Irreducibility as a Shared Logical Structure: Prime Numbers and Causal Order as Dual Instances A Companion Paper to the Inverse Projection Principle
Year: 2026
Type: preprint
Abstract: This paper identifies and formalizes a shared logical structure between two apparently unrelated domains: the prime numbers in mathematics and causal order in relativistic physics. We argue that both instantiate what we call irreducibility: a structural property of elements that cannot be decomposed into simpler constituents within their respective systems, and that generate all other elements of those systems through composition. In arithmetic, the prime numbers are the irreducible elements of the multiplicative structure of the integers: every integer factors uniquely into primes (the Fundamental Theorem of Arithmetic), but primes themselves admit no such factorization. In relativistic physics, causally connected events are the irreducible elements of the causal order of spacetime: every causal chain decomposes into elementary causal links, but these links themselves are not further decomposable without violating the causal structure. We formalize this parallel using the categorical framework of the companion paper (Abuhelweh, 2025 — The Inverse Projection Principle, DOI: 10.5281/zenodo.19814499), showing that in both cases the irreducible elements are anomalous in a lower-dimensional representational space and regular in a higher-dimensional extension space, and that the projection map discards the structural information that makes them irreducible. We further propose that irreducibility is not merely a formal analogy between the two domains but a shared logical invariant: a structural property that appears whenever a generative system is embedded in a representational space that cannot fully represent its generative logic. The philosophical implication is that prime numbers and causal order may be dual projections of a single deeper organizational principle — one whose full character is not yet visible in either mathematics or physics alone.
Author: Abuhelweh, Mohammad
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164001973