Symmetric-obstruction virtual class is the degree-zero Aluffi class

We first prove the following local presentation, rather than assume it. Let X be a finite-type complex scheme with a symmetric perfect obstruction theory $E\to L_X$ of amplitude[−1,0]. Use the symmetric structure defined at the start of the Statement. Then Zariski locally at any closed point p there is a closed embedding X→M into a smooth scheme and a regular almost-closed one-form ω on M with $X=Z(\omega )$, such that the POT with its symmetric structure is isometric to $[T_M{\mathrm{\mid }}_X\to {\mathrm{\Omega }}_M{\mathrm{\mid }}_X]\to [I\mathrm{/}{I}^2\to {\mathrm{\Omega }}_M{\mathrm{\mid }}_X]$ induced by ω. For a DM stack the same conclusion is étale-local after taking an affine atlas.

Start with an affine closed embedding $X\to A^N$. Let $n=di{m}_C({\mathrm{\Omega }}_X\otimes \kappa (p))$ be its embedding dimension. Choose N−n elements of its ideal whose independent linear terms span the kernel of ${\mathrm{\Omega }}_A^N\otimes \kappa (p)\to {\mathrm{\Omega }}_X\otimes \kappa (p)$. Their zero scheme is smooth of dimension n near p by the Jacobian criterion; shrink it to obtain M, still containing X. This produces an embedding with Ω_M|_X⊗κ(p)→Ω_X⊗κ(p) an isomorphism.

Shrink X to an affine neighbourhood on which E is a complex of finite free modules. If its differential has positive rank at p, a unit minor allows row/column operations splitting off a block $[O^r\to O^r]$ with identity differential. Cancel it and repeat. The resulting two-term model has differential zero at p. These cancellations are homotopy equivalences and transport the derived symmetric structure and the POT map. Since h⁰(E)≅Ω_X, the surviving E⁰ has rank n. The symmetric quasi-isomorphism, after derived tensor with κ(p), identifies the two zero-differential dual complexes, so $E^{-1}$ also has rank n.

The truncated cotangent complex for the smooth embedding is [I/I²→Ω_M|_X]. A map from the bounded finite free complex E to this complex is represented by a chain map on our affine neighbourhood; equality of such derived morphisms is chain homotopy. This is the existing finite-projective derived-morphism property applied over the affine coordinate ring. Its degree0 map φ⁰:E⁰→Ω_M|_X is an isomorphism at p by the preceding fibre calculation, hence an isomorphism after shrinking. Identify E⁰ with Ω_M|_X and write its differential as $a:E^{-1}\to {\mathrm{\Omega }}_M{\mathrm{\mid }}_X$.

Represent θ by a chain map with components ${\theta }_{-1}:E^{-1}\to T_M{\mathrm{\mid }}_X$ and ${\theta }_0:{\mathrm{\Omega }}_M{\mathrm{\mid }}_X\to (E^{-1})\vee$. In the dual-and-shift sign convention where the differential of E∨[1] is a∨, the dual map has components θ₀∨ and θ_{−1}∨. Their difference from θ is null-homotopic by self-duality. Average the two chain maps, defining λ₀=(θ₀+θ_{−1}∨)/2 and λ_{−1}=λ₀∨. This remains a chain map, is homotopic to θ, and is strictly self-dual. At p the differential is zero, so the self-duality homotopy says the two maps already agree on fibres. Thus λ₀(p)=θ₀(p) is invertible; shrink once more so λ₀ is invertible. Its dual identifies $E^{-1}$ with $T_M{\mathrm{\mid }}_X$. The chain-map equation now says that $a:T_M{\mathrm{\mid }}_X\to {\mathrm{\Omega }}_M{\mathrm{\mid }}_X$ is self-dual. Under this identification θ is the canonical symmetric pairing, not just an abstract quasi-isomorphism.

Write ${\phi }^{-1}:T_M{\mathrm{\mid }}_X\to I\mathrm{/}{I}^2$ for the degree−1 POT map. It is surjective. Indeed the h⁰ isomorphism says im(a)=im(d:I/I²→Ω_M|_X). For any z∈I/I² choose e with a(e)=d(z). Then z−φ^{−1}(e)∈kerd; surjectivity on $h^{-1}$ supplies e'∈kera with φ^{−1}(e')=z−φ^{−1}(e). Hence z=φ^{−1}(e+e').

