Minimizing foliations: counterexamples and common-volume rigidity

We study minimizing graph foliations for the periodic action
$\int(\frac12|\nabla u|^2-V(x,u))\,dx$ in arbitrary base dimension $n\ge2$.
We first construct nonconstant potentials with ordinary smooth minimizing foliations at every slope. The failure of closed rational holonomy can be placed at any prescribed rational vector.
We then prove an exact common-volume rigidity theorem: a prescribed set of at least two slopes forces $V_u=0$ under one common positive invariant volume if and only if it is not contained in a Euclidean sphere with rational centre. Every excluded configuration is realized within the ordinary all-slope class. Phase locking and a mean-removing shear yield the criterion, and a spectral--Poisson construction describes all compatible families. We obtain optimal finite tests and an intrinsic obstruction at the rational centre of a full-rank family.
The appendices establish rational-cell and quantitative rigidity results and extend the compatibility calculation to uniformly convex kinetics and nonlinear elliptic selection. The global implication from rational identity holonomy alone is not asserted.