Prior work and the interpolation approach

The geometry of plane linear systems connects this problem with the cones of effective and nef divisor classes on blowups. Ciliberto–Harbourne– Miranda–Roé (Ciliberto et al. 2013) develop these connections and compare several forms and extensions of Nagata’s conjecture. For homogeneous multiplicities, Roé (Roé 2001) obtained the uniform estimate \(d>m(\sqrt{r-1}-\pi/8)\) for \(r\ge10\) by specializing to configurations with infinitely near points. Harbourne (Harbourne 2001) sharpened rational lower bounds and proved further cases in restricted multiplicity ranges. For ten very general points, degeneration methods give \(d/m\ge117/37\) for every nonempty homogeneous system: Eckl’s Seshadri estimate and Ciliberto–Dumitrescu–Miranda–Roé’s analysis of plane systems give this bound in their respective formulations (Eckl 2011; Ciliberto et al. 2011). Such bounds explain the importance of controlling arbitrarily large multiplicities at each fixed nonsquare \(r\).

Dumnicki–Harbourne–Küronya–Roé–Szemberg (Dumnicki et al. 2017) study an analogue for very general monomial valuations, while Nivoche (Nivoche 2021) relates the conjecture to transcendental interpolation in \(\mathbb C^n\). These perspectives emphasize that the obstruction is not merely the number of imposed conditions, but how vanishing conditions interact with the geometry of the ambient surface. More recently, Ciliberto–Miranda–Roé (Ciliberto et al. 2026, Theorem 1) constructed irrational square-zero nef rays on the blowups at every \(r\ge10\) very general points. Those constructions do not determine the equal-weight multipoint Seshadri constant. Nagata’s conjecture singles out the class with equal exceptional coefficients, whose nefness gives that constant’s exact value.

The proof below uses two established approaches to this interaction. Ciliberto–Miranda (Ciliberto and Miranda 1998) develop degenerations of plane linear systems, and Evain (Evain 2007) studies their infinitesimal limits when fat points, meaning points equipped with specified orders of vanishing, approach a divisor. Dumnicki (Dumnicki 2007) encodes interpolation by finite diagrams of monomial exponents and their evaluation matrices. In particular, the proof of his Proposition 13 converts a one-point derivative matrix into polynomial evaluation on the exponent diagram. Our final rank problem has the same evaluation interpretation, with a different set of allowed polynomial degrees supplied by a collinear collision.

The elliptic stage uses the classical multiplicative theta product and its residue-class Laurent bases; the conventions used here are recorded in Rosengren (Rosengren 2020, arXiv version, Section 1.2 and Exercise 1.6.5). We prove the needed product identities, basis formulas and convergence estimates in the normalization required by the argument.

Proof strategy

Closed incidence loci turn a failure at very general points into a system of curves existing at every configuration. A product over cyclic permutations makes the multiplicities equal; write \(d\) and \(m\) for the resulting degree and common multiplicity. Moving the points toward a smooth curve gives a nonzero polynomial on its normal line bundle, with order at least \(m\) at arbitrary normal displacements. For square \(r\), its two highest coefficients already exclude \(d/m\le\sqrt r\), including equality (Section 2).

For nonsquare \(r\), we use a cubic and set nine normal displacements to zero. Dividing out the forced coefficient zeros and changing the fiber coordinate give a finite-dimensional space of polynomial sections whose first \(m+1\) coefficient bundles have equal degree (Section 3). On each fixed elliptic curve, the remaining \(q=r-9\) points collide along a horizontal coordinate line. After \(b<m\) fiber derivatives, a section must then have base order at least \(q(m-b)\); thus a particular finite jet map has a nonzero kernel. A separate degeneration gives explicit normalized theta bases whose expressions in logarithmic base and fiber coordinates \((x,y)\) tend to \(e^{Kx+jy}\). Here \(j\) is the fiber degree and \(K\) a Laurent exponent. Their mixed derivatives give the limiting matrix \[\bigl(K^\ell j^b\bigr),\qquad 0\le b<m,\quad 0\le\ell<q(m-b),\] with columns indexed by exponent pairs \((j,K)\) (Section 4). Only these numerical matrices are compared across curves; the argument does not choose a simultaneous family of kernel vectors.

