Proof. Expand \(\ell_0(z\ell_0)^k\) using \([z\partial_z,z]=z\) and \([\mu_X,\mathcal R_X]=\mathcal R_X\). The resulting coefficients are polynomials in the first Hodge index, composed with Chern-class powers. The three blocks of its quadratic Hamiltonian give the last three types of operation under normal ordering; the dilaton shift gives the special unit marking. The block with both variables on the positive polarization has its only relevant negative shift at the primary pair, giving the classical \(\mathcal R_X^{k+1}\)-pairing. This is also the three-block decomposition in the cited coordinate formula. Since the Hamiltonian is quadratic, at most two existing labels are involved. Super quantization uses the same calculation with graded derivatives and permutations. ◻

Proof. Perfection follows from Poincaré duality on the smooth projective \(Y_v\) and \(D_e\). The edge summands pair with one another, not with themselves. Ordinary cup-commutativity gives graded symmetry with respect to total degree. The projection formula says that \(\gamma\) is transpose to \(\rho\) before the shift signs. If \(x\) is in a component summand of degree \(k\) and \(y\) is in a \(b\) summand of degree \(2n-k-1\), the two terms of \[ B_Y(dx,y)+(-1)^k B_Y(x,dy)=0 \tag{9}\] are \((-1)^{k+1}\int \rho x\cup y\) and \((-1)^k\int \rho x\cup y\). They cancel. The case with the \(b\) summand first follows in the same way, and all other cases vanish. This proves the chain identity.

We specify the compatibility with nearby integration below, in 9; it uses the multiplicative comparison and trace of Fujisawa. The residue calculation identifying that trace pairing with (8) is (Fujisawa 2014, Lemma 6.13 and proof of Theorem 8.11), after the sign choices in (6). Thus this compatibility does not require choosing representatives for the groups in (7).

A perfect chain pairing identifies a finite complex with its shifted dual. Under this identification its inverse-pairing tensor corresponds to the identity chain map. It is therefore closed. Finally, (8) is block diagonal by vertices and by edges, so its inverse has the asserted support. All inverse tensors here and below use the categorical Koszul convention. ◻

Proof. A restriction preserves Hodge type. A Gysin map raises both Hodge indices by one, exactly as does the twist in its source \(b\) summand. Thus both arrows of (6) preserve the first index. The last assertion follows by applying a chain map to a closed tensor. Notice that the total cohomological degree operator does not have this property: \(d\) raises total degree by one. ◻

Proof. Restriction of a cochain indexed by all injective ordered tuples to increasing tuples commutes with the Čech differential, because every face of an increasing tuple is increasing. It commutes with the internal differential and preserves Čech degree and the two filtrations. Consequently restricting the explicit residue map of (Fujisawa 2014, Definition 5.24 and Lemma 5.25) gives a bifiltered chain map into the ordered model. This conclusion only uses the formula and its chain identity, not a quasi-isomorphism assertion for an unrestricted tuple complex.

Evaluate at \(u=0\) and pass to relative log forms. The comparison square in (Fujisawa 2014, proof of Theorem 5.29) remains commutative after this restriction: its equality is an equality of residue maps on each intersection. Its other arrows are the Steenbrink comparison with relative log de Rham cohomology, the ordered relative-log Čech resolution (Fujisawa 2008, Proposition 4.19), and the polynomial-log comparison (Fujisawa 2008, Lemma 5.6 and Corollary 5.7). All three induce isomorphisms on cohomology. Therefore so does \(\phi_Z\). The rational comparison is obtained by the same restricted Koszul-residue formula, and it has the same compatibility with Tate factors as the complex comparison. The source is a cohomological mixed Hodge complex, and the ordered target is a weak cohomological mixed Hodge complex by (Fujisawa 2008, Proposition 5.8). Its cohomology carries the mixed Hodge structure of (Fujisawa 2008, Theorem 5.9), and its weight spectral sequence degenerates at \(E_2\) by (Fujisawa 2008, Lemma A.6). Thus the cohomological isomorphism is one of mixed Hodge structures. Weight degeneration and strictness now give the asserted isomorphism of second pages, exactly as in (Fujisawa 2014, Corollary 5.30). ◻

Proof. Write \(f\) for the component assignment of a projection. For a source increasing tuple \(J=(j_0,\ldots,j_k)\), the component of the pullback of a Čech cochain \(\eta\) is \[ (f^*\eta)_J= \begin{cases} f_J^*\eta_{f(j_0),\ldots,f(j_k)},& f(j_0)<\cdots<f(j_k),\\ 0,&\text{some projected vertices repeat}. \end{cases} \tag{12}\] In the second case the tuple is degenerate. Formula (12) commutes with the Čech differential: if a projected tuple has exactly one adjacent repetition, deleting either member produces the same restriction with opposite signs; all other faces remain degenerate. More repetitions give zero on both sides, and distinct tuples give the usual pullback identity. Weak monotonicity ensures that repetitions are adjacent. This also makes (12) compatible with the Alexander–Whitney product: a repeated adjacent pair lies in at least one of the two overlapping subtuples used by that product.

On each intersection use the pullback of absolute log forms and set \(f^*u=u\). The morphism is over the same base, so \(f^*d\log s=d\log s\); it commutes with \(\nabla\). Ordinary forms pull back to ordinary forms, while a local log generator pulls back to a sum of log generators and an ordinary form. Thus log weight and form degree do not increase. Every nonzero term of (12) retains the same Čech degree, so (11) proves preservation of both filtrations.

Finally, take residues. Ordinary corrections to a logarithmic generator have zero residue, and the remaining maps only use its orders along the incident divisors. A unit change of a local boundary equation does not change these residue maps. This proves the asserted dependence and locality. ◻

Proof. Let \(\ell\) be the first Chern class of a relatively ample line bundle. On a connected nearby component of dimension \(N\), its \(N\)th power is nonzero and spans top cohomology. Its integral is the sum of its integrals over the components of the corresponding central fiber, by specialization of a top intersection number. The residue trace has exactly this value by its stated normalization. Hence the two traces agree. The argument applies separately to disconnected total families. After a finite base change, which does not change the assertion, connected components of smooth fibers can be treated separately.

Multiplicative comparison identifies cup products. Applying the identical traces to cup products identifies the pairings. Equivalently, the explicit residue calculation quoted in 4 gives component integration and opposite-column integration on the same double stratum, with no cross-stratum terms. This also identifies the induced inverse pairings. No comparison between two constructions of polarizations on the full limiting mixed Hodge structure is needed for this argument. ◻

Proof. The central restriction of \(\Gamma\) defines a morphism from \(\mathbb Q(-c)\) into central-fiber cohomology, and its image in nearby cohomology is obtained by the functorial specialization map. For the ordinary central-fiber Čech complex, the weight-\(2c\) part of degree \(2c\) is represented by \[\bigl(\Gamma|_{Z_v}\bigr)_v\in\bigoplus_v H^{2c}(Z_v).\] The restrictions agree on every double intersection, because they come from one class on the total space. Thus this tuple is a cycle in the first ordinary weight differential. Higher Čech-degree terms in total degree \(2c\) have smaller weight.

The map from the ordinary Čech model to the nearby log model sends this tuple to the corresponding zero-log, zero-\(u\) component restrictions. It is a morphism of filtered complexes and gives the usual specialization on cohomology. Taking its weight-\(2c\) graded part gives the asserted representative. Strictness of morphisms of mixed Hodge structures ensures that passing to this graded part loses no part of the induced map from the pure source. This statement concerns that induced graded map; it does not assert that an arbitrary representative in the entire limiting complex has no lower-weight terms.

For such an extending source correspondence, apply the same argument to each pure cohomology group of its fixed source, with the prescribed weight shift. Exposing its inputs before restriction and applying them afterwards is the projection formula. This also treats preintegrated admissible homogeneous singles. ◻

Proof. Work in a local flat trivialization. All factors except the Hodge-index projectors are constant. The derivative of a projector onto a summand of the Hodge decomposition has only off-diagonal blocks: differentiation of \(\Pi_p^2=\Pi_p\) gives \(\Pi_p(d\Pi_p)\Pi_p=0\), and differentiation of \(\Pi_p\Pi_q=0\) gives the other diagonal blocks. In a derivative of the closed diagram, exactly one such factor has been replaced by an off-first-index block. At every other vertex the prescribed Hodge indices balance. Summing these balances over the diagram leaves the nonzero index change at the exceptional factor, so the resulting scalar contraction is zero. This proves local constancy; a function on the finite index spectrum is a linear combination of projectors.

