Triangle solitaire

The paper proves a complete orbit classification for triangle solitaire via the filling map ϕ, a unique decomposition of ϕ(P) into non-touching triangles, and the excess e(P), culminating in canonical representatives vi+P_{ni,ki} with sum identities (Theorems 1–3) . The candidate solution reaches the same classification and invariants, but by a different normal-form procedure: (i) fill a boundary line of each component and (ii) monotonically raise interior pegs to a right-justified, upward-closed configuration matching P_{n,k}. This provides an alternative potential-function argument to the paper’s figure-guided “merge lines, fetch excess, push right” method. The only substantive gap is in Phase I: when attempting to fill the rightmost hole x on the boundary, the argument implicitly assumes the left neighbor y of x can be realized concurrently with the vertex u above it; this requires an explicit lemma (or a reference to the paper’s merging argument) to ensure such a sequence of triangle moves exists without reducing the already-filled suffix. With this standard patch, the model’s proof aligns with the paper’s result.