The relation problem

Mumford initiated the systematic study of natural classes in the Chow ring of the moduli space of curves [39]. Faber’s calculations and geometric constructions on powers of the universal curve led to a proposed description of the tautological ring of the smooth unpointed space \(\mathcal M_g\), including its Gorenstein property [16]. Faber and Zagier later gave explicit hypergeometric formulas for relations among the kappa classes. Pandharipande–Pixton proved these relations in Chow using stable quotients and formulated their completeness on \(\mathcal M_g\) [43].

On the compactification, a presentation must also account for boundary strata and their intersections. Arbarello–Cornalba and Graber–Pandharipande developed the forgetful and boundary calculus [2], [21]; in genus zero, Keel gave a presentation of the entire Chow ring by boundary divisors [27]. Pixton’s 2012 graph formula extends the Faber–Zagier system to \(\overline{\mathcal M}_{g,n}\) and proposes a complete set of relations [47]. Pandharipande–Pixton–Zvonkine proved the cohomological validity of this system using the \(3\)-spin CohFT [44]; Janda proved its Chow validity using equivariant Gromov–Witten theory of \(\mathbb P^1\) [25]. These results establish relations, while completeness asks whether any further relations exist.

We fix the original relation space precisely. Let \(S_{g,n}\) be the strata algebra of formal decorated stable graphs, with decoration degree truncated at the dimension of each vertex factor. Its realization is \[q_{g,n}\colon S_{g,n}\longrightarrow A^*(\overline{\mathcal M}_{g,n};\mathbb Q), \qquad [\Gamma,\gamma]\longmapsto\xi_{\Gamma*}\gamma.\] The gluing pushforward generators are unnormalized, as in [44]. Let \(\mathcal P_{g,n}\) be the span obtained by placing one of Pixton’s graph-formula relations at a vertex of an outer stable graph, placing arbitrary strata classes at the other vertices, and gluing. Pixton’s Proposition 1 shows that this span is already an ideal, stable under the gluing and forgetting pushforwards and pullbacks [47]. Thus it includes these formal consequences from the outset. The smaller \(0/1\)-marking seed system recovers the same span under the operations described in [44]. Section 2 gives the full recipe and all conventions.

Writing \(R^*(\overline{\mathcal M}_{g,n})=\mathop{\mathrm{im}}q_{g,n}\), the Chow completeness question is whether \[\ker q_{g,n}=\mathcal P_{g,n}\] for every stable pair. The known validity theorem gives the inclusion \(\mathcal P_{g,n}\subseteq\ker q_{g,n}\). Our example makes it strict.

The counterexample

The construction uses \(g\) colors and one three-label factor for each increasing triple of colors. Each factor is a product of two rational-tail divisor sums, corrected by a class supported on two-node rational bridges. We multiply these factors and alternate over relabelings that preserve the color of each slot. Definition 5 specifies the resulting formal class \(Y\).

The same class gives the rational-cohomological consequence. To specify that formulation, write \(q^{\mathrm{CH}}_{g,n}=q_{g,n}\) and define, in formal degree \(d\), \[q^H_{g,n}=\operatorname{cl}\circ q^{\mathrm{CH}}_{g,n} \colon S^d_{g,n}\longrightarrow H^{2d}\!\left(\overline{\mathcal M}_{g,n}(\mathbb C);\mathbb Q\right),\] where \(\operatorname{cl}\) is the rational cycle-class map for the complex Deligne–Mumford stack. Its image is the cohomological tautological ring \(RH^*=\operatorname{cl}(R^*)\). Completeness in both \(R^*\) and \(RH^*\) is explicitly recorded as Pixton’s conjecture in [9].

The theorem and corollary give negative answers to the Chow and rational-cohomological completeness questions for the full original system. The large parameters leave ample room in the support and color estimates; they are not proposed as minimal parameters.

Completeness is distinct from the Gorenstein property. The latter asks for nondegenerate multiplication pairings in complementary degrees, whereas completeness asks for all relations among the generators. Pixton proved that \(R^*(\mathcal M_{25})\) is not Gorenstein while establishing Faber–Zagier completeness there [48]. Canning–Larson–Schmitt proved completeness together with non-Gorenstein behavior on the compact-type spaces \(\mathcal M^{\mathrm{ct}}_6\), \(\mathcal M^{\mathrm{ct}}_{5,2}\), and \(\mathcal M^{\mathrm{ct}}_7\) [10]. Our relation lies on the full stable compactification.

There is also a distinction between completeness and comparison of methods for producing relations. Janda’s comparison theorem identifies the systems arising from cancellation of reconstruction singularities for convergent semisimple CohFTs with nonempty discriminant [26]. It concerns the relations generated by those reconstructions; it does not identify the entire Chow relation kernel. The comparison throughout this paper is with the explicitly fixed span \(\mathcal P_{g,n}\).

Proof and methods

The proof compares two algebras of cubic coefficients. A formal super Frobenius algebra can carry independent odd cubic coefficients while satisfying the identities needed to annihilate the original Pixton span. The actual Chow correspondences of a curve satisfy an additional nilpotence identity obtained from a surface Quot scheme. The first algebra proves nonmembership; the second proves vanishing. Figure 1 separates these two assertions about \(Y\).

Conventions

All varieties, stacks, and families are over \(\mathbb C\), and Chow groups have rational coefficients. On a smooth space we identify Chow classes with operational classes when pulling them to a test space. Products of stacks are understood in rational Chow theory [30]. We use proper rational pushforward for relative Deligne–Mumford maps and its compatibilities from [4]. Smooth Deligne–Mumford stacks have the operational identification in [4]; for smooth quotient stacks, see also [13]. Every formal series calculation is truncated at a specified coefficient, so only finitely many terms occur.

Parity is distinct from Chow codimension. In a super tensor product, interchanging homogeneous factors of parities \(p,q\) gives \((-1)^{pq}\). In particular, on an odd vector space the super-symmetric tensors are ordinary alternating tensors. All finite averaging and polarization arguments take place in characteristic zero.

Decorated graphs

A stable graph \(\Gamma\) of type \((g,n)\) is a connected graph with \(n\) labeled legs and genera \(g_v\geq0\) at its vertices, satisfying \[g=\sum_{v}g_v+|E(\Gamma)|-|V(\Gamma)|+1, \qquad 2g_v-2+n_v>0.\] Here \(n_v\) counts all incident half-edges, including legs. Its gluing map is \[\xi_\Gamma\colon\prod_{v\in V(\Gamma)} \overline{\mathcal M}_{g_v,n_v}\longrightarrow\overline{\mathcal M}_{g,n}.\] At every half-edge let \(\psi\) denote the cotangent class. At a vertex define \[\kappa_j=\pi_*(\psi_{n_v+1}^{j+1}),\qquad \kappa_0=2g_v-2+n_v,\] where \(\pi\) forgets an additional marking.

The vector space \(S_{g,n}\) is spanned by formal symbols \([\Gamma,\gamma]\), modulo graph isomorphism, with \(\gamma\) a monomial in the half-edge \(\psi\)’s and vertex \(\kappa_j\)’s for \(j\geq1\). At each vertex its decoration degree is at most \(3g_v-3+n_v\). The total degree is \(|E(\Gamma)|+\deg\gamma\), and \[q_{g,n}[\Gamma,\gamma]=\xi_{\Gamma*}(\gamma).\] There is no automorphism denominator in this definition of \(q_{g,n}\). The per-vertex degree bound is the explicit convention in [44].

We use the usual decorated-graph intersection product. More explicitly, take common refinements with specified contractions to fixed representatives of the input graphs, requiring every refined edge to survive in at least one contraction. Divide by the source symmetries preserving those contractions. For each additional use of an overlapping edge insert its normal factor \(-\psi-\psi'\); pull the leg and flag decorations to their components and split kappa classes additively. Specified contractions record the lifts to the gluing parameter spaces, so this convention gives the product of the unnormalized pushforward generators. The construction is the transverse and self-intersection rule for node cuts [21].

Terms exceeding any vertex dimension are set to zero. For multiplication and restriction to a further graph splitting, an excessive decoration remains excessive: the combined dimensions on the pieces decrease, so at least one piece still exceeds its dimension. Forgetting pullback of an undecorated graph is obtained by adding the new leg successively at each vertex. These conventions lift the corresponding Chow operations and do not impose any unproved relation. The preceding dimension argument is not an assertion about forgetting, which raises a vertex dimension. Compatibility of forgetting with the formal detector is checked separately in Section 3.

The exact Pixton span

Define \[A(T)=\sum_{m\geq0}\frac{(6m)!}{(3m)!(2m)!}T^m,\qquad B(T)=\sum_{m\geq0} \frac{6m+1}{6m-1}\frac{(6m)!}{(3m)!(2m)!}T^m.\] For a commuting parity variable \(\zeta^2=1\), put \[\widehat C_{3j}(T,\zeta)=T^jA(\zeta T),\qquad \widehat C_{3j+1}(T,\zeta)=\zeta T^jB(\zeta T).\] Indices congruent to \(2\) modulo \(3\) are not used. For a series \(F\) write \[\{F\}=\sum_{m\geq0,\ \epsilon\in\{0,1\}} [F]_{T^m\zeta^\epsilon}K_{m,\epsilon}T^m.\] Give every vertex of \(\Gamma\) its own variable \(\zeta_v\), with \(\zeta_v^2=1\). Define the operator \[\widehat\kappa_\Gamma \left(\prod_{i=1}^{\ell}K_{e_i,\epsilon_i}\right) = \sum_{\tau\in\mathfrak S_\ell} \prod_{\substack{c\text{ a cycle}\\\text{of }\tau}} \left( \sum_{v\in V(\Gamma)} \kappa^{(v)}_{\sum_{i\in c}e_i} \zeta_v^{\sum_{i\in c}\epsilon_i} \right).\] The empty product maps to \(1\), and the operator acts coefficientwise in \(T\). The permutation-cycle expression is Faber’s formula, as recorded in [2].

