Proof. We recall the coefficient issue because it prevents an unwarranted descent hypothesis in the applications. At torsion coefficient level, a constructible object on \(Y_{\bar\eta}\) descends to a finite extension of \(K\); the same holds for any specified finite diagram of objects and maps. Geometric nearby cycles of those descents agree after further finite extensions. Hence the torsion functor is defined on the filtered union of these categories, independently of all such choices. Applying the adic formalism and extension of coefficients gives the geometric functor above. This construction is the specialization functor of (Gaitsgory and Raskin 2025, secs. 4.1.1–4.1.5). Different reductions of an adic object can require different extensions of \(K\); the construction does not replace this tower by a single finite extension.

The constructibility and exchange assertions are recorded in (Gaitsgory and Raskin 2025, sec. 4.1.6). For the general traits used here, they follow from (Hansen and Scholze 2023, Theorem 4.1 and Corollary 4.2), in the rational-adic coefficient setting (C) of that paper. Indeed, the integral closure \(V\) of \(R\) in \(\overline K\) is a rank-one absolutely integrally closed valuation ring. On a separated finitely presented \(V\)-scheme, restriction to the generic fiber identifies universally locally acyclic complexes with arbitrary constructible generic-fiber complexes; its inverse is \(Rj_*\). Nearby cycles are \(i^*Rj_*\). This applies to \(E\) itself, and thus directly to objects without common finite descent. Perverse exactness follows from this equivalence and the relative perverse \(t\)-structure of (Hansen and Scholze 2023, Theorem 6.7): generic restriction and special restriction are exact for their fiberwise perverse structures. Finite extensions of the original trait give the same \(V\). On a smooth stack, smooth pullback of relative dimension \(d\) commutes with \(\Psi\) and with \([d]\). The scheme assertions therefore descend, and give both local constructibility and perverse exactness on the stack. The proper assertion is checked after smooth base change to schemes, or algebraic spaces, for a representable proper map.

For the last assertion choose the lattice and reduce modulo a power of its uniformizer. After the permitted trait extension its extension is lisse on \(Z\). The lisse tensor formula for nearby cycles, after pullback to \(Y\), gives (10) at this finite coefficient level. Its construction is natural in reductions, maps, and further trait extensions. Passing to the adic system and then inverting the coefficient uniformizer proves the assertion. This proof permits arbitrary \(g\): it uses lisseness of the extended factor, not a base-change theorem for \(\Psi\) along \(g\). ◻

Proof. We give the ordinary-level argument; the unlevelled version omits the condition at \(x\). This is the parabolic form of the isotropy argument for the global nilpotent cone (Ginzburg 2001, Lemma 7); compare the separable-lifting criterion and its bundle-stack application in (Arinkin et al. 2020, Appendix D.1.5 and D.1.8).

Let \(Z\) be an irreducible component of \(\Lambda_D\) with its reduced structure, and put \(F=k(Z)\). Its generic point specifies a family \((\mathcal P,\beta,\phi)\) on \(X_F\). By (Drinfeld and Simpson 1995, Theorem 2), after a finite separable extension \(F'/F\) the bundle is trivial over the generic point of \(X_{F'}\). Hypothesis [hyp:nilpotent], applied to the field \(F'(X)\), gives a parabolic subgroup whose nilradical contains the value of \(\phi\) there. In characteristic zero the same assertion is the rational parabolic consequence of Jacobson–Morozov. Choose the standard parabolic \(P\) of this type. For the split group \(G\), the scheme of parabolics of that type is \(G/P\); the rational parabolic therefore gives, in the generic trivialization, a section of \(\mathcal P/P\) over \(F'(X)\). This is the required generic \(P\)-reduction and does not require a choice of rational conjugating element. Properness of \(G/P\) extends it to a reduction \(\mathcal Q\) on the whole smooth curve \(X_{F'}\). Since its nilradical bundle is a subbundle of \(\mathop{\mathrm{ad}}(\mathcal P)\), the generic containment extends to \[\phi\in H^0\bigl(X_{F'}, \mathcal Q\mathbin{\times}^{P}\mathop{\mathrm{Lie}}R_u(P) \otimes\omega_{X_{F'}}(x)\bigr).\]

Consider the stack \(\mathcal Y\) of a \(P\)-bundle together with a Borel flag at \(x\) in the relative position of \((\mathcal Q_x,\beta)\) just obtained. This stack is smooth. Indeed \(\mathop{\mathrm{Bun}}_P\) is smooth by the vanishing of the second cohomology of its deformation bundle on a curve, and the relative-position locus over it is the associated bundle with fiber a \(P\)-orbit in \(G/B\). That orbit is smooth: its stabilizer is an intersection of a parabolic and a Borel, with the usual smooth torus and root-group description.

The pullback of \(\phi\) to a cotangent vector on \(\mathcal Y\) is zero. To check this without suppressing the level condition, deformations on \(\mathcal Y\) are governed by the locally free sheaf \(\mathcal E\) of sections of \(\mathop{\mathrm{ad}}(\mathcal Q)\) whose value at \(x\) lies in the intersection with the Borel Lie algebra. Pairing \(\phi\) with \(\mathcal E\) has no pole at \(x\), by the original residue condition. It is zero generically because the nilradical of a parabolic is the annihilator of its Lie algebra. Hence the induced section of \(\mathcal E^*\otimes\omega\) is zero everywhere. Serre duality proves the asserted vanishing on deformation spaces.

Shrink to a smooth dense open of \(Z\). The finite separable extension \(F'/F\) spreads to a finite étale cover of a smaller such open, and the reduction spreads to a map from that cover to \(\mathcal Y\). The canonical one-form of \(T^*D\) pulls back to zero there: it evaluates the given covector on the induced deformation of the bundle, and this deformation factors through \(\mathcal Y\). Since the cover is separable, the one-form already vanishes on the smooth dense open of \(Z\). Its exterior differential, the symplectic form, also vanishes there. An isotropic tangent space in \(T^*D\) has dimension at most \(d\), proving (16). The separability in this argument is essential; a purely inseparable cover would not detect vanishing of differential forms.

Products and smooth pullbacks preserve this dimension bound: on cotangent bundles a smooth pullback adds exactly the relative dimension to the dimension of the cone. The description of the enhanced and quotient cones above therefore proves the remaining cases. In characteristic zero the same argument can start with any reduced irreducible subvariety of the cone, rather than only an irreducible component. It gives the stated isotropy, which is likewise preserved by products and smooth pullbacks. ◻

Proof. We construct the Levi from the semisimple part of \(a(0)\), choose a central cocharacter that pairs nontrivially with that part, and then remove the remaining off-Levi coefficients of the series. The first two steps require some care in positive characteristic.

The semisimple part and its centralizer. Embed \(G\) as a closed subgroup of some \(\operatorname{GL}(W)\). In positive characteristic \(\mathfrak g\) is closed under the restricted \(p\)-power, which is matrix \(p\)-th power in this representation. All eigenvalues of \(a(0)\) belong to a finite subfield of \(k\). For a sufficiently large power \(p^r\) that fixes these eigenvalues and kills every nilpotent Jordan block, matrix power \(a(0)^{p^r}\) is its semisimple part \(s\). Consequently both \(s\) and \(n=a(0)-s\) belong to \(\mathfrak g\) and \([s,n]=0\). In characteristic zero this is the usual Jordan decomposition in an algebraic Lie algebra.

To identify \(s\) and \(n\) intrinsically, fix a maximal torus \(T\), a simple system \(\Delta\), and compatible root vectors \(e_\alpha,e_{-\alpha}\) with \([e_\alpha,e_{-\alpha}]=d\alpha^\vee\). Invariance gives \[ \kappa(d\alpha^\vee,H)=c_\alpha\,d\alpha(H), \qquad c_\alpha=\kappa(e_\alpha,e_{-\alpha})\ne0 \quad(H\in\mathfrak t). \tag{18}\] The nonzero constant follows from the perfect pairing of opposite \(T\)-weight spaces. Hypothesis [hyp:form], applied to the Levi \(T\), makes \(\kappa|_{\mathfrak t}\) nondegenerate. Since \(G\) is simply connected, the simple coroots form a basis of \(X_*(T)\); their differentials are therefore a basis of \(\mathfrak t\). Equation (18) shows that the simple-root differentials are linearly independent. The same equation, applied to any root, shows that its differential is nonzero.

We claim that matrix semisimplicity agrees with the toral definition in the theorem, and that matrix nilpotence agrees with the orbit-closure definition. The proper incidence morphism \[G\mathbin{\times}^{B}\mathfrak b\longrightarrow\mathfrak g\] is surjective. Its differential is surjective at a toral element outside the finitely many root-differential hyperplanes, so its closed image contains a dense open. Inside \(\mathfrak b\), write an element as \(H+u\). Successive conjugations by root groups, in increasing root height, remove every root coefficient with \(d\alpha(H)\ne0\): the coefficient in question changes by a nonzero multiple of the root-group parameter, while changes to other coefficients occur in higher heights. The remaining term \(u_0\) commutes with \(H\) and is a nilpotent matrix. Thus \(H+u_0\) is matrix semisimple only if \(u_0=0\). Likewise a matrix nilpotent in \(\mathfrak b\) has zero toral projection, since a faithful representation is upper triangular on \(B\) and its toral differential is injective. Contraction by a strictly dominant cocharacter sends the nilradical to zero. This proves that matrix nilpotence is equivalent to the orbit-closure definition; the reverse implication also follows directly from the closed matrix nilpotent cone. The same argument applies inside any Levi.

We may now conjugate \(s\) into \(\mathfrak t\). Since \(a(0)\) is not nilpotent, \(s\ne0\). Hypothesis [hyp:centralizer] makes its scheme-theoretic centralizer a Levi \(M=Z_G(s)\). We have \(s\in\mathfrak z(\mathfrak m)\) and \(n\in\mathfrak m\). Conjugate \(n\) within \(M\) into a Borel nilradical of \(\mathfrak m\), using the preceding argument. Torus weights then show that \(n\) is orthogonal to \(\mathfrak z(\mathfrak m)\); this conclusion is unchanged by the conjugation.

Choosing an integral central cocharacter. To detect \(s\) by a cocharacter, we need the differentials of central cocharacters to span \(\mathfrak z(\mathfrak m)\). We verify this before using the nondegeneracy of \(\kappa\) on the center. Choose the simple system \(\Delta\) so that \(M\) corresponds to a subset \(\Delta_M\subset\Delta\). Its Lie center is \[\mathfrak z(\mathfrak m) =\bigcap_{\alpha\in\Delta_M}\ker(d\alpha)\subset\mathfrak t.\] There are no additional central root vectors, since each root differential is nonzero. The integral map \[X_*(T)\longrightarrow\mathbb Z^{\Delta_M}, \qquad \nu\longmapsto (\langle\alpha,\nu\rangle)_{\alpha\in\Delta_M}\] has full row rank after reduction modulo \(p\) in positive characteristic, by the linear independence of the simple-root differentials proved above. Its Smith normal form therefore has no nonunit elementary divisor divisible by \(p\). It follows that the kernel of its differential is spanned over \(k\) by the differentials of its integral kernel. The latter kernel is exactly \(X_*(Z(M))\). Thus the Lie center is spanned by differentials of actual central cocharacters. In characteristic zero the same conclusion follows by tensoring the integral kernel with \(k\).