Shrink the smooth ambient neighbourhood and choose regular functions $x_1,\dots ,x_n$ giving an étale coordinate map $M\to A^n$; use their dual coordinate tangent basis. Lift the n images of φ^{−1} in this basis arbitrarily to $f_1,\dots ,f_n\in I$. They generate I/I², so $I=(f_i)+I^2$. Nakayama applied to $I\mathrm{/}(f_i)$ at p and then shrinking the ambient neighbourhood gives $I=(f_i)$. Thus the one-form $\omega =\mathrm{\Sigma }{f}_{i}d{x}_i$ has zero scheme exactly X. The matrix of dφ^{−1}=a is the Jacobian $(\mathrm{\partial }{f}_i\mathrm{/}\mathrm{\partial }{x}_j)$ modulo I. Its symmetry means dω∈IΩ²_M, so ω is almost closed. The chain map, its symmetric form and its map to the truncated cotangent complex are precisely those constructed, proving the isometry of POTs. All operations commute with étale restriction; for a DM stack first apply this argument to its affine étale chart.

We next prove the arc statement used below. Let B be an integral finite-type C-algebra, f∈B nonzero, and Q:B→K a field-valued point with Q(f)=0. There are a field extension L/K and a C-algebra map δ:B→Ls, specializing to Q after K→L, with $\delta (f)=s^m$ for some $m>0$.

To prove this, take L an algebraic closure of K. Because C is algebraically closed, the finite-type integral C-algebra B is geometrically integral; hence $B_L$ remains integral. The induced point $Q_L$ is L-rational. Noether normalization gives a finite dominant map Spec $B_L\to A_L^d$. The image of V(f) is a proper closed subset: finiteness preserves dimension, while a nonzero principal ideal in a domain has height-one minimal primes, so V(f) has dimension d−1. Choose a line through the image of $Q_L$ not contained in that image. Its inverse image is cut out by d−1 equations and is finite over the line. Every minimal component through $Q_L$ has dimension at least1 by the principal ideal height bound and the dimension formula, and at most1 by finiteness. Thus there is an integral curve through $Q_L$ not contained in V(f).

Normalize that curve using the existing finite normalization theorem. At a point above $Q_L$ its local ring is a DVR: the supplied normalization is a regular curve over L. Its residue field is L. Its completion is Lt: constants provide the coefficient-field inclusion, and successive subtraction of residue coefficients times powers of a uniformizer constructs the unique power-series expansion. The nonzero f maps to $u(t)t^m$ with u(0)≠0 and $m>0$. Since L is algebraically closed of characteristic0, u has a formal m-th root: choose its constant root and solve each next coefficient recursively, dividing by the nonzero scalar m times the (m−1)-st power of that root. The invertible parameter change $s=tu(t{)}^{1\mathrm{/}m}$ makes $f=s^m$. Composing gives δ. This proves the arc lemma without assuming a dominating-DVR or Cohen-structure theorem for an arbitrary local algebra.

Work étale locally on M, so $M=Spec$ A with A an integral smooth C-algebra and étale coordinates $x_1,\dots ,x_n$. Write $\omega =\mathrm{\Sigma }{f}_{i}d{x}_i$ and $I=(f_1,\dots ,f_n)$. The cone is Spec gr_I A, embedded in Spec $A[p_1,\dots ,p_n]=T\ast M$ by $p_i\mapsto f_i$ mod I². This is a graded map, hence its support is invariant under fibrewise scaling.

Consider the extended Rees algebra $D=A[t,I\mathrm{/}t]\subset A[t,t^{-1}]$. It is an integral finite-type C-algebra, with D/(t)=gr_I A and $D[t^{-1}]=A[t,t^{-1}]$. Its fraction field therefore has transcendence degree n+1. Since t is nonzero, each minimal prime of (t) has height1 by the principal ideal theorem (height0 is impossible in a domain). The affine-domain dimension formula makes every component of D/(t) have dimension n. Thus the cone is pure-dimensional of dimension n. The trivial $I=0$ case gives the zero section and is immediate. Let Q:gr_I A→K be the generic point of any reduced irreducible cone component. The canonical one-form on T*M is $\alpha =\mathrm{\Sigma }{p}_{i}d{x}_i$ and its symplectic form is dα=Σdp_i∧dx_i.

