Givental toric complete-intersection mirror theorem

Equivariant Chow operations and degenerate targets. If $Y$ is empty its cohomology-valued series are zero. If $X$ has dimension zero, every effective curve class is zero and the asserted identity is the common classical term with zero corrections. Hence suppose $\dim X>0$ and $Y\mathrm{\ne }\mathrm{\varnothing }$. Fix a proper finite-type complex DM stack $M$ with torus $T=({\mathbb{C}}^{\ast }{)}^r$ action and a global equivariant two-term POT, with fixed virtual rank on the component under discussion. In any specified Chow degree use $U_N=({\mathbb{C}}^N\setminus 0{)}^r$ and the mixed quotient $(M\times U_N)\mathrm{/}T$, shifted by $r(N-1)$. Two good pairs compare on their product by vector-bundle pullback and localization away from an increasing-codimension bad locus. For a GL framing use the product with the full-rank matrix good pair and the explicit common affine-torsor replacements of [F3]. Thus the actual scheme proper pushforward, operational Chern and cone/zero-section Gysin operations define compatible equivariant rational-stack operations. The comparison is through the proved Chow isomorphisms of [F3], not restriction to an étale chart.

The actual fixed substack. An equivariant Hilbert-graph presentation embeds each stable-map stack used below into a smooth finite-type DM quotient: its framed quasi-projective scheme embeds equivariantly in a Grassmannian, and the open set with finite stabilizers is smooth; remove the invariant boundary so the graph locus is closed there. At a fixed geometric point a finite torus isogeny lifts the action to its étale formal chart. Exactness of torus weights chooses compatible homogeneous regular parameters; the zero scheme of its infinitesimal vector fields is cut out by the parameters of nonzero weights and is formally smooth. These intrinsic vector fields define the same closed fixed locus on overlapping charts. Intersecting with $M$ therefore represents its geometric fixed substack $F$, closed and hence proper. The finite-isogeny construction of [F5] makes the fixed coherence unique after one integer divisible by all geometric inertia orders: the connected group of pairs $(t,x\simeq tx)$ is a torus mapping isogenously to $T$, and the indicated power map lifts through it. Unramified inertia makes the geometric coherence and cocycle unique in families. Thus it is the saturated fixed convention of [F5]. Rational character spaces identify under finite isogeny, so equalities after inverting nonzero character classes descend to the original torus. No missing represented-fixed-functor hypothesis is imposed on the stable-map stack.

On $U=M\setminus F$, repeat the actual generic-gerbe support construction in the geometric proof in [F5]12–16 for cycle supports on [U/T]. On a dense generic support the stabilizer image is a proper subtorus up to a finite group. A nonzero character χ, after a positive multiple, is trivial on that stabilizer and its line becomes trivial on the generic gerbe. The operational first Chern class $c1(L_{\chi })$ consequently takes any cycle of that support to a cycle supported in its lower-dimensional complement: take a rational trivialization on the dense open and its divisor. Noetherian induction proves that a product of nonzero character classes annihilates each class of $A_{\ast }^T(U)$. Rational Chow of finite inertia introduces no additional nonzero-degree generic character class. After inverting those character classes, localization for the closed immersion i:F→M proves surjectivity of $i_{\ast }$.

The weight-zero part $E_{\mathrm{f}\mathrm{i}\mathrm{x}}\to L_F$ is a POT by the actual invariant lifting argument in the geometric proof in [F5]23. Write the moving part $E_{\mathrm{m}\mathrm{o}\mathrm{v}}=[A\to B]$, degrees−1,0, and $V=B^{\vee }$, $W=A^{\vee }$. The weight decomposition is exact on the fixed stack. The fixed map and the projection $E_{\mathrm{m}\mathrm{o}\mathrm{v}}\to A[1]$ give a compatible triangle $E{\mathrm{\mid }}_F\to E_{\mathrm{t}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{e}}:=E_{\mathrm{f}\mathrm{i}\mathrm{x}}\oplus A[1]\to B[1]$. The corresponding cotangent triangle gives a map $B[1]\to L_{F\mathrm{/}M}$; the exact cohomology diagram gives B↠I/I². Thus $C_{F\mathrm{/}M}$ embeds in V. These are statements about the geometric cotangent maps and are independent of the homology theory.

Define $i^{!}$ on equivariant Chow by deformation to the normal cone, its closed inclusion in V and zero-section Gysin: $i^{!}=0_V^{!}$ $j_{\ast }$ σ. This is the actual bivariant operation of the scheme Gysin construction [F6], transported through the good pairs and affine replacements of step 1.1. For a cycle $i_{\ast }\alpha$ from F the Rees specialization is the zero-section cycle, giving $i^{!}{i}_{\ast }\alpha =c_{\mathrm{t}\mathrm{o}\mathrm{p}}(V)\cap \alpha$ by the proved zero-section excess identity.

On coherently fixed F, the projective-bundle formula on the good pairs gives $A_{\ast }^T(F)=A_{\ast }(F)\otimes Sym(t^{\ast })$, with the ordinary degree convention in each component. For a moving bundle, its equivariant top Chern class has nonzero product-of-weights leading term; its remaining terms have positive ordinary codimension. Their powers vanish above dim F, so a finite geometric series gives its inverse after character localization. Hence $c_{\mathrm{t}\mathrm{o}\mathrm{p}}(V)$ is invertible and $c_{\mathrm{t}\mathrm{o}\mathrm{p}}(V{)}^{-1}{i}^{!}$ is a left inverse to $i_{\ast }$. Together with step 3.1 this proves concentration without an unproved equivariant Chow localization theorem.

