The Hubble Operator a Rigorous Mathematical Treatment

This document presents a novel theoretical framework called PCP-Lattice Theory, which seeks to resolve the Hubble tension—the discrepancy between early and late universe measurements of cosmic expansion—by unifying quantum gravity with computational complexity. The author, Moses Kelley, introduces a rigorous Hubble operator acting on a dual-lattice Hilbert space, where the expansion rate is governed by the mathematical constraints of the PCP theorem (Probabilistically Checkable Proofs). A central pillar of the theory is a "non-trivial mathematical coincidence" where the PCP error bound $(ξ \approx 0.41)$ matches known corrections in EPRL spin-foam models, suggesting a deep duality between geometric physics and information verification. By accounting for logarithmic dual-lattice evolution and a holographically reduced effective dimension $(d_{eff} \approx 0.186)$, the model predicts a local Hubble value of $73.01 \text{ km/s/Mpc}$, successfully aligning with local SH0ES measurements. Ultimately, the text functions as a formal proof that the universe’s large-scale expansion may emerge from the computational cost of maintaining consistency across the quantum fabric of spacetime.