Proof. Fix integral homomorphisms \(f_1,\ldots,f_b\) forming a real basis of \(\mathop{\mathrm{Hom}}(G,G')_{\mathbb R}\). For each \(j\), ample domination of \(f_j^*L'\) by a power of \(L\) gives \(\widehat h_{L'}(f_j(x))\le C_j\widehat h_L(x)\) for all algebraic points \(x\). The same inequality holds on \(\mathcal H(G)\), by its quadratic-form interpretation on every finite-dimensional span. Hence \[\left\|\sum_j t_j f_j(x)\right\| \le \left(\sum_j |t_j|\sqrt{C_j}\right)\|x\|.\] The constants are independent of the fields of definition of the points. This proves the assertion about operator norms.

The underlying real Lie representation identifies with the real extension of the representation on rational homology. It is faithful, also after scalar extension from \(\mathbb Q\) to \(\mathbb R\). The polarization adjoint belongs to rational homomorphisms, and therefore extends to real homomorphisms. For a real homomorphism \(f\), the operator \(f^*f\) is nonnegative and self-adjoint on \(\mathop{\mathrm{Lie}}(G)\). Choose a real polynomial \(p\) that is zero at zero and equals \(1/t\) at every positive eigenvalue \(t\) of \(f^*f\). Then \[f^\dagger=p(f^*f)f^*\] is a real homomorphism and \(f^\dagger f=P_{(\ker f)^\perp}\), \(ff^\dagger=P_{\operatorname{im}f}\). Faithfulness makes these identities identities of real homomorphisms, so they also hold on the height spaces. The last assertion follows by applying the fixed bounded operator \(f^\dagger\). ◻

Proof. If the point heights are bounded, use \(A\) itself as the bounded factor. Otherwise take a subsequence with \(H_i=\|x_i\|^2\to\infty\). Among fixed quotient homomorphisms \(\pi:A\to Q\) for which \(\|\pi(x_i)\|^2=o(H_i)\) on some subsequence, choose one with maximal \(\dim Q\), and restrict to that subsequence. The zero quotient is allowed. A quotient of dimension \(\dim A\) is an isogeny and cannot have this property, by height comparison. Poincaré reducibility provides an isogeny \(A\to Q\times W'\) whose first component is \(\pi\); harmless fixed isogenies of the targets can be absorbed here.

Write \(p'_i\) for the \(W'\) coordinate. Suppose a fixed nonzero homomorphism \(g\) on \(W'\) satisfies \[\liminf_{i\to\infty}\frac{\|g(p'_i)\|}{\sqrt{H_i}}=0.\] Take a further subsequence on which this ratio tends to zero. The homomorphism whose coordinates are \(\pi\) and \(g\) on the complementary factor has image of dimension \(\dim Q+\dim g(W')>\dim Q\), and its point heights are \(o(H_i)\). Quotienting by the identity component of its kernel changes its image only by an isogeny. This contradicts the maximality of \(Q\). Thus \(W'\) is a high factor with scale \(H_i\) and the required nondegeneracy for every fixed homomorphism.

If \(Q=0\), stop. Otherwise continue on \(Q\) with the points \(\pi(x_i)\). If their heights are bounded, this is the low factor; if not, extract a diverging scale and repeat. At each step the remaining quotient has smaller dimension and its squared heights are little-oh of the preceding scale. The process therefore terminates. Reverse the order of the extracted factors, so that the scales are increasing. Composing the finitely many fixed isogenies gives \(\alpha\). Fixed height comparisons preserve all the asserted bounds and ratios. ◻

Proof. Let \(X\subseteq A\) be irreducible and contained in no proper torsion coset. If \(\dim A=0\), there are no torsion cosets of codimension \(\dim X+1\), so there is nothing to prove. Suppose otherwise that \(X\cap A^{[\dim X+1]}\) is Zariski dense. Since \(\overline{\mathbb Q}\) is countable, choose a sequence in this intersection that eventually avoids every fixed proper closed subset of \(X\) defined over \(\overline{\mathbb Q}\). Apply Lemma 7 and replace \(A,X\) and the sequence by their isogeny images. Dimensions of subvarieties and codimensions of torsion cosets are preserved by this finite map. Its image \(X\) is still contained in no proper torsion coset. We have now written \(A=\prod_{\ell\in I}W_\ell\) and \(x_i=(p_{i\ell})_\ell\).

For each \(i\), choose a torsion coset \(B_i+\tau_i\) containing \(x_i\) with \(\mathop{\mathrm{codim}}_A B_i>\dim X\). After extraction its codimension is constant. In \(\mathop{\mathrm{Lie}}(A)\) let \[U_i=\operatorname{diag}(H_{i\ell}^{-1/2})\mathop{\mathrm{Lie}}(B_i).\] The orthogonal projector \(P_i\) onto \(U_i\) is a real endomorphism by Lemma 4, applied to the scaled subgroup inclusion. All these projectors have the same rank. A subsequence converges to an orthogonal projector \(P\) of the same rank. Moreover \(P\in\mathop{\mathrm{End}}(A)_{\mathbb R}\), because this finite-dimensional subspace is closed in the space of real linear maps on \(\mathop{\mathrm{Lie}}(A)\). Put \(U=\operatorname{im}P\). Then \[ \mathop{\mathrm{codim}}_{\mathop{\mathrm{Lie}}(A)}U>\dim X, \qquad \left\|(1-P)(p_{i\ell}/\sqrt{H_{i\ell}})_\ell\right\| \longrightarrow0. \tag{6}\] For the second assertion, torsion vanishes in the height space, so \(1-P_i\) kills the indicated normalized vector exactly. Those vectors are bounded, and Lemma 4 converts convergence of the projectors into degree-independent operator-norm convergence.

To obtain the local fiber bounds in the sequence theorem, we retain copies of all slower coordinates. For each \(\ell\in I\) set \[A_\ell=\prod_{h\le\ell}W_h, \qquad Y_\ell=\operatorname{pr}_{\le\ell}(X), \qquad y_{i\ell}=(p_{ih})_{h\le\ell}.\] Regard \(W_\ell\) as the last coordinate in \(A_\ell\). Let \(m_\ell=\dim Y_\ell-\dim Y_{\ell^-}\), where \(Y_{\ell^-}\) is the preceding projection and is a point for the first index. These integers are nonnegative and sum to \(\dim X\). The local dimension of the fiber of \(Y_\ell\to Y_{\ell^-}\) equals \(m_\ell\) on a nonempty open subset. The sequence eventually lies in that subset: the inverse image in \(X\) of its complement is a fixed proper closed set. Thus (3) holds after discarding finitely many terms.

For a high factor, every slower coordinate is \(o(\sqrt{H_{i\ell}})\) in norm. A fixed homomorphism on \(A_\ell\) nonzero on its last \(W_\ell\) coordinate is the sum of a nonzero homomorphism on that coordinate and fixed homomorphisms on the slower ones. The triangle inequality and Lemma 7 prove (4). If a low factor occurs, its image of \(X\) is contained in no proper torsion coset: the inverse image of such a coset is a finite union of proper torsion cosets, one of which would contain irreducible \(X\). The same pullback argument proves eventual avoidance for the low sequence.

Finally let \(j:A\to\mathcal A\) insert each \(W_\ell\) into the distinguished last coordinate of \(A_\ell\), and let \(\rho:\mathcal A\to A\) be the coordinate retraction. Put \(V=j(U)\). Since \(\rho j=1\), the real endomorphism \(jP\rho\) is idempotent with image \(V\), and \[\mathop{\mathrm{codim}}_{\mathop{\mathrm{Lie}}(W)}V=\mathop{\mathrm{codim}}_{\mathop{\mathrm{Lie}}(A)}U>\dim X =\sum_\ell m_\ell.\] Write \(q_i^0=(p_{i\ell}/\sqrt{H_{i\ell}})_\ell\) for the normalized vector before enlargement, and \(q_i\) for the enlarged vector in Theorem 6. Their difference has only copied slower coordinates. With product norms, \[\|q_i-jq_i^0\|^2 =\sum_\ell\sum_{h<\ell}\frac{\|p_{ih}\|^2}{H_{i\ell}} =O\!\left(\sum_\ell\sum_{h<\ell} \frac{H_{ih}}{H_{i\ell}}\right)\longrightarrow0.\] If \(\Phi\) is the complementary projector for \(V\), then \(\Phi j=\Phi j(1-P)\). Equation (6) and the displayed error bound therefore give \(\|\Phi q_i\|\to0\). We have constructed (5), a contradiction to Theorem 6. ◻

Proof. Suppose such subvarieties and integers exist and restrict to the given subsequence. Let \(\pi_\ell:A_\ell\to A'_\ell=A_\ell/C_\ell\) be the quotient, let \(W'_\ell=\pi_\ell(W_\ell)\), and let \(e_\ell\) be the dimension of the image of \(C_\ell\) in \(D_\ell=A_\ell/W_\ell\). We first construct fixed irreducible parents \(Y'_\ell\subseteq A'_\ell\) for the projected points with local \(W'_\ell\)-fiber bounds \[ m'_\ell=m_\ell+e_\ell-a_\ell. \tag{8}\]

Fix one factor and temporarily omit its index. In \(Y\) let \(S\) be the intersection of the open locus where local fiber dimension over \(A/W\) is at most \(m\) and the closed locus where local fiber dimension over \(A/C\) is at least \(a\). Local fiber dimension is upper semicontinuous, so \(S\) is locally closed and contains every marked point under consideration. Every nonempty fiber of \(S\to A/C\) has dimension at least \(a\). Indeed, through each of its points there is an irreducible component of the original \(C\)-fiber of dimension at least \(a\). That whole component belongs to the closed high-fiber locus, and its intersection with the open low-fiber locus is a nonempty open subset of the same dimension.