To exclude the kernel, a total count of exponent pairs is not enough. We group them into horizontal rows of fixed \(K\), with \(j\) varying. Interpolation one row at a time gives full column rank if no row has more than \(m\) points and at most \(q(m-t+1)\) rows have at least \(t\) points. The exponent pairs lie in a polygon whose slice-width profile supplies exactly these bounds. Since \(d/m\) is rational and \(r\) is nonsquare, the strict gap \(d/m<\sqrt r\) permits a contraction of that polygon; taking a common power of the original curve absorbs the integer endpoint error in the row count. Full rank persists for small positive degeneration parameter, contradicting the kernel on each fixed curve (Section 5).

A universal equal-multiplicity system

For positive integers \(d,m\), write \(\mathsf U_r(d,m)\) for the property \[ \begin{gathered} \text{For every }(p_1,\ldots,p_r)\in U_r\text{ there is a nonzero form}\\ S\in H^0(\mathbb P^2,\mathcal O_{\mathbb P^2}(d)) \quad\text{such that}\quad \operatorname{mult}_{p_i}(S)\ge m\quad(1\le i\le r). \end{gathered} \tag{1}\] Here and below the multiplicity of a form is the multiplicity of its zero divisor.

Specialization to a normal bundle

Let \(B\subseteq\mathbb P^2\) be a smooth irreducible plane curve of degree \(k\ge1\), and fix a nonzero homogeneous equation \(G_B\) for it. Put \[L=\mathcal O_B(1),\qquad M=L^k.\] The equation identifies the normal bundle of \(B\) in \(\mathbb P^2\) with \(M\), as in the formula for an effective Cartier divisor (The Stacks Project Authors 2026, Tag 0B3P). In a local frame \(e\) for \(\mathcal O_{\mathbb P^2}(1)\), the local equation \(t=G_B/e^k\) identifies a normal vector with its derivative under \(t\). For \(w\in M_a\), the expression \(f_j(a)w^j\) is an element of \(L_a^d\) when \(f_j\) is a section of \(L^{d-kj}\). Thus the polynomial in the next proposition is intrinsically a section of the pullback of \(L^d\) to \(\operatorname{Tot}(M)\).

The square case

We use two basic facts about smooth projective curves. A nonzero section of a line bundle has an effective zero divisor whose degree is the degree of the bundle. In particular, a degree-zero line bundle has a nonzero section only if it is trivial. Also, for a smooth complex projective curve of positive genus, \(\operatorname{Pic}^0(B)\) is a complex torus; its subgroup of elements of any fixed finite order is finite, so its torsion subgroup is countable (The Stacks Project Authors 2026, Tag 0C1Z).

Together with Lemma 2, the proposition proves Theorem 1 for square \(r\), including the strict inequality and unequal multiplicities.

Nine zero displacements on a cubic

For the remaining argument we will apply Proposition 3 with \(k=3\) and with zero normal displacements at nine distinct points of a smooth plane cubic. Write \(P=a_1+\cdots+a_9\). Extend the coefficient list by \(f_j=0\) for \(j>\lfloor d/3\rfloor\). At each \((a_i,0)\), expansion in the fiber variable shows that multiplicity at least \(m\) implies \[ f_j\in H^0\bigl(B,L^{d-3j}(-(m-j)P)\bigr) \qquad(0\le j<m). \tag{3}\] Indeed, in a local base coordinate \(\zeta\) centered at \(a_i\), the coefficient of \(\zeta^a w^j\) must vanish whenever \(a+j<m\). Thus \(f_j\) has base order at least \(m-j\) at every point of \(P\). The coefficient bundles in (3) all have degree \(3(d-3m)\), independent of \(j\). These conditions hold regardless of the additional points and normal displacements chosen in the proposition.

