Symbolic Geometry of Phase Transitions - A Curvature Framework for Critical Phenomena

This manuscript presents a geometric reformulation of classical Landau mean-field theory for continuous phase transitions. We introduce a curvature–strain coordinate pair (κ, τ) as reduced descriptors of the mean-field free-energy landscape and define the diagnostic ratio Γ = κ / τ as a compact stiffness–deformation indicator along equilibrium branches. The model is expressed through an auxiliary quartic potential Φ ( k , τ ) = a κ 2 + b τ 2 − c κ τ + d τ 4 whose stability is governed by standard convexity and Hessian criteria. By completing the square and eliminating κ variationally, the formulation reduces to the usual τ-only Landau free-energy form and reproduces the corresponding mean-field criticality condition c ² = 4 ab . In this representation, loss of convexity in Φ provides a transparent geometric diagnostic for bifurcation and metastability boundaries, while the associated response denominators reproduce the familiar mean-field susceptibility divergence and the finite heat-capacity discontinuity ( α = 0 ) when mapped back to thermodynamic variables. We emphasize that the framework does not introduce new critical exponents or physics beyond the Landau mean-field fixed point; its purpose is interpretive, providing a concise curvature-space view of stability and universality. The domain of validity is delineated via the Ginzburg criterion, with renormalization-group effects identified as the source of deviations in the fluctuation-dominated regime. Keywords