A height in \(\mathrm{Aff}(q)\) agrees with one affine function of the picture at every body, and so on every closed neighbourhood; it lies in \(L(q)\). The map \((\alpha , \beta , \gamma ) \mapsto (\alpha x_w + \beta y_w + \gamma )_w\) is linear, with image \(\mathrm{Aff}(q)\). At an admissible picture it is injective, since an affine function that vanishes at three non-collinear picture points of a closed neighbourhood is zero. So \(\dim \mathrm{Aff}(q) = 3\), and \(\dim L(q) \geq 3\) by the containment.

Admissibility reads the picture at the two ends of each edge and on each closed neighbourhood, and the lifting space on each closed neighbourhood; all of these consist of bodies of \(G\).

As \(z \in L(q)\), there are \(a, b, c \in K\) with \(z_w = a x_w + b y_w + c\) for every \(w\) in the closed neighbourhood of \(v\). For each such \(w\), the point \(p_w\) is therefore the image of the homogeneous picture point \((x_w, y_w, 1)\) under the linear map \((\xi , \eta , \omega ) \mapsto (\xi , \eta , a\xi + b\eta + c\omega , \omega )\) from \(K^3\) to \(K^4\). The four points \(p_{a_v}, p_{b_v}, p_{c_v}, p_w\) lie in its image, of dimension at most three. So they are linearly dependent, and \(n_v \cdot p_w = \det [p_{a_v}, p_{b_v}, p_{c_v}, p_w]\) vanishes.

The hinge at \(uv\) of the panel-hinge framework that defines the rank of \(p\) is the meet of the planes \(p_u^{\perp }\) and \(p_v^{\perp }\). It is the image of \(p_u \vee p_v\) under the polarity (lemma 25.9), an invertible linear map of the screw space over every field. So the two rigidity matrices have the same rank (lemma 28.35). Over an admissible picture, adjacent picture points differ. A relation \(\lambda \, p_u + \mu \, p_v = 0\) then has \(\lambda + \mu = 0\) in the last coordinate and \(\lambda \, (q_u - q_v) = 0\) in the first two, so \(\lambda = \mu = 0\). Hence \(p_u\) and \(p_v\) are linearly independent, and their join is nonzero.

The three chosen configuration points at \(v\) are linearly independent, since their picture points are, so \(n_v \neq 0\). And \(n_v\) is orthogonal to \(p_w\) for every \(w\) in the closed neighbourhood of \(v\) (lemma 30.7).

The two ends of an edge \(uv\) lie in the closed neighbourhoods of both \(u\) and \(v\). So \(p_u\) and \(p_v\) are orthogonal to \(n_u\) and to \(n_v\): the line \(p_u \vee p_v\) lies in both panels, and it passes through both points by construction. Taking \(w = v\) gives the incidence \(p_v \cdot n_v = 0\), and \(p_v \neq 0\) since its last coordinate is \(1\). Adjacent points are linearly independent, and every hinge is nonzero (lemma 30.9). By the same lemma the rank of \(F_p\) is the rank of \(p\).

Suppose that \(z \in L(q)\). Admissibility gives at each body \(v\) three members of its closed neighbourhood with non-collinear picture points. Their configuration points are linearly independent, as their picture points are, so their plane normal is nonzero. It is orthogonal to the points of the whole closed neighbourhood (lemma 30.7). So these points lie in a plane.

Conversely, suppose that the points of every closed neighbourhood lie in a plane. At a body \(v\), let \(n \neq 0\) be orthogonal to them, so that \(n_0 x_w + n_1 y_w + n_2 z_w + n_3 = 0\) for every \(w\) in the closed neighbourhood of \(v\). If \(n_2\) were zero, \((n_0, n_1, n_3)\) would be orthogonal to the homogeneous picture points of the three members that admissibility gives, which are linearly independent, and \(n\) would vanish. So \(n_2 \neq 0\), and solving for \(z_w\) shows that \(z\) agrees with an affine function of the picture on the closed neighbourhood. Hence \(z \in L(q)\).

If \(z_w = a x_w + b y_w + c\) on the closed neighbourhood of \(v\), the vector \((a, b, -1, c)\) is orthogonal to its points. The points of the three members above are linearly independent, so the vectors orthogonal to them form a line. Hence every nonzero vector orthogonal to the closed neighbourhood is a multiple of \((a, b, -1, c)\), and its height coordinate is nonzero: the plane is not vertical.

The hinge at an edge \(uv\) with normals \(p\) is the image of the line \(p_u \vee p_v\) under the polarity (lemma 25.9). So the rigidity matrix with normals \(p\) has the rank of the body-hinge framework whose hinge at \(uv\) is \(p_u \vee p_v\), as in the proof of lemma 30.9, and likewise for \(p'\). Both ends of an edge of \(G\) are bodies of \(G\). So \(p'_u \vee p'_v = g(p_u) \vee g(p_v)\) is the image of \(p_u \vee p_v\) under the invertible linear map of the screw space induced by \(g\), and the two ranks agree (lemma 28.35).

Write \(a_w = \alpha x_w + \beta y_w + \gamma \). The linear map \((\xi , \eta , \zeta , \omega ) \mapsto (\xi , \eta , t\zeta + \alpha \xi + \beta \eta + \gamma \omega , \omega )\) of \(K^4\) is invertible, since \(t \neq 0\). At every body \(w\) it maps the point \((x_w, y_w, z_w, 1)\) of \((q, z)\) to the point \((x_w, y_w, t z_w + a_w, 1)\) of \((q, tz + a)\). So lemma 30.13 applies.

A height \(z\) lies in \(L(q)\) exactly when every body \(v\) has a vector \(h_v\) with \(z_w = h_v \cdot (x_w, y_w, 1)\) for every \(w\) in its closed neighbourhood (definition 30.2). Over an admissible picture the heights of a solution determine each \(h_v\), as three members of the closed neighbourhood of \(v\) have non-collinear picture points. So there the solutions form a space of dimension \(\dim L(q)\).

Let \(\pi \) be a projection onto \(\ker M(s_0)\). The stacked matrix \([M(s_0); \pi ]\), with the rows of \(M(s_0)\) above those of \(\pi \), is injective, as \(\pi \) fixes every vector of \(\ker M(s_0)\). Let \(H\) be a left inverse of it, and let \(S(s) = H\, [M(s); \pi ]\). This is a square matrix of polynomials with \(S(s_0) = 1\). Put \(D = \det S\). Where \(D(s) \neq 0\) the stack \([M(s); \pi ]\) is injective, so \(\pi \) maps \(\ker M(s)\) injectively into \(\ker M(s_0)\). If moreover \(\dim \ker M(s) \ge \dim \ker M(s_0)\), this map is onto, and some \(x \in \ker M(s)\) has \(\pi x = x_0\). Then \(S(s)\, x = H\, [0; x_0] = S(s_0)\, x_0 = x_0\). By Cramer’s rule \(X(s) = \operatorname {adj} S(s)\, x_0\) equals \(D(s)\, x\), so it lies in \(\ker M(s)\). And \(X(s_0) = x_0\), as \(S(s_0) = 1\).

The lifting system. Over an admissible picture \(q\) the solutions of the lifting system \(M(q)\) form a space of dimension \(\dim L(q)\) (definition 30.15). As \(q_0\) is a main picture, this dimension is at least \(\dim L(q_0)\) at every admissible picture \(q\).

The rank. The rank of a configuration is the rank of the panel-hinge framework whose panel normals are its points. Over an admissible picture its hinges are nonzero (definition 23.1), since two points \((x, y, z, 1)\) over distinct picture points are linearly independent (IsAdmissiblePicture.supportExtensor_ne_zero). So lemma 23.120 gives a polynomial \(Q\) in the coordinates of the points, nonzero at \((q_0, z_0)\), such that the rank is at least \(6(|V(G)| - 1) - \operatorname {def}(\tilde G)\) wherever \(Q\) is nonzero. Wherever every hinge is nonzero, the rank is at most this value (lemma 20.16).

The polynomials \(P\) and \(R\). Some polynomial \(A\) in the picture coordinates, nonzero at \(q_0\), has only admissible non-roots. For instance, take the product of one coordinate difference for each pair of adjacent bodies, in a coordinate in which they differ at \(q_0\), and, for each body, the determinant of the homogeneous picture points of three members of its closed neighbourhood that are independent at \(q_0\) (IsAdmissiblePicture.exists_mvPolynomial). Lift \(z_0\) to a solution \(x_0\) of \(M(q_0)\), and let \(D\) and \(X\) be as in lemma 30.17, for \(M\) at \(q_0\) and \(x_0\), with \(Z(q)\) the heights of \(X(q)\). Let \(P\) be the product of \(A\), \(D\) and \(Q\) at the configuration \((q, Z(q))\). Then \(P(q_0) \neq 0\). A picture \(q\) with \(P(q) \neq 0\) is admissible, and \(Z(q) \in L(q)\). For such \(q\), let \(R\) be \(Q\) at \((q, \cdot )\), a polynomial in the heights with \(R(Z(q)) \neq 0\). At every \(z \in L(q)\) with \(R(z) \neq 0\) the two bounds above give \((q, z)\) the rank \(6(|V(G)| - 1) - \operatorname {def}(\tilde G)\).

Each condition of the lifting system of \(H\) reads the picture at a member of the closed neighbourhood of a body of \(H\), and these are bodies of \(H\) (as in lemma 30.5).

Admissible pictures exist (Graph.exists_isAdmissiblePicture): place the bodies at distinct points of the parabola \(\{ (t, t^2)\} \), as after definition 30.1. So \(\ell _0\) exists, and it is attained at a main picture \(q_0\).

At any main picture \(q_0\), apply lemma 30.17 to the lifting system \(M(q)\) (definition 30.15) at \(q_0\) and the zero solution. It gives a polynomial \(D\), nonzero at \(q_0\), such that wherever \(D(q) \neq 0\) the solutions of \(M(q)\) form a space of dimension at most \(\dim L(q_0)\). Let \(P\) be the product of \(D\) and a polynomial nonzero at \(q_0\) whose non-roots are admissible. It exists since the admissible pictures form a Zariski-open set, as noted after definition 30.1 (IsAdmissiblePicture.exists_mvPolynomial). At a non-root \(q\) of \(P\), \(q\) is admissible and \(\dim L(q) \le \dim L(q_0)\), so \(q\) is a main picture, giving the openness around \(q_0\). Taking \(q_0\) to be the main picture of the first paragraph, \(\dim L(q_0) = \ell _0\), gives the claim.

As \(K\) is infinite, the nonzero polynomial \(P\) of definition 30.8 has a non-root \(q\). This picture is admissible, and some \(z \in L(q)\) gives \((q, z)\) the rank \(6(|V(G)| - 1) - \operatorname {def}(\tilde G)\). Apply lemma 30.11 to \((q, z)\).

Let \(z_1, z_2 \in L\) be points at which the first and the second polynomial are nonzero. On the line \(z_1 + t\, (z_2 - z_1)\), \(t \in K\), each polynomial restricts to a polynomial in \(t\). The first restriction is nonzero at \(t = 0\) and the second at \(t = 1\), so their product is a nonzero polynomial in one variable. As \(K\) is infinite, it has a non-root \(t\). The point \(z_1 + t\, (z_2 - z_1)\) lies in \(L\), and both polynomials are nonzero there.

The first claim is lemma 30.23. The second is lemma 30.26, applied to the linear subspace \(L(q)\) of the heights. For the third, the general configuration attains by lemma 30.20, and then \(G\) has an adjacent-distinct pencil realization (lemma 30.25).

Define \(\Phi (h)_w = h_w(q_w)\). Let \(v\) be a body. For a neighbour \(u\) of \(v\), \(h_u(q_u) = h_v(q_u)\), as \(h_u - h_v\) vanishes at \(q_u\). So on the closed neighbourhood of \(v\), \(\Phi (h)\) agrees with the affine function \(h_v\), and \(\Phi (h) \in L(q)\).

Conversely, let \(z \in L(q)\). For each body \(v\), let \(h_v\) be an affine function that agrees with \(z\) on the closed neighbourhood of \(v\). Both ends of an edge \(uv\) lie in both closed neighbourhoods, so \(h_u - h_v\) vanishes at \(q_u\) and \(q_v\). Hence \((h_v)_v \in F(q)\), and \(\Phi \) maps it to \(z\).

If \(\Phi (h) = 0\), then \(h_v\) vanishes at the picture point of every member of the closed neighbourhood of \(v\). Three of these points are not collinear, as \(q\) is admissible, so \(h_v = 0\). Hence \(\Phi \) is an isomorphism.

Let \(e_0, \dots , e_3\) be the standard basis of \(K^4\), with \(e_2\) the height direction, and write \(e_{ij} = e_i \vee e_j\). The plane \(z = 0\) is spanned by \(e_0\), \(e_1\) and \(e_3\). So at \(z = 0\) every hinge \(p_u \vee p_v\) lies in \(W_\Pi = \operatorname {span}(e_{01}, e_{03}, e_{13})\), the span of the lines of that plane. Split \(\bigwedge ^2 K^4 = W' \oplus W_\Pi \) with \(W' = \operatorname {span}(e_{02}, e_{12}, e_{23})\), and write a screw as \(m_v = (m'_v, m^\Pi _v)\). The hinge condition \(m_u - m_v \in \operatorname {span}(p_u \vee p_v)\) (definition 19.3) splits into \(m'_u = m'_v\) and \(m^\Pi _u - m^\Pi _v \in \operatorname {span}(p_u \vee p_v)\). As \(G\) is connected, the first condition makes \(m'\) constant, which gives three dimensions.

For the second, identify \(W_\Pi = \bigwedge ^2 \hat\Pi \), where \(\hat\Pi = \operatorname {span}(e_0, e_1, e_3)\), with the affine functions on \(K^2\) by \(a \vee b \mapsto \det (a, b, \cdot )\), evaluated at \((x, y, 1)\). The line \(p_u \vee p_v\) becomes a nonzero affine function vanishing at \(q_u\) and \(q_v\). As \(q_u \ne q_v\), its multiples are all the affine functions vanishing at these two points. So the second condition says that \(h_u - h_v\) vanishes at \(q_u\) and \(q_v\), where \(h_v\) is the affine function of \(m^\Pi _v\). Its solutions are \(F(q)\).