Removing the nine prescribed coefficient zeros

For this construction, let \(d,m\) be any positive integers with \(d>3m\). Let \(B\) be a smooth plane cubic, let \(L=\mathcal O_B(1)\) and \(M=L^3\), and choose nine distinct points with divisor \(P=a_1+\cdots+a_9\). Write \(\sigma_P\) for the canonical section of \(\mathcal O_B(P)\) with zero divisor \(P\), and put \[A=M\otimes\mathcal O_B(-P),\qquad T=L^{d-3m},\qquad u_0=\lfloor d/3\rfloor.\] Thus \(\deg A=0\) and \(\deg T=3(d-3m)>0\). Multiplication by \(\sigma_P\) defines a bundle map \(A\to M\), written \[w=\sigma_P v,\] which is invertible over \(B\setminus P\). Define coefficient spaces inside \(H^0(B,T\otimes A^{m-j})\) by \[ V_j= \begin{cases} H^0(B,T\otimes A^{m-j}),&0\leq j\leq m,\\[1mm] \sigma_P^{j-m}H^0(B,L^{d-3j}),&m<j\leq u_0. \end{cases} \tag{4}\] Negative powers denote dual powers. The target in the second line is correct because \[L^{d-3j}\otimes\mathcal O_B((j-m)P)=T\otimes A^{m-j}.\] Multiplication by \(\sigma_P^{j-m}\) is injective, as can be checked on \(B\setminus P\). Let \(\mathcal W\) be the space of polynomial sections on \(\operatorname{Tot}(A)\) \[\widetilde F(v)=\sum_{j=0}^{u_0}g_jv^j,\qquad g_j\in V_j,\] valued in the pullback of \(T\otimes A^m=L^d(-mP)\). It is a finite-dimensional space: its coefficient blocks are spaces of sections on the compact curve \(B\), and distinct powers of the fiber coordinate make their sum direct.

The nine zero displacements are now built into \(\mathcal W\). The remaining points impose ordinary multiplicities at scalar fiber coordinate \(1\), as follows.

We next realize these coefficient spaces on explicit elliptic curves. Their bases will retain the bundle multipliers, not merely their degrees; those multipliers determine the Laurent exponents in the limit.

Theta functions and plane cubics

For \(0<\tau<1\), let \[X_\tau=\mathbb C^*/\tau^{\mathbb Z} \simeq\mathbb C/(2\pi i\mathbb Z+(\log\tau)\mathbb Z).\] This is a connected compact complex torus of dimension one. For \(n\in\mathbb Z\) and \(\gamma\in\mathbb C^*\), define \(D(n,\gamma)\) as the holomorphic line bundle obtained from \(\mathbb C^*\times\mathbb C\) by the action generated by \[(z,t)\longmapsto(\tau z,\gamma z^{-n}t).\] In its covering-space frame, sections are holomorphic functions with \[f(\tau z)=\gamma z^{-n}f(z),\] and ordinary multiplication gives the tensor identifications \[D(n,\gamma)\otimes D(n',\gamma') =D(n+n',\gamma\gamma').\]

Define \[ \Theta_\tau(z) =\prod_{p=0}^{\infty}(1-\tau^p z) \prod_{p=1}^{\infty}(1-\tau^p/z). \tag{5}\] We write \(\Theta\) when \(\tau\) is fixed. On every compact annulus, the sums of the absolute deviations of the factors from \(1\) are bounded by geometric series. The products therefore converge locally uniformly on \(\mathbb C^*\), and reindexing gives \[\Theta(\tau z)=-z^{-1}\Theta(z).\] Its zeros are precisely the points of \(\tau^{\mathbb Z}\), all simple: exactly one factor vanishes at each, while the remaining product is nonzero. Thus \(\Theta(z/a)\) is a section of \(D(1,-a)\) with divisor the single point represented by \(a\in\mathbb C^*\). Products of \(n\) such sections, with the parameters having product \((-1)^n\gamma\), show that \(D(n,\gamma)\) has degree \(n\) for \(n>0\). Tensor products and duals give the formula for every integer \(n\); in degree zero, use \(D(0,\gamma)=D(1,\gamma)\otimes D(1,1)^{-1}\).

