# Verification and scope of the ridge routing results

The accepted mathematical scope is an all-dimensional orientation-dependent example for the unsigned ridge-routing sandpile, together with exact thin-case group, recurrence, avalanche and labeled-period formulas. The model is inherited from the frozen upstream attempt and credited. The broad AIM Question 22 is not counted as resolved.

The six-facet example has relative boundary determinant one. Its two fitting orientations give cyclic group orders d^6-d^3-d and d^6-2d for every integer d >= 2, realized by actual simplicial joins rather than by an abstract threshold substitution. At d=2 the orders are 54 and 60. The thin-case abstract group is orientation independent, but a fixed ridge in the supplied projective-plane example has addition periods 63 and 126 under the two orientations.

The manuscript supplies the general proofs. Detailed originating-researcher self-audit is preserved in the research dossier; no known gap remains within the stated theorem scope. Neither independent review nor proof-assistant formalization has occurred.

Exact regression reports are bundled with their checker source. The first suite passed 460 Smith-form/determinant comparisons, 2391 inverse-column period comparisons, 87140 one-chip stabilizations under two legal orders, 337 Markov recurrence graphs on 16596 stable configurations, and 1291 directly iterated recurrent addition periods. The extension suite passed 4802 Hall matching/absorption comparisons across 1770 small hypergraphs, 180 explicit unimodular similarities, two symbolic cyclic presentations, 24 actual simplicial orientation-lift checks through dimension nine, and all 1296 spanning root trees of the chosen projective-plane triangulation.

An initial extension assertion compared expanded and factored polynomial expressions structurally. Expanding their difference fixed the checker; the failed and successful records are both retained. No theorem was weakened to make a test pass.

Cardon and Tuckfield's cycle-and-chain decomposition and difference-vector method are prior art, not a new Jordan-form result. Directed global-sink sandpile facts are taken from Holroyd and coauthors. Bernardi and Klivans supply the rooted-forest and fitting-orientation definitions. Related orientability and gradient results are discussed by Contreras and Tawfeek. Novelty of the precise synthesis is undetermined after a bounded primary-source comparison; no priority certificate follows from publication.

This preprint is AI-assisted, self-audited and unrefereed. The manuscript discloses the assistance in prose. No canonical torsor on all ambient forests, identification with cellular homology torsion, unrestricted threshold extension or resolution of the whole AIM construction agenda is claimed.