The virtual-cone comparison is $i^{!}[M{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=[F,E_{\mathrm{t}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{e}}{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=c_{\mathrm{t}\mathrm{o}\mathrm{p}}(W)\cap [F,E_{\mathrm{f}\mathrm{i}\mathrm{x}}{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$. Here is the exact comparison construction. Use the actual intrinsic deformation of M, and on F×P1 form $\nu :E{\mathrm{\mid }}_F\otimes O(-1)\to E{\mathrm{\mid }}_F\oplus E_{\mathrm{t}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{e}}$ with ν=(z0 id,z1 u). Its mapping cone maps to the corresponding cotangent cone. Locally, for $F=\mathrm{Spec}(A\mathrm{/}I)$ inside Spec(A/J), the ideal $K=(I,J\mathrm{/}t)$ in A[t,J/t] has K/K²=((J/IJ)[t]⊕(I/I²)[t])/(tj,−jbar); the differential is (j,i)↦dj+t di. Eliminating the identity cotangent summand of cone((t id,f)) gives this same module. It includes possible t-torsion. Functoriality in universal derivations and smooth-embedding product refinements glues this identity to the intrinsic normal-sheaf stacks, exactly as the geometric proof in [F5]31–34. The global mapping cone has a two-term bundle presentation because its degree−2 term injects through (z0 id,z1 id), with locally free Euler-sequence cokernel, as in the geometric proof in [F5]45.

Apply the actual Chow specialization/cone-pushforward/zero-section operation to this normal family. Refined parameter fibres commute with these pushed operations by the scheme bivariant Gysin theorem and the stable-map transport117–126,134–141. In particular this uses the surjection of Rees algebras after an arbitrary base change and compares the pushed classes; it NEVER identifies raw normal cones under a nonflat base change. At z1=0 the cone cancels $E{\mathrm{\mid }}_F$ and leaves $E_{\mathrm{t}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{e}}$. At z0=0 it is $E{\mathrm{\mid }}_F\oplus B[1]$. Interchanging its two zero-section operations gives $i^{!}[M{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$ at this fibre. The two refined fibre classes agree: their Cartier divisors on P1 differ by a rational function; proper pushforward and zero-section Gysin preserve that rational equivalence. Finally the extra zero-obstruction summand A[1] contributes exactly $c_{\mathrm{t}\mathrm{o}\mathrm{p}}(A^{\vee })=c_{\mathrm{t}\mathrm{o}\mathrm{p}}(W)$ by the zero-section excess formula. This proves7.1 using Chow operations themselves, not a K-to-Chow transformation.

Combining steps step 6.1–step 8.1 gives $[M{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=i_{\ast }([F{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}\mathrm{/}{e}_T(N^{\mathrm{v}\mathrm{i}\mathrm{r}}))$ with $N^{\mathrm{v}\mathrm{i}\mathrm{r}}=V-W$ and $e_T(N^{\mathrm{v}\mathrm{i}\mathrm{r}})=c_{\mathrm{t}\mathrm{o}\mathrm{p}}(V)\mathrm{/}{c}_{\mathrm{t}\mathrm{o}\mathrm{p}}(W)$. Decompose F into connected rank-constant components before using each Euler factor; do not assume one global virtual dimension. Perfect twists follow by operational projection formula, and proper equivariant pushforward is available without localization because M is proper. The formula therefore applies in the original full torus variables, checked after a generic character/isogeny if convenient.

All needed marked graph spaces. For ${\bar{\mathcal{M}}}_{0,n}(X,\beta )$ and $G_{\beta }={\bar{\mathcal{M}}}_{0,0}({\mathbb{P}}^1\times X,(1,\beta ))$, the Artin stack of prestable domains is smooth for every $n$: the nodal hypersurface resolution has no degree-two obstruction and the normalization calculation gives its tangent and automorphism complex. The relative universal-differential and trace construction in [F3] uses $(R{\pi }_{\ast }{f}^{\ast }{T}_X{)}^{\vee }$ and the absolute cotangent triangle over that Artin base. Stability of the map makes infinitesimal automorphisms zero and cancels the possible degree-one bundle, giving amplitude $[-1,0]$ also for positive target degree with $n=0,1,2$. The line ${\omega }_C(\text{markings})\otimes f^{\ast }{\mathcal{O}}_X(3)$ and a sufficiently divisible power give the same framed Hilbert-graph presentation as [F3]; for graph spaces use the product ample line. All maps and sections are equivariant. Global equivariant resolutions exist concretely on the framed quasi-projective presentation: twist an equivariant coherent presentation by that ample line, choose finite generators and then their finite-dimensional rational $T\times \mathrm{G}\mathrm{L}$ spans, and truncate the resulting resolution using the proved perfect amplitude. The syzygy at the truncation is locally free. The affine strictification in [F3] provides strict maps when required. For these stacks the virtual ranks are respectively $\dim X-3+n+c_1(T_X)\cdot \beta$ and $\dim X+c_1(T_X)\cdot \beta$. Nonconvexity is retained through $H^1(C,f^{\ast }{T}_X)$; none of these groups is set to zero. No nonexistent stable-curve stabilization with two markings is used.

Put $E=\oplus L_a$. For a genus-zero tree every $L_a$ has nonnegative degree on each component. Leaf-removal of the normalization sequence gives $H1(C,f\ast L_a)=0$ and surjectivity of evaluation at any chosen smooth marking. Hence $E_{\beta }={\pi }_{\ast }f\ast E$ is locally free and commutes with every base change; $E_{\beta ,x}^{\mathrm{\prime }}=\ker (E_{\beta }\to e{v}_x\ast E)$ is also locally free. The same proof holds on graph spaces. It provides honest equivariant Euler twists and the normalization $e(E_{\beta })=e(E_{\beta ,x}^{\mathrm{\prime }})e{v}_x\ast e(E)$. No division by e(E) is made after nonequivariant specialization.