By Hypothesis [hyp:form] the restriction of \(\kappa\) to this center is nondegenerate, and by the integral-kernel calculation its central cocharacter differentials span it. Some \(\lambda\in X_*(Z(M))\) therefore satisfies \[\kappa(a(0),d\lambda) =\kappa(s,d\lambda)\ne0.\] On \(\mathfrak g/\mathfrak m\) the operator \(\mathop{\mathrm{ad}}(s)\) is invertible and \(\mathop{\mathrm{ad}}(n)\) is nilpotent and commutes with it. Their sum is invertible, giving the middle assertion of (17).

Putting the formal series in the Levi. It remains to put the whole series in \(\mathfrak m[[t]]\). Decompose \(\mathfrak g\) into the zero and nonzero eigenspaces of \(\mathop{\mathrm{ad}}(s)\), so that the first summand is \(\mathfrak m\). Suppose inductively that the non-\(\mathfrak m\) coefficients below order \(t^r\) have been removed. Conjugation by a group element congruent to \(1+t^rY\) modulo \(t^{r+1}\) changes the offending coefficient by \([Y,a(0)]\). The invertibility just proved supplies \(Y\) that removes it. Such a group element exists by smoothness of \(G\). The successive changes converge in \(G(k[[t]])\), proving the first assertion without changing the constant term. ◻

Proof. Represent \(A\) by its Higgs field or pair of fields. On a curve where it is not generically nilpotent, choose an unmarked point \(y\) at which the value is not nilpotent. In a parameter \(t\) centered at \(y\) and a formal frame, write this field as \(a(t)\,dt\). Lemma 4 supplies \(M\) and \(\lambda\) satisfying (17). Make the modification by \(\lambda(t)\), with the convention that its adjoint output lattice is \[L'=\operatorname{Ad}(\lambda(t))L, \qquad L=\mathfrak g[[t]],\qquad J=L\cap L'.\] Because \(\lambda\) centralizes \(M\), the field \(a(t)\,dt\) is regular in both lattices. It therefore transports to a global output covector \(A'\) with the same residue conditions at every marked point.

We record explicitly the tangent spaces and their pairing. Gauge transformations on the input formal disc move \(L'\), whereas those on the output move \(L\). Quotienting in each case by the common stabilizer gives \[ T_hH_{\mathrm{in}}=L/J, \qquad T_hH_{\mathrm{out}}=L'/J. \tag{20}\] Both \(L\) and \(L'\) are their own annihilators for the pairing \(\operatorname{Res}_{t=0}\kappa(-,-)\,dt\). More explicitly, if \(\mathfrak g=\bigoplus_{j\in\mathbb Z}\mathfrak g_j\) is the integer weight decomposition for \(\lambda\), then \[ \begin{aligned} L/J&=\bigoplus_{j>0} \mathfrak g_j\otimes k[[t]]/(t^j),\\ L'/J&=\bigoplus_{j<0} \mathfrak g_j\otimes t^jk[[t]]/k[[t]]. \end{aligned} \tag{21}\] Opposite weights pair perfectly under \(\kappa\), and the coefficient of \(t^i\) pairs with that of \(t^{-1-i}\). Thus \[ (L/J)\times(L'/J)\longrightarrow k, \qquad (D,C)\longmapsto\operatorname{Res}\kappa(D,C)\,dt \tag{22}\] is a perfect pairing of \(m\)-dimensional vector spaces.

To compute the covector identity, first hold the output and leg fixed. A tangent vector represented by \(C\in L'\) changes the input bundle by that principal part. Serre duality evaluates \(p^*A\) on it as \(\operatorname{Res}\kappa(a(t),C)\,dt\), which vanishes because \(a(t)\in L'\). The degree-zero smooth cotangent exact sequence for \(q\) implies that \(p^*A\) is the pullback of a covector on output bundle times leg. This argument is valid for stack covectors: the pullback in that exact sequence is injective, and its image consists precisely of covectors annihilating the relative tangent space.

The bundle component is \(A'\). It can be tested on principal parts at points away from \(y\) and the markings, where the input and output are identified. Such tests span the deformation space: sufficiently large pole divisors supported at those points kill the first cohomology of the deformation bundle, and the principal-parts exact sequence then surjects onto that first cohomology. For the moving-point component, hold the output fixed and replace \(t\) by \(t-z\) in the modification. The logarithmic derivative of \(\lambda(t-z)\) at \(z=0\) is \(-d\lambda/t\). Consequently, up to the common sign determined by the gluing convention, \[ \xi=\pm\kappa(a(0),d\lambda)\,dz\ne0. \tag{23}\] These computations prove (19). In the simultaneous quotient case they may first be made with both enhancements retained and then descended along the smooth torus quotient; the modified curve is disjoint from all gluing data.

It remains to prove the two nondegeneracy assertions. The perfect pairing above pairs tangents to the two fixed-side fibers; its \(\mathop{\mathrm{ad}}(a(t))\)-twist will be the derivative of each section. Move in \(H_{\mathrm{in}}\) in direction \(D\in L/J\) and transport a test vector \(C\in L'/J\) with the moving lattice. Differentiating its pairing with the fixed input covector gives \[ \operatorname{Res}\kappa(a(t),[D,C])\,dt =\operatorname{Res}\kappa([a(t),D],C)\,dt, \tag{24}\] up to the same harmless choice of sign. The expression is independent of lifts of \(D\) and \(C\): \(\mathop{\mathrm{ad}}(a(t))\) preserves \(L\), \(L'\) and \(J\). It preserves every \(\lambda\)-weight space because \(a(t)\in\mathfrak m[[t]]\) and \(\lambda\) is central in \(M\). For \(j\ne0\), \(\mathfrak g_j\) lies in the noncentralizer summand \(\mathfrak g/\mathfrak m\); hence the constant term of this operator is invertible on \(\mathfrak g_j\). It is therefore invertible on each of the truncated modules in (21). The perfect residue pairing (22) makes (24) perfect as well. This is exactly the derivative of the first section in the statement at its zero.

For the section on \(H_{\mathrm{out}}\), keep the output fixed, move \(L\) using \(C\in L'/J\), and transport \(D\in L/J\). The resulting derivative is the transpose of (24), up to sign. It too is perfect. A section of a rank-\(m\) vector bundle on a smooth \(m\)-dimensional fiber whose derivative at a zero is invertible has an isolated reduced zero there, proving both assertions. All truncation lengths in this proof are integer cocharacter weights; none is inverted in \(k\). ◻

Proof. The Hecke eigenrelation first gives a vanishing after proper pushforward. The two nondegenerate zeros of Proposition 5 then have different roles: one isolates a single stalk in that pushforward, and the other gives local coordinates in which the equation is the original hypersurface equation plus a split quadratic form. A projective compactification will recover the cycles of the original hypersurface from those of the stabilized one.

1. The modified hypersurface and the eigenrelation. If \(df_d\) is nonzero on the tangent space relative to \(b\), the map \((b,f):D\to\mathcal B\times\mathbb A^1\) is smooth near \(d\) in the field case, and \(\{f=0\}\to\mathcal B\) is smooth near \(d\) in the trait case. Smooth base change proves the assertion. We may therefore assume \(df_d=b^*A\) for a covector \(A\notin\Lambda\) at \(b(d)\).

Choose the modification \(h\) supplied by Proposition 5, at a leg \(y\). Write \(H\) for its exact-position Hecke stratum and \[p:H\longrightarrow\mathcal B,\qquad o:H\longrightarrow\mathcal B,\qquad q=(o,\operatorname{leg}):H\longrightarrow\mathcal B\times L, \qquad \widetilde p=(p,\operatorname{leg}).\] Both \(q\) and \(\widetilde p\) are smooth of relative dimension \(m\). At \(h\) the proposition gives \[ p^*A=q^*(A',\xi),\qquad \xi\ne0. \tag{25}\] Use the two fibers named in that proposition: \[H_{\mathrm{in}}=\widetilde p^{-1}(b(d),y),\qquad H_{\mathrm{out}}=q^{-1}(o(h),y).\] On \(H_{\mathrm{in}}\), the restriction of \(p^*A\) to the \(q\)-relative tangent has a nondegenerate zero at \(h\); on \(H_{\mathrm{out}}\), the restriction of \(o^*A'\) to the \(\widetilde p\)-relative tangent does too. Each fiber retains the identification with its fixed-side bundle.

In the trait case these are special-fiber statements; take the corresponding relative Hecke stratum over \(S\). In the field case introduce the trait in the function line with parameter \(z\), extend all spaces constantly, and impose \(f=z\). We will apply vanishing cycles over this trait. Thus the two cases have a common notation: \(e=0\) in the trait case and \(e=z\) in the field case, and the operation to be computed is, respectively, \(\Psi\) and \(\Phi\).

Take two copies \(H_1,H_2\) of \(H\) and form \[ W_0=H_2\times_{p,\mathcal B}D, \qquad P=H_1\times_{q,\mathcal B\times L,q}W_0. \tag{26}\] Thus the two modifications have the same output bundle and leg, while the input of the second lies in the chart \(D\). The defining square is \[\begin{CD} P @>{\operatorname{pr}_1}>> H_1\\ @V{\operatorname{pr}_2}VV @VV{q}V\\ W_0 @>{q_2}>> \mathcal B\times L, \end{CD} \qquad W_0\longrightarrow D\longrightarrow\mathcal B,\] where \(q_2\) is the output-and-leg map of the second modification and the composite on the right is its input map. Let \(c:P\to D\) be the projection through \(W_0\). Set \(W=\{f=e\}\subset W_0\) and \(P_c=\{f(c)=e\}\subset P\). The distinguished points are \(w=(h,d)\) in \(W_s\) and \(a=(h,h,d)\) in \((P_c)_s\). Products here are over the trait when one is present.

The output-bundle map \(W\to\mathcal B\) is smooth near \(w\). Indeed, \(W_0\to\mathcal B\times L\) is smooth, and (25) says that the cut differential, on directions relative to the output bundle, has the nonzero leg component \(\xi\). The eigenisomorphism therefore implies that the cycles of the pulled-back Hecke transform vanish at \(w\). In the trait case this is smooth base change from \(\Psi F=0\), followed by the lisse tensor compatibility. In the field case \(W\to\mathcal B_S\) is smooth over the function trait, where \(\mathcal B_S\) is the constant stack. Pullback of the constant family with coefficient \(F\) therefore has zero vanishing cycles, and tensoring by the lisse eigenvalue preserves this assertion. The map to the bundle stack is what is smooth here; no smoothness of \(W\to\mathcal B\times L\) is needed.

2. Isolating one stalk in the proper pushforward. Replace \(H_1\) in (26) by its proper Schubert bound \(\overline H_1\). Choose the Satake representation whose kernel is the intersection complex of this bound. Proper base change identifies the cycles of the preceding transform with the proper push of the cycles of its Satake integrand. If \(T\) is the proper fiber over \(w\) and \(C\) is the restriction of those cycles to \(T\), we obtain \[ R\Gamma(T,C)=0. \tag{27}\] We next isolate the contribution of \(a\); no assertion about the boundary of the Schubert variety is required. In the field case the Satake integrand is defined on the entire constant family and then restricted to the cut \(f(c)=z\); \(C\) is the restriction of its vanishing cycles. In the trait case we instead start with the geometric-generic integrand and take its nearby cycles.

