The Uniqueness of Area as a Measure of Causal Inaccessibility

We consider local, diffeomorphism-invariant functionals I[Σ] on closed, spacelike codimension-2 surfaces Σ in Lorentzian spacetimes. Subjecting these functionals to the dynamical constraints imposed by the Raychaudhuri equation and the requirement of binary causal connectivity, we demonstrate that causal inaccessibility reduces to area scaled by gravity: I = η A on marginal and trapped surfaces. We perform an explicit falsification test, attempting to construct alternative functionals involving higher-order curvature terms or non-local dependencies; we show that any such deviation violates the fundamental causal structure of the spacetime, rendering the Area Law the unique stable solution. This geometric result is derived dynamically and is independent of thermodynamic postulates. The proportionality constant η is a fundamental geometric scaling factor. By requiring that the functional satisfy the physical constraint of the First Law of black hole mechanics, we calibrate this scale to the value η=(4G)⁻¹, thereby matching the geometric scale of the functional to the thermodynamic scale of General Relativity. In this sense, information is geometry, quantified by gravity. The author acknowledges the use of large language models (LLMs) as an aid in drafting, code structuring, and linguistic refinement of this manuscript. The underlying theoretical physics, research design, and data analysis were formulated and verified by the author.