The source-normalized twisted series. For the bundle $E={⨁}_a{L}_a$, set $E_{\beta }={\pi }_{\ast }{f}^{\ast }E$ and $E_{\beta ,1}^{\mathrm{\prime }}={\pi }_{\ast }(f^{\ast }E(-x_1))$. The genus-zero tree calculation in step 11.1 gives the exact sequence $0\to E_{\beta ,1}^{\mathrm{\prime }}\to E_{\beta }\to {\mathrm{ev}}_1^{\ast }E\to 0$. Define $S(q,z)=1+{\sum }_{\beta \mathrm{\ne }0}{q}^{\beta }({\mathrm{ev}}_1{)}_{\ast }(e(E_{\beta ,1}^{\mathrm{\prime }})\mathrm{/}[z(z-{\psi }_1)]\cap [M_{0,1}(X,\beta ){]}^{\mathrm{v}\mathrm{i}\mathrm{r}})$. It represents the corrected two-point covector of [F1] in the Euler pairing $(a,b{)}_E={\int }_{X}abe(E)$. Indeed the forget-marking-2 map has a section divisor $D$ and ${\psi }_1={\pi }^{\ast }{\psi }_1+D$, with $D^2=-{\pi }^{\ast }{\psi }_{1}D$. Binomial expansion gives ${\pi }_{\ast }{\psi }_1^m={\psi }_1^{m-1}$ for $m\ge 1$, whereas ${\pi }_{\ast }1=0$. The evaluation and $E_{\beta ,1}^{\mathrm{\prime }}$ pull back, so summing ${\sum }_m{\psi }_1^m{z}^{-m-1}$ gives $1\mathrm{/}[z(z-{\psi }_1)]$. The section-divisor relation follows by writing the nodal contraction at the collision of the forgotten marking, exactly the universal dualizing-line calculation in [F3]. Therefore $S=1+O(z^{-2})$, and the full divisor/unit exponential gives the source-normalized $J$ covector. Test class and descendant are both at marking 1; no transposition of the two-point column is made.

Construct the graph universal line without the separate regular-contraction theorem. Let g:C→P1 be the graph parameter map and f:C→X, with universal projection π. For a T-linearized line L on X define $P_L=\det$ $R{\pi }_{\ast }f\ast L\otimes (\det R{\pi }_{\ast }(f\ast L\otimes g\ast O_{{\mathbf{P}}^1}(-1)))$^{-1}. These are determinant lines of the actual perfect curve pushforwards already constructed in the stable-map construction [F3]. Choose the C*-linearization of O(−1) with weights −z at0 and0 at∞. On the C*-fixed graph locus of degrees β0,β∞ attached at0,∞, the normalization exact sequence cancels the two identical terms on the infinity tree. On the zero tree the determinant difference is $(\chi (f\ast L\mathrm{\mid }tree)-1)z=(L\cdot \beta 0)z$; the parameter component contributes the node fibre $L_x$. Therefore $c1(P_L)=e{v}_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}\ast c1(L)+(L\cdot \beta 0)z$. This is an equality of equivariant line restrictions, including families, obtained from determinant of the actual normalization triangle.

For divisor basis $p_i=c1(L_i)$, define $G(q,s,z)={\sum }_{\beta }{q}^{\beta }$ $p_{G\beta \ast }(e(E_{\beta })\exp ({\sum }_i{s}_{i}c1(P_{L_i}))\cap [G_{\beta }{]}^{\mathrm{v}\mathrm{i}\mathrm{r}})$. Every coefficient in q and s is a proper equivariant pushforward of a nonlocalized algebraic class. It therefore lies in Q[z,λ,λ'] and is polynomial in the torus parameters. This proves the needed polynomiality before any fixed-point computation; it is not an axiom and does not require X convex.

Graph pairing factorization. Under rotation of the parameter ${\mathbb{P}}^1$, a fixed graph map has one parameter component mapped constantly to $X$, with trees of degrees ${\beta }_0,{\beta }_{\mathrm{\infty }}$ attached at 0 and $\mathrm{\infty }$. A missing degree-zero tree is interpreted as the evaluation factor $X$. Normalization gives the fixed obstruction complex as the two tree complexes minus $T_X$ at their common node, hence its fixed virtual class is their diagonal refined Gysin product by the explicit labelled-node comparison in [F3]. At 0 the moving smoothing line is the parameter tangent of weight $z$ tensored with the branch tangent of first Chern class $-\psi$, so its Euler factor is $z-\psi$; the parameter component contributes the factor $z$. At $\mathrm{\infty }$ they are $-z-\psi$ and $-z$. These are exactly the denominators of the two series in step 12.1. Normalization of $f^{\ast }E$ subtracts the common node fibre, so $e(E_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}})=e(E_{{\beta }_0})e(E_{{\beta }_{\mathrm{\infty }}})\mathrm{/}e(E_x)$; replacing both factors by their evaluation kernels leaves one factor $e(E_x)$, the Euler pairing. The determinant line in step 13.1 restricts to $p+(p\cdot {\beta }_0)z$. The localization formula of step 9.1 therefore gives $G(q,s,z)=(S(q{e}^{zs},z),e^{ps}S(q,-z){)}_E$. This equality includes the degree-zero end conventions and all finite stabilizer factors because it is the rational-stack virtual pushforward. Step step 14.1 proves that every coefficient of this paired series is polynomial in the equivariant parameters.

Cox charts and the Euler sequence. Let $\mathrm{\Sigma }$ be the smooth complete projective fan of [F11], with rays $v_1,\dots ,v_N$ in lattice $N_X$ of rank $n$. The exact sequence $M_X\to {\mathbb{Z}}^N\to \mathrm{Pic}(X)\to 0$ sends a character $m$ to $(⟨m,v_j⟩{)}_j$ and the coordinate basis to the boundary divisors. To see exactness, trivialize a line on the open torus (a Laurent polynomial ring is factorial); its rational trivializing section has divisor supported on the boundary. A principal relation supported there restricts on the torus to a unit, hence to a scalar monomial. Put $G=\mathrm{Hom}(\mathrm{Pic}(X),{\mathbb{C}}^{\ast })$, of rank $k=N-n$. For each maximal cone the rays are a lattice basis, so the divisor characters outside it are a basis of $\mathrm{Pic}(X)$. On the open set where those coordinates are nonzero, $G$ normalizes them uniquely to 1; the remaining coordinates give the corresponding affine fan chart. Gluing gives the principal quotient $U_{\mathrm{\Sigma }}\mathrm{/}G=X$. Every line bundle has a torus linearization from its boundary divisor, by these same monomial transition functions. Differentiating the free action and descending gives the exact bundle sequence $0\to {\mathcal{O}}_X^{\oplus k}\to {⨁}_j{\mathcal{O}}_X(D_j)\to T_X\to 0$, equivariantly. For an invariant orbit line $e$ from fixed point $\alpha$ to $\gamma$, a nonconstant fixed cover is $t\mapsto t^n$ with deck group ${\mu }_n$. Pulling this sequence to the cover computes its full moving $H^0-H^1$, so negative toric divisor degrees are not discarded.

