Geometric QCD I: The Hodge-Dual Surface and Quark Confinement

This is the first of two papers presenting a geometric framework for Planar QCD ($N_c \to \infty$). In this part, we establish the kinematic foundation of the theory by constructing the unique stable vacuum of the loop equation. We demonstrate that the Makeenko-Migdal loop equation admits a solution of the form $W[C] = W_{pert}[C] \exp{-\kappa S[C]}$, provided $S[C]$ is a specific minimal surface possessing a self-dual area derivative. We prove that such a surface exists and corresponds to the Hodge-dual projection of a minimal surface in $\mathbb{R}^3 \otimes \mathbb{R}^4$. Crucially, this confinement mechanism relies on the self-duality of the area derivative -- a property that exists exclusively in four dimensions. This geometric constraint ensures stability only in $D=4$, distinguishing the resulting theory from standard string models which require higher critical dimensions. We relate the string tension parameter $\kappa$ to the gluon condensate via the Operator Product Expansion. The dynamical quantization of the Fermi string on this rigid surface and the resulting meson spectrum are derived in the companion paper \cite{Migdal2026GeometricQCDII}.