Semantic Phase Transitions in Observation Geometries

This article develops a mathematically explicit, architecture-agnostic framework for semantic phase transitions in learned representations. The core idea is to model “semantic units” (concepts, features, internal symbols) as balls in a metric measure space, called an observation geometry, and to study how their overlaps, density, and quality control capacity and emergent behaviour. On the static side, the paper introduces a semantic Hamiltonian with a chemical-potential term and a convex overlap penalty. Under only Ahlfors-type regularity (no exact homogeneity), it proves a capacity inequality linking semantic count, average overlap, and an energy budget, and derives an interaction-limited scaling lawNmax⁡≲B1/(2−γ)Nₘₐₓ B1/(2-γ)Nmax≲B1/(2−γ)for small-overlap potentials U(o)≳oγU(o) o^(o)≳oγ. This yields universal constraints on admissible semantic-capacity exponents, independent of specific architectures or optimisers. On the probabilistic side, the work analyses a one-dimensional Poisson interval model, obtaining an exact cluster-density formula κ(λ)=λe−2λrκ(λ)=λ e-2λ rκ(λ)=λe−2λr with a critical density λc=1/(2r)λ_c = 1/(2r)λc=1/(2r) separating dilute, critical, and condensed regimes. In higher dimensions, semantic centres are modelled by Poisson Boolean models and random geometric graphs, and known continuum-percolation and sharp-threshold results are used to define and constrain task events such as coverage, connectivity, and bounded-depth compositional paths. Finally, the paper proposes a mapping from geometric/energetic parameters (λ,r,p,μ,β,γ)(λ,r,p,μ,β,γ)(λ,r,p,μ,β,γ) to model size, effective intrinsic dimension, compute, and data quality, leading to semi-quantitative, falsifiable constraints on scaling exponents and emergent-task thresholds. The resulting framework is intended as a universal low-level reference model that any detailed theory of neural scaling laws and emergent abilities must respect.