Context and earlier obstructions

The original Lax question concerned homogeneous polynomials in three variables (Lax 1958). The determinantal theorem of Helton and Vinnikov (Helton and Vinnikov 2007), together with the equivalence explained by Lewis, Parrilo, and Ramana (Lewis et al. 2005), 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 (Brändén 2011) 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 (Kummer 2016) 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 (Kummer et al. 2015), exclude all finite lifts.

Positive representation results cover substantial classes. Brändén (Brändén 2014) proved spectrahedrality for elementary symmetric hyperbolicity cones. Building on the representability criteria of Helton and Nie (Helton and Nie 2010), Netzer and Sanyal (Netzer and Sanyal 2015, 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 (Scheiderer 2025) subsequently obtained second-order-cone lifts for hyperbolicity cones with Nash-smooth boundary. Recent preprints prove spectrahedrality for five-variable hyperbolic cubics (Netzer 2026) and for strictly hyperbolic polynomials (Kummer and Netzer 2026). 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 (Scheiderer 2018) 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 (Scheiderer 2018, sec. 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 (Lasserre 2001); 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 (Gouveia et al. 2013). We do not need to invoke their factorization theorem.

The construction and proof strategy

We begin with a matrix-valued homogeneous quadratic map \(Q:\mathbb R^m\to\mathop{\mathrm{Sym}}_r\) that is positive semidefinite at every real input. Here \(\mathop{\mathrm{Sym}}_r\) 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: \[E_Q=\{(A,t,y): A\succeq0,\ \mathop{\mathrm{ran}}Q(y)\subseteq\mathop{\mathrm{ran}}A, \ t\ge\mathop{\mathrm{tr}}(A^\dagger Q(y))\}.\] 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 6; 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 \[G(y)=I_N\otimes Y(y),\qquad Q(y)=U^\top G(y)gG(y)U.\] 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 \((1,s)\), \(s\in\mathbb R^7\), give a degree-four polynomial patch \(x(s)\) in \(E_Q\). A positive definite moment functional \(\Lambda\), applied to \(x(c+\varepsilon s)\), produces points violating the trace inequality of \(E_Q\) by a negative term of order \(\varepsilon^4\). At the same time their \(A\) coordinates are bounded below by a positive multiple of \(\varepsilon^4 I\).

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 \(O(\varepsilon^{16})\) 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.

The rank stratum and the seed matrix

Let \(x=(x_0,x_1,\ldots,x_7)\) and put \[H=\mathbb R[x]_2=L\oplus P, \qquad L=\langle x_0^2,x_0x_1,\ldots,x_0x_7\rangle, \qquad P=\mathbb R[x_1,\ldots,x_7]_2.\] 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 \[ h=\dim H=36,\qquad d_*=20,\qquad k=h-d_*=16, \qquad q=d_*-\dim L=12. \tag{7}\]

Let \(\mathcal H\subset\mathop{\mathrm{Sym}}(H)\) be the space of Hankel matrices \[ Y(p,p')=y(pp'),\qquad p,p'\in H, \qquad y\in\mathbb R[x]_4^*. \tag{8}\] Multiplication \(\mathop{\mathrm{Sym}}^2H\to\mathbb R[x]_4\) 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 (8); 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 6 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 \(\mathop{\mathrm{rank}}Y=d_*\), \(\ker Y\cap L=0\), and the projected kernel \(K_Y=\operatorname{proj}_P(\ker Y)\) satisfies the two multiplication conditions of Lemma 4. Positivity is not part of the definition of \(\mathcal X\). Define \[\pi:\mathcal X\longrightarrow\mathop{\mathrm{Gr}}(k,P), \qquad \pi([Y])=K_Y.\]

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 \[ J_Y=\{p(s)+b(p)(s)+a(p):p\in K_Y\}, \tag{10}\] where \(b:K_Y\to\mathbb R[s]_1\) takes values in homogeneous linear forms and \(a:K_Y\to\mathbb R\) 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 \[ b_t(p)=b(p)-\partial_t p, \qquad a_t(p)=a(p)-b(p)(t)+p(t), \tag{11}\] where \(\partial_t p\) is the directional derivative of \(p\) in direction \(t\). Pushforward by \(s\mapsto\lambda s\), with \(\lambda>0\), gives \[ b\longmapsto\lambda b, \qquad a\longmapsto\lambda^2 a. \tag{12}\] 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 \(b\perp\mathop{\mathrm{ran}}\Delta_{K_Y}\). Explicitly, its parameter is \[t=(\Delta_{K_Y}^*\Delta_{K_Y})^{-1}\Delta_{K_Y}^*b.\] Next choose the unique positive dilation for which \[ \|b\|^2+\|a\|^2=1. \tag{13}\] 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 6.