The multiplicative theta convention and residue-class bases are classical; see Rosengren (Rosengren 2020, arXiv version, Section 1.2 and Exercise 1.6.5). We include the proof to fix all multipliers and normalizations.

Choosing the cubic and marked points

We now fix a nonsquare integer \(r\geq10\) and a hypothetical universal system \(\mathsf U_r(d,m)\) with \(d/m\leq\sqrt r\). The rationality of \(d/m\) and Lemma 5 give \(3<d/m<\sqrt r\). Set \[ \begin{gathered} q=r-9,\qquad \rho=3(d/m-3),\qquad \rho_*=3(\sqrt r-3),\\ 0<\rho<\rho_*,\qquad 2\rho_*+\frac{\rho_*^2}{9}=q. \end{gathered} \tag{7}\] Put \[h=3(d-3m)=m\rho>0,\qquad u_0=\lfloor d/3\rfloor\geq m.\]

Set \(x_0=1/2\) and choose a rational number \(\Delta\) such that \[ 0<\Delta<x_0,\qquad \rho+\frac{\Delta\rho}{9}<\rho_*. \tag{8}\] The strict slope gap leaves an interval of such choices. Rationality of \(\Delta\) will keep the Laurent exponent intervals away from integral endpoints.

Choose nine distinct real numbers \(x_i\) with \[ \begin{gathered} 0<x_i<x_0,\qquad \sum_{i=1}^{9}(x_0-x_i)=\Delta,\\ \eta=\frac{\sum_{i=1}^{9}x_i+\rho_*}{3},\qquad z_i=\tau^{x_i}\ (1\leq i\leq9),\qquad z_0=\tau^{x_0}. \end{gathered} \tag{9}\] For example, take \(x_i=x_0-i\Delta/45\). The quantities \(\rho,\Delta,x_i,\eta\) can be kept unchanged under a common multiple of \((d,m)\), whereas \(h,u_0\) and the coefficient spaces depend on the actual pair. We keep this pair fixed while taking the collision on each curve and the subsequent limit in \(\tau\).

For each \(0<\tau<1\), use Lemma 8 to identify \(X_\tau\) with a plane cubic \(B\), with the holomorphic identification \[L=\mathcal O_B(1)=D(3,\tau^\eta),\qquad M=L^3.\] Let \(a_i\) be the point represented by \(z_i\) and put \(P=a_1+\cdots+a_9\). These points and the point represented by \(z_0\) are distinct: their exponents lie in \((0,1)\), and two lifts represent the same point only if their exponents differ by an integer.

In the covering-space frame, the section \[\Pi(z)=\prod_{i=1}^{9}\Theta(z/z_i)\] has divisor \(P\) and gives an identification \[\mathcal O_B(P)=D\left(9,-\tau^{\sum_i x_i}\right)\] under which the canonical section \(\sigma_P\) is represented by \(\Pi(z)\). Thus the bundles already used in the construction are \[A=M\otimes\mathcal O_B(-P),\qquad T=L^{d-3m}.\] Since \(3\eta=\sum_i x_i+\rho_*\), we have \[A=D(0,-\tau^{\rho_*}),\qquad \deg T=h,\qquad T\otimes A^m=L^d\otimes\mathcal O_B(-mP).\] These identifications use the covering-space frames and their tensor products. We retain the displayed powers of \(\tau\) as written, without replacing the multiplier presentations by isomorphic ones.

A cubic equation identifies the normal bundle of \(B\) with \(M=L^3\). Its differential at each point surjects onto the normal fiber, so every scalar specified in the chosen covering-space frame is an allowed normal displacement in Proposition 3. Rescaling the cubic equation does not restrict those choices. All constructions are made for each fixed \(\tau\); their algebraic dependence on \(\tau\) is not required.

Denote the resulting polynomial space by \(\mathcal W_\tau\). By Lemma 7, its first \(m+1\) coefficient blocks have degree and dimension \(h\). For \(j>m\), the dimension is \(3(d-3j)\) when \(d-3j>0\), and is one when \(d-3j=0\). In the last case the source is \(H^0(B,\mathcal O_B)=\mathbb C\), since \(B\) is compact and connected.