Put \(E=A/(W+C)=A'/W'\). The map from a fiber of \(S\to E\) to \(A/W\) has image in an \(e\)-dimensional coset, and all of its nonempty fibers have dimension at most \(m\). Consequently every fiber of \(S\to E\) has dimension at most \(m+e\). Let \(Q\) be the constructible image of \(S\) in \(A'\). The fibers of \(S\to Q\) have dimension at least \(a\), so the dimension inequality for fibers, applied over every point \(t\in E\), gives \[\dim Q_t\le m+e-a.\] This can also be seen by applying the fiber dimension theorem to each irreducible locally closed stratum of \(Q_t\): one of the finitely many strata in its inverse image dominates it with generic fiber dimension at least \(a\). In particular \(m+e-a\ge0\) when a marked point exists.

Partition \(Q\) into finitely many irreducible locally closed subsets. After taking a subsequence, the projected points lie in one such subset \(Q^\circ\). Set \(Y'=\overline{Q^\circ}\). Because \(Q^\circ\) is open in its closure, its local germ at any of the selected points equals the germ of \(Y'\). The local fiber dimension of \(Y'\to E\) at that point is therefore at most \(m+e-a\). Performing this construction in each of the finitely many factors, with successive subsequences, proves (8). All these parents are defined over \(\overline{\mathbb Q}\).

The new parents provide the local fiber bounds. We next check the small-projection condition and the strict codimension inequality. Let \(\pi=\prod_\ell\pi_\ell\) and \(V'=\pi(V)\subseteq\mathop{\mathrm{Lie}}(W')\). This is the image of a real homomorphism, so Lemma 4 supplies its orthogonal projector. If \(\Phi'\) is the complementary projector, then \[\Phi'\pi=\Phi'\pi\Phi,\] since \(\pi(V)=V'\). Thus the projected normalized points satisfy the required small-projection condition. The height upper bounds are preserved by the fixed quotient homomorphisms.

The gain in fiber bounds compensates for the rank lost under projection: \[\begin{align*} \mathop{\mathrm{codim}}_{\mathop{\mathrm{Lie}}(W')}V' &=\dim W-\dim C+\sum_\ell e_\ell -\dim V+\dim(V\cap\mathop{\mathrm{Lie}}(C))\\ &=\mathop{\mathrm{codim}}_{\mathop{\mathrm{Lie}}(W)}V+\sum_\ell e_\ell -\mathop{\mathrm{rank}}(\Phi|_{\mathop{\mathrm{Lie}}(C)})\\ &>\sum_\ell(m_\ell+e_\ell-a_\ell) =\sum_\ell m'_\ell. \tag{9}\end{align*}\] Here \(\dim(W\cap C)=\dim C-\sum e_\ell\) and \(\ker(\Phi|_{\mathop{\mathrm{Lie}}(C)})=V\cap\mathop{\mathrm{Lie}}(C)\).

It remains to obtain nonzero distinguished factors and check the scale conditions. Discard each factor with \(W'_\ell=0\). In such a factor \(V'\) has zero coordinate, so deletion does not change the codimension inside \(\mathop{\mathrm{Lie}}(W')\) and only decreases the sum of the nonnegative bounds \(m'_\ell\). The small-projection condition on the remaining product follows by composition with its coordinate projection. At least one factor survives, since otherwise (9) would say that zero is strictly greater than a sum of nonnegative integers. The scales on the remaining ordered list still have successive ratios tending to zero. For a surviving high factor, a homomorphism nonzero on \(W'_\ell\) composes with \(\pi_\ell\) to one nonzero on \(W_\ell\), proving (4). If the low factor survives, then \(W'_0=A'_0\) and its torsion-coset avoidance follows by pulling a proper torsion coset back under \(\pi_0\).

We have obtained counterexample data of smaller total ambient dimension because \(C\ne0\). If the low factor is present both before and after projection, or absent both before and after, this contradicts minimality. If it has disappeared, it contradicts the already-established all-high case. This proves the exclusion. ◻

Proof. We will replace the subgroup supplied by Ax’s theorem by a product subgroup. The degeneration used for this replacement cannot decrease the tangent-intersection dimension. Its \(g\)-rank must also be bounded by the rank appearing in Ax’s inequality. We arrange this comparison first by choosing factor scalings that maximize the rank on every abelian subgroup; only then do we apply Ax’s theorem.

Write \(D(\lambda)=\operatorname{diag}(\lambda_\ell)\) for \(\lambda\in(\mathbb C^\times)^I\). For every abelian subvariety \(B\subseteq A\), the rank of \(gD(\lambda)|_{\mathop{\mathrm{Lie}}(B)}\) attains its maximum on a nonempty Zariski open subset of \((\mathbb C^\times)^I\). There are only countably many abelian subvarieties of \(A\): their rational homology groups determine distinct rational subspaces of \(H_1(A,\mathbb Q)\). The Baire category theorem therefore gives one \(\lambda\) at which all these ranks are maximal. Fix such a \(\lambda\) and set \(h=gD(\lambda)\).

At a smooth point of \(Z\), its tangent space is the direct sum of the tangent spaces of its factors. It is consequently preserved by \(D(\lambda)\). The failure of tangent injectivity for \(g\), which is equivalent to \(P_Z(g)=0\), thus also holds for \(h\) at every smooth point. This is the same product property underlying (11).

Choose a very general smooth point \(z\in Z\). We require that the rank of \(h\) on \(T_zZ\) be maximal, and that, for every abelian product subgroup \(C\subseteq A\), the differential of \(Z\longrightarrow A/C\) have its generic rank at \(z\). These requirements exclude only countably many proper closed subsets of \(Z\), so they can be imposed simultaneously. On a sufficiently small logarithmic chart around \(z\), the map obtained by applying \(h\) to the logarithm has constant rank less than \(\dim Z\). Its fiber through \(z\) therefore has a positive-dimensional irreducible analytic germ. Let \(J\) be a sufficiently small irreducible representative in the logarithmic chart, let \(U_1\) be the Zariski closure of \(\exp(J)\), and let \(z+C^*\) be the smallest coset containing \(U_1\). Writing \(\widetilde z\) for the logarithm of \(z\), we have \[J\subseteq\widetilde z+ \bigl(\mathop{\mathrm{Lie}}(C^*)\cap\ker h\bigr).\]

We use Ax’s theorem (Ax 1972), in the precise analytic-subset formulation (Habegger and Pila 2016, Theorem 5.3). If an irreducible analytic subset \(K\) of an open subset of a translate of a complex vector space \(U\) has exponential image with Zariski closure \(B\), that theorem gives \[\dim\langle B\rangle_{\mathrm{geo}}-\dim B \leq\dim U-\dim K,\] where \(\langle B\rangle_{\mathrm{geo}}\) denotes the smallest abelian coset containing \(B\). Applied to \(J\) and \(U=\mathop{\mathrm{Lie}}(C^*)\cap\ker h\), it gives \[\dim U_1\geq \mathop{\mathrm{rank}}\bigl(h|_{\mathop{\mathrm{Lie}}(C^*)}\bigr)+\dim J >\mathop{\mathrm{rank}}\bigl(h|_{\mathop{\mathrm{Lie}}(C^*)}\bigr).\] Since \(U_1\subseteq Z\cap(z+C^*)\) contains \(z\), the Zariski tangent space at \(z\) gives \[ \dim\bigl(T_zZ\cap\mathop{\mathrm{Lie}}(C^*)\bigr) \geq\dim U_1 >\mathop{\mathrm{rank}}\bigl(h|_{\mathop{\mathrm{Lie}}(C^*)}\bigr). \tag{13}\] No smoothness assumption on \(U_1\) at \(z\) is needed for this inequality.

We now replace \(C^*\) by a product direction. Order the index set \(I\) and choose strictly increasing integers \(w_\ell\). Put \(S=\mathop{\mathrm{Lie}}(C^*)\) and \(F_\ell=\bigoplus_{j\geq\ell}\mathop{\mathrm{Lie}}(A_j)\). A basis of \(S\) adapted to the filtration \(S\cap F_\ell\) shows that the Grassmannian limit \[S_0=\lim_{t\longrightarrow0} \operatorname{diag}(t^{w_\ell})S =\bigoplus_{\ell\in I}\operatorname{pr}_\ell(S\cap F_\ell)\] exists. Indeed, after dividing an adapted basis vector by the power of \(t\) corresponding to its first nonzero coordinate, its limit is the projection to that coordinate. Each summand is the Lie algebra of an abelian subvariety \(C_\ell\subseteq A_\ell\): take the identity component of the intersection of \(C^*\) with the product subgroup corresponding to \(F_\ell\), and project to \(A_\ell\).

