The Minimalist Cosmological Equation: From the Q′-Reduced Dimensional Structure to Cosmological Distance Relations and Type Ia Supernova Tests

This work develops a minimalist cosmological framework originating from the model first introduced by Nianjiu Qiu in Physics Teaching (China), No. 7 (2001), pp. 36–37. The present study extends that foundation through a Q′-reduced dimensional formulation, cosmological distance relations, and Type Ia supernova tests. Starting from Q = 1/(4πG), M = QV/t², and ρ = Q/t², the model gives r ∝ t^(2/3), H(t) = 2/(3t), H₀t₀ = 2/3, and q = 1/2 under mass conservation. The corresponding redshift–time and distance relations are derived, including t(z) = t₀/(1+z)^(3/2) and, for the spatially flat case, d_L(z) = 3ct₀[(1+z) − √(1+z)]. Type Ia supernova tests are performed using Pantheon+, DES-Dovekie, and Union3 with their supplied covariance or precision information. An empirical time-domain correction, τ = α[1 − (1+z)^(-3/2)]^p, is examined. A joint fit gives α = 0.430742 and p = 1.259617 and substantially reduces residual structure relative to the uncorrected baseline. Information-criterion comparisons favor the common two-parameter correction over separate dataset-specific corrections. The correction is treated as an empirical diagnostic rather than evidence for a uniquely identified physical mechanism. The paper distinguishes dimensional structure from dynamical assumptions, conditional absolute calibration from supernova-only constraints, and proper particle-horizon distance from its comoving coordinate. Possible physical interpretations, observational systematics, and requirements for future out-of-sample predictive tests are discussed. The present 2026 work distinguishes the original 2001 foundational model from the new derivations, dimensional analysis, cosmological distance formulation, and supernova data analysis reported here.