Closed Universe and Endless Cycle: Cycle Equations in the ΨD Framework A Counter-Language Derivation of the Bang, Deceleration, Halt, and Turnaround

This note formalizes a cyclic-universe scenario on the axioms of the main article; the ground–inventory laws and the thermal thresholds it uses are internalized in Appendix A (Propositions Z-1…Z-6). Four postulates are laid down: the universe is a closed system (the total budget is conserved) ; the state space is bounded by two walls — the lower wall is saturation (no singularity), the upper wall is the causality bound (no infinite expansion) ; between the two walls the only behavior is oscillation; and at the turning points the content is reduced to d1. From these postulates six phases are proved in sequence by derivativeintegral (Propositions S-1…S-6; energy conservation is checked at every phase): the trigger of the initial bang is wall reflection — no energy is created, the reserve begins to discharge; expansion is the flow from reserve to active; deceleration is the integral of the inventory’s gravity (~7. 7 billion years) ; re-acceleration begins at the ρinv = 2ρgnd threshold — the ΔkH = log₄3 step — (z ≈ 0. 63) ; the halt is the point where four physical indicators coincide with the wall; the collapse is the counter-reversal mirror of the equation and, through the summoning effect of the cosmic web, flows along filament lines into the nodes. The thermal law is a single equation: Θ = Θmax·2^− (k − kmin) — the expansion half cools and locks the patterns, the collapse half heats by the same law and unlocks them in reverse order (ascent d1→d2→d3, descent d3→d2→d1). The d1: d2: d3 ratios come from the combinatorics of binary folding (4: 2: 1) and are the same in every cycle. The bang and halt moments are inverse-mirror conjugates. The upper wall is derived in this version: the horizon–ceiling theorem together with the cosmic reading of B3 yields kmax = kΛ = 202. 676 N — the magnitude lock, the cosmic coincidence, and the halting wall unite in a single number. The scenario is at level V; the absolute value of kΛ is left open O; the entropy question, the effective wall potential, and the thermal closure are closed in Appendix B (Propositions K-1…K-9).