The screw space of the plane is \(K^3\). The hinge of \(P_q\) at an edge \(uv\) is the meet of the lines with normals \((x_u, y_u, 1)\) and \((x_v, y_v, 1)\) (definition 23.1). It is a nonzero vector orthogonal to both normals, since \(q_u \ne q_v\). So a relative screw is a multiple of the hinge exactly when it is orthogonal to both normals, which is the condition defining \(F(q)\). The rank follows by rank–nullity and lemma 30.30.

By lemma 30.33 and lemma 30.30, it suffices to show that the motions of \(P_q\) have dimension at least \(3 + \operatorname {def}_2(G)\). Each body of \(P_q\) has the screw space \(K^3\), and each hinge is nonzero, as adjacent picture points differ. Let \(P\) be a partition of \(V(G)\). The screw assignments that are constant on each part form a space of dimension \(3|P|\). On it the edges inside a part impose no condition, and each crossing edge imposes at most two. So the motions have dimension at least \(3|P| - 2d_G(P) = 3 + [3(|P| - 1) - 2d_G(P)]\) for every \(P\), and hence at least \(3 + \operatorname {def}_2(G)\). This is the relative deficiency bound at \(D = 3\) (lemma 20.16).

The rank of \((q, 0)\) is that of \(F_{(q,0)}\) (lemma 30.9). Its motions have dimension \(3 + \dim F(q) = 3 + \dim L(q)\) (lemma 30.31, lemma 30.30). By rank–nullity in the \(6|V(G)|\)-dimensional space of screw assignments, the rank is \(6|V(G)| - 3 - \dim L(q)\). The bound and its equality case follow from lemma 30.34.

The first equivalence is the exact rank of theorem 30.35. By lemma 30.34 and lemma 20.7, every admissible picture \(q'\) has

At \(q' = q\) this gives the second equivalence.

Now suppose that \(\dim L(q) \le 3 + \operatorname {def}_3(G)\). Then \(\dim L(q') \ge \dim L(q)\) at every admissible picture \(q'\), so \(q\) is a main picture (definition 30.18). By the exact rank, the flat configuration \((q, 0)\) has rank at least \(6(|V(G)| - 1) - \operatorname {def}_3(G)\), and its height \(0\) lies in \(L(q)\). So it is a witness for lemma 30.20, and the general configuration of \(G\) attains.

By lemma 30.34, \(3 + \operatorname {def}_2(G) \le 3\), so \(\operatorname {def}_2(G) \le 0\). Both deficiencies are nonnegative, and \(\operatorname {def}_3(G) \le \operatorname {def}_2(G)\) (lemma 20.7), so both vanish. Then \(\dim L(q) = 3 + \operatorname {def}_3(G)\), and corollary 30.36 applies.

The support extensor is linear in each of its two normals (definition 23.1), in the second by antisymmetry (panelSupportExtensor_smul_right). A nonzero multiple of a hinge spans the same line, and the hinge constraint depends only on that line (definition 19.3). Equal motion spaces give equal row spans (lemma 19.25).

By theorem 23.54 at \(d = 2\) (\(D = 3\)), \(G\) has a panel-hinge realization of rank \(3(|V(G)| - 1) - \operatorname {def}_2(G)\). As \(K\) is infinite, three nonzero polynomials in the normals have a common non-root \(n\): the rank polynomial of this realization (lemma 23.120), a polynomial whose non-roots are in general position (definition 23.9), and the product of the third coordinates of the normals. Dividing each normal \(n_v\) by its third coordinate gives the normals \((x_v, y_v, 1)\) of a picture \(q'\). The framework at \(q'\) has the rank of the framework at \(n\), at least \(3(|V(G)| - 1) - \operatorname {def}_2(G)\) (lemma 30.38). Its hinges are nonzero (lemma 23.11). So the rank polynomial of the framework at \(q'\), composed with the chart \(q \mapsto (x_v, y_v, 1)\), is nonzero at \(q'\). It is the required \(P\).

The bodies and the pairs of adjacent bodies are unchanged, as in the proof of lemma 23.42, and hence so are the closed neighbourhoods (closedNbhd_embedEdges) and the dimensions of the lifting spaces, which the main pictures read (finrank_liftingSpace_embedEdges).

Let \(P\) be the product of two nonzero polynomials. The non-roots of the first are main pictures (lemma 30.23). The second is the polynomial of lemma 30.39, which needs two bodies. Here \(G\) has at least three, since a closed neighbourhood of a body lies in \(V(G)\) (Graph.closedNbhd_subset_vertexSet). (The formal proof applies lemma 30.39 to \(G\) with its edges relabelled, lemma 23.42 and lemma 30.41.)

Let \(q\) be a non-root of \(P\). The plane framework \(P_q\) has rank at least \(3(|V(G)| - 1) - \operatorname {def}_2(G)\), and exactly \(3|V(G)| - \dim L(q)\) (lemma 30.33). So \(\dim L(q) \le 3 + \operatorname {def}_2(G)\). The reverse inequality is lemma 30.34. As \(q\) is a main picture, \(\ell _0 = \dim L(q) = 3 + \operatorname {def}_2(G)\) (definition 30.18).

By theorem 30.43, some main picture \(q\) has \(\dim L(q) = 3 + \operatorname {def}_2(G) = 3 + \operatorname {def}_3(G)\). So the general configuration of \(G\) attains (corollary 30.36).

\(z'\) agrees with \(z\) on \(V(G')\). A closed neighbourhood in \(G'\) lies in the closed neighbourhood of the same body in \(G\), where \(z\) agrees with an affine function of the picture. So \(z'\) agrees with that affine function on it too. The configuration points at a body of \(G'\) read only its picture point and its height. For the third, substituting zero for the heights off \(V(G')\) in a polynomial and then evaluating at \(z\) is the same as evaluating the polynomial directly at \(z'\), which already vanishes there.

The hinge at an edge is built from the normals at its two ends, and swapping them negates it (definition 23.1); the hinge constraint reads only the line it spans (definition 19.3). So the motion spaces agree, and equal motion spaces give equal row spans (lemma 19.25).

Given \(z^1 \in L_{G_1}(q)\), extend it across \(V(G_2)\) by the affine function of its closed neighbourhood at \(v\). Every closed neighbourhood of \(G\) other than \(N_G[v]\) lies in one side, and \(N_G[v] = N_1[v] \cup N_2[v]\) carries that affine function on both, so the extension lies in \(L_G(q)\) and restricts to \(z^1\). The other side is symmetric.

We follow the three parts above. Take a picture \(q\) off the zero sets of the polynomials of the attainment at \(G[V_1]\) and at \(G[V_2]\), and of the main-picture polynomial of \(G\) (lemma 30.23). For \(i = 1, 2\) some polynomial \(R_i\) in the heights is nonzero somewhere on \(L_{G[V_i]}(q)\), and \(G[V_i]\) has its target at every height of \(L_{G[V_i]}(q)\) off the zero set of \(R_i\). Read at the restriction to \(V_i\), \(R_i\) is a polynomial in the heights of \(G\) (lemma 30.46). It is nonzero somewhere on \(L_G(q)\), since the restriction is onto (lemma 30.48). So some \(z \in L_G(q)\) makes both nonzero (lemma 30.26).

At \((q, z)\) the rank of each side is its target (lemma 30.47), and the rank of \(G\) is their sum (lemma 19.26). The deficiencies add (lemma 20.29), and \(|V_1| + |V_2| = |V(G)| + 1\), so

Hence lemma 30.20 applies.

Given \(z^1 \in L_{G_1}(q)\), take an affine function \(h\) agreeing with \(z^1\) on the closed neighbourhood \(N_{G_1}[a]\) (definition 30.2), and extend \(z^1\) by \(h\) on every body outside \(G_1\). The body \(a\) is the only body of \(G_1\) with an edge leaving \(G_1\): the path’s own bodies and \(b\) lie outside \(G_1\), so only the path’s first edge, at \(a\), can cross. So the closed neighbourhood in \(G\) of a body of \(G_1\) other than \(a\) lies in \(G_1\), where the extension is \(z^1\); the closed neighbourhood of \(a\), or of a body outside \(G_1\), meets \(G_1\) only inside \(N_{G_1}[a]\), so the extension equals \(h\) on all of it. The extension is therefore affine on every closed neighbourhood of \(G\): it lies in \(L_G(q)\) and restricts to \(z^1\). The other restriction is symmetric.

We find \(q\) and \(z\) as in the proof of theorem 30.49, with lemma 30.50 for the restrictions. For \(G[V_2]\) we apply it from \(b\), along the reversed path. We count the ranks and the deficiencies directly, rather than apply the step one bridge at a time. That would ask a graph with one path body as its only body to attain, and no picture of such a graph is admissible.

Write \(S_j = V_1 \cup \{ x_1, \dots , x_j\} \). Every edge off the path lies inside \(V_1\) or inside \(V_2\), so the only edge of \(G\) leaving \(S_j\) is the path edge from \(S_j\) to the next path body, or to \(b\) when \(j = k\). The last of these edges is a bridge between \(G[S_k]\) and \(G[V_2]\). It adds \(D - 1 = 5\) to the rank (lemma 19.24; every hinge is nonzero at an admissible picture) and \(1\) to the deficiency (lemma 20.27). Inside \(G[S_{j+1}]\) the path body \(x_{j+1}\) is joined to \(S_j\) by a single edge. By the same two lemmas, it adds at least \(5\) to the rank and exactly \(1\) to the deficiency. Compared with the two sides together, the term \(6(|V(G)| - 1)\) is larger by \(6(k + 1)\) and the deficiency by \(k + 1\), so

The rank at \((q, z)\) is at least this, so lemma 30.20 applies.

\(G[W]\) is simple, and it is connected and of minimum degree at least two by lemma 20.23.

An edge of \(G/H\) is an edge of \(G\) not inside \(W\). Two such edges with the same ends in \(G/H\) would give a body outside \(W\) adjacent to two bodies of \(W\), or two parallel edges of \(G\), so \(G/H\) is simple. A path of \(G\) maps to a walk of \(G/H\), so \(G/H\) is connected. At \(r\), the \(2\)-edge-connectivity of \(G\) gives two edges leaving \(W\). Their ends outside \(W\) are distinct: a body at the end of both would be adjacent to two bodies of \(W\), or joined to one by two parallel edges. At a body outside \(W\) the contraction is injective on the closed neighbourhood. So every closed neighbourhood of \(G/H\) has at least three members, and \(G/H\) has minimum degree at least two.

At a body \(v\) whose closed neighbourhood meets \(W\), the heights \(z(t)\) on that neighbourhood agree with the affine function of the picture \(q(t)\) with coefficients \((h_{v,1},\ h_{v,2},\ t\, h_{v,3} - (1 - t)(h_{v,1} x_r + h_{v,2} y_r))\); at the other bodies they agree with \(h_v\). At \(t \neq 0\) the substitution inverts: divide the core heights, and the corrected constant coefficients, by \(t\).

(1) As \(H\) is induced, the closed neighbourhood of a core body \(c\) in \(H\) is its closed neighbourhood in \(G\) met with \(W\). The rows of \(M(t)\) there keep the weight \((x_w, y_w, 1)\) at every \(t\), so \(\rho \) sends the heights of a solution into \(L_H(q)\).

(2) By rank–nullity it suffices that the solutions of \(M(0)\) with \(\rho = 0\) form a space \(N\) with \(\dim N + 3 \le \dim L_{G/H}(q)\). A solution in \(N\) has zero heights on \(W\), so by the rows of (1) its plane at every core body is zero: three members of each closed neighbourhood in \(H\) have independent picture points. Send a solution in \(N\) to the height of \(G/H\) that agrees with it outside \(W\) and vanishes at \(r\). This height lies in \(L_{G/H}(q)\), by the rows at \(t = 0\) (read after definition 30.55). At a neighbour \(u\) of \(W\), the members of its closed neighbourhood outside \(W\) lie on a plane through \((q_r, 0)\), which is the point of \(r\) in the image. The planes of the core are zero, so the rows at the core put every neighbour of \(W\) at height zero, and the height vanishes on the closed neighbourhood of \(r\) in \(G/H\).

The map is injective. If the image of a solution vanishes, every plane outside \(W\) vanishes, since three members of its closed neighbourhood in \(G/H\) have independent picture points; and the constant coefficient of a neighbour of \(W\) vanishes by its row at a neighbour in \(W\). The image vanishes on the closed neighbourhood of \(r\), three members of which have independent picture points, so the only affine height in the image is zero. As \(\mathrm{Aff}(q)\) is three-dimensional and lies in \(L_{G/H}(q)\) (lemma 30.3), this gives the bound.

For each body \(v\) of \(G/H\), let \(a_v\) be an affine function with which \(z\) agrees on the closed neighbourhood of \(v\) in \(G/H\). As \(z_r = 0\), the plane of \(a_r\) passes through \((q_r, 0)\), and it passes through the point of every neighbour of \(W\). Put the core on this plane: give each body \(c\) of \(W\) the height \(a_r(q_c)\), and give \(h_c\) the coefficients of \(a_r\). Give each body \(v\) outside \(W\) the height \(z_v\), and give \(h_v\) the coefficients of \(a_v\) if \(v\) is not a neighbour of \(W\). At a neighbour \(u\) of \(W\), the closed neighbourhood of \(u\) in \(G/H\) contains \(r\), so the plane of \(a_u\) passes through \((q_r, 0)\); give \(h_u\) the linear coefficients of \(a_u\). Then the rows at \(t = 0\) hold, as read after definition 30.55. The constant coefficient of \(h_u\) is fixed by the row of \(u\) at its one neighbour in \(W\).

At the collapsed placement each row of \(G \setminus E(H)\), with the columns of \(W\) deleted, is the corresponding row of \(G/H\) with the column of \(r\) deleted (as in lemma 23.98), and deleting the one column of \(r\) loses no rank (lemma 19.35). So there are \(N\) such rows, independent at the collapsed placement, and lemma 23.93 gives a polynomial nonzero there off whose zero set they stay independent. This is the comparison of KT eqs. (6.5) and (6.9)  [ KT11 , pp. 674–675 ] .

At a body \(c\) of \(W\) the point of \((q(t), z(t))\) is \(((1 - t)\, x_r + t\, x_c,\ (1 - t)\, y_r + t\, y_c,\ t\, z_c,\ 1)\), the image of \((x_c, y_c, z_c, 1)\) under the linear map

of \(K^4\), which is invertible as \(t \neq 0\). So lemma 30.13 applies.

The pieces. \(H\) satisfies the standing hypotheses (lemma 30.53, at \(d = 3\)). By lemma 30.54, \(G/H\) is simple and satisfies them too. As \(H\) is rigid, \(G/H\) has the deficiency of \(G\) (lemma 29.22, at \(d = 3\)).

