The Closed Universe and the Infinite Cycle: Cycle Equations in the ΨD Framework

This note formalizes a cyclic-universe scenario on the axioms of the main paper; the groundinventory laws and the thermal thresholds are internalized in Appendix A (Propositions Z-1…Z6). 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 (prohibition of the singularity), the upper wall is the causality bound (prohibition of infinite expansion); between the two walls the only behaviour is oscillation; at the turning points the content is reduced to d1. From these postulates six phases are proved in sequence by derivative–integral methods (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 gravitational pull (~7.7 billion years); re-acceleration begins at the threshold ρenv = 2ρground — at the rung ΔkH = log₄3 — (z ≈ 0.63); the halt is the point at which four physical indicators coincide with the wall; collapse is the counter-reversal mirror of the equation and, through the summoning effect of the cosmic web, flows along the 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 the patterns 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 the halt are inverse-mirror conjugates. The upper wall is derived in this version: the horizon–ceiling theorem together with the cosmic reading of B3 gives kmax = kΛ = 202.676 [N → SD+N; Appendix G, G-4] — 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 closed in Appendix G: the saturation–thermal–horizon chain is scalecovariant — no input-free absolute address emerges from the chain, the absolute address is necessarily a counter reading (degeneracy theorem, G-5) — and the address theorem derives kΛ = kH + ΔkH = 202.674 from a single input (H₀): the observational chain lies inside the band 202.664–202.676 (G-4); the entropy question, the effective wall potential and the thermal closure are closed in Appendix B (Propositions K-1…K-9); the remaining [N] and [O] items are tied to ledger closures in Appendix G (Propositions G-1…G-8): the triple cross-check 0.2697 / 0.2718 / 1 0.273, the exact constants of the thermal channel 45/π³ and 8ζ(3)/(3π), Ωr as a closed function of (H₀, T₀) and the derivation of the canonical (w₀, wa) pair from the quenching rule; the real-scale full-cycle run closes by resolving the regime split onto the walls (G-10: upper hard-limit, lower threshold vc = 0.8172c) and the residue of absoluteness reduces to a single amplitude (G-9: A = kΛ − kmin = 105.63); Appendix H gives a closed-form candidate for the window part of the amplitude (ΩDE = e^W(4ln2)/4 = 0.68634, zero parameters) and merges the series-wide absoluteness into a single number (Ntop).