For an edge whose two half-edges lie at \(v,w\), with cotangent symbols \(x,y\), set \[\Delta_e= \frac{ A(\zeta_vxT)\zeta_wB(\zeta_wyT) +\zeta_vB(\zeta_vxT)A(\zeta_wyT) +\zeta_v+\zeta_w }{(x+y)T}.\] This is a power series by \[A(T)B(-T)+A(-T)B(T)+2=0.\] The same rule applies to a loop.

No converse to Proposition 4 is used. In particular, same-vertex multiplication of a relation does not enlarge the span, whereas an arbitrary new Chow relation is not assumed to belong to it.

The corrected cubic and its alternation

For the rest of the construction fix \[ g=10^{60},\qquad D_*=\binom g3,\qquad n=3D_*. \tag{3}\] Form a word \(\mathbf i\) in the colors \(1,\ldots,g\) by concatenating all increasing triples of colors in lexicographic order. Its consecutive three-slot blocks will be denoted \(J\). Each color occurs \(\binom{g-1}{2}\) times.

For a set of labels \(T\) of size at least two, let \(D_T\) be the one-edge class cutting off a genus-zero tail with precisely those labels. For distinct labels \(a,b\), put \[\bar D_{ab}=\sum_{T\supseteq\{a,b\}}D_T.\] We also need a class supported on rational bridges, not tails.

Every operation in Definition 5 is finite. Each cubic factor has codimension two, so \[\deg Y=2D_*<3g-3+n.\] The pair \((g,n)\) is stable. The automorphism weights in \(\beta_J^{\mathrm{st}}\) specify coefficients of this particular class; they do not change the unnormalized generators of \(S_{g,n}\).

The proof of Theorem 1 now divides into two independent claims. Proposition 6 proves formal nonmembership. The remaining sections establish the geometric vanishing in Theorem 36.

An isotropic cubic Frobenius algebra

The state space has the curve pairing familiar from Frobenius-algebra and CohFT constructions [1, 45]. The exterior coefficient ring below allows an independent cubic on every triple of colors. We check associativity and the handle formula directly, since these particular identities govern the graph test.

Let \[R=\Lambda_{\mathbb Q}\bigl(\theta_{ijk}:1\leq i<j<k\leq g\bigr),\] with each generator odd. Let \(W\) be a purely odd symplectic vector space of dimension \(2g\), with complementary Lagrangians \(W_+\) and \(W_-\). Choose bases \(a_1,\ldots,a_g\) and \(b_1,\ldots,b_g\) such that \(\langle a_i,b_j\rangle=\delta_{ij}\). Extend the symbols \(\theta_{ijk}\) alternately to all triples of indices. There is an alternating, odd-coefficient three-form \(\Phi\) on \(W\) defined by \[\Phi(a_i,a_j,a_k)=\theta_{ijk}, \qquad \Phi(w_1,w_2,w_3)=0 \quad\text{if some }w_i\in W_-.\]

Consider the free supermodule \[V=R\otimes_{\mathbb Q}\bigl(\mathbb Q1\oplus W\oplus\mathbb Qe\bigr),\] where \(1,e\) are even. Its trace, denoted by \(\int\), takes \(e\) to \(1\) and is zero on \(1,W\). Give \(V\) the even pairing which is hyperbolic on \(\mathbb Q1\oplus\mathbb Qe\), the prescribed symplectic pairing on \(W\), and zero between these two summands. Declare \(1\) to be the unit, \(e^2=eW=0\), and determine the remaining products by \[\int w_1w_2=\langle w_1,w_2\rangle, \qquad \int w_1w_2w_3=\Phi(w_1,w_2,w_3).\] Equivalently, the product of two standard odd vectors is its symplectic pairing times \(e\), plus the vector dual to the cubic functional in the third input. The latter vector has odd coefficients and belongs to \(R\otimes W_-\).

Write \(\mu\) for multiplication and \(\Delta\) for its adjoint coproduct.

Set \[\lambda=(2g-2)e.\] For a nonempty finite label set \(L\), write \(\Delta_L:V\to V^{\otimes L}\) for the coproduct adjoint to iterated multiplication; for a singleton it is the identity. Coproducts are coassociative and satisfy the projection formula. In particular, writing \(\Delta_{ij}(1)\) with units in all other slots, we have \[\begin{align*} \Delta_{ij}(1)\Delta_{ik}(1)&=\Delta_{ijk}(1), \tag{5}\\ \Delta_{ij}(1)^2&=-\Delta_{ij}(\lambda). \tag{6}\\ \int_j\Delta_{ij}(v)&=v. \tag{7}\end{align*}\] The minus sign in (6) will match the normal bundle of a rational-tail boundary divisor.

Compression of rational tails

Call a graph a rational-tails graph if it is a tree with one genus-\(g\) vertex, called its root, and all other vertices rational. Every branch away from the root has nonempty labeled descendants. Such a graph has no automorphisms fixing the labels: the descendant label sets distinguish its branches, recursively.

We initially work without the degree cutoff, discard all graphs which are not rational-tails graphs, and allow genuine genus-zero Chow identities on rational parameter factors. This is only an auxiliary computation; we will prove that its output respects the cutoff in \(S_{g,m}\). We use the genus-zero Chow boundary presentation of [27].

For each label \(i\), put \[\bar\psi_i=\psi_i-\sum_{\substack{T\ni i\\ |T|\geq2}}D_T.\] If the labels are ordered \(1,\ldots,m\), let \(L_x\) be the pullback of \(\psi_x\) from the space retaining just \(1,\ldots,x\). Thus \[L_x=\psi_x- \sum_{\substack{T\ni x,\ |T|\geq2\\T\setminus\{x\}\subseteq\{x+1,\ldots,m\}}} D_T.\] For \(j\geq1\), define \[\bar\kappa_j=\kappa_j-\sum_{x=1}^m L_x^j.\] The leading, graphless change of variables is \(\bar\psi_i=\psi_i\) and \(\bar\kappa_j=\kappa_j-\sum_i\psi_i^j\), which is an invertible polynomial change. Hence the bar monomials form a complement to the boundary subspace before truncation.

Define \(t_m\) on that complement by \[ t_m\left(\prod_i\bar\psi_i^{d_i} \prod_{j\geq1}\bar\kappa_j^{c_j}\right) = \begin{cases} \lambda_1^{d_1}\cdots\lambda_m^{d_m}, &\text{if every }c_j=0,\\ 0,&\text{otherwise}. \end{cases} \tag{8}\] Here the tensor on the right has a unit in every unused slot. For a nontrivial rational-tails graph, integrate each non-root decoration on its genus-zero parameter space, retaining only its top-degree integral. Apply \(t_a\) to the decorated root with its \(a\) incident slots. Finally duplicate each root slot into the descendant labels of its branch by \(\Delta_L\). Every branch attached to the root has at least two descendant labels, so \(a<m\); this is a recursive definition. For \(m=0\), the empty tensor is \(R\), the unit maps to \(1\), and positive bar-kappa monomials map to zero.

Boundary operations and the test

We first show how the test treats multiplication by bar classes and rational-tail divisors. We then prove the forgetting identities that will reduce the number of labels in the relation argument. All tensor products, contractions, and permutations below use the super sign convention.

Forgetting markings

The forgetting identities are needed in both directions: pullback will remove a zero-index leg from the Pixton recipe, and pushforward will test a tensor slot against the two even basis vectors.

The zero-leg identity

We require one precise identity for the recipe in Definition 3. Package the partition insertions with formal variables \(s_\ell\), one for each allowed positive part, using exponential coefficients for repeated parts, and put \[H(T,\zeta)=\widehat C_0(T,\zeta) -\sum_\ell s_\ell\widehat C_\ell(T,\zeta).\] The variables \(s_\ell\) are purely formal; all subsequent coefficient extractions involve finitely many of them. The logarithm below is expanded jointly in \(T\) and the \(s_\ell\), so its constant term at \(T=s_\ell=0\) is defined even though \(H(0,\zeta)\) depends on the partition variables. The cycle exponential formula for \(\widehat\kappa_\Gamma\) rewrites the vertex characteristic as \[ \exp\{-\log H\}_{\kappa,\zeta_v}. \tag{21}\] Here a monomial \(T^d\zeta^\epsilon\) in \(-\log H\) is replaced by \(\kappa_dT^d\zeta_v^\epsilon\), including \(d=0\). For the edges use \(U(x,x')=T\Delta_e\), so the coefficient of total degree \(T^r\) is the degree-\(r\) recipe. Denote this packaged series, with its usual parity extractions, by \(\mathcal R(s;a)\).