The picture. As \(H\) and \(G/H\) satisfy the standing hypotheses, Jackson and Jordán’s equality applies to both (theorem 30.43). Take a picture \(q\) off the zero sets of five nonzero polynomials: those of the attainment and of the equality at \(H\) and at \(G/H\), and the main-picture polynomial of \(G\) (lemma 30.23). Then \(q\) is admissible for \(H\) and for \(G/H\), and both have attaining heights over \(q\). Also \(\dim L_H(q) = 3 + \operatorname {def}_2(H)\) and \(\dim L_{G/H}(q) = 3 + \operatorname {def}_2(G/H)\). Subtracting the constant height at \(r\) does not change the rank of \(G/H\) (lemma 30.14). So \(G/H\) has an attaining height \(z'\) with \(z'_r = 0\).

The kernel at the end of the curve. Let \(\rho \) take a vector of unknowns to its core heights, as in lemma 30.57. The polynomial section through the zero solution at \(t = 0\) (lemma 30.17, with the parameter \(t\)) gives \(\dim \ker M(t) \le \dim \ker M(0)\) for all but finitely many \(t\). For all but finitely many \(t\), also, \(q(t)\) is a main picture of \(G\), as the main-picture polynomial of \(G\) along the curve is nonzero at \(t = 1\), where \(q(1) = q\). At a \(t \neq 0\) with both properties,

(lemma 30.34, lemma 30.56). By lemma 30.57, Jackson and Jordán’s equality at \(H\) and \(G/H\), and additivity,

So \(\dim \ker M(0) = 3 + \operatorname {def}_2(G)\), and the second chain is an equality. Hence \(\rho (\ker M(0))\), which lies in \(L_H(q)\), has the dimension of \(L_H(q)\): the core heights of the solutions of \(M(0)\) are all of \(L_H(q)\).

Two open conditions. The polynomial of the attainment at \(H\) over \(q\) is nonzero somewhere on \(L_H(q)\). Composed with \(\rho \), it is therefore nonzero somewhere on \(\ker M(0)\). Let \(Q\) be the polynomial of lemma 30.59 for the rank \(\operatorname {tgt}(G/H)\) of \(G/H\) at \((q, z')\), and read it at the configuration \((q(0), z(0))\) of a solution. It is nonzero at the solution that agrees with \(z'\) off \(W\) (lemma 30.58), as the configuration of that solution at \(t = 0\) is the collapsed placement of \((q, z')\). So some \(x_0 \in \ker M(0)\) makes both polynomials nonzero (lemma 30.26).

The curve. Apply lemma 30.17 to \(M(t)\), at \(t = 0\) and \(x_0\). It gives a polynomial \(D\) in \(t\) with \(D(0) \neq 0\), and a vector \(X(t)\) of polynomials in \(t\) with \(X(0) = x_0\). At every \(t \neq 0\) with \(q(t)\) a main picture of \(G\) the kernel does not jump: \(\dim \ker M(0) = 3 + \operatorname {def}_2(G) \le \dim L_G(q(t)) \le \dim \ker M(t)\). So \(X(t)\) solves \(M(t)\) at every such \(t\) with \(D(t) \neq 0\), and its heights \(z(t)\) lie in \(L_G(q(t))\) (lemma 30.56).

A good parameter. Take \(t\) off the zero sets of \(t\), of \(D\), of the main-picture polynomial of \(G\) along the curve, and of the two polynomials of the open conditions along \(X(t)\), which are nonzero at \(t = 0\). Then \(\rho (X(t))\) lies in \(L_H(q)\) (lemma 30.57(1)), and it is an attaining height of \(H\) over \(q\). So \(H\) has rank \(6(|W| - 1)\) at \((q, \rho (X(t)))\), and hence at \((q(t), z(t))\) (lemma 30.60, lemma 30.47). As \(Q\) is nonzero at \((q(t), z(t))\), the rows of the other edges there, with the columns of \(W\) deleted, have rank at least \(\operatorname {tgt}(G/H)\). By lemma 23.86 the rank of \(G\) at \((q(t), z(t))\) is at least \(6(|W| - 1) + \operatorname {tgt}(G/H) = \operatorname {tgt}(G)\), since \(|V(G/H)| = |V(G)| - |W| + 1\) (lemma 21.51) and \(G/H\) has the deficiency of \(G\). So lemma 30.20 applies.

The core \(H\) satisfies the standing hypotheses (lemma 30.53, at \(d = 2\)). Since \(\operatorname {def}_3(H) \le \operatorname {def}_2(H) = 0\) (lemma 20.7), both deficiencies of \(H\) vanish. So \(H\) is rigid, and its general configuration attains (corollary 30.44). Contracting a core of planar deficiency zero does not change the planar deficiency (lemma 29.22, at \(d = 2\)). So \(\operatorname {def}_2(H) + \operatorname {def}_2(G/H) = \operatorname {def}_2(G)\), and theorem 30.61 applies.

By admissibility three members of the closed neighbourhood of \(v\) have linearly independent homogeneous picture points, so they are all its members, and any three values at them are taken by exactly one affine function.

The closed neighbourhood of an interior body consists of the body and its two neighbours on the path, and every height is affine on it (lemma 30.64). This gives (1). For (2), the closed neighbourhood in \(G\) of a body of \(V_1\) is its closed neighbourhood in \(G[V_1]\), together with \(x_1\) at \(a\) and \(x_k\) at \(b\). Take affine functions \(h_a\) and \(h_b\) agreeing with \(z^1\) on the closed neighbourhoods of \(a\) and \(b\) in \(G[V_1]\). Set \(z_{x_1} = h_a(q_{x_1})\) and \(z_{x_k} = h_b(q_{x_k})\), which is possible since \(k \ge 2\) makes \(x_1 \neq x_k\). Then (1) applies.

Expand \(p \vee p'\) in the basis of \(W'\) and \(W_\Pi \) used in the proof of lemma 30.31: the coefficients on \(W_\Pi \) are the \(2 \times 2\) minors of \(\bar p\) and \(\bar p'\), and those on \(W'\) pair the heights with the planar parts.

By lemma 30.66 the coordinates of each join are polynomials in the picture at fixed heights, and in the heights at a fixed picture. A finite family of vectors is linearly independent exactly when some maximal minor of its coordinate matrix is nonzero; take for \(P\), or \(R\), one such minor.

By lemma 30.66 the joins have integer coordinates, and in each of the four cases their coordinate matrix has a maximal minor equal to \(\pm 1\), which is nonzero in every characteristic.

The hinge at an edge \(uw\) spans the line of the meet of the planes \(p_u^{\perp }\) and \(p_w^{\perp }\). The polarity, an invertible linear map of the screw space, maps the join \(p_u \vee p_w\) to that meet (lemma 30.9). So the meets at \(g_1, \dots , g_m\) are independent and lie in the span of the hinges there. The screw space has dimension six.

The configuration. Every closed neighbourhood of \(G\) has three members, so at an admissible picture \(q\) every height lies in \(L_G(q)\) (lemma 30.64). Put the bodies at the points of lemma 30.68. For \(m \le 5\), put \(a, x_1, \dots , x_k, b\) at \(Y_0, \dots , Y_{m-1}\), so that the joins at the \(m\) edges are those of the closed \(m\)-gon. For \(m \ge 6\), put \(a, x_1, x_2, x_3\) at \(Y_0, \dots , Y_3\), the bodies \(x_4, \dots , x_k\) at \(Y_4\) and \(b\) at \(Y_5\). Then the joins at the edges \(a x_1\), \(x_1 x_2\), \(x_2 x_3\), \(x_3 x_4\), \(x_k b\) and \(ba\) are those of the closed hexagon. Let \(z\) be the heights of these points, and let \(P\) be the polynomial in the picture that lemma 30.67 gives for these \(\min (m, 6)\) joins at the heights \(z\). Take a picture \(q\) off the zero sets of \(P\) and of the main-picture polynomial of \(G\) (lemma 30.23). Then \(q\) is a main picture of \(G\), and the joins are independent at \((q, z)\).

The count. The cycle is the edge \(ab\) with the open ear \(a - x_1 - \cdots - x_k - b\) on \(\{ a, b\} \), and every hinge is nonzero at the admissible picture \(q\). The edge has rank \(5\) and relative screws spanned by its hinge (lemma 19.30 and lemma 19.31, at \(k = 0\)). So by lemma 19.32 the rank at \((q, z)\) is \(5 + 5(k + 1) + \lambda - 6 = 5m - 6 + \lambda \), where \(\lambda \) is the dimension of the span of all the hinges. By lemma 30.70, \(\lambda \ge \min (m, 6)\). The partition into single bodies has value \(6(m - 1) - 5m = m - 6\) (definition 20.3), so \(\operatorname {def}(\tilde G) \ge \max (0, m - 6)\). Hence \(\operatorname {tgt}(G) \le \min (6m - 6, 5m) \le 5m - 6 + \lambda \), and by lemma 30.20 the general configuration of \(G\) attains.

The hexagon. Let \(q^*\) put every body of \(V_1\) at \((1, 0)\), the bodies \(x_1, x_2, x_3\) at \((0, -1)\), \((0, 0)\), \((-1, 0)\), the bodies \(x_4, \dots , x_{k-1}\) at \((0, 0)\) and \(x_k\) at \((-1, -1)\). Let \(z^*\) be \(1\) at \(x_2\) and \(x_3\) and \(0\) elsewhere. The points of \(a, x_1, x_2, x_3, x_4, x_k\) are then \(Y_0, \dots , Y_5\) (lemma 30.68), and \(x_{k-1}\) is at \(Y_4\) too. So the joins at the six ear edges \(a x_1\), \(x_1 x_2\), \(x_2 x_3\), \(x_3 x_4\), \(x_{k-1} x_k\) and \(x_k b\), distinct since \(k \ge 5\), are those of the closed hexagon. Let \(P\) be the polynomial in the picture that lemma 30.67 gives for these joins at the heights \(z^*\); it is nonzero at \(q^*\).

The picture and the height. Take a picture \(q\) off the zero sets of \(P\), of the polynomial of the attainment at \(G[V_1]\), and of the main-picture polynomial of \(G\) (lemma 30.23). The height \(z^*\) vanishes on \(V_1\) and at \(x_1\) and \(x_k\). So it lies in \(L_G(q)\) (lemma 30.65(1)), and the six joins are independent at \((q, z^*)\). The restriction maps \(L_G(q)\) onto \(L_{G[V_1]}(q)\) (lemma 30.65(2)). So the heights of \(L_G(q)\) at which \(G[V_1]\) has its target contain a nonempty Zariski-open subset (lemma 30.46). So do the heights at which the six joins stay independent, as \(z^*\) is one of them (lemma 30.67, in the heights). The two subsets meet (lemma 30.26); let \(z\) be a height in both.

The count. At \((q, z)\) the rank of \(G[V_1]\) is its target (lemma 30.47). The hinges of the ear span the screw space (lemma 30.70), so \(\rho + \Lambda \) is the whole screw space. Every hinge is nonzero at the admissible picture \(q\), so the rank of \(G\) exceeds that of \(G[V_1]\) by \(5(k + 1)\) (lemma 19.32). The target grows by at most \(5(k + 1)\), as in the opening of this subsection (lemma 20.31). So by lemma 30.20 the general configuration of \(G\) attains.

The body \(c\) is a cut vertex: the sides \(V_1\) and \(V_2 = \{ c, x_1, \dots , x_k\} \) meet in \(c\), and every edge of \(G\) lies inside one of them. As \(G\) has no loop, \(G[V_2]\) is the cycle \(c - x_1 - \cdots - x_k - c\) on \(k + 1 \ge 3\) bodies. It attains by theorem 30.71, with the edge \(x_1 c\) in the role of \(ab\). So the general configuration of \(G\) attains by theorem 30.49.

Expand both sides in the coordinates of the four points, with the flat coordinates of lemma 30.66.

Pair a vanishing combination \(\sum c_{ij}\, p_i \vee p_j = 0\) with the opposite edge \(p_k \vee p_l\). By lemma 30.76 the pairing of two edges is \(\pm \det (p_0, p_1, p_2, p_3) \neq 0\) when they are opposite and \(0\) when they share a point, so every \(c_{ij}\) vanishes. The space \(\bigwedge ^2 K^4\) has dimension six.

A line through \(y\) or through \(y'\) meets the line \(y y'\), so the pairing vanishes (lemma 30.76). Extend \(y, y'\) to a basis \(y, y', e, e'\) of \(K^4\). Five of the six joins of lemma 30.77, all but \(e \vee e'\), lie in the two stars.

The first claim is a computation in the flat coordinates of lemma 30.66. On \(\bigwedge ^2 \hat\pi _h\) the coordinate \(\sigma \) is determined by \(\tau \), so \(\tau \) is injective there.

For \(u, v \in K^3\) let \(\hat u \in \hat\pi _h\) and \(\hat v \in \hat\pi _{h'}\) be the points over them. In the flat coordinates of lemma 30.66,

We test it on the unit vectors \(e_1, e_2, e_3\) of \(K^3\), and write \(\delta = h - h'\) and \(\tau = \tau (c)\).

Suppose \(h \neq h'\), say \(\delta _i \neq 0\). At \(u = v = e_i\) the pairing is \(\delta _i \tau _i\), so \(\tau _i = 0\). For \(j \neq i\), the sum of the pairings at \((e_i, e_j)\) and at \((e_j, e_i)\) is \(\delta _i \tau _j + \delta _j \tau _i\), so \(\tau _j = 0\). So \(\tau = 0\), and the pairing at \((e_j, e_l)\) is \(\sigma (c) \cdot (e_j \times e_l)\). These cross products are the unit vectors up to sign, so \(\sigma (c) = 0\).

Suppose \(h = h'\). As \((h \cdot u)(\tau \cdot v) - (h \cdot v)(\tau \cdot u) = (h \times \tau ) \cdot (u \times v)\), the pairing is \(\bigl(\sigma (c) + h \times \tau \bigr) \cdot (u \times v)\), and the cross products \(u \times v\) span \(K^3\). So \(\sigma (c) = -\, h \times \tau (c)\).

For the consequence, if \(h \neq h'\) then \(\rho = 0\). If \(h = h'\) then \(\rho \) and the \(f_i\) lie in \(\bigwedge ^2 \hat\pi _h\), of dimension at most three (lemma 30.79).

