Hyperbolicity Cones Without Semidefinite Lifts

1 Introduction

A real homogeneous polynomial p on a finite-dimensional real vector space is hyperbolic with respect to e if p(e)>0 and, for every x, all roots of t\mapsto p(te-x) are real. Its closed hyperbolicity cone is

Gårding proved that this is a convex cone [4]. The determinant on real symmetric matrices, with direction the identity, gives the positive semidefinite cone as a basic example. Hyperbolicity cones also support logarithmic self-concordant barriers, as shown by Güler [6], and hence form a natural setting for convex optimization beyond semidefinite programming.

A spectrahedron is a set defined by a finite affine linear matrix inequality. A spectrahedral shadow, or a set with a finite semidefinite lift, is a set S\subset\R^d of the form

where a and the common size of the real symmetric matrices are finite. Equality in this definition is exact; taking the closure of the displayed projection is not part of the definition. The generalized Lax conjecture asserts that every hyperbolicity cone is a spectrahedron. The weaker Projected Lax Conjecture, formulated by Netzer and Sanyal [17], asks whether every hyperbolicity cone is a spectrahedral shadow.

Thus both conjectures have a negative resolution. The assertion excludes every finite choice in (1.1), with arbitrary real coefficients and any number of auxiliary variables. The construction is an existence argument in high dimension, not a lower bound on the size of a particular representation.

1.1 Context and earlier obstructions

The original Lax question concerned homogeneous polynomials in three variables [13]. The determinantal theorem of Helton and Vinnikov [8], together with the equivalence explained by Lewis, Parrilo, and Ramana [15], gave an affirmative answer: a ternary hyperbolic polynomial has a definite symmetric determinantal representation. The distinction between a representation of a polynomial and a representation of its cone becomes essential in higher dimension. Brändén [2] constructed a real-zero polynomial for which no positive integral power has a definite determinantal representation. This does not exclude another polynomial with the same positivity region. Indeed, Kummer [9] proved that the related specialized Vámos polynomial has a spectrahedral hyperbolicity cone. Nor does failure of a specified sums-of-squares relaxation, such as those studied by Kummer, Plaumann, and Vinzant [11], exclude all finite lifts.

Positive representation results cover substantial classes. Brändén [3] proved spectrahedrality for elementary symmetric hyperbolicity cones. Building on the representability criteria of Helton and Nie [7], Netzer and Sanyal [17, Theorem 1.1] proved that a hyperbolicity cone is a spectrahedral shadow when each nonzero boundary point is a smooth point of its defining hyperbolic polynomial. Scheiderer [19] subsequently obtained second-order-cone lifts for hyperbolicity cones with Nash-smooth boundary. Recent preprints prove spectrahedrality for five-variable hyperbolic cubics [16] and for strictly hyperbolic polynomials [10]. These results concern special families or impose regularity hypotheses; they do not yield an exact finite lift for every hyperbolicity cone.

For general convex semialgebraic sets, Scheiderer [18] established nonexistence of semidefinite lifts by local positivity obstructions. His finite-dimensional sum-of-squares pullback criterion and local arguments explain how a putative lift constrains functions near a smooth point. He explicitly asked whether these methods could settle the hyperbolicity-cone question [18, Section 5.4]. The argument here is closely related to that approach, but tests analytic Gram identities directly by a finite positive moment functional. Moment and sum-of-squares semidefinite methods are classical [12]; our functional need not be integration against a measure. The general relationship between cone lifts and positive factorizations is developed by Gouveia, Parrilo, and Thomas [5]. We do not need to invoke their factorization theorem.

1.2 The construction and proof strategy

We begin with a matrix-valued homogeneous quadratic map Q:\R^m\to\Sym_r that is positive semidefinite at every real input. Here \Sym_r denotes the real symmetric r-by-r matrices. Section 2 associates to Q a homogeneous polynomial p_Q and proves its hyperbolicity by a symmetric determinant identity on each line in the input space. It also identifies an exact affine section and projection of its closed cone:

The symbol A^\dagger denotes the Moore–Penrose inverse. It is enough to choose Q so that E_Q has no finite lift.