The product tangent space \(T_zZ\) is preserved by all the displayed diagonal scalings. Upper semicontinuity of intersection dimension on the Grassmannian therefore gives \[\dim(T_zZ\cap S_0)\geq\dim(T_zZ\cap S).\] Set \(R=\mathop{\mathrm{rank}}(h|_S)\). By the choice of \(\lambda\), for every \(t\neq0\), \[\mathop{\mathrm{rank}}\left(g\bigm|_{\operatorname{diag}(t^{w_\ell})S}\right) =\mathop{\mathrm{rank}}\bigl(g\operatorname{diag}(t^{w_\ell})|_S\bigr)\leq R.\] The condition that the restriction of a fixed linear map have rank at most \(R\) is closed on the Grassmannian. Hence \(\mathop{\mathrm{rank}}(g|_{S_0})\leq R\). Together with (13), this proves \[\dim\left(T_zZ\cap\bigoplus_\ell\mathop{\mathrm{Lie}}(C_\ell)\right) >\mathop{\mathrm{rank}}\left(g\bigm|_{\bigoplus_\ell\mathop{\mathrm{Lie}}(C_\ell)}\right).\]

By the choice of \(z\), the dimension on the left is the generic fiber dimension of \(Z\longrightarrow A/\prod_\ell C_\ell\). Local fiber dimension at every point is at least this generic value. Since both \(Z\) and the subgroup are products, local fiber dimensions add over the factors. This proves (12) for every point of \(Z\). ◻

Proof. Choose an irreducible component \(D_\ell\) of \(Z_\ell\cap(y_\ell+C_\ell)\) through \(y_\ell\) of dimension \(d_\ell\). Write its smallest containing coset as \(y_\ell+B_\ell\); then \(B_\ell\subseteq C_\ell\). Among the subvarieties of \(Y_\ell\) containing \(D_\ell\) and having geodesic defect at most \(\delta_{\mathrm{geo}}(D_\ell)\), choose one, denoted \(E_\ell\), of maximum dimension. Such a choice is possible because \(D_\ell\) is an admissible choice and dimensions are bounded integers. It is geodesic-optimal: a proper enlargement of no greater defect would contradict its choice.

Write \(\langle E_\ell\rangle_{\mathrm{geo}}=y_\ell+H_\ell\). The inclusion \(D_\ell\subseteq E_\ell\) implies \(B_\ell\subseteq H_\ell\). The defect comparison gives \[\dim H_\ell-\dim B_\ell \leq\dim E_\ell-d_\ell.\] Rank increases by at most the dimension added to the domain. Therefore \[\begin{align*} \mathop{\mathrm{rank}}\left(g\bigm|_{\bigoplus_\ell\mathop{\mathrm{Lie}}(H_\ell)}\right) &\leq \mathop{\mathrm{rank}}\left(g\bigm|_{\bigoplus_\ell\mathop{\mathrm{Lie}}(B_\ell)}\right) +\sum_\ell(\dim H_\ell-\dim B_\ell)\\ &\leq \mathop{\mathrm{rank}}\left(g\bigm|_{\bigoplus_\ell\mathop{\mathrm{Lie}}(C_\ell)}\right) +\sum_\ell(\dim E_\ell-d_\ell)\\ &\leq\sum_\ell\dim E_\ell-\varepsilon. \end{align*}\] Since \(E_\ell\subseteq Y_\ell\cap(y_\ell+H_\ell)\) contains \(y_\ell\), this implies (14). Moreover, a positive-dimensional \(D_\ell\) forces \(H_\ell\neq0\).

By (Habegger and Pila 2016, Proposition 6.1), the directions of the geodesic closures of geodesic-optimal subvarieties of a fixed \(Y_\ell\) belong to a finite set. Take this set as \(\mathcal H_\ell\). The cited proposition applies to arbitrary subvarieties of complex abelian varieties and imposes no condition on the translating points. Thus the sets are independent of \(y_\ell\), \(Z_\ell\), \(C_\ell\), and \(g\). ◻

Proof. Suppose first that the first assertion fails. Choosing successively smaller neighborhoods and larger indices produces a subsequence, relabeled by \(i\), maps \(g_i\longrightarrow\Phi\), and subproducts \(Z_i=\prod_\ell Z_{i\ell}\) through the marked points such that \(P_{Z_i}(g_i)=0\). Lemma 10 gives product subgroups \(C_i=\prod_\ell C_{i\ell}\) satisfying \[\sum_\ell\dim_{y_{i\ell}} \bigl(Z_{i\ell}\cap(y_{i\ell}+C_{i\ell})\bigr) \geq\mathop{\mathrm{rank}}\bigl(g_i|_{\mathop{\mathrm{Lie}}(C_i)}\bigr)+1.\] The sum on the left is positive. Apply Lemma 11 with \(\varepsilon=1\) and pass to a subsequence on which all the resulting directions \(H_\ell\) are fixed. Their product \(H\) is nonzero. Passing again to a subsequence, we may also fix the integers \[a_\ell=\dim_{y_{i\ell}} \bigl(Y_\ell\cap(y_{i\ell}+H_\ell)\bigr).\] They are bounded between \(0\) and \(\dim Y_\ell\). We obtain \(\mathop{\mathrm{rank}}(g_i|_{\mathop{\mathrm{Lie}}(H)})\leq\sum_\ell a_\ell-1\). Since a rank upper bound is closed under limits of linear maps, \[\mathop{\mathrm{rank}}(\Phi|_{\mathop{\mathrm{Lie}}(H)})\leq\sum_\ell a_\ell-1.\] This contradicts the assumed exclusion of (15).

For the second assertion, suppose that there are arbitrarily large indices \(i\) and subproducts \(Z_i\) through the marked points with \(P_{Z_i^r}(\Delta\Phi)=0\). Apply Lemma 10 to the product whose factors are indexed by both \(\ell\in I\) and \(j\in\{1,\ldots,r\}\). For a fixed \(i\), suppress \(i\) from the notation, write \(C_j=\prod_\ell C_{j\ell}\) for the resulting subgroup in copy \(j\), and set \[a_j=\sum_\ell\dim_{y_{i\ell}} \bigl(Z_{i\ell}\cap(y_{i\ell}+C_{j\ell})\bigr), \qquad E_j=\Phi(\mathop{\mathrm{Lie}}(C_j)).\] The consecutive-difference map on \(\bigoplus_j E_j\) has kernel the diagonal copy of \(\bigcap_j E_j\). Its rank is consequently \(\sum_j\dim E_j-\dim\bigcap_j E_j\), and the lemma gives \[ \sum_{j=1}^r a_j> \sum_{j=1}^r\dim E_j-\dim\bigcap_{j=1}^r E_j. \tag{16}\] There is an index \(j\) such that \(C_j\neq0\) and \(a_j\geq\dim E_j\). Indeed, if some \(C_j=0\), then the intersection term vanishes and (16) forces a strict surplus in another copy. If all \(C_j\) are nonzero and every \(a_j<\dim E_j\), the integral deficits sum to at least \(r=\dim\mathcal A+1\), whereas \(\dim\bigcap_jE_j\leq\dim\mathcal A\), again contradicting (16).

Select one such copy for each \(i\). If its \(a_j\) is zero for any index, then its nonzero product subgroup is contained in \(\ker\Phi\). That one fixed subgroup satisfies (15) for every index, since all local fiber dimensions are nonnegative. This is impossible. Thus the selected sum is positive. Apply Lemma 11 with \(g=\Phi\) and \(\varepsilon=0\). It supplies a nonzero product direction in a fixed finite set and satisfying the parent inequality. Passing to a fixed direction along an infinite subsequence contradicts (15) once more. ◻

Proof. Put \(n=\dim\mathcal A\) and \(q=\mathop{\mathrm{rank}}_{\mathbb Z}\mathop{\mathrm{End}}(\mathcal A)\), and choose an integral basis \(e_1,\ldots,e_q\). The polynomial \[p_Z(x_1,\ldots,x_q) =P_Z\left(\sum_{j=1}^q x_je_j\right)\] has degree \(2d\le 2n\): pullback of a Hermitian form is quadratic in the real coordinates of the map. At integral arguments it is an integer, because it is the top intersection number of the pullback of \(L\) by an actual endomorphism.

Set \(K=2n\). In the multivariate Newton expansion, \[p_Z(x)=\sum_{|\alpha|\le K}(\Delta^\alpha p_Z)(0) \prod_{j=1}^q\binom{x_j}{\alpha_j},\] all finite differences are integers. Since \(\prod_j\alpha_j!\) divides \(K!\), every coefficient of \(p_Z\) has denominator dividing \(Q=K!\). This denominator is independent of \(Z\).

