Emergent Dark Energy and Cyclic Dynamics from Dual-Sector Energy Exchange: Phenomenological Framework and First Observational Comparison

Version 1. 1 — 20 March 2026 (extends v1. 0 from 17 March 2026) New in this version: numerical background integration via Runge–Kutta (DOP853), first χ² fit to 29 cosmic chronometer H (z) measurements (χ²/dof = 0. 59), χ² contour map over (λₑff, H₀), parameter uncertainty estimates, Hubble tension discussion, and a portable Python script reproducing all figures. --- I present a phenomenological toy model in which dark energy and cyclic cosmological evolution emerge from energy exchange between two coupled Friedmann–Robertson–Walker sectors. The observable sector x and a hidden partner sector y interact through Q = λHₓ ρᵧ, producing an effective dark energy component with equation of state wₑff = −1 − λρᵧ/ (3ρde) < −1 without phantom fields, driven by a closed feedback loop between energy density, expansion rate, and energy transfer. Combined with a loop-quantum-cosmology bounce, the model admits non-singular cyclic dynamics. In this version (v1. 1), the background equations are integrated numerically. The observable sector is decomposed into standard matter (Ωₘ = 0. 30, fixed) and a coupling-driven dark energy component. A χ² fit to 29 cosmic chronometer H (z) measurements yields best-fit effective coupling λₑff ≡ λ·r₀ = 0. 30, H₀ = 69. 6 km/s/Mpc, and χ²/dof = 0. 59. A χ² contour map over (λₑff, H₀) shows the model prefers H₀ between the Planck value and the SH0ES measurement, suggesting a possible connection to the Hubble tension. The model predicts wₑff (z=0) < −1, consistent with recent hints from DESI 2024.