A proof-based mathematical preprint reports that the free square contains infinitely many free extreme points at matrix level 2, with no two related by unitary equivalence. Their first-level matrix-convex hulls are exactly the ellipses inscribed in the square, giving the result a concrete geometric description.
The study concerns matrix convex sets, free spectrahedra and free extreme points. Its main question is how the abundance of free extreme points in bounded real free spectrahedra can be understood, particularly beyond simplices.
The result extends across polygons
The paper states the same kind of infinite-family result for every bounded convex d-gon with d at least 4. For each polygon in that class, its maximal matrix convex set contains infinitely many level-2 free extreme points modulo unitary equivalence.
The polygon construction combines the geometry of ellipses inscribed in the polygon with projective equivalence between quadrilaterals. The work uses geometric constructions and proofs rather than statistical estimation or sampling.
For a bounded convex d-gon with d at least 4, the polygon’s ordinary extreme points are not crucial free extreme points of its maximal matrix convex set.
A geometric lift preserves the key status
The preprint defines a graded lift by taking a first-level face, forming its matrix affine hull, and intersecting that hull with the surrounding closed matrix-convex set.
For a bounded real free spectrahedron with a g-dimensional first-level face in standard position, the induced set is itself a bounded free spectrahedron. Under those conditions, a point is a free extreme point in the induced set exactly when it is a free extreme point in the graded lift and in the original spectrahedron.
A broader corollary applies to any eligible g-dimensional face of a bounded real free spectrahedron: the free extreme points of the graded lift are exactly the free extreme points of the full spectrahedron.
Related statements for polytopes
For a closed bounded polytope with a two-dimensional face that is not a simplex, the extreme points of that face are not crucial free extreme points of the maximal matrix convex set. Under the same nonsimplicial-face condition, the maximal matrix convex set has infinitely many free extreme points modulo unitary equivalence.
The paper also gives an obstruction involving polar duality. If 0 lies in the interior of a polytope, the minimal matrix convex set over its polar is not a free spectrahedron.
The claims are mathematical, and bounded
These are theorem-level statements about specified mathematical structures, not measurements from an empirical sample. The results establish infinitude, but do not provide finite counts, growth rates or probabilistic estimates for the number of free extreme points.
The polygon theorem is stated for bounded convex d-gons with d at least 4. The authors state that the constructed extreme points do not have arbitrarily large size, and the paper works over the real field; corresponding conclusions for complex free spectrahedra or arbitrary matrix convex sets are not established.
The preprint does not settle whether bounded real free spectrahedra outside simplices can have only finitely many free extreme points. It also does not show that every bounded real free spectrahedron has infinitely many such points at level 2, or that the construction reaches arbitrarily large levels or sizes.
The manuscript is an arXiv preprint, version 1, dated 25 August 2026. The authors disclose using ChatGPT by OpenAI for exposition and parts of argument development, while stating that they independently verified the mathematics and take responsibility for errors.
Paper data and sources
Original title: Graded face lifts and free extreme points of free spectrahedra
Authors: Eric Evert, Jack Graham
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text