Bifurcation formula for transition paths in stochastic dynamical systems by spectral flow

The paper explicitly proves the σ–bifurcation (spectral–flow) decomposition sf(Iσ(0)) = sf(A(σ,1)) + sf(a(σ)) under the endpoint compatibility m+(a(σi)) = n(σi) by a two–parameter homotopy in (σ,r) combined with a clean index identity m−(xσ,Tσ) = m−_{Tσ}(xσ) + n(σ) and a separate treatment of the scalar path a(σ) (Theorem 4.4). This is clearly documented via equations (3.12), (3.25)–(3.26), Lemma 4.2, Lemma 4.3, and the final statement (4.9) in the PDF . By contrast, the candidate solution hinges on an unsubstantiated Step 2: it asserts that for r>0 the Schur complement Sσ(r)=Cσ(r)−Bσ* Aσ(r)−1 Bσ is a uniformly positive scalar so that sf(Iσ^r)=sf(I_{T,σ}^r). The paper does not assume nor prove such a positivity for Sσ(r); indeed, Sσ(r) depends on Aσ(r)−1 and need not stay positive along σ under the paper’s hypotheses (see the block structure I(r)=[A(r) B; B* C(r)] in (3.10), where only C(r)>0 is immediate) . The paper instead avoids this pitfall by introducing the coupling parameter ϵ in Iσ(r,ϵ), computing sf in ϵ to obtain the {0,1}-correction n(σ), and then invoking homotopy invariance (Theorem 3.2) to assemble the decomposition rigorously . Therefore, while the candidate’s final formula matches the paper, the presented proof is not valid under the stated assumptions.