The difficulty is to force such an obstruction while keeping Q(y) positive semidefinite for every y. Sections 3 and 4 address this constraint. Let H be the 36-dimensional space of quadratic forms in eight variables. Each linear functional y on quartic forms determines a Hankel matrix Y(y) on H by Y(y)(f,g)=y(fg). We construct a rank-twenty positive semidefinite Hankel matrix whose kernel has strong multiplication properties. After normalizing translations and dilations, its geometry yields a twenty-dimensional subspace W\subset H and a finite set of projective classes of nonzero symmetric forms on W. The class of every nonzero limit of inverses of compressions to W arising in the separation argument belongs to this set. The seed matrix is established by the finite-field certificate in Appendix A; the passage from its rank computations to a real positive semidefinite matrix is part of the proof.

For a sufficiently large integer N, finiteness lets us choose a tuple w=(w_1,\ldots,w_N)\in W^N with \sum_i w_i^TCw_i\ne0 for every representative C of one of these classes. We then choose U\subset w^\perp\subset H^N so that convergence to w of vectors (I_N\otimes Y(y))u, u\in U, would force one of these pairings to vanish. Thus w lies outside the closure of those Hankel images. A symmetric form g can be chosen nonnegative on every such image and negative at w. Using a basis matrix also denoted by U, we set

Nonnegativity on the Hankel images proves that Q is admissible. The construction uses m=330 and r=20N-1, where one may take N>10\cdot36^2.

Section 5 turns the negative direction of g into a local obstruction. Evaluation functionals at points (1,s), s\in\R^7, give a degree-four polynomial patch x(s) in E_Q. A positive definite moment functional \Lambda, applied to x(c+\eps s), produces points violating the trace inequality of E_Q by a negative term of order \eps^4. At the same time their A coordinates are bounded below by a positive multiple of \eps^4 I.

If a finite lift existed, semialgebraic selection would provide analytic lift coordinates and analytic Gram factors on an open part of this patch. Applying \Lambda to their Taylor polynomials almost preserves the lift inequality. A mixing argument on the common support of the feasible pencil values repairs the small error without a strict feasibility assumption. It yields feasible points within O(\eps^{16}) of the tested patch, too close to remove its order-four violation. The finite positive extension and finite-jet approximation lemmas isolate reusable parts of this obstruction. Together with the finite inverse-limit construction, they supply the proof of Theorem 1.1.

2 A determinant construction and its affine section

The first step is to place the obstruction in a tractable affine section. We prove hyperbolicity for every admissible quadratic matrix and identify the section exactly, including its singular boundary.

Let Q:\R^m\to\Sym_r have homogeneous quadratic entries and satisfy Q(y)\succeq0 for every y. Put n=r+1. For W=\left(\begin{smallmatrix}a&v^\top\\v&T\end{smallmatrix}\right) \in\Sym_n, define

The polynomial used throughout the paper is

The adjugate in this formula is the classical adjugate, so the formula is polynomial even when X is singular.

The Gram factorization in Proposition 2.1 is chosen separately for each y. In particular, (2.3) is not asserted to be a linear matrix inequality jointly in X,Z,y. The closed cone K_Q is convex by Gårding’s theorem [4].

For a positive semidefinite matrix A, let A^\dagger denote its Moore–Penrose inverse. Define

We will construct Q for which E_Q has no such lift. The rest of the argument therefore concerns (2.5), not the much larger determinant in (2.2).

3 Hankel matrices and finitely many inverse limits

We construct a positive semidefinite Hankel matrix of rank 20 and a 20-dimensional subspace W with a finiteness property: certain nonzero limits of inverses of compressions to W have only finitely many projective directions. The construction has two parts. A rank calculation, proved in Appendix A, supplies a point at which a kernel-projection map has sufficiently large derivative. We then divide out translations and dilations, choose a finite fiber of the resulting map, and use that fiber to classify the inverse limits.

3.1 The rank stratum and the seed matrix

Let x=(x_0,x_1,\ldots,x_7) and put

Subscripts denote homogeneous degree. Monomials are our coordinate bases, and these bases are orthonormal whenever we take an orthogonal complement or compression. Setting x_0=1 identifies H with polynomials of degree at most two in s=(s_1,\ldots,s_7). In this identification, L consists of the constant and linear terms and P consists of the quadratic terms. Write

Let \mathcal H\subset\Sym(H) be the space of Hankel matrices

Multiplication \Sym^2H\to\R[x]_4 is onto, so y\mapsto Y is injective and \dim\mathcal H=\binom{11}{4}=330. We also call Y a moment matrix. This terminology refers only to (3.2); no representing measure is assumed. In the chart x_0=1, the functional y acts on polynomials in s of degree at most four.

