INTERACTING GEODESICS ON DISCRETE MANIFOLDS

The paper defines the evolution T = BA with A(x) = −x′ (partner across a wall) and B(+x) = −L^{k−l}(x), B(−x) = +L^{l−k}(x) where k,l count positive/negative particles over the same facet, and L is the left rotation on frames; see the model definition in Section 1.12 and surrounding text , the frame/partner setup in Sections 1.4–1.6 , and the summary statement (1.17.4) asserting “If X(x) = 0 mod (q+1) for all x∈P, then T^2(X)=X” . However, the paper merely states this fact without a written proof. The candidate solution supplies a short and correct proof: after A, each fiber’s signed excess is 0 mod (q+1), so B acts as a sign flip since L^{q+1} = id on each fiber; hence T = B∘A equals the partner pushforward τ_*, and T^2 = id because τ is an involution. This argument is fully consistent with the paper’s definitions and hypotheses, but the paper does not present it explicitly.