1. By I1 over \(\mathbb C\) there are finitely many orbits \(o^1,\dots,o^m\), locally closed, partitioning \(Y_{\lambda,\mathbb C}\). Finitely many equations define them, so they are defined over a finitely generated \(R_\lambda\subset\mathbb C\) (BBD 6.1.7; EGA IV §8).
2. 'The \(o^j_{R_\lambda}\) partition \(Y_{\lambda,R_\lambda}\)' holds on the generic fibre, hence over a dense open (EGA IV §8–9, as used in BBD 6.1.7).
3. Homogeneity of fibres: \(a_j:H\times o^j\to o^j\times o^j\), \((h,y)\mapsto(hy,y)\), is surjective on the generic fibre. Its image is constructible (Chevalley) and contains the generic fibre, so it contains the preimage of a dense open of \(\operatorname{Spec}R_\lambda\). Over that open each \(o^j_{\bar s}\) is a single orbit; nonempty after shrinking (\(o^j_{R_\lambda}\to\operatorname{Spec}R_\lambda\) is dominant, generic flatness).
4. Disjoint orbits give distinct IC sheaves with constant coefficients. They are simple equivariant objects at \(\bar s\). Under I2 at \(\bar s\) there are no others (BBD 4.3.1), so \(\beta_{\bar s}\) is bijective. Closure order: \(\overline{o^j}\) spreads, and the closure of a fibre equals the fibre of the closure on a dense open (EGA IV 9.5).
5. \(\operatorname{Spec}R_\lambda\to\operatorname{Spec}R\) is dominant of finite type between integral schemes. By Chevalley the image of a dense open contains a dense open \(V\). For a closed point \(s\in V\), the fibre is nonempty of finite type over the finite field \(k(s)\), so it has a closed point with finite residue field, which embeds into \(\overline{\mathbb F}_p\) compatibly. So \(\bar s\) lifts.
6. Density: if \(\mathcal P\) is dense and \(V\neq\varnothing\) is open, then \(\mathcal P\cap V\) is dense (irreducibility: every nonempty open \(W\) meets \(V\) in a nonempty open, which meets \(\mathcal P\)). Hence lifting and shrinking to finitely many dense opens never destroys density. For \(\dim R=1\) the proper closed subsets are finite, so dense = infinite.

1. Finite-type model. By (Dict)(d) and the shearing \([g,y]\mapsto(gI,gy)\) (trace.html, step 4), at the level \(\lambda'=(N',M'')\)

$$G_w|_{Y_{N'}}=\pi_{\lambda'}^*\,f_{w*}K_w,\quad Z_w=\{(gI,y)\in\mathrm{Fl}_{\le w}\times Y_{\lambda'}:\ g^{-1}y\in L^+X\},\quad f_w=\mathrm{pr}_2,\quad K_w=\mathrm{pr}_1^*\mathrm{IC}_w|_{Z_w}.$$

\(f_w\) is proper because \(\mathrm{Fl}_{\le w}\) is projective. \(Z_w\to\mathrm{Fl}_{\le w}\) has smooth fibres (a level of \(L^+X\), \(X\) smooth), so \(K_w\) is \(\mathrm{IC}(Z_w)\) up to shift. \(S\) is pulled back to \(\lambda'\).

1. *Models over \(R\).* \(\mathrm{Fl}_{\le w}\) is the image of the Demazure resolution \(D_w\to\mathrm{Fl}\), which is defined over \(\mathbb Z\) (cited: Faltings 2003; Pappas–Rapoport 2008 §8). The image of a proper map commutes with taking fibres set-theoretically. So the geometric fibres of \(\mathrm{Fl}_{\le w,R}\) are the Schubert varieties \(\mathrm{Fl}_{\le w,\bar s}\), and \(\mathrm{IC}_w(\bar s)=j_{!*}\) of the constant sheaf on the cell. \(Z_{w,R}\) is cut out by the same incidence condition, with \(N',M''\) uniform by (Sp)(ii).
2. Forget equivariance. For connected \(H_{\lambda'}\), \(\mathrm{For}\) is t-exact. It sends \(\mathrm{IC}(o,\mathrm{triv})\) to the simple \(\mathrm{IC}(\bar o)\), and these are pairwise non-isomorphic. It sends \(\mathrm{IC}(o,\xi)\), \(\xi\) nontrivial irreducible, to \(\mathrm{IC}(\bar o,\mathrm{For}\,\xi)\). Since \(\pi_1(o)\to\pi_0(\mathrm{Stab})\) is onto, \(\mathrm{For}\,\xi\) has no constant constituent. Hence \(e_{S,i}(w)=[{}^pH^i(f_{w*}K_w):\mathrm{IC}(\bar o_S)]\) in \(\mathrm{Perv}(Y_{\lambda'})\), a statement about schemes.
3. Comparison. Choose a stratification \(T\) of \(Y_{\lambda'},Z_w,\mathrm{Fl}_{\le w}\) (refining the orbits and the Schubert cells) and finite sets \(L\) of irreducible lisse sheaves such that \(\mathrm{IC}_w,K_w,f_{w*}K_w,S\) lie in \(D^b_{T,L}\) and \(D^b_{T,L}\) is stable under \(f_{w*}\), \(j_!\), \(j_*\) (refinement allowed: BBD 6.1.10, paragraph after (6.1.10.1)). Spread \((T,L)\), \(f_w\) over a finitely generated \(A\supseteq R_{\lambda'}[1/\ell]\).

