An interesting fact about the Leech lattice is discussed by Tony Smith: the neighbours of the origin in 24 dimensional surreal number space form a Leech lattice. Recall that the surreal numbers give a tree structure to the reals, and as we know, trees are more fundamental than real numbers.
Speaking of canonical constructions for the reals, recall that in using associahedra, we came across integral arguments for the Riemann zeta function. But the full Riemann hypothesis requires us to understand the function on the complex plane. The surreal tree is a binary logic construction of the reals. A Space Time complexification could give us a notion of complex number.