The Complete Cycle of the Universe: Bounce, Ascent, Halt, Mirror Descent, and Collapse — Derivative–Integral Proofs in Doubling-by-Two Cubic Geometry (ΨD)

This paper constructs the complete cycle of the universe within the ΨD (Dimensional Flow Cosmology) framework: the universe is not born from a singular point and does not expand forever; it traces a closed cycle between two walls. The method is restricted to three tools — doubling-by-two cubic geometry, the energy–potential ledger (E (d) = (d+1) ·εP, U (d) = (3−d) ·εP, E + U = 4εP), and discrete derivative–integral operations; time is not a dimension in any proof — all derivatives are counter derivatives (n). The cycle variable is the conversion fraction φ: W = 3φ·εP generates gravity, U = 3 (1−φ) ·εP drives expansion, W + U = 3εP, and the swap φ → 1−φ exchanges the shares exactly — ascent and descent are two readings of one ledger. Nine theorems are proved. C1: the collapse rebounds from the density ceiling — Vₘin = M·ħG²/c⁵ > 0, Tc ≤ Tbounce ≤ 4·TP. C2: R″ ∝ (1 − 2φ) — deceleration at the mirror point (φ = 1/2), halt at full conversion (φ = 1). C3: the descent is the ascent read backwards; ∮dR = 0. C4: T·R is conserved; Tc = εP/ (kB·ln2), Tf = εP/ (kB·W (4ln2) ) open a hysteresis band; the relic distribution at freezing is the hidden-channel biased binomial: f₀ = 0. 4640, f₁ = 0. 3688, f₂ = 0. 1412, f₃ = 0. 0260 at A₀ = 0. 2094. C5–C6: the walls have their equations and the nine-step cycle closes, ledger conserved at every step. C7: Nₚart = Mc²/ (4εP) ≈ 1. 15 × 10⁶⁰. C8: ΔScycle = 0. 8983·Nₚart·kB > 0 — the arrow of time is the band itself. C9: the budget–debt balance gives Rₘax = (4/3) ·GM/c² ≈ 9. 9 × 10²⁵ m — two-thirds of the budget’s Schwarzschild radius — closing Tₕalt ≈ 10⁻⁸ K and Vₘax/Vₘin ≈ 5 × 10¹²¹, and reducing the mass input to a geometric count, Nₚart = (3/16) ·Rₘax/ℓP. All derivations are verified by 56/56 programmatic checks (Appendix A). The framework is budget-closed and falsifiable: w₀ = −27/32 = −0. 844 and ΩDE: ΩDM: Ωb = 0. 6881: 0. 2634: 0. 0485 are internal outputs, with no external tuning.