The Myrion Number System

This paper defines a family of finite-dimensional real algebras, termed Myrion algebras,that extends the complex numbers through an ordered sequence of basis elements. The squareof each higher basis element descends to the preceding element, while products of distinctnon-scalar basis elements follow the same descending-axis rule. Each finite-dimensionalmember is closed under its defining operations, unital, commutative, and distributive, and thetwo-dimensional member is isomorphic to the complex numbers. Every member of dimensiongreater than two is non-associative. For an element x, multiplication defines a real-linear operator Lx : y 7→ x ⋆ y.The set: Σn = {x ∈ Mn : detLx = 0}is introduced as the singular division locus. Elements outside this locus possess a uniquemultiplicative inverse, whereas elements on it may have no inverse or non-unique inversesolutions. In dimension two the singular division locus reduces to the single point 0, recoveringthe familiar exceptional element of complex division. In higher dimensions it becomes ahomogeneous algebraic set through the origin. The paper further proves that, for everyfinite n ≥ 2, the squaring map Sn(x) = x ⋆ x is surjective, so every element of Mn has atleast one square root within the same Mn. Enlargement of the ambient Myrion algebramay nevertheless reveal additional roots; any finite-support square root can extend at mostone basis level above the highest occupied level of the element being rooted. The paperderives coordinate multiplication formulae, proves the principal algebraic properties, givesexplicit low-dimensional multiplication operators, and discusses path dependence caused bynon-associativity. Possible relevance to structured neural computation is outlined, particularlywhere fixed or learned association trees and distance from the singular division locus mayprovide useful inductive biases. No empirical machine-learning claims are made.