The mathematical proof is grounded in the Weyl Curvature Hypothesis. Entropy in the universe is stored gravitationally in the “clumpiness” of space (measured by Weyl Curvature). At the Big Bang, the Weyl Curvature was zero (perfectly smooth). At the Heat Death, once all mass has evaporated via Hawking Radiation, the “anchors” that maintain this curvature vanish.
- The Reset Formula: At the crossover boundary, the metric $$g_{ab}$$ undergoes a conformal rescaling:$$\hat{g}_{ab} = \Omega^2 g_{ab}$$
- The Result: This rescaling “stretches” the high-entropy boundary until the Weyl Curvature is forced back to zero. By shedding the constraint of mass, the universe sheds its history. The “Start” and “End” are not two different points in time, but the same geometric boundary viewed through different scales.