Title The Mathematically-Structured Dimensional Unit System (MSDUS): A Foundational Reformulation of Physical Algebra for Guaranteed Dimensional Safety and Numerical Stability
Abdulsalam Al-Mayahi · 2025
Abstract (OSF version)Modern computational science is built upon a critical flaw: the disconnection between abstract number and physical dimension. This weakness has produced catastrophic failures, from the loss of the Mars Climate Orbiter to the Ariane 5 explosion, and continues to undermine reproducibility across physics, engineering, and AI.We introduce the Mathematically-Structured Dimensional Unit System (MSDUS), a universal algebraic framework that embeds dimensional awareness into the structure of computation. MSDUS defines every physical quantity as a tuple of numerical value and dimensional vector, enforcing provably sound rules that make dimensional safety automatic and violations mathematically impossible.We prove three theorems:Dimensional Soundness – every valid operation yields a physically meaningful dimension.Unit-System Independence – results are invariant under choice of measurement units.Automatic Homogeneity – dimensional consistency is enforced by construction.Beyond dimensional correctness, MSDUS integrates a φ–π–τ scaling architecture and adaptive numerical algorithms (Kahan summation, precision promotion) that maintain stability across 60+ orders of magnitude. Illustrative scenarios in fusion plasma control, climate modeling, and aerospace systems demonstrate MSDUS’s potential to prevent costly errors, accelerate discovery, and redefine computational reliability.MSDUS represents a paradigm shift from “numerically correct” to “physically coherent” computation, establishing a new foundation for scientific precision, AI discovery, and mission-critical engineering.🔖 KeywordsDimensional Analysis; Unit Systems; Numerical Stability; Computational Physics; Graded Algebra; Verification; Fusion Plasma; Climate Modeling; Aerospace Safety; Formal Methods📜 Description (OSF page text)The Mathematically-Structured Dimensional Unit System (MSDUS) is part of Abdulsalam Al-Mayahi’s broader scientific program built on the Union Dipole Theory (UDT) and the discovery of the Union Dipole Particle (UDP).MSDUS complements UDT’s τ-dynamics by providing a computational infrastructure where dimensional safety and numerical stability are guaranteed by design. Together with related platforms such as the τ-Math Engine, GBGL heavy-tail statistics, τ-BioCore for biophysics, and τ-Clock metrology frameworks, MSDUS extends UDT from a theoretical model into a practical, cross-disciplinary technology stack.By unifying dimensional algebra, fundamental constants, and precision computation, MSDUS aims to redefine the computational standards of 21st-century physics, engineering, and AI.⚖ LicenseCreative Commons Attribution 4.0 International (CC BY 4.0) — recommended to maximize citations, reuse, and visibility.