Before cutting, \(P\to\mathcal B\) by the first input bundle is smooth. At a point of \(T\) in its open-orbit neighborhood, failure of smoothness after cutting would mean that \(d(f\circ c)\) comes from the cotangent space of that first input bundle. On this fiber the same differential is the pullback of \((A',\xi)\) along \(q_1\). Since \(P\to H_1\) is smooth, its cotangent map is injective; thus failure would already give on \(H_1\) an equality \[q_1^*(A',\xi)=p_1^*A_1\] for some \(A_1\). On \(\widetilde p_1\)-relative tangents this forces \(o_1^*A'\) to vanish. The nondegenerate zero on \(H_{\mathrm{out}}\) shows that \(a\) is isolated among such points of \(T\). Consequently \(C\) vanishes on a punctured neighborhood of \(a\) in \(T\): there the cut map to the first input stack is smooth and the open-orbit Satake kernel is an invertible constant shift and twist. This smoothness is over the trait. In the field case the integrand therefore pulls back the constant \(F\)-family on \(\mathcal B_S\), so it has zero \(\Phi\), as required; the trait case uses \(\Psi F=0\). Here the open part of \(T\) is exactly \(H_{\mathrm{out}}\) for \(H_1\), since \(w\) fixes the second modification and \(d\).

Here is the precise consequence of this isolation. If a constructible complex on a proper scheme vanishes on a punctured neighborhood of a closed point, its contribution at that point is a direct summand of its global cohomology. To see this, let \(j:T\setminus\{a\}\to T\) and \(i:\{a\}\to T\). The restriction \(j^*C\) is zero near \(a\), so \((Rj_*j^*C)_a=0\). The localization triangle consequently splits as the direct sum of \(i_*C_a\) and \(Rj_*j^*C\); both connecting morphisms vanish by these disjoint supports. The same localization argument applies to algebraic spaces. Equation (27) now gives \[ C_a=0. \tag{28}\]

3. An exact split quadratic equation at the isolated point. We identify the local equation at this isolated point exactly. The diagonal \(W_0\to P\), given by making the two modifications equal, is a section of the smooth map \(P\to W_0\). Etale locally at \(a\), there is a map \(r:P\to D\) lifting the first-input map to \(\mathcal B\), whose restriction to this diagonal is the original chart point, including its identification with the first input. Indeed, the smooth morphism \(P\times_{\mathcal B}D\to P\) has that section on the diagonal. To extend it, choose relative smooth coordinates, giving an etale map from a neighborhood of its value to \(\mathbb A^\delta_P\), where \(\delta\) is the relative dimension of \(D\to\mathcal B\) at \(d\). The coordinates of the prescribed section are functions on the diagonal; lift these functions to a neighborhood in \(P\) and take their graph in \(\mathbb A^\delta_P\). Pullback of the etale map along this graph gives an etale neighborhood of \(P\) with a lift to \(P\times_{\mathcal B}D\). Retain the branch carrying the prescribed section along the diagonal. The resulting projection to \(D\) is \(r\), and the fiber product retains the specified stack isomorphism between \(b\circ r\) and the first input. Thus \(c\) records the second input everywhere, whereas \(r\) records the first input in this etale neighborhood; the two agree on the diagonal. The diagonal, being a section of the smooth morphism \(P\to W_0\) of relative dimension \(m\), is a regular immersion of codimension \(m\). Choose relative coordinates \(v_1,\ldots,v_m\) generating its ideal. Since \(r=c\) on it, there are functions \(u_i\) with \[ f(c)-f(r)=\sum_{i=1}^m u_i v_i. \tag{29}\] This is an equality of functions, not merely an equality of quadratic terms.

On the diagonal fiber with \(c=d\) and leg \(y\), which is the fixed-input, fixed-leg fiber of \(H\) through \(h\), the conormal coefficients \(u_i\) are the negatives of the restriction of \(p^*A\) to the \(q\)-relative tangent, in the frame dual to the \(v_i\). In fact variation of the first modification with the second fixed has \(dc=0\), whereas \(b\circ r\) is its first input, so differentiating (29) gives precisely this restriction. This calculation is independent of the chart-vertical part of \(dr\): \(df_d=b^*A\) annihilates \(\ker(db_d)\). The description holds along that entire diagonal fiber. The nondegenerate zero on \(H_{\mathrm{in}}\) shows that \(u_i(a)=0\) and that the derivatives of the \(u_i\) on that fiber form a basis. The map \(W_0\to D\times L\) is the smooth base change of \(\widetilde p\), of relative dimension \(m\). On the diagonal, \(r=c\) and the leg provide the other coordinates; the \(v_i\) provide its normal coordinates. Hence \[ (r,\operatorname{leg},u_1,\ldots,u_m,v_1,\ldots,v_m): P\longrightarrow D\times L\times\mathbb A^{2m} \tag{30}\] is etale near \(a\), with the evident relative interpretation over \(S\). The first-input integrand is \(r^*K\) up to its invertible constant shift and twist. Thus (28) says that its cycles vanish at the vertex of \[ f+\sum_i u_i v_i=e. \tag{31}\] For \(m=0\) this is already the required assertion, after removing the smooth leg factor.

4. Recovering the original hypersurface. We have proved vanishing after adjoining the split quadratic variables. To remove them, we use a proper family whose extra cohomology is supported precisely on the original cut \(\{f=e\}\). For \(m>0\), put \(B=D\times L\) and compactify (31) to the proper quadric family \[ \pi:Q=\left\{\sum_{i=1}^m u_i v_i+(f-e)w^2=0\right\} \subset\mathbb P^{2m}_B\longrightarrow B. \tag{32}\] In the field case \(B\) also carries the constant extension in \(z\). Every point above \((d,y)\) other than the vertex \([0:\cdots:0:1]\) is smooth for \(\pi\), since some \(u_i\) or \(v_i\) is nonzero. The cycles of \(\pi^*K\) there vanish by smooth base change; they vanish at the vertex by (30). Proper compatibility therefore gives zero cycles for \(R\pi_*\pi^*K\) at \((d,y)\).

We record why the extra cohomology of this projective quadric recovers the desired cut as a sheaf, including in characteristic two. Write \(i:Z=\{f=e\}\hookrightarrow B\). Hyperplane powers define a morphism \[ \bigoplus_{a=0}^{2m-1}E_B(-a)[-2a]\longrightarrow R\pi_*E_Q \longrightarrow i_*E_Z(-m)[-2m]\xrightarrow{+1}. \tag{33}\] To verify this triangle, first take its first arrow and call its cone \(J\). Away from \(Z\), the fibers are smooth odd-dimensional quadrics and their cohomology is freely generated by these hyperplane powers. Consequently \(J\) is supported on \(Z\). Over \(Z\) the entire family, including its hyperplane bundle, is the product with the split projective cone \(Q_0\) on the smooth quadric of dimension \(2m-2\). Removing its vertex gives an affine-line bundle over that even-dimensional quadric. Here the affine cells can be computed inductively: in a smooth split quadric of dimension \(N\), the open set where an isotropic coordinate is nonzero is \(\mathbb A^N\), and its complement is a point together with an affine-line bundle over the split quadric of dimension \(N-2\). Starting with the two-point quadric and the conic \(\mathbb P^1\) gives one cell in every dimension, with a second middle cell precisely when \(N\) is even. Apply the same decomposition to the cone. Localization and the absence of odd-dimensional cohomology give \[\begin{aligned} H^{2a}(Q_0,E)&=E(-a) &&(0\leq a\leq2m-1,\ a\ne m),\\ H^{2m}(Q_0,E)&=E(-m)^{\oplus2},\\ H^{\mathrm{odd}}(Q_0,E)&=0. \end{aligned}\] Every hyperplane power is nonzero. To check this without invoking duality on the singular cone, write \(C_0\) for its smooth quadric base and resolve the cone by \(\mathbb P_{\mathrm{lines}}(\mathcal O_{C_0}(-1)\oplus\mathcal O_{C_0})\). The hyperplane class pulls back to \(\zeta=c_1(\mathcal O(1))\) on this projective bundle. If \(h_0\) is the hyperplane class on \(C_0\), the projective-bundle formula gives \(\zeta^2=h_0\zeta\) and hence \(\int\zeta^{2m-1}=\int_{C_0}h_0^{2m-2}=2\). Thus the top hyperplane power on the cone is nonzero, and so are all its lower powers. This also covers \(m=1\), when \(C_0\) consists of two points and the resolution is the two projective lines. Consequently \(i^*J\) has just one cohomology sheaf, the constant sheaf \(E_Z(-m)\) in degree \(2m\). Proper base change gives this calculation over \(Z\) as a complex of sheaves, since the family there is the specified product. Localization gives \(J\simeq i_*i^*J\), proving (33). A choice of generator of the one-dimensional quotient identifies it with the displayed Tate line; its constancy, rather than that choice, is what we use.

All these assertions hold in characteristic two. The split even quadric has the same affine cells, and for \(f-e\ne0\) the only possible singular point of the odd quadric would have all \(u_i=v_i=0\), which does not lie on it. Hyperplane degree \(2\) is invertible in \(E\), irrespective of the characteristic of the geometric base. No division by \(2\) in coordinates or quadratic Morse lemma has been used.

Tensor (33) with \(K\) and use the projection formula. In the trait case take this triangle on the geometric generic fiber. The cycles of its first term vanish at \((d,y)\) because \(\Psi K=0\); those of its middle term vanish by the proper calculation above. Its last term therefore has zero nearby cycles there, and proper compatibility for \(i\) proves the claimed vanishing on \(\{f=0\}\). In the field case use the triangle on the whole \(z\)-family. Its first term has zero vanishing cycles because \(K\) is pulled from the constant family \(D\). The identical argument with \(\Phi\) proves \((\Phi_f K)_d=0\). Smooth base change removes the leg factor in both cases. The sheaf operations used throughout are those in Proposition 2. ◻

Proof of Theorem 6. Apply Lemma 8 on every smooth chart and every open subchart. It says that every function whose differential at a point is outside \(\Lambda_D\) is locally acyclic there relative to \(b^*F\). The cone includes the zero section, so these are precisely the nonzero directions that must be excluded. Weak singular support is the closure of the differentials of functions with nonzero vanishing cycles; over an infinite field closed-point tests suffice by constructibility. Barrett proves this description for the present \(E\)-coefficients and proves that weak singular support equals full singular support (Barrett 2024, secs. 1.4–1.5, Theorem (vii)), extending Beilinson’s torsion-coefficient theorem (Beilinson 2016, sec. 1.5). The hypersurface calculation therefore gives \(\relax(b^*F)\subset\Lambda_D\). The same calculation on charts with additional smooth parameters is available, so the conclusion is compatible with the smooth-chart definition of singular support on the stack. ◻

Proof. The case \(r=1\) is Lemma 8. For \(r>1\) consider the proper incidence projection \[\pi:J=\{(t,[a_1:\cdots:a_r])\in D\times\mathbb P^{r-1}: \textstyle\sum_i a_i f_i(t)=0\}\longrightarrow D.\] At \((d,[a])\) the differential of the defining function on a standard projective chart is \(\sum_i a_i df_i\); derivatives in the projective directions are zero since all \(f_i(d)\) vanish. It is nonzero and lies outside the pulled-back nilpotent cone. The map \(D\times\mathbb A^{r-1}\to\mathcal B\) on each such chart is smooth, so Lemma 8 proves that the nearby cycles of \(\pi^*K\) vanish on the whole fiber \(\mathbb P^{r-1}\) over \(d\). Proper compatibility gives \((\Psi R\pi_*\pi^*K)_d=0\).

