ΨD — Dimensional Flow Cosmology: From the Planckon Fabric to Particles, the Dark Sector and the Endless Cycle — A Unified Derivative Integral Theory on the Folding Cubic Geometry

This paper presents the ΨD (Dimensional Flow Cosmology) framework from beginning to end as a single chain of reasoning. At the base of the theory there is no continuous spacetime; instead there are countable, indivisible energy nodes at the Planck scale — planckons. All of physics is the conserved energy ledger of how these nodes organize in the folding cubic geometry (the climb from d0 to d3 and the unwinding from d3 back to d0): at every node the active share is E (d) = (d+1) εP, the reserve share is U (d) = (3−d) εP, and the total E + U = 4εP is conserved in every event. Time is not a dimension; it is the count of events, and all dynamics advances on the counter τ = n·tP. The Lorentz signature is not obtained by inserting a time axis into the metric by hand; it is derived from the invariant maximum speed of event transmission (c = ℓP/tP). On the ascent ladder energy turns into matter; on the descent ladder matter returns to energy: E = mc² is the shared expression of this two-way conversion. All particles, photons included, are patterns of the d3 fabric; charge, mass, spin and force are four integrals of the same phase field; the photon’s scale is given by E·N = 2πεP, and because massless spin-½ is forbidden, the neutrino is massive. The freeze-out threshold is derived as xf = W (4ln2), and the opening measure closes with zero input: A₀ = 0. 2094. In the dark sector, d0 is the isolated reserve, d1 is dark energy and d2 is dark matter; the closed forms ΩΛ/Ωc = 2. 613 and Ωb/Ωc = 0. 1843 agree with observation. The cosmological constant is a reading of the event counter: Λ·ℓP² = 3Ω_Λ/n². The black hole is singularity-free; the 1/4 coefficient of its entropy is derived from the hidden-share ratio, and the evaporation lifetime scales as m³. The universe is a closed system; the cycle extends to infinity by induction, and the cycle entropy resets at every closure. All derivations have been verified by programmatic checks, and a unified falsifiability list of eleven items is given.