If Q vanishes on every positive-degree generator, Q*(dα)=0 immediately. Otherwise apply the arc lemma to D and $f=t$ at the point D→D/(t)→K. It gives δ:D→Ls with $\delta (t)=s^m$. Restrict δ to A, calling the resulting map γ. For every $i,f_i(\gamma (s))=s^m{a}_i(s)$, with $a_i(0)=Q(f_i<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>I^2)$, and $x_i(\gamma (0))=Q(x_i<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>I)$. Here Q values are viewed in L. These identities are immediate from the relation $t\cdot (f_i\mathrm{/}t)=f_i$ in D, not an assumption that an arbitrary cone point is a tangent direction of a smooth path.

Use continuous formal differentials over C, with their explicit coefficient derivation into the completed module Ω¹_{L/C}s⊕Lsds. This avoids asserting an identification for unrestricted algebraic Kähler differentials of a formal power-series ring. Almost-closedness and $\gamma (I)\subset (s^m)$ imply $\gamma \ast (d\omega )=\mathrm{\Sigma }d(s^m{a}_i(s))\wedge d(x_i(\gamma (s)))$ ≡0 mod $s^m$. The coefficient of $s^{m-1}ds$ in this expression is $m\mathrm{\Sigma }{a}_i(0)d_L(x_i(\gamma (0)))$. Every other term has order at least m: terms from $s^{m}d{a}_i$ do, and in the term $m{s}^{m-1}{a}_{i}ds\wedge d{\gamma }_i$ only the constant coefficient of the non-ds part contributes at order m−1. Characteristic0 makes m invertible, hence $\mathrm{\Sigma }{a}_i(0)d_L(x_i(\gamma (0)))=0$.

Taking the exterior derivative in Ω_{L/C} gives $\mathrm{\Sigma }{d}_L{a}_i(0)\wedge d_L(x_i(\gamma (0)))=0$. By the leading-coefficient identities above this is exactly the pullback Q(dα) after K→L. The map Ω¹_{K/C}⊗K L→Ω¹{L/C} is injective in characteristic0: choose a transcendence basis for the finitely generated field K/C and extend it to one for L/C; its differentials remain linearly independent, and algebraic extensions are separable. The same applies to exterior squares. Therefore Q*(dα)=0 already over K. Each reduced component has dimension n and is generically isotropic in the symplectic2n-fold T*M, so it is Lagrangian. Its conicity was established by the grading. All statements are étale-local in M and hence glue.

Let X be a finite-type separated complex DM stack with a symmetric POT $E\to L_X$ and a chosen global two-term vector-bundle resolution $E=[E^{-1}\to E^0]$. Put $F=(E^{-1})\vee$ and H=(E⁰)∨. Work on each open-and-closed locus of constant bundle rank if needed. The vector-bundle stack is [F/H], and let $C_F\subset F$ be the inverse image of the intrinsic normal cone. By the given BF construction, $[X]vir=0_F^{!}[C_F]$. Symmetry E≅E∨[1] identifies $ob=h^1(E\vee )$ with h⁰(E)=Ω_X, the latter identification being the POT map. Thus the quotient F→ob gives a global epimorphism F→Ω_X. Symmetry also makes the virtual rank zero.

For an integral reduced closed substack Z⊂X of dimension d, define $K_Z\subset F{\mathrm{\mid }}_Z$ to be the closure of the kernel bundle of $F{\mathrm{\mid }}_{Z_{sm}}\to {\mathrm{\Omega }}_{Z_{sm}}$. This has dimension rank F: over the dense smooth locus the map is surjective and its kernel has rank rank F−d. The closure is taken with its reduced structure. Define the Nash blowup ν_Z:Z̃→Z intrinsically as the closure of the smooth-locus rank-d quotient of Ω_Z in its relative Grassmannian. Equivalently, use the relative Grassmannian of rank-d quotients of F|_Z and close the section $F{\mathrm{\mid }}_{Z_{sm}}\to {\mathrm{\Omega }}_{Z_{sm}}$; the result lies in the closed sub-Grassmannian of quotients factoring through Ω_Z. The two constructions are therefore identical, including their tautological rank-d quotient $Q_Z$. Relative Grassmannians and these closed substacks are constructed on an étale scheme presentation and descend with their projective morphisms. They are representable and projective over Z; no global smooth ambient embedding is required.

