The Cortical Informational Field Theory (CIFT) models mesoscopic cortical dynamics through a stochastic Ginzburg–Landau equation. Here we implement the geometrically correct operator — the Laplace–Beltrami operator on the cortical surface — and ask whether the qualitative phase structure of CIFT survives on the folded cortical manifold. Validated on a unit sphere (eigenvalue error <0.7%, curvature correction scaling (ξ/R)² with r=0.99), then implemented on fsaverage5 (10,242 vertices) with spectrally calibrated diffusion. The Laplace–Beltrami formulation preserves all qualitative features — phase transition, C2_emerg as order parameter, the C1→C2_emerg cascade — while introducing cortical renormalization: critical point shifts upward ~68%, emergent heterogeneity suppression is amplified, and multi-attractor dynamics for C1 emerge. ABIDE I multisite data provide an empirical calibration anchor.