Let \(i:Z=\{f_1=\cdots=f_r=0\}\hookrightarrow D\). On the geometric generic fiber hyperplane powers give the triangle \[\bigoplus_{a=0}^{r-2}E_D(-a)[-2a]\longrightarrow R\pi_*E_J \longrightarrow i_*E_Z(-(r-1))[-2(r-1)]\xrightarrow{+1}.\] Indeed, over \(D\setminus Z\) the incidence is a projective bundle of relative dimension \(r-2\), so the first arrow is an isomorphism there. Over \(Z\) it is the product \(Z\times\mathbb P^{r-1}\), and the cone has only its constant top cohomology sheaf. Proper base change and localization identify the cone as written, not merely its stalk dimensions. Tensor by \(K\). The first term has zero nearby cycles by \(\Psi K=0\), and the middle term by the preceding proper calculation. Proper compatibility for \(i\) proves the lemma. ◻

Proof of Theorem 7. Suppose that the closure of the nonzero-stalk locus meets \(D_s\). Shrink to an affine chart smooth of constant relative dimension \(n\) over \(S\). The closure of that locus in \(D_{\bar\eta}\) has finitely many irreducible components \(Z_{\bar\eta,\alpha}\). Constructibility gives, on each component, a dense open subset where the stalk of \(K\) is nonzero. After a finite extension of the trait, descend the components and the proper closed complements of these opens. We only descend these finite algebraic data, and continue to use \(K\) over \(\bar\eta\). Let \(Z_\alpha\subset D\) and \(E_\alpha\subset Z_\alpha\) be their horizontal closures. Components not meeting \(D_s\) may be removed.

Write \(j\) for the largest dimension of a generic component whose closure meets \(D_s\), and choose an irreducible component \(Z_0\) of the reduced special fiber of such a closure. A horizontal integral finite-type scheme over this excellent trait, with generic dimension \(j\), has every nonempty special-fiber component of dimension \(j\); its local dimension at a special-fiber closed point is \(j+1\). For these two assertions see, respectively, (The Stacks Project Authors 2026, Tags 0B2J and 02JU). The same dimension calculation shows that the special fibers of the horizontal exceptional closures \(E_\alpha\) have dimension at most \(j-1\) whenever the corresponding generic component has dimension at most \(j\).

Choose a general closed point \(d\) of \(Z_0\). We may arrange that \(Z_0\) is smooth at \(d\), that \(d\) is in none of the exceptional closures, and that every reduced special-fiber component of the support closure through \(d\) equals \(Z_0\) near \(d\). Distinct horizontal components may have this same special component; this causes no difficulty. By Proposition 3, \[\dim\Lambda_D\leq n.\] After shrinking a dense open in \(Z_0\), the fiber dimension theorem therefore gives \[ \dim(\Lambda_D)_d\leq n-j. \tag{34}\]

Choose a \(j\)-plane \(V\subset T_d^*D_s\) which avoids \((\Lambda_D)_d\setminus\{0\}\) and whose restriction to \(T_dZ_0\) is an isomorphism onto \(T_d^*Z_0\). Both are nonempty open conditions on the Grassmannian. When \(j>0\), for the first condition projectivize the cone in (34): it has dimension at most \(n-j-1\), and a general \(\mathbb P^{j-1}\) in \(\mathbb P^{n-1}\) is disjoint from it by the elementary incidence dimension count. The second is the usual complement-to-a-fixed-subspace condition. The base field is infinite, so the two opens meet. If \(j=0\), take \(V=0\) and use no cutting functions. Lift a basis of \(V\) to differentials of regular functions \(f_1,\ldots,f_j\) vanishing at \(d\). Their restrictions are parameters on the smooth scheme \(Z_0\). Put \(Y=\{f_1=\cdots=f_j=0\}\subset D\). Lemma 9 says \[ (\Psi(K|_{Y_{\bar\eta}}))_d=0. \tag{35}\]

We finish by showing directly that this stalk is nonzero. Near \(d\), the special fiber of the closed support closure intersected with \(Y\) is zero-dimensional. On any of its selected horizontal components \(Z_\alpha\) of generic dimension \(j\) through \(d\), the local ring has dimension \(j+1\), so its quotient by the \(j\) functions has dimension at least one (The Stacks Project Authors 2026, Tag 0B52). Its further quotient by the uniformizer is zero-dimensional. Thus this intersection has a generization of \(d\) in the generic fiber. Such a generization lies outside \(E_\alpha\), since \(d\) was chosen outside its closure, and \(K\) has a nonzero geometric stalk there.

For completeness, the passage from this generization to nearby cycles is local and finite. Let \(Z\subset Y\) be the reduced intersection \(Y\cap\bigcup_\alpha Z_\alpha\). Its generic fiber contains the nonzero-stalk locus of \(K|_{Y_{\bar\eta}}\), the preceding generization lies in \(Z\), and \(d\) is isolated in \(Z_s\). The morphism \(Z\to S\) is quasi-finite at \(d\). Take an affine neighborhood in \(Y\) and use the henselian decomposition of its closed support: the local ring of \(Z\) at \(d\) is a finite local \(R\)-algebra and defines an open-and-closed piece \(T\subset Z\) (The Stacks Project Authors 2026, Tag 04GH(3) and its proof). Its unique special-fiber point is \(d\). Remove the closed complement \(Z\setminus T\) from the ambient neighborhood. On the resulting neighborhood, which has the same nearby-cycle stalk at \(d\), the support closure is the finite \(S\)-scheme \(T\).

The generic restriction of \(K\) is the extension by zero from this finite closed support. Proper base change now identifies its nearby-cycle stalk at \(d\) with \[\bigoplus_{z\in T_{\bar\eta}} (K|_{Y_{\bar\eta}})_z.\] This is a finite direct sum of complexes of \(E\)-vector spaces, at least one of which is nonzero. Hence the sum is nonzero, contradicting (35). The argument includes \(j=0\), when there are no cutting functions and the original support already has the required finite local piece. This proves the theorem. ◻

Proof. The identification of spectral functions with excursion operators is (Nadler and Yun 2019, sec. 7.1 and Proposition 7.2.3). In the present free-group case its evaluation rule can be seen directly. Matrix coefficients span the coordinate ring of \(Y\). Exactness of invariants for the reductive group \(\widehat G\) shows that an invariant function is a finite sum of functions of the form \[ (g_1,\ldots,g_a)\longmapsto \operatorname{Tr}_{W} \left(A\circ\bigotimes_{i=1}^a V_i(g_i)\right), \qquad W=\bigotimes_{i=1}^a V_i,\quad A\in\operatorname{End}_{\widehat G}(W). \tag{45}\] Indeed \(E[Y]=\bigotimes_i E[\widehat G]\) is spanned by products of matrix coefficients. Each such product lies in the image of the equivariant map \(\operatorname{End}_E(W)\to E[Y]\) given by the trace in (45). A given invariant function belongs to the image of finitely many of these maps. The direct sum of their source spaces surjects onto a finite-dimensional \(\widehat G\)-stable image containing that function. Exactness of invariants lifts it to a tuple of invariant endomorphisms, since \((\bigoplus_\nu\operatorname{End}_E(W_\nu))^{\widehat G} =\bigoplus_\nu\operatorname{End}_{\widehat G}(W_\nu)\). This proves the asserted finite-sum expression.

The corresponding excursion inserts the invariant coevaluation tensor for \(W\otimes W^*\), transports the \(V_i\) factors around \(\gamma_i\), applies \(A\), and evaluates. The \(W^*\) factor may be retained at the base point as an additional identity leg. Naturality, convolution, and the unit constraints of the eigenstructure identify these maps with the same operations on the tensor local system of \(\sigma\). Their composite on the multiplicity space is precisely (45) evaluated at \(y\). Summing proves the assertion. This uses the maps in the coherent eigenstructure, not just the isomorphism classes of its fixed-point transforms. ◻

Proof. The group \(\widehat G\) is adjoint because \(G\) is simply connected. The stabilizer of \(y\) centralizes a dense subgroup and hence equals \(Z(\widehat G)=1\). Its orbit \(O\) is closed. To recall the closed-orbit criterion in this case, a one-parameter subgroup producing a limit under simultaneous conjugation forces every entry of the tuple to belong to its associated parabolic. Density forces that parabolic to be \(\widehat G\), so the one-parameter subgroup is central. The Hilbert–Mumford criterion therefore gives closedness. Thus \(O\simeq\widehat G\) and the equivariant bundle \(\mathscr E_V|_O\) has equivariant sections \(s_1,\ldots,s_b\) specified by any basis \(v_1,\ldots,v_b\) of \(V\) at \(y\): at \(g\cdot y\) their values are \(g v_1,\ldots,g v_b\).

Because \(O\) is closed in the affine variety \(Y\), restriction gives a surjection \(E[Y]\otimes V\to E[O]\otimes V\). Taking invariants remains exact in characteristic zero. The \(s_i\) therefore extend to equivariant regular sections of \(\mathscr E_V\) on \(Y\). Let \(\alpha_V\) have these sections as its columns. Every algebraic character of the semisimple group \(\widehat G\) is trivial, so \(\det(V)\) is trivial. After fixing a volume form, the determinant \(f_V=\det(\alpha_V)\) is an invariant function, nonzero at \(y\). The adjugate matrix defines \(\beta_V\) regularly on all of \(Y\). It is equivariant, either by the exterior-power construction of the adjugate or by the identity \(\beta_V=f_V\alpha_V^{-1}\) on the open set where \(f_V\ne0\). The adjugate identities give (46). ◻

Proof. For any central degree-zero natural endomorphism \(f\) of the identity, naturality for \({}^p\tau_{\le n}K\to K\), together with the universal property of truncation, gives \[ f_{{}^p\tau_{\le n}K}={}^p\tau_{\le n}(f_K). \tag{47}\] For example the map on the left is the unique endomorphism of \({}^p\tau_{\le n}K\) whose composite with its map to \(K\) is \(f_K\) composed with that map. The same argument for the other truncation gives \(f_{{}^p\tau_{\ge n}K}={}^p\tau_{\ge n}(f_K)\). Consequently invertibility of \(f_K\) implies invertibility on both truncations. This establishes the asserted restricted \(t\)-structure without any exactness assertion for arbitrary spectral operators.

