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Divisibility of the rectangular sandpile limit

Abstract

The direct limit of the rectangular sandpile groups with Shkolnikov’s signed reflected tiling maps is divisible: every element has a multiple-root at a later rectangular stage.

Mikhail Shkolnikov, Algebraic Limits of Sandpiles, arXiv:2506.09912v3 (2 September 2025), Section 3, p. 8. The paper states verbatim: “Let Γ(p, q) be the portion of the square lattice given by the intersection with an open rectangle (0, p) × (0, q), where p and q are positive integers.” It defines the rectangular monomorphism by the signed reflected formula and, after Proposition 5, states verbatim: “For instance, one may expect that it is a rational subtorus or a divisible group.”

Definition 1.1 (The circle-valued carrier).

Formalization. D5/S3/StatisticalMechanics/Sandpiles/ShkolnikovSandpileLimitDivisible.T (✓ std3).

Citation. Mikhail Shkolnikov (2025). Algebraic Limits of Sandpiles. DOI: 10.48550/arXiv.2506.09912. URL: https://arxiv.org/abs/2506.09912v3.

Commentary.

The carrier is the additive circle ℝ/ℤ.

Definition 1.2 (The open rectangular domain).

Formalization. D5/S3/StatisticalMechanics/Sandpiles/ShkolnikovSandpileLimitDivisible.InRect (✓ std3).

Citation. Mikhail Shkolnikov (2025). Algebraic Limits of Sandpiles. DOI: 10.48550/arXiv.2506.09912. URL: https://arxiv.org/abs/2506.09912v3.

Commentary.

InRect p q z means that both integer coordinates of z lie strictly between the corresponding rectangle boundaries.

Definition 1.3 (Rectangular sandpile functions).

Formalization. D5/S3/StatisticalMechanics/Sandpiles/ShkolnikovSandpileLimitDivisible.IsSandpile (✓ std3).

Citation. Mikhail Shkolnikov (2025). Algebraic Limits of Sandpiles. DOI: 10.48550/arXiv.2506.09912. URL: https://arxiv.org/abs/2506.09912v3.

Commentary.

A rectangular sandpile function vanishes outside the open rectangle and satisfies the four-neighbour harmonic equation at every point inside it.

Definition 1.4 (The reflected one-dimensional coordinate).

Formalization. D5/S3/StatisticalMechanics/Sandpiles/ShkolnikovSandpileLimitDivisible.unfold (✓ std3).

Citation. Mikhail Shkolnikov (2025). Algebraic Limits of Sandpiles. DOI: 10.48550/arXiv.2506.09912. URL: https://arxiv.org/abs/2506.09912v3.

Commentary.

The unfolded coordinate subtracts an even tile index and reflects an odd tile index, using integer division and remainder.

Definition 1.5 (The signed reflected embedding).

Formalization. D5/S3/StatisticalMechanics/Sandpiles/ShkolnikovSandpileLimitDivisible.embed (✓ std3).

Citation. Mikhail Shkolnikov (2025). Algebraic Limits of Sandpiles. DOI: 10.48550/arXiv.2506.09912. URL: https://arxiv.org/abs/2506.09912v3.

Commentary.

The embedding extends a function by zero and evaluates it at the reflected coordinates, multiplied by the sign determined by the two tile indices.

Definition 1.6 (Divisibility of the direct limit).

Formalization. D5/S3/StatisticalMechanics/Sandpiles/ShkolnikovSandpileLimitDivisible.claim (✓ std3).

Citation. Mikhail Shkolnikov (2025). Algebraic Limits of Sandpiles. DOI: 10.48550/arXiv.2506.09912. URL: https://arxiv.org/abs/2506.09912v3.

Commentary.

For every positive rectangle, harmonic function, and positive integer k, a later rectangle contains a harmonic function whose k-fold sum is the signed reflected image of the original function.

Theorem 1.7 (Divisibility theorem).

Proof. Machine-checked in Lean as D5/S3/StatisticalMechanics/Sandpiles/ShkolnikovSandpileLimitDivisible.result (✓ std3). ∎

Resolves. Problems/arxiv-2506-09912-shkolnikov-sandpile-limit-divisible (proved) by D5/S3/StatisticalMechanics/Sandpiles/ShkolnikovSandpileLimitDivisible.result.

Source. Repository-derived.

Acknowledgement. Mikhail Shkolnikov (2025). Algebraic Limits of Sandpiles. DOI: 10.48550/arXiv.2506.09912. URL: https://arxiv.org/abs/2506.09912v3.

Commentary.

The reversible row recurrence closes on a finite denominator subgroup. Its period supplies a taller rectangle, and multiplying the lifted rows by k recovers the signed reflected embedding of the original harmonic function.

References