Informational Geometry and the ln 2 Invariant

This paper introduces a previously unrecognised information–geometric invariant in gravitational systems: the fixed ratio between interior entropy and Bekenstein–Hawking horizon entropy, equal to the natural logarithm of two. This ln 2 invariant defines an information‑critical state in which each nat of interior entropy requires one bit of horizon capacity. We show that this invariant modifies the mass–radius relation of self‑gravitating systems, producing a universal mass‑boost factor and a corresponding density‑boost factor. When applied to the cosmic horizon, the invariant yields an information‑critical density that is 20.1% higher than the standard Friedmann critical density. This rescaling alters the interpretation of the cosmic energy budget and reveals a previously unrecognised information‑compression term. Within this framework, dark matter emerges as the geometric compression mass required to raise the density from the Schwarzschild value to the information‑critical value, while dark energy becomes the information deficit between interior entropy and horizon capacity. Both components arise from the same invariant rather than from independent physical fluids. The paper derives the corrected Schwarzschild relations, the information‑critical density of the universe, the informational Hubble rate, the informational age, and the informational Friedmann equation. It also identifies a universal saturation threshold for black holes and predicts that cosmic expansion halts when the universe reaches entropy saturation. This work establishes the ln 2 invariant as a structural feature of gravitational systems and provides a unified information‑theoretic correction to black hole thermodynamics, cosmological critical density, and the interpretation of dark components. A companion paper will apply this framework to the full cosmological expansion history and the numerical reconstruction of the observable universe.