Annihilating the relation span

The nonzero value on the candidate

We finish the proof of Proposition 6. Every graph in a bridge correction \(\beta_J^{\mathrm{st}}\) has nontrivial unmarked stabilization and is not a rational-tails graph. Its refinements, pullbacks, and products remain outside the support of the test. By (11) and (5), the remaining part of each factor of \(Y\) tests as \[\Delta_{ab}(1)\Delta_{ac}(1)=\Delta_{abc}(1).\] Pair slot \(j\) with the first-Lagrangian vector \(a_{i_j}\) prescribed by the color word in Definition 5. A block with increasing colors \(i<j<k\) gives \(\theta_{ijk}\). The product of the unsymmetrized factors therefore evaluates, up to one fixed super sign, to \[\Theta=\prod_{1\leq i<j<k\leq g}\theta_{ijk}\neq0,\] with the product ordered as the blocks.

Let \(G_{\mathbf i}\) be the stabilizer of the word. For \(\tau\in G_{\mathbf i}\), transporting its super permutation across the pairing of odd input slots contributes \(\operatorname{sgn}(\tau)\); the input word itself is unchanged. This cancels the explicit sign of that summand in \(Y\). No reordering of the exterior generators is required in this comparison: all summands are compared with the same ordered evaluation. Every color occurs \(\binom{g-1}{2}\) times, and hence \[ \bigl\langle t_n(Y), a_{i_1}\otimes\cdots\otimes a_{i_n}\bigr\rangle = \pm\left(\binom{g-1}{2}!\right)^g\Theta\neq0. \tag{28}\] Together with Lemma 12, this proves the proposition.

Correspondences and the curve projector

We use the rational correspondence category whose objects are smooth projective \(S\)-families and whose morphisms are \[\mathop{\mathrm{Hom}}(P,Q)=A^*(P\times_S Q),\] with all codimensions allowed. Composition is pullback to the triple product, intersection, and pushforward. Tensor product is relative product. Equivalently, this is the usual correspondence calculus with Tate twists forgotten. We take finite direct sums and split idempotents. The diagonal identifies every generating object with its dual; the evaluation for the curve is its intersection pairing followed by \(\pi_*\).

The morphisms that vanish away from a closed base subset of codimension at least \(a\) form a tensor ideal. Indeed, a finite union of such subsets has the same codimension bound, and restriction to its complement commutes with every operation just described. We can therefore first quotient by this ideal and then split idempotents. Morphisms between the original objects are unchanged by the latter completion. Denote either the unquotiented category or this completed quotient by \(\mathcal C\). Its tensor unit is the base \(S\). The endomorphisms of that unit are \(A^*(S)\), or their base-support quotient; positive-degree base classes remain part of the correspondence calculus unless the chosen quotient kills them.

The projection formula gives \[ \pi_*1_C=0,\qquad \pi_*e'_C=1,\qquad \pi_*((e'_C)^2)=a_C-a_C=0. \tag{31}\] Thus \(1_C,e'_C\) span a nondegenerate hyperbolic plane in the curve object. The two corresponding projectors sum to \(e'_{C,1}+e'_{C,2}\), so their orthogonal complement has the self-transpose projector \(\eta\). Write this complementary object as \(H_C\).

The categorical trace of the curve identity is \(\pi_*c_1(T_{C/S})=2-2b\), by diagonal self-intersection. Each of the two unit summands has trace one. Consequently \[ \dim_{\mathcal C}H_C=-2b. \tag{32}\] The remaining relation needed to model \(H_C\) by an odd symplectic space holds already in relative Chow.

Retaining arbitrary Chow correspondences

Let \(\mathcal R_b=\operatorname{Rep}^{\mathrm{odd}}(\mathop{\mathrm{Sp}}_{2b})\). This is the usual semisimple rational representation category of the split symplectic group, with parity prescribed by its central involution. Its standard representation \(W_b\) is odd. The ordinary alternating symplectic form is therefore a symmetric pairing in the super sense; its closed loop has value \(-2b\).

The first and second fundamental theorems for symplectic tensors present this category, after finite direct sums and idempotent completion, by pairing diagrams with that loop value and the sum-of-matchings relation on \(2b+2\) slots. In this formulation crossings on the standard odd space are minus the ordinary swap. See [50], or [32]. The diagram spaces and relations are rational, so the complex formulation descends by faithful scalar extension to \(\mathbb Q\). Complete reducibility, absolute simplicity, and tensor generation over \(\mathbb Q\) follow from [37]. In particular, the irreducibles occur as rational direct summands of tensor powers of the standard representation. Equations (32) and (33) thus give a symmetric tensor functor \[ F\colon\mathcal R_b\longrightarrow\mathcal C, \qquad F(W_b)=H_C. \tag{36}\] For \(b=0\), (33) says \(\eta=0\), which gives the same conclusion with \(W_0=0\).

It is essential not to assume that (36) is full. The target contains the small diagonal and many other correspondences. We record them by enlarging the coefficient algebra. The following reconstruction is a concrete split semisimple form of O’Sullivan’s tensor realization [42]. It applies to a symmetric tensor functor with this semisimple source and an arbitrary rational tensor target; the endomorphism ring of the target unit need not be a field. We give its coefficient construction explicitly.

Apply the lemma to \[V_b=\mathbb Q1\oplus W_b\oplus\mathbb Qe, \qquad F(V_b)\simeq C.\] In particular, a pure Chow class on \(C^k\) is a morphism \(1\to H_C^{\otimes k}\), so it is identified with an invariant in \(\mathfrak B\otimes W_b^{\otimes k}\). This identification retains the class itself, including any component invisible to cohomology. The small diagonal and the structure map supply multiplication, unit, and integration. Their correspondence identities give a supercommutative Frobenius algebra on \(\mathfrak B\otimes V_b\). The pairing is the one fixed by (31). In particular \[ \varphi(w_1,w_2,w_3)=\int_C w_1w_2w_3 \quad (w_i\in W_b) \tag{39}\] is the tensor of the small diagonal after projection by \(\eta\) in all three slots. Its coefficients are odd, and it is alternating in the ordinary underlying vectors \(w_i\). For a pointed curve over a field, this projection is the modified diagonal of Gross and Schoen [23]. The normalization here also incorporates the base corrections required for a family.

Colored operations and finite polarization

For an ordinary rational color space \(U\), define \[\mathcal A_k(U)= \left( A^*(C^k)\otimes U^{\otimes k}\otimes\mathrm{sgn}_k \right)_{\mathfrak S_k}.\] The product is pullback to disjoint curve slots followed by the coinvariant projection. With parity \(k\bmod2\) in arity \(k\), the direct sum of these spaces is supercommutative: interchanging blocks of \(k\) and \(\ell\) slots has sign \((-1)^{k\ell}\). The same definition applies to a collection of configuration spaces with equivariant forgetting maps, using their forgetting pullbacks for the product. We will use that version for weighted pointed curves.

A symmetric three-slot class gives an odd alternating entry \(f(i,j,k)\), multilinear in its colors. For the pure small diagonal, its scalar-color evaluation is \(\varphi(z(i),z(j),z(k))\). We use a fixed ordering of tensor factors throughout; reordering several entries introduces exactly their Koszul signs.

The surface bound

Let \(\pi_C:C\to S\) be a smooth projective family of connected curves of genus \(b\), with a section, over an irreducible smooth quasi-projective complex variety. Choose a rational divisor \(e_C\) of relative degree one, and use the corrected divisor \(e'_C\), projector \(\eta\), and odd alternating cubic form \(\varphi\) of Proposition 13. Set \[ \chi=(b-1)^2,\qquad s_K=8\chi,\qquad d=3,\qquad t_0=\chi+\frac{d(d-1)}2s_K-1=25\chi-1,\qquad m=3t_0. \tag{40}\] Throughout this section, the support quotient is the quotient by Chow classes that vanish after deleting a closed subset of \(S\) of codimension at least \(b-1\). Its coefficient superalgebra will be denoted by \(\mathfrak B=\mathfrak B_{S,b-1}\).

Isotropic tests alone do not separate arbitrary Chow tensors. In Lemma 27, we first fix an arbitrary color map, find a subspace of its domain with isotropic image, and use the exterior-ideal argument to annihilate the entire sufficiently long product, throughout modulo base support of codimension at least \(b-1\). Only after doing this for every color map do we apply the faithful test of Proposition 13.

We prove the localization relation underlying this theorem in the present section. Section 6 proves that its scalar coefficient is nonzero and thereby completes the proof.

Write \[X=C\times_S C,\qquad \pi_X:X\longrightarrow S,\qquad K=K_{X/S}.\] We write line bundles additively. On every fiber, \(K\) is ample, \(K^2=s_K\), and \(\chi(\mathcal O_X)=\chi\). The curve section gives a section \(o:S\to X\). The two summands of the target bundle and their torus weights are \[ V=L_1\oplus L_2=K\oplus\mathcal O_X,\qquad (w_1,w_2)=(0,1),\qquad A_0=L_1-L_2=K. \tag{43}\] Here the use of \(w_1,w_2\) for weights is confined to formulas concerning the target bundle; the tensor vectors retain their indices \(\alpha\). Let \(\mathcal Q\) be the relative Quot scheme of rank-zero quotients \[0\longrightarrow I'\longrightarrow V\longrightarrow Q \longrightarrow0\] with the following determinant and numerical data: \[ D=dK,\qquad \det Q=D\text{ in }\mathop{\mathrm{Pic}}(X/S),\qquad \int_{X_s}\mathop{\mathrm{ch}}_2(Q_s)=\nu+L_1D-\frac{D^2}{2}, \qquad \nu=t_0-1. \tag{44}\] All intersections in this display are fiber intersections. One may use the relative polarization \(K\) to specify the Hilbert polynomial. Projective representability and the Grassmannian construction are supplied by Grothendieck’s Quot theorem [40]. The determinant condition is a closed fiber of the separated relative Picard scheme [28], so \(\mathcal Q\) is projective over \(S\).