For subspaces of polynomial spaces, a product such as JH_0 denotes the linear span of all products of one element of each space. We use the following algebraic input.

The proof in Appendix A begins with two cyclic orbits of real points and deforms their moment matrix inside the rank-20 stratum. All rank conditions needed for that deformation are verified there by explicit minors.

Let \mathcal X be the set of projective classes [Y]\in\mathbf P(\mathcal H) such that \rank Y=d_*, \ker Y\cap L=0, and the projected kernel K_Y=\operatorname{proj}_P(\ker Y) satisfies the two multiplication conditions of Lemma 3.1. Positivity is not part of the definition of \mathcal X. Define

3.2 Removing translations and dilations

The fibers of \pi contain the orbits of affine translations and positive dilations in the s variables. We give explicit coordinates for removing these eight parameters. Because J_Y\cap L=0, its elements, in the affine chart, have a unique description

where b:K_Y\to\R[s]_1 takes values in homogeneous linear forms and a:K_Y\to\R is linear. These coefficient maps depend continuously and semialgebraically on [Y].

Pushforward of moments by s\mapsto s+t transforms a kernel polynomial by f(s)\mapsto f(s-t). Thus translation changes the coefficient maps by

where \partial_t p is the directional derivative of p in direction t. Pushforward by s\mapsto\lambda s, with \lambda>0, gives

Both actions are congruences on moment matrices. They preserve Hankel structure, rank, and \pi, and preserve positive semidefiniteness when it is present.

Equip the spaces of coefficient maps with the Euclidean inner products induced by the monomial bases. We normalize [Y] in two steps. First choose the unique translation for which b\perp\ran\Delta_{K_Y}. Explicitly, its parameter is

Next choose the unique positive dilation for which

For the translated coefficients, this amounts to solving \lambda^2\|b\|^2+\lambda^4\|a\|^2=1. The left side is strictly increasing from zero to infinity, by Lemma 3.3.

3.3 Selecting a finite fiber

We now show that the positive semidefinite part of \mathcal X has enough distinct projected kernels to avoid both infinite normalized fibers and limits at the boundary of \mathcal M. We use the standard dimension and fiber-dimension theorems for semialgebraic sets, as well as the fact that the frontier \overline S\setminus S of a semialgebraic set has smaller dimension than S; see [1]. A zero-dimensional semialgebraic set is finite.

At any T representing a point of \mathcal X, put J=\ker T and H_0=\{p:T(p,L)=0\}. The kernel of the derivative of the kernel-projection map, before projectivizing, is

Here the right side is an annihilator in \R[x]_4^*, identified with \mathcal H. To verify the identity, continue a vector j\in J along a differentiable curve of kernels. Its first variation j' must satisfy

The projected kernel has zero first variation precisely when the pure part of each j' lies in K. Subtracting an element of J then makes j' belong to L, without changing this equation. Thus the condition is Y'J\subset T(L). Conversely, that condition allows all the corrections to be chosen in L; it also implies Y'|_{J\times J}=0, so Lemma 3.2 gives an actual rank-stratum tangent. Finally, H_0=T(L)^\perp, which proves the annihilator description.

At the seed matrix of Lemma 3.1, the space in (3.8) has dimension 9. The rank stratum before projectivizing has dimension 194, so the derivative of \pi has rank 194-9=185. Projectivization removes one kernel direction, namely scaling, and leaves this derivative rank unchanged. Near the seed matrix, rank remains 20 and its nonzero eigenvalues remain positive. Its L block remains positive definite as well. Consequently the \pi-image of matrices with these positivity properties has dimension at least 185. It has dimension at most 185 because \pi factors through \mathcal M.

The seed’s final condition can also be retained in this neighborhood. Indeed, an invertible 321-row product minor for JH_0 remains invertible nearby. On the other hand, scaling, translations, and dilation always give nine independent tangent directions in (3.8): the eight affine directions have independent kernel variations by Lemma 3.3, whereas scaling has zero kernel variation. Thus \dim(JH_0)\le321 throughout this neighborhood.

