Sharp Energy Barriers and Discontinuous Shape Selection in Elastic Cells

How much stored energy is necessary for an elastic cell to lose a deformation direction, and which shape first minimizes its area? For a triangle of equal Hookean links, we derive the complete minimum-area envelope and prove a discontinuous change from equilateral to isosceles optimal shapes. The global switch occurs at energy 0.08301573938 kL², strictly before the known local shear-softening threshold 3kL²/32. A sextic polynomial determines the global crossing exactly. For unequal lengths and stiffnesses, the collapse energy is the squared reference triangle slack divided by twice the sum of link compliances; a sharp slack bound and optimal stiffness allocation follow. Strictly convex nonlinear links reduce to three one-dimensional dual problems. A complementary directional compliance theorem gives the exact rank-loss barrier and attaining metrics for arbitrary positive quadratic metric energies, retaining anisotropy and a nonidentity reference metric. A barrier at the boundary of the calibrated region prevents unjustified constitutive extrapolation along trajectories. Below the corresponding collapse barrier, positive definite damping preserves orientation and restores the reference geometry of the exact models while exporting only their finite initial energy. Independent numerical optimizations and exact polynomial checks audit the results. A proposed planar experiment measures work separately from geometry. Established spectral, convex-optimization and local-bifurcation results are identified explicitly; the numerical outputs are synthetic and no physical experiment is reported.