The derived obstruction theories

The determinant constraint changes both the obstruction space and the virtual dimension of \(\mathcal Q\). We first construct its tangent complex as the fiber of the determinant differential. The surjectivity of that differential on obstruction spaces will give a two-term theory. The same construction on the torus-fixed factors will then express their virtual classes as virtual pullbacks from one rank-one Quot scheme.

Fixed loci and their virtual classes

The equivariant embedding and resolution hypotheses of virtual localization [20] hold here. The Grassmannian construction of relative Quot is equivariant and embeds the determinant fiber in a smooth scheme over the smooth base \(S\). One can construct a global equivariant two-term presentation of its perfect obstruction theory using a linearized ample line. For completeness, over an affine base sufficiently positive twists generate its degree-zero cohomology and kill the higher cohomology of its degree-minus-one cohomology. Lifting generators gives a bundle mapping to the complex and surjective on degree-zero cohomology; its derived fiber is a bundle by the Tor-amplitude criterion. Further Serre vanishing lifts the map to the two-term embedding cotangent presentation. Torus linear reductivity makes the choices equivariant. For a nonaffine \(S\), the same argument may be made after a Jouanolou affine-bundle pullback [3]. The resulting identity descends by equivariant Chow homotopy invariance and smooth pullback of the base-changed obstruction theories. Subsequent dimension arguments are on the original base \(S\).

A torus-fixed subsheaf of \(L_1\oplus L_2\) splits into its two weight subsheaves, also in families. Kernels and quotients in the two weights are flat, since they are direct summands of flat families. Thus a fixed point is a pair of rank-one arrows, subject to the total determinant constraint. Their direct-sum morphism of derived enhancements identifies its tangent triangle with the weight-zero part of (46), compatibly with the cotangent comparison. Normalizing a determinant cancels all character lines pulled from the base, so the based condition does not acquire a moving term. The moving virtual normal complex is \[ \bigoplus_{i\ne j}\mathbf R\mathop{\mathrm{Hom}}(I'_i,Q_j) \tag{48}\] with weight \(w_j-w_i\).

We use the divisor-and-residual-point factorization established in the relative setting by [17]. We spell out its compatibility with the determinant and nonreduced bases. Each rank-one kernel has a unique factorization \[ I'_i=L_i(-\mathcal C_i)\mathcal I_{Z_i}, \qquad D_i=[\mathcal C_i]\text{ in }\mathop{\mathrm{Pic}}(X/S), \qquad D_1+D_2=D. \tag{49}\] Here \(\mathcal C_i\) is a relative effective Cartier divisor and \(Z_i\) is a finite-flat point family. This holds over nonreduced parameter schemes. To see this, let \(T\) be such a parameter scheme and untwist the target line. The \(T\)-flat kernel ideal \(I\) has a length-one locally free resolution, by the fiberwise projective-dimension bound. Its determinant is a line bundle on \(X_T\), commuting with arbitrary base change. The maximal-minor map \(I\to\det I\) restricts on every surface fiber to the inclusion of a rank-one torsion-free sheaf into its double dual. It is therefore fiberwise injective with zero-dimensional cokernel. Since both its source and target are \(T\)-flat, the local flatness criterion makes the map injective and its cokernel \(\mathcal T\) \(T\)-flat. Properness and the fiber support make \(\mathcal T\) finite over \(T\).

The inclusion into the original target line extends uniquely across that finite support: on a smooth relative surface a finite-flat module has grade at least two, as follows by lifting a fiber regular sequence with flat successive quotients. Thus its Hom and first Ext into the structure sheaf vanish. Applying Hom to \(0\to I\to\det I\to\mathcal T\to0\) gives the extension. The extension is fiberwise an injection of lines with Cartier quotient; the local flatness criterion makes its divisor relative Cartier. It gives the required point ideal and divisor, uniquely and compatibly with base change.

Let the point lengths be \(r,s\). The Chern character of the two quotients gives \[ r+s=\nu+(A_0-D_1)D_2. \tag{50}\] The fixed scheme has only finitely many numerical choices. Its summand Hilbert polynomials are locally constant. In addition, the determinant line of each perfect kernel is a base-change-compatible line family, so its fiber Hilbert polynomial is locally constant. The residual point length is the difference between the Euler characteristics of this determinant and its kernel, and is therefore locally constant as well. Finite type now gives finiteness of these choices. All divisor and Picard moduli used in a choice may be restricted to the required finite-type quasi-projective pieces. For instance, lines of a fixed Hilbert polynomial form an open in the moduli of stable rank-one sheaves; rigidification at \(o\) represents their Picard classes.

Choose a high index with \(KD_i>K^2\), which exists since \(KD_1+KD_2=3K^2\), and call the other index low. Write \(P=D_{\mathrm{low}}\). For this numerical choice, denote the fixed locus by \(\mathcal F\) and the unrestricted low rank-one Quot scheme by \(\mathcal Q_{\mathrm{low}}\). Forgetting the high arrow gives a map \[f:\mathcal F\longrightarrow\mathcal Q_{\mathrm{low}}.\] Its relative derived tangent is \[\operatorname{fib}\left\{ \mathbf R\mathop{\mathrm{Hom}}(I'_{\mathrm{high}},Q_{\mathrm{high}}) \longrightarrow\mathbf R\pi_{X*}\mathcal I_o[1]\right\}.\] Lemma 20, now with rank one, gives amplitude \([0,1]\). The transitivity comparison above is therefore a compatible triple. Consequently \[ [\mathcal F]^{\mathrm{vir}} =f^![\mathcal Q_{\mathrm{low}}]^{\mathrm{vir}}. \tag{51}\] Here \(f^!\) is a bivariant virtual pullback. We will use its action on cycle representatives of the low virtual class, including cycles supported on special divisor loci.

Obstruction Euler factors of point Hilbert schemes also enter the surface Quot calculations of Oprea and Pandharipande [41]. We establish the relative comparison with varying determinant and arbitrary irregularity needed here.

Let \[\rho:\mathcal Q_{\mathrm{low}} =\operatorname{Div}_{P}(X/S)\times_S X^{[r_{\mathrm{low}}]}_{/S} \longrightarrow\operatorname{Div}_{P}(X/S)\] be the smooth projection. Combining point excess with (51) gives the fixed-cycle formula \[ [\mathcal F]^{\mathrm{vir}} =f^!\left( e(F_{\mathrm{low}})\cap \rho^*[\operatorname{Div}_{P}(X/S)]^{\mathrm{vir}} \right). \tag{53}\] This formula separates the two remaining tasks. We first determine where the low divisor virtual cycle can be supported. Over the surviving support, the high map \(f\) becomes smooth, and its virtual pullback will supply the second point Euler factor.

Support of the low divisor class

Put \(y=KP/s_K\). The virtual relative dimension of the divisor scheme is \[ \frac{P(P-K)}2. \tag{54}\] If \(0<y<1\), this is at most \(-(b-1)\). To check the integer bound, put \(q=b-1\). The positive intersection \(p=KP\) is a multiple of \(2q\): each of the two canonical divisors on the product has this fiber degree. Thus \[2q\leq p\leq8q^2-2q,\qquad \frac{P(P-K)}2 \leq\frac{p^2/(8q^2)-p}{2} \leq-q+\frac14\] by Hodge index and convexity in \(p\). The left side is integral, proving the claim. The absolute divisor virtual cycle consequently has a representative of dimension at most \(\dim S-(b-1)\). The image of its support in \(S\) has codimension at least \(b-1\). In (53), smooth pullback by \(\rho\) and cap product with \(e(F_{\mathrm{low}})\) preserve this base support. So does \(f^!\): it is bivariant and commutes with restriction and proper pushforward [34]. Although these pullbacks can increase the dimension of a cycle, they do not enlarge the locus over which it lies in \(S\). Every marked fixed contribution above this low cycle therefore vanishes in our support quotient.

On the Picard open where \(H^2(P)=0\), consider instead the morphism from the divisor enhancement to the derived based Picard scheme. The divisor exact sequence shows that its obstruction map surjects onto \(H^2(\mathcal O_X)\). Hence its relative tangent has amplitude \([0,1]\), and the transitivity construction again gives a compatible virtual-pullback triple. The absolute virtual dimension of that Picard theory is \[\dim S+1-\chi<0.\] Its virtual class is zero, so the restricted divisor virtual class is zero as well. For \(y>1\), the condition \(H^2(P)=0\) is automatic: otherwise duality makes \(K-P\) effective, contrary to ampleness. For \(y=1\), a nonzero section of \(K-P\) has zero intersection with \(K\), and therefore has empty divisor. Thus the exceptional line is exactly \(K\). Chow localization expresses the divisor virtual class as a pushforward from the canonical linear system \(|K|\). For \(y=0\), an effective divisor is empty; its tangent complex is zero and its class is the base fundamental class.

