{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0112",
  "title": "Orientation Dependence and Exact Groups for Ridge-Routing Sandpiles",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A fitting orientation of a simplicial rooted forest specifies an unsigned chip-firing rule on its nonroot ridges. We show that the resulting sandpile group can depend on that orientation: in every dimension d >= 2, a six-facet rooted forest with relative boundary determinant one admits cyclic groups of orders d^6-d^3-d and d^6-2d. In contrast, if each ridge belongs to at most two facets, the abstract group is independent of the fitting orientation. We give its cyclic decomposition and exact formulas for recurrent states, the identity, single-chip avalanches and individual addition periods. Even in this thin case, labeled addition periods can change. The proofs use an integral cycle-and-chain splitting and standard directed-sandpile theory. These results analyze a specified routing model; they do not settle the general AIM request for a higher-dimensional chip-firing model or establish a canonical torsor. The baseline model and known linear algebra are credited. AI-assisted, self-audited and unrefereed; novelty and absolute priority remain undetermined.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "keywords": [
    "sandpile groups",
    "chip firing",
    "simplicial forests",
    "fitting orientations",
    "integer presentations",
    "AIM-COMBINATORICS-0112"
  ],
  "manuscript_version_date": "2026-10-05",
  "publication_date": "2026-10-05",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-05",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0112/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0112/paper.pdf?v=e34ffe4fa877",
  "doi": "10.5281/zenodo.23166220",
  "zenodo_record_url": "https://zenodo.org/records/23166220",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete results for a specified ridge-routing model associated with AIM-COMBINATORICS-0112 and Question 22 of the 2013 AIM chip-firing list. This does not settle the general request for a higher-dimensional chip-firing model, construct a canonical torsor on all ambient forests, or establish a general cellular-torsion correspondence. The orientation counterexample is to invariance of this model, not to the existence of other models. The thin-case formulas and all-dimensional counterexample have written proofs supported by reproducible exact checks. AI-assisted, self-audited and unrefereed; no independent review or proof-assistant formalization is claimed. Novelty and absolute priority remain undetermined.",
  "files": {
    "paper.pdf": {
      "sha256": "e34ffe4fa877713a9c5a0570e8e644c8f90a26bfd55b59a8f2210b00049481a7"
    },
    "source.zip": {
      "sha256": "127e65af1bfd83ba5eba34811844f7c9aa8a928515ff1d937dc2658f301a7c8f"
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    "verification_report.md": {
      "sha256": "242eb1f465427961efe9aebcd1f185ab8acb8648757502cb52a79b0559ae45d3"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}