The coefficients also have absolute value at most \(C D_Z\) for a fixed \(C\). Indeed, on the fixed interpolation grid \(\{0,\ldots,K\}^q\), the semipositive forms \(g^*\omega_L\) are bounded above by a fixed multiple of \(\Omega=\sum_\ell\operatorname{pr}_\ell^*\omega_{L_\ell}\). Consequently \[0\le p_Z(x)\le C_0^d\int_Z\Omega^d =C_0^d\frac{d!}{\prod_\ell d_\ell!}D_Z.\] Fixed-grid interpolation gives the coefficient bound. There are only finitely many possible dimension tuples, so the constants are uniform. These integrations can be taken over the regular locus, and the same argument therefore includes singular subvarieties.

Here is the real-algebraic step which turns these bounds into a lower bound for a positive minimum. Let \(p_c(x)\) be the universal polynomial of degree at most \(K\), whose coefficients form the free parameter vector \(c\). Apply quantifier elimination over the real numbers to \[\exists x\,\bigl(x\in\mathcal B\ \text{and}\ t=p_c(x)\bigr),\] leaving \(c\) and \(t\) free. The resulting formula is a Boolean combination of sign conditions on finitely many polynomials \(R_j(c,t)\in\mathbb Z[c,t]\); denominators from the rational box have been cleared. Quantifier elimination with free variables, including its coefficient-ring formulation, is recalled in (Basu 2017, arXiv version, Section 2.5.2, Theorem 2.27).

Specialize to the coefficient vector \(c(Z)\), and put \(m_Z=\min_{\mathcal B}p_Z>0\). At least one specialized polynomial \(R_j(c(Z),t)\) which is not identically zero must vanish at \(m_Z\). Otherwise the signs of every nonzero specialized polynomial would be constant in a neighborhood of \(m_Z\), and the identically zero ones would have constant sign zero there. The formula describing the range would then be locally constant. This is impossible: it holds at \(m_Z\) and fails immediately to its left.

Let \(b=\max_j\deg_c R_j\) and \(h=\max_j\deg_t R_j\). Multiplication by \(Q^b\) makes each specialization an integer polynomial, of degree at most \(h\) and coefficient size at most \(C_1D_Z^b\). Choose a nonzero one which vanishes at \(m_Z\) and remove its maximal power of \(t\). Its constant coefficient is now a nonzero integer. If \(m_Z\le1\), its root equation implies \[1\le \sum_{k=1}^h |a_k|m_Z^k \le hC_1D_Z^b m_Z.\] For \(m_Z>1\) a lower bound is immediate. Enlarging fixed integer constants gives (17). The case \(d=0\), where \(p_Z=1\), is included directly. ◻

Proof. We first fix a positivity neighborhood and the constant in Dill’s inequality. For an arbitrary approximation accuracy, we then balance the height scales by integer multipliers and construct the required bundles. The final comparison will determine how small the accuracy must be.

Suppose that all-high data violate Theorem 6, and choose a counterexample with \(\dim\mathcal A\) minimal. The projection exclusion and Proposition 12 then apply. Every parent \(Y_\ell\) has positive dimension: if it were a point, the positive normalized lower bound (4), applied to the identity homomorphism, would contradict \(H_{i\ell}\to\infty\). Choose a closed rational box \(\mathcal B\) which contains \(\Phi\) in its interior and lies in the neighborhood supplied by Proposition 12. After discarding finitely many indices, \(P_Z(g)>0\) on this box for every subproduct through \(y_i\) in the fixed parents. Lemma 13 gives constants \(\theta,\omega_1\) valid for all these products.

If there is only one factor, use two subsequences \(n_i,m_i\to\infty\) with \(H_{n_i}/H_{m_i}\to0\). They exist by choosing \(H_{m_i}\ge i^2H_{n_i}\). On the two copies use the maps \(g\oplus g\) and the external product target polarization. For subvarieties \(Z_1,Z_2\) through the corresponding points, \[P_{Z_1\times Z_2}(g\oplus g) =\binom{\dim Z_1+\dim Z_2}{\dim Z_1}P_{Z_1}(g)P_{Z_2}(g).\] Thus the required bound holds with \(\theta^2\) and the same \(\omega_1\). The block diagonal limiting projector still sends the normalized point pairs to zero. From now on there are at least two factors. In the duplicated case all approximations remain in this block diagonal family; no new minimality argument in the larger ambient is needed. We relabel the doubled data and replace \(\theta\) by \(\theta^2\) in this case, retaining the notation already introduced.

Choose a positive integer \(t_1\), fixed independently of the rational approximation, so large that \[t_1\sum_\ell\operatorname{pr}_\ell^*\omega_{L_\ell}-g^*\omega_L\] is positive definite for every \(g\in\mathcal B\) (or for the indicated block diagonal family). Compactness of the box makes this possible. With the parent embeddings and \(\theta,\omega_1,t_1\) fixed, Theorem 14 has a fixed multiplicative constant \(c_1\).

Let \(0<\eta<1/2\). The following construction is available for every such \(\eta\), with \(c_1\) already fixed. Choose a rational homomorphism \(f/e\) in the box with \(\|f/e-\Phi\|\le\eta\), where \(e>0\) is even and \(f\) is an integral homomorphism divisible by two.

We want the multiplied points in every factor to have the same height scale: \(v_{i\ell}\sqrt{H_{i\ell}}\) should be close to one number \(R_i\). At the same time, the multiplication maps must have polynomial formulas whose coefficient heights grow at most quadratically in \(v_{i\ell}\). Both requirements are met by a dyadic family. Choose an integer \(N>2/\eta\) and put \(U=\{N,N+1,\ldots,2N\}\). The numbers \(u2^k\), with \(u\in U\) and \(k\ge0\), approximate every real number at least \(N\) to relative error at most \(\eta\). Taking \(R_i\ge N\max_\ell\sqrt{H_{i\ell}}\), choose \[ v_{i\ell}=u2^k,\qquad u\in U,\quad k\ge0, \tag{20}\] so that \[1-\eta\le\frac{v_{i\ell}\sqrt{H_{i\ell}}}{R_i}\le1+\eta.\] We verify the coefficient-height requirement below.

On the ambient product set \[\mathcal Q_0=\left(\boxtimes_\ell L_\ell^{t_1e^2}\right) \otimes(f^*L)^{-1}.\] The choice of \(t_1\) makes its invariant curvature positive definite, hence makes this bundle ample (Genestier and Ngô 2009, Theorem 1.1.4). Writing \(e=2e_0\) and \(f=2f_0\), symmetry identifies it with the fourth power of the ample bundle \[\left(\boxtimes_\ell L_\ell^{t_1e_0^2}\right) \otimes(f_0^*L)^{-1}.\] It is therefore globally generated (Genestier and Ngô 2009, Theorem 2.2.3). Fix a generating system \(z_1,\ldots,z_J\) for \(\mathcal Q_0\).

Let \([v_i]\) denote factorwise multiplication by the chosen integers, and define \[a_{i\ell}=e^2v_{i\ell}^2,\qquad N_{a_i}=\boxtimes_\ell L_\ell^{a_{i\ell}},\qquad \mathcal M_i=(f\circ[v_i])^*L\big|_{\prod Y_\ell},\qquad \mathcal P_i=N_{a_i}^{t_1}.\] The bundle \(\mathcal M_i\) is nef. Symmetry identifies \(\mathcal P_i\otimes\mathcal M_i^{-1}\) with \([v_i]^*\mathcal Q_0|_{\prod Y_\ell}\), which is generated by the pullbacks of the \(z_j\).

Choose a coordinate section \(\sigma\) of the target \(L\) which is nonzero at \(f([v_i]y_i)\). Multiplication by \((f\circ[v_i])^*\sigma\) gives the injection \[\mathcal P_i\otimes\mathcal M_i^{-1} \hookrightarrow N_{a_i}^{t_1}.\] It is an isomorphism on the nonvanishing open set of that section, which contains \(y_i\). Thus we may take \(t_2=t_1\) in Theorem 14. Before pullback, each product \(z_jf^*\sigma\) is a section of \(\boxtimes_\ell L_\ell^{t_1e^2}\). Projective normality of the ambient factor embeddings, together with the external-product formula for global sections, represents these sections by multihomogeneous polynomials of multidegree \((t_1e^2,\ldots,t_1e^2)\). There are only finitely many systems, one for each choice of target coordinate \(\sigma\).

We next check that the dyadic choice (20) gives the uniform coefficient bound required by Dill. Symmetry and projective normality give polynomial coordinate systems of degree four for \([2]\) and of degree \(u^2\) for each \([u]\), \(u\in U\). Their compositions give the required degree \(v_{i\ell}^2\) formulas.

Fix a number field containing the ambient data, \(f\), and the chosen generating systems. At a finite place use the maximum coefficient norm, and at an infinite place use the sum of absolute coefficient values. If \(b_w(k)\) is the logarithm of the maximum of one and the coefficient norm after \(k\) iterations of the degree-four system, submultiplicativity gives \[b_w(k+1)\le4b_w(k)+C_w.\] Thus \(b_w(k)=O_w(4^k)\). Composition with one of the finitely many \([u]\) gives \(O_w((u2^k)^2)\). At finite places the constants vanish outside a fixed finite set. Substitution in the finitely many fixed systems representing \(z_jf^*\sigma\) now bounds their joint coefficient height by \[C_{f,e,U}\sum_\ell a_{i\ell}.\] Consequently we may fix \(\delta_\ell=\max(1,C_{f,e,U})\) for every \(\ell\), independently of \(i\) and of the chosen nonvanishing coordinate. The generator count \(J\) is also fixed. Dependence on the approximation and on \(U\) changes only the height thresholds.