Write \(R = \rho + \operatorname {span} F\). For \(t \neq 0\), \(y \vee (y + t u) = t\, (y \vee u)\) and \((y + t u) \vee y' = v_t\) with \(v_t = y \vee y' + t\, (u \vee y')\). So \(W_t\) contains \(R\), \(y \vee u\) and \(v_t\). A vector \(v\) outside a subspace \(S\) has \(v + t w \notin S\) for one of any two distinct nonzero values of \(t\), since otherwise \(S\) contains \(w\) and then \(v\). As \(K\) is infinite, two such values exist.

If \(y \vee y' \in R\), then \(W = R\), and \(\dim W_t \ge \dim \bigl(R + K\, (y \vee u)\bigr) = \dim W + 1\) for every \(t \neq 0\).

Otherwise \(\dim W = \dim R + 1\), and \(y \vee y' \notin R + K\, (y \vee u)\), since a relation \(y \vee y' = r + c\, (y \vee u)\) with \(r \in R\) has \(c \neq 0\) and puts \(y \vee u\) in \(W\). So some \(t \neq 0\) has \(v_t \notin R + K\, (y \vee u)\), and \(\dim W_t \ge \dim R + 2 = \dim W + 1\).

The lifting space of \(G[V_1]\) and the closed neighbourhoods of \(a\) and \(b\) in \(G[V_1]\) read the picture only on \(V_1\), where \(q(s)\) is \(q_1\) (lemma 30.5). So \(z(s)\) agrees with an affine function of the picture on the closed neighbourhood in \(G\) of every body of \(V_1\). At \(a\) this neighbourhood is the one in \(G[V_1]\) together with \(x_1\), and \(z(s)\) agrees with \(h_a\) on it. At \(b\) it is the one in \(G[V_1]\) together with \(x_k\), which differs from \(x_1\) as \(k \ge 2\), and \(z(s)\) agrees with \(h_b\) on it. At any other body of \(V_1\) it lies inside \(V_1\), where \(z(s)\) is \(z_1\). As \(z(s)\) vanishes off \(V(G)\), lemma 30.65(1) applies.

Every coordinate of a point of the configuration is a constant, a coordinate of \(s\), or, at \(x_1\) and \(x_k\), affine in the picture coordinates of \(s\). This proves the polynomial claim. On \(V_1\) the coordinates are those of the base data.

Every coordinate of \(q(s)\) is a constant or a coordinate of \(s\) (definition 30.82), so substituting them for the picture’s coordinates in \(P\) gives a polynomial in \(s\).

For the rank bound, substitute the points into the polynomial of lemma 23.120, which asks for the nonzero hinges. For the span, take a basis of \(\rho \) together with \(F\). The coordinates of its members are polynomial in \(s\) (lemma 30.66), and \(n\) of them are linearly independent at the given parameter. They stay independent wherever a maximal minor of their coordinate matrix, nonzero there, does not vanish.

The hexagon. Let \(q^*\) put \(a\), \(x_1\), \(x_2\) and \(b\) at \((0, 0)\), \((-1, -1)\), \((1, 0)\) and \((0, -1)\), and every other body at \((0, 0)\). At the heights \(0\) the points of \(a, x_1, x_2, b\) are then \(Y_4, Y_5, Y_0, Y_1\) (lemma 30.68). So the joins at the three ear edges \(a x_1\), \(x_1 x_2\) and \(x_2 b\) are \(Y_4 \vee Y_5\), \(Y_5 \vee Y_0\) and \(Y_0 \vee Y_1\). These are three joins of the closed hexagon, and they are linearly independent. Let \(P\) be the polynomial in the picture that lemma 30.67 gives for these joins at the heights \(0\); it is nonzero at \(q^*\). The picture \(q^*\) need not be admissible.

The picture and the height. Take a picture \(q\) off the zero sets of \(P\), of the polynomial of the attainment at \(G[V_1]\), and of the main-picture polynomial of \(G\) (lemma 30.23). The heights \(0\) lie in the linear space \(L_G(q)\), and the three joins are independent at \((q, 0)\). We find the height as in the proof of theorem 30.72. The heights of \(L_G(q)\) whose restriction to \(V_1\) gives \(G[V_1]\) its target contain a nonempty Zariski-open subset (lemma 30.65(2), lemma 30.46). So do the heights at which the three joins stay independent (lemma 30.67, in the heights). The two subsets meet (lemma 30.26); let \(z\) be a height in both.

The count. At \((q, z)\) the rank of \(G[V_1]\) is its target (lemma 30.47), and every hinge is nonzero at the admissible picture \(q\). The hinges of the ear span a space of dimension at least three (lemma 30.70), so \(\dim (\rho + \Lambda ) \ge 3\). As \(\operatorname {def}(\tilde G[V_1]) \le \operatorname {def}(\tilde G)\), this gives \((\star )\) at \(k = 2\). So lemma 19.32 and lemma 30.20 apply.

Write \(f = \operatorname {def}(\tilde G[V_1])\). The graph \(G''\) has the open ear \(a - x_1 - x_3 - x_4 - b\) on \(V_1\), and the same \(G[V_1]\). We choose the configuration in the two stages of the opening of this subsection. The second stage takes a general ear datum twice: once for \(G''\), and once for \(G\) after \(x_2\) is put back. The stages and the count are the same in the proof of theorem 30.89 (Graph.X0Attains.of_openEar_splitOff). Only the third condition on the ear data, and its use in putting \(x_2\) back, are specific to four interior bodies.

The first stage: base data. Take a picture \(q^*\) off the zero sets of the main-picture polynomial of \(G\) (lemma 30.23) and of the polynomials of the attainment at \(G[V_1]\) and at \(G''\). In \(L_{G''}(q^*)\) the heights at which \(G''\) has its target contain a nonempty Zariski-open subset. So do the heights whose restriction to \(V_1\) gives \(G[V_1]\) its target (lemma 30.65(2) for the ear of \(G''\), lemma 30.46). The two subsets meet (lemma 30.26); let \(z^*\) be a height in both. The base data (definition 30.82) are the restriction \((q_1, z_1)\) of \((q^*, z^*)\) to \(V_1\), and the affine functions \(h_a\) and \(h_b\) with which \(z^*\) agrees on the closed neighbourhoods of \(a\) and \(b\) in \(G''\). These contain the closed neighbourhoods of \(a\) and \(b\) in \(G[V_1]\), as definition 30.82 asks, and also \(x_1\) and \(x_4\). So \((q^*, z^*)\) is the configuration of one ear datum \(s^*\). The rank of \(G[V_1]\) and its relative screws \(\rho \) depend only on the hinges at the edges of \(G[V_1]\), and the ear data do not move the points on \(V_1\) (lemma 30.47, lemma 30.5). So at every ear datum, in the frameworks of \(G\) and of \(G''\) alike, \(G[V_1]\) has its target and the relative screws \(\rho \) (Graph.exists_earBase_splitOff).

The second stage, for \(G''\). We ask three conditions of the ear data. Each holds off the zero set of a polynomial that is nonzero at one ear datum (lemma 30.84, lemma 30.85). The first two are that \(G''\) has rank at least its target and that the picture is admissible for \(G''\). Both hold at \(s^*\), where the hinges of \(G''\) are nonzero. The third is that the six joins of the points of \(x_1\), \(x_3\), \(x_4\) and \(b\) span \(\bigwedge ^2 K^4\). Let \(q_b\) and \(z_b\) be the picture and the height of \(b\). Put \(x_1\) over \(q_b + (0, 1)\) on the plane of \(a\), \(x_4\) over \(q_b + (1, 0)\) on the plane of \(b\), and \(x_3\) over \(q_b\) at the height \(z_b + 1\). The points of \(x_1\), \(x_3\), \(x_4\) and \(b\) are then linearly independent, so the third condition holds there (lemma 30.77). Take ear data \(y\) off all three zero sets, and let \(y_1\), \(y_3\) and \(y_4\) be the points of \(x_1\), \(x_3\) and \(x_4\). Then \(G''\) has rank at least its target and \(G[V_1]\) has its target. So by the ear rank formula at \(G''\), with the joins as hinges (lemma 19.32, lemma 30.9),

has \(\dim W \ge 4 + f - \operatorname {def}(\tilde G'')\).

Putting \(x_2\) back. Keep the other bodies of \(y\) and give \(x_2\) an affine point \(x\). As \(x_2\) is a middle body of the ear, this is an ear datum of \(G\). Let \(\Lambda \) be the span of the joins of its ear. Every line through an affine point \(y'\) is \(y' \vee u\) with \(u\) of last coordinate \(0\), since \(y' \vee v = y' \vee (v - v_4 y')\). If \(W\) does not contain every line through \(y_1\), take such a \(u\) with \(y_1 \vee u \notin W\). Then \(y_1 + t u\) is affine for every \(t\), and for some \(t \neq 0\) the point \(x = y_1 + t u\) has \(\dim (\rho + \Lambda ) \ge \dim W + 1\) (lemma 30.81). The same holds at \(y_3\). Suppose that \(W\) contains every line through \(y_1\) and every line through \(y_3\). Then it contains \(y_1 \vee y_3\), \(y_1 \vee y_4\), \(y_1 \vee p_b\), \(y_3 \vee y_4\) and \(y_3 \vee p_b\), and also \(y_4 \vee p_b\), a join of the ear of \(G''\). These are the six joins of \(y_1, y_3, y_4, p_b\), which span \(\bigwedge ^2 K^4\) by the choice of \(y\). So \(W = \bigwedge ^2 K^4\), and we take \(x = y_1\): the join \(y_1 \vee x\) vanishes, but \(\rho + \Lambda \) contains \(W\). Either way \(\dim (\rho + \Lambda ) \ge \min (\dim W + 1, 6)\) (exists_insertion_four).

The configuration of \(G\). The ear datum just built may have a vanishing hinge, but the span bound of lemma 30.85 asks for none. So \(\dim (\rho + \Lambda ) \ge \min (\dim W + 1, 6)\) holds off the zero set of a polynomial in the ear data of \(G\) that is nonzero there. The condition that the picture be a main picture of \(G\) is polynomial too, and nonzero at \(s^*\) (lemma 30.84). Take an ear datum off both zero sets. Its picture is a main picture of \(G\), so the hinges are nonzero, and its heights lie in \(L_G\) (lemma 30.83). As at every ear datum, \(G[V_1]\) has its target and the relative screws \(\rho \).

The count. \(\operatorname {def}(\tilde G'') \le \operatorname {def}(\tilde G)\) (lemma 21.34), and \(\operatorname {def}(\tilde G) \ge f + k - 5 = f - 1\) (lemma 20.31). So \(\min (\dim W + 1, 6) \ge 5 + f - \operatorname {def}(\tilde G)\), which is \((\star )\) at \(k = 4\). By lemma 19.32 and lemma 30.20 the general configuration of \(G\) attains.

We follow the proof of theorem 30.88 (Graph.X0Attains.of_openEar_splitOff), with the open ear \(a - x_1 - x_3 - b\) of \(G''\). Write \(f = \operatorname {def}(\tilde G[V_1])\). The first stage gives the base data, the ear datum \(s^*\) and the relative screws \(\rho \) (Graph.exists_earBase_splitOff). Write \(\hat\pi _a = \hat\pi _{h_a}\) and \(\hat\pi _b = \hat\pi _{h_b}\) for the homogeneous points of the planes of \(a\) and \(b\). At every ear datum, \(p_a\) and the point \(y_1\) of \(x_1\) lie in \(\hat\pi _a\), and the point \(y_3\) of \(x_3\) and \(p_b\) lie in \(\hat\pi _b\).

The case split on \(\rho \), made before the ear data are chosen. In the first case \(\langle c, u \vee v \rangle = 0\) for every \(c \in \rho \), \(u \in \hat\pi _a\) and \(v \in \hat\pi _b\). In the second, some \(c \in \rho \) pairs to a nonzero value with some \(u_0 \vee v_0\). We then ask one more condition of the ear data, \(\langle c, y_1 \vee y_3 \rangle \neq 0\). For \(s, s' \in K^3\) let \(\hat s \in \hat\pi _a\) and \(\hat s' \in \hat\pi _b\) be the points over them. The map \((s, s') \mapsto \langle c, \hat s \vee \hat s' \rangle \) is bilinear and not zero. The vectors \((0, 0, 1)\), \((1, 0, 1)\) and \((0, 1, 1)\) span \(K^3\), so it is nonzero at a pair of them. The points over them are affine, so some ear datum puts \(x_1\) and \(x_3\) there (exists_klein_liftPlane_affine_ne_zero). Let \(c^\perp \) be the hyperplane of screws pairing to zero with \(c\). The condition says \(\dim \bigl(c^\perp + K\, (y_1 \vee y_3)\bigr) = 6\). So it holds off the zero set of a polynomial in the ear data that is nonzero at that ear datum (lemma 30.85).

The second stage, for \(G''\). Take ear data \(y\) off the zero sets of the polynomials that ask the rank of \(G''\) to be at least its target and the picture to be admissible for \(G''\). In the second case, take \(y\) also off the zero set for \(\langle c, y_1 \vee y_3 \rangle \). As in the four-body step, the ear rank formula at \(G''\) (lemma 19.32) shows that

has \(\dim W \ge 3 + f - \operatorname {def}(\tilde G'')\). As \(x_1 x_3\) is an edge of \(G''\) and the picture is admissible for \(G''\), the pictures of \(x_1\) and \(x_3\) differ. So \(y_1\) and \(y_3\) are linearly independent, and \(\operatorname {star}(y_1) + \operatorname {star}(y_3)\) has dimension at least five and pairs to zero with \(y_1 \vee y_3\) (lemma 30.78).

A span containing both stars. We show that \(W\) contains both stars only if \(W = \bigwedge ^2 K^4\). In the first case \(\dim W \le 3\) (lemma 30.80). When \(h_a = h_b\) this uses that the three joins of the ear of \(G''\) are lines of the one plane \(\hat\pi _a = \hat\pi _b\) (lemma 30.79). So \(W\) does not contain both stars (not_star_sup_star_le_of_klein). In the second case \(c\) lies in \(W\), and outside \(\operatorname {star}(y_1) + \operatorname {star}(y_3)\), as it pairs to a nonzero value with \(y_1 \vee y_3\). So a \(W\) containing both stars has dimension six (eq_top_of_star_sup_star_le).