Write $i_Z:Z\to X$ for the closed inclusion. Set $c^M[Z]=i_{Z\ast }{\nu }_{Z\ast }(c(Q_Z\vee )\cap [\tilde{Z}])$ in A_*(X), and extend linearly to cycles. In particular its zero-dimensional part is $i_{Z\ast }{\nu }_{Z\ast }(c_d(Q_Z\vee )\cap [\tilde{Z}])$. In formulas below this closed-inclusion pushforward is implicit; equivalently first calculate the supported Gysin class in A₀(Z) and then push it to A₀(X). This is the intrinsic Nash definition of the Chern–Mather class.

First record explicitly how different presentations compare. For a bundle presentation F→[F/H] of the obstruction stack, its coherent quotient is $ob=coker(H\to F)$. On the big étale site, let $c_v$ be the sheaf of isomorphism classes of the intrinsic cone stack. The closed immersion of that stack into [F/H] makes $c_v$ a subsheaf of ob. The square $C_F\to F$ over $c_v\to ob$ is cartesian as sheaves. Indeed, after a test-scheme base change, equality in the quotient sheaf means equality after an étale cover up to translation by a section of H. The cone $C_F$ is invariant under that translation. Membership in its inverse image is therefore exactly membership in $c_v$. Conversely a member of $c_v$ has a local lift to $C_F$ by the definition of the quotient presentation. This verifies the equality as functors on all test schemes and hence its scheme-theoretic meaning. Formation of the coherent cokernel commutes with base change by right exactness, so this argument uses the big quotient sheaf, rather than falsely asserting left exactness of tensor product.

Suppose another local bundle G presents the same obstruction stack (under an isomorphism of POTs over $L_X$), with cone $C_G$ and quotient G→ob. On an affine étale scheme chart, form $P=F{\times }_{ob}G$ in coherent modules and choose a finite free bundle Q mapping onto P. Both Q→F and Q→G are surjective bundle maps, hence have vector-bundle kernels. Commutativity with the quotient maps and the preceding cartesian description show that their scheme-theoretic inverse images of $C_F$ and $C_G$ coincide, say D⊂Q. In particular $D\to C_F$ and $D\to C_G$ are vector bundles obtained by base change. They give bijections of irreducible components and preserve their generic multiplicities. To see the latter directly, near a generic component the added variables give a polynomial algebra over the artinian generic local ring followed by localization at its generic prime; a composition series of that artinian ring stays a composition series with the same length after the faithfully flat residue-field extension. The component base in X also remains the same. No Chow comparison is being inferred from this local observation.

Apply the proved local symmetric-POT presentation: after shrinking an affine étale chart, there is a smooth scheme M of dimension n and an almost-closed one-form ω with zero scheme X, and an isometry of POTs over $L_X$ with $[T_M{\mathrm{\mid }}_X\to {\mathrm{\Omega }}_M{\mathrm{\mid }}_X]$. For this presentation G=Ω_M|_X and $C_G=C_{X\mathrm{/}M}$, embedded by the generators supplied by ω. The proved almost-closed-form argument says every reduced irreducible component of $C_G$ is conic Lagrangian and has dimension n.

Here is the conormal identification, including the generic-projection point. Let D₀ be such a reduced irreducible component and Z the closure of its image in M. In characteristic zero the extension of function fields for D₀→Z is separably generated. On dense smooth loci the differential of this dominant morphism therefore has rank dim Z (equivalently choose a separating transcendence basis, then use the nonvanishing differential minor). Thus its tangent image is $T_Z$. The connected group $G_m$ preserves each irreducible component of a conic set, so the fibre Euler vector field is tangent to D₀. If σ=dα is the cotangent symplectic form and α the tautological one-form, then ι_Euler σ=α. Since σ restricts to zero on D₀, so does α. At a generic point (z,ξ), this says ξ annihilates the entire tangent image $T_{zZ}$. Hence D₀ lies generically in the conormal bundle of $Z_{sm}$. Both have dimension n and are irreducible; D₀ is the closure of that conormal bundle.

The common bundle Q above has a single composite $Q{\mathrm{\mid }}_{Z_{sm}}\to {\mathrm{\Omega }}_X{\mathrm{\mid }}_{Z_{sm}}\to {\mathrm{\Omega }}_{Z_{sm}}$. Therefore the inverse images of the two kernel bundles in F and G are precisely the same kernel in Q. The common-cone comparison consequently identifies each reduced component of $C_F$ with $K_Z$. Its generic multiplicity is the multiplicity of the corresponding normal-cone component, with the same base Z. It follows that globally $[C_F]={\mathrm{\Sigma }}_Z{m}_Z[K_Z]$.