For every subproduct \(Z\) through \(y_i\), the scaling identity for \(P_Z\) and (17) give \[(\mathcal M_i^{\dim Z}\cdot Z) =P_Z(f/e)\prod_\ell a_{i\ell}^{\dim Z_\ell} \ge\theta^{-1}\prod_\ell (\deg Z_\ell)^{-\omega_1}a_{i\ell}^{\dim Z_\ell}.\] This verifies the last geometric hypothesis of Dill’s theorem.

Height comparison and choice of accuracy.

All geometric and coefficient conditions have now been checked for each fixed \(\eta\). The balancing of the multipliers will make the source height large and the image height small after division by \(e^2R_i^2\).

Let \(\Xi_i\) be the full monomial coordinates for \(\mathcal P_i\), and let \(\mathcal Z_i\) be the pulled-back generating system just constructed. Then \[ h(\Xi_i(y_i))-h(\mathcal Z_i(y_i)) =\widehat h_L(f([v_i]y_i)) +O_{f,e}\left(1+\sum_\ell a_{i\ell}\right). \tag{21}\] Indeed, the monomial height is exactly \(t_1\sum_\ell a_{i\ell}h(y_{i\ell})\). Multiplying all generator values by the same nonzero injection section does not change their projective height. Their height is therefore the fixed generator height of \(\mathcal Q_0\) at \([v_i]y_i\). As \(\mathcal Q_0\) is symmetric, this equals \[t_1\sum_\ell a_{i\ell}\widehat h_{L_\ell}(y_{i\ell}) -\widehat h_L(f([v_i]y_i))+O_{f,e}(1).\] The bounded difference between projective and canonical heights on each fixed factor proves (21).

By the high-scale condition applied to the identity, there are constants \(\kappa_\ell>0\), independent of the approximation, such that \[h(y_{i\ell})\ge\kappa_\ell H_{i\ell}\] for all sufficiently large \(i\). The normalized vectors \(q_i\) are bounded. Operator-norm comparison for real homomorphisms therefore gives a constant \(C\) independent of \(\eta\), \(f/e\), and the multipliers: the maps \(f/e\) lie in the fixed box, and the balancing ratios lie in \([1/2,3/2]\). Let \(D_i\) be the diagonal real endomorphism with entries \(v_{i\ell}\sqrt{H_{i\ell}}/R_i\). In the real height space, \[\frac{f([v_i]y_i)}{eR_i}=(f/e)D_iq_i.\] Writing this as the sum of \(\Phi q_i\), \((f/e-\Phi)D_iq_i\), and \(\Phi(D_i-1)q_i\) gives \[ \frac{\sqrt{\widehat h_L(f([v_i]y_i))}}{eR_i} \le C\eta+o(1). \tag{22}\]

For the weights, separation of successive scales gives \[\frac{a_{i\ell}}{a_{i,\ell+1}}\longrightarrow\infty, \qquad \frac{\sum_\ell a_{i\ell}}{e^2R_i^2} \le (1+\eta)^2\sum_\ell H_{i\ell}^{-1}\longrightarrow0.\] Moreover \[ \frac{\sum_\ell a_{i\ell}h(y_{i\ell})}{e^2R_i^2} \ge(1-\eta)^2\sum_\ell\kappa_\ell \ge\frac14\sum_\ell\kappa_\ell. \tag{23}\] Once \(\eta\), the approximation, and \(U\) are fixed, all the height thresholds in Theorem 14 are fixed. The projective heights tend to infinity and the weight ratios cross the fixed ratio threshold, so (19) applies for all sufficiently large \(i\). Dividing it by \(e^2R_i^2\) and using (21)–(23) yields \[\frac14\sum_\ell\kappa_\ell \le c_1 C^2\eta^2+o(1).\] The construction and this inequality hold for every \(0<\eta<1/2\), whereas \(c_1\), \(C\), and the \(\kappa_\ell\) are independent of \(\eta\). Choose \(\eta\) so that \(c_1C^2\eta^2<\frac14\sum_\ell\kappa_\ell\). Keeping its associated \(f,e,U\) fixed and letting \(i\) tend to infinity gives the contradiction. All constants depending on this approximation enter only fixed thresholds or errors which vanish after normalization. ◻

Proof. Let \(q=\binom{N+D}{D}\) and form the rows of evaluations of the degree-\(D\) monomials at points of \(\mathcal S\). Each row is a projective vector of height at most \(DT\). Select a maximal independent set of these rows; there are at most \(q\) of them. Their kernel is the required polynomial space. Choose pivot columns and give the free variables their standard basis values. The resulting kernel vectors have entries which are ratios of minors of the selected matrix. The determinant height inequalities bound their coefficient heights by \(O_{N,D}(T)\).

Although the rows can have entries in an extension of arbitrarily large degree, the resulting vectors lie in \(F\). Indeed the kernel is \(\operatorname{Gal}(\overline{\mathbb Q}/F)\)-invariant, and its normalized basis with the chosen free coordinates is unique. Each automorphism therefore fixes that basis. Both the number of rows and all height estimates are independent of their fields of definition. ◻

Proof. Linear equations defining the linear span of \(Z\) supply its normal directions in \(\mathbb P^N\). We may therefore work in that span and assume \(Z\) nondegenerate. Put \(d=\dim Z\) and \(c=N-d\). There is nothing to prove if \(c=0\), and the hypersurface equation proves the case \(c=1\).

Suppose \(c\geq2\) and let \(z\) be smooth. Choose a hyperplane \(H\) containing the projective tangent space at \(z\). Since \(Z\) is nondegenerate, \(H\) does not contain \(Z\). The closed join of \(z\) and \(Z\) has dimension at most \(d+1\), and its intersection with \(H\) has dimension at most \(d\). A general \((c-2)\)-plane \(\Lambda\) in \(H\) therefore avoids this join and the tangent space. Projection from \(\Lambda\) gives a morphism \[\pi:Z\longrightarrow Z'\subseteq\mathbb P^{d+1}.\] It is finite because \(\pi^*\mathcal O(1)=\mathcal O_Z(1)\) is ample. It has just one point over \(\pi(z)\) and is immersive at \(z\). The scheme-theoretic fiber has length one: its tangent space is zero and its residue field is the ground field. For the finite local algebra over the local ring of \(Z'\) at \(\pi(z)\), Nakayama’s lemma now says that \(1\) generates. Since \(Z'\) is the scheme-theoretic image, the local map is an isomorphism. In particular \(Z'\) is smooth at \(\pi(z)\).

The hypersurface \(Z'\) has degree at most \(D\). Its equation pulls back to an equation of \(Z\) whose differential at \(z\) defines \(H\): the inverse image of the tangent hyperplane is the span of \(\Lambda\) and the tangent space of \(Z\), which is \(H\). Varying \(H\) supplies all conormal directions. ◻

Proof. Suppose the first assertion fails for fixed \(D,B\). Choose violating hypersurfaces along an infinite subsequence, pass to a fixed factor and a fixed degree, and denote their fields by \(F_i\). Lemma 16, applied to the conjugates of \(y_{i\ell}\) over \(F_i\), replaces each equation by one of height \(O(H_{i\ell})\) which still vanishes at the marked point and does not vanish identically on \(Z_{i\ell}\). Here we used the height bound in the original sequence data, rather than any bound on the degree of the marked point.

Take a geometric irreducible component through \(y_{i\ell}\) of the intersection with this hypersurface. Its dimension is strictly smaller than \(d_\ell\), and its degree is bounded by ordinary Bézout. Its conjugates over \(F_i\) are also components of this intersection. The degree bound therefore bounds the number of conjugates, and Galois descent defines the chosen component over an extension of \(F_i\) of bounded degree.

We check the dense height condition on that component. Use the standard projective adelic metrics, with Fubini–Study metrics at infinity. For effective cycles use the additive absolute arithmetic intersection degree. The normalized height of an \(n\)-dimensional irreducible variety is this degree divided by the product of \(n+1\) and its geometric degree. The dense height condition, the fundamental height inequality, and the bounded geometric degrees therefore bound the arithmetic degree of \(Z_{i\ell}\) by \(O(H_{i\ell})\). The arithmetic divisor intersection formula bounds the arithmetic degree of its intersection with a degree-bounded polynomial by the polynomial degree times the original arithmetic degree, plus the geometric degree times the absolute weighted sum of logarithmic sup norms of that polynomial. The latter sum is at most its coefficient height plus a constant depending only on the degree and the projective space. Thus the intersection has arithmetic degree \(O(H_{i\ell})\). Additivity and nonnegativity give the same bound for the selected component. Dividing by its positive geometric degree and dimension factor, and using the other direction of the successive minima inequality, gives a dense set of points on it of height \(O(H_{i\ell})\); see (Zhang 1995, Theorem 1.10), or (Yuan and Zhang 2025, Theorem 5.3.3) with absolute normalization. The arithmetic divisor estimate used here is the usual intersection formula for a section, equivalently the arithmetic divisor formula of (Bost et al. 1994, Proposition 3.2.1(iv), Equation (3.2.2)); see also the divisor identity in the proof of (Yuan and Zhang 2025, Corollary A.4.3). We have produced an admissible product of smaller total dimension, a contradiction.

