Infinitesimal Discrete Probability Geometry – A Nonstandard Axiomatic Framework

Abstract

We introduce a closed axiomatic framework, Infinitesimal Discrete Probability Geometry (IDPG), built on nonstandard analysis and the Loeb measure construction. The theory defines a hyperfinite probabilistic universe equipped with an internal finitely additive measure, extended to a standard σ‑additive probability via the Loeb completion. Within this structure, probability, geometry, and information theory are unified through a normalized infinitesimal measure. Classical probabilistic laws—normalization, independence, and the Borel‑Cantelli lemma—emerge naturally from the hyperfinite foundation. An information‑theoretic layer is derived as a structural consequence of the probability measure. The framework is extended to encompass quantum amplitudes and gravitational curvature, deriving the Born rule, interference, and discrete Einstein equations from the same hyperfinite foundation. The resulting system provides a consistent bridge between infinitesimal geometry, discrete probability, quantum mechanics, gravity, and asymptotic probabilistic phenomena in infinite systems.

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