The finite edge coefficient. For an integer $b$ define $P_b(u,z)={\prod }_{m=1}^b(u+mz{)}^{-1}$ if $b\ge 0$, and $P_b(u,z)={\prod }_{m=b+1}^0(u+mz)$ if $b<0$. Put $H_{d,\alpha }(z)={\prod }_a{\prod }_{m=1}^{L_a\cdot d}(v_a(\alpha )+mz){\prod }_j{P}_{D_j\cdot d}(u_j(\alpha ),z)$. For an edge $e=\alpha \to \gamma$ with tangent weight $w=u_j(\alpha )$ and cover order $n$, put $\eta =w\mathrm{/}n$ and $b_s=D_s\cdot e$, $l_a=L_a\cdot e$. The finite coefficient is $C_{\alpha e}(n)=\frac{{\prod }_a{\prod }_{m=1}^{n{l}_a}(v_a(\alpha )-m\eta )}{(n-1)!{\eta }^{n-1}}{\prod }_{s\mathrm{\ne }j}{P}_{n{b}_s}(u_s(\alpha ),-\eta )=n{<mrow><mi\; mathvariant="normal">R</mi><mi\; mathvariant="normal">e</mi><mi\; mathvariant="normal">s</mi></mrow>}_{z=-w\mathrm{/}n}{H}_{ne,\alpha }(z)\mathrm{.}$ Here the factor $u_j(\alpha )+nz$ is removed before taking the residue; its derivative is $n$. On the cover, a line of degree $n{b}_s\ge 0$ has weights $u_s(\alpha )-m\eta$, $0\le m\le n{b}_s$, while for $n{b}_s\le -2$ its $H^1$ has the opposite-exponent weights in the intervening negative range. The tangent Euler sequence, the node evaluation and its smoothing remove the endpoint and zero tangent weights. The remaining factors are the displayed $P_{n{b}_s}$, the cover-coordinate factors ${\prod }_{m=1}^{n-1}(w-m\eta )=(n-1)!{\eta }^{n-1}$, and the convex kernel factors ${\prod }_{m=1}^{n{l}_a}(v_a-m\eta )$. The degree-$n$ deck group contributes $1\mathrm{/}n$ to the residue; equivalently the recursion coefficient is $n$ times that residue. Thus virtual localization yields $S_{\alpha }=R_{\alpha }(q,z^{-1})+{\sum }_{e,n\ge 1}{q}^{ne}\frac{C_{\alpha e}(n)}{nz+w}{S}_{\gamma }(q,-w\mathrm{/}n)\mathrm{.}$ The cotangent class on a contracted marked component has weight zero and is nilpotent on its finite-dimensional fixed moduli, so it contributes the polynomial remainder in $z^{-1}$. This computes the coefficient from the full moving complex, including negative-degree $H^1$, instead of assuming a convex target.

The finite-dimensional toric comparison. Write $d_j=D_j\cdot d$. Replace Cox coordinate $j$ by the coefficient space $H^0({\mathbb{P}}^1,\mathcal{O}(d_j))$, taken as zero for $d_j<0$, with its original $G$ character on each coefficient. Use the chamber of an ample divisor of $X$. A point is stable exactly when its nonzero coefficient-character set contains one of the original chamber bases: this is the original fan-chart monomial criterion, with repeated identical columns, so it introduces no new stability walls. Each such basis has determinant $\pm 1$ and normalizing the chosen nonzero pivot coefficients gives a smooth affine quotient chart. Its quotient $X_d$ is therefore smooth. It is projective: the functional $D\mapsto A_X^{n-1}D$ is strictly positive on each nonzero boundary divisor, so every nonconstant coefficient monomial has positive weight under this functional; the degree-zero invariant ring is $\mathbb{C}$. The ample-character semigroup ring is finitely generated: triangulate its rational polyhedral cone and reduce lattice points to the finitely many points in the fundamental parallelepipeds. Its Proj is the quotient, projective over $\mathbb{C}$. Define the obstruction bundle $W_d$ from the coefficient representations $H^1({\mathbb{P}}^1,\mathcal{O}(d_j))$, and $V_d$ from $H^0({\mathbb{P}}^1,\mathcal{O}(L_a\cdot d))$, with the induced rotation and torus weights. Set $[Y_d]=e(W_d)e(V_d)\cap [X_d]$. This is an actual Euler-weighted class on a smooth projective quotient; it is not being identified with the stable-map stack.

Polynomiality of the hypergeometric pairing. The rotation/target-torus fixed points of $X_d$ choose a chamber basis $\alpha$ and coefficient indices $r_i$ between 0 and their corresponding $d_i$. They split $d=d_0+d_{\mathrm{\infty }}$ with both endpoints in the dual cone of that basis. Normalize the pivot coefficients as in step 18.1. The remaining coordinate tangent weights are the factors $u_s+mz$; $W_d$ restores the $H^1$ factors. At a fixed point the two finite product ranges join to $-D_s\cdot d_{\mathrm{\infty }}\le m\le D_s\cdot d_0$ for a toric coordinate and $-L_a\cdot d_{\mathrm{\infty }}\le m\le L_a\cdot d_0$ for a bundle summand. The shared $m=0$ factors are respectively the tangent Euler denominator and the single Euler-pairing numerator $e(E_{\alpha })$; the pivot zero weights are omitted before forming the tangent Euler class, with unimodular Jacobian 1. Hence cancellation of these ranges gives ${\sum }_d{q}^d{\int }_{[Y_d]}{e}^{ps}=(H(q{e}^{zs},z),e^{ps}H(q,-z){)}_E\mathrm{.}$ This is equality coefficient by coefficient of finite products obtained from the isolated-fixed-point formula of step 9.1 with the trivial smooth POT and the specified Euler twists. The left side is a nonlocalized proper equivariant integral on $X_d$, so every coefficient is polynomial in $z$ and the torus parameters. This proves the second polynomiality identity without identifying polynomial-map and stable-map spaces.