One may equivalently make the entire contraction on the associated Hodge graded bundles. The canonical extensions of the Hodge filtrations in a semistable degeneration are locally free, and flat morphisms of variations extend as filtered morphisms. Their graded tensor contraction therefore extends with the same scalar value. Finally, the limiting objects are mixed Hodge structures. Tensor products, duals, and morphisms are strict for their weight filtrations. Taking weight gradeds of the closed contraction, whose shifts cancel at the scalar output, leaves its value unchanged. Hodge and weight gradeds can be taken successively; the resulting double graded spaces retain the first index including every Tate shift. ◻

Proof. Over a field, the Künneth map identifies the cohomology of a finite tensor product with the tensor product of the cohomologies. The inverse-pairing cycle represents the inverse of the induced perfect pairing, since the chain-level evaluation and coevaluation identities descend to cohomology. The tensor product of all inserted cycles is a cycle. A chain functional annihilates its boundaries, so its value depends only on that cohomology class. Applying chain maps to its legs preserves this argument. Koszul symmetry gives the same statement for loops: the resulting contraction is the categorical supertrace. No splitting of cycles modulo boundaries enters the calculation. ◻

Proof. First use 11 to pass to the Hodge and weight gradeds of nearby cohomology. For every coevaluation use the cycle \(\Delta_{C_Y}\) in the small complex (5); first-index factors are chain maps by 5. Extending cup classes give chain maps with their usual bidegree shifts.

For a tensor with \(r\) exposed evaluation slots, use its own ordered configuration model. Transfer each input by \(\phi_Y\) and pull it to that model by 8. Multiply the inputs in \(K_Z\), multiply by the specialization from 10, and apply \(\Theta_Z\). All these operations induce chain maps on first pages, and \(\Theta_Z\) is a chain functional by (13). The construction represents the desired cohomological evaluation by multiplicative comparison and 9. Use separate models for separate evaluation tensors; only their exposed slots are linked. For a classical cup tensor, use \(K_Y\) itself, cup all its inputs there, and apply \(\Theta_Y\). The resulting closed diagram is therefore a finite tensor contraction of chain functionals and cycle coevaluations. By 12 it equals the original weight-graded contraction. An isomorphism on second pages in (10) is sufficient: it identifies the induced functionals and pairings, while all maps used to perform the calculation go forward.

Now expand these actual chain maps and tensors into their residue and component summands. A block of the inverse pairing uses one vertex or edge. A comparison summand uses an incidence of strata. A product summand uses intersecting strata in the ordered Čech diagram. A trace summand uses one incidence flag. Finally, a closed stratum of the ordered configuration model projects in each target coordinate to one component or to an adjacent double stratum. These descriptions prove that each summand is supported in a union of bounded incidence neighborhoods of the exposed positions.

The bounds are uniform. A model with \(r\) recorded factors has relative dimension \(nr\) and consequently bounded lengths of nonempty incidence flags. The input weight indices are bounded by the fixed number of legs; by (11), only bounded powers of \(u\) can occur in the relevant weight and total degree. For preintegrated source inputs, these are the degrees of the image after the Gysin, descendant, and pushforward shifts. That image lies in the cohomology of the retained configuration, whose degree range is bounded by its dimension. Thus the same bound applies regardless of the number of ordinary auxiliary inputs. Each slot contributes finitely many local index types. Increasing the number of components may increase the number of summands, but does not enlarge the support or index ranges of any one summand. The total number of exposed slots and tensor vertices was fixed in advance. This proves (i) and (ii).

Every formula just used consists of ordinary maps on these strata, the induced normal data, and universal index signs and Tate factors. Unit changes in equations disappear after residue. Hence it is unchanged when the specified local diagrams are unchanged, proving (iii). At no step did we choose a representative of a surviving nearby class or test whether two positions lie in the same connected smooth component. Disconnected spaces are handled by direct sums of their complexes and componentwise traces. Thus their cohomological connectedness is already encoded by the differential and its closed Casimir.

Lastly, preintegration changes the coefficient correspondence rather than its exposed arity. Its actual extending class and its component restrictions are still the ones supplied by 16. Applying a new cohomological operation to one of its markings requires exposing that slot; it then contributes one further position. This gives the final assertion. ◻

Proof. Near a collection of double-locus points, write the fiber product as \[x_i y_i=s,\qquad 1\leq i\leq m=|I|,\] omitting smooth coordinates along the double loci. For a permutation \(\sigma\) of the recorded labels, an ordered chart has coordinates \(a_0,\ldots,a_m\) and equations \[ x_{\sigma(i)}=\prod_{k=0}^{i-1}a_k, \qquad y_{\sigma(i)}=\prod_{k=i}^{m}a_k, \qquad s=\prod_{k=0}^{m}a_k. \tag{15}\] These are the charts for the subdivision of the cube by ordered coordinates. The simplices are unimodular. Thus the total space is smooth and the central fiber is reduced with simple normal crossings. Factors away from the double locus are treated by making the appropriate coordinates invertible. The projections are monomial in these charts and hence logarithmic.

The charts have an intrinsic interpretation: put the recorded points on a common expansion and contract precisely those exceptional levels containing none of them. Ratios of normal coordinates on a retained level record the relative positions of its points. Changing a local equation of \(D\) multiplies such a coordinate by a unit and changes the normal coordinate by the transition function of its divisor line bundle. Consequently the descriptions glue globally. They are the configuration spaces above, whose projectivity is supplied by the cited construction.

The divisor \(a_j=0\) in (15) puts the first \(j\) marks on one side of a cut and the remaining marks on the other. Gluing the distinguished divisors of the two expanded pairs on the right of (14) gives a configuration in this component. All exceptional levels remain occupied, so this gluing does not require a further contraction. Conversely, a stable configuration in the indicated component has a unique cut with exactly \(I_0\) on the \(X_0\) side. Two distinct such cuts would enclose an exceptional level containing no recorded point. Cutting there recovers the two factors of (14). Scheme theoretically, on \(a_j=0\) the remaining coordinates separate into \(a_0,\ldots,a_{j-1}\) and \(a_m,\ldots,a_{j+1}\). These are the ordered charts of the left relative configuration and the right relative configuration with reversed order. The smooth coordinates along \(D\) accompany their respective labels. Further vanishing coordinates describe deeper strata inside the two factors. This also proves that the constructions commute with base change and are inverse as morphisms. There is no additional fiber product over \(D\): the divisors of the targets are glued, whereas the recorded interior positions have independent projections to \(D\). If \(I_0\) or \(I_1\) is empty, its relative configuration is the unexpanded pair with no recorded positions, represented by \(\mathrm{pt}\).

For disconnected sides the collapsed evaluation of a point determines its connected component of \(X_i\). Restricting this locally constant choice gives the last assertion. In particular, no independent expansion is chosen for each connected component within a side. ◻

Proof. The untwisted target stack with a chosen global cut normalizes its normal-crossings boundary with pure degree one (Abramovich et al. 2013, Proposition 7.4.1). Base-changing the map stack and its relative obstruction theory to this normalization gives the virtual cut class by the virtual pushforward comparison used in (Abramovich and Fantechi 2016, secs. 5.4–5.6). This target-stack statement does not depend on the source graph or on the number of components of \(D\). For a twist index \(R\), the reduced twisted split stack has degree \(1/R\) over the corresponding nonreduced boundary; hence this step contributes \(R\) to the virtual class.

Normalize the source at its nodes mapping to the selected cut. A rootless component remains wholly on one side, including when it maps into an expanded level. On fixed labelled data the split-map stack is the fiber product of the two relative-map stacks over the matching root evaluations and the single target-gluing stack. Relative to this common target-expansion base, the AF map-deformation complex on a disjoint union is a direct sum. Its source-normalization triangle has exactly the normal complex of the matching-root diagonal as the difference between split and glued complexes (Abramovich and Fantechi 2016, Proposition 5.16); there is no term at a rootless component. Thus the compatible obstruction theories give the refined diagonal pullback in (18). The common-refinement comparison for disconnected source maps (Abramovich and Fantechi 2016, Proposition 4.14 and its proof) keeps one expansion per side; it does not duplicate the target deformation for each source component. Although the published degeneration formula indexes connected glued graphs, its target normalization and this labelled source-normalization comparison do not use that condition. When both sides are nonempty, connectedness ensures that every side vertex has a root. Here rootless vertices and empty sides are retained, and source-component labels are kept until their finite quotient is taken.