Fix this T and K for the rest of the paper, and set

Thus H=W\oplus K is an orthogonal decomposition, \dim R=q=12, and \dim W=d_*=20.

Each normalized class [\bar Y]\in\mathcal M_K determines a nonzero symmetric matrix on W, up to scale. Let \bar A be the compression of a representative \bar Y to W. The kernel of \bar Y is a graph over K with lower terms in L, so W is a complement of that kernel. The same nondegeneracy argument used in Lemma 3.2 shows that \bar A is invertible, even if \bar Y is indefinite. Define C_{\bar Y}\in\Sym(W) to have block form

If its R,R block were zero, the invertible operator \bar A^{-1} would map R into L, which is impossible because 12>8. Thus \mathcal C is a finite nonempty set of projective classes of nonzero symmetric matrices on W.

3.4 The inverse-limit property

To connect the normalized fiber to arbitrary limiting sequences, we first need a rank correction that remains small even when some matrix entries diverge. All norms below are fixed Euclidean operator norms; convergence is equivalent in any choice of such norms.

We retain the positive semidefinite matrix T, the subspaces K, R=K^\perp\subset P, and W=L\oplus R, and the finite set \mathcal C from (3.10). These are the geometric data used in the next construction.

4 Separation from Hankel images

Retain the Hankel matrix T, the spaces K, R=K^\perp\subset P, and W=L\oplus R from the preceding section. In particular, \dim W=d_*=20 and H=W\oplus K orthogonally. Let \mathcal C be the finite set of projective classes of nonzero symmetric operators on W supplied by Lemma 3.7. We will construct a vector w and a subspace U such that w is outside the closure of all images of U obtained by applying one Hankel matrix componentwise. A quadratic form separating w from those images will then give the required positive semidefinite quadratic matrix Q.

Fix an integer N>10h^2, where h=\dim H=36, and put

We equip H^N with the sum of the Euclidean inner products on its factors. For a subspace F\subset H, write P_F for orthogonal projection onto F; the notation P_F^N denotes its componentwise action on H^N. The two subspaces in the third condition below will represent directions in which eigenvalues diverge or have finite nonzero limits. Their different projection constraints arise by taking limits of bounded Hankel images of unit tuples.

Fix w and U as in Lemma 4.1. Notice that the P^N component of w is nonzero: otherwise all its components would lie in L and could not span W. We now use the third property of the lemma to control every possible limit of Hankel images, including limits produced by unbounded matrices.

The preceding separation has a useful quadratic consequence. The set \mathcal S_U is closed under multiplication by any real scalar. Hence its closed set of directions on the unit sphere contains neither w nor -w. Compactness gives a number \delta>0 such that

Here the projections act on H^N, and w has unit length. Define the symmetric operator

For z\in\mathcal S_U, its quadratic form satisfies z^Tgz\ge\|z\|^2-\|P_{\R w}z\|^2\ge0, whereas w^Tgw=-1. We have therefore obtained

We can now define the quadratic matrix used in the rest of the proof. Use the monomial coordinates to identify \R[x_0,\ldots,x_7]_4^* with \R^m, where m=330. For such a functional y, let Y(y) be its Hankel matrix on H, so Y(y)(p,p')=y(pp'), and set

Choose a basis of U and also write U for the hN-by-r matrix whose columns are these basis vectors. Define

The entries of G(y) are linear in y, so Q is a real symmetric r-by-r matrix with homogeneous quadratic entries. For every a\in\R^r, the vector Ua belongs to the subspace U, and (4.6) gives

Thus Q(y)\succeq0 for every real y. The data retained for the next section are T, w, U, and g, together with this admissible map Q; the strict inequality at w in (4.6) will produce the obstruction to a finite semidefinite lift.

5 A local obstruction to a semidefinite lift

We prove that the quadratic matrix map constructed in Section 4 has an epigraph set E_Q with no finite semidefinite lift. The proof has two parts. First, a polynomial patch in E_Q admits a positive moment functional whose application at small scales violates the defining inequality of E_Q to order four. Second, any finite lift would force the resulting points to lie much closer to E_Q than this violation allows. The second part is a local sum-of-squares obstruction, in the sense of the methods developed by Scheiderer [18, Section 4]; here we give a direct proof using finite Taylor polynomials of Gram factors.

