We present the nonperturbative solution of the loop equation in quenched QCD (one quark loop in full gluon vacuum, including nonplanar graphs). This solution relies on a specific local minimum of the Plateau problem -- one that is additive over the closed parts of the bounding loop formed at self-intersections. This surface applies to large loops, leading to quark confinement via a factor $\exp(-κS[C])$ multiplying the perturbative Wilson loop $W_{pert}[C]$. Crucially, the 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 four dimensions, distinguishing the resulting spectrum from standard string models, which are stable only in higher embedding dimensions. We compute the high-energy meson spectrum resulting from this novel confinement mechanism. The result is a usual linear Regge trajectory with the slope $α' = \frac{1}{2 πσ}$ in our normalization of string tension $σ= 2 \sqrt{2} κ$. However, there are no string modes to be added to the spectrum in our solution, which amounts to the static linear potential for the quark pair. As we argue, the fluctuations of ``flux tube'' between quarks are accounted for in the gluon diagrams in $W_{pert}[C]$, and do not contribute to the confining force. This eliminates the problems of quantization of string in four dimensions.
