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Directional Geometry and Anisotropy in the Partition Graph

Fedor B. Lyudogovskiy

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.

Directional Geometry and Anisotropy in the Partition Graph

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.

Paper Structure

This paper contains 23 sections, 15 theorems, 64 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Canonical reference sets and directional observables
  3. Canonical reference sets
  4. Distance observables, shells, and neighborhoods
  5. Connectivity
  6. Directional classes of moves
  7. Edgewise direction relative to a reference set
  8. Tangential and transverse motion
  9. Named directional classes and combined signatures
  10. Monotone paths and directional corridors
  11. Monotone paths
  12. Directional corridors and pathwise transport
  13. Anisotropy and interaction with established morphology
  14. Height orientation
  15. Directional equivalence and its failure
  16. ...and 8 more sections

Key Result

Proposition 2.4

For every $n \ge 1$, the partition graph $G_n$ is connected.

Figures (3)

  • Figure 1: Two shell structures on the same embedding of $G_{10}$, shown with identical vertex positions to isolate the difference between axial and spinal shells. The two shell structures are close but not identical; vertices where the two shell distances differ are marked by black rings.
  • Figure 2: Same-shell and cross-shell edge behavior for two different reference sets. In each panel, blue edges remain inside a shell and orange edges cross between adjacent shells.
  • Figure 3: Canonical axial, spinal, and framework corridors from the same starting vertex in $G_{12}$, defined by the lexicographically minimal descent rule.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Corollary 3.3
  • proof
  • ...and 41 more