The Economy of the Lie Squad

We investigate the mechanisms by which natural systems encode data across multi-dimensional spaces. Integrating principles from information theory, probability, and geometry, we propose that certain Lie Groups govern these encoding processes. We first demonstrate that evenly distributed information becomes computationally unsolvable in higher dimensions. If we do not notice the "curse of dimensionality," it is because nature likely uses geometric positional notation at a rudimentary level. By extending the definition of representational cost to m dimensions using Benford’s Law, we identify a cost minimum at powers of Euler’s number (е^m). We introduce the "Lie Squad" (B3, F4, G2, A2, A1, and E6), a set of six compact simple Lie groups whose irreducible representations coincide with this ideal cost when m matches the group’s algebraic rank. These irreps facilitate a fundamental, rank-invariant number system based on balanced ternary, uniquely encoding integers as the difference of two natural numbers in bijective notation. Finally, we examine the Weyl orders of the Lie Squad members to show that Weyl divisors yield a logarithmic scale consistent with Benford’s Law and the universal number system proposed.