Emergent Dark Energy from Dual-Sector Energy Exchange: Phenomenological Background-Level Framework and First H(z) 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), decomposition of the observable sector into standard matter and a coupling-driven dark-energy component, first χ² fit to 29 cosmic chronometer H(z) measurements (χ²/dof = 0.59), χ² contour map over (λ_eff, H₀), parameter degeneracy discussion, and a portable Python script reproducing all figures. --- I present a phenomenological two-sector Friedmann–Robertson–Walker model in which an interaction term Q = λH_x ρ_y transfers energy from a hidden sector y into an effective dark-energy component in the observable sector x. At the homogeneous background level, this generates an effective equation of state w_eff = −1 − λρ̃_y/(3ρ̃_de), so that w_eff < −1 whenever the coupling is active. This statement is purely a property of the background equations and should not be interpreted as a result on perturbative stability. Separately, when the model is supplemented by a loop-quantum-cosmology bounce prescription, it admits non-singular cyclic solutions. A first comparison with 29 cosmic chronometer H(z) measurements indicates that the model is not obviously inconsistent with this dataset at background level, with effective coupling λ_eff ≡ λr₀ ≈ 0.52, H₀ ≈ 70.8 km/s/Mpc, w_y = −1.00, and χ²/dof = 0.59. Because the reduced χ² lies well below unity and only diagonal chronometer errors are used, this result should be interpreted as a first consistency check rather than as a statistically robust parameter determination. The fitted H₀ lies in an intermediate range relative to commonly quoted CMB- and distance-ladder determinations, but no statement about the Hubble tension is made. No perturbation analysis, BAO, CMB, or supernova comparison is included. This remains an exploratory phenomenological proof of concept.