Orientation Dependence and Exact Groups for Ridge-Routing Sandpiles
Manuscript 5 October 2026 · Online 5 October 2026
Abstract
A fitting orientation of a simplicial rooted forest specifies an unsigned chip-firing rule on its nonroot ridges. In every dimension \(d\ge2\), a six-facet rooted forest with relative boundary determinant one admits cyclic sandpile groups of orders \(d^6-d^3-d\) and \(d^6-2d\) for two fitting orientations.
When every ridge belongs to at most two facets, the abstract group is independent of the fitting orientation. We give its cyclic decomposition and exact recurrent-state, identity, single-chip avalanche and addition-period formulas. Labeled addition periods can still change.
The proofs use an integral cycle-and-chain splitting and standard directed-sandpile theory. The baseline routing model and known linear algebra are credited.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Scoped theorem and orientation counterexample
- Categories
- math.CO · math.AT
- Manuscript
- 5 October 2026
- Online release
- 5 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Orientation Dependence and Exact Groups for Ridge-Routing Sandpiles,” EulerSolve Research Papers, AIM-COMBINATORICS-0112, 2026. https://doi.org/10.5281/zenodo.23166220.
BibTeX
@misc{Ferudun2026RidgeRouting,
author = {Ferudun, Alper},
title = {Orientation Dependence and Exact Groups for Ridge-Routing Sandpiles},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-combinatorics-0112/},
doi = {10.5281/zenodo.23166220},
note = {AIM-COMBINATORICS-0112; unrefereed preprint}
}