Dynamic Dark Energy and Hubble Tension Resolution via Redshift-Dependent Topological Coupling in the Paper and Pen Framework

This paper extends The Paper and the Pen framework to resolve the Hubble tension (H_{0}) and characterize the dynamical nature of dark energy without introducing fine-tuned scalar fields. By expanding the internal space topological coupling functional into a redshift-dependent dynamical variable \mathcal{F}(k,z)=(k_{eff}(z)/k_{stable})^{\beta}, where k_{eff}(z)=k_{0}(1+z)^{\xi} we model the continuous boundary-layer interaction of quantum record states across the expanding Hubble horizon. We derive a parameter-free dark energy equation of state w(z)=w_{0}+w_{a}\frac{z}{1+z} matching recent observational bounds from DESI and Euclid while smoothly reconciling the early-universe Cosmic Microwave Background (H_{0}\approx67.4~km/s/Mpc) with late-universe local distance ladder measurements (H_{0}\approx73.0~km/s/Mpc).