Functional Coherence, Constant Curvature, and the Liouville Equation in Two Dimensions

This work examines a mathematical structure common to effective two-dimensional systems described by a sufficiently regular positive scalar field f > 0. It is shown that, in two dimensions, the introduction of the logarithmic potential φ = log f naturally leads to the scalar quantity κ = (Δ log f)/f, which satisfies a structural identity valid independently of any dynamical assumption. This identitary level is completely general, follows solely from the positivity of f and the elliptic geometry of two dimensions, and does not, by itself, constitute a physical law. The original contribution of the article lies in the explicit and operational formulation of functional coherence as an independent physical hypothesis, defined in a measurable and falsifiable manner as the spatial homogenization of κ in a specified norm. In contrast to approaches that directly postulate the Liouville equation or constant curvature as starting assumptions, this work demonstrates that these structures arise only as necessary consequences when functional coherence is satisfied, and not as prior postulates. Under this single additional hypothesis, it is shown that the structural identity necessarily closes to the Liouville equation Δφ = κ₀ e^φ, implying a conformal metric of constant Gaussian curvature. This implication is established through independent geometric, variational, and spectral stability arguments, with regularity assumptions, stability criteria, and explicit failure conditions carefully stated. Outside the coherent regime, spatial variability of κ prevents any global geometric closure, requiring descriptions that remain dependent on the specific microscopic mechanism. The results are strictly restricted to two dimensions and to stationary or dissipative-relaxation regimes, and no extrapolation to higher dimensions or general dynamics is proposed. The originality of the work therefore does not reside in the Liouville equation itself, nor merely in identifying common patterns across 2D systems, but in identifying functional coherence as a minimal, operationally testable, and model-independent structural criterion that determines when Liouville solutions and constant-curvature geometries become mathematically unavoidable.