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

In three-dimensional space the number of dimensions does not change. This article presents the ΨDframework, which describes the transformations of matter into energy, energy into strings, strings intoPlanck particles, and their reverses, in three-dimensional space by means of a single scalar field 𝑑 ∈ (0,3).The energy law is derived from the number of vertices of the 𝑑-simplex: 𝐸 = (𝑑 + 1)Λ4. With theaccompanying potential 𝑈 = (3−𝑑)Λ4,theledgerprinciple𝐸+𝑈 = 4Λ4 isconserved,andthedynamicsreduce to a single workdifferential: d𝑊 = Λ4 d𝑑. Thegeometrickineticfunction𝐾 = 9𝑀2P/[𝑑2(3−𝑑)2]reduces the flow to a logistic curve and the potential to an exponential tail of slope 𝜆 = 1. The frameworkcarries a complete cosmic cycle: a non-singular beginning; inflation requiring no separate inflaton —with amplitude-independent shape predictions 𝑛𝑠 = (1 − 1/𝑁)2 ≈ 0.965, 𝑟 = 8/𝑁2 ≈ 0.0025, 𝛼𝑠 ≈−2/𝑁2; a beginning-independent dark-energy attractor 𝑤0 = −0.843, 𝑤𝑎 ≈ −0.22 (𝑤0 consistent withDESI DR2; ∼2𝜎 tension in 𝑤𝑎) and a phantom ban 𝑤 ≥ −1; the end of acceleration, a gravitationalturnaround, a collapse ladder with 𝑘Λ4 thresholds, and a new Big Bang at 𝑑 → 0. Dark matter consists ofconfined Planck-phase regions that cannot cross the unfolding threshold; under stepwise confinement, anequal split gives the parameter-free upper value ΩDM/Ω𝑏 = 7 (the observed 5.36 ⇒ 𝑝 ≈ 0.46 per level).Energy conservation is proved at the algebraic, covariant, and cyclic levels; the net work per cycle is zero:∮ d𝑊 =0.