Infinitesimal Discrete Probability Geometry – A Nonstandard Axiomatic Framework

Abstract

We present a closed axiomatic framework, Infinitesimal Discrete Probability Geometry (IDPG), built on nonstandard analysis and the Loeb measure construction. Within a hyperfinite probabilistic universe, IDPG introduces a discrete curvature deviation, constructed from local measure imbalance, and establishes its exact asymptotic order relative to the local entropy gradient at finite N; we show that this order coincides with a naive O(N^-2α) correspondence only at a single parameter value (α=1), and that the entropy gradient is otherwise asymptotically subdominant to the curvature deviation rather than predictive of it. IDPG does not claim that classical Ricci curvature is derived from Shannon entropy in general; it examines a specific, precisely-bounded relationship between two quantities within this hyperfinite measure geometry, jointly formulating probability, geometry, and information theory through a normalized infinitesimal measure. Main results: (i) the exact order of the entropy-curvature deviation at finite N, and where it is and is not informative; (ii) a discrete Bianchi identity for the linearized field on the flat hyperfinite torus; (iii) O(ε²) convergence of the discrete geometric action for the linearized field to its continuum limit. A proposed graph generalization of (i) was tested and its claimed identity withdrawn; a valid graph-level generalization remains open. Classical probabilistic laws (normalization, independence, Borel-Cantelli) are recovered as consistency checks of the measure layer. An information-theoretic layer is derived via the uniqueness theorem for entropy. Computational verification in 1D and 2D and of the Bianchi identity in 2D supports internal consistency within the tested parameter range; limitations of that range are detailed in the paper. The extension to the full nonlinear metric, and identification with classical geometric field theory, is an open problem for future work. IDPG is compatible with standard quantum mechanical amplitudes as a special case. It is a theoretical contribution for mathematical discussion, not an empirical or algorithmic claim. What is and is not established, separating the constructed language from claims tested and withdrawn, is listed explicitly in the Conclusion.

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