Near the point represented by \((z,v)=(z_0,1)\), use coordinates \[ z=z_0e^x,\qquad v=e^y. \tag{10}\] Choose the \(x\)-disk small enough to map injectively to \(B\) and avoid \(P\), and the \(y\)-disk small enough that \(e^y\) is injective. In the fixed covering-space frame of \(T\otimes A^m\), the scalar expression is \[\widetilde F(x,y)=\sum_{j=0}^{u_0}g_j(z_0e^x)e^{jy}.\] Here \(g_j(z)\) denotes the holomorphic function in its covering-space frame; all summands take values in the same bundle.

For every fixed \(\tau\in(0,1)\) and every \(q\) distinct positions \(\xi_1,\ldots,\xi_q\) sufficiently near zero, apply Proposition 6 in this disk and the fixed frame of \(A\). It gives a nonzero \(\widetilde F\in\mathcal W_\tau\) of multiplicity at least \(m\) at each \((x,y)=(\xi_\alpha,0)\), since \(v=1\) corresponds to \(y=0\). The existence is pointwise in \(\tau\) and in the tuple of positions; the collision will use no simultaneous choice of sections.

Collision on a fixed surface

The collision below belongs to the broader study of limits of fat-point conditions; compare Evain (Evain 2007, sec. 2 and 6). We prove the particular local constraint directly.

For each fixed \(\tau\), use the coordinates and the frame in (10). If \(F\in\mathcal W_\tau\) has coefficients \(g_j\), write its scalar expression in this chart as \[H_F(x,y)=\sum_{j=0}^{u_0}g_j(z_0e^x)e^{jy}.\] Define the finite index set \[\mathcal I=\{(\ell,b)\in\mathbb Z_{\geq0}^2: 0\leq b<m,\quad 0\leq\ell<q(m-b)\}\] and the linear map \(J_\tau:\mathcal W_\tau\longrightarrow\mathbb C^{\mathcal I}\) whose coordinates are \[ (J_\tau F)_{\ell,b} =\left.\partial_x^\ell\partial_y^b H_F(x,y)\right|_{(0,0)}, \qquad 0\leq b<m,\quad 0\leq\ell<q(m-b). \tag{11}\] The value bundle is expressed in one fixed local frame when these derivatives are taken.

The rows in (11) describe a local ideal. If the distinct centers are \((\xi_a,0)\) and \(Q(x)=\prod_{a=1}^q(x-\xi_a)\), write a local holomorphic function as \(\sum_{b\geq0}h_b(x)y^b\). The multiplicity conditions require \(Q^{m-b}\) to divide \(h_b\) for \(b<m\). When the centers collide, the corresponding limiting conditions are divisibility by \(x^{q(m-b)}\), or membership in \((x^q,y)^m\). Their number is \[|\mathcal I|=\sum_{b=0}^{m-1}q(m-b)=\frac{qm(m+1)}2.\] Lemma 9 establishes these conditions for a nonzero limit. It remains to compute the matrix of \(J_\tau\) in bases suited to \(\tau\to0^+\).

A common chart and normalized theta sections

Although the tori vary, one polydisk in \((x,y)\) is a valid chart for all sufficiently small positive \(\tau\). Fix \(R\) with \(0<R<\pi/2\), and let \[\delta_P=\min_{1\leq i\leq9} \{x_0-x_i,\,1-(x_0-x_i)\}>0.\] For small enough \(\tau\), require \(|\log\tau|>2R\) and \(\delta_P|\log\tau|>R\). If \(z_0e^x\) and \(z_0e^{x'}\), with \(|x|,|x'|<R\), represent the same point of \(X_\tau\), then \[x'-x=a\log\tau+2\pi\mathrm{i}b\qquad(a,b\in\mathbb Z).\] The bounds on the real and imaginary parts force \(a=b=0\). Thus this disk maps injectively to \(X_\tau\). If a point of the disk represented a point of \(P\), its real coordinate would satisfy \[\operatorname{Re}x=(x_i-x_0+a)\log\tau \qquad\text{for some }i\text{ and }a\in\mathbb Z,\] whose absolute value is at least \(\delta_P|\log\tau|>R\). Hence the disk avoids \(P\). The covering-space frame trivializes \(A\) there, and \(v=e^y\) is injective for \(|y|<R\). Thus (10) gives the common polydisk \(|x|,|y|<R\) around the marked point on \(\operatorname{Tot}(A)\). We continue to use the covering-space frames of Section 3.

