Directional Geometry and Anisotropy in the Partition Graph

TL;DR

This paper introduces a formal directional framework for the partition graph G_n, analyzing anisotropy and directional structures using reference sets, and provides computational tools for visualization and analysis.

Contribution

It formalizes anisotropy in G_n through directional fields, proves structural non-equivalence of reference sets, and develops a computational atlas for directional analysis.

Findings

Distinct reference sets induce unique directional fields.

Every vertex admits at least one inward geodesic corridor.

Bounded neighborhoods are accessible via monotone inward corridors.

Abstract

We develop a directional formalism for the partition graph G_n based on several canonical reference sets: the main chain, the self-conjugate axis, the spine, and the boundary framework. For each such set S, the graph distance d_S induces a shell structure and a local trichotomy of edges into inward, outward, and level classes. Passing from edges to paths, we define directional corridors as monotone inward geodesics toward a chosen reference set and prove that every vertex admits at least one. We then prove a structural non-equivalence theorem: for connected G_n, two nonempty reference sets induce the same edgewise directional field if and only if the difference of their distance functions is constant; in particular, distinct reference sets induce distinct directional fields. This gives a first precise formalization of anisotropy in G_n. We also show that every bounded neighborhood of a reference set is accessible by a monotone inward corridor, which gives a directional interpretation to previously established controlled regions around the axis, the spine, and the framework. Finally, we complement the strict theory with a computational atlas illustrating edgewise directional statistics, directional mixing, local invariant drift, and corridor-based transport profiles.