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 (Bochnak et al. 1998). 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 \[ \{Y'\in\mathcal H:Y'(J)\subset T(L)\} =(JH_0)^\perp. \tag{14}\] Here the right side is an annihilator in \(\mathbb R[x]_4^*\), 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 \[Y'j+Tj'=0.\] 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 5 gives an actual rank-stratum tangent. Finally, \(H_0=T(L)^\perp\), which proves the annihilator description.

At the seed matrix of Lemma 4, the space in (14) 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 (14): the eight affine directions have independent kernel variations by Lemma 6, 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 \[ R=K^\perp\subset P,\qquad W=L\oplus R. \tag{15}\] 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 5 shows that \(\bar A\) is invertible, even if \(\bar Y\) is indefinite. Define \(C_{\bar Y}\in\mathop{\mathrm{Sym}}(W)\) to have block form \[ C_{\bar Y}= \begin{pmatrix}0&0\\0&(\bar A^{-1})_{R,R}\end{pmatrix} \quad\hbox{on }W=L\oplus R, \qquad \mathcal C=\{[C_{\bar Y}]:[\bar Y]\in\mathcal M_K\}. \tag{16}\] 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 \(W\).

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 (16). These are the geometric data used in the next construction.

A polynomial patch and its moment deformation

Let \(v(s)\in H\), for \(s\in\mathbb R^7\), 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 \[ V(s)=I_N\otimes v(s)\in\mathbb R^{36N\times N},\qquad M(s)=U^TV(s)\in\mathbb R^{r\times N}. \tag{28}\] The moment identity gives \(G(y_s)=V(s)V(s)^T\). In the following lemma, \(E_Q\) is the set defined in Proposition 3.

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.

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.

The contradiction

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 \[ \omega=(0,5,1,9,2,8,4,6)\pmod {10}. \tag{45}\] 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 \[ t_a(m)= \begin{cases} 1+m(a),&\operatorname{wt}(m)=0,\\ 0,&\operatorname{wt}(m)\ne0. \end{cases} \tag{46}\] In particular \(T(a)_{u,v}=t_a(uv)\) for quadratic monomials \(u,v\). Formula (46) 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 \[C_i(a)= \begin{pmatrix} 1&\cdots&1\\ u_1(a)&\cdots&u_{|I_i|}(a) \end{pmatrix}, \qquad (u_1,\ldots,u_{|I_i|})=I_i.\] 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 \(\mathop{\mathrm{rank}}T(a)=20\) 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 \[ T_1|_{J\times J}=0, \tag{47}\] 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 \(K\), \[ \mathop{\mathrm{rank}}D=302, \qquad \mathop{\mathrm{rank}}\begin{pmatrix}D&0\\D_1&D\end{pmatrix}\ge623 =2\cdot302+19. \tag{48}\] 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 (48).

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 6, which uses only these hypotheses and works over \(\mathbb C\) as well as over \(\mathbb R\). 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, \[ \mathop{\mathrm{rank}}D\le330-9=321. \tag{49}\]

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 \[ \mathop{\mathrm{rank}}D\le330-20-8=302. \tag{50}\] In particular, attaining rank at least \(302\) there is an open condition equivalent to equality.

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 \[ a=(1,6,6,0,3,0,2,5). \tag{51}\] 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 \[ E_i=(1,2)\quad\text{except that}\quad E_0=E_8=(1,3). \tag{52}\] 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 \[ \begin{gathered} 0000,0011,0023,0045,0067,0126,0137,0225,0334,0447,\\ 0466,0556,0577,1111,1123,1145,1167,1244,1257,1346,\\ 1355,2233,2247,2266,2345,2367,3356,3377,4446,4455,\\ 4567,4777,5557,5666,6677. \end{gathered} \tag{53}\] Define \(T_1\) by setting its quartic moments outside weight zero equal to zero and its moments in the order (53) equal to \[ (2,0,0,0,3,6,1,4,1,6,0,3,0,5,2,2,0,0,4,0,6, \underbrace{0,\ldots,0}_{14}). \tag{54}\] The weight-zero block of products in \(\mathop{\mathrm{Sym}}^2J\) has seventeen columns. Its transpose times the vector (54) is the zero vector. Products of every other weight pair trivially with \(T_1\), so (47) 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 \[ (J_i')_{E_i}= -\bigl(T_{E_{-i},E_i}\bigr)^{-1} (T_1)_{E_{-i},I_i}J_i. \tag{55}\] 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 \[ (N_i')_{L_i}= -\bigl(T_{L_{-i},L_i}\bigr)^{-1} (T_1)_{L_{-i},I_i}N_i. \tag{56}\] 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 \(T(L,N_i)=0\).