For the second assertion apply Lemma 16 to the dense set of bounded-height points on \(Z_{i\ell}\). Its evaluation kernel is the space of equations of \(Z_{i\ell}\), hence is defined over \(k_i\). In degrees up to the bounded degree of \(Z_{i\ell}\), these bases span the conormal at every smooth point by Lemma 17.

Suppose a marked point were singular for infinitely many indices. Choose an affine chart containing that point. Among the Jacobian minors of size equal to the codimension, formed from the bounded equations just obtained, some minor is nonzero generically on the variety. Every such minor vanishes at a singular point: otherwise the corresponding equations would cut out a smooth germ of the same dimension containing the germ of the reduced variety, which would itself then be smooth. Homogenizing a suitable minor gives a bounded-degree hypersurface over \(k_i\) containing the marked point and not the variety. This contradicts the first assertion. The codimension-zero case has no singular points.

Finally \(H_{i0}=1\). The low-factor equations just constructed have bounded degrees and bounded coefficient heights over fields of bounded degree. Northcott’s theorem gives a finite set of possible coefficient vectors. The conormal assertion at a smooth point shows that each \(Z_{i0}\) is an irreducible component of a zero locus of these bounded-degree equations. There are only finitely many such zero loci and components. This proves the last assertion. ◻

Proof. If \(d_0=0\), the fixed low variety is a point \(p\), so \(y_{i0}=p\) for every \(i\). Eventual avoidance implies that \(p\) lies in no proper torsion coset of \(A_0\).

First \(V\) projects onto \(\mathop{\mathrm{Lie}}(A_0)\). Otherwise let \(P\) be the orthogonal projector onto the orthogonal complement of that projection. This is a nonzero element of \(\mathop{\mathrm{End}}(A_0)_{\mathbb R}\). Writing \(\pi_0\) for the low projection, the identity \(P\pi_0=P\pi_0\Phi\) and the small-projection hypothesis give \(P(p)=0\) in the real height space. But evaluation at \(p\) is injective on \(\mathop{\mathrm{End}}(A_0)_{\mathbb Q}\): a nonzero rational endomorphism taking \(p\) to torsion places \(p\) in a proper torsion coset. Tensoring this injection with \(\mathbb R\) proves the same assertion for real endomorphisms. This contradicts \(P\ne0\). If there are no high factors, surjectivity already contradicts the strict codimension hypothesis.

Put \(L=\mathop{\mathrm{Lie}}(A_0)\), \(H=\bigoplus_{\ell\geq1}\mathop{\mathrm{Lie}}(W_\ell)\), and write \(a=\dim L\), \(b=\dim H\), and \(c=\mathop{\mathrm{codim}}_{L\oplus H}V\). For a positive integer \(h\) define \[V_* = \{(v_1,\ldots,v_h)\in H^h: \text{there is }v_0\in L\text{ with }(v_0,v_j)\in V \text{ for every }j\}.\] Kernels and images realize this as the image of a real homomorphism, hence of a real idempotent. The space before forgetting \(v_0\) has dimension \(a+h(b-c)\), since \(V\to L\) is surjective. The kernel of forgetting \(v_0\) has dimension \(t=\dim(V\cap(L\oplus0))\). Consequently \[\mathop{\mathrm{codim}}_{H^h}V_* = hc-a+t\geq hc-a.\] Since \(c>\sum_{\ell=0}^s m_\ell\), a sufficiently large fixed \(h\) makes this strictly greater than \(h\sum_{\ell\geq1}m_\ell\).

Choose \(h\) subsequences of indices so that, in the ordered list of their high factors, the largest scale in one copy divided by the smallest scale in the next tends to zero. This is possible recursively because all the high scales tend to infinity. Take the corresponding high marked points, retaining their original parents and fiber bounds. All the high nonvanishing hypotheses remain valid on these subsequences.

Their normalized tuples satisfy the small-projection condition for \(V_*\). Write \(A_+=\prod_{\ell\ge1}A_\ell\) and consider the fixed real homomorphism whose Lie action is \[T:\mathop{\mathrm{Lie}}(A_0)\oplus\mathop{\mathrm{Lie}}(A_+)^h\longrightarrow\mathop{\mathrm{Lie}}(\mathcal A)^h, \qquad T(v_0,v_1,\ldots,v_h) =\bigl(\Phi(v_0,v_j)\bigr)_{j=1}^h.\] Let \(K=\ker T\) and let \(\pi_+\) forget the low coordinate. Since \(\ker\Phi=V\subseteq L\oplus H\), we have \(\pi_+(K)=V_*\). Let \(\Phi_*\) be the orthogonal projector with kernel \(V_*\) in the full high ambient space \(\mathop{\mathrm{Lie}}(A_+)^h\). The generalized inverse from Lemma 4 gives the identity \[\Phi_*\pi_+=\Phi_*\pi_+T^\dagger T,\] because \(1-T^\dagger T\) projects onto \(K\), whose image under \(\pi_+\) is killed by \(\Phi_*\). This identity also holds in the real height spaces. At the tuple formed from \(p\) and the selected normalized high points, \(T\) tends to zero. Applying the fixed bounded operator \(\Phi_*\pi_+T^\dagger\) therefore proves the required small-projection condition. The high points need only lie in \(\mathcal H(A_+)^h\); the codimension inequality is measured in \(H^h\). We have obtained forbidden all-high sequence data, contrary to Theorem 15. ◻

Proof. We first compute two distinct limiting measures geometrically. We then show that a vanishing subsequence of the arithmetic intersection quotients would force those same measures to agree.

The two measures and their geometric limits.

Put \(X_i=Z_i^r\), \(u=rd\), and let \(p_0:\mathcal A^r\to A_0^r\) denote projection to the low coordinates. For \(j=1,2\), define a probability measure on \(A_0^r(\mathbb C)\) by \[\mu_{i,j} =\frac{(p_0)_*\big(c_1(\bar E_{i,j})^u\big|_{X_i}\big)} {(E_{i,j}^u\cdot X_i)}.\] Curvatures in this formula are taken at the chosen complex embedding. Both measures are supported on the fixed low product \(Z_0^r\). The geometric comparison follows the product-versus-pullback argument of the Bogomolov method (Ullmo 1998, sec. 4); the consecutive-difference construction is explained in (Zhang 1998b, sec. 3), following Faltings.

Let \(\omega=c_1(\bar L)\) at the fixed complex embedding. For a real homomorphism \(g\), let \(\beta_1(g)\) be the invariant form on \(\mathcal A^r\) obtained by summing the pullbacks of \(\omega\) under \(g\) in the \(r\) separate copies. Let \(\beta_2(g)\) be the sum of the pullbacks under the consecutive differences of those outputs. These definitions use only linear maps on Lie algebras and therefore make sense also for \(g=\Phi\).

On a product tangent space, block scaling multiplies the top form by \(B_i^r\). For any smooth real function \(f\) on \(A_0^r(\mathbb C)\), multiplication by \(p_0^*f\) does not alter that pointwise identity. It follows that \[ \mu_{i,j}(f) =\frac{\displaystyle\int_{Z_i^r}p_0^*f\, \beta_j(\varphi_i)^u} {\displaystyle\int_{Z_i^r}\beta_j(\varphi_i)^u}. \tag{28}\] No multiplication map on points or change of the test function is being used in this identity.

Write \(Q_i=\prod_{\ell=1}^s Z_{i\ell}\), allowing a point when there are no high factors. Integration of an invariant form along \(Q_i^r\) gives an invariant form on \(A_0^r\): in invariant coordinates its coefficients are integrals of fixed invariant forms over \(Q_i^r\), hence depend only on its homology class. Choose one of the products \(Q_i\) as a fixed representative \(Q\). Integrating \(\beta_j(g)^u\) over \(Q^r\) produces a smooth invariant form \(\gamma_j(g)\) of bidegree \((rd_0,rd_0)\) on \(A_0^r\). Its coefficients are polynomial in those of \(g\). Thus \(\gamma_j(\varphi_i)\to\gamma_j(\Phi)\) smoothly, and (28) shows that \(\mu_{i,j}\) converges to the probability measure \(\mu_j\) given by \[\mu_j(f) =\frac{\displaystyle\int_{Z_0^r}f\,\gamma_j(\Phi)} {\displaystyle\int_{Z_0^r}\gamma_j(\Phi)}.\] Both denominators are positive by the tangent positivity already used in (26). Singularities cause no problem in these integrations: one may integrate over resolutions, and the resulting densities on \((Z_0^{\mathrm{sm}})^r\) are smooth.