Apply the spectral action to the two maps of Lemma 15. They give natural maps \[K^{\oplus b}\xrightarrow{\alpha_{V,K}} \mathsf H_{V,u}(K) \xrightarrow{\beta_{V,K}}K^{\oplus b}.\] The composites are the indicated actions of \(f_V\). Symmetric monoidality identifies the action of any \(f\in S\) on \(\mathsf H_{V,u}(K)\) with \(\mathsf H_{V,u}(f_K)\). Thus \(\mathsf H_{V,u}\) preserves \(\mathscr D_S\), and \(f_V^{-1}\beta_{V,K}\) is an inverse to \(\alpha_{V,K}\) there. In particular \[ \mathsf H_{V,u}|_{\mathscr D_S} \simeq \operatorname{id}_{\mathscr D_S}^{\oplus b}. \tag{48}\] This proves \(t\)-exactness at \(u\). Locally constant Hecke transport along a path from \(u\) to any other point gives the same conclusion there. The zero representation acts by zero. Compositions of fixed-point Hecke functors, including convolutions at one point, are consequently \(t\)-exact and preserve \(\mathscr D_S\). ◻

Proof of Proposition 13. Pass temporarily to Betti realization, with coefficients in \(E\). Theorem 6 puts \(\mathcal F\) in the nilpotent category. Its parameter remains Zariski dense after restriction from the profinite fundamental group to topological loops. Indeed the latter have dense image in the former; any polynomial equation on the matrices is closed for the adic topology after passing to a finite coefficient extension containing its coefficients. An equation vanishing on topological monodromy therefore vanishes on the entire adic image.

Use this tuple \(y\) in Lemma 15. Lemma 14 gives \(f_{V,\mathcal F}=f_V(y)\operatorname{id}_{\mathcal F}\), so \(\mathcal F\in\mathscr D_S\). All its perverse truncations and cohomologies belong to this same subcategory by Lemma 16. We next extend fixed-point exactness to the moving system and explain its compatibility with collisions.

For \(K\in\mathscr D_S\) and a finite set \(I\), put \(m=|I|\) and consider \[\mathrm T_{I,\boldsymbol V}(K) =\mathsf H_{I,\boldsymbol V}(K)[m].\] On a contractible parameter patch as in (43), its unshifted value is the exterior product of a fixed-point transform and the constant sheaf of the patch. That transform is a succession of fixed-point Hecke operators; at a collision it is their convolution. By Lemma 16 it has the same perverse bounds as \(K\). The constant sheaf shifted by \(m\) is perverse on the smooth complex \(m\)-fold parameter space. Testing on smooth charts therefore proves that \(\mathrm T_{I,\boldsymbol V}\) is \(t\)-exact on \(\mathscr D_S\). In particular, the images of a truncation triangle for \(K\) are the truncation triangle for \(\mathrm T_{I,\boldsymbol V}(K)\). Its universal property supplies natural comparison isomorphisms \[ {}^pH^j\bigl(\mathsf H_{I,\boldsymbol V}(K)[m]\bigr) \simeq \mathsf H_{I,\boldsymbol V}({}^pH^jK)[m]. \tag{49}\]

These comparisons also hold for successive Hecke operations when the input already carries other leg variables. Locally in all the parameter variables it is a constant family, so the same fixed-point exactness argument applies, with the shift equal to the total dimension of the retained smooth parameter space. This proves the comparison for convolution diagrams as well as for individual functors.

More explicitly, for a surjection \(\pi:I\twoheadrightarrow J\) let \(\Delta_\pi:U^J\to U^I\) be the collision diagonal and put \(W_j=\bigotimes_{i\in\pi^{-1}(j)}V_i\). Fusion gives the ordinary pullback identification \[ (\operatorname{id}\times\Delta_\pi)^* \mathsf H_{I,\boldsymbol V}(K) \simeq \mathsf H_{J,\boldsymbol W}(K). \tag{50}\] On the locally constant families under consideration, the functor \[(\operatorname{id}\times\Delta_\pi)^*[\,|J|-|I|\,]\] is perverse \(t\)-exact: in product charts it replaces the perverse constant sheaf on an \(|I|\)-dimensional smooth base by the perverse constant sheaf on an \(|J|\)-dimensional smooth base. Applied to (50), this proves compatibility of (49) with diagonal restriction. The full-product assertion in (43) is essential here; constancy only at pairwise distinct legs would not provide this comparison.

Now let \(\mathcal V_{I,\sigma}=\boxtimes_{i\in I}(V_i)_\sigma\). Shift the original eigenisomorphism by \(m\) and apply \({}^pH^j\). Exterior product with the lisse sheaf \(\mathcal V_{I,\sigma}[m]\) is \(t\)-exact, so (49) gives \[\mathsf H_{I,\boldsymbol V}({}^pH^j\mathcal F)[m] \simeq ({}^pH^j\mathcal F\boxtimes\mathcal V_{I,\sigma})[m].\] Canceling the common shift yields precisely the relative-normalized eigenisomorphism \[ \mathsf H_{I,\boldsymbol V}({}^pH^j\mathcal F) \simeq {}^pH^j\mathcal F\boxtimes\mathcal V_{I,\sigma}. \tag{51}\] There is no moving-leg shift in this formula.

To verify coherence, apply these functorial truncation comparisons to each diagram for the original eigenstructure. Naturality in representations, the unit, and permutations commute with the comparisons by their construction from truncation. The relative exactness for successive operations gives convolution compatibility; the dimension-adjusted diagonal comparison gives every fusion compatibility, including iterated collisions. On a diagram over \(U^I\), shift each entire vertex by \(|I|\) once: successive modifications at a shared leg do not introduce further leg parameters. Every vertex is locally the exterior product of a composite of fixed-point exact functors applied to \({}^pH^j\mathcal F\) with a lisse multiplicity space. Thus all these shifted vertices lie in the perverse heart. After a collision to \(U^J\), the shift \(|J|-|I|\) puts all vertices in the corresponding target heart. For two objects \(P,Q\) of any of these hearts, \(\pi_r\operatorname{Map}(P,Q)=\operatorname{Hom}(P,Q[-r])=0\) for \(r>0\). Their mapping spaces are therefore discrete; positive Ext groups do not affect this assertion. The resulting commutative diagrams consequently determine the required coherent constraints. In particular choices of the auxiliary spectral frames introduce no choices into (51).

Finally these constructions give eigenmaps in geometric adic sheaves, not only after realization. Start with the adic truncation triangles and apply the adic Hecke functors. Betti realization, which preserves perversity and detects isomorphisms on finite-type charts, verifies the perverse bounds just proved. The universal property of truncation then gives (49) in the adic category itself. The same reasoning applies to successive operations and diagonal pullbacks. Applying it to the original adic eigenmaps therefore constructs (51) and all its compatibilities there. Singular support remains in \(\Lambda\) by the perverse dévissage already used, or by Theorem 6 applied to these eigenmaps.

Every assertion has been checked on finite-type smooth charts and is compatible with further smooth pullback. Thus no uniform cohomological bound on the bundle stack has entered the proof. If \(\mathcal F\) is nonzero, its restriction to some such chart is nonzero and bounded. Some perverse cohomology on that chart is nonzero, and smooth perverse pullback identifies it with a restriction of \({}^pH^j\mathcal F\). This proves the final assertion. ◻

Proof. The cotangent description in Section 3 identifies the nilpotent cone on \(\mathcal A^+\) with the smooth pullback of the cone on \(\mathcal A\): a nilpotent residue lying in a Borel Lie algebra has zero toral component. In particular every covector in this cone annihilates the tangent to a \(T\)-fiber.

Take a smooth finite-type chart \(D\to\mathcal A\) and trivialize the pulled-back torsor, after further localization on \(D\). The singular support on \(D\times T\) lies in \(C_D\times 0_T\), where \(C_D\subset T^*D\) is the ordinary-level cone of Proposition 3. Choose a microlocal stratification of \(D\) whose conormals contain \(C_D\). The singular-support criterion for constructibility then applies to the product stratification \(\{S_\alpha\times T\}\); this is precisely the criterion used in the proof of (Nadler and Yun 2019, Proposition 6.3.2). On a contractible patch of \(T\), restriction to a slice is an equivalence for sheaves constructible with respect to this product stratification. Consequently the complex, including its extension data between strata, is pulled back from the slice there. The descriptions agree on overlaps by locally constant continuation. They establish the asserted condition on the torsor and after every further base change. In particular \(X_*(T)\) acts by continuation on the stalk cohomology of the restriction to each fiber. ◻

Proof. First take a local system with fiber \(W\), and write \(M_1,\ldots,M_r\) for its commuting monodromy operators in a basis of \(\Gamma\). For fixed output, the integrand has monodromies \(z_i\otimes M_i^{-1}\) on \(R_T\otimes_E W\). Ordinary cohomology of \(T\) is computed by the cohomological Koszul complex \[K^\bullet\bigl(z_1M_1^{-1}-1,\ldots,z_rM_r^{-1}-1; R_T\otimes_E W\bigr).\] Untwisting the diagonal \(\Gamma\)-action identifies this with the Koszul complex for \(z_1-1,\ldots,z_r-1\) on the free module with underlying fiber \(W\). Its only cohomology is \(W\) in degree \(r\). The shift in (52) therefore makes the integral \(W\) in degree zero, and the residual \(z_i\) acts by \(M_i\).

There is a useful geometric description of this identification. Remove the contractible real radial directions of \(T\). The free local system is the direct image with finite supports from the universal cover \(\mathbb R^r\) of the compact torus. Integration over that compact torus becomes compactly supported integration over \(\mathbb R^r\). On the cover the other factor is pulled back from its fiber by continuation from the zero lift. The orientation identifies \(R\Gamma_c(\mathbb R^r,E)[r]\) with \(E\). This construction also works for complexes and relatively over a stratified base, so it proves naturality and base change.

For two enhancement differences \(d_1,d_2\), convolution replaces them by their product \(d_2d_1\). On their real covers this is addition of lifted angles. The change from the two lifted differences to the first lifted difference and the lifted total difference preserves product orientation. Integrating in this order agrees with the two successive integrations; the continuation of the input follows the sum of the same two paths. This proves compatibility with the unit convolution isomorphism, including its shifts and \(R_T\)-action. ◻

Proof. Let \(T_c\) be the compact real form of \(T\), and let \(A\simeq\mathbb R^r\) be its positive real radial subgroup. A reduction of the structure group from \(T\) to \(T_c\) exists on the paracompact charts. Equivalently, quotient the torsor by \(A\): \[P\xrightarrow{\rho}\overline P=P/A\xrightarrow{\bar\pi}C.\] Here \(\overline P\) is a topological quotient, and we use topological nearby cycles, defined by the universal cover of the punctured disc. The first map has contractible real fibers, and \(\bar\pi\) is a locally trivial compact-torus bundle, hence proper. These are factorizations over the full disc, including its center.

The hypothesis implies that \(F=\rho^*\overline F\) for a sheaf \(\overline F\) on \(\overline P|_{\Delta^*}\). This may be checked in a torsor chart, where it is the restriction equivalence for the product stratification and a contractible radial factor. Such local descriptions glue because that restriction equivalence is fully faithful. In a trivialization extending across zero, nearby cycles are computed in the product of a radial patch with a neighborhood in \(\overline P\). Contracting the radial patch gives \[\Psi F=\rho_0^*\Psi\overline F, \qquad R\rho_*\rho^*\overline F=\overline F.\] These identities hold also after base change and after tensoring by a factor from the target. Now \(b\bar\pi\) is proper, so ordinary proper base change gives (54). It also proves the other asserted formulas after the radial factors have been removed. Every identification is the natural exchange map, as can be seen using restriction and direct image from the universal cover of \(\Delta^*\). They therefore compose for composite maps. The same argument works for products of enhancement tori. ◻

Proof. The trace in (53) applies monodromy to the \(Z(V)\) factor alone. We must therefore construct the central eigenisomorphism together with its tensor compatibility, retaining the monodromy on each factor. We first specialize one moving action, then compare successive actions on a common correspondence, and finally identify the individual monodromies and evaluate the trace.