Equality of the recursions. At the edge pole $z_0=-w\mathrm{/}n$ the toric weights satisfy $u_s(\gamma )=u_s(\alpha )-(D_s\cdot e)w$ and $v_a(\gamma )=v_a(\alpha )-(L_a\cdot e)w$: these are the endpoint weights of the same linearized line on the orbit ${\mathbb{P}}^1$. In the finite factors of step 17.1 the shift $m\mapsto m+n{D}_s\cdot e$ and its bundle analogue therefore cancel all ranges depending on the remaining degree. Their residue is exactly ${<mrow><mi\; mathvariant="normal">R</mi><mi\; mathvariant="normal">e</mi><mi\; mathvariant="normal">s</mi></mrow>}_{z=z_0}{H}_{d+ne,\alpha }(z)=\frac{C_{\alpha e}(n)}{n}{H}_{d,\gamma }(z_0)\mathrm{.}$ If the shifted degree lies outside the endpoint dual cone, a zero endpoint factor kills the proposed residue; conversely every nonzero pole corresponds to one such endpoint and cover. The two exchanged pivot divisor degrees on $e$ are 1 and all shared pivot degrees are 0, so this finite-interval cancellation accounts for every pole. A dual chamber cone lies in the effective cone, and subtracting the edge in a surviving residue stays in that endpoint cone. Partial fractions thus give precisely the recursion of step 17.1 for $H$, with the same $C_{\alpha e}(n)$ and a Laurent-polynomial remainder. The nonnegative anticanonical-minus-bundle degree in the Statement makes this a polynomial in $z^{-1}$: after finite cancellation each degree has nonpositive large-$z$ order. The two constructed series now have the common edge recursion and the separate polynomial paired series required for uniqueness.

Convex zero-locus comparison and its exact scope. The graph-space package is established by step 20.1; we now construct its required geometric zero-locus specialization. Put $E={⨁}_a{L}_a$, $r=\mathrm{rank}$ E and $n_X=\dim$ X, and let s be the regular defining section with smooth zero locus i:Y→X. For $\beta \mathrm{\ne }0$ fix $M=overline$ $M_{0,2}(X,\beta )$, with universal curve pi and map f. Set $R_{\beta }=r+c1(E)\mathrm{.}\beta$, $v_X=n_X-1+c1(T_X)\mathrm{.}\beta$ and $v_Y=v_X-R_{\beta }$. These ranks are constant on the fixed-class stacks. For a Y-class gamma with $i_{\ast }\gamma =\beta$, adjunction gives $dimY-1+c1(T_Y)\mathrm{.}\gamma =v_Y$, so the same rank applies to every contributing gamma. These fixed-class obstruction theories therefore define homogeneous classes of the displayed ranks.

For a geometric stable-map domain C, its irreducible components are copies of P1 and its dual graph is a tree. Every $f^{\ast }{L}_a$ has nonnegative degree on each component. Remove an end component and use the normalization exact sequence at the attaching node: H1 on the end is zero and its evaluation at that node is onto. Induction gives $H1(C,f^{\ast }{L}_a)=0$. Root the tree at a marked point $x_k$ and repeat with $f^{\ast }{L}_a(-x_k)$; all removed ends still have nonnegative degree, and the final root has degree at least -1. Thus $H1(C,f^{\ast }E)=H1(C,f^{\ast }E(-x_k))=0$ for $k=1,2$.

The universal direct image $R{\pi }_{\ast }{f}^{\ast }E$ is perfect of Tor-amplitude [0,1]. In a finite-free local model its last differential is surjective on every geometric fibre by step 22.1, hence is split surjective on a neighbourhood by the finite-free criterion; contracting this split pair leaves a locally free module in degree0. It follows that $E_{\beta }={\pi }_{\ast }{f}^{\ast }E$ is a vector bundle of rank $R_{\beta }$ and commutes with arbitrary base change. The same argument for $f^{\ast }E(-x_k)$, followed by the evaluation exact sequence, gives the actual bundle exact sequence $0\to E_{\beta ,k}^{\mathrm{\prime }}\to E_{\beta }\to e{v}_k^{\ast }E\to 0$, where $E_{\beta ,k}^{\mathrm{\prime }}={\pi }_{\ast }(f^{\ast }E(-x_k))$ has rank c1(E).beta.

Pull s to the universal curve and push it forward to a section $s_{\beta }$ of $E_{\beta }$. After every base change T→M, step 23.1 identifies its restriction with the actual section $f_T^{\ast }s$ on the universal curve $C_T$. This section is zero if and only if the universal map factors through the scheme-theoretic zero locus Y. Stability and automorphisms are unchanged under the closed immersion Y→X. Consequently the zero stack $Z(s_{\beta })$ is exactly the stack of all Y-maps whose pushed-forward curve class is beta, including families and nilpotents. We denote its closed map to M by j. Its virtual class is the sum of the classes on its Y-class components, not a choice of a single class gamma.

The smooth regular zero locus has the exact tangent sequence $0\to T_Y\to i^{\ast }{T}_X\to i^{\ast }E\to 0$, whose last arrow is ds. Applying $R{\pi }_{\ast }$ over the common prestable-domain base, using step 23.1 and dualizing, gives $E_{\beta }^{\vee }\mathrm{\mid }Z\to E_X\mathrm{\mid }Z\to E_Y\to E_{\beta }^{\vee }\mathrm{\mid }Z[1]$. The arrows agree with the universal-differential obstruction morphisms because they are obtained by applying the differential of the same universal map and the actual ds. The Artin-domain complex is common on both sides; adding it and performing the stable-map infinitesimal-automorphism cancellation gives the same triangle for the absolute POTs. Thus the induced section-cut obstruction theory is the genuine Y stable-map theory. It is not merely an abstract equality of ranks.

Here the marked domain has genus0 and two markings. The public stable-curve-range conclusion in [F15] is not invoked to define a nonexistent stabilization to overline $M_{0,2}$. Instead use [F18] and the actual universal-differential construction in [F15] over the smooth Artin stack of prestable two-pointed nodal curves. The arbitrary-base nodal trace constructs $(R{\pi }_{\ast }{f}^{\ast }{T}_X{)}^{\vee }\to L_{M\mathrm{/}prestable}$; stable-map infinitesimal-automorphism vanishing cancels the possible degree1 term of the Artin cotangent complex, yielding the absolute amplitude[-1,0] POT and rank $v_X$. These steps do not use stabilization. The same construction gives $E_Y$ and rank $v_Y$. Only evaluations and cotangent-line Chern classes are used below. At $\beta =0$ the J-function uses its prescribed classical term, not a virtual integral on an unstable two-point constant-map stack.

