Dimensional Flow Cosmology: The Linear Energy Ledger, the Work Principle, and the Complete Cosmic Cycle

This paper presents the ΨD framework: a dynamical dimensional parameter d ∈ (0,3) drives the transformation matter → energy → string → Planck while the topological three-dimensionality of space is preserved—what flows is not the metric but the effective organizational dimension of physical form. The framework rests on a minimum-resource principle: each effective dimension is realized by the most economical Euclidean object, the d-simplex of d + 1 nodes (the tetrahedron’s 4 vertices against the cube’s 8), not the hypercube capacity 2ᵈ. This fixes a linear energy ledger E = (d + 1)ρ★, U = (3 − d)ρ★, conserved as E + U = 4ρ★, with a constant work quantum dW = ρ★ dd bridging the fractional gap between integer dimensions. Energy is thus the count of minimal Planck nodes; it gravitates and is falsifiable. The single amplitude ρ★ sets only the scale (ρ★ = ρΛ,0/4, the status of Λ in ΛCDM), leaving every dimensionless prediction invariant. A geometric kinetic function makes the flow logistic with an exponential tail of slope λ = 1, yielding a complete cosmic cycle: a non-singular beginning, inflation with no separate inflaton (nₛ ≈ 0.965, r ≈ 0.0025), a thawing dark-energy attractor w₀ = −0.843, wₐ ≈ −0.22 with a structural phantom ban w ≥ −1 and era attractor w∞ = −2/3, then turnaround, a collapse ladder, and a new Big Bang at d → 0. Dark matter is Planck-phase regions trapped at d̃ = 0, giving ΩDM/Ωb = 5 (observed 5.36, within 7%). The model fits DESI DR2 BAO at least as well as ΛCDM and is sharply testable with Euclid and LSST/Rubin.