Global analysis of a conformal phase–metric flow with coercive potential

We study a conformal phase–metric flow on a closed surface (Σ,g0), where a metric g=e^{2u}g0 and a phase field φ:Σ→R evolve by the L^2-gradient flow of an energy combining Dirichlet terms, a coercive potential V(φ), and a geometric coupling G(u,φ,∇u,∇φ) inspired by the Phase–Metric Flows (PMF) programme. Under structural assumptions ensuring coercivity and strong parabolicity in a small-coupling regime, we prove global existence of classical solutions and uniform H^1-type bounds. The energy is a strict Lyapunov functional, converges along trajectories, and every solution admits a non-empty ω-limit set consisting of stationary states. When V and G are real-analytic and a Łojasiewicz–Simon inequality holds at an equilibrium, we obtain convergence to equilibrium with an explicit algebraic rate, and exponential decay in the non-degenerate case. We also analyse a curvature-coupled model G ~ λ φ R(g), derive explicit coercivity and parabolicity thresholds, and outline numerical scenarios based on a variational semi-implicit scheme. This conformal model provides a controlled two-dimensional laboratory for global phase–metric coupling, as an analytic foundation for the broader PMF programme rather than a definitive phase–metric theory. Positioning and status (December 2025)This preprint belongs to the PDE-controlled technical core of the Phase–Metric Flows (PMF) programme (discrete → 1D → 2D conformal prototypes). After this release, the programme enters a temporary pause while a journal article is prepared based on the analytic components. No algorithmic/computational claim is established in the present paper; conceptual notes (ICA / Bohr-type coupling / Meta-Geodesy) are programmatic and intentionally separated from the results proved here. The model choice is guided by minimal-order geometric coupling and by explicit coercivity/parabolicity thresholds enabling full PDE control.