A PERSISTENTLY SINGULAR MAP OF Tn THAT IS C2 ROBUSTLY TRANSITIVE BUT IS NOT C1 ROBUSTLY TRANSITIVE.

The paper constructs an endomorphism H = F ∘ f on T^n starting from a linear expanding map A, with a carefully engineered critical set S_f that is a codimension-one submanifold, predominantly fold points, and a robust unstable cone-field; then a local diffeomorphism F flattens the image of S_f near f(p) so that H(S_H) locally lies in a hyperplane. It proves: (i) non-collapse of open sets in a C^2-neighborhood (Theorem 4.3) and uses cone expansion plus a C^0-covering argument to obtain C^2-robust transitivity (Theorem 4.4); and (ii) a C^1-small perturbation near a fold collapses an open set into an invariant codimension-one subset, destroying transitivity (Lemma 4.4 and Theorem 4.2). These steps appear explicitly in the text and proofs, e.g., the construction H = F ∘ f and SH = Sf (Remark 4.2) with the flattening near f(p) (Corollary 4.5) ; the C^1-local flattening near fold points (Lemma 4.4) and the non-robustness in C^1 (Theorem 4.2) ; non-collapse in C^2 (Theorem 4.3) and the robust transitivity argument via Lemmas 4.6–4.8 (Theorem 4.4) ; robustness of unstable cones (Lemma/Proposition 3.1) and the Jacobian structure (Eq. (3.2)) ; classification of fold points in the critical set (Theorem 3.4 and Cor. 3.2) . The candidate solution outlines the same blueprint: build f with a persistent critical set and robust unstable cones; compose with a small diffeomorphism to control the image of the critical set; prove non-collapse in C^2 and use cone expansion plus a covering property for transitivity; and use a C^1-local flattening near a fold to kill transitivity. Minor naming differences (an auxiliary diffeomorphism Ξ in the candidate, versus F in the paper) do not affect substance. One proof step that would benefit from more detail in the paper is the third bullet of Lemma 4.8, where the C^0 argument ensuring g^m(B_j) = T^n is sketched via cellular approximation and boundary proximity; strengthening this part would improve rigor, but it does not undermine the overall construction and conclusions.