Simple Lie Groups as Information Spaces

We investigate mechanisms by which natural systems store data across multi-dimensional spaces, integrating principles from information theory, probability, algebra, and geometry. We propose that simple Lie groups (SLGs) govern these encoding processes and act as basic information spaces. We start by showing that Euclidean space turns computationally unsolvable in higher dimensions. If we do not notice the "curse of dimensionality," it is likely because nature uses geometric positional notation at a rudimentary level. Using Benford's law, we extend the definition of radix economy to m physical dimensions. The new function measures a system's storage cost and has interesting properties. It grows as (Log_r(N))^m, rather than as N^m, providing radix r and rank m invariance, and attaining a "natural" minimum at r = e^m. This cost accommodates a homogeneous distribution of the space with a centripetal bias, avoiding the curse, and diverges when we abandon the positional system (i.e., as r->1 or r->∞) or when the dimensionality explodes (as m->∞). We review the SLGs central to modern geometry and physics. We note that they have irreducible representations of order s ≈ e^m and storage cost sorted by s, as expected. We suggest a universal notation based on balanced ternary to uniquely encode integers as the difference of two natural numbers in bijective notation. Structure and information are coupled so that the unit vectors of an s-dimensional irrep each have a label used by the radix r as an alphabetic symbol of the proposed symmetric bijective system to represent data. Finally, we examine the Weyl order of some of these SLGs, showing that Weyl divisors give rise to a logarithmic scale consistent with Benford's Law with radix r = s. This means that an SLG is autonomous to serve as an information space of non-Euclidean geometry.