The moving action and its specialization. Choose a coordinate disc \(\Delta\) around \(x\), with \(x=0\). Consider the correspondence \(\mathfrak H_\Delta\) whose points consist of a leg \(z\in\Delta\), two enhanced bundles \((P_0,\eta_0)\) and \((P_1,\eta_1)\), and an isomorphism \[P_0|_{X\setminus\{z\}}\simeq P_1|_{X\setminus\{z\}}.\] The two enhancements at \(x\) are independent, including when \(z\ne0\). Write \(p\) for the input map and \(o\) for the output-and-leg map. We use bounded versions of this correspondence throughout.

In a frame of \(P_0\) that takes \(\eta_0\) to the standard enhanced flag, the local family is the Gaitsgory family of (Dhillon and Taylor 2025, sec. 6). Its fiber for \(z\ne0\) is \[\mathop{\mathrm{Gr}}_G\times G/N_B,\] and its fiber at zero is \(LG/\mathrm{Iw}^0\). For \(V\in\mathop{\mathrm{Rep}}(\widehat G)\), let \(K_V\) on the punctured family be the Satake kernel on the Grassmannian factor, externally tensored with \(\mathbf 1_T\) on the locally closed subspace \(T=B/N_B\subset G/N_B\), extended by zero. Thus its support requires the two ordinary flags to agree under the modification, while its universal local system records their enhancement difference. By the local construction, \[ \Psi K_V=Z(V). \tag{57}\] We use unsuspended geometric nearby cycles on relative kernels; the moving-curve perverse shift and its inverse in perverse nearby cycles have both been removed. This is the normalization in which the moving tensor-unit Hecke functor is exterior product with \(E_\Delta\).

These descriptions descend from the frames to the global correspondence. Indeed changes of input frame are regular loop changes whose value at \(x\) lies in \(N_B\). Off zero the Satake factor has its positive-loop equivariance and the flag factor has the corresponding \(N_B\)-invariance. The descent isomorphisms and their cocycle identities specialize by smooth pullback. At zero their group is \(\mathrm{Iw}^0\), which is exactly the required equivariance for (57). On a fixed bound all this can be carried out with finite jets: use a sufficiently thick neighborhood of the sum of the graphs of \(x\) and \(z\), so the same frame space works at collision. These frame spaces are smooth. Off zero, descent of morphisms uses the specified Satake and flag equivariance. At collision the jet quotients of \(\mathrm{Iw}^0\) are unipotent, hence contractible in Betti topology; restriction along their group nerves is fully faithful. Thus the descent retains the maps of kernels, not only their isomorphism classes. Gluing bundles off the disc identifies these local descriptions with the global ones.

Over \(\Delta^*\) form the action \[A_V(Q)=o_*\bigl(Q\,\widetilde\boxtimes K_V\bigr),\] where the twisted product includes the fixed relative normalization. On the support of \(K_V\), ordinary flags agree. Hence the correspondence is the usual enhancement-preserving Hecke correspondence together with a varying input enhancement. Its \(T\)-integration is the unit calculation of Lemma 18. It gives, with its functorial unit identification, \[ A_V(Q)\simeq \mathsf H_V(Q)|_{\Delta^*} \simeq Q\boxtimes V_\sigma|_{\Delta^*}. \tag{58}\]

We next justify specialization of the direct image in this equation. The obstruction to properness is entirely the input enhancement. Forgetting that enhancement, while retaining its ordinary flag, is a \(T\)-torsor over the whole bounded correspondence over \(\Delta\). Its quotient is proper over output and leg: after fixing the output bundle and enhancement, a bounded input modification is in a proper Grassmannian bound, and the ordinary input flag is a point of the proper flag bundle over it. This description uses full flags over the bound and remains valid at \(z=0\).

Both factors of the integrand are relatively monodromic in this input-enhancement torsor. For \(Q\) this is the base-change-stable condition in Lemma 17, or the hypothesis of the present lemma. For \(K_V\) over \(\Delta^*\), its ordinary-flag condition and Grassmannian factor are independent of the enhancement; its remaining factor is a local system in the enhancement difference. Thus their product satisfies Lemma 19. It follows that nearby cycles commute with \(o_*\). This argument uses one radial quotient extending across zero, so it also proves the asserted comparison at collision, where properness of \(o\) itself is unavailable.

On the correspondence, nearby cycles of the integrand are its twisted product with \(Z(V)\). To check this assertion, pass to the input frames just described. There the input is pulled back from the bundle factor and the kernel from the local degeneration. The nearby-cycle exterior-product map is an isomorphism. More generally the same statement holds when the input is a family \(D\) over \(\Delta^*\) that is \(Q\) tensored with a lisse complex there: in the untwisted notation of a frame chart, it is the natural map \[ p_0^*\Psi D\otimes\Psi K_V \longrightarrow \Psi(p^*D\otimes K_V). \tag{59}\] Trivialize the lisse factor on the universal cover of \(\Delta^*\) to reduce to the same exterior-product calculation.

For precision concerning coefficients, these are calculations in weakly constructible Betti sheaves on finite-dimensional bounds. Adapt a complex stratification to the kernels and input. The local Milnor data have finite triangulations, and the exterior comparison is computed by cellular cochains on nearby fibers in product neighborhoods, using the same lift to the universal cover of the puncture. Each complex has finitely many cells; the usual Künneth map therefore works for the vector-space stalks of the universal local system as well. There is no inverse limit of finite-rank adic systems in this assertion. The computation is natural in maps of the factors and respects their monodromy; smooth frame changes preserve it. This is also the local calculation used in (Dhillon and Taylor 2025, Proposition 6.3.1).

Write \(A_V\) for the punctured-family action above and \[B_V(R)=Z(V)\star_x R\] for its collision action. For an input family \(D\) to which (59) applies, let \[ c_V(D):B_V(\Psi D)\longrightarrow\Psi A_V(D) \tag{60}\] be the tensor-exchange map followed by the inverse of the push exchange. In frame notation it is the composite \[o_{0,*}(p_0^*\Psi D\otimes Z(V)) \longrightarrow o_{0,*}\Psi(p^*D\otimes K_V) \xrightarrow{\sim}\Psi o_*(p^*D\otimes K_V).\] The twists in this formula are the same on both sides. The preceding calculations prove that \(c_V(D)\) is an isomorphism for \(D\) constant with value \(Q\), and for \(D=Q\boxtimes L\) with \(L\) a finite-rank local system on \(\Delta^*\).

For constant input, compose \(c_V(Q)\) with nearby cycles of (58). This gives (55) and identifies \(m_V\) with \(M_{V,\partial}\), since \(Q\) is constant in the disc direction. We now compare these maps for tensor products. Compatibility with total nearby-cycle monodromy would only identify the product of the factor monodromies; the trace requires each factor separately.

Successive actions and the tensor comparison. For a second representation \(W\), the eigenisomorphism identifies \(A_W(Q)\) with \(Q\boxtimes W_\sigma|_{\Delta^*}\), so it is among the inputs for which \(c_V\) is an isomorphism. We first record how this comparison treats an extra lisse factor. For \(D=Q\boxtimes L\), set \(L_\partial=\Psi L\). Projection formula and the exterior comparison give the following commutative square: \[ \begin{CD} B_V\bigl(\Psi(Q\boxtimes L)\bigr) @>{c_V(Q\boxtimes L)}>> \Psi A_V(Q\boxtimes L)\\ @V{\sim}VV @VV{\sim}V\\ B_V(Q)\otimes L_\partial @>{c_V(Q)\otimes\mathrm{id}}>> \Psi A_V(Q)\otimes L_\partial. \end{CD} \tag{61}\] Indeed, on the universal cover of \(\Delta^*\) the factor \(L\) is constant. There both routes are the tensor-exchange map for \(Q\) and \(K_V\) tensored with its fiber, followed by the same push exchange. Finite rank permits that fiber to pass through all the direct images. The construction is equivariant for deck transformations, so the square descends, and it is natural in every local-system endomorphism of \(L\). In particular it respects its boundary monodromy as an endomorphism on this factor alone: that monodromy commutes with the cyclic fundamental group of \(\Delta^*\) and hence is an endomorphism of \(L\), not merely an automorphism of its nearby fiber.

It remains to identify the resulting successive comparison with the one for the convolution of kernels. Let \(\mathfrak H^{(2)}_{W,V}\) parametrize two bounded modifications at \(z\in\Delta\), \[(P_0,\eta_0)\longrightarrow(P_1,\eta_1) \longrightarrow(P_2,\eta_2),\] with each enhancement independent. The first bound supports \(K_W\) and the second supports \(K_V\). There are two forgetting maps. The map \(f:\mathfrak H^{(2)}_{W,V}\to\mathfrak H_V\) forgets \((P_0,\eta_0)\) and the first modification; it is the base change of the output map for \(\mathfrak H_W\). Forgetting only \(\eta_0\), while retaining its ordinary flag \(\beta_0\), factors \(f\) as a \(T\)-torsor followed by a proper map, over the whole disc. For its projection formula the second kernel is pulled back from \(\mathfrak H_V\).

The other map \(g:\mathfrak H^{(2)}_{W,V}\to\mathfrak H_{V\otimes W}\) forgets \((P_1,\eta_1)\) and composes the two modifications; take a large enough bound on the target to contain their composites. First forget only \(\eta_1\), retaining \(P_1\) and its ordinary flag \(\beta_1\). This is again a \(T\)-torsor on the entire family. The remaining map is proper. For fixed endpoints and composite modification, the possible bounded \(P_1\) form the closed convolution fiber inside a proper Grassmannian bound: the two relative-position bounds are closed conditions. Choosing \(\beta_1\) adds a proper \(G/B\)-bundle. This description is valid also at \(z=0\), where the composite is an ordinary local modification with the endpoint enhancements retained.

The integrand \(Q\,\widetilde\boxtimes K_W \,\widetilde\boxtimes K_V\) is monodromic in the torus forgotten by either map. For \(f\) this is the already proved input-enhancement calculation, with the second kernel pulled back from the target. For \(g\) the factor \(Q\) is independent of \(\eta_1\). Over \(\Delta^*\), changing \(\eta_1\) changes the two enhancement differences inversely; the two universal local systems therefore remain a local system in that variable. Their ordinary-flag conditions and Grassmannian factors do not depend on \(\eta_1\). This proves relative monodromicity in the actual intermediate-enhancement torsor, in frames, without requiring an equivariant structure on \(Q\). Lemma 19 now applies to \(f\) and \(g\), their base changes, and the required projection formulas. The same uniform radial quotient applies to their specialized integrands by the nearby-cycle tensor comparison.

For \(z\ne0\), the two unit integrations give a convolution isomorphism \[b:A_VA_W(Q)\xrightarrow{\sim}A_{V\otimes W}(Q),\] where tensor factors are written in action order. Forgetting the intermediate bundle first gives exactly this map: on the two torus covers, replace the two lifted enhancement differences by the first lifted difference and their lifted total difference. The coordinate change preserves product orientation. Integration in the first variable is the free-unit convolution map of Lemma 18, and its continuation agrees with the successive unit integrations.

