The Coloured Sand Theorem: the p-part of the sandpile group of the Hamming graph, for every prime p dividing q and every dimension d

Rafael Amichis Luengo

The Coloured Sand Theorem: the p-part of the sandpile group of the Hamming graph, for every prime p dividing q and every dimension d

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatorics math.GR — Group Theory math.RT — Representation Theory

Contributed by Rafael Amichis Luengo ↗

Version 1 / Oct 15, 2026 / CC BY 4.0

Abstract

Let

H(d,q)=K_q^{d}

be the Hamming graph and

K(H(d,q))

its sandpile group (critical group, Jacobian). For primes

p\nmid q

its

p

-part was determined by Jacobson, Niedermaier and Reiner in 2003, who wrote that the other primes appear "to be difficult, due to bad behavior of its

p

-primary structure". We determine the Sylow

p

-subgroup of

K(H(d,q))

for every prime

p

dividing

q

and every

d

, by an explicit rule. After the splitting of Jacobson, Niedermaier and Reiner, the group is a sum of cokernels of operators

q(c+H)+U

on the Boolean lattice (Theorem A). These split over the indecomposable tilting modules

T(\lambda)

of

SL_2

in characteristic

p

that occur in a tensor power of the natural module. The contribution of

T(\lambda)

consists of explicit small cyclic factors and of

2^{|I|}

large ones, where

I

is the set of positions of the nonzero base-

p

digits of

\lambda+1

below the leading one. The large ones are obtained by pooling explicit integers, built from binomial valuations and base-

p

carries, into a non-increasing sequence (Theorems I and J). The proof writes down explicit

\mathbb{Z}_p

-lattices for all tilting modules of

SL_2

, one move per base-

p

digit; it reduces each contribution to the Smith form of an explicit

2^{|I|}\times2^{|I|}

matrix, and computes that form as in our proof for the hypercube (The Dancing Sand Theorem), the case

p=2

. We also give closed formulas that need no pooling: for

p>d

with ballot numbers (B), for the exponent with every prime (C), and for the number of cyclic factors (D). This proves Conjecture I of version 3 of this paper. Every result of the paper, Theorems A–J, is checked in Lean 4 with Mathlib, for every prime, and the Lean project was rebuilt from its sources on GitHub's machines, on Mathlib's published build. This work has not been refereed by a human expert.

Provenance statement

The author, Rafael Amichis Luengo, does not write proofs. As in his earlier paper on the hypercube (The Dancing Sand Theorem), his part was to unblock the work: when the proof was stuck he gave an image in plain words, which was translated piece by piece into something that could be counted, and the translation was sealed before any measurement. Of the 27 images he gave for the Dancing Sand, 19 became the statement or the mechanism of a result and 6 served in part (record/THE_IMAGES.md of the Dancing Sand repository). The proofs of Theorems I and J reuse that machinery for odd primes: the unique cheapest matching of the natural minor is his «orphan», the floor is the flat that settles its account with its mirror, and the carries are his climb «in the air». The route to Theorems I and J began with the author. When Conjecture I was open, he insisted that the Dancing Sand had to hold the key to closing this problem entirely, because he knew what its images had solved. The auditor checked, and he was right: the key was the dance of the Dancing Sand Theorem, §3–§4. A constructor carried the dance to every odd prime, a cold reader checked the proof, and Lean checked it by machine. The dance itself was an auditor's proposal; the author's part here was knowing where to look, and the images whose mechanisms the proof reuses. The attached file HOW_THE_COLOURED_SAND_WAS_BUILT.md tells this story and says what belongs to whom. The text and the proofs were written by instances of Claude (Anthropic) working in separate roles from written missions: constructors that proposed and wrote proofs, an auditor that re-derived every step and ran checks with its own code, and cold readers with no access to the audits. The proofs of Theorems A–H were re-derived by an independent reader; the proofs of Theorems I and J were written by one reader, re-derived by a second and read cold by a third; the Lean certificate and version 5 were read cold by a fourth reader, and the changes of version 6 by a fifth. Every correction is listed in §13 of the paper. Computations were used as checks, never as steps of a proof. Every theorem of the paper, Theorems A–J, was proved in Lean 4 with Mathlib, for every prime, by Aristotle (Harmonic), from pieces written from the paper, compiled on the author's machine and audited: 36 pieces, 254 files, 46 787 lines, no sorry, standard axioms only; the whole project was rebuilt on GitHub Actions. Theorem I and the blocks of Theorem J are proved in Lean for explicit dance lattices; their identification with the tilting lattices is proved in the paper and not in Lean. No human expert has refereed this work. Zenodo record (paper, Lean certificate, Lean project, checks): https://doi.org/10.5281/zenodo.23327608 (version 6: https://doi.org/10.5281/zenodo.23400922). Repository with the full record: https://github.com/tretoef-estrella/coloured-sand-theorem. This TeX source is the source of version 6 of the paper, unchanged.

AI provenance filesHOW_THE_COLOURED_SAND_WAS_BUILT.md · 4 KB

Tools used

Anthropic: Claude Code
Lean: LeanVersion v4.28.0 (with Mathlib)
SageMath: SageMath
Harmonic: Aristotle
Python Software Foundation: Python 3 (NumPy)

References

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Harmonic, Aristotle, https://aristotle.harmonic.fun.link
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Version history

v1Submitted by Rafael Amichis LuengoInitial depositCurrentOct 15, 2026