1 · The essentials
Can every shape in a particular family defined by polynomials be described using a finite matrix condition, perhaps after adding extra variables? The main manuscript shows that the answer is no for some of these shapes, called hyperbolicity cones. For its examples, no exact description of this kind works, however many extra variables are added, as long as their number is finite. Allowing arbitrary real numbers in the description does not remove the obstruction. A Lean-formalized companion constructs a particular cone that cannot be described by a matrix condition directly in its original coordinates. Another companion shows that this particular cone can nevertheless be described after extra variables are added. The distinction between failing in the original coordinates and failing even with extra variables is therefore essential.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine trying to describe a shape on a table by casting the shadow of an object in a larger space. One companion cone cannot be described directly in the required way but can arise as such a shadow; the main manuscript shows that some cones cannot arise this way from any allowed finite-dimensional object.
Why it matters
The main manuscript rules out exact finite affine semidefinite lifts as a universal description of hyperbolicity cones. This gives a negative answer to the Projected Lax conjecture and, consequently, to the generalized Lax conjecture. The companion manuscripts also explain why ruling out a direct matrix representation is not enough to rule out a representation with auxiliary variables.
2 · Going further
The central question is whether every hyperbolicity cone is a spectrahedral shadow: whether membership can be expressed by requiring a finite real symmetric affine matrix expression to be positive semidefinite, with auxiliary variables allowed. The main manuscript shows that some closed hyperbolicity cones admit no such exact representation. Its obstruction allows arbitrary real coefficients and any finite number of auxiliary variables, rather than merely excluding a particular choice of coefficients or a particular lift size.
The Lean-formalized companion constructs a homogeneous polynomial of degree in real variables. Its hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality in the original coordinates. This supplies a formalized negative result for the geometric Generalized Lax conjecture. The supplied abstract states the construction and its nonrepresentation conclusion but does not describe the mechanism of the argument.
The third manuscript constructs an exact semidefinite lift of that same explicit -variable cone. The lift uses a homogeneous real symmetric pencil of size and auxiliary variables, and it represents the entire closed cone, including every point with singular . Thus this cone is nonspectrahedral but does have a semidefinite lift; it is not an example of the stronger obstruction asserted in the main manuscript. The abstracts establish how the claims relate, but do not supply the argument behind the main manuscript’s no-lift result.
Useful notions
- Homogeneous polynomials
- Convex cones
- Real symmetric matrices and positive semidefiniteness
- Linear matrix inequalities
- Projection and auxiliary variables
What was known before / what is new
Before
The source identifies the Projected Lax conjecture as the assertion that every hyperbolicity cone is a spectrahedral shadow. It also identifies the generalized Lax conjecture as asserting that every hyperbolicity cone is spectrahedral. No further historical background or earlier partial results are supplied.
New
The main manuscript shows that some closed hyperbolicity cones have no exact finite affine semidefinite lift at all. The Lean-formalized companion provides an explicit polynomial whose cone has no direct finite homogeneous matrix representation. The third manuscript shows that this explicit cone nevertheless admits an exact lift, separating the direct-representation obstruction from the stronger no-lift obstruction.
Limits and assumptions
The main negative result concerns some closed hyperbolicity cones, not all of them, and excludes exact finite affine semidefinite lifts. It does not state a conclusion about approximate or infinite-dimensional descriptions. The family has a Lean formalization, but the supplied verification scope is the nonspectrahedral construction in Manuscript 2, not the stronger no-lift result in Manuscript 1 or the lift construction in Manuscript 3. The lift sizes in Manuscript 3 are not claimed to be minimal. The abstracts identify no further open problem and do not provide enough detail to explain the underlying constructions.
Mini glossary
- Homogeneous polynomial —
- A polynomial whose terms all have the same total degree.
- Positive semidefinite —
- A real symmetric matrix is positive semidefinite when its quadratic form is nonnegative on every real vector.
- Linear matrix inequality —
- A condition requiring a symmetric matrix expression depending linearly or affinely on variables to be positive semidefinite.
- Spectrahedral —
- A set is spectrahedral when it has a description by a finite linear matrix inequality in its original coordinates.
- Semidefinite lift —
- A description of a set by a finite linear matrix inequality using auxiliary variables, which are then projected away.
- Spectrahedral shadow —
- A set obtained by projecting a spectrahedral set, equivalently a set admitting a semidefinite lift.
Keywords : Hyperbolicity cones · Spectrahedral shadows · Semidefinite lifts
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Hyperbolicity Cones Without Semidefinite Lifts
05/10/2026We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.
A nonspectrahedral hyperbolicity cone
24/09/2026We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.
An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone
05/10/2026We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X. Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.