The Functional Equation Δlog f = κf and Its Exact Reduction to Liouville and JT Constant-Curvature Geometry

This work presents a unified mathematical formulation of the functional equationΔ log f = κ f, expressed through the logarithmic field φ = log f and the functional curvature κ = e^(−φ) · Δφ. In two dimensions, the reparametrization f = e^(2u) transforms this relation into the Liouville equation, which characterizes conformal metrics of constant curvature. Replacing the Laplacian Δ with the two-dimensional d’Alembert operator □ yields the dynamical Jackiw–Teitelboim (JT) equation, showing that both stationary and time-dependent constant-curvature geometries arise from the same underlying field. The article develops a closed functional structure based on the set {f, φ, χ, ψ, κ}.The coherence flux is defined by χ = ‖∇φ‖, and the structural support is defined by ψ = χ² · e^(−φ). Both quantities are direct differential projections of φ. The structural identity f = χ² / ψ shows that observable amplitudes result from the balance between gradient intensity and internal structural resistance, without requiring additional assumptions. A key result is that the functional curvature κ(x) = Δ log f(x) / f(x) can be computed directly from experimental data whenever f represents a measurable density or intensity. This makes constant functional curvature a practical diagnostic of coherence in real physical systems. The article illustrates how this criterion appears in optical solitons, Laguerre–Gaussian modes, Thomas–Fermi Bose–Einstein condensates, and radial reductions of stationary black-hole geometries. The work also shows that the same logarithmic field φ that generates geometric curvature governs the informational structure of gradients through Fisher information. This reveals a common functional origin linking geometric organization, coherence, and gradient-based information in two-dimensional systems. Overall, the paper provides a self-contained and rigorous presentation of how curvature, coherence, information, and dynamics emerge from a single functional structure defined by φ = log f. The formulation relies solely on differential identities and reparametrizations and applies broadly across diverse physical scenarios where f is a positive, measurable field.