Hyperbolicity cones: matrix descriptions, lifts, and obstructions

Source and verification

This article explains family 095 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Hyperbolic polynomials and convex cones

A homogeneous real polynomial $p$ is hyperbolic in a direction $e$ if $p(e)>0$ and every polynomial $t\mapsto p(te-x)$ has only real roots. Its closed hyperbolicity cone consists of those $x$ for which all these roots are nonnegative.

The determinant is the basic example. On real symmetric matrices, $p(X)=\det X$ is hyperbolic in the direction $I$, and its cone is the positive-semidefinite cone. Write $X⪰0$ for positive semidefiniteness and $X\succ 0$ for positive definiteness.

A spectrahedral cone has a description by a linear matrix inequality in its original variables:

$$K=\{x:A(x)⪰0\}.$$

A semidefinite lift allows auxiliary variables:

$$K=\{x:\exists y,A(x)+B(y)⪰0\}.$$

The second class is potentially larger. The existential quantifier takes a projection of a higher-dimensional feasible set. Exact equality matters: replacing a projection by its closure changes the assertion.

These definitions use convexity and linear algebra. The polynomial constructions below can be read as inputs without developing a separate algebraic-geometry theory.

🏷️ Three manuscripts with different conclusions

The family contains three logically compatible results:

• The September manuscript constructs a hyperbolicity cone that is not spectrahedral in its original coordinates. (OpenAI, 2026)
• One October companion gives an explicit exact semidefinite lift of that same cone. (OpenAI, 2026b)
• The other October manuscript constructs different hyperbolicity cones with no finite semidefinite lift. (OpenAI, 2026c)

The first statement rules out every original-variable matrix size, but does not rule out auxiliary variables. It therefore cannot alone establish the third statement.

🏷️ The explicit nonspectrahedral example

The September construction uses $X,Z\in {\mathrm{Sym}}^4$ and $y\in {\mathbb{R}}^3$. Thus the ambient dimension is $10+10+3=23$. A positive matrix-valued quadratic form $Q(y)$ is inserted into a linear map ${\Phi }_y$ on symmetric matrices, and the defining polynomial is

$$\begin{array}{c}p(X,Z,y)=\det M(X,Z,y),\\ M(X,Z,y)\\ =(\det X)Z-{\Phi }_y(\mathrm{adj}X).\end{array}$$

The adjugate makes this a polynomial even when $X$ is singular. The manuscript claims that it has degree $20$, is hyperbolic at $(I_4,I_4,0)$, and has a nonspectrahedral closed cone. It also removes a determinant factor to obtain degree $16$ with the same cone. No minimality of dimension or degree is asserted. (OpenAI, 2026a)

On the open region $X\succ 0$, its geometry has the more transparent description

$$Z⪰{\Phi }_y(X^{-1}).$$

Matrix inversion suggests a Schur-complement representation, but a positive quadratic matrix field need not possess the factorization that would make such a representation immediate.

The obstruction uses a nonnegative biquadratic form $b(z,y)$ with a rigid property: every bilinear form $\ell (z,y)$ satisfying ${\ell }^2\le b$ everywhere is zero. If an arbitrary matrix pencil represented the cone, rescaling and rank-one limits would produce kernel projections satisfying an exact norm identity. Differentiating those projections on a constant-rank neighborhood would generate a nonzero bilinear square dominated by $b$, a contradiction.

This excludes arbitrary pencils, rather than merely showing that the displayed polynomial is not itself a determinant of the desired kind.

🏷️ Why that example still has a lift

The exact-lift companion constructs a $100\times 100$ linear matrix inequality with $307$ auxiliary variables for the same cone. These are explicit construction sizes, not claims of optimality. The equality includes singular $X$ and does not require adding a closure afterward. (OpenAI, 2026b)

The method restricts $y$ to planes parametrized by an angle. On these slices, $Q$ has a finite trigonometric Gram representation, producing a smaller matrix inequality. A finite-dimensional positive functional packages the slice information into the final matrix.

The converse direction is essential. Positivity of a truncated functional does not automatically provide a representing measure for every desired purpose. The source proves a bounded-degree harmonic matching statement that supplies exactly the information needed for the cone. That proof establishes feasibility in both directions, including the boundary.

The result illustrates why obstruction to an original-coordinate matrix inequality and obstruction to all lifts require different arguments.

🏷️ The stronger no-lift construction

The other October manuscript starts from a different positive quadratic matrix field. An affine section and projection of its hyperbolicity cone gives a matrix epigraph involving a pseudoinverse:

$$\begin{array}{c}A⪰0,\mathrm{range}Q(y)\subseteq \mathrm{range}A,\\ t\ge \mathrm{tr}(A^{†}Q(y)).\end{array}$$

The range condition is indispensable at singular $A$; the pseudoinverse formula alone would allow points that are not limiting feasible epigraph data. (OpenAI, 2026c)

The source constructs a positive functional with a finite rank certificate and a local positivity obstruction. Under a hypothetical finite lift, matrix factorizations along a small parameter patch would impose a much stronger approximation property. The proof compares the order of the obstruction, of size ${\epsilon }^4$, with a proposed correction of order ${\epsilon }^{16}$. For small $\epsilon$, such a correction cannot remove the obstruction.

The finite-field computation is used as a certificate for a real rank statement, not as the definition of the real convex cone. Those certificate computations were not executed for this article. The source’s transfer from local positivity to the impossibility of every finite lift is the substantive research claim.