The universal split target over the two side expansion stacks is one \(\mu_R\)-gerbe, of degree \(1/R\) (Abramovich et al. 2013, Proposition 7.4.2); this cancels the preceding \(R\) (Abramovich and Fantechi 2016, Lemma 5.15). The separate comparison from rigidified root evaluations to the ordinary diagonal has degree \(\prod_jc_j\) (Abramovich and Fantechi 2016, Proposition 5.18). These are rootwise factors even if the roots lie on different connected components of \(D\); the target-twist factor remains global.

If a side source graph is empty, every positive-length expansion on that side has an unstabilized level-scaling automorphism. Its stable relative-map stack is therefore the point represented by the unexpanded pair, with zero relative obstruction complex and virtual class \([\mathrm{pt}]\). For an empty root profile choose \(R=1\); the target gerbe is trivial, \(D^0=\mathrm{pt}\), and both contact and root-permutation factors are one. A nonempty rootless source component on a bubble is retained on its side and is not confused with this empty unit. Finally, forgetting the ordered root labels is an \(r!\)-cover and gives the denominator in (18). Quotienting the fixed source-component labels by permutations of identical components gives their separate stack weights. ◻

Proof. We have already constructed the extending class. We prove the assignment-specific statement by a Cartier divisor calculation, then apply virtual splitting on each resulting boundary divisor.

On a chart of the stack of expanded targets let \(t_0,\ldots,t_\ell\) be the untwisted smoothing parameters, so that \(s=t_0\cdots t_\ell\). At an ordinary recorded point the normal coordinate along its level is a unit. Order the recorded points by their levels. Evaluation into (15) has the following form: each \(e_I^*a_j\) is a unit times the product of the \(t_k\) belonging to the interval between the two successive recorded levels, with the two end intervals used for \(j=0,m\). Points on the same level give unit ratios. Consequently, as divisor line bundles with their defining sections, \[ e_I^*(a_j=0) =\sum_{k\in J_j}(t_k=0), \tag{20}\] where \(J_j\) consists exactly of the cuts putting the recorded marks on the sides specified by \(a_j=0\). There is order one on each of these divisors in untwisted target coordinates and order zero on the other cut divisors. Empty unrecorded levels simply contribute more factors to the same interval product. On a normalized splitting divisor, the connected component of a side containing a recorded mark is also discrete. Selecting that choice gives precisely the refinement from the side assignment to \(\epsilon\).

Here the notation in (20) means the factorization of the pulled-back section and its line bundle. It does not assert that this section is regular on every component of \(\mathcal M\). The virtual boundary intersections below are refined pullbacks of the boundary divisors on the expansion stack. Their additivity under this factorization therefore also applies to components of the map space supported over \(s=0\).

The equations change by units under normal-coordinate transitions, so (20) is an equality of the corresponding divisor line bundles and sections, not just an equality on an open set of maps. In particular, it controls refined intersection also when the evaluation image lies in a deeper central stratum. The map from each normalized cut divisor to \(Q_\epsilon\) is obtained by cutting the expansion and stabilizing the recorded positions on its two sides. The uniqueness argument in 14 shows that this is the map occurring in (14).

Refined Cartier pullback commutes with proper pushforward. Applied to (17), it identifies the homological class corresponding to \(i_\epsilon^*\Gamma_I\) with the pushforward of the virtual intersection with the left side of (20). Intersect the right side one boundary at a time. The virtual splitting identities in the proof of the degeneration formula identify each such intersection with the matching relative virtual classes and their refined root diagonal. The required identity for disconnected sources and an empty side is 15, applied before evaluation pushforward. The additional assertion beyond the numerical degeneration formulas (Abramovich and Fantechi 2016, secs. 5.3–5.9) and (Argüz et al. 2023, Theorem 2.3) is the retained correspondence with its recorded assignment and the full labelled source graph, including rootless components.

A target level is retained when stabilized by the union of its source components; after cutting, each side is stabilized separately. Bounded expanded-map stacks supply properness and finite type. The target-cut normalization and Cartier equation (20) depend only on the target expansion and the recorded assignment, so they are unchanged by the source graph.

Passing to unlabelled disconnected sources divides both sides by the finite permutation group of isomorphic components; intrinsic map automorphisms remain in the stack virtual classes. Two identical closed components on one side, for example, contribute \(1/2!\), while the two labelled assignments with one on each side cancel that quotient. Labelling the relative roots and dividing by \(r!\) gives the convention of (19). This proves the formula as a correspondence class before further cohomology insertions or numerical integration, with the aligned evaluation in \(Q_\epsilon\) retained.

For clarity about multiplicities, a twisted node parameter \(\tau\) has untwisted parameter \(t=\tau^r\). Pulling back the untwisted component equation therefore gives its twisting multiplicity. The gerbe degree in gluing and the root-pairing normalization in the twisted splitting identity combine with it to give the ordinary contact coefficient in (19). One must not replace this calculation by an assertion of multiplicity one in source-node or twisted coordinates. We have used multiplicity one only in the untwisted target boundary equation (20).

Cutting and gluing a target leave the curve unchanged in a neighborhood of every ordinary marking. Hence its ordinary cotangent line and all powers of that line restrict as used in \(\Theta_\eta\). Preintegrated extending classes restrict by the projection formula. For constant-source Hodge inputs the equality is first made with the source slots exposed and then contracted, as described above. Finally, with no recorded positions the sole component equation is \(s=0\) and all cuts occur; the same proof gives the usual formula. ◻

Proof. For groups consisting of single connected source components, (22) is the relative disconnected product rule (Abramovich and Fantechi 2016, Proposition 4.14). We describe its comparison with groups retained, since the additional evaluation assertion cannot be obtained merely by quoting numerical factorization.

Let \(\mathcal T\) denote the stack of expansions of the pair, with the twisting choices used in the transverse formulation when needed. Form the auxiliary stack \(\mathcal T'\) of one common expansion \(\mathcal X\) and partial contractions \[\mathcal X\longrightarrow\mathcal X_a, \qquad 1\leq a\leq u,\] where each \(\mathcal X_a\) is an expansion, and every exceptional level of \(\mathcal X\) is retained by at least one contraction. The twists and partial untwistings are included as in (Abramovich and Fantechi 2016, secs. 4.7.3–4.7.4); alternatively they may be put under a common dominating twisting choice. The comparison with the product expansion stack has pure degree one: both stacks have the same dense locus of unexpanded targets. The forgetful map from \(\mathcal T'\) to the common expansion stack is the same local etale comparison as in that construction.

The square \[ \begin{matrix} \mathcal K_\Xi &\xrightarrow{\ \varepsilon\ }& \prod_a\mathcal K_{\Xi^{(a)}}\\ \big\downarrow &&\big\downarrow\\ \mathcal T'&\longrightarrow&\mathcal T^{u} \end{matrix} \tag{24}\] is cartesian in the transverse expanded-map comparison. Indeed, given maps from the separate groups and a specified common refinement, pull each map back along \(\mathcal X\to\mathcal X_a\). Transversality makes this pullback a curve with the canonical trivial cylinders at the levels contracted in \(\mathcal X_a\). The markings lift uniquely, and the disjoint union of the lifted groups is a map to the common expansion. The requirement that each level is retained by at least one group gives stability of the combined map. Conversely, separating a map and contracting the levels unstable for an individual group recovers this construction. No step uses connectedness inside a group.

The relative map obstruction complex on a disjoint union is the direct sum of the complexes on its groups. Under the partial curve contraction \(c:C'_a\to C_a\) above, the contracted components are trivial rational cylinders and \(Rc_*\mathcal O_{C'_a}=\mathcal O_{C_a}\). The target-relative complex in the transverse comparison pulls back to the corresponding complex on \(C'_a\). Projection formula therefore identifies the relative obstruction theory upstairs with the pullback of the direct-sum theory on the product. This is exactly the obstruction-theory comparison of (Abramovich and Fantechi 2016, Lemmas 4.16–4.18), with a disjoint union in place of each individual connected source. Its degree-one virtual pushforward argument applied to (24) proves (22).

Consider an ordinary marking belonging to group \(a\). A level containing its image cannot be trivial for that group: the nontrivial scaling of a fiber of that level cannot fix an interior marked point while preserving the map. Consequently the curve contractions used in separation are isomorphisms near all ordinary marks of the retained group. They preserve their cotangent lines. They also preserve their evaluation positions on the remaining levels. Stabilizing these recorded positions before or after separation gives the same configuration, because in either case the final expansion contracts precisely the levels empty of those positions. Thus \(e\) and each \(\Theta_a\) factor through (21). Projection formula and proper pushforward give (23).