We have reduced the low divisor support to \(0\) or \(K\), and hence the high line to \(3K\) or \(2K\). On a neighborhood of either support the high direction is smooth: Kodaira vanishing gives its projective bundle of sections, together with its smooth point Hilbert scheme. The relative virtual tangent class there is \[ [\mathbf R\pi_*\mathcal O(\mathcal C_{\mathrm{high}})] -1-[F_{\mathrm{high}}]+[T_{X^{[r_{\mathrm{high}}]}/S}]. \tag{55}\] Indeed, the unrestricted rank-one arrow class subtracts \(\mathbf R\pi_*\mathcal O_X\), while fixing the based determinant subtracts \(\mathbf R\pi_*\mathcal I_o[1]\); together these subtract exactly one. The remaining point-ideal calculation is that in Lemma 21. The projective Euler sequence identifies actual tangent minus (55) with \([F_{\mathrm{high}}]\).

For a smooth morphism with a relative perfect obstruction theory, the obstruction sheaf is locally free and its virtual pullback is smooth pullback capped with its top Chern class. This follows by locally splitting the cotangent comparison onto the locally free relative cotangent; equivalently, the intrinsic cone is \([0/T_{\mathrm{rel}}]\) and zero-section excess gives the Euler factor. The argument works on arbitrary base cycles, not just the fundamental cycle; compare [6] and [34]. Here the obstruction bundle has K-class \([F_{\mathrm{high}}]\), so its Euler class is \(e(F_{\mathrm{high}})\). Let \(\mathcal D\subset\operatorname{Div}_{P}(X/S)\) be the empty-divisor locus \(S\) or the canonical linear system \(|K|\). Base-change \(f\) to \(\mathcal D\times_S X^{[r_{\mathrm{low}}]}_{/S}\), and use the same letters \(f,\rho\) for the resulting maps. For every cycle \(\alpha\) on this product, the high virtual pullback is therefore \[ f^!(\alpha)=e(F_{\mathrm{high}})\cap f^*(\alpha). \tag{56}\] In particular, apply it to \(\alpha=e(F_{\mathrm{low}})\cap\rho^*\beta\), where \(\beta\) is a cycle on \(\mathcal D\) supplied by Chow localization for the low divisor virtual class. Bivariant proper-pushforward compatibility then returns the result to \(\mathcal F\). All operational insertions restrict to the actual linear systems on those cycles; no formula for the low cycle supported on \(|K|\) is needed. These linear systems are smooth over \(S\): for \(K\), constant cohomology dimensions follow from the product of curves and base change, and for \(2K,3K\) from Kodaira vanishing. Their universal divisor classes are \(D_i+x_i\), where \(x_i\) is the hyperplane class.

Marked localization and the short-length exclusion

Adjoin \(m=3t_0\) independent surface positions to \(\mathcal Q\). At each position insert \(\mathop{\mathrm{ch}}_2^{\mathbb G_m}(Q)\), cap with the smoothly pulled virtual class, and push to \(X^m_{/S}\). This push is proper. The torus acts trivially on the output. Before localization the total codimension is \(2m-v\). An ordinary output component of codimension \(4t_0\) would therefore have equivariant exponent \[(2m-v)-4t_0=2t_0-v<0.\] It is absent. Thus the ordinary codimension-\(4t_0\) part of the localized fixed sum is zero after setting the equivariant parameter equal to one.

On a fixed branch the insertion is a sum, for \(i=1,2\), of \[ [Z_i]+(L_i+w_i)(D_i+x_i)-\frac{(D_i+x_i)^2}{2}. \tag{57}\] Here \([Z_i]\) denotes the incidence cycle on the corresponding surface position. Test every surface slot by the even tensor \[ z_\gamma=\sum_{\alpha,\beta}\gamma_{\alpha\beta} w_\alpha\otimes w_\beta. \tag{58}\] This is a tensor test in the coefficient algebra, not a numerical intersection test alone. Since a unit curve class pairs to zero with a pure \(W_b\) input, a line-only insertion that survives this test uses a positive divisor degree on both curve axes. Hence each such position supplies at least two ordinary codimensions, from powers of the two curve canonical classes.

Suppose \(k\) of the positions use line-only insertions. Extract these \(2k\) divisor codimensions by the projection formula. The remaining \(m-k\) incidence positions lie in the union of the two point subschemes, of total length \(r+s\). Their pushed support therefore has at most \(r+s\) distinct surface positions and has codimension at least \[2(m-k-r-s)\] in those factors. Operational incidence classes vanish away from their incidence supports, so this support assertion applies to successive caps with arbitrary low cycle multipliers. If \(r+s<t_0\), the displayed codimension is strictly larger than the available \(4t_0-2k\), and the tested contribution is zero.

By (50), the four remaining ordered divisor pairs have lengths \[\begin{array}{c|cccc} (D_1,D_2)&(0,3K)&(3K,0)&(K,2K)&(2K,K)\\ \hline r+s&\nu+3s_K&\nu&\nu&\nu-s_K . \end{array}\] Only \((D_1,D_2)=(0,D)\) survives. On this branch the low divisor is empty and its virtual class is the base class. The calculation is on the product of the two point Hilbert schemes and \(|D|\), with both point Euler factors and the ordinary fundamental cycle. We have \[ N=r+s=\nu+ds_K,\qquad \dim(|D|/S)=\chi+\frac{d(d-1)}2s_K-1=t_0,\qquad x_1=0,\quad x_2=x. \tag{59}\]

A relative marked Hilbert-scheme recursion

The surviving fixed contribution must be expanded as a Chow class on \(X^m_{/S}\), with every output retained. Relative EGL recursion for surface pushforwards to a base appears in [29]; the same work also treats one and two marked curve outputs [29]. The construction below retains an arbitrary finite set of surface outputs. This is the form that lets the codimension bound identify the Chow tensor left by localization.

Apply Lemma 22 to the surviving fixed branch. For the surface product of curves, all the allowed output decorations expand into powers of the two curve canonical classes. Consider a connected equality block containing \(k\) outputs. For \(k=1\), pure-slot testing requires at least two decorating codimensions, one on each axis. For \(k=2\), a bare diagonal on either curve axis pairs two isotropic inputs and is zero. Decorations are therefore needed on both axes, in addition to the codimension-two surface diagonal. For \(k\geq3\), the equality block already costs \(2(k-1)\) codimensions. These lower bounds are summarized in Table 2. Every surviving block costs at least \(4/3\) codimensions per output, with equality only for an undecorated three-output block.

The total selected codimension is \(4t_0\) on \(3t_0\) outputs. It follows that only undecorated triple blocks occur. No line-only insertion survives. There is no remaining positive horizontal codimension: every closed surface component must have top relative degree, and every base bundle contributes only its scalar characteristic term. Integration over \(|D|\) therefore extracts exactly \(x^{t_0}\). In particular, the scalar multiplying the triple blocks can be computed over a point with the same numerical characteristic data, using the fixed polynomial procedure of Lemma 22.

The scalar interface

To specify this scalar without an independence assumption on diagonal classes, use the polynomial procedure just fixed. For a formal surface input \(z\), write a connected component carrying \(k\) marked inputs and decoration \(\alpha\) as the moment \[M_k(\alpha)=\int_X z^k\alpha.\] Give unmarked moments their actual top intersection numbers. Give marked moments value zero except for \(M_3(1)=1\). Thus this evaluation retains precisely bare triple blocks. In the generating series of Section 6, the variable \(q\) records total point length, \(p\) is the exponential variable for marked positions, and \(x\) is the high hyperplane class. Let \(a_{N,t_0}\) be the coefficient defined in (65), with the moments evaluated in this way and \(N\) as in (59). Equivalently, \((3t_0)!\,a_{N,t_0}\) is the sum of the coefficients of all bare triple partitions in the labeled marked pushforward. The polynomial procedure is fixed before imposing relations among the diagonal classes of a particular surface. Section 6 evaluates this same representative: its moment-additivity argument is what permits numerical surface calculations to determine the selected scalar. Thus the passage to those calculations does not replace the marked Chow identity by numerical universality.

The moving factor

For a vector bundle \(E\), write \(e_a(E)\) for its Euler class after adding \(a\) to each Chern root. If \(a\) has invertible constant term, this convention extends multiplicatively to virtual bundles and perfect complexes. Inverses are expanded in positive Chow degree. Let \[M_{r,s}=X^{[r]}\times X^{[s]},\] with universal finite subschemes \(Z_1,Z_2\) on \(M_{r,s}\times X\). Derived Hom expressions in this subsection include pushforward along the surface \(X\).

