Nonuniqueness of Model Potentials and Weak Geodesic Rays

We establish two nonuniqueness phenomena on ${\mathbb{C}\mathbb{P}}^1$. First, distinct normalized positive-mass model potentials can have the same non-pluripolar Monge–Ampère measure. By the contact-set formula, their zero-contact sets agree ${\omega }_{\mathrm{F}\mathrm{S}}$-almost everywhere. Moreover, both have zero Lelong number at every point and identical multiplier ideal sheaves at every positive scale. Second, we construct distinct maximal test curves whose zero-contact sets agree pointwise at every parameter. Their inverse Legendre transforms are distinct bounded weak geodesic rays from zero with the same pointwise right initial tangent.