The first tangent integral supplies smooth points \(z\in Z_0\) and \(q\in Q\) such that \(\Phi\) is injective on \(T_zZ_0\oplus T_qQ\), where the tangent spaces are translated into the ambient Lie algebras. At the diagonal low tuple \((z,\ldots,z)\), the density \(\gamma_1(\Phi)|_{Z_0^r}\) is strictly positive. Indeed, at \((q,\ldots,q)\) the separate-output form is positive on the full product tangent space, and it remains positive on a nonempty open set of high tuples, whose integration gives a positive density.

At this same low tuple the density \(\gamma_2(\Phi)|_{Z_0^r}\) is zero. Choose a nonzero tangent vector \(v\in T_zZ_0\), possible because \(d_0>0\). For every smooth high tuple, the tangent vector which has low part \((v,\ldots,v)\) and zero high parts is killed by every consecutive difference. The top power of the semipositive form \(\beta_2(\Phi)\) therefore vanishes on that full product tangent space; its integral over the high factors vanishes as well.

After their positive normalizations, the two smooth densities still differ at \((z,\ldots,z)\). Their difference has a fixed positive sign on a sufficiently small neighborhood in the smooth locus. Choose a nonnegative smooth function \(f_*\) on \(A_0^r(\mathbb C)\), supported in a coordinate neighborhood of this point and positive on a smaller neighborhood. We have proved \[ \lim_i\bigl(\mu_{i,1}(f_*)-\mu_{i,2}(f_*)\bigr)>0. \tag{29}\] Figure 1 records this comparison on the low product.

Small points from a vanishing arithmetic intersection. We now suppose, contrary to the proposition, that the nonnegative quotient in (27) tends to zero along a subsequence. We recall the two arithmetic inequalities used here and in the small-section construction below. For a nef adelic bundle \(\bar D\) on a projective variety \(X\) of dimension \(n\), with \(D\) geometrically big, the fundamental inequalities read \[ \frac{e_1(X,\bar D)}{n+1} \le \frac{(\bar D^{n+1}\cdot X)}{(n+1)(D^n\cdot X)} \le e_1(X,\bar D). \tag{30}\] Here \(e_1\) is the essential minimum of the point height. We use the nef, geometrically big form of these inequalities (Yuan and Zhang 2025, Theorem 5.3.3); the ample projective case is (Zhang 1995, Theorem 1.10). We also use the arithmetic Siu inequality \[ \widehat{\operatorname{vol}}(\bar D-\bar N) \ge (\bar D^{n+1}\cdot X) -(n+1)(\bar D^n\bar N\cdot X) \tag{31}\] for nef adelic bundles, with the same absolute normalization (Yuan and Zhang 2025, Theorem A.5.1(2)). These statements apply after a projective resolution; pullback preserves the intersection numbers and essential minima used here. Resolution in characteristic zero can be performed over the original field \(k_i\), by smooth-center blowups of a smooth projective ambient variety, with centers disjoint from the original smooth locus of \(X\) (Włodarczyk 2005, Theorems 1.0.2–1.0.3 and Section 5.7). Thus using a resolution requires no extension of the bounded-degree field \(k_i\). A bundle of positive arithmetic volume has a nonzero section of some positive power with supremum norm at most one at every place. Its point height is therefore nonnegative outside a proper closed subset.

By (30) and (26), each \(Z_i\) has a dense set of points of \(\bar E_i\)-height at most \(\epsilon_iM_i^2\), where \(\epsilon_i\to0\). Products of these sets give a dense subset \(\mathcal S_i\subset X_i(\overline{\mathbb Q})\) such that \[ h_{\bar E_{i,1}}(x),\ h_{\bar E_{i,2}}(x) \le C_r\epsilon_iM_i^2\qquad(x\in\mathcal S_i). \tag{32}\] For the difference bundle this follows from \(\|a-b\|^2\le2\|a\|^2+2\|b\|^2\). We will use these common dense sets to show that, for every smooth real function \(f\) on \(A_0^r(\mathbb C)\), \[ \mu_{i,1}(f)-\mu_{i,2}(f)\longrightarrow0. \tag{33}\] Applied to \(f_*\), this will contradict (29). The required metric variation must be uniform in both the varieties and their defining fields.

Uniform bounds for metric variation. Let \(v_i\) be the complex place of \(k_i\) determined by the chosen embedding, and put \(\lambda_i=2/[k_i:\mathbb Q]\). Thus \(\lambda_i\) is bounded below by a positive constant. Write \(\overline{\mathcal O}_{v_i}(f)\) for the trivial bundle whose negative logarithmic norm is \(p_0^*f\) at this embedding, its conjugate at the conjugate embedding, and zero at all other places. Write \(\bar{\mathbf 1}\) for the trivial bundle with negative logarithmic norm one at every archimedean place and zero at the finite places. Its absolute point height is one.

Fix \(\eta>0\). For either \(j\), abbreviate \(\bar D_i=\bar E_{i,j}|_{X_i}\), \(\mu_i=\mu_{i,j}\), and set \[a_i=\lambda_i\mu_i(f)-\eta,\qquad \bar U_i=\overline{\mathcal O}_{v_i}(f)-a_i\bar{\mathbf 1}.\] In particular \(|a_i|\le\|f\|_\infty+\eta\). There are nef bundles \(\bar U_{i,1},\bar U_{i,2}\), pulled back from the low ambient product, such that \[ \bar U_i=\bar U_{i,1}-\bar U_{i,2},\qquad \bar U_{i,2}=C\bar H+K\bar{\mathbf 1}, \tag{34}\] where \(\bar H\) is a fixed canonical symmetric ample bundle on \(A_0^r\), and \(C,K\) are fixed positive constants. To obtain this, choose an integer \(C\) so that \(C c_1(\bar H)+dd^cf\) is semipositive, and then choose \(K>\|f\|_\infty+\sup_i|a_i|\).

We spell out why this is nef in the model sense required by (31). Start with a relatively ample projective model of a power of \(H\), give it the invariant archimedean curvature, and add a constant so that it is nef. Normalized pullbacks by multiplication, on graph models, converge to \(\bar H\); their archimedean curvatures are all the same invariant form. Adding the metric in (34) leaves every vertical curve degree unchanged. At infinity the new curvature is semipositive by the choice of \(C\). The added negative-log-norm function is nonnegative at each archimedean place by the choice of \(K\), so the arithmetic degree of every horizontal integral curve can only increase. This verifies the nef criterion for hermitian models: semipositive curvature and nonnegative degrees on all integral arithmetic curves. The perturbed nef models converge to \(\bar U_{i,1}\). The construction also applies to \(\bar U_{i,2}\), and its constants are independent of \(i\) and of the field extension.

Every arithmetic mixed intersection of total order \(u+1\) formed from \[\bar D_i,\qquad M_i^2\bar U_{i,1},\qquad M_i^2\bar U_{i,2}\] is bounded by \[ O(M_i^2B_i^r), \tag{35}\] with one constant for all these finitely many types of intersections. For a direct proof, let \(p_{a\ell}\) be the coordinate projection from \(\mathcal A^r\) to factor \(\ell\) in copy \(a\), and put \[\bar T_i=\sum_{a=1}^r\sum_{\ell=0}^s b_{i\ell}^2p_{a\ell}^*\bar L_\ell, \qquad \bar S_i=\bar D_i+M_i^2\bar U_{i,1} +M_i^2\bar U_{i,2}+\bar T_i\] on \(X_i\). This is nef and geometrically ample. Remove the factors \(b_{i\ell}\) on each product tangent space. The remaining forms for \(D_i\) are bounded by (24); those for \(M_i^2U_{i,a}\) are bounded because their geometric parts use only the low coordinates, where \(b_{i0}=M_i\). Consequently \[(S_i^u\cdot X_i)=O(B_i^r).\] On the products of the dense factorwise sets of height \(O(H_{i\ell})\), all the point heights just displayed are \(O(M_i^2)\). For \(D_i\), use boundedness of \(\varphi_i\) to get \[h_{\bar D_i}(x)\ll \sum_{a,\ell} b_{i\ell}^2 h_{\bar L_\ell}(x_{a\ell})=O(M_i^2).\] The same estimate holds for \(\bar T_i\). The heights of \(\bar U_{i,1},\bar U_{i,2}\) are bounded on these sets because the low point heights and the metric shifts are bounded. Thus \(e_1(X_i,\bar S_i)=O(M_i^2)\), and the upper inequality in (30) yields \((\bar S_i^{u+1}\cdot X_i)=O(M_i^2B_i^r)\). Mixed intersections of nef adelic bundles are nonnegative. Each term in (35) occurs with a positive coefficient in this last self-intersection, which proves the assertion. Absolute normalization has introduced no field-dependent constant.