The source exposition of the product comparison assumes every connected component carries a marking or a relative root, expressly for simplicity (Abramovich and Fantechi 2016, sec. 4.7.1). Its proof of the product-class identity extends directly to stable labelled components with neither. A level occupied by a positive-degree map is retained when contracting it would destroy transversality or stability. A rootless closed component of collapsed degree zero cannot have a nonconstant fiber-only part: a complete nonconstant curve in a ruled fiber meets its boundary, and predeformability propagates this to a root or positive collapsed degree. Thus the remaining unmarked component is constant and has genus at least two by stability. The cartesian common-refinement square, etale expansion comparison, and direct-sum obstruction theory in (Abramovich and Fantechi 2016, Lemmas 4.16–4.18) depend on these stable maps and their universal curves, not on the presence of a mark or root. Their degree-one virtual pushforward proves (22) for these components as well. The empty source graph is the unit \([\mathrm{pt}]\).

Finally, a connected source curve mapping into a disconnected target lies in one connected target component. We may collect all such source components into the desired target groups. The same argument applies, keeping one expansion and one aligned evaluation for each group until (23) is used. With cohomology inputs of odd parity, the products and permutations in this argument are taken in the graded tensor category. This inserts the usual Koszul signs and does not change any of the virtual pushforward equalities. ◻

Proof. Choose a homogeneous basis of \(H^*(T)\) and use the projective bundle basis \(1,H_f,\ldots,H_f^{r-1}\) on \(E\), where \(H_f=c_1(\mathcal O_E(1))\). For root order \(e\geq1\) and \(0\leq b<r\), use the variable corresponding to \[ \tau_{r(e-1)+b}(i_*\gamma). \tag{28}\] At a possibly different root order \(e'\), pushforward by the relative evaluation has codimension beyond \(\gamma\) equal to \[ r(e-e')+b. \tag{29}\] Indeed, writing \(t=\dim T\), the relative virtual dimension with one ordinary and one full-contact relative marking is \(t+r-1+re'\). Subtract the insertion codimension \(r+\mathop{\mathrm{codim}}\gamma\) and its descendant exponent, and compare with \(\dim E=t+r-1\). The projection formula factors out \(\gamma\), so a negative value in (29) forces zero even if the degree of \(\gamma\) itself is large. Thus orders \(e'>e\) do not occur.

At \(e'=e\), the output is a nonzero scalar times \(H_f^b\gamma\) plus terms of smaller fiber power with additional base degree. To compute the scalar, restrict to a point of \(T\). A connected genus-zero domain with constant base projection has \(H^1(C,\mathcal O_C)\otimes T_tT=0\); horizontal deformations supply the base factor and no horizontal obstruction. The calculation is therefore the relative invariant of \((\mathbb P^r,\mathbb P^{r-1})\). The exact formula is \[ \left\langle\tau_{re-1-j}(\mathrm{pt})\mid H_f^j\right\rangle^{ (\mathbb P^r,\mathbb P^{r-1})}_{0,e[\mathrm{line}]} =\frac{1}{e^{r-j}((e-1)!)^r}, \qquad 0\leq j\leq r-1. \tag{30}\] This is (Hu et al. 2008, Theorem 7.1), for one ordinary marking and one relative marking of full contact \(e\), using the ordinary marked-point cotangent line. To match the expansion convention used here, apply the AF-to-Li virtual pushforward of (Abramovich et al. 2014, Theorem 1.1.2). That comparison takes the relative coarse source without contracting its components (Abramovich et al. 2014, sec. 4.1), so it preserves the ordinary cotangent line.

For this particular scalar, the jet calculation of (Hu et al. 2008, Lemmas 7.2–7.4) can be performed on Li’s algebraic relative-map space. Impose evaluation at a point \(p\notin\mathbb P^{r-1}\), and denote its ordinary marking by \(x\). The first \(e-1\) jets at \(x\) form an algebraic section of a bundle with successive quotients \((L_x^{\otimes k})^{\oplus r}\), \(1\leq k<e\). Its top Chern class is \(((e-1)!)^r\psi_x^{r(e-1)}\). The component containing \(x\) cannot be constant: stability would give at least two branches, each forced to contain the unique relative root, contradicting the genus-zero tree. At a zero of the jet section, a generic hyperplane through \(p\) has intersection multiplicity at least \(e\) at \(x\), so this component uses the full degree \(e\). Every remaining rubber branch would have to carry an outer root by contact conservation. The single-root and genus-zero conditions therefore leave one full-contact attachment; any remaining unmarked rubber cylinders are unstable. Thus the zero locus has unexpanded target and maps \(w\mapsto a w^e\). Near it the point-constrained moduli have the smooth polynomial chart \([(a_1,\ldots,a_e),\ a_e\ne0\,/\mathbb C^*]\), with \(a_k\in\mathbb C^r\). The jet equations are regular there, so their localized Euler class meets the virtual cycle with multiplicity one. The zero locus is a \(\mu_e\)-gerbe over \(\mathbb P^{r-1}\) and \(L_x^{\otimes e}=\mathcal O(1)\) on it. Its remaining integral is \(e^{-(r-j)}\), giving (30). This argument only compares the stated one-root fiber invariant.

With \(j=r-1-b\), the leading scalar is \(e^{-(b+1)}((e-1)!)^{-r}\).

Order the blocks by root order and, within each block, by fiber power. The preceding vanishing and nonzero diagonal give a triangular matrix with invertible diagonal blocks. Corrections with positive base degree are triangular in fiber power; alternatively their repeated composition is nilpotent because the base has bounded cohomological degree. This proves the assertion in both parities. ◻

Proof. Write \(Z_{\mathrm{cl}}(\mathbf t,\hbar)\) for the disconnected factor formed by connected components without relative markings. The relative product rule in 19 gives the same factor at every profile, so it may be divided out before selecting rows. In fiber degree, a component without roots has degree zero: its intersection with the relative hyperplane is zero. At \(\mathbf t=0\) its disconnected series is \(1+O(\hbar)\), because unmarked constant maps in genus zero or one are unstable. It is therefore invertible over \(\mathbb C((\hbar))\); its inverse is well defined coefficientwise in \(\mathbf t\). At a fixed auxiliary monomial, only finitely many coefficients enter multiplication or division by this series.

After this normalization every connected component has at least one root. For a profile \(\lambda\) with \(\ell(\lambda)\) roots, its lowest possible genus exponent is \(-\ell(\lambda)\). Equality holds exactly when every connected component has one root and genus zero. Its coefficient is therefore, up to the nonzero finite labeling factor, the signed product of the connected genus-zero single-root series associated with the entries of \(\lambda\).

Let these single-root coordinates be \(f_1(\mathbf t),\ldots, f_s(\mathbf t)\) in homogeneous bases. By 20, choose \(s\) auxiliary variables with invertible super Jacobian and set the others to zero. The formal inverse function theorem applied to \(f_i-f_i(0)\) identifies the completed super power series algebras in these variables. Substitution by the \(f_i\) is therefore injective on the algebra of polynomials in even variables and exterior variables in odd degree. Nonzero constant terms in the even \(f_i\) merely translate the polynomial variables. It follows that the finitely many signed monomials corresponding to a basis of \(\mathcal F_{\leq H}(E)\) are linearly independent.

For completeness, finite coefficient detection here involves no compactness assumption on formal series. Intersect the kernels of successively more coefficient functionals on the finite-dimensional span of these monomials. If their intersection were nonzero, it would contain a nonzero series with every coefficient zero. The decreasing sequence of subspaces stabilizes, so a finite set of coefficients already has zero kernel. Choose a square full-rank submatrix of those coefficient rows. Multiply each column of the corresponding full Gromov–Witten matrix by \(\hbar^{\ell(\lambda)}\). Its constant matrix is the just chosen invertible coefficient matrix, and the remaining entries have positive \(\hbar\)-valuation. It is invertible over \(\mathbb C[[\hbar]]\); undoing the column shifts gives a Laurent inverse.