We prove the section-cut virtual-class identity with the actual conormal operation. Use the framed matrix-scheme replacement of [F15] in the dimensions at issue, then its two affine-bundle torsors. Their Chow pullbacks are injective isomorphisms with the stated dimension shifts, and the POT is pulled back with the corresponding smooth tangent terms. On the resulting affine scheme M' the proof in [F15] supplies an embedding into a smooth quasi-projective scheme P and a strict conormal model $[F^{\vee }\to {\mathrm{\Omega }}_P\mathrm{\mid }{M}^{\mathrm{\prime }}]\to [I\mathrm{/}{I}^2\to {\mathrm{\Omega }}_P\mathrm{\mid }{M}^{\mathrm{\prime }}]$, with $F^{\vee }\to I\mathrm{/}{I}^2$ onto. Its class is $c(M^{\mathrm{\prime }},P,F)[P]=0_F^{!}[C_{M^{\mathrm{\prime }}\mathrm{/}P}]$. Smooth tangent shifts are retained throughout the calculation and cancel on returning to M.

There is an explicit ambient extension of the pulled-back $E_{\beta }$. Since M' is affine, its finite projective module has finitely many generators, giving a quotient $O_{M^{\mathrm{\prime }}}^N\to E_{\beta }$ and a classifying map $M^{\mathrm{\prime }}\to Gr(R_{\beta },N)$. Replace the embedding by its graph in $P\times Gr(R_{\beta },N)$. The universal quotient bundle extends $E_{\beta }$ to that smooth ambient. Apply the same two affine-bundle torsors of [F15] to this ambient; its inverse image is now a smooth affine ambient P'', and its closed centre M'' is affine. The extended vector bundle is locally free on P''. Exactness of global sections of quasi-coherent sheaves on P'' lifts the section $s_{\beta }$ from M'' to a section $\tilde{s}$ on P''. On this new affine centre rerun the strict conormal construction of [F15] for the pulled-back fixed POT and its new smooth ambient; include the actual smooth tangent summands introduced by the affine-bundle torsors. The embedding/refinement comparison of [F15] identifies its class with the original one. This construction uses no unproved global extension of a bundle or of a section.

On M''×A1 form $W=Z(t\ast s_{\beta })$, a closed subscheme of P''×A1. If I is the ideal of M'' and $({\tilde{s}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\tilde{s}}_R)$ are the local components of the lifted section, the ideal of W is $J=I+(t\ast {\tilde{s}}_1,\mathrm{.}\mathrm{.}\mathrm{.},t\ast {\tilde{s}}_R)$. The existing map $F^{\vee }\to I\mathrm{/}{I}^2$, followed by $I\mathrm{/}{I}^2\to J\mathrm{/}{J}^2$, together with the section components, gives a surjection (F $\oplus$ $E_{\beta }{)}^{\vee }\mathrm{\mid }W\to J\mathrm{/}{J}^2$. Its differential is the actual differential of these equations relative to A1. It therefore gives the strict relative POT and the conormal operation c(W,P''×A1,F $\oplus$ $E_{\beta }$). The actual $R{\pi }_{\ast }(ds{)}^{\vee }$ map from $E_{\beta }^{\vee }$ and the lifted ambient-section differential are lifts of the same $d(s_{\beta })$ to this fixed POT. On the affine zero locus at $t=1$, using derived restrictions of the fixed POT, $Cone(E_M\to L_M)$ has cohomology in degrees at most -2 and $E_{\beta }^{\vee }$ is finite projective. Therefore $\mathrm{Hom}(E_{\beta }^{\vee },Cone)=\mathrm{Hom}(E_{\beta }^{\vee },Cone[-1])=0$: these two derived lifts exist uniquely and are homotopic. The induced cone POT is consequently the actual theory from step 25.1. If the lift of the section changes by elements of I, the models are related by the corresponding triangular conormal homotopy/refinement from [F15]; the construction is independent of that choice.

Let alpha be the proper pushforward to M''×A1 of c(W,P''×A1,F $\oplus$ $E_{\beta })[P^{\mathrm{\prime }\mathrm{\prime }}\times A1]$. The pushforward exists because W is closed in M''×A1; no properness of the affine replacement over a point is needed. For $a=0$ or1, the base inclusion a→A1 is regular and P''×A1 is smooth over A1. The exact bivariant cone identity of [F16] gives refined restriction equal to the conormal operation for the fibre. Proper-pushforward compatibility therefore identifies $a^{!}\alpha$ with the pushforward of that fibre's virtual class. This does not assert that normal cones commute with nonflat base change and does not require W→A1 flat.

Homotopy invariance for the vector bundle M''×A1→M'' identifies its refined zero and one fibres on every Chow class, so $0^{!}\alpha =1^{!}\alpha$. At $t=0$ the underlying fibre is M'', and the $E_{\beta }$ summand of its conormal map is zero. Its cone factors through F inside F $\oplus$ $E_{\beta }$. The vector-subbundle excess identity of [F16] thus gives $0^{!}\alpha =c_{R_{\beta }}(E_{\beta })$ cap $[M^{\mathrm{\prime }\mathrm{\prime }}{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$. At $t=1$ the conormal model is the section-cut theory, identified with the actual Y-map theory by step 25.1, so $1^{!}\alpha =j_{\ast }[Z{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$ on the same replacement. We have proved $j_{\ast }[Z{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=c_{R_{\beta }}(E_{\beta })$ cap $[M^{\mathrm{\prime }\mathrm{\prime }}{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$ there.