On the surviving fixed component, the two kernels are \[I'_1=L_1\mathcal I_{Z_1},\qquad I'_2=L_2(-D-x)\mathcal I_{Z_2},\] where \(x\) is the hyperplane class of \(|D|\). Apart from the factor \((1-x)^{\chi(B)}\), the inverse moving Euler class is \[ \begin{aligned} R(x)={}& \frac{e_1\bigl(((K+A_0)^{[r]})^\vee\bigr)} {e_{1-x}\bigl(((K-B)^{[r]})^\vee\bigr)\, e_{1-x}(B^{[s]})\,e_{-1+x}((-B)^{[r]})}\\ &{}\times e_{1-x}\bigl(R\mathop{\mathrm{Hom}}(\mathcal O_{Z_1},\mathcal O_{Z_2}(B))\bigr)\, e_{1-x}\bigl(R\mathop{\mathrm{Hom}}(\mathcal O_{Z_1},\mathcal O_{Z_2}(K+B))\bigr). \end{aligned} \tag{62}\] Here and below only scalar terms of bundles pulled from the original base can contribute, by Proposition 23. In particular the omitted inverse Euler factor of \(R\pi_*(-A_0)\), at weight one, has scalar term one.

To check (62), the forward moving arrow has virtual tangent \[R\mathop{\mathrm{Hom}}(\mathcal I_{Z_1},-A_0) - R\mathop{\mathrm{Hom}}(\mathcal I_{Z_1},\mathcal I_{Z_2}(B)),\] at weights \(1\) and \(1-x\), respectively. Expand each point ideal as \(\mathcal O-\mathcal O_Z\). The constant term of the second Hom gives \((1-x)^{\chi(B)}\). Surface duality identifies \[R\mathop{\mathrm{Hom}}(\mathcal O_{Z_1},B) \quad\text{with}\quad ((K-B)^{[r]})^\vee[-2]\] in the convention relevant to the Euler class, and likewise for \(-A_0\). This gives the first numerator and the first two denominator factors of (62). The reverse moving arrow is \[R\mathop{\mathrm{Hom}}(\mathcal I_{Z_2},\mathcal O_{Z_1}(-B))\] at weight \(-1+x\). Its point term gives the final denominator. Duality changes its cross term to the remaining numerator term. The cross Hom has virtual rank zero, so changing both its dual and the sign of its weight introduces no rank sign.

The fixed obstruction Euler classes on this component are \[e\bigl((K^{[r]})^\vee\bigr), \qquad e_x\bigl(((K-D)^{[s]})^\vee\bigr).\] The sign of the second shift is worth recording: the line in the obstruction bundle is \(K-D-x\) before taking the dual, and hence has twist \(+x\) afterwards.

Marked moments and shifted additivity

Temporarily replace \(K\) and \(K-D\) in these two fixed Euler classes by independent divisor variables \(L,H\). Also separate the Euler shift \(x_e\) from the moving shift \(x_m\). For a surface class \(z\), let \[\mu_r(z)= (\operatorname{pr}_{X^{[r]}})_* \bigl([Z_1]\operatorname{pr}_X^*z\bigr),\] and define \(\mu_s(z)\) similarly. Consider \[ \Theta(q,p,x_e,x_m) = \sum_{r,s\geq0}q^{r+s} \int_{M_{r,s}} e\bigl((L^{[r]})^\vee\bigr)\, e_{x_e}\bigl((H^{[s]})^\vee\bigr)\, R(x_m)\, \exp\bigl(p(\mu_r(z)+\mu_s(z))\bigr). \tag{63}\] The exponential records labeled output insertions with their usual factorial denominators.

We use the fixed marked recursion of Proposition 23 to regard each finite coefficient of (63) as a polynomial in decorated moments \[M_k(\alpha)=\int_X z^k\alpha.\] Here \(\alpha\) is a monomial in \(L,H,A_0,D,K,c_2(T_X)\). More precisely, carry out the nested-Hilbert recursion while retaining the marked output slots. Each connected component of the final equality graph is integrated over one copy of \(X\); if it contains \(k\) marked slots and decoration \(\alpha\), replace that integration by \(M_k(\alpha)\). This fixes a polynomial representative before imposing numerical relations between moments. The recursion and its projective-bundle signs are those of [14]. The version with arbitrary retained outputs used here is proved in Lemma 22. This is the interface that keeps the calculation in relative Chow throughout.

Let \(\mathcal E\) denote the following evaluation: \[ \mathcal E(M_0(\alpha))=\int_X\alpha,\qquad \mathcal E(M_3(1))=1,\qquad \mathcal E(M_k(\alpha))=0 \quad(k>0,\ (k,\alpha)\neq(3,1)). \tag{64}\] Unmarked integrations in this formula are their top-degree numerical integrals; other degrees give zero. The scalar normalization agreed with Proposition 23 is \[ a_{N,t_0} = [q^N p^{3t}x^t]\, (1-x)^{\chi(B)} \mathcal E\bigl(\Theta(q,p,x,x)\bigr) \big|_{L=K,\ H=K-D}. \tag{65}\] Indeed, \(\mathcal E\) retains precisely the partitions into bare triples. Passing from the exponential series to labeled outputs multiplies by the nonzero common factor \((3t)!\). Every such partition has the same tested value, up to the common Koszul sign for grouping the two curve axes. Those factors have already been divided out in Proposition 23.

The next observation allows us to compute (65) using ordinary numerical surface calculations, without assuming that all formal moments are independent.

Restriction to two curves

We first claim that \[ c(q,0,x_m)=0. \tag{68}\] Represent very ample \(L,H\) by smooth transverse curves. At \(x_e=0\), the two Euler classes in (63) restrict the integral to \(L^{(r)}\times H^{(s)}\), with sign \((-1)^{r+s}\). These are the regular zero loci of the tautological sections: the zero schemes are the Hilbert schemes of points on the two smooth curves, of the expected codimensions \(r,s\). A point-class perturbation of the constant \(z=u\) restricts to zero on both curves. The Vandermonde test in the preceding proof therefore annihilates the bare marked coefficient. The coefficient of the single moment \(M_3(1)\) is a universal scalar, independent of the numerical line classes, so this proves (68) identically.

To determine \(c_H\), set both shifts to zero and take \[z=u+\tau Z\] for a divisor \(Z\) on the surface. Write \(x',y'\) for the numerical point divisors on \(L^{(r)},H^{(s)}\). After the Euler restrictions, the one-body factors of (62), including their signs, are \[ c_{-1}\bigl((A_0+2D)^{[r]}\bigr)^{-1} \,(-1)^s c_{+1}(B^{[s]})^{-1}. \tag{69}\] Here \(c_a(E)=\sum_j a^j c_j(E)\). For verification, on the ordered cover of a symmetric power, a tautological line bundle \(M^{[r]}\) has a filtration with roots \[M_i-\delta_i,\qquad \delta_i=\sum_{j<i}[\Delta_{ij}].\] Two line classes pulled from the same curve position multiply to zero. Applying this to the low ratio in (62) reduces its factor at position \(i\) to \[\frac{1}{1+\delta_i-(A_0+2D)_i},\] which gives the first term in (69).

Put \(I=LH\). At each of the \(I\) transverse intersections, the two universal finite subschemes meet by independently imposing their point-containment divisors. Their derived Hom has the class of the structure sheaf of the product of those divisors, twisted by the determinant normal of the first universal subscheme. Numerically its class is \[e^{x'}(1-e^{-x'})(1-e^{-y'}).\] Its Euler class at weight one is therefore \[\frac{(1+x')(1-y')}{1+x'-y'}.\] The two cross Hom terms give the combined factor \[ \left(\frac{(1+x')(1-y')}{1+x'-y'}\right)^{2I}. \tag{70}\] This calculation also follows by resolving the two transverse containment divisors and taking their tensor-product resolution; the twists by surface lines have numerical degree zero at the fixed intersection point. The incidence exponential becomes \[ \exp\bigl(pu(r+s)+p\tau(ZL\,x'+ZH\,y')\bigr). \tag{71}\]

We recall the symmetric-power integral needed to sum these terms. For a curve of genus \(h\), a line of degree \(d_1\), and its tautological bundle on the \(r\)-th symmetric power, \[ c_a(\mathrm{line}^{[r]}) =(1-a x')^{r-d_1-1+h} \exp\left(\frac{a\vartheta}{1-a x'}\right), \qquad \int (x')^{r-j}\frac{\vartheta^j}{j!}=\binom hj. \tag{72}\] Here \(\vartheta\) is the pulled-back Jacobian polarization. For completeness, the universal divisor sequence and GRR give \[\mathop{\mathrm{ch}}(\mathrm{line}^{[r]}) =d_1+1-h+ (r-d_1-1+h+\vartheta)\exp(-x').\] Indeed the universal divisor class has terms \(r[\mathrm{pt}]+x'+\ell\), where the mixed class satisfies \(\int_{\mathrm{curve}}\ell^2/2=-\vartheta\). Converting this Chern character to the total Chern class gives the first formula in (72). For the second, pull to the ordered cover and expand the polarization in a symplectic basis. Intersecting the point divisors first leaves \(j\) factors; choosing \(j\) distinct symplectic pairs contributes one after the factorials, giving \(\binom hj\).

Let \[d_1=(A_0+2D)L,\qquad d_2=BH,\qquad 2h_1-2=L(L+K),\qquad 2h_2-2=H(H+K).\] By (72), low-curve integration of the first factor in (69), times any series in \(x'\), is coefficient extraction against \[ [ (x')^r]\, (1+x')^{d_1+1-2h_1-r}(1+2x')^{h_1}. \tag{73}\] High-curve integration has the analogous expression after replacing \(y'\) by \(-y'\), which absorbs the sign \((-1)^s\). Accordingly (70) becomes \[\left(\frac{(1+x')(1+y')}{1+x'+y'}\right)^{2I},\] and the nonconstant part of (71) becomes \(\exp(p\tau(ZL\,x'-ZH\,y'))\).

