On Geometry, Arithmetics and Chaos

The paper defines a “Milnor fiber” for the dynamical system via the level set V(x,α)=η⃗ with ∥η⃗∥=η and uses Sard/Thom–Mather/Ehresmann ideas to claim local triviality along strata and introduces boundary, spherical, and tubular chaos; it then states the Classification Theorem (Theorem 6) essentially matching the model’s classification, but it provides no complete proof and contains a dimensional inconsistency just before Theorem 4 (n vs n−1) and significant gaps (e.g., reliance on unproven lemmas and external results) . By contrast, the model gives a coherent proof outline: it reduces to f=∑V_i^2, uses regular values and boundary transversality, and applies Ehresmann’s theorem to show triviality off a discriminant, yielding exactly the three-mode classification by the nature of the singular set (isolated/non-isolated/none) and locating all changes on the (n−1)-dimensional Milnor fiber; this aligns with the paper’s stated result and clarifies missing hypotheses .