Remarks on rigidity properties of conics

The paper’s Theorem 1 proves that if the local “polar-duality” associated with a strictly convex smooth curve γ preserves incidence (i.e., B lies on the chord of contact from A implies the chord from B passes through A), then γ is an arc of a conic; the proof is carried out via equiaffine normalization, explicit Taylor expansions, and the conclusion that the affine curvature is locally constant, hence γ is a conic . By contrast, the model’s solution posits a local correlation T in the dual plane and asserts, without sufficient justification, that it is a polarity and that the fact “all tangents to the dual curve are self-conjugate for T” forces the dual curve to lie on a dual conic. Two key gaps are: (i) no rigorous derivation that the locally defined incidence-reversing map is a projective correlation/polarity; and (ii) the nontrivial step from “every tangent line to C is self-conjugate” to “C is an arc of a conic.” Hence, the paper is correct while the model’s proof is incomplete.