The elementary Lagrange identity used for each variable is \[ \sum_{r\geq0}Q^r [x^r]\frac{F(x)}{(1+x)^r} =F(w)\frac{1+w}{1+2w}, \qquad w(1+w)=Q,\quad w(0)=0. \tag{74}\] One proof is to express coefficient extraction as the formal residue at zero: summing the geometric series gives \(F(x)(1+x)/(x(1+x)-Q)\), whose small root is \(w\); its residue there is the right side. The identity holds with other formal variables in the coefficients, so it can be applied successively to \(x'\) and \(y'\). Using \(Q=q\exp(pu)\), both small roots equal \(w\). Equations (73)–(74) therefore give \[\begin{align*} \log\Theta(q,p,0,0) ={}& (d_1+d_2+4-2h_1-2h_2+4I)\log(1+w) \\ &+(h_1+h_2-2-2I)\log(1+2w) +p\tau(ZL-ZH)w . \tag{75}\end{align*}\] This equation is the genuine numerical evaluation with \(z=u+\tau Z\). It determines the universal coefficients used below.

The first hyperplane coefficient

Let \(w_0(1+w_0)=q\), with \(w_0(0)=0\), and let \(\mathcal D=q\partial_q\). By Lemma 25, the numerical evaluation of \(\log\Theta\) on a genuine moment vector equals that of its one-moment part: every component with at least two moment factors vanishes there. Its term of \(p\)-degree three is a universal scalar combination of the moments \(M_3(\alpha)\). The term involving \(M_3(H)\) is \(c_H(q)M_3(H)p^3/3!\). Since \[[u^2\tau]M_3(H)=3ZH,\] we can identify \(c_H\) by varying the very ample \(H\), with the other divisor classes fixed. Only degree-one decorations can contribute to \(u^2\tau\), and \(H\) is the only varying such decoration. For example, one may take arbitrarily large integral multiples of a fixed ample line for \(H\), with \(Z\) fixed ample. The first two terms of (75) contain no \(\tau\). The coefficient of \(p^3u^2\tau(ZH)\) in its last term is \(-\mathcal D^2w_0/2\). Comparing factorials yields \[c_H(q)=-\mathcal D^2w_0.\] Together with (67) and (68), this proves \[ [x]c(q,x,x) =\mathcal D^2w_0 =\frac{q(1+2q)}{(1+4q)^{3/2}}. \tag{76}\] There is no contribution from differentiating \(x_m\), because \(c(q,0,x_m)\) is identically zero.

The unmarked calculation is equally explicit. Set \(p=0\) in (75). Every finite coefficient is polynomial in the intersection numbers of \(L,H,A_0,D,K\), so the formula continues from very ample \(L,H\) to \(L=K,\ H=K-D\). This can also be seen by varying sufficiently large multiples of \(K\) in both variables and then using polynomial identity. The adjunction expressions for \(h_1,h_2\) are polynomial expressions in this continuation, not assertions that a negative line has a smooth effective representative. Substitution gives \[ \mathcal E(\Theta(q,0,0,0)) \big|_{L=K,\ H=K-D} = (1+w_0)^{ds_K} (1+2w_0)^{d(d+1)s_K/2}. \tag{77}\]

A residue with a definite sign

We now prove Proposition 24. By (66), the \(p^{3t}\)-coefficient is \(c(q,x,x)^t/(6^t t!)\) times the unmarked series. The series \(c(q,x,x)\) has zero constant term in \(x\). Thus the coefficient of \(x^t\) uses its linear term in every factor, and uses only the constant terms in \(x\) of the unmarked series and of \((1-x)^{\chi(B)}\). Equations (65), (76), and (77) give \[ a_{N,t_0} = \frac{1}{6^t t!} [q^{ds_K-1}] (1+w_0)^{ds_K}(1+2q)^t(1+4q)^E, \qquad E=\frac{d(d+1)s_K}{4}-\frac{3t}{2}. \tag{78}\]

Put \(v=1+2w_0\), so that \(q=(v^2-1)/4\). The coefficient residue in (78) becomes \[ 2^{ds_K-t-1} \operatorname*{Res}_{v=1} \frac{(1+v^2)^t v^{2E+1}}{(v-1)^{ds_K}}\,dv. \tag{79}\] For the present parameters, \[ds_K=24\chi,\qquad t=25\chi-1,\qquad E=\frac{3-27\chi}{2},\qquad \chi=(b-1)^2\geq16.\] Hence the rational differential in (79) has finite poles only at zero and one. Its rational coefficient has degree \(2-\chi\leq-14\) at infinity, so its residue at infinity is zero. The residue at zero is \[[v^{27\chi-5}] (1+v^2)^{25\chi-1}(1-v)^{-24\chi}.\] This is strictly positive: the second factor has positive coefficients in every nonnegative degree, the first has nonnegative coefficients and constant term one, and \(27\chi-5\geq0\). The residue at one is its negative and therefore is nonzero. Both scalar prefactors in (78)–(79) are nonzero. This proves \(a_{N,t_0}\neq0\).

Finally, Proposition 23 gives \(a_{N,t_0}T_\gamma^{t_0}=0\) for the isotropic cubic contraction \(T_\gamma\) appearing in Theorem 18. The scalar is a nonzero rational number and is invertible in the coefficient algebra. Thus \(T_\gamma^{t_0}=0\), which completes the proof of that theorem.

From isotropic tests to arbitrary color products

The surface theorem now gives nilpotence for every isotropic tensor test. To use it in the product defining the counterexample, we need a vanishing statement for arbitrary color inputs. We return to a smooth genus-\(b\) curve family, choosing a rational divisor of relative degree one and its pure projector as in Proposition 13. We use the operation algebra \(\mathcal A_k(E)\) of Section 4 for an ordinary rational color space \(E\). A pure cubic entry is the class of the projected small diagonal in \(\mathcal A_3(E)\) with a specified triple of color inputs; products use disjoint curve slots.

We will fix an arbitrary color map \(E\to W_b\), apply the surface theorem to a subspace with isotropic image, and use the following exterior-algebra lemma to annihilate every sufficiently long product. The final application of faithful polarization will then give a Chow identity with the same base-support bound.

The smooth-family argument now gives vanishing of long products of pure cubic entries for arbitrary color inputs, modulo the stated base-support bound. The remaining task is to compare the compact candidate with these pure entries and control the errors along the boundary.

Weighted configurations and cut charts

Let \((h,B)\) be stable heavy data: \(B\) is a finite ordered set and \(2h-2+|B|>0\). Write \[X_{h,B}(k)= \overline{\mathcal M}_{h,(1^B,\epsilon^k)}, \qquad 0<k\epsilon\leq1,\] with \(X_{h,B}(0)=\overline{\mathcal M}_{h,B}\). Here and below any positive rational weights of total at most one give the same moduli problem for the light labels. Lights can coincide with one another, but not with a heavy marking. A rational component with one node and only light markings cannot be stable; a rational component with two nodes and positive light mass is stable. These observations also show the independence from the individual light weights.

The stacks \(X_{h,B}(k)\) are smooth and proper, and reduction of weights and forgetting labels give the usual stabilization morphisms [24]. The one-light space is the universal curve over \(\overline{\mathcal M}_{h,B}\): when its coarse position is a node or a heavy marking, a unique three-special-point rational component is inserted. Thus sections and the relative dualizing line define natural one-light classes.

Fix a stable graph \(\Gamma\) of the heavy type. At its vertex \(v\) let \(h_v\) be the genus and let \(B_v\) comprise the original heavy markings and the incident half-edges, all ordered. Put \[\mathcal B_\Gamma=\prod_v\overline{\mathcal M}_{h_v,B_v}, \qquad \mathcal S_\Gamma=\prod_v\mathcal M_{h_v,B_v}.\] The cut chart with \(k\) lights is the disjoint union \[ X^\Gamma(k)= \coprod_{a:\{1,\ldots,k\}\to V(\Gamma)} \prod_{v\in V(\Gamma)}X_{h_v,B_v}\bigl(a^{-1}(v)\bigr). \tag{80}\] It has a proper gluing map to the weighted space of the original heavy type. Assignments are part of the indexing, so a class supported on the assignment of specified labels to a vertex means zero on the other components of (80).

Refining a graph gives further proper cut charts. These charts cover the preimage of the corresponding heavy boundary. Indeed, a node of the light-forgotten curve may have a chain above it before stabilization. Choose a node on that chain and cut it, replacing its branches by heavy flags. The resulting pieces remain weighted stable; gluing recovers the original curve. Several prescribed nodes can be cut independently. The map on heavy bases is finite, as for the ordinary gluing of stable curves.

Cutting and light forgetting commute. The heavy flag replacing a node branch retains exactly the stability contribution of that branch. Thus stabilizing on the separate pieces and then gluing gives the same curve as forgetting and stabilizing after gluing. This compatibility will permit operations on unconsumed light labels to pass through any later cut.