The explicit minors

We now give the minors that establish the two product conditions on \(K\) and the rank increase in (48). 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 \[u_{jj'}v_{kk'} \quad\text{to row}\quad \operatorname{sort}(jj'kk').\] Order columns lexicographically by the weight \(i\), the column number of \(u\), and the column number of \(v\). For products from \(\mathop{\mathrm{Sym}}^2J\), 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 \(\mathop{\mathrm{Sym}}^2K\). 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 (55), replacing the second by its derivative in (56), and adding the two resulting product columns. In weights \(0,\ldots,5\), the dimensions of \(D\) are respectively \[35\times48,\quad31\times42,\quad35\times47,\quad 31\times43,\quad35\times47,\quad31\times42.\] 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 \[ \begin{split} Z={}&D_1[R',S']-D_1[R',S]E^{-1}A -BE^{-1}D_1[R,S']\\ &\hspace{27mm}+BE^{-1}D_1[R,S]E^{-1}A. \end{split} \tag{57}\] 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 (57) by the same rule. For the \(G_2\) minor in weight zero, the signed pivots are \[(-2,-2,-5,-1,4,-1,1,5).\] For \(E\) in weight zero they are \[ \begin{gathered} 1,4,1,4,-4,-1,-4,-5,-5,-4,-4,6,-4,4,6,-3,\\ -3,-1,6,6,4,5,-5,2,1,5,2,5,6,4,4,4. \end{gathered} \tag{58}\] 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.

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 (51) 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 (55) 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 \(\mathop{\mathrm{Sym}}^2J\) 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 (54) 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 (55) and (56). 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 \[(x_2,x_3),\quad(x_4,x_5),\quad(x_6,x_7) \quad\text{interchanged, with }x_0,x_1\text{ fixed},\] 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 \[ \begin{array}{c|rrrrrrrrrr|r} w&0&1&2&3&4&5&6&7&8&9&\text{sum}\\\hline \mathop{\mathrm{rank}}G_1&17&11&16&11&16&11&16&11&16&11&136\\ \mathop{\mathrm{rank}}G_2&8&8&9&8&9&8&9&8&9&8&84\\ \mathop{\mathrm{rank}}D&32&28&32&29&32&28&32&29&32&28&302\\ \mathop{\mathrm{rank}}\left(\begin{smallmatrix}D&0\\D_1&D\end{smallmatrix}\right) &65&58&66&60&66&58&66&60&66&58&623 \end{array} \tag{59}\] Here \(136=\dim\mathop{\mathrm{Sym}}^2K\) and \(84\) is the dimension of the pure cubic space. The first two rows prove the required injectivity and surjectivity. Equation (50) then turns the third row into equality, and the fourth row gives (48).

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 \[a_1\in\mathbb R,\qquad a_3=\overline{a_2},\quad a_5=\overline{a_4},\quad a_7=\overline{a_6}\] 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 (59).

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.

Deformation to the seed matrix

We have obtained a real positive semidefinite orbit matrix \(T\) and a real Hankel tangent \(T_1\) satisfying (48). Independence of the products in \(\mathop{\mathrm{Sym}}^2K\) implies independence of those in \(\mathop{\mathrm{Sym}}^2J\) by taking highest chart degree. Lemma 5, or its Schur-complement proof, therefore shows that the rank-\(20\) Hankel locus is smooth near \(T\), with tangent condition (47). 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 \[D(0)=\begin{pmatrix}I_{302}&0\\0&0\end{pmatrix}.\] Writing the transformed derivative in the same blocks as \(\left(\begin{smallmatrix}A&B\\C&F\end{smallmatrix}\right)\), elementary block elimination gives \[\mathop{\mathrm{rank}}\begin{pmatrix}D(0)&0\\D'(0)&D(0)\end{pmatrix} =604+\mathop{\mathrm{rank}}F.\] Thus \(\mathop{\mathrm{rank}}F\ge19\). 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 \(\mathop{\mathrm{rank}}D(t)\ge321\) there. The upper bound (49) applies along the arc and gives equality. Taking one such \(T(t)\) proves Lemma 4.