CONTINUATION SHEAVES IN DYNAMICS: SHEAF COHOMOLOGY AND BIFURCATION

The paper’s Theorem 9.10 states exactly the obstruction the model proves: if Λ is a contractible manifold and H^k(Λ; A_φ) ≠ 0 for some k>0, then a bifurcation point exists. The paper’s proof proceeds by contradiction: no bifurcation implies stability at every point (Def. 9.1), which by Lemma 9.3 makes A_φ locally constant; on a simply connected, locally path-connected space (e.g., a contractible manifold) every locally constant sheaf is constant (Prop. 9.6); for a constant sheaf on a locally contractible space, sheaf cohomology agrees with singular cohomology (Prop. 9.8), hence vanishes in positive degrees on a contractible manifold, contradicting H^k ≠ 0. The candidate’s argument follows the same chain: no bifurcation ⇒ local conjugacy to a constant family via the conjugacy invariance framework (Thm. 8.7) ⇒ A_φ locally constant ⇒ constant on contractible Λ ⇒ higher cohomology vanishes ⇒ contradiction. Hypotheses and technicalities (e.g., component-preserving conjugacies, compact metric phase space) match the paper’s setup. Therefore, both are correct and essentially the same proof. See the paper’s statements of Def. 9.1, Lemma 9.3, Prop. 9.6, Prop. 9.8, and Thm. 9.10 for the exact ingredients used , and the construction of A_φ via pullback and booleanization is precisely as in Section 8 . The conjugacy invariance used by the model is Theorem 8.7 .