This working paper develops the mathematical core of Resolution Cosmology by deriving the full Schwarzschild spacetime metric from informational and thermodynamic principles, without invoking the Einstein field equations. Starting from the postulate that proper time is accumulated resolution, combined with Landauer’s bound, the Unruh effect, and the Newtonian limit, the framework shows that the Tolman temperature relation emerges as a consistency condition rather than an input from General Relativity. This closes the circularity present in earlier drafts and yields both the weak-field and strong-field Schwarzschild metric purely from resolution dynamics. The paper also sketches a cosmological extension in which dark energy is interpreted as the late-time acceleration of geometric resolution as thermal cost declines, naturally dissolving the Coincidence Problem and leading to testable predictions for the evolution of the dark energy equation-of-state parameter w(z). This document is intended as a technical companion to the main “Resolution Cosmology v5.2” framework and is released as a work in progress, open to comment and critique.