Variational Backbone and Regime Closures IV: Observational Projection and Cosmology Model Cards (Fiber-Volume Entropy, Two-Time Structure, Dark-Sector Interpretation)

Part IV is an implementation layer for the fixed-density single-functional frameworkof Parts I–III. It separates “projection” into (i) a reading selection (R1/R2/R3) for thesame energy E and (ii) an observational channel Πδobs : X → Yδ(typically irreversible).The channel induces an observational equivalence ∼Π, and operational physical space isXδphys := X/∼Π.Irreversibility is quantified by the fiber-weight entropy Sobs(y) = kB log µ((Πδobs)−1(y)),stated in renormalized form when needed. The “almost one” claim refers only to principalpart rigidity under the ABC contract: up to declared equivalences, the derivative-quadraticbackbone is fixed, and with a fixed reading protocol the equation skeleton is fixed as well.IAS compares observer cards by induced distinguishability; operational gauge is fiberpreserving symmetry of the declared channel. A minimal cosmology rail is given by anFRW summary card and a single-mode relaxation variable ηcos(t), with a normalized bridgeρdark(a) = ρdark,0 (ηcos(a0)/ηcos(a))p(default p = 1). A short worked illustration (late-timecorridor template and a schematic w(z) plot) is included to make the validation rail concrete(Section 5.2.1)