Global Dynamics of Ordinary Differential Equations: Wall Labelings, Conley Complexes, and Ramp Systems

The paper proves Theorem 15.0.1 by constructing a compatibility class of geometrizations G3 for the Janus complex that map specific (N−1)-faces to carefully defined internal/external transversal manifolds Mi_ε and Me_ε near equilibria, and by reducing the remaining faces to earlier F1/F2 transversality cases; the alignment is then verified case-by-case for all N ∈ N(F3) (see Theorem 15.0.1 and its execution in §15.3 for N=2 and N=3, including Defs. 15.3.1–15.3.2 and the role of Mi_ε, Me_ε, and S̄(ξ) surfaces) . By contrast, the candidate’s construction replaces all Janus faces by uniform ε-offsets of axis-aligned walls, justified via a global Lipschitz bound and presumed uniform wall-normal margins. This ignores the core F3 features: the internal/external Lyapunov-level manifolds used inside equilibrium cells and the F2 compatibility required for pseudo-opaque 2-cycles. In particular, many boundary faces of elements of N(F3) lie on S̄(ξ) and on Mi_ε/Me_ε, not on wall offsets, and are handled via Theorems 10.0.1, 13.0.1, and 15.1–15.2 in the paper, not by wall-offset arguments . The candidate’s approach therefore fails to establish alignment on precisely those F3-specific boundary components. The downstream consequences R1–R3 (Morse decomposition, equality of Conley indices, and validity of a connection matrix) are correctly stated but only hold once the alignment granted by the paper’s G3 is in place (cf. Chapter 3 and the Introduction) .