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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.25488 |
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| _version_ | 1866910092807372800 |
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| author | Lyudogovskiy, Fedor B. |
| author_facet | Lyudogovskiy, Fedor B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25488 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Directional Geometry and Anisotropy in the Partition Graph Lyudogovskiy, Fedor B. General Mathematics 05C12, 05C75, 05A17 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. |
| title | Directional Geometry and Anisotropy in the Partition Graph |
| topic | General Mathematics 05C12, 05C75, 05A17 |
| url | https://arxiv.org/abs/2603.25488 |