The Paradigm of Sedenionic Probability: A Formulation of Continuous Order

The axiomatic foundation of probability theory, formalised by A. N. Kolmogorov in 1933, situates probability as a real-valued measure governing uncertainty on measurable spaces. My subsequent developments broadened this framework to complex amplitudes (CPP) and quaternionic representations (QPP), while my Octonionic Probability Paradigm (OPP) introduced an eight-dimensional non-associative structure in which probabilistic interaction acquires geometric depth.

The present work advances this trajectory through the introduction of the Sedenionic Probability Paradigm (SPP), a framework in which probability is embedded in the sixteen-dimensional algebra of the sedenions. Unlike prior extensions, the sedenionic setting incorporates both non-associativity and the presence of zero divisors, thereby enriching the algebraic landscape within which probabilistic phenomena unfold. A probabilistic configuration in SPP is represented as a sedenionic element:

                                                      S = R + M = R + (M₁ + M₂ + … + M₁₄ + M₁₅)

where the real component R encodes aggregate magnitude while the imaginary components in M capture layered relational, dynamical, and structural features of uncertainty.

SPP reframes classical stochastic constructs as projections of higher-dimensional deterministic structures. By introducing sedenionic descriptors of knowledge, interaction, and intrinsic variability, familiar probabilistic models – such as Markov processes, limit laws, and diffusion systems – are reinterpreted as shadows of richer algebraic dynamics. This perspective allows randomness to be viewed not merely as unpredictability, but as partial observability of a fully structured sixteen-dimensional evolution.

In computational contexts, the paradigm induces a transformation of probabilistic algorithms. Monté Carlo methods and sampling techniques, when reformulated within the sedenionic framework, exhibit extended state spaces and internal correlations that are invisible in lower-dimensional formulations. Canonical experiments, including discrete random trials, gain additional interpretive layers that reflect the internal coupling between sedenionic components. Furthermore, Monte Carlo procedures applied in this manuscript to the coin problem are reformulated as sedenionic algorithms.

In essence, the Sedenionic Probability Paradigm positions probability as a multidimensional algebraic process: one that integrates magnitude, direction, interaction, and structural memory within a unified formalism. This work establishes the conceptual and mathematical foundations of SPP, articulates its operator framework, and demonstrates its potential across theoretical and computational domains, thereby opening new pathways for the evolution of probabilistic reasoning.