A closed minimal surface in the unit sphere has a squared second fundamental form — a measure of how much the surface curves away from being totally geodesic. Simon conjectured that these curvature values can’t be arbitrary: there are gaps, forbidden intervals where no minimal surface can live. The first gap, between 0 and 2/3, was established decades ago. The second gap is known. The third gap, in the interval [5/3, 9/5], remained open.

Ding, Ge, and Li close this third gap entirely, including the endpoint cases. Their tools: refined third-order Simons-type integral identities that capture curvature information invisible to second-order methods, and new lower bounds for higher-order curvature terms. At the endpoints 5/3 and 9/5, rigidity holds — any minimal surface with exactly that curvature value must be a specific known surface. In the interior of the interval, quantitative gap estimates sharpen the forbidden zone.

The phenomenon is discretization without discreteness. The curvature of a minimal surface in a sphere is a continuous quantity — it can in principle take any positive real value. But the constraint of minimality, combined with the compactness of the sphere, forces the realized values into a discrete-looking spectrum. The gaps are not imposed by a lattice or a quantization condition. They emerge from the interplay between the global constraint (minimality) and the ambient geometry (the sphere). Each gap proved is one more piece of evidence that the spectrum of minimal surface curvatures is sparser than the real line — structured by geometric rigidity rather than algebraic discreteness.