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 (H₀) 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 F(k,z)=(keff(z)/kstable)β, where keff(z)=k₀(1+z)ξ 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)=w₀+wₐz/1+z matching recent observational bounds from DESI and Euclid while smoothly reconciling the early-universe Cosmic Microwave Background (H₀≈67.4~km/s/Mpc) with late-universe local distance ladder measurements (H₀≈73.0~km/s/Mpc).