We use rational Chow classes operationally on these smooth stacks. For an ordinary color space \(U\), the arity-indexed operation algebra is \[ \bigoplus_{k\geq0} \left( A^*(X^\Gamma(k))\otimes U^{\otimes k} \otimes\operatorname{sgn}_k \right)_{\mathfrak S_k}. \tag{81}\] The product pulls operations to disjoint sets of labels by forgetting and then multiplies them. Successive stabilization proves associativity; exchanging the two sets of labels gives the sign \((-1)^{k\ell}\). In particular, cubic and singleton operations are odd and pair operations are even. All expressions below are first identities of labeled classes and then identities in (81).

The corrected cubic

For three light labels define \[ F^0_{h,B} =[1=2=3]-\frac12\beta_{h,B} +\frac12\sum_{p\in B}\delta_{p,123}\psi_p \ \in A^2(X_{h,B}(3)). \tag{82}\] The coincidence class is the product of the divisors \([1=2]\) and \([1=3]\). The class \(\beta_{h,B}\) is defined by taking each stable one-edge graph of the heavy type, subdividing its edge by a genus-zero vertex carrying exactly the three lights, and summing the resulting gluing classes divided by their graph automorphism orders. The divisor \(\delta_{p,123}\) has a tail containing exactly the heavy mark \(p\) and the three lights; \(\psi_p\) is the heavy cotangent class on the weighted space. Each term in (82) is symmetric in the three light labels.

The local three-position calculation

Over \(\mathcal S_\Gamma\) let \(C_v\) denote the smooth core curve at vertex \(v\). For a fixed assignment of three labels, there is a coarse position map \[ c:X^\Gamma(3)|_{\mathcal S_\Gamma} \longrightarrow C_{a(1)}\times_{\mathcal S_\Gamma} C_{a(2)}\times_{\mathcal S_\Gamma} C_{a(3)}. \tag{83}\] All products in this subsection are over the indicated heavy base.

Denote the total first exceptional divisor at a triple flag locus by \(E_{p,123}\). Denote pair exceptional divisors, preferably pulled back from the corresponding two-light spaces, by \(E_{p,ij}\). The blowup formula for Chow groups [19] shows that a codimension-two class on (83) has, in addition to its main pullback, terms of the following forms: \[ E_{p,ij}\,A_{p,ij},\qquad E_{p,123}\,D_p,\qquad a_p E_{p,123}^{\,2}. \tag{84}\] Here \(A_{p,ij}\) is a divisor class on the remaining curve over the base, \(D_p\) is a base divisor class, and \(a_p\) is a scalar on each connected component of the base. For clarity, if \(E_{p,ij}^{\mathrm{str}}\) is the strict pair exceptional divisor and \(E_{p,ij}^{\mathrm{pull}}\) is pulled back from two labels, then \[E_{p,ij}^{\mathrm{pull}} =E_{p,ij}^{\mathrm{str}}+E_{p,123}.\] Multiplying the difference by a divisor on the remaining curve restricts that divisor, on \(E_{p,123}\), to the flag section. It is therefore a pulled base divisor. Replacing the strict version by the pulled version changes only the triple-exceptional-times-base-divisor term, and introduces no new scalar multiple of \(E_{p,123}^2\). Formula (84) includes assignments to different vertices, in which some of the centers are simply absent.

When a raw class \(F^0\) is pulled from an ancestor chart, its main component is the small diagonal if all three labels are assigned to one relevant descendant vertex, and is zero otherwise. Away from flags this is immediate. The coincidence, bridge, and heavy-tail corrections not seen there project to loci where all three coarse positions are flags, of codimension three; they cannot alter a main codimension-two class.

For the same pulled raw class, the divisor coefficients \(A_{p,ij}\) in (84) are supported on flag sections of the remaining curve, up to base divisor terms. To see this, remove all flags from that remaining curve. Neither the raw class nor its main diagonal pullback meets the pair exceptional locus over this open. Intersecting with the exceptional divisor and pushing to its center recovers its coefficient, with the usual negative self-intersection sign for a codimension-two blowup center. The coefficient is therefore zero on this open. Chow localization on the curve over the base gives the assertion.

Controlled pure lifts

For a smooth heavy core \(\pi:C\to S\), choose a degree-one rational class \[e_C= \begin{cases} [p],&\text{if a heavy flag $p$ has been chosen},\\ c_1(\omega_{C/S})/(2h-2),&\text{if $B=\varnothing$}. \end{cases}\] The second case occurs only for \(h\geq2\). As in the curve tensor construction, put \[ e_C'=e_C-\frac12\pi^*\pi_*(e_C^2), \qquad \eta_C=[\Delta_C]-e'_{C,1}-e'_{C,2}. \tag{88}\] Write \(\Delta_C^{\mathrm{pure}}\) for the small diagonal projected by \(\eta_C\) in each of its three slots. Expanding the three projectors shows that \[ \begin{gathered} [\Delta_{123}]-\Delta_C^{\mathrm{pure}} \quad\text{is a sum of pair--singleton products},\\ \text{with singleton factors }1_C\text{ or }e_C'. \end{gathered} \tag{89}\] No numerical equivalence is used in this expansion: it is the decomposition of the diagonal correspondence into the pure projector and the two hyperbolic projectors.

Advancing support with an arbitrary multiplier

Boundary-support methods also appear in tautological vanishing results such as [22], where support is measured by rational components. Here we lower the dimension of the image in the heavy base, using operational vanishing over smooth test families. We first make this induction precise. A term in it consists of active operations, pulled to disjoint sets of light labels, acting on a cycle on the full labeled chart. The cycle need not factor over the labels or over the vertices.

The operational hypothesis in this lemma is essential. In its applications below, it follows from identities over smooth test families, rather than from identities separately on geometric fibers. For a pure operation, Proposition 31 identifies its restriction with a pullback from a smooth curve power. Vanishing of its Chow class on that power is therefore operational vanishing and can be pulled further to any cycle multiplier. The formation of the projector and the projected diagonal commutes with these base changes: the relative diagonals and the integrations along smooth proper curve powers have the requisite flat or refined-Gysin base-change identities. For residual operations, the explicit base-divisor factors in Proposition 31 give the required open vanishing. The same is true of the controlled singleton relations below.

There is also no restriction on the multiplier hidden in our use of sign coinvariants. In characteristic zero, signed averaging identifies coinvariants with the corresponding signed invariant subspace. A factorization \(L*H\) of operations becomes a sum over shuffles of products of the separately averaged \(L\) and \(H\), pulled to disjoint slot sets. In each summand, apply \(L\) first to the multiplier. The factor \(H\) remains an averaged operation on its own active slots; it is pulled equivariantly through later cuts before it acts. This proves that the lemma applies repeatedly, even after the multiplier has become an arbitrary cycle produced by an earlier step. More explicitly, every later factorization of \(H\) is first an equality of its signed-average representatives on its own active slots. Pull that equality to the full chart and only then cap with the multiplier. No permutation is applied to the multiplier, and no invariance of the multiplier is required. Restriction to an assignment component or pullback through a further cut preserves this equality.

Initially the heavy base has dimension \(3g-3\). Every advancement strictly lowers the base-image dimension. Thus on any branch of the finite expansions there can be at most \[ 3g-2 \tag{91}\] advancements before the support is empty.

The elementary vanishing tests

On a current chart, Proposition 31 expands an ancestor entry into pure entries at current vertices, products of a pair operation with a controlled singleton, and residual cubic operations. Call the pair-times-singleton terms noise entries. Perform these expansions one entry at a time. If a residual summand is chosen, apply Lemma 33 to it immediately, at the cost of that one entry. We therefore never retain residual entries while counting the pure and noise entries.

The pure entries at any one vertex always use the same chosen symmetric class \(G_{h,B}\). This is important both for the finite coefficient bound and for the common-color test below.

A repeated-color test at a flagged vertex

The surface test of Lemma 27 is needed only while one vertex has almost all the genus. On every other chart a less expensive advancement follows from the flag-centered curve algebra.

The finite color budget

Set \[ K=10^5,\qquad m_0=2000g^{2/3},\qquad C_K=\binom{2K}{3}. \tag{93}\] These are integers. Reserve \(4K+20\) disjoint sets of \(m_0\) basis colors each, and let \(S_0\) be all the remaining colors. This is possible because \[\frac{(4K+20)m_0}{g} =8.0004\cdot10^{-12}<0.1; \qquad |S_0|>0.9g.\] For a current graph put \[b=\max_v h_v,\qquad l=g-b.\] Under refinement the maximum vertex genus cannot increase, so \(l\) cannot decrease. The number of edges cannot decrease either. Unmarked stability gives \[ |V|\le2g-2,\qquad |E|\le3g-3,\qquad R_\Gamma\le10g. \tag{94}\]

An advancement by a residual entry, by a noise test, by a small-genus test, or by Lemma 35 will be called ordinary. These are descriptions of the cost in entries, not additional geometric assumptions.

The two regimes below use different information about the remaining colors. While \(l<K\), we remove whole entries without changing the three colors of any retained entry; the reserved blocks can therefore be counted by their original triples. We use the repeated-color rewrites only once \(l\ge K\). They can change which colors occur together in an entry, but preserve the total multiset of color occurrences before the advancing factor is removed. In that regime we count occurrences in \(S_0\). Since \(l\) cannot decrease, we never need an original-triple count after making such a rewrite.

Completion