Discrete gradient methods for port-Hamiltonian differential-algebraic equations

The paper rigorously proves the local canonical form (Theorem 5.16) and its pHDAE consequence (Corollary 5.17) using a two–step construction: (i) a constant-rank, analytic reduction U⊤EΠ = [Ir E12; 0 0] (Corollary F.2) and (ii) a diffeomorphism built via a first-order PDE Dx2ψ1 = (Dx1ψ1)E12 that zeroes the off-pivot block, yielding Ẽ = [Ip 0 0; 0 0 0; 0 E32 I] and a specified Hamiltonian H̃1; see Theorem 5.16 and the Appendix F proof sketch that implements these steps and then derives (41a)–(41b) . The candidate’s outline omits the essential PDE-based construction and contains a false claim that if ∇H(x⋆)=0 then H∘φ is locally constant; it also does not justify how left-multiplication alone enforces the required zero structure without the right-side Jacobian change. The paper’s argument is therefore correct and complete, while the model’s is flawed/incomplete (e.g., it relies on Step 1 and Step 3 that do not hold as stated) .