Comparison against a low-factor test function. The metric intersection formula gives \[(\bar D_i^u\bar U_i\cdot X_i) =(D_i^u\cdot X_i)\big(\lambda_i\mu_i(f)-a_i\big) =\eta(D_i^u\cdot X_i).\] Apply (31) to the two nef bundles \(\bar D_i+tM_i^2\bar U_{i,1}\) and \(tM_i^2\bar U_{i,2}\), where \(t>0\) is rational. Expansion and (35) give \[\begin{align*} \widehat{\operatorname{vol}}(\bar D_i+tM_i^2\bar U_i) \ &\ge (\bar D_i^{u+1}\cdot X_i) +(u+1)tM_i^2\eta(D_i^u\cdot X_i) -O(t^2M_i^2B_i^r). \end{align*}\] The first term is nonnegative. By (26), a sufficiently small fixed \(t>0\), depending on \(f,\eta\) but not on \(i\), makes the right side positive for all sufficiently large \(i\). Hence the perturbed height is nonnegative outside a proper closed subset of \(X_i\).

Repeat this argument for both \(j\) and for \(f\) and \(-f\). Choose \(x_i\in\mathcal S_i\) outside the four resulting proper closed subsets. Let \(\operatorname{av}_{i,x_i}(f)\) be the average of \(f\) on the low projections of the conjugates of \(x_i\) over \(k_i\) at the chosen embedding. Then \[h_{\overline{\mathcal O}_{v_i}(f)}(x_i) =\lambda_i\operatorname{av}_{i,x_i}(f).\] Divide the perturbed height inequalities by \(tM_i^2\), and use (32). For \(j=1,2\) this yields \[\left|\lambda_i\operatorname{av}_{i,x_i}(f) -\lambda_i\mu_{i,j}(f)\right| \le\eta+o(1).\] Since \(\inf_i\lambda_i>0\) and \(\eta\) is arbitrary, this proves (33). This is the only point in this metric-variation argument that needs the bounded degrees of the fields \(k_i\); the degrees of the points \(x_i\) are unrestricted.

Taking \(f=f_*\) in (33) contradicts (29). This proves (27). ◻

Proof. By Proposition 20 and (26), there are constants \(\delta,C>0\) with \[(\bar E_i^{d+1}\cdot Z_i)\ge\delta M_i^2B_i, \qquad (E_i^d\cdot Z_i)\le CB_i.\] Choose \(0<c<\delta/((d+1)C)\). The constant bundle \(cM_i^2\bar{\mathbf 1}\) is nef, and (31) shows that \(\bar E_i-cM_i^2\bar{\mathbf 1}\) has positive arithmetic volume. After pulling back to the resolution, take a nonzero section of a positive power with norms at most one for this shifted metric. Restoring the original metric gives exactly (36). ◻

Proof. Put \(c_\ell=\mathop{\mathrm{codim}}_{\mathbb P^{d_\ell}}J_\ell\) and \(c=\sum_\ell c_\ell>0\). Both index loci are defined over \(k\), so every conjugate of \(J\) is again a common component. All conjugates have the same factor codimensions. Choose general linear spaces \(\Lambda_\ell\simeq\mathbb P^{c_\ell}\) and put \(\Lambda=\prod_\ell\Lambda_\ell\). Each conjugate of \(J\) meets \(\Lambda\) transversely in \(\prod_\ell\deg J_\ell\) reduced points. We may choose the slices so that all these sets are disjoint and none of their points belongs to any other component of \(\mathcal I_\sigma\).

Here is the incidence justification for this simultaneous choice. The intersection of a component \(J'\) with another component of \(\mathcal I_\sigma\) is a proper closed subset of \(J'\), hence has dimension less than \(\dim J'\). For a fixed point, the condition that it belong to \(\Lambda\) has codimension \(\sum_\ell(d_\ell-c_\ell)=\dim J'\) in the product of the Grassmannians parametrizing the slices. Thus a general product slice misses every such proper subset. There are only finitely many components and conjugates. The same argument, together with generic transversality in each factor in characteristic zero, avoids the singular loci. It follows that the indicated \(q\prod_\ell\deg J_\ell\) points are isolated in \(\Lambda\cap\mathcal I_\sigma\). They all belong to the higher index locus.

Multiply each derivative defining \(\mathcal I_\sigma\) by all monomials of the missing multidegree, so that every resulting equation has multidegree \(D\). This operation does not change their common zero set: at any point, some monomial of each specified nonnegative multidegree is nonzero. Near a point of the higher index locus, differentiating by weighted order less than \(\sigma\) leaves weighted order at least \(\epsilon\). Multiplying by a monomial cannot lower this order. Restriction to \(\Lambda\) is factorwise and cannot lower it either.

Choose \(c\) general linear combinations of these restored equations so that their restrictions to \(\Lambda\) have isolated intersections at every indicated point. This is possible because the common base locus is isolated there: successive general combinations avoid the finitely many associated components of positive dimension through these finitely many points. In completed local coordinates \(t_{\ell,1},\ldots,t_{\ell,c_\ell}\) at one such point, their ideal is contained in the monomial ideal generated by monomials of weighted order at least \(\epsilon\). The local intersection multiplicity is therefore at least the number of monomials of weight less than \(\epsilon\).

In particular consider monomials for which every exponent in the \(\ell\)th block is strictly less than \(\epsilon D_\ell/(2c)\). Their total weight is less than \(\epsilon/2\), and their number is at least \[ \left(\frac{\epsilon}{2c}\right)^c \prod_\ell D_\ell^{c_\ell}. \tag{51}\] Indeed there are \(\lceil x\rceil\ge x\) nonnegative integers strictly less than a positive real number \(x\).

On the other hand, the sum of the isolated intersection multiplicities of \(c\) divisors of multidegree \(D\) on \(\Lambda\) is at most \[ \left(\sum_\ell D_\ell H_\ell\right)^c\cdot\Lambda =\frac{c!}{\prod_\ell c_\ell!}\prod_\ell D_\ell^{c_\ell}, \tag{52}\] where \(H_\ell\) denotes the hyperplane class from the \(\ell\)th factor. The bound remains valid if other intersection components have positive dimension. Namely, perturb the divisors in their basepoint-free complete linear systems to make the intersection proper; at each original isolated complete intersection the local length is conserved, and its contribution is part of the total degree (52). There are \(q\prod_\ell\deg J_\ell\) of the isolated points counted above. Combining their individual multiplicity bound with the total bound gives \[q\left(\prod_\ell\deg J_\ell\right) \left(\frac{\epsilon}{2c}\right)^c \prod_\ell D_\ell^{c_\ell} \le \frac{c!}{\prod_\ell c_\ell!} \prod_\ell D_\ell^{c_\ell}.\] The powers of the multidegrees cancel, proving (50) independently of \(D\). For example, after harmless enlargement the constant is bounded by \(N!(2N/\epsilon)^N\).

Finally, the stabilizer of \(J\) in the absolute Galois group of \(k\) has index \(q\). Galois descent defines \(J\) over its fixed field, a degree-\(q\) extension of \(k\). Each \(J_\ell\) is the image of \(J\) under a factor projection, so it is defined over the same field. All factor degrees are positive integers; hence (50) bounds them separately as well. ◻

Proof. The finite projections and norms above give \(G_i\) and \(z_i\) with (47). Pass to constant dimensions of the projective factors and let \(N\) be their sum. Apply the preceding component construction with \(\eta=\eta_1\). Its \(N+2\) levels and their gap \(\epsilon\) are fixed independently of \(i\). It gives a proper component \(J_i\) through \(z_i\) common to two adjacent index loci; the adjacent pair may depend on \(i\).

For at least two factors, (48) eventually implies (49), so the product theorem gives \(J_i=\prod_\ell J_{i\ell}\). With one factor this conclusion requires no theorem. Lemma 23 then gives a common field of definition \(k_i'\) with \([k_i':k_i]\) bounded and all \(\deg J_{i\ell}\) bounded. The finitely many possible choices of adjacent levels do not affect these bounds.

Because \(J_i\) is proper, at least one factor \(J_{i\ell}\) is proper in \(\mathbb P^{d_\ell}\). After extraction fix this index \(\ell\). A proper projective variety of bounded degree admits a nonzero homogeneous equation of bounded degree over its field of definition. One can obtain it by projecting generally over that field to a hypersurface in a projective space of one larger dimension than the variety and pulling back its equation; the image degree does not exceed the original degree. Let \(R_i\) be such an equation for \(J_{i\ell}\) over \(k_i'\).

Pulling back \(R_i\) through the linear projection \(\pi_{i\ell}\) gives a bounded-degree hypersurface over \(k_i'\) containing \(y_{i\ell}\). It does not contain \(Z_{i\ell}\), because that projection is surjective onto \(\mathbb P^{d_\ell}\) and \(R_i\) is nonzero. The degrees \([k_i':\mathbb Q]\) are bounded, since the degrees \([k_i:\mathbb Q]\) were bounded in choosing the minimal products. This is exactly the hypersurface excluded by Proposition 18, a contradiction. ◻

Completion of the proof of Theorem 6. Theorem 15 handles the case with only unbounded scales. For a counterexample with a low factor, choose one of minimal ambient dimension. The projection and tangent arguments apply in this minimal counterexample. Proposition 18 then provides its auxiliary products. If \(d_0=0\), Lemma 19 produces an all-high counterexample, already impossible. If \(d_0>0\), Proposition 20 and Lemma 21 provide the small sections; Proposition 22 gives their positive index; and Proposition 24 contradicts minimality of the auxiliary products. Thus there is no counterexample with a low factor either. ◻