From Distinguishability Geometry to Holographic Entropy Bounds

Why should the maximum entropy contained within a region of space scale with its boundary area rather than its volume?

In this work we explore a geometric route to holographic entropy bounds based on statistical distinguishability. Physical configurations are treated as points on a Fisher information manifold, where distances quantify how well different states can be operationally distinguished.

Under finite statistical resolution, the number of distinguishable configurations within a region defines a distinguishability capacity. Geometric constraints on this capacity arise naturally from the isoperimetric inequality, which relates interior volume to boundary measure on the Fisher manifold.

We show that this constraint limits the growth of distinguishability inside a region relative to the size of its boundary. When gravitational consistency conditions—specifically the Bekenstein bound and gravitational collapse limits—are imposed, the normalization of this bound becomes fixed by the Planck scale, yielding the familiar holographic entropy scaling

S_max ≈ A / (4ℓ_P²).

This perspective suggests that holographic entropy bounds may arise from general geometric principles governing distinguishability and information flow, rather than from microscopic state counting alone.