This equality is an equality of cycles: coefficients are determined at generic points and are verified after étale chart pullback, which is injective for cycles on substacks. It is not descent of rational equivalence or of a Chow-group equality. The argument also proves that $C_F$ is pure of dimension rank F. The canonical constructible cycle of X is, with Behrend's sign convention, $c_X={\mathrm{\Sigma }}_Z(-1{)}^{\dim Z}{m}_Z[Z]$.

Indeed on a local smooth embedding this is exactly its definition from the normal-cone components and their projection multiplicities. Those coefficients are independent of the embedding. To check this explicitly, compare two embeddings in their product. Locally near any point, the map from X to the second smooth ambient extends to an étale neighbourhood in the first ambient: choose étale coordinates in the second ambient, lift their restrictions from the quotient defining X, and use the étale lifting property. Subtract the resulting graph from the second set of coordinates. The ideal of X in the product then has the form $(I,y_1,\dots ,y_s)$ in an étale neighbourhood of $A[y_1,\dots ,y_s]$, and its associated graded ring is $(g{r}_{I}A)[Y_1,\dots ,Y_s]$. Thus passing to the product adds a vector-bundle factor to the normal cone, preserving components, image supports and generic lengths. Apply this to each projection. This proves the required comparison locally at the generic points of all components, which determines the coefficients. Étale base change itself commutes with the associated graded ideal because it is flat. Equivalently, the local normal-cone description just established computes the intrinsic canonical cycle on every such chart. Thus the displayed global cycle uses the genuine canonical $c_X$, rather than a cycle defined to force the answer. On Z̃ there is an exact sequence of vector bundles $0\to N_Z\to {\nu }_Z\ast F\to Q_Z\to 0$.

Its left-hand bundle has rank rank F−d. Its total space is a closed substack of ν_Z*F; its map η_Z to F|_Z is proper, since ν_Z is proper and properness is preserved under base change and closed immersion. Its image is $K_Z$. Over $Z_{sm}$ it is the kernel bundle defining $K_Z$, so η_Z is generically an isomorphism, with the same generic stabilizers in the stack case. Accordingly ${\eta }_{Z\ast }[N_Z]=[K_Z]$ in the given rational Chow theory; there is no stabilizer-degree factor. Proper compatibility of the refined zero-section Gysin map and the excess formula for the displayed bundle inclusion give, globally, $0_F^{!}[K_Z]={\nu }_{Z\ast }(0_{{\nu }_Z\ast F}^{!}[N_Z])={\nu }_{Z\ast }(c_d(Q_Z)\cap [\tilde{Z}])=(-1{)}^d{c}_0^M[Z]$.

These formulas hold on the possibly singular Nash space: Chern classes are operational, and the excess formula is the vector-subbundle formula; smoothness of Z̃ is not used. In the DM case all maps used here are the representable projective maps and vector bundles just constructed. The given rational Chow, proper pushforward and refined-Gysin compatibilities apply directly, not by gluing a scheme Chow equality on an atlas. Sum this identity with the global coefficients $m_Z$. The defining global BF equality now gives $[X]vir=0_F^{!}[C_F]={\mathrm{\Sigma }}_Z{m}_Z(-1{)}^{\dim Z}{c}_0^M[Z]=c_0^M(c_X)$. This establishes the global virtual-class/degree-zero-Aluffi comparison in precisely the stated global-resolution/rational-Chow scope.

Using the constructibility and normalization proved below, Eu[Z] is the local Euler-obstruction function extended by zero, with value 1 on the dense smooth locus of Z. Such functions form an integral basis of constructible functions: subtract the generic value on each top-dimensional irreducible support and induct on the dimension of the remaining support. This also proves uniqueness. Consequently the transformation denoted $c^{SM}$ may be defined on this basis by $c^{SM}(Eu[Z])=c^M[Z]$. This is the usual Euler-obstruction/Nash construction of that class; no proper-pushforward theorem is needed for this construction or for the identity being proved here. By ${\nu }_X=Eu(c_X)$ and ${\alpha }_X=c^M(c_X)$, the established equality becomes $[X]vir=({\alpha }_X{)}_0=c_0^{SM}({\nu }_X)$. It does not establish the separate theorem $\int c^{SM}(\nu )=\chi (X,\nu )$. That proper Euler-index assertion remains a separate supplier for the weighted-Euler theorem; it has not been inferred from the definition of $c^{SM}$.