Finally, each of the finitely many normalized rows used above is a finite linear combination of unnormalized coefficient rows, with Laurent coefficients coming from \(Z_{\mathrm{cl}}^{-1}\). Enlarge the row collection by these finitely many contributing original rows. Their span still has full rank, so a square subset of original rows has an invertible Laurent matrix. Thus all rows can be taken to be genuine tests by finitely many ordinary markings. Perfectness of \(G_{\hbar}\) gives the equivalent assertion for inserted states. ◻

Proof. In the ordinary configuration the projected degree on \(D\) is bounded by the ample counting degree. The section relation (25), together with either specified section degree, then fixes the remaining numerical fiber degree. In the blowdown cap, the relative hyperplane is relatively ample over \(S\); a large multiple of the ample counting class on \(S\) plus this hyperplane is ample. Its intersection is bounded by \(C d+a\), where \(a\) is the relative contact total. This proves (i).

In the exceptional left cap, \(-c_1(N)\) is relatively ample on \(D\to S\). For a sufficiently large integer \(A\), \[A q|_D-c_1(N)\] is ample on \(D\). The same argument, or ampleness of \(q|_D\) in the ordinary case, gives \(c_1(N)\cdot\alpha\leq C_0q(\alpha)\). Since \(a=h+c_1(N)\cdot\alpha\geq0\), one also has \(c_1(N)\cdot\alpha\geq-h\). Thus the projected ample degree is at most \(Ad+h\). The section degree \(h\) then fixes the remaining fiber coordinate on \(P\). An ample class on \(P\) has bounded degree, proving (ii).

A bound on an integral ample degree bounds the homology classes of effective curves, not just their numerical images. One way to see this is to choose a Kahler metric representing that ample class: the integral of every fixed smooth real two-form over a holomorphic curve is bounded in absolute value by a constant times its area. Apply this to finitely many representatives of a basis of real cohomology, and use the integral homology lattice and its finite torsion subgroup. Hence only finitely many integral classes occur. In particular all extra divisor exponents have finite support.

For a fixed auxiliary row, let \(m\) be its number of ordinary marks. The number of nonconstant connected components is bounded by their total degree against the chosen integral ample class. Constant connected genus-zero components require at least three ordinary marks if they have no roots, and degree-zero components cannot have positive total contact. Their number is consequently bounded by \(m\). These are the only connected components contributing a negative power of \(\hbar\). Thus each entry has a lower bound on its \(\hbar\)-valuation. For a specified total exponent, genera are bounded; unmarked constant components of genus at least two contribute positive powers, while marked genus-one constants consume ordinary marks. Consequently the disconnected exponential and the finite matrix products are well defined coefficientwise. This also proves (iii) for finite lists of rows. ◻

Proof. At \(q=0\), projection to the base of \(Q\) is zero: that base is \(D\) in the ordinary configuration and \(S\) in the blowdown configuration, and the counting class there is ample. Choose the finite invertible fiber matrix \(M_0\) of 21. Apply the same rows at all counting degrees to obtain \[M(q)=M_0+qM_1+\cdots+q^d M_d \pmod{q^{d+1}}.\] Its entries are well defined by 22. If \(B=M_0^{-1}(M(q)-M_0)\), then \[ M(q)^{-1}=\left(\sum_{j=0}^{d}(-B)^j\right)M_0^{-1} \pmod{q^{d+1}}. \tag{31}\] This is a finite sum, and uses no negative \(q\)-powers. The target classes in a \(Q\) row are already bounded by the incoming contact. No independent fiber-degree selection on the smooth segment containing \(Q\) is required. ◻

Proof. First take projected class \(\alpha=0\). Then (25) says \(a=h\). The rank-one case of 21, restricted to total contact \(h\), supplies an invertible block of ordinary rows, with no output at other contact totals. Normalize these rows by that inverse.

If \(q(\alpha)=0\) but \(\alpha\ne0\) in the exceptional configuration, \(\alpha\) is vertical over \(S\). The line bundle \(-N\) is ample on every fiber, and hence \[ c_1(N)\cdot\alpha<0,\qquad a<h. \tag{32}\] Thus the full \(q=0\) matrix at bounded imposed flux is block triangular in contact, with precisely the invertible fiber blocks on the diagonal. Correcting successively downward in contact inverts it: its strictly contact-decreasing part is nilpotent on the finite range \(0,\ldots,H\). This step includes empty-profile errors. For example, over a point with rank-two normal bundle, \(D=\mathbb P^1\) and \(P\) is the Hirzebruch surface \(\mathbb F_1\). A projected class of degree \(m\) has \(a=h-m\), so \(0\leq m\leq h\); the term \(m=h\) indeed has empty boundary.

We now correct in increasing \(q\)-order. Suppose a remaining error has coefficient \(q^j\) and contact \(a\). Use imposed flux \(h=a\) and the already inverted \(q=0\) triangular block to remove it. Any new error of positive additional counting degree \(k\) has contact at most \(a+C_0k\), by 22. Starting with contact at most \(H\) at order zero proves inductively the sharper bound \[ a\leq H+C_0j \quad\text{for errors corrected at order }q^j. \tag{33}\] At each order the contact-decreasing corrections terminate by (32); only \(d+1\) counting orders are inspected. There are therefore finitely many required imposed fluxes and finitely many coefficient rows, and only nonnegative powers of \(q\) have been used.

A row introduced with coefficient divisible by \(q^j\) has imposed flux at most \(H+C_0j\). If its cap class has counting degree \(k\) and \(j+k\leq d\), its outgoing contact is at most \(H+C_0j+C_0k\leq H+C_0d\). This argument is an estimate on each row’s curve classes; it does not use cancellation or its insertion values. Altering auxiliary labels, including removing a label to a classical factor, leaves it valid as long as the external degree conditions are unchanged. ◻

Proof. Choose a homogeneous basis \(u_i\) of \(\mathcal F_{\leq H}(D)\) and its algebraic dual \(\epsilon_i\). Express the original gluing as \[G_{\hbar}(x,y) =\sum_i \epsilon_i(x)\,G_{\hbar}(u_i,y), \qquad x\in\mathcal F_{\leq H}(D),\] understood as a categorical contraction in the displayed order. If another order is chosen, insert its Koszul sign. By 23, finite \(Q\) tests realize each \(\epsilon_i\) on the incoming bounded domain, with the new seam gluing included. By 24, finite \(P\) tests realize the state \(u_i\) on the entire outgoing space: in particular they cancel every extraneous outgoing profile through order \(d\). Compose these tests with the actual gluing tensors. Perfectness of \(G_{\hbar}\) absorbs all contact orders, permutation denominators, and genus shifts into the external coefficients. Thus the two new gluings have exactly the weight of the original single gluing, rather than its square. The stated formula proves the desired identity.

All inversions and products used are finite at the given bounds, with the completions justified in 22. The outgoing estimate is the summandwise estimate of 24. Equality before contraction with any outside state makes tensoring and independent replicas immediate. The empty basis vector has been included throughout [cap:span,cap:P]; no nonempty-profile qualification is implicit. ◻

Proof. At the original left end, an ordinary ample counting class bounds the initial matching flux. In the blowup case, if \(E_U\) is the exceptional divisor, choose \(A\) such that \(Aq-E_U\) is ample. For an effective class on this end, \[E_U\cdot\beta\leq A q(\beta).\] Its matching flux is \(E_U\cdot\beta\geq0\), so both flux and ample degree on \(U\) are bounded. In a neck, the increase in matching flux is at most \(C_0\) times its counting degree, by (25). Nonnegative counting degrees on the complete chain have total at most \(d\).

Choose the incoming bound for the first switch accordingly. If that switch is retained, flux continues under the same neck estimate. If it is replaced, 25 gives a bound on the new outgoing flux before cancellation, depending only on the chosen incoming bound and \(d\). Take the larger of these two bounds, propagate to the next switch, and repeat. Since there are finitely many switches, this produces finite bounds for all subset replacements without a circular choice of rows.

For an exceptional neck with two matching seams, let its section degrees be \(k_l,k_r\geq0\). The already proved flux bounds and (25) give a lower bound on \(c_1(N)\cdot\alpha\). The ample class \(Aq|_D-c_1(N)\) consequently bounds the projected degree; either section degree fixes the remaining fiber coordinate. On the final \(V\), the same reasoning uses its recorded value \(k_r\) and bounded incoming \(k_l\), even if the recorded value is negative. On inserted ends it uses the imposed \(h\) or the bounded relative contact, as in 22. Ordinary ends are bounded by their ample counting degree. This exhausts the pieces and proves the effective-degree assertion.

