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 — V_min = M·ħG²/c⁵ > 0, T_c ≤ T_bounce ≤ 4·T_P. 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; T_c = εP/(k_B·ln2), T_f = εP/(k_B·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: N_part = Mc²/(4εP) ≈ 1.15 × 10⁶⁰. C8: ΔS_cycle = 0.8983·N_part·k_B > 0 — the arrow of time is the band itself. C9: the budget–debt balance gives R_max = (4/3)·GM/c² ≈ 9.9 × 10²⁵ m — two-thirds of the budget’s Schwarzschild radius — closing T_halt ≈ 10⁻⁸ K and V_max/V_min ≈ 5 × 10¹²¹, and reducing the mass input to a geometric count, N_part = (3/16)·R_max/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 Ω_DE : Ω_DM : Ω_b = 0.6881 : 0.2634 : 0.0485 are internal outputs, with no external tuning.