Creating hyperbolic-regular singularities in the presence of an S1-symmetry

The paper proves, for its explicit S1-invariant family Ft, that 2-, 3-, and 4-stacked hyperbolic-regular fibres occur, twisted stacked tori do not appear in this particular family, and k ≤ 13 when t1 ≠ 0. These claims are stated in Theorem 1.3 and supported by detailed constructions and the upper-bound argument in Theorem 6.8 (which bounds the number of hyperbolic points in a reduced fibre by 12, implying k ≤ 13 via the loop-chain correspondence) . The model reaches the same end-conclusions for this family but makes two substantive errors: (i) it wrongly asserts that twisted stacked tori cannot occur in any proper S1-system, contradicting the paper’s general bouquet–mapping-torus correspondence (and the explicit twisted example illustrated in Figure 2.2) ; and (ii) it miscounts the bound by claiming “at most 13 saddles,” whereas the paper proves “at most 12 hyperbolic points,” which is the tight ingredient leading to k ≤ 13 . The existence examples for k = 2, 3, 4 in the paper (Examples 6.2, 6.3, 6.6) directly align with the part of the model’s solution that asserts such examples exist in the given family .