Putting \(x_2\) back, and the configuration of \(G\). We argue as in the proof of theorem 30.88. Some affine point \(x\) for \(x_2\) has \(\dim (\rho + \Lambda ) \ge \min (\dim W + 1, 6)\), where \(\Lambda = \operatorname {span}(p_a \vee y_1,\ y_1 \vee x, x \vee y_3,\ y_3 \vee p_b)\) (exists_insertion_three). It lies on a line through \(y_1\) or \(y_3\) that is not in \(W\) (lemma 30.81), or at \(y_1\) when \(W = \bigwedge ^2 K^4\). This bound and the main-picture condition are polynomial conditions on the ear data of \(G\) (lemma 30.84, lemma 30.85). At a common non-root the picture is a main picture of \(G\), so the hinges are nonzero, and the heights lie in \(L_G\) (lemma 30.83); \(G[V_1]\) has its target and the relative screws \(\rho \).

The count. \(\operatorname {def}(\tilde G'') \le \operatorname {def}(\tilde G)\) (lemma 21.34), and \(\operatorname {def}(\tilde G) \ge f + k - 5 = f - 2\) (lemma 20.31). So \(\min (\dim W + 1, 6) \ge 4 + f - \operatorname {def}(\tilde G)\), which is \((\star )\) at \(k = 3\). By lemma 19.32 and lemma 30.20 the general configuration of \(G\) attains.

At a common zero \((z, h)\) of the two functionals, \(h_a\) and \(h_b\) take the same values at \(q_a\) and at \(q_b\). As \(z_a = h_a \cdot (q_a, 1)\) and \(z_b = h_b \cdot (q_b, 1)\), the height \(z\) agrees with \(h_a\) on the closed neighbourhood of \(a\) in \(G'\) together with \(b\), and with \(h_b\) on that of \(b\) together with \(a\). At every other body \(v\), the closed neighbourhood in \(H\) lies in the one in \(G'\), on which \(z\) agrees with \(h_v\). So \(z \in L_H(q)\). At a picture admissible for \(G'\) a solution is determined by its heights, and the solutions form a space of dimension \(\dim L_{G'}(q)\) (definition 30.15). So the common zeros of the two functionals form a space of dimension at most \(\dim L_H(q) \le \dim L_{G'}(q) - 2\), and the two are independent.

As \(\dim D(L) \ge 2\) and the constant affine functions form a line, some \(z_1 \in L\) has \(D(z_1)\) nonconstant. One of the first two coefficients of \(D(z)\) is a linear polynomial in \(z\) nonzero at \(z_1\). So some \(z_0 \in L\) has both \(A(z_0) \neq 0\) and \(D(z_0)\) nonconstant (lemma 30.26). As \(\dim D(L) \ge 2\), some \(u \in L\) has \(D(u)\) not a multiple of \(D(z_0)\).

For \(q \in K^2\) put \(w(q) = D(u)(q)\, z_0 - D(z_0)(q)\, u \in L\); then \(D(w(q))\) vanishes at \(q\). An affine function vanishing on the zero line of the nonconstant \(D(z_0)\) is a multiple of it, so some \(q_0\) on that line has \(D(u)(q_0) \neq 0\), and there \(w(q_0) = D(u)(q_0)\, z_0\). So \((q, s) \mapsto A(s\, w(q))\) is a polynomial, nonzero at \(q_0\) and \(s = D(u)(q_0)^{-1}\). Its product with \(B\) is a nonzero polynomial, and at a non-root \((q, s)\) the point \(z = s\, w(q)\) of \(L\) has \(D(z) \cdot (q, 1) = 0\), \(A(z) \neq 0\) and \(B(q) \neq 0\).

The smaller graph. Let \(H'\) be \(H\) with \(x\) suppressed: the splitting-off \(H_x^{ab}\) (definition 21.10) whose new edge \(ab\) carries the label of the edge \(ax\). It is \(G'\) together with the edge \(ab\), simple since \(a\) and \(b\) are not adjacent, and \(\operatorname {def}_2(H') + 2 \le \operatorname {def}_2(G')\) (lemma 21.36, at \(D = 3\)).

The incidence. Take a picture \(q'\) off the zero sets of three nonzero polynomials: \(B\), the polynomial of the attainment at \(G'\), and the polynomial of Jackson and Jordán’s equality at \(H'\) (theorem 30.43). The equality applies to \(H'\), as its closed neighbourhoods contain those of \(G'\), which have three members at a picture admissible for \(G'\). Then \(\dim L_{H'}(q') = 3 + \operatorname {def}_2(H') \le \dim L_{G'}(q') - 2\) (lemma 30.34). So on the solutions of the lifting system of \(G'\) at \(q'\), the two functionals of lemma 30.91 are independent, and the linear map \((z, h) \mapsto h_a - h_b\) has rank at least two. Apply lemma 30.92 to this map. As \(A\) take the product of the two functionals and the polynomial in the heights of the attainment at \(G'\). Each factor is nonzero somewhere on the solutions, so the product is too (lemma 30.26). For the polynomial on \(K^2\) take \(B\) as a polynomial in the picture point of \(x\), with the picture fixed at \(q'\) elsewhere; it is nonzero at \(q'_x\). This gives a solution \((z', h')\) for \(G'\) with \(A(z', h') \neq 0\), and a picture point \(q_x\) with \(B \neq 0\) at which \(h'_a - h'_b\) vanishes.

The solution for \(H\). Let \(q\) be \(q'\) with \(x\) moved to \(q_x\). Then \(B(q) \neq 0\), and \(G'\) reads the same picture at \(q\) as at \(q'\) (lemma 30.5). Extend \((z', h')\) to \(x\). The height at \(x\) is \(h'_a \cdot (q_x, 1)\), which equals \(h'_b \cdot (q_x, 1)\). The coefficient vector at \(x\) is that of the affine function through the three points of its closed neighbourhood \(\{ a, x, b\} \) (lemma 30.64). This is a solution of the lifting system of \(H\) at \(q\), as the closed neighbourhoods of \(a\) and \(b\) in \(H\) are those in \(G'\) with \(x\) added. Read \(A\) on the solutions for \(H\) through their restriction to \(V_1\) (lemma 30.46). It is nonzero at this solution. Wherever it is nonzero, both functionals are nonzero, and the heights give \(G'\) its target (lemma 30.47).

The configuration. Apply lemma 30.93 to \(G\), with \(B\) the main-picture polynomial of \(G\) (lemma 30.23). It gives a main picture \(q\) of \(G\), and at \(q\) a solution of the lifting system of \(G\) with \(A \neq 0\). Its heights \(z \in L_G(q)\) give \(G[V_1]\) its target (lemma 30.47).

The count. The closed neighbourhood of \(x_1\) is \(\{ a, x_1, b\} \), and its picture points are not collinear, as \(q\) is admissible. So \(p_a\), \(p_{x_1}\) and \(p_b\) are linearly independent, and so are the two joins \(p_a \vee p_{x_1}\) and \(p_{x_1} \vee p_b\). Every hinge is nonzero at \(q\), and \(\dim (\rho + \Lambda ) \ge 2\) (lemma 30.70). By the hypothesis on the deficiencies this is \((\star )\) at \(k = 1\), so lemma 19.32 and lemma 30.20 apply.

Write \(W = \rho + \operatorname {span}(p_a \vee y, y \vee p_b)\). If \(W = \bigwedge ^2 K^4\), take \(x_1 = x_2 = y\). Otherwise note that \(\pi _a \neq \pi _b\), as \(p_b \notin \pi _a\), so the two planes meet in a line through \(y\); let \(u_0\) be a direction of it, of last coordinate \(0\). The points \(p_a\), \(y\), \(u_0\), \(p_b\) are linearly independent: both plane equations vanish at \(y\) and \(u_0\), the equation of \(\pi _a\) vanishes at \(p_a\) but not at \(p_b\) and that of \(\pi _b\) the other way round, and \(y\) is affine while \(u_0\) is not. Their six joins span \(\bigwedge ^2 K^4\) (lemma 30.77), and \(W\) contains \(p_a \vee y\) and \(y \vee p_b\), so one of \(y \vee u_0\), \(u_0 \vee p_b\), \(p_a \vee u_0\), \(p_a \vee p_b\) lies outside \(W\).

Put \(x_1 = y + t u\) and \(x_2 = y + \sigma (p_b - y)\), with \(\sigma \neq 1\) and \(u\) a direction in \(\pi _a\) of last coordinate \(0\). For \(t \neq 0\), the span of \(\rho \) and the three joins contains \(y \vee p_b\) (a nonzero multiple of \(x_2 \vee p_b\)), \(u \vee x_2\), and \(p_a \vee y + t\, (p_a \vee u)\). So if \(u \vee x_2 \notin W\), some \(t \neq 0\) gives dimension at least \(\dim W + 1\), since a vector outside a subspace stays outside it after adding \(t\) times another for some \(t \neq 0\) (as in the proof of lemma 30.81). It remains to choose \(u\) and \(\sigma \) with \(u \vee x_2 \notin W\). If \(y \vee u_0 \notin W\), take \(u = u_0\) and \(\sigma = 0\). Otherwise, for \(\sigma \notin \{ 0, 1\} \), the join \(u \vee x_2\) is modulo \(W\) a nonzero multiple of \(u_0 \vee p_b\) when \(u = u_0\), and of \(p_a \vee u_0\) for the same curve with the two planes exchanged. For \(u = p_a - y\) and \(\sigma \notin \{ 0, 1\} \) it is modulo \(W\) a nonzero multiple of \(p_a \vee p_b\).

Write \(f = \operatorname {def}(\tilde G[V_1])\), and compute ranks with the joins as hinges (lemma 30.9). The graph \(G_1\) has the open ear \(a - x_1 - b\) on \(V_1\) and the same \(G[V_1]\). It is simple, and its ends are not adjacent.

The configuration of \(G_1\). Apply lemma 30.93 to \(G_1\). As \(B\) take the product of the polynomial of the attainment at \(G_1\) and the main-picture polynomial of \(G\) (lemma 30.23). At the picture \(q\) that the lemma gives, the non-roots of \(A\) form a nonempty Zariski-open set of solutions of the lifting system of \(G_1\). As \(q\) is off the zero set of the polynomial of the attainment at \(G_1\), the solutions whose heights give \(G_1\) its target contain another such set. The two sets meet (lemma 30.26). Take a solution in both, with heights \(z\). Let \(\pi _a\) and \(\pi _b\) be the planes of its coefficient vectors at \(a\) and \(b\), and \(y\) the point of \(x_1\). Then \(y \in \pi _a \cap \pi _b\), as \(x_1\) is adjacent to \(a\) and to \(b\) in \(G_1\). Also \(p_b \notin \pi _a\) and \(p_a \notin \pi _b\), and \(G[V_1]\) and \(G_1\) have their targets. So the ear rank formula at \(G_1\) (lemma 19.32) gives

Putting \(x_2\) back. By lemma 30.95 there are affine points \(y_1 \in \pi _a\) and \(y_2 \in \pi _b\) with \(\dim (\rho + \Lambda ) \ge \min (s + 1, 6)\), where \(\Lambda = \operatorname {span}(p_a \vee y_1,\ y_1 \vee y_2, y_2 \vee p_b)\).

The configuration of \(G\). Take as base data the restriction of \((q, z)\) to \(V_1\), with the affine functions of \(\pi _a\) and \(\pi _b\) (definition 30.82). The points \(y_1\) and \(y_2\) at \(x_1\) and \(x_2\) are the configuration of one ear datum of \(G\). So the bound \(\dim (\rho + \Lambda ) \ge \min (s + 1, 6)\) holds off the zero set of a polynomial in the ear data that is nonzero there (lemma 30.85). The main-picture condition of \(G\) is a polynomial condition on the ear data too, nonzero at the ear datum with picture \(q\) (lemma 30.84). At a common non-root the picture is a main picture of \(G\), so the hinges are nonzero, and the heights lie in \(L_G\) (lemma 30.83). The rank of \(G[V_1]\) and \(\rho \) are those of the base data (lemma 30.47, lemma 30.5).

The count. \(\operatorname {def}(\tilde G_1) \le \operatorname {def}(\tilde G)\) (lemma 21.34), and \(\operatorname {def}(\tilde G) \ge f + k - 5 = f - 3\) (lemma 20.31). So \(\min (s + 1, 6) \ge 3 + f - \operatorname {def}(\tilde G)\), which is \((\star )\) at \(k = 2\). By lemma 19.32 and lemma 30.20 the general configuration of \(G\) attains.

By lemma 23.120 at \(n(0)\) there is a polynomial \(Q\) in the normal coordinates, nonzero at \(n(0)\), such that the rank is at least its value at \(n(0)\) wherever \(Q\) is nonzero. The polynomial \(t \mapsto Q(n(t))\) in one variable is nonzero at \(t = 0\), so it has finitely many roots, and off them the bound holds.

The panel support extensor is bilinear and alternating (definition 23.1), so up to sign the hinges at \(ax\) and \(xb\) are \(s\, C\) and \((1 - s)\, C\), where \(C \neq 0\) is the hinge at \(ab\) in \(G''\); every other edge has the same hinge in \(G\) and in \(G''\). Since \(S_x - S_b = (S_x - S_a) + (S_a - S_b)\), a screw assignment \(S\) is therefore a motion of \(G\) if and only if it is a motion of \(G''\) with \(S_x - S_a \in K C\). As \(x\) is not a body of \(G''\), the map \(S \mapsto S_x - S_a\) takes the motions of \(G''\) onto the screw space, so the motions of \(G\) have codimension \(D - 1\) in those of \(G''\). The row spaces are the annihilators of the motion spaces (lemma 19.25), so the ranks differ by \(D - 1\).

Suppose some solution \(y_0\) has \(h_a \neq h_b\). The affine function \(h_a - h_b\) vanishes at \(q_a\) and \(q_b\), and \(q_a\), \(q_b\), \(q_x\) are not collinear, as \(q\) is admissible for \(G\). So the functional \(\varphi (z, h) = (h_a - h_b) \cdot (q_x, 1)\) is nonzero at \(y_0\).