Bases for the polynomial spaces

We now apply Lemma 10 to the actual coefficient spaces of \(\mathcal W_\tau\). For \(0\leq j\leq m\), their bundle is \[T\otimes A^{m-j} =D\bigl(h,(-1)^{m-j} \tau^{(d-3m)\eta+(m-j)\rho_*}\bigr).\] Since \(\eta-3x_0=(\rho_*-\Delta)/3\), its interval parameter is \[U_j=\frac h9(\rho_*-\Delta)+(m-j)\rho_*.\] For \(m<j\leq u_0\) with \(d-3j>0\), the space in (4) is the image of the sections of \[L^{d-3j}=D\bigl(n_j,\tau^{(d-3j)\eta}\bigr), \qquad n_j=3(d-3j)=h-9(j-m)>0,\] under multiplication by \(\Pi^{j-m}\). The interval parameter for this source bundle is \[U_j=\frac{n_j}{9}(\rho_*-\Delta).\] These parameters are irrational. Indeed, after substituting \(\rho_*=3(\sqrt r-3)\), the coefficient of \(\sqrt r\) in either expression is the positive integer \(d-3j\), and the remaining terms are rational because \(\Delta\in\mathbb Q\). In particular, \[ U_j\notin\mathbb Z\qquad (0\leq j\leq u_0,\ d-3j>0). \tag{13}\]

The finite exponent set is \[ \begin{split} \mathcal S={}& \bigcup_{j=0}^{m} \left\{(j,K):K\in\mathbb Z,\quad U_j<K<U_j+h\right\} \\ &{}\cup \bigcup_{\substack{m<j\leq u_0\\d-3j>0}} \left\{(j,K):K\in\mathbb Z,\quad U_j<K<U_j+n_j\right\} \\ &{}\cup \left\{(j,0):m<j\leq u_0,\quad d-3j=0\right\}, \qquad n_j=h-9(j-m). \end{split} \tag{14}\] All indices \(j\) in this definition are integers. The last set is empty unless \(3\) divides \(d\), and otherwise consists of the single pair \((d/3,0)\). Because \(d>3m\), that pair has \(j>m\).

For a pair \((j,K)\) in the first union, choose the normalized section from Lemma 10 in \(H^0(B,T\otimes A^{m-j})\) as the coefficient of \(v^j\). For a pair in the second union, choose the normalized section in \(H^0(B,L^{d-3j})\), multiply it by \(\Pi^{j-m}\), and use the result as the coefficient of \(v^j\). For the last union, use \(\Pi^{j-m}\) as the coefficient of \(v^j\): the source space is \(H^0(B,L^0)=\mathbb C\), with basis the constant section \(1\). Denote the resulting fiber polynomials by \(E_{j,K,\tau}\).

Each choice gives a basis of its coefficient block. For \(j>m\), multiplication by \(\Pi^{j-m}\) is injective, since \(\Pi\) is nonzero on \(B\setminus P\), and its image is the block by definition. The different powers \(v^j\) form a direct sum. Hence \[\{E_{j,K,\tau}:(j,K)\in\mathcal S\}\] is a basis of \(\mathcal W_\tau\) for every \(\tau\in(0,1)\). There are no duplicate pairs, and the index set is independent of \(\tau\).

The block dimensions also give the useful identity \[|\mathcal S|=\dim\mathcal W_\tau =\binom{d+2}{2}-9\binom{m+1}{2}.\] To see this, sum the dimensions \(3(d-3j)\) of \(H^0(B,L^{d-3j})\) for \(d-3j>0\) and add one when \(3\) divides \(d\). The result is \(\binom{d+2}{2}\). For \(j<m\), the coefficient condition in (3) reduces the block dimension by \(9(m-j)\); their sum is \(9\binom{m+1}{2}\).

