A Textbook Case of Pentagram Rigidity

The paper proves the full equivalence in Theorem 1.2 by (i) explicitly computing T3 on the 4-fold symmetric family and identifying the invariant Ψ(x,y)=(x−y)(x^2−y^2−1)/(xy), (ii) analyzing the level curves E_λ of Ψ as nonsingular cubics (elliptic curves) with real locus two loops, (iii) showing T3^2 acts as a nontrivial translation in the flat (torus) coordinates, and (iv) applying a “minor subset” lemma for translations on tori to force exit from the convex region unless one lies on the dihedral loci (the diagonal Δ: x=y and the unit circle S^1), together with explicit 1D dynamics on those loci (on Δ: x↦(1+x)/(1+2x), fixing x=±1/√2) to finish both “if” and “only if” directions . By contrast, the model proves only the “if” directions on the two dihedral subloci and then asserts global features (e.g., a universal radial contraction |u_j|→1 and monotone growth of a “dihedral defect” δ) that are incompatible with the paper’s torus-translation picture: on E_λ with λ≠0, T3 (or T3^2) is an isometry/translation on a compact loop, so no global contraction/expansion of a non-constant observable (like |u| or |δ|) can occur for all points. Hence the model’s claimed global contraction and δ-expansion are false, and its proof is incomplete, while the paper’s proof is complete and correct .