Differential-algebraic systems with dissipative Hamiltonian structure

The paper proves that an extended Hamiltonian DAE KP ż = LS z has differentiation index at most two and that any index-two algebraic constraints originate from singularity of P. It does so by deriving condensed forms that separate Dirac constraints (from K) and show index-two constraints linked to P’s singularity; e.g., z2 = 0 and z2 = K21 ż1 appear after block reductions, with the latter expressing an index-two relation and arising from the Lagrange side. The candidate solution reaches the same conclusions via left-right equivalences that put (K,L) into a block form, eliminate the Dirac constraints S2 z = 0, and reduce on the constraint manifold to a semi-explicit DAE with an invertible algebraic block D, implying index ≤ 2; it also correctly argues that index two requires P to be singular. The logical structure and transformations substantially mirror the paper’s condensed-form reductions, with no substantive contradictions.