Solitaire of Independence

The paper rigorously proves that for the triangle solitaire on Z^2 every finite pattern P can be transformed, in polynomial time, into a canonical normal form that is a disjoint union of non-touching translates of P_{n,k}, with a move sequence of cubic length in |P|. It also gives explicit constructive procedures and tight cubic bounds for worst-case orbits. The candidate solution reaches the same normal form with an O(|P|^3) bound via a different constructive approach (base seeding, helper-pair boundary walking, staircase escorts). While largely consistent with the paper’s framework, the model leaves some steps informal (creation/control of the exterior helper pair, independence of components during the whole run, and a minor adjacency convention), but these are plausibly repairable without changing the main conclusion.