Rational-slope rigidity with identity holonomy through finite-amplitude resonant cells

We distinguish ordinary and identity-holonomy minimizing foliations for
$u''=-V_u(x,u)$ on $\T^2$.  Under the ordinary deck-invariant definition,
$V(u)=-\frac12\sin^4(\pi u)$ admits minimizing foliations of every rotation
number although $V_u\not\equiv0$; its zero-rotation holonomy is nontrivial.
If, for each rational rotation, there exists a minimizing foliation with
identity primitive holonomy, we prove $V_u\equiv0$ by reducing to, and
classifying, phase-flat gauge-fixed rational cells of a periodic quadratic
Lagrangian.  The phase-space completion is obtained directly from order,
compactness, circle translation numbers, and area preservation.