The Dancing Sand Theorem — the 2-part of the sandpile group of the hypercube, for every n

Let K(Q_n) be the sandpile group (critical group, Jacobian) of the n-dimensional hypercube graph Q_n. Its odd part was determined by Bai in 2003 (Linear Algebra Appl. 369 (2003), 251–261); its 2-part has been open since then, and the latest work on it (Gao, Marx-Kuo, McDonald, Yuen, Comm. Algebra 52 (2024)) determines its n − 1 largest cyclic factors. We determine the whole 2-part of K(Q_n) for every n, by an explicit rule. The group splits over the indecomposable tilting modules T(λ) of SL_2 in characteristic 2 that occur in V^⊗n, with their multiplicities. The contribution of T(λ) at the shift i = (n − λ)/2 consists of explicit small cyclic factors and of 2^|I| large ones, where I is the set of binary digits of λ + 1 below the leading one. The large ones are obtained by pooling a sequence of explicit integers — binomial valuations and carries — into a non-increasing sequence, in the manner of isotonic regression. The proof reduces each contribution, by elimination inside explicit 2-adic lattices, to the Smith form of an explicit 2^|I| × 2^|I| integer matrix, and computes that Smith form from the valuations of its entries. It needs a unique cheapest matching for an upper bound and an inductive potential for a lower bound. Consequences. We recover Bai's counts and the theorems of Gao, Marx-Kuo, McDonald and Yuen on the largest factors, and we prove their Conjectures 4.14 and 5.4 and the formula for the (n+1)-th factor stated in the 2019 poster of three of them. We also prove the fold law: the 2-part of the sandpile group of the folded cube Q_n/⟨(1,…,1)⟩ is 2·Syl_2 K(Q_n), the group obtained by lowering every exponent by one. With the results above this gives the whole 2-part of the sandpile group of the folded cube, and, with the known odd part, its whole sandpile group. §1.0 of the paper is a one-table summary of every result and where it is proved; §1.8 states what is ours to the best of our knowledge, after the search of the literature it describes, and credits every ingredient we did not invent; §13 gives the exact status of every result. Status and method. Every result has a written proof, re-derived step by step by a reader other than its author; the chains of the main theorem and of the fold law, and then the text, were read cold by readers with no access to the audits (Appendix A of the paper). The mathematics was developed with extensive use of AI systems (Claude, Anthropic), working under the author's direction as constructors, auditors and cold readers; this is disclosed in the paper. This work has NOT been refereed by a human expert and is offered for expert review. There is no formal verification. The full working record — missions, reports, audits, cold readings, engines and logs, including the routes that failed — is available from the author. Files. THE_DANCING_SAND_THEOREM_v8.pdf: the paper (50 pages). THE_DANCING_SAND_THEOREM_v8.md: its Markdown source, the same text. dancing-sand-theorem-checks.zip: the folder gate/ named in §12.1 — the Python code of every check listed there, each with its control where §12.1 gives one — with the watchdog under which they run, a script that runs them all, and the logs of a complete run made from the folder as published (instructions in its README). Version 8 (6 October 2026) is the first public version. Versions 1–7 were not made public; the first was read cold as a whole, and the changes of each later one were read cold in turn (Appendix A).