We retain the spaces H=L\oplus P, R=K^\perp\subset P, and W=L\oplus R, the positive semidefinite Hankel matrix T, and the data N,w,U,g of the preceding sections. In particular,

Write J=\ker T. We use that J\cap L=0 and that the projection of J to P is K. As in Section 4, the symbol U also denotes a 36N-by-r matrix whose columns form a basis of the subspace U\subset H^N. By Lemma 4.1, orthogonal projection induces an isomorphism

Moreover, w\in W^N has components spanning W, and w^Tgw<0. The admissible quadratic matrix map is Q(y)=U^TG(y)gG(y)U, as defined in (4.7).

To keep track of repeated blocks, for an operator S on H we write S_N=I_N\otimes S, an operator on H^N. The same convention applies to maps between subspaces of H. We order the coordinates of H^N as L^N\oplus P^N when using two-by-two block matrices. Subscripts LL, LP, and PP on an operator on H^N refer to these blocks.

5.1 A polynomial patch and its moment deformation

Let v(s)\in H, for s\in\R^7, be the column of homogeneous quadratic monomials evaluated at (1,s). Thus its entries have affine degrees zero, one, and two. Let y_s be evaluation at (1,s) on homogeneous quartics, and define

The moment identity gives G(y_s)=V(s)V(s)^T. In the following lemma, E_Q is the set defined in Proposition 2.2.

For a linear functional \Lambda on polynomials in seven variables through degree four, its moment matrix is the symmetric matrix indexed by monomials of degree at most two, with (\alpha,\beta) entry \Lambda(s^{\alpha+\beta}). Under the affine chart x_0=1, this is precisely a Hankel matrix on H. Applying \Lambda to a vector or matrix of polynomials always means applying it entrywise.

The preceding lemma supplies points outside E_Q whose violation has a controlled leading term. We next show that a semidefinite lift would force any such moment deformation of a polynomial patch to be arbitrarily well approximated, to a prescribed finite order, by feasible points.

5.2 Moment tests of analytic lift data

We first record why moments given only through degree four can be used to test Taylor polynomials of higher degree. The assertion concerns finite positive moment matrices and does not require a representing measure.

The next lemma provides simultaneous analytic lift variables and Gram factors. Only a nonempty open part of the polynomial patch is needed.

5.3 The contradiction

A A rank certificate for the Hankel seed

We prove Lemma 3.1. Recall that

where L consists of the quadratic monomials divisible by x_0. We seek a positive semidefinite Hankel matrix T of rank 20 whose restriction to L is positive definite. Its projected kernel K=\operatorname{proj}_P(\ker T) must have independent pairwise products and span all pure cubics after multiplication by pure linear forms. The remaining requirement is

inside the 330-dimensional quartic space.

Our starting matrix is the moment matrix of twenty points arranged in two cyclic orbits. At that matrix the last product space has dimension 302. We exhibit a tangent direction for which nineteen independent product directions appear to first order. All nonvanishing assertions are certified below by explicit minors over \mathbb F_7; we then explain why these certificates produce a real positive semidefinite matrix with the required properties.

A.1 Cyclic orbit matrices and the required rank increase

For the algebraic construction work first over \mathbb C, keeping x_0 as the homogenizing coordinate. Give the eight variables the respective weights

Let \zeta be a primitive tenth root of unity. For a=(1,a_1,\ldots,a_7), take the two orbits of (1,\ldots,1) and a under the diagonal action x_j\mapsto\zeta^{\omega_j}x_j. Give every orbit point weight 1/10, and let T(a) be the resulting quadratic moment matrix. If m is a quartic monomial, its moment is

In particular T(a)_{u,v}=t_a(uv) for quadratic monomials u,v. Formula (A.2) has integer coefficients in the parameters and also defines the matrices used in the finite-field calculation; no tenth roots of unity in \mathbb F_7 are needed.

Write I_i for the ordered quadratic monomials of weight i and set

The only nonzero block in column weight i of T(a) is C_{-i}(a)^TC_i(a). If every C_i(a) has rank two, then \rank T(a)=20 and its kernel is the direct sum of the nullspaces J_i=\ker C_i(a). Thus J=\ker T(a) has dimension 16.

