $-2/3$. The crossing redshift is z^*=40.0642610636, a definite prediction free of free parameters; the peak of the dark energy density coincides with the crossing point, with ρDE/ρDE₀=4.038. The present-day theoretical value is w₀=-0.7465, in agreement with the DESI DR2 value -0.762±0.060; however, the crossing redshift differs from the DESI median $z=0.405$ by a factor of $29.2$, and this discrepancy is uniquely determined by postulates A5 (scale factor--field strength relation) and A8 (origin of the model), with no free parameter available to adjust it. If this evolution is required to come from a scalar field equation, the potential is uniquely determined to be V=[(u+2) N+u³]/[2u³(u+2)], tending to the linear form V≈54(φ lP-1) at the cold-death limit; when the global effective density is inserted into the local scalar field formulas, the kinetic term flips sign at the crossing point, which shows that a completion requires the construction of an averaging transformation between the global and local observers. This theory is a late-Universe theory for z100, and ``Λ is decreasing at the present day'' is the central prediction, one that does not depend on the crossing point and can be tested independently by next-generation surveys.