At zero the DT convolution comparison gives \[b_0:B_VB_W(Q)\xrightarrow{\sim}B_{V\otimes W}(Q).\] It is the action of the specific monoidal kernel map \(Z(V)\star Z(W)\simeq Z(V\otimes W)\) of (Dhillon and Taylor 2025, sec. 6.3): in the successive frames, (Dhillon and Taylor 2025, Lemma 6.3.2) identifies the nearby cycles of the twisted product of the two kernels with their specialized twisted product, and pushing by \(g\) is precisely their construction of this map. Exterior comparison with \(Q\) and frame descent preserve that map.

The equality of the two ways to compare actions is the square \[ \begin{CD} B_VB_W(Q) @>{b_0}>> B_{V\otimes W}(Q)\\ @V{c^{\mathrm{seq}}_{V,W}(Q)}VV @VV{c_{V\otimes W}(Q)}V\\ \Psi\bigl(A_VA_W(Q)\bigr) @>{\Psi(b)}>> \Psi A_{V\otimes W}(Q), \end{CD} \tag{62}\] where \[c^{\mathrm{seq}}_{V,W}(Q) =c_V(A_W(Q))\circ B_V(c_W(Q)).\] All domains in this expression are identified by \(\Psi Q=Q\) for the constant input family. To prove the square, on \(\mathfrak H^{(2)}_{W,V}\) take the natural map from the product of the pulled-back nearby cycles of \(Q,K_W,K_V\) to the nearby cycles of their twisted product. It is an isomorphism by the DT comparison just cited and the exterior comparison with \(Q\). Push this map first through \(f\) or first through \(g\). Through \(f\), base change and projection formula identify it with \(c_V(A_W(Q))\circ B_V(c_W(Q))\). Through \(g\), they identify it with the action of the DT monoidal map followed by \(c_{V\otimes W}(Q)\). These identifications give the same map after the final output push: pullback-and-tensor exchange composes for iterated tensors, push exchange composes for composite maps, and their intervening projection-formula square commutes. Equivalently these are the exchange squares for restriction and direct image from the punctured-disc cover. The uniform torus factorizations above justify every exchange, so this verifies (62) also for the nonproper maps.

Individual monodromy and the trace. Apply this square using the eigenisomorphism \(A_W(Q)\simeq Q\boxtimes W_\sigma|_{\Delta^*}\). The second input is covered by (61): it is still monodromic along the enhancement, and its extra factor is lisse on \(\Delta^*\). The single-step map \(c_W(Q)\) identifies \(m_W\) with \(M_{W,\partial}\). Applying \(B_V\) to that identification and then (61) carries this endomorphism to the \(W_{\sigma,\partial}\) factor alone. This is legitimate independently of simultaneous disc monodromy, since the latter square is natural in the local-system endomorphism \(M_{W,\partial}\) of \(W_\sigma|_{\Delta^*}\). For the outer factor \(m_V\), the bottom row of (61) is \(c_V(Q)\) tensored with the identity of \(W_{\sigma,\partial}\), so its single-step identification gives \(M_{V,\partial}\otimes\mathrm{id}\). Thus under the action comparison the two individual endomorphisms are respectively \(\mathrm{id}\otimes M_{W,\partial}\) and \(M_{V,\partial}\otimes\mathrm{id}\).

The convolution square and the original tensor compatibility of the eigenisomorphisms show that these successive identifications are the one for \(V\otimes W\). Naturality of all maps of kernels proves compatibility with representation morphisms, including coevaluation and evaluation. The unit compatibility is exactly Lemma 18. Iterating the same square gives any number of factors, and permutation and fusion maps are the corresponding Satake kernel maps transported by these natural exchanges.

We may therefore evaluate (53) on \(Q\). Its coevaluation and evaluation become the ordinary insertion and contraction for \(V_{\sigma,\partial}\), and the middle map becomes \(M_{V,\partial}\) on its indicated factor. The resulting scalar is \(\operatorname{Tr}(M_{V,\partial})\). On the other hand, the right side of (53) acts through the \(R_T\)-action identified in Lemma 18. This proves (56) and the lemma. ◻

Proof. By Proposition 13, a nonzero perverse cohomology of the given eigencomplex is again an eigenobject. Denote it by \(Q\). Its singular support is nilpotent by Theorem 6; hence its Betti realization satisfies Lemma 17. Lemma 20 gives, for every algebraic representation \(V\) of \(\widehat G\), \[ \chi_V(\text{enhancement monodromy})=(\dim V)\,\mathrm{id}_Q, \tag{63}\] because the boundary monodromy of \(V_\sigma\) is unipotent.

Restrict to any torsor fiber, and to any finite-dimensional stalk cohomology space of \(Q\) on it. The commuting enhancement monodromies have joint generalized eigencharacters \(a:\Gamma\to E^\times\), that is, points \(a\in\widehat T(E)\). Equation (63) says \(\chi_V(a)=\chi_V(1)\) for every \(V\). In characteristic zero the representation characters span \(\mathcal O(\widehat T)^W\) and separate semisimple conjugacy classes. Thus \(a\) and \(1\) have the same image in the finite Weyl quotient \(\widehat T/W\). Its fiber through \(1\) is the singleton \(\{1\}\), so \(a=1\). Every enhancement monodromy operator is therefore unipotent on every stalk cohomology space. This argument requires no uniform bound on Jordan blocks as the point of the bundle stack varies.

Return to geometric adic sheaves and set \(F=q_!Q\). The morphism \(q\) has finite type and fixed relative dimension, so \(F\) is locally bounded constructible. We prove \(F\ne0\) by one fiber. Choose a point of \(\mathcal A^+\) with nonzero stalk, and let \(a\) be its image in \(\mathcal A\). On the fiber \(T=q^{-1}(a)\) the restriction \(Q_T\) has locally constant finite-dimensional cohomology sheaves. Choose the largest \(j\) for which \(\mathcal H^j(Q_T)\ne0\), and write \(W\) for its fiber. This is a nonzero finite-dimensional representation of \(\Gamma\) with commuting unipotent monodromies. Its coinvariants \(W_\Gamma\) are nonzero. Indeed the augmentation ideal of \(E[\Gamma]\) acts nilpotently on \(W\): its finitely many commuting nilpotent generators have a common finite nilpotence bound on this particular space. If \(W_\Gamma\) vanished, successive powers of that ideal would give \(W=0\).

Poincaré duality on the torus identifies \[H_c^{2r}(T,\mathcal H^j(Q_T))\simeq W_\Gamma(-r)\ne0.\] In the compact-support hypercohomology spectral sequence, the term of bidegree \((2r,j)\) has no incoming differential, since all higher cohomology sheaves vanish, and no outgoing differential, since compactly supported cohomology vanishes above degree \(2r\). It yields a nonzero class in \(H_c^{2r+j}(T,Q_T)\). Comparison with Betti sheaves and base change for \(q_!\) now show \(F_a\ne0\).

Finally, away from \(x\) the Hecke correspondence transports both the ordinary flag and its enhancement. Thus the enhanced correspondence is the pullback of the ordinary one along \(q\), and the bounded output map is proper. Base change and the projection formula give natural isomorphisms \[\mathsf H_{I,(V_i)}(q_!Q) \simeq(q\times\mathrm{id}_{U^I})_! \mathsf H_{I,(V_i)}(Q) \simeq q_!Q\boxtimes\mathop{\boxtimes}_{i\in I}(V_i)_\sigma.\] The same correspondence calculation works on the successive modification spaces and on every collision diagonal in \(U^I\). The exchange maps are natural, so they preserve the unit, tensor, permutation, and fusion diagrams. Hence \(F\) is a nonzero coherent Hecke eigencomplex on \(\mathcal A\). A nonzero perverse cohomology of \(F\) is an eigensheaf by Proposition 13, and its singular support is in \(\Lambda_{\mathrm{par}}\) by Theorem 6. ◻

Proof. We use the characteristic-zero part of the Gaitsgory–Raskin theorem, namely the equivalence \[\operatorname{Shv}_{\mathrm{Nilp}}(\mathop{\mathrm{Bun}}_G(C)) \simeq \mathop{\mathrm{IndCoh}}_{\mathrm{Nilp}} \bigl(\mathop{\mathrm{Loc}}^{\mathrm{restr}}_{\widehat G}(C)\bigr),\] its \(\mathop{\mathrm{QCoh}}\)-linearity, and the preservation of compact objects by the inclusion of the automorphic nilpotent category (Gaitsgory and Raskin 2025, Main Theorem 1.3.9(ii), Lemma 1.3.7, and Theorem 1.1.7). Its scope includes geometric \(\overline{\mathbb Q}_\ell\)-sheaves and arbitrary algebraically closed characteristic-zero base fields (Gaitsgory and Raskin 2025, sec. 2.3). Auxiliary choices in that construction, such as a theta characteristic, can be made over \(K\).

We identify the external Satake labels with our Hecke input–output convention. If our label \(V\) corresponds externally to \(\alpha(V)\) for a symmetric tensor autoequivalence \(\alpha\) (for example a Chevalley relabeling), we use the external spectral point \(\tau\circ\alpha^{-1}\). Thus, in the convention used below, universal evaluation at the point denoted \(\tau\) is exactly \(V\mapsto V_\tau\).

Put \(\mathcal S=\mathop{\mathrm{Loc}}^{\mathrm{restr}}_{\widehat G}(C)\) and let \(i_\tau:\mathop{\mathrm{Spec}}E\to\mathcal S\) be the point defined by \(\tau\). We first check that the ind-coherent skyscraper \(\delta_\tau=i_{\tau,*}^{\mathop{\mathrm{IndCoh}}}E\) is a nonzero compact object with nilpotent singular support. The connected components of the restricted stack are indexed by semisimple parameters, and a component containing an irreducible parameter has a unique isomorphism class of geometric points (Arinkin et al. 2020, Proposition 3.7.2 and Corollary 3.7.5). Density makes \(\tau\) irreducible: a reduction to a proper parabolic would put its image in that parabolic. Moreover, \(\widehat G\) is adjoint, so \[\operatorname{Aut}(\tau)=Z_{\widehat G}(\operatorname{im}\tau) =Z(\widehat G)=1.\] The formal tangent complex at this point is \(R\Gamma(C,\mathop{\mathrm{ad}}(\tau))[1]\) (Arinkin et al. 2020, sec. 21.2.1). Its degree \(-1\) cohomology vanishes by the displayed centralizer calculation. The invariant perfect form on the characteristic-zero semisimple Lie algebra and Poincaré duality give \[H^2(C,\mathop{\mathrm{ad}}(\tau)) \simeq H^0(C,\mathop{\mathrm{ad}}(\tau))^\vee(-1)=0.\] Thus there are neither infinitesimal automorphisms nor obstructions, and this one-point formal component is a smooth formal polydisc with tangent space \(H^1(C,\mathop{\mathrm{ad}}(\tau))\).

For completeness, on the completion of a smooth affine space at the origin, ind-coherent sheaves identify with ind-coherent sheaves on that space supported at the origin. The coherent skyscraper is compact in this category, and extension to the disjoint union of formal components preserves compactness; this is the formal-completion description used in (Arinkin et al. 2020, secs. 21.1.7–21.1.10). On a smooth formal component the space of obstruction directions is zero. Hence the spectral singular support of \(\delta_\tau\) is zero, in particular nilpotent. Here spectral singular support refers to ind-coherent obstruction directions, rather than to the microlocal support of a constructible skyscraper.

