2026/05/15 by Rowan Brad Quni-Gudzinas · 1 voice
paper · doi:10.5281/zenodo.20213043
openalex publication_date 2026/05/15 · openalex created_date 2026/05/16 · openalex updated_date 2026/07/01
We propose that the hierarchical aspects of reality have the mathematical structure of a rooted tree with a monotone height function. Specifically, cophenetic distance d(x, y) = h(lca(x, y))—the height of the lowest common ancestor of two leaves—provides a unified formal framework for nested equivalence relations, emergent distance metrics, and the ontic/epistemic distinction. We prove that cophenetic distance satisfies the ultrametric inequality d(x, z) ≤ max(d(x, y), d(y, z)) and derive triadic rigidity as a necessary consequence: for any three items, the two largest pairwise distances are equal. These theorems are verified computationally (all checks passed). We build structural bridges: (1) the tree’s height function is a natural resolution parameter, with the root corresponding to the coarsest resolution and the first cut as the onset of structure; (2) successive differentiation is the generative mechanism by which branches form, resolving the static/dynamic tension; (3) the tree structure extends to cosmology as a cosmic timeline, to linguistics as the noun/verb distinction, and to epistemology through bounded consilience claims. We address seven objections. All academic citations have been verified.