For fixed rows, their total ordinary arity is finite. The genus argument in 22 now applies to every piece, and to their finite products with inverse-matrix coefficients. Thus extraction at fixed powers of all replica parameters is legitimate. Modifying or removing an auxiliary insertion changes neither a selected degree nor a section-difference relation; it can only change a finite marking bound. The same geometric and Laurent bounds therefore remain valid. ◻

Proof. In the ordinary configuration start with the given two-piece family. After a finite base change, resolve the chain singularity; in a transverse chart this is the projective semistable resolution of \(xy=t^a\). It inserts the required copies of \(C\). The resolution is obtained by blowups, so is projective, and its total space is smooth. The same construction applied to deformation to the normal cone of the divisor \(D\subset U\), or \(D\subset V\), gives the capped segments with smooth fibers \(U\) and \(V\). Deformation to the normal cone of either section of \(C\) gives the middle segments with smooth fiber \(C\).

For the blowup configuration, deformation to the normal cone of \(D\subset U\) supplies the main chain, with smooth fiber \(U\). Deformation of \(M\) to the normal cone of \(S\) has central fiber \(U\cup_D Q\) and gives the left capped segment. In \(Q\) the normal of its infinity divisor \(D\) is \(\mathcal O_D(1)=N^{-1}\); deformation to the normal cone of that divisor gives \(P\mid Q\), with smooth fiber \(Q\). The remaining segment is obtained by the divisor normal-cone degeneration of \(C\). Inserting further levels by the same base change and resolution gives any required length. Take disjoint unions of these families to realize several capped segments simultaneously.

Color the original chain alternately. At a cut, give each new cap the color opposite to its neighbor. Each color is consequently a disjoint union of smooth components, and the central fiber is a two-sided degeneration without triple intersections. This is the form to which the two-sided degeneration formula applies.

Away from a surgery the smooth pairs and normal bundles have not changed. Their logarithmic charts factor the base generator at the same double divisors. Units in these local equations do not change the graded residue maps. The ordered configuration spaces in 16 are constructed from these same divisors and normal bundles. Their projected incidence neighborhoods, and the pullback, cup and trace operations on those neighborhoods, therefore agree by 13. This assertion concerns the local operations before passage to cohomology and makes no choice of global representatives for surviving cohomology classes. ◻

Proof. This is the geometric application of [cap:identity,cap:uniform]; we spell out the bounds that are used in the cancellation. There are constants \(C_0,C_1\) such that, on effective classes, \[ \int_\alpha c_1(N)\leq C_0q(\alpha),\qquad D\cdot\beta_U\leq C_1q(\beta_U). \tag{37}\] Ordinarily these follow from ampleness on \(D\) and \(U\). For the blowup configuration, \(-N\) is relatively ample over \(S\), and minus the exceptional divisor is relatively ample over \(M\). Adding sufficiently large pullbacks of the chosen ample classes gives the inequalities. At the initial relative end the contact is nonnegative, so the second inequality bounds its total. Through an unchanged neck, (35) bounds any increase of contact by the counting degree spent on that neck.

At a surgically inserted \(P\), the preparation imposes one of a bounded list of nonnegative initial fluxes \(h\). Its outgoing total is \[ a=h+\int_\alpha c_1(N)\leq h+C_0d. \tag{38}\] This conclusion uses only the imposed degree condition. Altering a descendant, applying a grading or a Chern-class power to an auxiliary label, or removing labels to a classical operation does not change that condition. Choose the bound at the first switch from (37). Choose the next one using the largest of the unchanged-neck bound and all outgoing bounds in (38) from the preceding preparation. Continue from left to right, taking the maximum over the finitely many earlier choices. This proves uniformity over all subsets without requiring the identity property at a busy switch.

For clarity, the zero-counting-degree exceptional classes have not been removed in this argument. If \(\alpha\ne0\) is effective and \(q(\alpha)=0\) in the blowup configuration, it is vertical over \(S\) and \(\int_\alpha c_1(N)<0\). Such terms strictly lower the outgoing contact. They are included in the decreasing-contact part of the inverse construction in 25.

In the final \(V\) of the blowup configuration, let \(e\) be its recorded nonjoining section degree and let \(a\) be the incoming contact. Then \[\int_\alpha c_1(N)=e-a.\] The fixed \(e\) and bounded nonnegative \(a\) give the lower bound missing from a counting-degree bound alone. Together with relative ampleness, this bounds the effective horizontal degrees there. The imposed \(h\) supplies the corresponding bound at inserted ends. At a \(Q\) end, bounded incoming contact and bounded base degree bound the fiber degree. These are precisely the end conditions used in 26 to bound the effective degrees on all intervening pieces. Extra divisor fugacities consequently have finite support in the bounded blocks.

The cap inverse has nonnegative counting-degree powers and finitely many selected auxiliary coefficient rows at these orders. Its genus coefficients are Laurent series bounded below. The same holds for the virtual sums with these bounds: the number of negative-genus-power components is bounded at fixed degree and marking order, unmarked constant genus-zero and genus-one components being unstable. Higher genus unmarked constants have positive genus power. This proves the needed coefficientwise finiteness, including for separate genus variables. Finally 25 is an equality on the full bounded relative state space, so it gives assertion (iv) before any contraction with the outside data. ◻

Proof. Expand the fixed operator coefficient by 2. There are finitely many tensor operation patterns. Record the fixed original slots and every additional special GW slot. Record also any auxiliary slot moved to a classical operation or otherwise modified by this summand of \(L_k\). There are at most two such exceptional auxiliary slots: the linear part acts on one existing slot, the classical quadratic part can use two, and the second-order part creates its own pair of slots. A modified auxiliary class may be evaluated using its fixed, Gysin-shifted first Hodge index and the coherent Chern-class cup operation. Its support remains at its cap; it is nevertheless counted as a position.

All other auxiliary labels are ordinary insertions from constant cap sources. Integrate them into the GW correspondence, retaining only the exposed evaluations just listed. The construction of 16 applies with precisely that retained list; 19 permits the other group of labels to be preintegrated without altering it. The dimensions of the recorded products thus have a bound determined by the original expression and the two exceptional slots, independently of the number of auxiliary labels.

Now apply 13. Every first-page Casimir summand has a single local vertex or edge support, even when its two legs enter different replicas. All pulls, cups and traces involve bounded incidence neighborhoods of the recorded slots. Finitely many operation patterns on these bounded products give a uniform bound on their number and on a radius \(r_0\). Increase the bound to include the finitely many projected positions from all replicas and all classical operations. With the stated switch separation, each such neighborhood can affect at most one switch; without this separation one could instead multiply by a fixed bound for its number of vertices. Counting inserted-cap positions at their insertion switch therefore gives a constant \(B\) of the required kind. The argument does not count an ordinary preintegrated cap marking as a recorded factor, so neither the number of rows nor their arities enters \(B\). ◻

Proof. At an idle switch there is no recorded position on either inserted cap, and no auxiliary label there is modified by \(L_k\). By 28, all retained local component and stratum data agree on the two sides of the comparison. The component restriction formula of 16 is applied with these assignments fixed. Each color is a disjoint union of ends, necks, and caps, so every connected component of a side source lies over one connected component of that color. In each color, collect all source components mapping to the retained outside pieces into one group, with all of that color’s recorded positions. Use separate groups for the cap pieces at \(j\). The retained group is allowed to be disconnected.

Apply 19 with this grouping. Its common-refinement comparison preserves the full joint evaluation of the retained group into its aligned configuration space. The retained ordinary marks stabilize their own levels, so their cotangent classes are preserved as well. Projection formula integrates the ordinary cap groups, which have no recorded positions. The outside pairs and their recorded data agree on the two sides; hence this leaves the same outside relative correspondence, with two uncontracted contact-state inputs at \(j\) in each replica. The only remaining difference is the bilinear kernel contracting these states. With \(j\) uncut it is the ordinary contact kernel \(G_{\hbar_r}\) of 25. With \(j\) cut it is the sum of the two cap kernels with their prepared coefficients and degree conditions. By 25, these kernels are equal on every allowed incoming state and on every compatible outgoing state. 30 ensures that those domains include all terms under consideration, including histories in which other switches are busy.

The comparison is made separately in each replica, with its own \(q_r\), \(\hbar_r\), states and contact joins. Tensor the resulting equalities in the fixed order. Equality on the full relative state spaces permits subsequent contraction with arbitrary outside tensors, including tensors correlating different replicas. It never requires gluing a contact from one replica to a contact in another. All contact-order factors, finite permutation quotients, and one genus factor per joined root are already present in \(G_{\hbar_r}\). Consequently changes in domain connectedness or in the distribution of handles require no supplementary sign or condition. The only signs for odd labels are the categorical tensor signs fixed throughout.

