What question did this study set out to answer?

To establish a fundamental lower bound on adversarial perturbations in probability systems using geometric methods.

January 14, 2026Open Access

The Velado Bound: A Geometric Lower Bound on Adversarial Detectability via Fisher-Rao Information Geometry

Key Points

To establish a fundamental lower bound on adversarial perturbations in probability systems using geometric methods.
Proved a lower bound using Fisher-Rao distance and semantic impact.
Derived the bound with the Cauchy-Schwarz inequality on Fisher-Rao manifold.
Validated findings empirically on an H100 GPU across multiple trials.
Found that the product of geodesic distance and semantic impact meets a specific geometrical lower bound.
Confirmed the existence of a 'forbidden quadrant' where high-impact, low-detectability attacks are not possible.
Empirical tests showed the forbidden region remained empty in 100,000 trials.

Abstract

We prove a fundamental lower bound on adversarial perturbations in conservation-constrained probability systems. For states x, x' on the probability simplex Sₙ lying in different entropy bands, the product of Fisher-Rao geodesic distance D and semantic impact I satisfies D·I ≥ κ, where κ = (ΔH) ²/M emerges from simplex geometry. The bound is derived via Cauchy-Schwarz inequality on the Fisher-Rao manifold and establishes a geometrically forbidden region where high-impact, low-detectability attacks cannot exist. We call this the Adversarial Uncertainty Principle. Empirical validation on H100 GPU confirms the forbidden quadrant D0. 5 remains empty across 10⁵ trials. Priority: U. S. Provisional Application No. 63/958, 306, filed January 12, 2026.

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