We compare the solution spaces of the three lifting systems at \(q\). First, every solution for \(G'\) is one for \(G''\). Indeed, the closed neighbourhoods of \(a\) and \(b\) in \(G''\) gain \(b\) and \(a\) respectively, and by hypothesis \(h_a \cdot (q_b, 1) = h_b \cdot (q_b, 1) = z_b\) and \(h_b \cdot (q_a, 1) = z_a\). Second, restriction to \(V_1\) maps the solutions for \(G\) injectively into the solutions for \(G'\) on which \(\varphi \) vanishes. A restricted solution is a solution for \(G'\), since each closed neighbourhood in \(G'\) lies inside the one in \(G\). And \(\varphi \) vanishes on it, since \(x\) lies in the closed neighbourhoods of both \(a\) and \(b\) in \(G\), so \(h_a\) and \(h_b\) both take the value \(z_x\) at \(q_x\). The map is injective: a solution for \(G\) that vanishes on \(V_1\) has \(z_x = h_a \cdot (q_x, 1) = 0\), so \(h_x\) vanishes at the three non-collinear points \(q_a, q_x, q_b\), and \(h_x = 0\) (as in lemma 30.64). Since \(\varphi (y_0) \neq 0\), the solutions for \(G\) form a space of smaller dimension than the solutions for \(G'\), and hence than those for \(G''\). At a picture admissible for a graph, the solutions form a space of dimension \(\dim L(q)\) (definition 30.15). So \(\dim L_G(q) {\lt} \dim L_{G''}(q)\), a contradiction.

If \(\varphi _0(W) \neq 0\) for some \(W \in L\), take \(w = \varphi _0(W)^{-1}\bigl(\psi (W)\, y_0 - \psi (y_0)\, W\bigr)\). Then \(\varphi _0(y_0 + t w) = -t\, \psi (y_0)\) and \(\psi (y_0 + t w) = \psi (y_0)\). (One finds this \(w\) by solving the condition on the line \(y_0 + K W\) at each \(t\). The solution is rational in \(t\), and as the condition is homogeneous it can be rescaled to clear the denominator.) Otherwise \(\varphi _0\) and \(\psi \) vanish on \(L\), and \(w = 0\) will do.

At a body \(v\) of \(V_1\), the closed neighbourhood in \(G\) is the one in \(G[V_1]\), with \(x\) added when \(v\) is \(a\) or \(b\). The height agrees with \(h_v\) on it: at \(x\), because \(h_a\) and \(h_b\) agree at \((q_x, 1)\). The closed neighbourhood of \(x\) is \(\{ a, x, b\} \), and every height agrees with an affine function on three points (lemma 30.64).

The counts. The new edge \(ab\) lowers by five the value of every partition of \(V_1\) that separates \(a\) and \(b\), and by hypothesis the other partitions are already five below \(\operatorname {def}(\tilde G[V_1])\). So \(\operatorname {def}(\tilde G'') + 5 \le \operatorname {def}(\tilde G[V_1])\) (lemma 21.36, at \(D = 6\)). With \(\operatorname {def}(\tilde G[V_1]) \le \operatorname {def}(\tilde G) + 4\) (lemma 20.31, at \(k = 1\)) this gives \(\operatorname {def}(\tilde G'') + 1 \le \operatorname {def}(\tilde G)\). Also \(\operatorname {def}_2(G'') \le \operatorname {def}_2(G)\) (lemma 21.34, at \(d = 2\)).

The picture. As \(a\) and \(b\) are not adjacent, \(G''\) is simple and each of its closed neighbourhoods has as many members as in \(G\), at least three. So Jackson and Jordán’s equality applies to \(G''\) (theorem 30.43). Take a picture \(q\) off the zero sets of three nonzero polynomials: that of the attainment at \(G''\), that of the equality at \(G''\), and the main-picture polynomial of \(G\) (lemma 30.23). Then \(q\) is admissible for \(G''\) and a main picture of \(G\), some \(z'' \in L_{G''}(q)\) gives \(G''\) its target rank, and

(lemma 30.34). Let \(y_0 = (z'', h)\) be a solution of the lifting system of \(G''\) at \(q\) with heights \(z''\). It is a solution for \(G[V_1]\), and \(h_a - h_b\) vanishes at \((q_a, 1)\) and \((q_b, 1)\), since \(ab\) is an edge of \(G''\).

The special position. If some solution for \(G[V_1]\) has \(h_a - h_b\) nonzero at \((q_a, 1)\) or at \((q_b, 1)\), then for all but at most one \(s\) it is nonzero at \((1 - s)(q_a, 1) + s\, (q_b, 1)\). Otherwise every solution for \(G[V_1]\) has \(h_a = h_b\) (lemma 30.99). Fix \(s \neq 0, 1\) accordingly, put \(q^0_x = (1 - s) q_a + s\, q_b\), and let \(\varphi _0(z, h) = (h_a - h_b) \cdot (q^0_x, 1)\) and \(\psi (z, h) = (h_a - h_b) \cdot (q_x - q^0_x, 0)\). By lemma 30.100 there is a solution \(w\) for \(G[V_1]\) such that \(y(t) = y_0 + t w\) satisfies \(\varphi _0(y(t)) + t\, \psi (y(t)) = 0\) for every \(t\).

The curve. Let \(q(t)\) be \(q\) with the picture of \(x\) moved to \(q^0_x + t\, (q_x - q^0_x)\), so that \(q(1) = q\). As \(q(t)\) agrees with \(q\) on \(V_1\), \(y(t)\) is a solution for \(G[V_1]\) at \(q(t)\) (lemma 30.22), and the identity above says that its \(h_a - h_b\) vanishes at \((q(t)_x, 1)\). Let \(z(t)\) be the height of \(y(t)\) on \(V_1\), extended by \(h_a \cdot (q(t)_x, 1)\) at \(x\). The points of the configuration \((q(t), z(t))\) are polynomials in \(t\). At \(t = 0\) they are at the special position: \(p_x = (1 - s) p_a + s\, p_b\), because \(h_a\) takes the values \(z''_a\) and \(z''_b\) at \(q_a\) and \(q_b\). Also \(p_a\) and \(p_b\) are independent, as \(q_a \neq q_b\). Every hinge of \(G\) is nonzero at \(t = 0\): \(q^0_x\) differs from \(q_a\) and \(q_b\) as \(s \neq 0, 1\), and every other edge is an edge of \(G''\). So at \(t = 0\) the rank of \(G\) is that of \(G''\) at \((q, z'')\) plus five (lemma 30.98, lemma 30.47).

A good parameter. By lemma 30.97, the rank of \(G\) at \((q(t), z(t))\) is at least its value at \(t = 0\) for all but finitely many \(t\). The main-picture polynomial of \(G\) at \(q(t)\) is a polynomial in \(t\), nonzero at \(t = 1\), so \(q(t)\) is a main picture of \(G\) for all but finitely many \(t\). Take \(t\) outside both finite sets. Then \(z(t) \in L_G(q(t))\) (lemma 30.101), and \((q(t), z(t))\) has rank at least

as \(|V(G)| = |V_1| + 1\). By lemma 30.20 the general configuration of \(G\) attains.

Both \(P\) and the partition into singletons are separated by \(Z\), so each splits at \(Z\) with the same cut term (lemma 20.26). On \(G[Z]\) the partition \(P\) has one part and value zero, and the singletons have value \(s_D(Z)\). The second statement follows by induction on the number of bodies, since \(s_D(Z) \ge 0\) and the rest is the statement at \(G[V(G) \setminus Z]\). The first follows from one such split and the second statement at \(G[V(G) \setminus Z]\).

The number of parts grows by one exactly when \(u\) is alone, and the crossing edges gain the \(t\) edges at \(u\). For the consequence, let \(P\) partition \(W \cup \{ u\} \). If \(u\) shares a part, the value of \(P\) is at most that of \(P|_W\), which is at most zero. If \(u\) is alone, both of its edges into \(W\) cross, and the value is at most \(0 + D - 2(D - 1) {\lt} 0\).

Take \(Y \subseteq X\) minimal with \(|Y| \ge 2\) and \(s_D(Y) \le 0\). Every nonempty proper subset of \(Y\) then has nonnegative singleton value, a singleton having value zero. A partition of \(Y\) other than \(\{ Y\} \) has a part \(Z \subsetneq Y\), and its value is at most \(s_D(Y) - s_D(Z) \le 0\) (lemma 30.103); the partition \(\{ Y\} \) has value zero.

(1) Let \(P'\) merge the parts of \(P\) that meet \(W\) into one part. If \(m\) parts of \(P\) meet \(W\) and \(e\) edges join distinct ones among them, merging them lowers the value by \(D(m - 1) - (D - 1)e\) (lemma 27.7). The \(e\) edges include the edges of \(G[W]\) crossing \(P|_W\), so this quantity is at most the value of \(P|_W\) in \(G[W]\), which is at most zero.

(2) The merged deficiency is a maximum over fewer partitions. For the equality, merge a partition attaining \(\operatorname {def}(\tilde G)\) along \(W\) by (1); the result has \(a\) and \(b\) in one part.

Let \(P\) be a partition of \(X\), with values \(v_2(P)\) and \(v_3(P)\) in \(G[X]\). If \(P\) has a part \(Z\) with \(2 \le |Z| {\lt} |X|\), then \(v_2(P) \le s(X) - s(Z) \le 0\) (lemma 30.103), so \(v_3(P) \le 0\). The partition \(\{ X\} \) has \(v_3 = 0\). For the partition into singletons, \(v_2 = s(X) \le 1\) gives \(2e(X) \ge 3|X| - 4 \ge 5\), so \(e(X) \ge 3\) and \(v_3 = 2s(X) - e(X) {\lt} 0\).

(1) Let \(P\) be a partition of \(Y\), with values taken in \(G[Y]\). Let \(P'\) be a partition coarser than \(P\) with \(W\) inside one part \(Q\) and \(v_3(P) \le v_3(P')\) (lemma 30.106). If \(Q = Y\) then \(v_3(P') = 0\). Otherwise \(v_2(P') \le s(Y) - s(Q) \le 0\) (lemma 30.103), so \(v_3(P') \le 2v_2(P') \le 0\).

(2) Suppose not, and take \(X \subseteq V(G)\) with \(W \subseteq X\) and \(s(X) {\lt} s(W)\), of least size. Then \(X \ne W\), and \(s(W) \le s(Q)\) for every \(Q \supseteq W\) with fewer bodies than \(X\). If \(X\) avoids \(X_0\), then (1) at \(Y = X\) makes \(G[X]\) rigid, against the maximality of \(W\). Otherwise \(X_0 = \{ x\} \) with \(x \in X\); put \(Y = X \setminus \{ x\} \), which contains \(W\). Adding \(x\) to \(Y\) changes the singleton value by \(3 - 2t\), where \(t \le 2\) is the number of edges from \(x\) into \(Y\) (lemma 30.104), and \(t \le 1\) if \(Y = W\). So \(Y \ne W\), since otherwise \(s(X) \ge s(W) + 1\). Then \(s(Y) \ge s(W)\), as \(Y\) has fewer bodies than \(X\). Also \(s(X) \ge s(Y) - 1\), so \(s(X) = s(W) - 1\) and \(s(Y) = s(W)\). Now (1) at \(Y\) makes \(G[Y]\) rigid, a set avoiding \(X_0\) that strictly contains \(W\), against the maximality of \(W\).

(1) The parts and the crossing edges correspond under \(c\).

(2) By lemma 30.103 and sparsity, \(\operatorname {def}_2(G[W]) \le s(W)\). Also \(s(V(G)) \le \operatorname {def}_2(G)\), as \(s(V(G))\) is the value of the singletons. A partition of \(G/G[W]\) pulls back by (1) to a partition of \(V(G)\) with \(W\) inside one part \(Z\). The deleted edges of \(G[W]\) lie inside \(Z\), so the pull-back has the same value in \(G\). That value is at most \(s(V(G)) - s(Z) \le s(V(G)) - s(W)\) (lemma 30.103). Adding the three bounds gives the claim.

(1) An edge \(ab\) would give the triangle \(\{ x, a, b\} \) singleton value zero. Let a partition of \(V_1\) with \(a\) and \(b\) in one part \(Z\) attain the merged deficiency. Its value plus \(s(Z)\) is at most \(s(V_1) \le \operatorname {def}_2(G[V_1])\) (lemma 30.103), so it suffices that \(s(Z) \ge 2\). Adding \(x\) to \(Z\) lowers the singleton value by at least one, as \(x\) has at least two edges into \(Z\) (lemma 30.104). As \(s(Z \cup \{ x\} ) \ge 1\) by sparsity, \(s(Z) \ge 2\).

(2) Suppose otherwise, and let \(P\) attain the merged deficiency \(m\), with \(Q\) the part of \(a\) and \(b\). Refining \(Q\) by a partition \(R\) of \(Q\), relabelled apart from the other parts of \(P\), changes the value of \(P\) by the value of \(R\) in \(G[Q]\) (lemma 20.26). Since \(\operatorname {def}_3(G[V_1]) \le m + 4\), every such \(R\) has value at most \(4\) in \(G[Q]\), and at most \(0\) if it keeps \(a\) and \(b\) together.

We show that \(G[Q \cup \{ x\} ]\) is rigid, against the hypothesis. Let \(R'\) be a partition of \(Q \cup \{ x\} \), and compare its value with that of its restriction to \(Q\) (lemma 30.104). If \(x\) is alone, at least its two edges to \(a\) and \(b\) cross, and the value drops by at least \(10 - 6 = 4\), to at most \(0\). If \(x\) shares a part, each edge from \(x\) to another part costs \(5\). When \(a\) and \(b\) are apart, at least one such edge exists, and the value is at most \(4 - 5 {\lt} 0\). When they are together, the value is at most \(0\).

Otherwise the part \(Z\) of \(a\) and \(b\) in a partition attaining the planar merged deficiency has \(s(Z) \le 1\) (lemma 30.103). The pair \(\{ a, b\} \) has singleton value \(3\), as \(a\) and \(b\) are not adjacent (lemma 30.104), so \(|Z| \ge 3\), and \(Z\) is rigid (lemma 30.107). Then \(\operatorname {def}_3(G; a, b) = \operatorname {def}_3(G)\) (lemma 30.106).

(1) Each clause of definition 30.112, and each alternative in clauses (5) and (7), states the further conditions of one step, with \(\mathcal P\) at the graphs that the step assumes to attain. The steps are equal deficiencies (corollary 30.44), the cut vertex and the chain of bridges (theorem 30.49, theorem 30.52), the cycle (theorem 30.71), the open ears (theorem 30.72, theorem 30.88, theorem 30.89, theorem 30.86, theorem 30.96, theorem 30.94), the split-off (theorem 30.102) and the two contractions (corollary 30.62, theorem 30.61). As every graph satisfying \(\mathcal P\) attains, the step applies.

(2) Strong induction on the number of bodies, by (1) with \(\mathcal P\) the property of being smaller than \(G\).