Finally the histories are summed using the extended degree tests, with the compensating shifts of the cap coefficients. Their coefficientwise sums exist by 30. One has not chosen individual homology classes to match across different smoothings. The equality is therefore one of the full weighted virtual correspondences needed as input to the remaining local operations, as asserted. ◻

Proof. For a smooth projective target \(Y\), set \[\mu^{\mathrm{tot}}_Y\big|_{H^{p,q}(Y)} =\frac{p+q-\dim Y}{2},\qquad \nu_Y\big|_{H^{p,q}(Y)}=\frac{p-q}{2}.\] Thus \(\mu_Y=\mu_Y^{\mathrm{tot}}+\nu_Y\). Divisors, line-bundle Chern classes, and toric and equivariant parameters have Hodge difference zero. A Gromov–Witten correspondence preserves total Hodge difference: its algebraic virtual class, descendant powers, and evaluation pushforward have diagonal Hodge type. Consequently the transformations built from these correspondences and characteristic classes are equivariant for Hodge difference.

There is a small parameter issue in applying this observation. Fundamental solutions at an arbitrary auxiliary parameter are Hodge-equivariant as families, with the corresponding action on the parameter. We need an intertwining identity on cohomology at a fixed parameter. Choose the auxiliary base parameter \(\tau_B=0\) and retain only the toric divisor parameters. Such a specialization is permitted: the total descendant potentials are independent of the auxiliary parameters, and a single auxiliary value suffices in the construction (Coates et al. 2024, sec. 4, footnote 11). It imposes no restriction on the descendant variables in (46).

For completeness, this equivariance can be checked on the final nonequivariant transformation rather than on separate singular localization factors. In the notation of (Coates et al. 2024, sec. 4.4), the transformation is the product of the matrix \(T\) with the inverse of the stationary-phase matrix of the oscillating integral \(\mathcal I\); write \(\mathcal I_{\mathrm{as}}\) for that matrix. Both matrices have columns indexed by a toric basis class times a homogeneous base class \(\phi_b\). On this common column space define \(\nu_{\mathrm{col}}\) to have eigenvalue \((p_b-q_b)/2\) when \(\phi_b\) has type \((p_b,q_b)\). The toric basis classes have Hodge difference zero.

At \(\tau_B=0\), every column of the base fundamental solution has the Hodge difference of its input, because its coefficients are algebraic Gromov–Witten correspondences. The toric operations producing \(T\) preserve this difference. The construction of \(\mathcal I_{\mathrm{as}}\) uses the same base solution and differentiates it only in the line-bundle Chern-class directions. These directions, the toric differential operators, and the scalar stationary-phase coefficients all have Hodge difference zero. Consequently \[\nu_E T=T\nu_{\mathrm{col}},\qquad \nu_{B^{\sqcup s}}\mathcal I_{\mathrm{as}} =\mathcal I_{\mathrm{as}}\nu_{\mathrm{col}}.\] These identities include the columns indexed by odd classes. Forming \(T\mathcal I_{\mathrm{as}}^{-1}\) and taking its nonequivariant limit \(M\) gives \[ \nu_E M=M\nu_{B^{\sqcup s}}. \tag{47}\] The classical grading identity proved in (Coates et al. 2024, sec. 4.4, Equations (42) and (44)) is \[\left(z\partial_z+\frac12+\mu_E^{\mathrm{tot}} +\frac{c_1(E)\cup}{z}\right)M =M\left(z\partial_z+\frac12+\mu_{B^{\sqcup s}}^{\mathrm{tot}} +\frac{c_1(B^{\sqcup s})\cup}{z}\right).\] Adding (47) gives the same identity with \(\mu\) in place of \(\mu^{\mathrm{tot}}\). Since \(M\) commutes with multiplication by \(z\), it intertwines all the classical positive operators obtained from \(l_0(zl_0)^k\).

We use the central normalization for this new grading. For a target \(Y\) of dimension \(d\), the scalar in \(L_0\) is \[ C_Y=\frac{\chi(Y)}{16}-\frac14\mathop{\mathrm{str}}(\mu_Y^2) =\frac1{48}\int_Y\bigl((3-d)c_d(Y) -2c_1(Y)c_{d-1}(Y)\bigr). \tag{48}\] The equality is the Hodge-index Riemann–Roch identity, and the first-Hodge grading equation \(L_0Z_Y=0\) is the usual Hori equation (Getzler 1999, Theorem 2.1 and Proposition 2.6). For a zero-dimensional target the second Chern-number term is omitted. No half-total-degree grading equation is being asserted here.

Quantization is in the super symplectic space; the resulting cocycle uses supertrace. Conjugation of the first-Hodge quantized operators by \(\widehat M\) can differ from the target operators only by scalars. For \(k=0\) the scalar difference is zero, because both operators annihilate the same nonzero potential in (46), by their first-Hodge grading equations. For \(k\ne0\) the commutator with \(L_0\) forces the scalar difference to vanish. This is precisely the argument of (Coates et al. 2024, Proposition 1.3), now applied with the first-Hodge operators and (48). In particular one need not, and in general cannot, identify \(C_E\) with \(sC_B\): the quantization cocycle accounts for their difference. Equation (46) now transfers all constraints. The same argument applies at each stage of a tower. ◻

Proof. The codimension-two blowup formula gives \[ H^k(\widetilde M,\mathbb Q) =p^*H^k(M,\mathbb Q)\oplus j_*\pi^*H^{k-2}(S,\mathbb Q). \tag{54}\] The second summand has Hodge shift \((1,1)\). The only possible nonambient summands in (54) therefore occur for \(k=n\). Put \(\xi=c_1(\mathcal O_E(1))\), with \(N_{E/\widetilde M}=\mathcal O_E(-1)\). The push-pull and self-intersection formulas give, in complementary degrees, \[\begin{align*} \int_{\widetilde M}p^*u\,j_*\pi^*a&=0,\tag{55}\\ \int_{\widetilde M}j_*\pi^*a\,j_*\pi^*b &=-\int_E\xi\,\pi^*(ab)=-\int_Sab. \tag{56}\end{align*}\] For the first equality, the integrand on \(E\) is pulled back from \(S\) and has no fiber hyperplane factor. These identities prove nondegeneracy of (53).

The subspace \(B_0\) is stable under multiplication by the ambient hyperplane and by \(E\). Indeed the projective bundle relation expresses powers of \(\xi\) through Chern classes of \(N_{S/M}\), and those Chern classes are ambient; the pushforward of \(S\) in \(M\) is ambient as well. Equivalently the blowup ring formula closes on the two ambient summands in (53). For small positive rational \(\epsilon\), \(p^*H-\epsilon E\) is ample. Since \(H^{n-2}(\widetilde M)\) is contained in \(B_0\), so is its image under this Lefschetz operator. That image is the nonprimitive part of \(H^n(\widetilde M)\). Orthogonality then implies that \(J=B_0^\perp\) is primitive middle cohomology, with its Hodge–Riemann polarization.

When \(n=2\) and \(S=\{s_1,\ldots,s_m\}\), the exceptional part of \(B_0\) is generated by \(E_1+\cdots+E_m\). Formula (56) has matrix \(-I_m\) on the individual exceptional classes. The omitted differences, with coefficient sum zero, are primitive for \(p^*H-\epsilon\sum_iE_i\). They are included in \(J\), not discarded. This also proves directly that the ambient constant on a zero-dimensional center suffices in (53).

It remains to describe the lifts needed for transport. Pullbacks of ambient classes on \(M\) extend by pullback through the normal-cone families and their inserted ruled levels. Each exceptional generator in (53) is \(E\) multiplied by an ambient pullback. In the normal-cone expansion along \(E\), the divisor \(E\) follows into the nonjoining section of the final ruled component. Use the class Gysin-pushed from that section, multiplied by the corresponding ambient class. It is disjoint from the joining boundary, and hence its restriction there is zero. Give it zero restriction on all other pieces. These are the specializations of the indicated absolute classes, as can be seen from the strict transform of \(E\times\mathbb A^1\) in the deformation to its normal cone. The same construction after inserting additional levels places the class at the unchanged final section. At a surgery it remains on the segment containing that end, and is zero on segments from which the end was removed. Thus all retained component restrictions agree in the comparisons required by 29. ◻

Proof. If a positive-dimensional complete intersection has dimension different from two, its integral curve homology is detected by the hyperplane degree. For curves this follows from \(H_2(Y,\mathbb Z)=\mathbb Z\); for dimension at least three it follows from the integral Lefschetz theorem. There is no torsion ambiguity. Complete-intersection surfaces are simply connected, so Poincare duality and the universal coefficient theorem also show that their \(H_2\) is torsion free.

