The Flapping Birds in the Pentagram Zoo

The paper proves all three claims of Theorem 1.1—strict star-shapedness of birds, the nesting Δk(P) ⊂ PI, and the invariance Δk(Bk,n) = Bk,n—via a coherent geometric framework built on the “soul” S (intersection of oriented (k+1)-diagonal half-planes) and “feathers,” plus a degeneration argument and a duality-based factorization of Δk. Specifically: Theorem 3.1 shows S has nonempty interior, S ⊂ PI, and every x ∈ S is a strict star-center (establishing Statement 1) ; Theorem 4.1 controls intersections of feathers and implies Δk(P) is embedded and lies inside PI (Statement 2) ; Statement 3 follows first from Δk(Bk,n) ⊂ Bk,n via the Degeneration Lemma together with χk ∘ Δk = χk, and then from Δk−1 = Dk+1 ∘ Δk ∘ Dk+1 (on PolyPoints/PolyLines) plus planarity of Dk+1(P) on birds to obtain Δk−1(Bk,n) ⊂ Bk,n, yielding equality (Section 6) . By contrast, the model’s solution depends on several incorrect or unproved assertions: (i) it assumes all 2n lines Aj := Pj,j+k+1 and Bj := Pj+1,j−k are pairwise non-parallel “by k-niceness,” which is not part of the k-nice definition (only four specific lines per index are required to be distinct) ; (ii) it identifies K(P) = ⋂j(HjA ∩ HjB) with the interior of Δk(P) and then infers star-shapedness from sector separation without supplying the necessary combinatorial/angle control (the paper instead uses a Helly/interlacing-diagonals argument to control configurations and relies critically on n > 3k, cf. Lemma 3.6) ; (iii) it asserts “on the open set of k-nice polygons the higher pentagram map Δk is invertible with inverse Δ−k,” whereas the paper’s correct inversion uses projective duality and the factorization Δk−1 = Dk+1 ∘ Δk ∘ Dk+1, with additional work to keep planarity and niceness on birds (Section 6.2) . The model’s path-lifting and invariance arguments therefore lack essential ingredients that the paper provides (e.g., Theorem 4.1’s embeddedness/nesting and the degeneration/duality machinery), and several key steps are unsupported or false. The paper’s statements and proofs match the theorem’s claims, and the triangulation/spiral-path structure further corroborates nesting across iterates (Section 7) .