Extend a path through the body along bodies of degree two as far as possible (Graph.IsOpenEar.exists_maximal). An interior body has both of its neighbours on the path. So if an end had degree two, its other neighbour would either extend the path or be the other end. In the second case the path and that edge form a cycle. Either the cycle is all of \(G\), and every body of \(G\) has degree two, or the cycle meets the rest of \(G\) only at the other end, which is then a cut vertex. So both ends have degree at least three.

(1) The ends of the chain keep at least two neighbours in \(V_1\), and every other body of \(V_1\) keeps all of them. In \(G - x_1\) no body of \(V_1\) other than \(b\) has a neighbour outside \(V_1\). So if \(G[V_1]\) were disconnected, a part of it that avoids \(b\) would be left by no edge of \(G - x_1\), which is connected (Graph.Connected.induce_of_gate).

(2) The new edge is not parallel to an old one, since \(u\) and \(w\) are not adjacent, and \(u\) and \(w\) each exchange the edge to \(v\) for the new edge. A set of bodies left by no edge of \(G_v^{uw}\) contains both or neither of \(u\) and \(w\), and then, with \(v\) added in the first case, it is left by no edge of \(G\).

By the cycle clause. Extend a path through a body as far as possible along bodies of degree two (Graph.IsOpenEar.exists_maximal). As every body has degree two, the path stops only when its ends are adjacent. The path with that edge is then closed under adjacency, and so it is all of \(G\), as \(G\) is connected.

(1) By the clause of a chain of bridges. As \(G\) is connected but not 2-edge-connected, some nonempty proper set \(V'\) of bodies is left by exactly one edge; its sides are \(V'\) and \(V(G) \setminus V'\). Grow a path from that edge, keeping the two sides: an end of degree two has its other edge inside its own side, since that edge is not the one leaving \(V'\), so the path extends into that side, which loses the end. The path stops when both ends have degree at least three. Each side is then connected, since its end is its only body with a neighbour outside it. The end loses exactly one edge, so it keeps degree at least two, and every other body keeps all its neighbours (Graph.IsX0Graph.induce_of_gate).

(2) By the cut-vertex clause at \(v\). As \(G - v\) is disconnected, it has a nonempty set \(C\) of bodies, closed under adjacency in \(G - v\) and not all of its bodies. The sides are \(C \cup \{ v\} \) and \(V(G) \setminus C\). The body \(v\) sends at least two edges into each side, since a single one would be a bridge, and every other body keeps all its neighbours. Each side is connected, since \(v\) is its only body with a neighbour outside it (Graph.IsX0Graph.induce_of_gate).

By the ear clause or the split-off clause of definition 30.112 at the usable chain, on \(V_1\). The graphs \(G[V_1]\) and \(G\) with an interior body suppressed have fewer bodies than \(G\). Both satisfy the standing hypotheses (lemma 30.116). For the second, the two neighbours of the suppressed body must not be adjacent. When \(k \ge 2\), one of them is an interior body, adjacent only to the bodies next to it on the chain. When \(k = 1\), they are \(a\) and \(b\), which are not adjacent as \(G\) is sparse (lemma 30.110).

Where \(\operatorname {def}_3(G[V_1]; a, b) = \operatorname {def}_3(G[V_1])\), some partition attaining \(\operatorname {def}_3(G[V_1])\) has \(a\) and \(b\) in one part. So \(\operatorname {def}_3(G[V_1]) \le \operatorname {def}_3(G)\) (lemma 20.32), the bound that the ear steps with one or two interior bodies assume. At \(k = 1\) sparsity also gives \(\operatorname {def}_2(G[V_1]; a, b) + 2 \le \operatorname {def}_2(G[V_1])\) (lemma 30.110).

By the contraction clause at a maximal planar-rigid set \(W\). It is proper, since \(\operatorname {def}_2(G) {\gt} 0\). A body outside \(W\) with two neighbours in \(W\) would enlarge it (lemma 30.104, at \(d = 2\)). So no such body exists, and \(G/G[W]\) is smaller than \(G\) (lemma 30.54, lemma 21.51).

(1) Sparsity at \(V(G)\) gives \(2|E(G)| \le 3|V(G)| - 4\), so the sum of \(3 - \deg v\) over the bodies is at least \(4\); each term is at most one, and positive only at degree two.

(2) With \(m\) bodies of degree two, their \(2m\) edges are distinct, so \(|E(G)| \ge 2m\), while \(2|E(G)| \ge 2m + 3(|V(G)| - m)\). Hence \(5|E(G)| \ge 6|V(G)|\), and removing \(x\) removes one body and two edges.

The chains. As \(G\) has no planar-rigid set, it is sparse (lemma 30.105). So it has bodies of degree two (lemma 30.122), and each of them lies on a chain (lemma 30.115). No chain has three or more interior bodies, as such a chain is usable. Let a chain \(a - x_1 - x_2 - b\) on \(V_1\) have two. Its ends are adjacent. Otherwise, as the chain is not usable, \(\operatorname {def}_3(G[V_1]; a, b) \neq \operatorname {def}_3(G[V_1])\), and lemma 30.111 at \(G[V_1]\) would make it usable. So \(a\), \(x_1\), \(x_2\) and \(b\) induce a \(4\)-cycle, which is tight and so rigid (lemma 30.107). The body of a chain with one interior body lies in a rigid set, since otherwise the chain would be usable (lemma 30.110).

When \(\operatorname {def}_3(G) {\gt} 0\). Take a rigid set given by a chain, and a maximal rigid set \(W\) containing it. Then \(W \neq V(G)\). A body outside \(W\) has at most one neighbour in \(W\), as otherwise it would enlarge \(W\) (lemma 30.104). By lemma 30.108 with \(X_0\) empty, \(s(W) \le s(X)\) for every \(X \supseteq W\), and lemma 30.109 gives the inequality.

When \(\operatorname {def}_3(G) = 0\). We take a body \(x\) of degree two and a rigid set that avoids it. If two bodies of degree two are adjacent, they lie on a chain with two interior bodies, whose \(4\)-cycle is rigid. The \(4\)-cycle contains two bodies of degree two, and at least four bodies have degree two (lemma 30.122). So some body \(x\) of degree two lies off it. Otherwise let \(x\) be any body of degree two. Then \(s_6(V(G) \setminus \{ x\} ) \le 0\) (lemma 30.122), so \(V(G) \setminus \{ x\} \) contains a rigid set (lemma 30.105).

Let \(W\) be maximal among the rigid sets that avoid \(x\) and contain the one taken. The body \(x\) has at most one neighbour in \(W\). A neighbour of degree two would have only one neighbour in \(W\), while every body of a rigid set has at least two neighbours in it (lemma 20.23). Two neighbours \(a\) and \(b\) of degree at least three would make \(a - x - b\) a chain on \(V(G) \setminus \{ x\} \). As \(W\) is rigid and contains \(a\) and \(b\), the merged deficiency of \(G - x\) at \(a\) and \(b\) would equal its deficiency (lemma 30.106), and the chain would be usable. Every other body outside \(W\) has at most one neighbour in \(W\), by the maximality of \(W\) (lemma 30.104). Lemma 30.108 with \(X_0 = \{ x\} \) and lemma 30.109 give the inequality.

Let \(G\) satisfy the standing hypotheses. We take the cases of the table in order. If \(G\) is not 2-edge-connected, or \(G - v\) is disconnected for some body \(v\), then \(G\) reduces by lemma 30.118. If every body has degree two, it reduces by lemma 30.117. If \(\operatorname {def}_2(G) = \operatorname {def}_3(G)\), it reduces by clause (1) of definition 30.112. Otherwise \(\operatorname {def}_2(G) {\gt} 0\), as \(\operatorname {def}_3(G) \le \operatorname {def}_2(G)\) (lemma 20.7) and a deficiency is never negative. Then a planar-rigid set gives a reduction (lemma 30.121).

In the remaining case \(G\) is sparse (lemma 30.105), 2-connected, and has a body of degree other than two. A usable chain gives a reduction (lemma 30.120). If there is none, theorem 30.123 gives a core \(W\). Then \(G[W]\) and \(G/G[W]\) are smaller than \(G\) (lemma 30.53, lemma 30.54, lemma 21.51), and \(G\) reduces by clause (7) at \(W\).

The rest is theorem 30.113. A simple 2-edge-connected graph on at least three bodies is connected (definition 20.20), since no edge leaves the vertices of a component. Its minimum degree is at least two (lemma 20.21), so it satisfies the standing hypotheses.

A simple two-edge-connected multigraph on at least three bodies is connected (definition 20.20), since no edge leaves the vertices of a component. Its minimum degree is at least two (lemma 20.21), so it satisfies the standing hypotheses (definition 30.45). Its general configuration attains (theorem 30.124). One attaining configuration is such a realization (lemma 30.25). No hypothesis on the number of edge labels is used.

At a nondegenerate realization the normals of a closed hub-neighbourhood of \(v\) are independent and all orthogonal to the nonzero point of \(v\), so there are at most three of them.

Let \(x\), \(y\), \(z\) be a triangle with hubs \(y\) and \(z\). At a nondegenerate realization the normals of \(y\) and \(z\) are independent. The points of \(x\), \(y\) and \(z\) are orthogonal to both, so they lie in a two-dimensional space \(W\), which the independent points of \(y\) and \(z\) span. If \(x\) is not a hub, its closed neighbourhood is \(\{ x, y, z\} \), and its three points cannot be independent. If \(x\) is a hub, its normal is orthogonal to \(W\), so it lies in the span of the normals of \(y\) and \(z\), and the closed hub-neighbourhood of \(x\), which contains \(x\), \(y\) and \(z\), has dependent normals.

At a nondegenerate realization the normals of \(a\) and \(b\) are linearly independent, since both lie in the closed hub-neighbourhood of \(x\). The point of a body lies in the panel of each of its neighbours, as noted after definition 29.23. So the points of \(x\), \(y\) and \(z\) are orthogonal to both normals, and three independent points would lie in a two-dimensional space. Over a picture at which \(q_x\), \(q_y\) and \(q_z\) are not collinear, the configuration points of \(x\), \(y\) and \(z\) are independent whatever the heights, since dropping the height coordinate maps them to independent homogeneous picture points.

Equal planes. Give the new body \(x\) of \(G_{uw}\) (definition 30.128) any picture point. A height of \(G\) whose planes at \(u\) and \(w\) agree extends to \(G_{uw}\): the height of \(x\) is the value of the common plane at its picture point. The closed neighbourhoods of \(u\) and \(w\) gain \(x\), which lies on that plane, and the plane agrees with the heights on the closed neighbourhood \(\{ x, u, w\} \) of \(x\). This extension is injective.

The count. We show that \(\operatorname {def}_2(G_{uw}) \le \operatorname {def}_2(G) - 1\). Compare the value of a partition of the bodies of \(G_{uw}\) with that of its restriction to \(V(G)\), which is at most \(\operatorname {def}_2(G)\). If \(x\) is alone in its part, there is one more part and there are two more crossing edges, so the value is lower by \(2 \cdot 2 - 3 = 1\). Otherwise the number of parts is the same, and the value is lower by \(2\) for each of \(u\) and \(w\) outside the part of \(x\). It remains that \(x\), \(u\) and \(w\) share a part. Then the value is that of the restriction, in which \(u\) and \(w\) share a part \(X\). As \(G\) has no planar-rigid set, every set of at least two bodies has singleton value at least one (lemma 30.105). So the restriction has value at most \(s(V(G)) - s(X) \le \operatorname {def}_2(G) - 1\) (lemma 30.103).

The equality. At a picture general for both graphs, Jackson and Jordán’s equality (theorem 30.43) gives \(\dim L_G(q) = 3 + \operatorname {def}_2(G)\), and the lifting space of \(G_{uw}\) has dimension \(3 + \operatorname {def}_2(G_{uw}) \le 2 + \operatorname {def}_2(G)\). So the heights of \(G\) with equal planes at \(u\) and \(w\) form a proper subspace of \(L_G(q)\). The polynomial of the equality at \(G_{uw}\) involves the picture point of \(x\). Fixing that point at one non-root turns it into a nonzero polynomial in the picture coordinates of \(G\), and \(P\) is its product with the polynomial of the equality at \(G\).

The configuration is a pencil realization with adjacent points distinct (lemma 30.11). A body that is not a hub has at most three members in its closed neighbourhood, and admissibility puts three with non-collinear picture points in it. So the points of its closed neighbourhood are independent.

Each normal is a nonzero multiple of \((h_0, h_1, -1, h_2)\), where \(\zeta = h_0 \xi + h_1 \eta + h_2\) is the plane of the body (lemma 30.12). So normals are linearly independent exactly when the planes are affinely independent. At a hub \(v\) with one hub neighbour this is the first condition. Let \(v\) have two hub neighbours \(w_1\) and \(w_2\), and let \(d_i\) be the difference of the planes of \(w_i\) and \(v\), an affine function. It is nonzero by the first condition, and it vanishes at \(q_v\) and at \(q_{w_i}\), where both planes take the height of the body. As \(q_v\), \(q_{w_1}\) and \(q_{w_2}\) are not collinear, \(d_2\) does not vanish at \(q_{w_1}\), nor \(d_1\) at \(q_{w_2}\). Evaluating a linear relation between \(d_1\) and \(d_2\) at \(q_{w_1}\) and at \(q_{w_2}\) shows that \(d_1\) and \(d_2\) are independent, so the three planes are affinely independent. The closed hub-neighbourhood of a body that is not a hub consists of at most two of its neighbours, and for two this is the second condition.

Every closed hub-neighbourhood has at most three members (lemma 30.126). Take a picture \(q\) off the zero sets of three kinds of nonzero polynomial: one witnessing that the general configuration of \(G\) attains (theorem 30.124), whose non-roots are admissible; for every three distinct bodies, the determinant of their homogeneous picture points; and for every two distinct bodies, the polynomial of lemma 30.130.

Over \(q\), consider the pairs \((z, h)\) of a height \(z \in L(q)\) and, at every body \(v\), an affine function \(h_v\) that agrees with \(z\) on the closed neighbourhood of \(v\). They form a linear space. The attaining heights contain the non-roots of a polynomial in \(z\) that is nonzero at some pair. For two distinct bodies \(u\) and \(w\), some coordinate of \(h_u - h_w\) is nonzero at the pair that lemma 30.130 gives. One pair makes all these finitely many polynomials nonzero, by lemma 30.26 and induction on their number. At it the planes of any two distinct bodies differ, so the configuration \((q, z)\) is a nondegenerate pencil realization (lemma 30.131). Its height \(z\) is an attaining height, so the realization has the deficiency rank (lemma 30.9).