Let D denote a matrix whose columns are the quartic products of a basis of J with a basis of H_0. For a Hankel variation T_1 satisfying

write D_1 for the first-order product variation obtained by differentiating the equations for bases of J and H_0. We give explicit formulas below and later realize T_1 as the velocity of a smooth arc. We shall establish, simultaneously with the two product conditions on K,

These ranks do not depend on the choices of differentiable bases. Indeed, a change of the derivatives of those bases changes D_1 by a matrix of the form AD+DB, which is removed by block row and column operations in the second matrix of (A.4).

We first record the upper bounds that give the target ranks their meaning. They apply in characteristic zero wherever T has rank 20, J\cap L=0, and K spans all pure cubics after multiplication by pure linear forms. The seven translations and the dilation in the chart x_0=1 have independent derivatives on the kernel graph, by the proof of Lemma 3.3, which uses only these hypotheses and works over \mathbb C as well as over \R. Recall the obstruction to a dependence involving dilation: it would allow a translation making the kernel equal to the homogeneous space K. Cubic spanning would then annihilate all chart moments of degrees three and four, forcing rank at most 8+1=9 instead of 20.

A translation or dilation preserves the projected kernel K. Continue each kernel vector with its highest homogeneous part fixed, so that its first derivative lies in L. Differentiating the kernel equation therefore gives T'(0)J\subset T(L), so its quartic functional annihilates JH_0. Scaling T gives one further independent annihilator, since scaling has zero kernel derivative whereas the eight affine motions have independent kernel derivatives. Hence, throughout the rank-20 locus under consideration,

At an orbit matrix, scaling belongs to the twenty-dimensional family obtained by varying the individual weights of its twenty points. Rank 20 means that their evaluation functionals are independent already on quadratics, and hence on quartics by multiplication by x_0^2. Each such quartic evaluation annihilates JH_0, and all weight variations have zero kernel derivative. The eight affine motions remain independent modulo these twenty directions. Thus at the orbit matrix

In particular, attaining rank at least 302 there is an open condition equivalent to equality.

A.2 Bases and first derivatives modulo seven

All entries in the remainder of this subsection and the next are in \mathbb F_7. Use the specialization

Index monomials by nondecreasing strings of variable indices, ordered lexicographically, both globally and within each weight. For example, 23 means x_2x_3 and 0011 means x_0^2x_1^2. Every row and column number below starts at one within its indicated block.

For C_i, take pivot columns

Use the nullspace basis having an identity matrix on the remaining coordinates, in their order. The quadratic blocks and resulting columns of J_i are as follows. A digit string in the third column records one entire column vector in the order of I_i.

Each specified two-column pivot is invertible. Thus every C_i has rank two and T has rank 20. The block on L is also invertible: in the order 00,01,\ldots,07 its determinant is 6.

For later calculations, the ordered list of weight-zero quartics is

Define T_1 by setting its quartic moments outside weight zero equal to zero and its moments in the order (A.9) equal to

The weight-zero block of products in \Sym^2J has seventeen columns. Its transpose times the vector (A.10) is the zero vector. Products of every other weight pair trivially with T_1, so (A.3) holds.

We specify a basis N_i for the weight-i part of H_0. There is one coordinate of L in each weight except 3 and 7. Denote its within-block index by L_i; thus L_0=1 corresponds to 00. If weight -i has no L coordinate, let N_i be the identity matrix. Otherwise let N_i be the nullspace basis of the single row T_{L_{-i},I_i}, using pivot L_i and identity on the free indices. Its pivot is nonzero because T|_L is invertible.

The following formulas specify derivatives of both bases. Put J_i' equal to zero off the pivot rows E_i, and set

If N_i is an identity matrix, put N_i'=0. In the other cases, put N_i' equal to zero off row L_i, and set

These are the differentiated kernel equations with the free coordinates held fixed. For J_i', tangency ensures that solving the two selected equations also solves all the equations TJ_i'+T_1J_i=0: the image of T_1J_i lies in the image of T, and the two selected rows give coordinates on that image in the relevant weight block. The formulas for N_i' follow by differentiating T(L,N_i)=0.

