The sharp SAT/UNSAT phase transition in random ellipsoid fitting

Let x1,ldots,xn be independent standard Gaussian vectors in ℝd. An ellipsoid fit is a matrix S succeq 0 such that xi^top S xi =d for every i, so that all the points lie on the boundary of the centered ellipsoid x : x^top S x = d. Saunderson, Parrilo and Willsky conjectured that, as n,d → ∞, this semidefinite feasibility problem undergoes a sharp transition at n ~ d2/4. We prove this conjecture. If lim sup n/d2 = α^* <1/4, then, with probability tending to one, an ellipsoid fit exists; moreover, one can choose S with all eigenvalues in a fixed interval [λ_- , λ_+] subset (0,∞) depending only on α^*. Conversely, if lim inf n/d2 > 1/4, then, with probability tending to one, no ellipsoid fit exists, without any spectral restriction. Our proof builds on the Gaussian-equivalence framework developed by Bandeira and Maillard (2025) and closes the two gaps left open in their work: establishing exact fitting and removing the operator-norm constraint. On the satisfiable side, the new ingredients are a head-tail decomposition of the dual vector, exact correction of the sparse head constraints, and a Gaussian comparison principle for the low-influence tail. On the unsatisfiable side, we split a candidate into a low-rank spectral head and a Schatten-3 diffuse bulk, Gaussianize the bulk conditionally on the head, and apply a projected Gordon escape argument. The threshold is governed by the statistical dimension d(d+1)/4 of the positive semidefinite cone.