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 square
of each higher basis element descends to the preceding element, while products of distinct
non-scalar basis elements follow the same descending-axis rule. Each finite-dimensional
member is closed under its defining operations, unital, commutative, and distributive, and the
two-dimensional member is isomorphic to the complex numbers. Every member of dimension
greater 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 unique
multiplicative inverse, whereas elements on it may have no inverse or non-unique inverse
solutions. In dimension two the singular division locus reduces to the single point 0, recovering
the familiar exceptional element of complex division. In higher dimensions it becomes a
homogeneous algebraic set through the origin. The paper further proves that, for every
finite n ≥ 2, the squaring map Sn(x) = x ⋆ x is surjective, so every element of Mn has at
least one square root within the same Mn. Enlargement of the ambient Myrion algebra
may nevertheless reveal additional roots; any finite-support square root can extend at most
one basis level above the highest occupied level of the element being rooted. The paper
derives coordinate multiplication formulae, proves the principal algebraic properties, gives
explicit low-dimensional multiplication operators, and discusses path dependence caused by
non-associativity. Possible relevance to structured neural computation is outlined, particularly
where fixed or learned association trees and distance from the singular division locus may
provide useful inductive biases. No empirical machine-learning claims are made.