A.3 The explicit minors

We now give the minors that establish the two product conditions on K and the rank increase in (A.4). All product matrices use the following convention. In weight w, multiply a column u from block i by a column v from block w-i, adding

Order columns lexicographically by the weight i, the column number of u, and the column number of v. For products from \Sym^2J, retain only pairs for which the first index (i,\text{column number}) is at most the second. Thus each unordered product occurs once, with its ordinary polynomial coefficients.

Let G_1 be this matrix for J\cdot J, restricted to quartic rows with no zero index. It is the product matrix for \Sym^2K. Let G_2 be the product matrix of J with the seven coordinate vectors x_0x_j, 1\le j\le7, restricted to rows with exactly one zero index. Removing that zero identifies it with the product map from K times pure linear forms to pure cubics. Rows and columns are numbered anew after the restriction to a weight and to these row sets. In all the tables, a colon denotes the inclusive range.

The following square minors use all columns of G_1 and all rows of G_2, respectively. The specified omissions determine the other index set completely.

The number of columns of G_1 in weights 0,\ldots,5 is 17,11,16,11,16,11; the number of rows of G_2 is 8,8,9,8,9,8. Thus these determinants give precisely the full column and full row ranks needed in these weights.

For D, use first factors from J and second factors from H_0, with the preceding product order. Form D_1 by replacing the first factor by its derivative in (A.11), replacing the second by its derivative in (A.12), and adding the two resulting product columns. In weights 0,\ldots,5, the dimensions of D are respectively

The next table specifies a square submatrix E=D[R,S] in each weight by listing the omitted rows and columns; R,S always denote the retained indices.

To obtain the bordered minors, use the indices R,S in both diagonal positions of \left(\begin{smallmatrix}D&0\\D_1&D \end{smallmatrix}\right). Add rows R' in its second block row and columns S' in its first block column. Set A=D[R,S'] and B=D[R',S]. Eliminating the two copies of E leaves the corner

The corner matrices are the following, with rows separated by semicolons when their entries are listed.

In particular the resulting bordered minor has size 2|R|+1 in weight zero and 2|R|+2 in each of weights one through five. The bordered determinant is, up to its row and column ordering sign, (\det E)^2\det Z, as follows directly by eliminating the two diagonal copies of E. Thus the table proves the asserted bordered lower bounds without requiring an upper bound on the rank of D in the finite field.

For completeness, every entry in these tables can be checked with the following fixed elimination rule. At column n, interchange row n with the first row at or below it whose entry t in that column is nonzero. Divide the new row by t and subtract the appropriate multiples from every other row, always reducing modulo seven. The determinant is the product of the pivots, with one minus sign for each interchange. Appended columns give the inverse actions in (A.13) by the same rule. For the G_2 minor in weight zero, the signed pivots are

For E in weight zero they are

Together with the monomial order, the basis rules, and the moment vectors above, this specifies the finite calculation using matrices with at most thirty-five quartic rows in each weight.

A.4 Characteristic zero and simultaneous real data

We now pass from the explicit modular minors to actual orbit matrices and tangent directions in characteristic zero. First lift (A.7) to the same integer vector. The chosen pivots of the C_i, the entries used as pivots for N_i, and the 2\times2 blocks in (A.11) are all invertible modulo seven. Hence the corresponding bases and inverse formulas are defined over \mathbb Z_{(7)}, the rational numbers whose denominators are not divisible by seven.

There is one point to address before lifting the bordered minors: the first variation must remain tangent exactly, not just modulo seven. The weight-zero part of \Sym^2J has dimension seventeen. Its seventeen product columns are independent, since their restriction to the pure rows contains the nonzero 17\times17 minor of G_1 above. Consequently tangency of a weight-zero functional gives seventeen independent equations on its thirty-five moment coordinates. Choose as pivot coordinates the seventeen quartic rows of that minor. Lift the remaining eighteen coordinates of (A.10) to integers and solve these seventeen equations over \mathbb Z_{(7)}. The solution reduces to precisely the displayed moment vector. It therefore defines an exact characteristic-zero tangent whose basis derivatives reduce to (A.11) and (A.12). Every nonzero determinant in the preceding tables thus proves that its defining rational function is nonzero in characteristic zero.

It remains to obtain all weights and a real positive semidefinite matrix simultaneously. We spell out the genericity argument to make clear that the tangent need not be chosen separately for different weights. The permutation

negates every weight and preserves the family of orbit matrices. The nonzero minors for weights 1,2,3,4 therefore also give nonempty rank conditions for weights 9,8,7,6, respectively.