For a complete-intersection surface with all linear equations eliminated, the very general rational algebraic cohomology is ambient unless the surface is \(\mathbb P^2\), a quadric, a cubic, or an intersection of two quadrics. This is the complete-intersection Noether–Lefschetz statement used in (Argüz et al. 2023, sec. 5.2). In the nonexceptional case, ambient degree therefore distinguishes all effective classes on a very general fiber. A class with a nonzero primitive part is not algebraic there and cannot carry a stable map. Deformation invariance makes its ordinary invariant zero also on any special smooth fiber, with the curve class and insertions transported locally. An effective disconnected total is itself an algebraic class, so this argument applies to disconnected coefficients as well.

For the exceptional surfaces, all of \(H^2\) has type \((1,1)\). Their primitive integral lattice is negative definite by Hodge–Riemann. The monodromy, which preserves that lattice and the hyperplane class, is consequently finite. The local monodromy of (51) is also unipotent, since the total space is smooth and the central fiber is semistable. It is therefore the identity. In these exceptional-surface splittings the seam \(D\) is a line or a smooth conic, hence \(H^1(D,\mathbb Q)=0\); the projective-plane base case needs no splitting. The Mayer–Vietoris sequence consequently identifies \(H^2(U\cup_D\widetilde M,\mathbb Q)\) with the kernel of the difference of the two restrictions to \(H^2(D,\mathbb Q)\). The invariant cycle theorem, with trivial monodromy, supplies a central-fiber lift of every rational divisor class on the nearby surface. Its image in this kernel is a matching pair \[ (\delta_U,\delta_{\widetilde M})\in H^2(U,\mathbb Q)\oplus H^2(\widetilde M,\mathbb Q),\qquad \delta_U|_D=\delta_{\widetilde M}|_D, \tag{57}\] modulo the opposite boundary-divisor relation. One way to see the Hodge type of these representatives is to use the column-zero weight-complex description: restriction and Gysin are Hodge morphisms, and strictness permits a type-\((1,1)\) representative for a pure invariant divisor class. Equivalently, one may take closures of invariant divisors in the resolved family.

These representatives extend coherently across surgery. Keep the class on each retained end, and on every ruled neck and cap take the pullback of the common class on \(D\). On each shortened smoothing these are the usual extensions in the deformation to a normal cone. Opposite boundary twists change a representative, but their contributions to the sum of degrees cancel at a matching contact. A basis of \(H^2(X,\mathbb Q)\) among these tests distinguishes all integral curve classes on the exceptional surface, since the latter lattice has no torsion.

Finally, for \(\widetilde M=\mathop{\mathrm{Bl}}_S M\), the integral curve lattice is \[H_2(\widetilde M,\mathbb Z) \simeq H_2(M,\mathbb Z)\oplus H_0(S,\mathbb Z).\] Retain the needed divisor degrees from \(M\) by pullback and the degrees against each connected component of the exceptional divisor. The latter are represented by the nonjoining sections at the final end of the blowup configuration; hence they are not altered by the cap bookkeeping at an internal surgery. The pullback classes distinguish the first summand, and exceptional intersection distinguishes the second, with nonzero diagonal coefficient on its fiber generators. If \(M\) is a nonexceptional surface we choose it very general; if it is exceptional we retain all its divisor degrees. The hyperplane degree and these additional tests are among the compatible degrees of 29. Finiteness at a fixed hyperplane and exceptional bound is part of that theorem. This proves the claim. ◻

Proof. Work locally in the smooth parameter space, using a topological trivialization of the cohomology and curve-class local systems. The ordinary Gromov–Witten evaluation tensors are flat morphisms of Hodge structures with fixed Tate shifts. The cup pairing and the ambient classes are flat, as are the ambient and exceptional subspaces in (53). Thus their orthogonal projectors are flat. Connected components of a zero-dimensional center may be labelled locally; their possible global permutation is irrelevant.

Fix a coefficient error and its Hermitian-square test from 36. All contractions in that test have balanced Hodge type. The only nonflat ingredients in its expression in a flat cohomology trivialization are first-Hodge-index projectors and functions of them. The derivative of such a projector is first-index off-diagonal: differentiating its idempotence and its constant eigenspace decomposition shows that its diagonal blocks vanish. Replacing one projector by its derivative in the balanced contraction changes the total first Hodge degree while leaving all other tensor morphisms at their fixed type. The resulting scalar is zero. The product rule therefore shows that the Hermitian-square scalar is locally constant. This is the same Hodge-balance argument used for numerical tests in the degeneration calculation.

The square vanishes on the very general fibers under consideration, so it vanishes throughout the connected smooth parameter space. Positivity in 36 proves vanishing of the error on each fiber, for all its primitive inputs. The ambient inputs follow from the same scalar argument without free primitive slots. If a curve class is nonalgebraic on a very general fiber, all its ordinary Gromov–Witten tensors are zero there; the preceding reasoning still applies. Coefficient extraction for the disconnected potential is legitimate at each fixed polarized degree and genus power by the finiteness conventions of 29. ◻

Proof of 1. A zero-dimensional smooth complete intersection is a disjoint union of points. The point constraints are the Witten–Kontsevich theorem (Witten 1991; Kontsevich 1992). For a disjoint union, the potential is the product and each Virasoro operator is the sum of the component operators, so the assertion follows in dimension zero.

Assume the assertion in dimensions below \(n\). In dimension \(n\), induct on (52). Complexity zero is projective space after eliminating linear equations. This is a toric bundle over a point, so 38 proves the base case. At positive complexity choose \(d_r\ge2\) and a splitting \(d_r=a+b\). Use (51). Both \(U\) and \(M\) have smaller complexity, and \(D,S\) have smaller dimension. Their full constraints are therefore available by induction.

We first establish the constraints for \(\widetilde M\). If \(S\) is empty, this is the assertion for \(M\) already known. Otherwise set \[E=\mathbb P_S(N_{S/M}),\qquad C_E=\mathbb P_E(\mathcal O_E\oplus\mathcal O_E(-1)),\qquad Q_S=\mathbb P_S(N_{S/M}\oplus\mathcal O_S).\] In the blowup configuration of 29, the main smoothing is \(\widetilde M\), and the three shorter target types are \(M,Q_S,C_E\). By (50), \(E\) and \(Q_S\) are projectivizations of sums of line bundles over \(S\), and \(C_E\) is one more such projectivization over \(E\). The constraints for \(S\) and 38 give the full constraints for \(Q_S\) and \(C_E\). Thus all shorter targets required by transport are known.

Apply 29 to obtain every tested positive constraint on \(\widetilde M\). Use the hyperplane degree pulled back from \(M\) and retain the additional degrees in 40, including individual exceptional degrees when the center is disconnected. The coherent subspace in 39 meets all the hypotheses of 36. Detection proves every positive constraint for \(\widetilde M\), with arbitrary insertions. The degree tests distinguish its relevant curve classes, and 41 includes special smooth choices of the nested intersections. The string and grading equations, or the string equation and Virasoro commutators with the normalization (48), supply the nonpositive constraints.

We now use the ordinary configuration for \(U\cup_D\widetilde M\). Its shorter targets are \(U\), \(\widetilde M\), and the ruled neck \[C_D=\mathbb P_D(\mathcal O_D\oplus N_{D/U}).\] The first two have just been treated and \(C_D\) has the constraints by 38 and the induction hypothesis for \(D\). All these statements include all descendants, so they remain valid after every allowed test or auxiliary insertion used by transport.

Take \(B_0=A_X\) on the smooth main target. Weak and hard Lefschetz show that \(B_0\) contains the nonmiddle cohomology and the nonprimitive middle part, and that \(B_0^\perp\) is primitive. Its classes are coherent: restrict ambient projective classes to the two ends, and pull their common restriction on \(D\) to every neck and cap. These are the natural extensions in the pencil and the normal-cone families. Theorem 29, followed by 36, proves the positive constraints for \(X\) for every coefficient measured by the compatible degrees. By 40 this gives the constraints per curve class on the very general fibers; in the exceptional surface cases all divisor degrees have been retained throughout. Lemma 41 then gives the conclusion on every smooth complete intersection of the chosen multidegree. The nonpositive constraints follow as in the blowup step.

Both induction parameters strictly decrease where invoked, and no restriction on genus or on the number or parity of primitive insertions was imposed. This proves the theorem. ◻