All the operations just used are the scheme operations transported through the matrix and affine-bundle replacements of [F15]. On common replacements, the explicit smooth-coordinate and strict-POT-refinement comparisons there identify the conormal morphisms, cycles and zero-section operations. Their Chow pullbacks are injective isomorphisms, including after the closed section base changes used above. Transporting the equality back therefore gives the global rational-stack identity $j_{\ast }[Z{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=c_{R_{\beta }}(E_{\beta })$ cap $[M{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$. In particular rational stabilizer multiplicities are retained. This is a global transported identity, not equality inferred only on an etale atlas.

The marked curve is unchanged under j, so both evaluation maps and cotangent lines pull back from M. Every ambient insertion here is a polynomial in restricted toric divisor classes and hence is an operational algebraic class. Proper projection formula applied to step 32.1 gives, for each ambient test a, ${\sum }_{i_{\ast }\gamma =\beta }{\int }_{[{\bar{\mathcal{M}}}_{0,2}(Y,\gamma ){]}^{\mathrm{v}\mathrm{i}\mathrm{r}}}\frac{{\mathrm{ev}}_1^{\ast }(e^{i^{\ast }\tau \mathrm{/}z}{i}^{\ast }a){\mathrm{ev}}_2^{\ast }(1)}{z-{\psi }_1}={\int }_{[M{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}}\frac{{\mathrm{ev}}_1^{\ast }(e^{\tau \mathrm{/}z}a){\mathrm{ev}}_2^{\ast }(1)e(E_{\beta })}{z-{\psi }_1}\mathrm{.}$ Expand the denominator formally; dimensional degree makes each integral coefficient finite. The ample-degree completions make the sum over pushed-forward Y-classes well-defined as in the Statement. No injectivity of i_* on effective curve classes is asserted.

In the source-normalized convention of [F19] the test insertion is at marking1, the same marking as psi1. By step 23.1, $e(E_{\beta })=e(E_{\beta ,1}^{\mathrm{\prime }})$ $ev{1}^{\ast }e(E)$. Consequently the normalized Euler-twisted vector is the ev1 pushforward with $e(E_{\beta ,1}^{\mathrm{\prime }})$; pairing this vector with a by ${\int }_X$ a*(-)*e(E) restores precisely the full $e(E_{\beta })$ in step 33.1. The unit remains at marking2. No unpaired-vector formula with the full $e(E_{\beta })$, no interchange of the two test markings, and no division by ordinary e(E) is used.

At curve class0 the classical identity ${\int }_Y$ $i^{\ast }(ab)={\int }_X$ ab e(E) identifies the Euler-twisted pairing with the specified ambient restriction pairing. At nonzero classes, step 33.1 identifies the corresponding paired descendant coefficients. Nondegeneracy of the ambient restriction pairing in the Statement thus yields $J_{E,noneq}={\mathrm{pr}}_{\mathrm{a}\mathrm{m}\mathrm{b}}({\rho }_i(J_Y))$. The conclusion is only the ambient component; it supplies no identification of primitive cohomology components.

For completeness the parameter specialization is also defined before inverting Euler classes. Choose a finite free torus approximation $U\to S=U\mathrm{/}T$ large enough in each degree and the associated smooth projective family $X_U\to S$, with the descended equivariant $E_U$. Its relative stable-map POT is constructed by the same universal differential, trace and Artin-domain cancellation as step 26.1; the fixed virtual ranks are unchanged. The smooth-family/base-Gysin comparison in [F15] gives the ordinary fixed-rank theory after pullback to a point of S. The Euler-weighted correlator is proper pushforward of the actual Euler/Chern and descendant insertions to S, so it is an equivariant Chow class, polynomial in the equivariant parameters in each relevant degree. Compatible approximations define the paired covector without a localization denominator.

Forgetting the torus action, equivalently the base fibre pullback in step 36.1, sends equivariant Chern and cotangent classes to their ordinary classes and gives the covector in step 33.1. Although an inverse equivariant twisted-pairing matrix can have parameter denominators, we first specialize these polynomial paired covectors and then pass to the pairing-radical quotient identified with ambient cohomology. Thus their nonequivariant specialization is exactly the vector in step 35.1 and is not defined by setting parameters to zero in an inverse Euler class. A T-invariant defining section of Y is unnecessary: only X and E were used equivariantly, and the ordinary zero-locus comparison uses the given regular section.

Finally the equivariant hypergeometric coefficient specializes to the stated $I_Y$ by finite-product cancellation. Nefness makes every E numerator finite with nonnegative upper bound. A positive toric divisor degree contributes denominator factors $u_j+mz$ with $m\ge 1$, invertible as formal Laurent z series; a negative toric divisor degree contributes only finite numerator factors after cancellation. Therefore putting equivariant parameters to zero is defined coefficientwise and yields the stated ordinary hypergeometric coefficient. Together with step 37.1 this proves the convex quantum-Lefschetz ambient specialization used by the mirror comparison, under exactly its stated convex-bundle, smooth-zero-locus, pairing and completion conventions.

The weighted large-parameter terms. Let $\kappa =c_1(T_X)-c_1(E)$ and give $z$, the equivariant weights and divisor classes degree 1 and $q^d$ degree $\kappa \cdot d$. In each finite factor of step 17.1 count numerator and denominator powers of $z$. A negative toric divisor degree contributes a factor $u_j$ at $m=0$, lowering the large-$z$ order by one. Since $\kappa \cdot d\ge 0$, no positive $z$ power survives. Consequently $H=H_0+H_1\mathrm{/}z+O(z^{-2})$. The term $H_0$ is the scalar series ${\sum }_{\kappa \cdot d=0,D_j\cdot d\ge 0\mathrm{\forall }j}{q}^d{\prod }_a(L_a\cdot d)!\mathrm{/}{\prod }_j(D_j\cdot d)!$, with constant term 1. The coefficient $H_1$ is a degree-one polynomial in divisor and torus parameters: degrees with $\kappa \cdot d=1$ give a scalar, and degree-zero terms give a linear combination of the $u_j$ and $v_a$ from the next factor coefficient, including the single negative-coordinate zero factor. All coefficients are rational and have positive curve support apart from the unit of $H_0$.