More formally, take the Zariski open subset of the parameter space \mathbb C^7 on which all C_i have rank two, T|_L is invertible, and the seventeen weight-zero products are independent. This is a nonempty irreducible parameter space. The solutions of the seventeen tangency equations form a rank-eighteen algebraic vector bundle over it: locally an invertible seventeen-column minor solves for seventeen moment coordinates in terms of the other eighteen. Its total space is irreducible. Each condition certified above is a nonempty Zariski open subset of this space, as is its image under the weight-negating permutation, after restricting to the common nonempty parameter domain. Their finite intersection is nonempty. The same reasoning for the minors of G_1 and G_2, which depend only on a, gives both product conditions on K.

The resulting lower bounds in weights 0,\ldots,9 are

Here 136=\dim\Sym^2K and 84 is the dimension of the pure cubic space. The first two rows prove the required injectivity and surjectivity. Equation (A.6) then turns the third row into equality, and the fourth row gives (A.4).

Impose now the real structure in which x_0,x_1 are real and the pairs (x_2,x_3), (x_4,x_5), (x_6,x_7) are conjugate. Its fixed parameter set

is a real form of \mathbb C^7 and is Zariski dense. One way to see this is to write each conjugate pair as (u+iv,u-iv): this is an invertible complex linear change of coordinates from a real coordinate space, and a polynomial vanishing on that real space is zero. The weight-zero tangent equations at a fixed real parameter are preserved by the real structure. Their solution space is therefore the complexification of its real fixed subspace, which is likewise Zariski dense. A nonempty open condition in the tangent bundle is attained over a nonempty open set of parameters, and within each such fiber it is an open condition on a linear space. We can thus choose simultaneously a parameter in the real form and a real tangent satisfying every rank condition in (A.15).

For these parameters all twenty orbit points are real in the corresponding real coordinates. Indeed multiplication by \zeta^{\omega_j} respects the conjugate pairs and multiplies the weight-five coordinate by \pm1. Since every point has positive weight 1/10, the moment matrix is real positive semidefinite. Its rank is 20, and its nonsingular restriction to L is positive definite. Choose ordinary real coordinates on this fixed real form, keeping x_0 and the pure-variable span, and relabel them x_0,\ldots,x_7. The induced polynomial change of coordinates preserves L\oplus P and multiplication, and hence the total product and first-variation ranks above. Thus these are real Hankel data in the conventions of Section 3.

A.5 Deformation to the seed matrix

We have obtained a real positive semidefinite orbit matrix T and a real Hankel tangent T_1 satisfying (A.4). Independence of the products in \Sym^2K implies independence of those in \Sym^2J by taking highest chart degree. Lemma 3.2, or its Schur-complement proof, therefore shows that the rank-20 Hankel locus is smooth near T, with tangent condition (A.3). There is a real smooth arc T(t) on this locus with T(0)=T and T'(0)=T_1. For sufficiently small t, its twenty nonzero eigenvalues remain positive, its restriction to L remains positive definite, and the two product conditions on K(t) remain valid.

Choose smooth local kernel and H_0 bases along the arc and let D(t) be their product matrix. Apply fixed invertible row and column operations that put D(0) in rank normal form

Writing the transformed derivative in the same blocks as \left(\begin{smallmatrix}A&B\\C&F\end{smallmatrix}\right), elementary block elimination gives

Thus \rank F\ge19. Eliminate the invertible leading 302\times302 block of D(t). Its remaining Schur complement is tF+o(t), since both off-diagonal blocks vanish at t=0. A nonzero 19\times19 minor of F remains nonzero after division by t^{19} for every sufficiently small nonzero t. Hence \rank D(t)\ge321 there. The upper bound (A.5) applies along the arc and gives equality. Taking one such T(t) proves Lemma 3.1.

Reproducible calculation. The accompanying source file verify_certificate.py reconstructs the monomial blocks, bases, differentiated products, and every displayed minor using exact integer arithmetic modulo seven. It also verifies the ranks in weights six through nine at the very same specialization (A.7) and tangent (A.10). This supplies an independent reproduction of the finite calculation; the explicit minors and the characteristic-zero argument above establish the lemma without relying on the program.