Probability, once formalised as a scalar measure, has
undergone successive enlargements: complex amplitudes introduced interference,
quaternionic vectors encoded multi‑component epistemic states. The
Octonionic Probability Paradigm (OPP) inaugurates the next generative turn,
situating probability within the non‑associative eight‑dimensional
algebra of the octonions.
In OPP, a probabilistic state is not merely a number or
vector but an authored octonionic entity. Its components inscribe magnitude,
orientation, resonance, generative memory, and the twist of non‑associativity.
This construction reframes probability as a sovereign epistemic artefact – an authored compass that records the act of knowing alongside
the algebraic resonance of its unfolding.
Classical stochastic structures – limit theorems, Markov
chains, diffusion processes – are reinterpreted through octonionic dynamics. By
embedding chaotic factors and degrees of knowledge into the octonionic field,
convergence acquires new dimensions of determinism and relationality. Simulation
itself is transformed: Monté Carlo procedures become octonionic algorithms
whose outputs carry orientation, resonance, and non‑associative
trace, turning randomness into generative inscription.
The paradigm’s operational depth is exemplified through the
octonionic reformulation of Thomas Bayes’ problem. Here, stochastic
unpredictability is lifted into the octonionic domain, revealing how non‑associativity
encodes epistemic entanglement and relational depth. Probability ceases to be a
passive measure; it becomes a participatory geometry of uncertainty.
In sum, OPP declares uncertainty as an eight‑fold
phenomenon: directional, resonant, twisting, entangled, authored, and
generative. This book establishes the canonical principles, develops operator
machinery, and demonstrates applied exemplars, inviting scholars to engage with
the octonionic turn as both mathematical innovation and epistemic revolution.
Author(s) Details
Abdo Abou Jaoudé
Department of Mathematics and Statistics, Faculty of Natural and Applied
Sciences, Notre Dame University - Louaize, Lebanon.
Please see the book here :- https://doi.org/10.9734/bpi/mono/978-81-69006-87-3
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