🏷️ Key steps and proof outlines

The inverse-matrix description explains hyperbolicity

On $X\succ 0$, the determinant construction becomes a matrix epigraph involving ${\Phi }_y(X^{-1})$. The adjugate identity $\mathrm{adj}X=(\det X)X^{-1}$ shows that clearing denominators recovers the polynomial displayed above. To prove hyperbolicity, the source factors the positive quadratic field through a Gram representation on suitable data and relates the one-variable determinant to a symmetric block pencil. Real symmetry supplies real roots, while polynomial identities extend the conclusion through singular parameter values.

Degree counting is exact: $\det X$ has degree four, the adjugate has degree three, and ${\Phi }_y$ is quadratic in $y$. Each entry of $M$ has total degree five, so its determinant has degree twenty. Removing a common determinant factor requires checking that the connected positive region, hence the closed hyperbolicity cone, is unchanged; factor cancellation by itself does not establish that geometric fact. (OpenAI, 2026a)

Rank-one limits test every possible original-variable pencil

Suppose a finite symmetric pencil represented the cone. First remove its common kernel and normalize its positive blocks. The source then rescales the pencil while preserving the same inverse-matrix inequality. To isolate rank-one directions $u,v$, use

$$X_t=v{v}^T+t(I-v{v}^T),Z_t=u{u}^T+t(I-u{u}^T),$$

with $u,v$ unit and $t\to \infty$. Their inverses approach the corresponding rank-one projections. This extracts an exact norm identity involving the pencil’s off-diagonal part and kernel projections.

On a constant-rank neighborhood, those projections can be differentiated. The tangent identity forces a nonzero bilinear form whose square is dominated everywhere by the chosen nonnegative biquadratic form $b(z,y)$. The source’s initial form lemma proves that no such dominated square exists, by evaluating at its zeros and then using the remaining coefficient constraints. The contradiction concerns an arbitrary hypothetical pencil, not merely failure to factor the particular displayed polynomial.

Angular slices admit Gram factors that the full field lacks

The exact-lift companion restricts $y$ to two-dimensional planes parameterized by an angle $\theta$. On each slice, the field has a finite trigonometric Gram representation. A Schur-complement pencil therefore represents the slice exactly. For invertible $X$, the block determinant identity is immediate; the source then checks the closed-cone statement at singular $X$ as well. (OpenAI, 2026b)

A finite positive functional packages the angular slice data. In one direction, an actual cone point defines an evaluation functional, making feasibility transparent. The reverse direction is harder: a positive truncated functional need not automatically be integration against a measure with all desired identities. The degree-two matching lemma supplies precisely the harmonic identities needed by the output coordinates and the cone inequality. Its construction rotates a Gram factor unitarily while preserving its product with its adjoint.

This proves exact projection of the lifted spectrahedron onto the cone, including boundary points. The explicit matrix size and auxiliary-variable count come from the finite spaces of trigonometric coefficients and moment constraints. No limiting closure is appended after projection.

A no-lift obstruction must survive arbitrary auxiliary variables

The stronger companion uses a different quadratic field and an affine section whose boundary contains the pseudoinverse epigraph. At singular $A$, the range condition is forced by finite limiting energy: a component of $Q(y)$ outside the range of $A$ would require an inverse contribution in a zero-eigenvalue direction. This is why $\mathrm{tr}(A^{†}Q(y))$ alone is insufficient. (OpenAI, 2026c)

The source constructs a Hankel-type positive functional on polynomial data with a controlled rank stratum. A finite rank certificate ensures that prescribed products remain independent and that the local moment deformation has the required directions. Under any finite semidefinite lift, analytic factorization on a small parameter patch would force those deformations to satisfy an additional approximation property.

The local obstruction has a negative defect of order ${\epsilon }^4$. Although coordinate perturbations are $O({\epsilon }^{16})$, their effect on the functional is amplified by matrix inversion: $A_{\epsilon }⪰a{\epsilon }^{4}I$ gives $\Vert A_{\epsilon }^{-1}\Vert =O({\epsilon }^{-4})$. Applying the inverse resolvent identity on both sides of the perturbation gives an $O({\epsilon }^8)$ functional error. The resulting defect is $-\kappa {\epsilon }^4+O({\epsilon }^8)+o({\epsilon }^4)<0$ for sufficiently small $\epsilon$, so the perturbation cannot repair it. This yields the contradiction independently of the proposed lift size. The finite arithmetic certificate establishes a rank input; the analytic patch and order comparison carry the passage from that input to exclusion of all finite lifts. The certificate computations remain attributed to the source, as stated earlier.

See Also

Projection-body extrema uses matrix normalization in another convex optimization problem. Entropy and isotropic constants also requires care with singular limits of positive matrix fields.

🏷️ Manuscripts and verification scope

• Hyperbolicity Cones Without Semidefinite Lifts — pinned manuscript and companion source.
• A nonspectrahedral hyperbolicity cone — pinned manuscript and companion source.
• An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope covers the September degree-twenty nonspectrahedral cone. It does not cover the later exact lift, the different no-lift construction, or the degree-sixteen reduction. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A nonspectrahedral hyperbolicity cone.

🐻  OpenAI 2026b. An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone.

🐻  OpenAI 2026c. Hyperbolicity Cones Without Semidefinite Lifts.