Standard rational quiver vanishing cycles realize the critical coefficient package

The action and projection maps $G_{\mathbf{d}}\times X_{\mathbf{d}}\to X_{\mathbf{d}}$ are smooth and pull back $f_{\mathbf{d}}$ to the same function. Smooth commutation [F2] therefore identifies their pullbacks of ${\mathrm{\Phi }}_{\mathbf{d},z}$. The identity and cocycle equations are the equations of the inverse-image maps of the same support triangles, so these are actual Cartesian action-nerve coefficients, not unrelated equivariance isomorphisms. Constructibility and boundedness follow from [F2], and normalized equivariant duality [F5] gives $K$ as stated. Angular continuation commutes with every descent map, hence induces the specified monodromy on both coefficients.

Let a free frame $U$ have dimension $t$, put $X_U=X{\times }_{G}U$, and let ${\mathrm{\Phi }}_U=V_{f_U}{\mathbb{Q}}_{X_U}$. Pulling back to $X\times U$ identifies this with the pullback of $\mathrm{\Phi }$ by smooth commutation; the locally split descent of [F3]–[F4] identifies the coefficients downstairs. The representative of the intrinsic $K$ is $D_{X_U}{\mathrm{\Phi }}_U[-2t]$: normalized equivariant duality removes $2(t-g)$ and the further intrinsic shift removes $2g$. Its degree-$(-n)$ cohomology is therefore $H^{-n-2t}(X_U,D{\mathrm{\Phi }}_U)$. This is the critical analogue of the intrinsic index $n+2t$, and for standard coefficients it equals $H_{n+2g}^{\mathrm{B}\mathrm{M},G}(X;\mathrm{\Phi })$ in the original variety-normalized convention; here $H_j^{\mathrm{B}\mathrm{M},G}(X;\mathrm{\Phi })$ abbreviates $H_G^{-j}(X,{\omega }_X^G\otimes \mathrm{\Phi })$, which equals the dual-coefficient formula for these standard constant-input cycles by [F2].

The unshifted coefficient $\mathrm{\Phi }$ on the fixed presentation has bounded amplitude by [F2]. Its pullbacks to frame factors have the same amplitude. On a local product frame chart the unit comparison is the fixed coefficient complex tensored with the fibre cochain complex; [F4]'s Stiefel gap tends to infinity, so its unit cone has lower bound tending to infinity after that fixed finite width. Thus every fixed-degree Hom and group comparison stabilizes. The normalized dual representative in step 2.1 has the same fixed downstairs bounds after removing the stated auxiliary shift; no bound on the unnormalized dual of growing-dimensional varieties is presumed. On a common product frame bundle both coefficient identifications are the identity on the same pullback from $X$, so they preserve morphisms, operations, monodromy and the triple-product cocycle. This checks the actual hypotheses for [F4] on these vanishing-cycle coefficients.

On ${\mathfrak{E}}_{\mathbf{a},\mathbf{b}}$, [F1] identifies the function with $f_{\mathbf{a}}⊞f_{\mathbf{b}}$. Smooth commutation and the actual TS isomorphism [F2] therefore identify its intrinsic coefficient ${\mathrm{\Psi }}_E$ with $q^{\ast }({\mathrm{\Phi }}_{\mathbf{a},z_1}⊠{\mathrm{\Phi }}_{\mathbf{b},z_2})$ on the indicated value-pair summand. In the induced smooth variety presentation of [F3], invariant-flag forgetting is proper. Apply $V$ to its unit $\mathbb{Q}\to R{p}_{\ast }\mathbb{Q}$ and use proper commutation [F2]; this constructs ${\mathrm{\Phi }}_{\mathbf{a}+\mathbf{b},z}\to R{p}_{\ast }{\mathrm{\Psi }}_E$. Its dual is $R{p}_{\ast }D{\mathrm{\Psi }}_E\to D{\mathrm{\Phi }}_{\mathbf{a}+\mathbf{b},z}$. All fixed-group and auxiliary shifts are the same on the two corners of this proper map, so it has intrinsic degree zero. No closed-restriction isomorphism or integral-identity inverse is used.

For the smooth map use exactly the factorization of [F3]: $[Y\mathrm{/}L]\stackrel{r}{\to }[Y\mathrm{/}P]\stackrel{q}{\to }[Z\mathrm{/}L]$, with $r$ an affine fibration of rank $u$ and $qr=v$ a vector-bundle projection of rank $e$. The intrinsic vanishing-cycle coefficient upstairs is the pullback along each such smooth map by [F2]. Duality and positive orientation give the ordinary smooth shifts $2u$ and $2e$, and homotopy invariance [F2] makes $r^{!}$ an isomorphism. Define $q^{!}=(r^{!}{)}^{-1}{v}^{!}$. Its shift is $2(e-u)=-2{\chi }_Q(\mathbf{b},\mathbf{a})$, and the dual coefficient on the extension stack is exactly $q^{\ast }(K_{\mathbf{a},z_1}⊠K_{\mathbf{b},z_2})[2r]$. This is the required generic smooth identification.

By [F6] only finitely many critical values occur in each dimension. At a point where $df\mathrm{\ne }0$, the holomorphic submersion product has zero half-space relative complex, so $V_{f-z}\mathbb{Q}$ is supported on $\mathrm{Crit}(f)\cap f^{-1}(z)$. For a sum of two smooth functions, its critical locus is the product of their critical loci. The total-value-$z$ support therefore splits into finitely many disjoint closed pieces with values $(z_1,z_2)$ and $z_1+z_2=z$; each piece is also open relative to that support. Closed-support localization gives the canonical direct-sum decomposition. Translating each function by its value and applying TS [F2] on every piece proves exactly the all-values identification of step 4.1. Thus nonzero critical values and nonhomogeneous potentials introduce no omitted summand or completion.

Proper coefficient maps compose because the unit of a composite is the composite of units; smooth base change sends that unit to the unit in the pulled-back diagram. The proper/smooth comparisons of [F2] consequently identify the two resulting coefficient maps. The parent flag construction [F3] realizes the relevant Cartesian diagrams on common finite models, and [F2]'s support-product naturality gives their TS and external-product compatibility. On a three-step flag both TS composites are the coefficient map for the same sum of three diagonal trace functions; positive complex orientations concatenate without a sign because their real degrees are even. The affine-fibration inverse comparisons have the same identities since they invert actual natural isomorphisms. Faithfulness [F4] and step 3.1 reflect these identities from a sufficiently large common model. This proves all composition, base-change, flag-pasting and monodromy coherence demanded by the generic datum.

In dimension zero $f=0$ and $V_0\mathbb{Q}=\mathbb{Q}$ by [F2]; the endpoint correspondences are identities, so their coefficient maps and orientations are identities. More generally if $W=0$, then $\mathrm{\Phi }=\mathbb{Q}$, $K$ is the intrinsic dualizing coefficient and the dual of $\mathbb{Q}\to R{p}_{\ast }\mathbb{Q}$ is the ordinary proper trace. The smooth operation of step 5.1 is then exactly that of [F3]. Every argument uses the same finite block matrices and counts every arrow, including loops and repetitions. Rank-zero blocks and the empty quiver are point or zero-rank instances. The preceding construction and these unit comparisons realize the full generic package, with AC exactly through [A1].