A scalar unit $f(q)$ of weighted degree zero preserves both the recursion and paired polynomiality: it commutes with the edge recursion, and multiplies the paired series by $f(q{e}^{zs})f(q)$, whose coefficients are polynomial in $z$. A scalar series $g(q,\lambda ,{\lambda }^{\mathrm{\prime }})$ of degree one, linear in the torus parameters and with zero constant term, similarly permits multiplication by $e^{g\mathrm{/}z}$. At an edge pole, the difference between its values at $z$ and $z_0$ is divisible by $nz+w$, giving only a polynomial-in-$z^{-1}$ remainder. The paired series is multiplied by $\exp ((g(q{e}^{zs})-g(q))\mathrm{/}z)$; each coefficient is polynomial because $e^{zd\cdot s}-1$ is divisible by $z$. These are direct formal identities, not assumptions about a coordinate change.

For degree-zero divisor coefficients ${\varphi }_i(q)$ with positive curve support, the operation $Z(q,z)\mapsto e^{{\sum }_i{p}_i{\varphi }_i(q)\mathrm{/}z}Z(q{e}^{\varphi (q)},z)$ also preserves both conditions. At an edge pole, $p_i(\alpha )-p_i(\gamma )=(e\cdot p_i)w$ cancels exactly the factor $e^{ne\cdot \varphi }$ introduced in $q^{ne}$; the residual difference is divisible by $nz+w$ as in step 40.1. For the paired series the operation gives $W(s+(\varphi (q{e}^{zs})-\varphi (q))\mathrm{/}z,q{e}^{\varphi (q)})$, again polynomial coefficientwise since every numerator difference is divisible by $z$. All substitutions and their inverses exist by positive ample filtration: determine each next coefficient recursively from lower filtration, with identity linear term. Thus the inverse operation preserves the same conditions as well.

Divide $H$ by the scalar unit $H_0$. Write the resulting degree-one coefficient uniquely as $g(q,\lambda ,{\lambda }^{\mathrm{\prime }})+{\sum }_i{p}_i{\varphi }_i(q)$, using the divisor basis and the equivariant weight parameters, where $g$ is scalar and linear in those parameters plus its degree-one ordinary scalar part. All these series have zero curve-class constant term. Set $Q_i=q_i{e}^{{\varphi }_i(q)}$ and define $\tilde{H}(Q,z)=e^{-(g+{\sum }_i{p}_i{\varphi }_i)\mathrm{/}z}H(q,z)\mathrm{/}{H}_0(q)$, using the inverse formal substitution. Steps 40.1–41.1 show that $\tilde{H}$ has the common recursion and polynomial paired series, and the construction gives $\tilde{H}=1+O(z^{-2})$. The same normalization holds for $S$ by step 12.1.

Uniqueness in the actual ample filtration. If $S$ and $\tilde{H}$ differ, choose a first positive ample degree and a curve monomial $q^d$ of that degree with discrepancy $R(z)$. An edge has positive ample degree, so the common recursion cancels every nonzero pole of this coefficient by lower-degree equality. Therefore $R$ is a finite polynomial in $z^{-1}$, divisible by $z^{-2}$. The $q^d$ difference of the paired polynomial series is $\delta (R)={\int }_X{e}^{ps}(e^{z(d\cdot s)}R(z)+R(-z))e(E)$, because all cross terms involve lower curve degree. The ample completion has finite support in every bounded degree, so this induction and coefficient extraction are legitimate even for classes of $\kappa$-degree zero.

Choose generic torus parameters and a generic divisor direction $s=t\mathrm{\ell }$ so the finitely many fixed-point weights $h_{\alpha }=p(\alpha )\cdot \mathrm{\ell }$ are distinct and $d\cdot \mathrm{\ell }>0$. If $z^{2r}R=A\mathrm{/}z+B+O(z)$ for $r\ge 1$, polynomiality implies $\delta (A\mathrm{/}z+B)=O(z)$. Its constant term is ${\sum }_{\alpha }{e}^{t{h}_{\alpha }}\frac{e(E_{\alpha })}{e(T_{\alpha }X)}(t(d\cdot \mathrm{\ell })A_{\alpha }+2B_{\alpha })=0.$ The exponential polynomials $e^{t{h}_{\alpha }}$ and $t{e}^{t{h}_{\alpha }}$ are linearly independent: successive application of the differential factors $({\mathrm{\partial }}_t-h_{\gamma }{)}^2$ isolates each one, and their two coefficients then vanish. The generic Euler factors are nonzero, so $A=B=0$. Taking $2r$ successively below the largest negative degree kills the two highest negative coefficients at every step. Since $R$ has no degrees 0 or −1, it is zero. Equality for generic parameters is equality of rational functions and hence gives $S=\tilde{H}$ in the original formal coefficient algebra.

The stated triangular mirror change. Put $f_0=\log H_0$ and $f_i={\varphi }_i$. After ordinary specialization put $h=g(q,0,0)$. Then $f_0,f_i$ have weighted degree zero, $h$ has degree one, and all have positive curve support. Multiplying $S(Q,z)=e^{-(g+p\varphi )\mathrm{/}z}H(q,z)\mathrm{/}{H}_0$ by the divisor/unit exponential shows exactly $I_Y(q,t_0,z)=J_Y^{X,\mathrm{a}\mathrm{m}\mathrm{b}}(Q,t_0+f_0(q)z+h(q),z),\log Q_i=\log q_i+f_i(q)\mathrm{.}$ Indeed the $p\varphi$ from $\log Q$ cancels its normalization factor, and $e^{f_0}=H_0$ cancels the scalar denominator. The convex comparison and parameter specialization in steps 33.1–38.1 identify the paired covectors with the ambient projection of ${\rho }_i(J_Y)$; no inverse ordinary Euler class is used. Thus this is precisely the weighted homogeneous triangular substitution and common $X$-curve-class completion in the Statement.

The construction proves the two packages [A1] and [A2] at the stated smooth projective toric, nef globally generated bundle, smooth regular zero-locus and ambient-pairing hypotheses. It supplies a formal genus-zero comparison, with nonconvex target obstruction groups retained. It asserts neither primitive-cohomology equality, analytic convergence, an all-genus theorem nor an HMS equivalence. The complete conditional public Statement is therefore obtained without replacing any missing local localization or Lefschetz argument by an unproved source theorem.