SENSITIVITIES IN COMPLEX-TIME FLOWS: PHASE TRANSITIONS, HAMILTONIAN STRUCTURE AND DIFFERENTIAL GEOMETRY

The candidate solution reproduces the paper’s main formulas for sensitivity and momentum, including Δz(z)=h(z)/h(z0)·Δz0 and p(z)=h(z0)/h(z)·p0, the conservation Δz·p≡Δz0·p0, and the momentum-sensitivity formula Δp(z)=((h′(z0)−h′(z))/h(z))p0Δz0+(h(z0)/h(z))Δp0, matching Eqs. (25), (23), and (32) in the paper, respectively . Geometrically, both show that for the metric g=(1/2)⟨·,·⟩/|h|^2 the real- and imaginary-time directions h and ih are geodesic (∇_h h=0, ∇_{ih} ih=0) and that the holomorphic splitting of sensitivities is parallel, as established via the paper’s Lemma 5.3/5.4 and Remark 5.5 . Both also identify flatness in complex-time coordinates (t1,t2), with constant metric coefficients and vanishing Christoffel symbols . The model’s proof strategy differs (Euler–Lagrange and direct complex identities vs. the paper’s Jacobian/connection computation) but yields the same conclusions. Minor issues in the paper are a missing explicit restriction to the zero-free set |h|>0 for the Riemannian metric and a factor-of-two inconsistency between the metric’s 1/2 normalization and the later statement gij,t=δij; the candidate explicitly notes the harmless time rescaling that resolves the normalization.