In the common local chart, the scalar expressions of this basis satisfy \[H_{E_{j,K,\tau}}(x,y)\longrightarrow e^{Kx+jy} \qquad((j,K)\in\mathcal S)\] locally uniformly, together with all fixed mixed derivatives. For the first union this is (12) times \(e^{jy}\). For the second union, the additional factor \(\Pi(z_0e^x)^{j-m}\) tends to \(1\) with all derivatives by Lemma 10; the exponent \(j-m\) is fixed. The same product assertion proves the claim for the degree-zero case with \(K=0\). As \(\mathcal S\) is finite, all of these convergences can be taken on one fixed smaller polydisk and for the same sufficiently small values of \(\tau\).

The limiting jet matrix

Evaluation of monomials in exponent pairs is a useful way to express interpolation rank. Dumnicki (Dumnicki 2007, proof of Proposition 13) uses this principle to relate a one-point multiplicity matrix to polynomial evaluation on a diagram. Here the normalized theta sections produce the exponent pairs directly, with the derivative indices prescribed by the preceding collision.

The factors \(\tau^{-x_0K}\) may tend to zero or infinity, but for each \(\tau>0\) they give invertible basis changes and preserve rank. Lemma 9 supplies a nonzero kernel separately for each fixed \(\tau\); Proposition 11 compares numerical matrices in the explicit bases. Thus no continuous choice of kernel vectors, or algebraic family of plane embeddings across \(\tau=0\), is required. To contradict Lemma 9, it remains to prove full column rank of the corresponding limiting matrix for a sufficiently large common multiple of \((d,m)\).

Interpolation by horizontal rows

The interpolation procedure treats larger rows first and multiplies each correction by factors vanishing on all previously treated rows. The number of earlier rows controls the degree in \(K\), while the current row size controls the degree in \(J\). Their joint constraint explains why a bound on the largest row, or on the total number of points, would not suffice.

For the exponent set \(\mathcal S\), it remains to prove the two row bounds in this lemma. We will obtain them by enclosing \(\mathcal S\) in a polygon and using its horizontal slice lengths to count the rows.

An enclosing polygon

Write \(a=\rho_*=3(\sqrt r-3)>0\), and set \[U_0=\frac h9(a-\Delta)+ma, \qquad X=\frac jm,\qquad Y=\frac{K-U_0}{m}.\] Thus \(U_0\) is the lower endpoint of the exponent interval for \(j=0\). For \(0<\lambda\le1\), and with \(t_+=\max\{t,0\}\), define \[ \mathcal D_\lambda =\bigl\{(X,Y)\in\mathbb R^2: X\ge0,\quad -aX\le Y\le-aX+\lambda a-9(X-\lambda)_+\bigr\}. \tag{17}\] By (8), we have \(\rho+\Delta\rho/9<a\). Choose \(0<\lambda<1\) sufficiently close to \(1\) that \[ \rho+\frac{\Delta\rho}{9}<\lambda a-9(1-\lambda). \tag{18}\] This choice depends only on \(r\), the ratio \(d/m\), and \(\Delta\). We will show that this inequality places \(\mathcal S\) inside \(m\mathcal D_\lambda+(0,U_0)\). The horizontal widths of this scaled, translated polygon will be at most \(m\lambda<m\), and the heights supporting a slice of length at least \(s\) will form an interval of length \(q(m\lambda-s)\) for \(0\le s\le m\lambda\).

Horizontal slices and integer rows

The length of a bounded interval is the difference of its endpoints; a singleton has length zero. When a required length is zero, we still require the slice to be nonempty.

For each integer \(K\), let \[N_K=\#\{j\in\mathbb Z:(j,K)\in\mathcal S\}.\] Only finitely many of these row sizes are nonzero. The slice profile in Figure 1 bounds both each row size and the number of rows of any given minimum size. The passage from lengths to integer counts adds at most one endpoint. We will absorb it with the size condition \[ q(1-\lambda)m\ge1. \tag{20}\]

Full rank and the strict inequality