Here is the needed radial/Segre comparison on an affine scheme chart. Embed the reduced integral support $Z$ locally in ${\mathbb{C}}^m$, and write $\mu :N\to Z$ for its Nash modification, $T$ for its rank-$d$ tangent bundle, and $F={\mu }^{-1}(p)$. For $d>0$ blow up the ideal of $F$ and write $b:B\to N$ and $D$ for the blowup and its exceptional Cartier divisor. The displacement vector $x-p$ determines a direction line $L$ on $B$ and is a section of it with divisor $D$, hence $L={\mathcal{O}}_B(D)$. Whitney B for the point stratum $p$ says that along $D$ the secant direction belongs to the limiting tangent plane. Hermitian projection therefore embeds $L$ into $b^{\ast }T$ on a neighbourhood of the compact exceptional fibre, with complex quotient $Q$ of rank $d-1$. The projected displacement has positive radial component on a small link. The controlled outward tangent field of [F5] lifts to the same Nash bundle by Whitney A and has positive radial component there too. Their straight-line homotopy stays nonzero on the link. The relative Thom product gives the radial obstruction as $\mathrm{deg}(c_{d-1}(Q)\cap [D])$, retaining the divisor multiplicities. Since $c(Q)=b^{\ast }c(T)\mathrm{/}(1+D)$ and $b_{\ast }[B]=[N]$, the projection formula identifies it with $\mathrm{deg}\{c(T{\mathrm{\mid }}_F)\cap s(F,N){\}}_0$. Capping Thom classes with top-dimensional complex cycles justifies this on singular $B$ and $N$; the divisor/flag comparison in [F5] identifies their algebraic Chern classes with the same topological classes. For $d=0$ both conventions give the reduced point. Thus the function in [F4] is exactly the radial Nash Euler obstruction.

Constructibility is local and follows with its actual hypotheses as follows. The Nash morphism over our affine chart is projective, and its source is quasi-projective. Apply the finite algebraic Whitney stratification supplier to this one morphism; over each target stratum the whole source restriction is proper. Apply the controlled-submersion supplier to that restriction in smooth ambient charts. For the normal data, take a small closed normal tube over a relatively compact stratum disk, with transverse radius boundary, and its inverse image under the proper Nash map. The projection to the stratum is a submersion on every source stratum, including the boundary strata: the Nash map is a stratified submersion and the target projection/radius pair is submersive by Whitney control. The inverse image of each closed tube slice is compact by properness. Applying the same supplier to this compact pair gives whole-fibre local products for the normal Nash pair and its link, rather than only for the inverse image of the stratum itself. In a holomorphic local projection to that stratum, Whitney A makes the projection of each limiting tangent plane onto its stratum tangent space surjective. A Hermitian right inverse splits off that horizontal complex tangent bundle. Along a contractible stratum disk, the remaining Nash bundle is homotopic to its restriction over one fibre: use the disk contraction in the proper local product and the bundle homotopy property. The normal outward link field transports under these controlled products; different choices are homotopic through fields with positive radial component. Relative Thom evaluation is invariant under those bundle homotopies. Consequently its integer value is locally constant on each target stratum. There are finitely many connected algebraic strata, so the function is algebraically constructible. On the smooth locus the Nash map is the identity and the complex radial local degree is $+1$. This proves both assertions used in the triangular basis argument.

In the stated quasi-projective DM-stack scope choose a finite affine étale atlas and apply these constructions on its finitely many pieces. The associated graded fibre ideals, projective Nash modification, quotient bundle and geometric-point degree commute with étale base change. Hence the functions and local comparisons agree on the overlap scheme charts. Constructible functions on the stack mean precisely these compatible étale-local integer-valued constructible functions; finiteness follows from the finite atlas. The canonical cycle coefficients were already glued as cycles in steps 24.1–25.1, and the Nash/Gysin calculation in 26.1–27.1 uses the stipulated rational DM Chow operations directly. Thus no descent of a scheme Chow equality or integration over a residual gerbe is inserted. The Eu-basis transformation of step 28.1 is therefore defined on exactly these constructible functions, completing the stated equality.