A parametrization. Let \(\Gamma \) be a graph whose closed hub-neighbourhoods have at most three members. For \(y_1, y_2, y_3 \in K^4\), let \(y_1 \times y_2 \times y_3\) be the vector with \((y_1 \times y_2 \times y_3) \cdot t = \det [y_1, y_2, y_3, t]\) for every \(t \in K^4\), as for the plane normal of definition 30.6. It is orthogonal to \(y_1\), \(y_2\) and \(y_3\), its coordinates are polynomials in theirs, and it is nonzero exactly when they are linearly independent. The parameters are a vector \(m_v \in K^4\) and three auxiliary vectors at every body \(v\). The normal of a hub \(v\) is \(m_v\). The point of a body \(v\) is \(p_v = y_1 \times y_2 \times y_3\), where \(y_1, y_2, y_3\) are the vectors \(m_w\) for \(w\) in the closed hub-neighbourhood of \(v\), in a fixed order, completed to three vectors by auxiliary vectors of \(v\). At a body that is not a hub, the closed neighbourhood has at most three members, and the normal is any nonzero vector orthogonal to their points. When it has three members with independent points, this normal is, up to a scalar, the product of their points, a polynomial in the parameters.

Each point is then orthogonal to the normals of the bodies of its closed neighbourhood: to those of hubs by construction, and to the others by the choice of their normals. At parameters where every point is nonzero, adjacent points are independent, and the points of the closed neighbourhood of every body that is not a hub are independent, these points and normals form a nondegenerate realization of \(\Gamma \). Its hinge at each edge is the line through the points of the ends. The normals of each closed hub-neighbourhood are independent, as factors of a nonzero product. Each of these finitely many conditions says that a polynomial in the parameters is nonzero. If the rank is \(r\) at some parameters, it is at least \(r\) off the zero set of a polynomial that is nonzero there. Indeed, the hinge of an edge \(uv\) is \(p_u \vee p_v\) up to a scalar, and the rows of the rigidity matrix at an edge can be taken linear in its hinge, so the minors of the rigidity matrix are polynomials in the parameters.

Conversely, let a nondegenerate realization be given, and let \(m_v\) be its normal at every hub \(v\). At each body \(v\), the normals of the closed hub-neighbourhood are independent and orthogonal to \(p_v\). Completing them by auxiliary vectors to a basis of \(p_v^{\perp }\) makes \(y_1 \times y_2 \times y_3\) a nonzero multiple of \(p_v\). So the realization comes from some parameters, with its own normals at the bodies that are not hubs, up to a nonzero scalar at each point. These scalars change neither the nondegeneracy nor the line of any hinge, so they do not change the rank.

(a) We parametrize the realizations of \(G\), whose closed hub-neighbourhoods have at most three members (lemma 30.126). The hubs of \(H\) are hubs of \(G\), and each closed neighbourhood or closed hub-neighbourhood in \(H\) lies in the one in \(G\). So every condition of nondegeneracy for \(H\) is one for \(G\), except at a hub \(h\) of \(G\) that is a body but not a hub of \(H\). There the points of the closed neighbourhood of \(h\) in \(H\) must be independent. If \(h\) has at most one neighbour in \(H\), these are the points of a body or of the two ends of an edge of \(G\), independent at every nondegenerate realization of \(G\). Let \(h\) have two neighbours \(u\) and \(w\) in \(H\). By hypothesis the closed hub-neighbourhood of \(h\) in \(G\) lies in \(\{ h, u, w\} \). Let \(m_h\), \(m_u\) and \(m_w\) be the first three standard basis vectors, and \(m_v\) the fourth at every other body. As no triangle of \(G\) contains two hubs (lemma 30.126), the auxiliary vectors can be chosen among the standard basis vectors so that \(p_h\), \(p_u\) and \(p_w\) are multiples of the fourth, third and second. So each of these conditions, and each condition of nondegeneracy for \(G\), holds off the zero set of a polynomial that is nonzero at some parameters: those just chosen, or those of a nondegenerate realization of \(G\). As \(K\) is infinite, these finitely many polynomials have a common non-root, and it gives the realization.

(b) The graph \(H\) is nondegeneracy-feasible, so loopless: the two ends of a loop are one body, whose concurrency point is not projectively distinct from itself. We parametrize the realizations of \(H\), and take parameters \(s_1\) and \(s_2\) that give the generic realization and the second one, up to scalars. The conditions of nondegeneracy, and the rank at \(s_1\) as a lower bound, hold off the zero sets of polynomials nonzero at \(s_1\). Each member of an \(S_i\), and each \(w_j\), is a hub, whose normal is a parameter, or its normal is the product of the three points of its closed neighbourhood. So at \(s_2\) these normals are those of the second realization, up to scalars. Hence the independence on each \(S_i\), and each condition \(p_{u_j} \cdot n_{w_j} \neq 0\), holds off the zero set of a polynomial nonzero at \(s_2\). A common non-root of all these polynomials gives a nondegenerate realization with the conditions, of rank at least the deficiency rank. The deficiency rank bounds the rank of every realization (lemma 20.16), so the realization is generic.

Let \(H = G - x\). An end of degree three is not a hub of \(H\), and its only neighbour in \(G\) outside \(H\) is \(x\), which is not a hub. So \(G\) has a nondegenerate realization whose restriction to \(H\) is nondegenerate (lemma 30.133(a)), and by hypothesis \(H\) has a generic realization. At the realization of \(G\), the normals \(n_a\) and \(n_b\) are independent, as \(a\) and \(b\) lie in the closed hub-neighbourhood of \(x\). Also \(p_a \cdot n_b\) and \(p_b \cdot n_a\) are not both zero. Otherwise the independent points of \(x\), \(a\) and \(b\) would all lie in the two-dimensional space \(n_a^{\perp } \cap n_b^{\perp }\).

We apply lemma 30.133(b) to \(H\). The sets are the closed hub-neighbourhoods in \(G\) of the bodies of \(H\), and \(\{ a, b\} \). The one pair is \((a, b)\) if \(p_a \cdot n_b \neq 0\), and \((b, a)\) otherwise. Every member of these sets is a hub of \(G\). A hub of \(G\) that is not a hub of \(H\) is an end of degree three, and its closed neighbourhood in \(H\) has three members. So \(H\) has a generic realization with three properties: the normals of every closed hub-neighbourhood in \(G\) are independent, \(n_a\) and \(n_b\) are independent, and \(p_a \cdot n_b\) or \(p_b \cdot n_a\) is nonzero.

As \(p_a \cdot n_b\) or \(p_b \cdot n_a\) is nonzero, \(p_a\) and \(p_b\) do not both lie on the line \(n_a^{\perp } \cap n_b^{\perp }\), where the panels of \(a\) and \(b\) meet. Nor are they proportional: each lies in its own panel, so both products would vanish. So this line is not the line through \(p_a\) and \(p_b\). Put \(p_x\) on the first line and off the second. Give \(x\) the plane through \(p_a\), \(p_x\) and \(p_b\), and give the edges \(ax\) and \(xb\) the lines through the points of their ends. This is a nondegenerate realization of \(G\). The normals of the closed hub-neighbourhoods in \(G\) of the bodies of \(H\) are independent, and so are those of the closed hub-neighbourhood \(\{ a, b\} \) of \(x\). The points of the closed neighbourhood \(\{ x, a, b\} \) of \(x\) are independent. A body of \(H\) that is not a hub of \(G\) is not \(a\) or \(b\), so it has the same closed neighbourhood in \(H\).

The new hinges \(p_a \vee p_x\) and \(p_x \vee p_b\) are independent, so the rank rises by at least six (lemma 19.32). As \(\operatorname {def}_3(G - x) \le \operatorname {def}_3(G)\), the rank of \(G\) reaches its deficiency rank, which bounds it (lemma 20.16).

Let \(H = G - \{ x_1, x_2\} \). Of the bodies of \(H\), only \(c\) can stop being a hub. If it does, it has degree four in \(G\), and its closed neighbourhood in \(H\) has three members. The neighbours it loses, \(x_1\) and \(x_2\), are not hubs. So \(H\) is nondegeneracy-feasible, by a realization whose normals are independent on the closed hub-neighbourhood in \(G\) of each body of \(H\) (lemma 30.133(a)). By hypothesis \(H\) has a generic realization. Every member of these closed hub-neighbourhoods is a hub of \(G\), so \(H\) has a generic realization with the same property (lemma 30.133(b), with these sets).

Put \(p_{x_1}\) and \(p_{x_2}\) in the panel of \(c\), with \(p_c\), \(p_{x_1}\) and \(p_{x_2}\) independent. Give \(x_1\) and \(x_2\) the panel of \(c\), and give the three edges of the triangle the lines through the points of their ends. This is a nondegenerate realization of \(G\). The closed hub-neighbourhoods in \(G\) of the bodies of \(H\) contain neither \(x_1\) nor \(x_2\), which are not hubs, so their normals are independent. That of \(x_i\) is \(\{ c\} \). The closed neighbourhood of \(x_i\) is \(\{ x_1, x_2, c\} \), with independent points. A body of \(H\) that is not a hub of \(G\) is not \(c\), so it has the same closed neighbourhood in \(H\) and is not a hub of \(H\).

Extend \(p_c, p_{x_1}, p_{x_2}\) to a basis of \(K^4\). The three hinges of the triangle are three of the six joins of pairs of basis vectors, hence independent. So the triangle is rigid, of rank twelve (lemma 23.77). The ranks of \(H\) and of the triangle add at the cut vertex \(c\) (lemma 19.26). The deficiencies add at \(c\) too (lemma 20.29), and that of the triangle is nonnegative. Hence the rank reaches the deficiency rank of \(G\), which bounds it (lemma 20.16).

Take a planar-rigid set \(W_0\) with as few bodies as possible. Then \(s(W_0) \le 0\), and every \(Z \subsetneq W_0\) with \(|Z| \ge 2\) has \(s(Z) \ge 1\) (lemma 30.105). Also \(|W_0| \ge 3\), since two bodies of a simple graph have singleton value at least one. Adding a body \(u\) to a set changes its singleton value by \(3 - 2t\), where \(t\) is the number of edges from \(u\) into the set (lemma 30.104). So every body of \(W_0\) has at least two neighbours in \(W_0\).

A triangle. Suppose two adjacent bodies \(y\) and \(u\) of \(W_0\) each have at most one neighbour in \(Z = W_0 \setminus \{ y, u\} \). Putting \(y\) and then \(u\) back into \(Z\) does not lower the singleton value: \(y\) adds at least \(3 - 2\), and \(u\), with at most two edges into \(Z \cup \{ y\} \), at least \(3 - 4\). So \(s(Z) \le s(W_0) \le 0\), and \(|Z| \le 1\). Hence \(W_0 = \{ y, u, w\} \), and \(s(W_0) \le 0\) makes it a triangle.

A body of degree two. Suppose every body of \(W_0\) is a hub. Then each has at most two hub neighbours, as its closed hub-neighbourhood has at most three members. So each has exactly two neighbours in \(W_0\). The argument above then applies at any edge of \(G[W_0]\), and \(W_0\) is a triangle of hubs, which is excluded. So some \(y \in W_0\) is not a hub. It has degree two, and both of its neighbours \(a\) and \(b\) lie in \(W_0\).

The two cases. If \(a\), say, is not a hub, it has degree two too, and the argument above at the edge \(ya\) gives a pendant triangle \(\{ y, a, w\} \). Its third body \(w\) has degree at least four. Indeed, at least two edges leave the triangle, as \(G\) is two-edge-connected on at least four bodies (definition 20.20), and all of them are at \(w\). If \(a\) and \(b\) are hubs, let \(X = W_0 \setminus \{ y\} \). As \(y\) has two edges into \(X\), \(s(X) = s(W_0) + 1 \le 1\). Also \(|X| \ge 3\): otherwise \(X = \{ a, b\} \), \(s(X) \le 1\) needs the edge \(ab\), and \(\{ y, a, b\} \) would be a triangle with two hubs. So \(X\) is rigid (lemma 30.107). As \(a\) and \(b\) lie in the rigid set \(X\) of \(G - y\), we have \(\operatorname {def}_3(G - y; a, b) = \operatorname {def}_3(G - y)\) (lemma 30.106). Putting \(y\) into the part of \(a\) and \(b\) does not change the value of a partition (lemma 20.32), which gives \(\operatorname {def}_3(G - y) \le \operatorname {def}_3(G)\).

As a simple two-edge-connected graph on at least three bodies, \(G\) satisfies the standing hypotheses (definition 30.45). If it has no planar-rigid set, it has a generic realization (theorem 30.132).

On three bodies \(G\) is a triangle: each body has degree at least two and, \(G\) being simple, at most one edge to each of the other two. A direct construction gives it a generic realization. The three bodies get linearly independent points and the plane through them as their common panel, and each edge gets the line through the points of its ends. No body is a hub, and the three points are independent, so the realization is nondegenerate. The three hinges are linearly independent, so the framework is infinitesimally rigid (lemma 23.77). Its rank is therefore the deficiency rank, as the triangle is rigid.

Otherwise \(G\) has at least four bodies. Its closed hub-neighbourhoods have at most three members, and no triangle contains two hubs (lemma 30.126). So \(G\) has a body of degree two whose neighbours are hubs and whose removal does not raise the deficiency, or a pendant triangle (lemma 30.136). Removing that body, or the two bodies of degree two of the triangle, leaves a simple graph with fewer bodies, at least one. By hypothesis it satisfies the conditioned pair, whose generic conjunct gives it a generic realization whenever it is nondegeneracy-feasible. So \(G\) has a generic realization (lemma 30.134, lemma 30.135).

We prove the contraction and split cases of theorem 29.30 as in the proof of theorem 29.33, with the generic statement replaced by the induction hypothesis. Let \(G\) be a loopless multigraph on at least three bodies, and suppose that every multigraph on fewer bodies, at least one, satisfies the conditioned pair.

If \(G\) is not two-edge-connected, it satisfies the conditioned pair by the cut-edge case of lemma 29.29. If it is two-edge-connected and not simple, its bare conjunct is lemma 29.32, and the other two are vacuous. At a simple two-edge-connected multigraph, the distinct conjunct is lemma 30.125, and the bare conjunct follows from it. The generic conjunct follows from the conditioned pair at the multigraphs with fewer bodies (theorem 30.137).

The first claim is theorem 30.124. The distinct statement is lemma 30.125. The generic statement is not proved from attaining configurations (lemma 30.127). It is the generic conjunct of the conditioned pair at a simple two-edge-connected multigraph on at least three bodies (theorem 30.138).