Let \(F\) be the inverse image of \(\delta_\tau\) under the equivalence. It is nonzero and compact in the nilpotent category, hence compact in the ambient automorphic category by the cited compactness theorem. Compact automorphic sheaves are bounded constructible after pullback to each finite-type smooth chart (Arinkin et al. 2020, Appendix F, Section F.2.2). This proves the required local finiteness.

It remains to explain why the construction supplies the whole eigenstructure. For \(A\in\mathop{\mathrm{QCoh}}(\mathcal S)\) the projection formula gives the natural module identity \[A\otimes\delta_\tau \simeq i_{\tau,*}^{\mathop{\mathrm{IndCoh}}}(i_\tau^*A).\] For a finite set \(I\) and representations \((V_i)_{i\in I}\), apply this identity to the universal evaluation object with its lisse dependence on \(C^I\). Its restriction to \(\tau\) is \(\boxtimes_{i\in I}(V_i)_\tau\). The universal evaluation action is the Hecke action (Arinkin et al. 2020, Main Theorem 14.3.2 and Section 14.3.3). \(\mathop{\mathrm{QCoh}}\)-linearity therefore transports the displayed identity to \[\mathsf H_{I,(V_i)}(F) \simeq F\boxtimes\Bigl(\boxtimes_{i\in I}(V_i)_\tau\Bigr).\] These are restrictions of one universal tensor-evaluation system. The unit, convolution, permutations, and restrictions to collision diagonals are consequently transported together, with their coherence relations. Converting the universal system to our relative Satake normalization gives the displayed formula without a moving-leg shift, as fixed in Section 2. ◻

Proof. On the first normalized component put \(t=u^n\). Locally on a test scheme \(S\), choose a frame on the root-disc cover in which the inertia acts by \(\lambda(\zeta)\). Such frames lift from the origin through all infinitesimal neighborhoods: the frame spaces are smooth and \(\mu_n\)-invariants are exact. A change of frame satisfies \[ h(\zeta u)=\lambda(\zeta)h(u)\lambda(\zeta)^{-1}. \tag{67}\] In particular \(h(0)\in T\). Passing to the invariant punctured-disc frame replaces \(h\) by \(g(t)=\lambda(u)^{-1}h(u)\lambda(u)\).

We check exactly which series \(g\) occur. For a positive root \(\alpha\) put \(j=\langle\alpha,\lambda\rangle\). The \(\alpha\)-root series in (67) and its conjugate have the forms \[u^j a_\alpha(u^n)\longmapsto a_\alpha(t), \qquad a_\alpha\in\mathcal O_S[[t]],\] whereas the \(-\alpha\)-root series have the forms \[u^{n-j}b_\alpha(u^n)\longmapsto t b_\alpha(t), \qquad b_\alpha\in\mathcal O_S[[t]].\] The toral series are simply series in \(t\). These formulas give the entire group, including its converse. Namely, \(h(0)\in T\) puts \(h\) in the formal big cell \(U^-TU_B\), in a fixed root order; the displayed formulas transform that factorization into precisely the big-cell factorization of \[I_B=\{g(t)\in G(\mathcal O_S[[t]]):g(0)\in B\}.\] Conversely every series in \(I_B\) has that factorization since its reduction belongs to \(B\subset U^-TU_B\), and the inverse formulas produce a regular equivariant \(h\). The calculation is valid over arbitrary test schemes and identifies the group functors.

Gluing to the complement of the point therefore identifies bundles of this root type with ordinary \(B\)-level bundles on \(X_1\). Moreover, evaluation \(h\mapsto h(0)\) becomes the torus projection \(I_B\to T\). Its kernel becomes the inverse image of \(U_B\) under \(I_B\to B\). Thus an identification of the bundle on the residual gerbe with the fixed type torsor corresponds exactly to an enhanced flag.

For the second branch the coordinate action is \(v\mapsto\zeta^{-1}v\). If its positive coordinate generator is denoted by \(\eta=\zeta^{-1}\), the fiber action is \(\lambda(\eta^{-1})\). The preceding calculation therefore uses \(-\lambda\) and gives the opposite Borel \(B^-\). This explains the sign and the choice of \(\mathcal A_2^+\). Both gerbe frame spaces have automorphism group \(T\) when expressed using the common node generator \(\zeta\).

A bundle on the nodal twisted curve is a pair of bundles on its normalization together with an identification on their residual gerbes. This is ordinary nodal gluing in the equivariant chart (64). Choosing identifications of both gerbe restrictions with the fixed type torsor makes that gluing identification automatic, and changing both choices by the same element of \(T\) leaves it unchanged. Descent gives (66); there is no additional quotient. If enhancements are instead written as right actions with the second identification reversed, the same quotient is written with inversion on the second torus. We retain the simultaneous-frame convention.

Finally a modification away from the node identifies the bundles near the gerbe, and hence transports the two frames and the gluing map. After pullback to the product it acts on exactly the indicated factor. ◻

Proof. The preceding construction gives the nonzero enhanced-flag eigencomplex \(F_1^+\). Proposition 21 applies because \(\sigma\) is dense and its boundary monodromy is unipotent. It supplies a nonzero locally constructible perverse eigenobject at ordinary \(B\)-level. Its singular support is nilpotent by Theorem 6. All steps used geometric adic sheaves; Betti realization enters only through the previously proved perversity and torus-descent statements. ◻

Proof. Use the compact image \(H\) and its quotients \(Q_r=H/H_r\) as above. The corresponding \(Q_r\)-torsor on \(U\) is tamely ramified at \(x\). Since its inertia comes from \(\mathbb Z_\ell(1)\), its inertia image has order a power of \(\ell\). Choose compatible indices \(d_r=\ell^{e_r}\), with \(d_r\mid d_{r+1}\), that kill these finite inertia images. Each torsor extends to a finite étale torsor over the root stack \(X(\sqrt[d_r]{x})\). These indices are invertible in \(R\), so the relative root stacks \(\mathscr X(\sqrt[d_r]{\mathfrak x})\) are smooth proper tame Deligne–Mumford stacks over \(R\).

Apply proper henselian invariance of finite étale covers to lift the torsors, their actions, and all transition maps to these relative root stacks (Rydh 2014, Theorem 4.5). The root-stack and tame-cover formulation is also given in (Lieblich and Olsson 2010, Proposition A.10, Theorem A.11, and Corollaries A.12–A.13). Restricting to \(\mathscr U\) and then to \(K_0\) gives a compatible tower of finite torsors, hence a continuous homomorphism \(\pi_1^{\mathrm{et}}(\mathscr U_{K_0})\to H\). These statements require the local ramification index to be prime to \(p\); they put no prime-to-\(p\) restriction on \(|Q_r|\).

Each special torsor on the root stack is connected and geometrically reduced, since its dense restriction to \(U\) is connected. The proper flat tame-coarse-space argument used for (69) gives connected geometric generic root-stack torsors. They are smooth connected curves as stacks; removing the inverse image of the marked gerbe preserves connectedness. The homomorphism therefore surjects onto every \(Q_r\), and compactness again gives image exactly \(H\).

The gerbe along \(\mathfrak x\) records the inertia map at each finite stage. Compatible prime-to-\(p\) roots of unity identify its special and generic generators, and full faithfulness in the lifting theorem preserves the entire action of \(\mu_{d_r}\). Thus the generic inertia homomorphism factors through \(\mu_{\ell^{e_r}}\) at every finite stage and agrees there with the original one. In particular every \(\mathbb Z_a(1)\) factor of characteristic-zero inertia for a prime \(a\ne\ell\), including \(a=p\), acts trivially. Passing to the inverse limit gives the full formula \[\rho_{K_0}(\gamma) =\exp\bigl(t_\ell(\gamma)N\bigr) \qquad(\gamma\in I_{\mathfrak x,K_0}),\] with the same regular nilpotent \(N\), up to the allowed change of generator. This also accords with the peripheral compatibility of specialization in (Lieblich and Olsson 2010, Lemma 3.5).

Finally every invariant-lattice reduction in an algebraic representation factors through some \(Q_r\). The corresponding lifted torsor is lisse on \(\mathscr U\) and has the required special fiber. The tower respects tensor operations, so these reductions give the claimed lisse specialization condition for the whole tensor local system. ◻

Proof of Theorem 1. Use Lemma 25 and identify \(K_0\) with \(\mathbb C\). Proposition 24 produces a nonzero locally constructible perverse eigensheaf \(M_{K_0}\) on \(\mathop{\mathrm{Bun}}_{G,B,\mathfrak x}(\mathscr X)_{K_0}\) with eigenvalue \(\sigma_{K_0}\). Set \[\mathscr A=\mathop{\mathrm{Bun}}_{G,B,\mathfrak x}(\mathscr X/R), \qquad M=\Psi M_{K_0}\quad\text{on }\mathscr A_k=\mathcal A.\] The relative bundle stack is smooth over \(R\): the unlevelled bundle stack is smooth by \(H^2\)-vanishing, and adding the flag is a smooth \(G/B\)-fibration. By Proposition 2, the geometric nearby cycles used here are locally constructible and perverse, without taking inertia invariants. The lattice extensions in Lemma 25 and Proposition 10 give \[\mathsf H_{I,(V_i)}(M) \simeq M\boxtimes\Bigl(\boxtimes_{i\in I}(V_i)_\sigma\Bigr)\] for every finite set of legs, with the stated naturality, unit, convolution, permutation, and fusion compatibilities.

It remains to prove that \(M\ne0\). As in Section 8.4, we extend a bundle with nonzero generic stalk by a bounded Hecke modification. Here the level datum to extend is an ordinary flag, so properness of \(G/B\) will complete the extension. Choose a \(K_0\)-valued point \((P,\beta)\) where \(M_{K_0}\) has nonzero stalk, a reference \(G\)-bundle \(P_0\) on \(\mathscr X\), and a smooth section \(y\) disjoint from \(\mathfrak x\). The existence of \(y\) follows by lifting a point of \(X\setminus\{x\}\) over the henselian base. Off \(y_{K_0}\), the bundles \(P\) and \((P_0)_{K_0}\) are isomorphic by the affine-curve triviality used above. A bounded Hecke space at \(y\), proper over the reference input, extends this modification after a finite trait extension. Its output extends the underlying bundle \(P\). The generic flag \(\beta\) then extends by properness of the associated \(G/B\)-bundle over the marked section. Thus the point with nonzero stalk extends to a section of \(\mathscr A\). Lifting this section in a finite-type smooth affine chart through its special value shows that the closure of the generic nonzero-stalk locus meets the special fiber on that chart.

Theorem 7 now proves \(M\ne0\). Its geometric assumptions hold for this smooth pointed family; the special-fiber cone dimension and transverse-modification statements are Propositions 3 and 5, under precisely the four characteristic hypotheses of Theorem 1.

Finally Theorem 6 applied to this locally constructible eigenobject gives \(\relax(M)\subset\Lambda\) on every smooth chart. At ordinary \(B\)-level the cotangent residue lies in \(\mathop{\mathrm{Lie}}R_u(B_\beta)\), so this cone is exactly \(\Lambda_{\mathrm{par}}\) from the theorem. The construction thus has all the required properties. All parameter lifts and nearby-cycle operations are geometric; no Frobenius structure is involved. ◻