On the dynamical evolution of randomness Part B: Geometrisation and the origin of convergence in LLN

TL;DR

This paper introduces a dynamical framework for understanding the Law of Large Numbers, modeling randomness as a structured, feedback-driven process that explains convergence, fluctuations, and geometric features without relying solely on statistical assumptions.

Contribution

It develops a novel dynamical formalism that derives statistical convergence from outcome space evolution, offering a mechanistic perspective on probability and randomness.

Findings

Classical LLN behavior emerges in large-number limit

Predicts deviations and fluctuations in early regimes

Reveals geometric asymmetries in outcome space

Abstract

In classical probability theory, the convergence of empirical frequencies to theoretical probabilities: as captured by the Law of Large Numbers (LLN): is treated as axiomatic and emergent from statistical assumptions such as independence and identical distribution. In this work, a novel dynamical framework is constructed in which convergence arises as a consequence of structured evolution in outcome space, rather than a statistical postulate. Through this formalism, statistical convergence is derived dynamically, revealing an internal structure to randomness and exposing entanglement between successive trials. The system recovers classical LLN behaviour in the large-number limit, while predicting deviations, transient fluctuations, and geometric asymmetries in the early regime. This work inaugurates a new paradigm: dynamical probability mechanics: in which randomness is modelled not as a sequence of disconnected stochastic events, but as a physically structured, feedback-driven process. The theory provides a novel explanatory layer beneath statistical laws and opens pathways toward a mechanistic foundation of probability itself.