• BBD 6.1.2 (A″), (B″), (C′): over \(\mathbb C\), étale \(\overline{\mathbb Q}_\ell\) and Betti \(\overline{\mathbb Q}_\ell\) agree on these objects (fully faithful, commuting with \(f_*,f^*,j_!,j_*\)).
• Lemma 6.1.9 and (6.1.10.1): after shrinking \(\operatorname{Spec}A\), for every closed point \(s\) there is an equivalence \(D^b_{T,L}(Y_{\lambda',\mathbb C},\overline{\mathbb Q}_\ell)\simeq D^b_{T,L}(Y_{\lambda',\bar s},\overline{\mathbb Q}_\ell)\). It respects perverse t-structures and commutes with \(f_*,f^*,j_!,j_*\), hence with \(j_{!*}\) and \({}^pH^i\) (BBD 6.1.10).

1. Under it \(\mathrm{IC}_w(\mathbb C)\mapsto\mathrm{IC}_w(\bar s)\), \(K_w\mapsto K_w(\bar s)\), \(f_{w*}K_w\mapsto f_{w*}K_w(\bar s)\), \({}^pH^i\mapsto{}^pH^i\), \(S\mapsto S_{\bar s}\) (step 2, A1-1). An exact equivalence of hearts preserves Jordan–Hölder multiplicities.
2. Coefficients. The objects come from \(\mathbb Q\)-sheaves (constant sheaves, \(j_{!*}\), \(f_*\)). Multiplicities in \(\mathrm{Perv}\) are unchanged by extending a characteristic-0 coefficient field, so Betti \(\mathbb C\) = Betti \(\overline{\mathbb Q}_\ell\). Alternatively, fix an abstract isomorphism \(\overline{\mathbb Q}_\ell\simeq\mathbb C\).
3. Let \(U_{S,w}\) be a dense open of \(\operatorname{Spec}R\) inside the image of the shrunken \(\operatorname{Spec}A\) (A1-1, step 5). For the group, replace \(\mathrm{IC}_w\) by \(\mathrm{IC}_x\boxtimes\mathrm{IC}_y\); the proof is unchanged. \(U_{S,w}\) depends on \((S,w)\): BBD gives no effective control of the bad set, and \(\mathcal S_I\), \(\widetilde W\) are infinite.

1. (1 ⇒ 3). By Proposition I(c) over \(\mathbb C\), each \(S\) is a summand \(S[-i]\) of some \(G_{w_S}(\mathbb C)\), so \(e_{S,i}(w_S)_{\mathbb C}\neq0\). Put \(\mathcal P_S=\mathcal P\cap U_{S,w_S}\), dense by A1-1 step 6. For \(\bar s\in\mathcal P_S\), A1-2 gives \(e_{S_{\bar s},i}(w_S)_{\bar s}\neq0\). By (DT) at \(\bar s\), \({}^pH^i(G_{w_S})\) is semisimple and \(G_{w_S}\) splits equivariantly, so \(S_{\bar s}[-i]\) is a summand. Take \(W_S=\{w_S\}\).
2. (3 ⇒ 2) is trivial.
3. (2 ⇒ 1), pigeonhole. \(\mathcal P_S=\bigcup_{w\in W_S}\mathcal P_{S,w}\), where \(\mathcal P_{S,w}\) is the set of points at which \(w\) is a witness. The closure of a finite union is the union of the closures, and \(\operatorname{Spec}R\) is irreducible, so some \(\mathcal P_{S,w}\) is dense.
4. A dense set meets the dense open \(U_{S,w}\); pick \(\bar s\) in the intersection. A summand \(S_{\bar s}[-i]\) of \(G_w(\bar s)\) makes \(S_{\bar s}\) a summand of \({}^pH^i(G_w(\bar s))\), so \(e_{S_{\bar s},i}(w)_{\bar s}\ge1\). This uses no (DT) at \(\bar s\).
5. By A1-2, \(e_{S,i}(w)_{\mathbb C}\ge1\). By (DT) over \(\mathbb C\), \(S[-i]\) is a summand of \(G_w(\mathbb C)\), so \(S\in\mathcal M_I(\mathbb C)\). Since \(S\) was arbitrary, \(\mathcal M_I=\mathcal S_I\), and Proposition I(c) gives 1.
6. Easy direction. Take \(U_S=U_{S,w_S}\) from step 1 (with (DT) at those points, or only the constituent statement without it). Intersecting infinitely many dense opens can lose every closed point: in \(\operatorname{Spec}\mathbb Z\), \(\bigcap_n\operatorname{Spec}\mathbb Z[1/n]\) has no closed point. So Betti generation does not formally give ℓ-adic generation at a single \(p\).

1. Suppose \(S\in\operatorname{Thick}\{G_w\}(\mathbb C)\). By Proposition I(b), \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\operatorname{Thick}(\mathcal M_I^+)\). By Proposition I(a), \(\operatorname{Thick}(\mathcal M_I^+)=\mathcal D_{\mathcal M_I^+}\), the objects whose perverse constituents lie in \(\mathcal M_I^+\).
2. \({}^pH^0(S)=S\) is simple, so \(S\in\mathcal M_I^+(\mathbb C)\): \(e_{S,i}(w)_{\mathbb C}\neq0\) for some \(w,i\).
3. A1-2 (Jordan–Hölder version, no (DT)) gives \(e_{S_{\bar s},i}(w)_{\bar s}\neq0\) for all \(\bar s\) in the dense open \(U_{S,w}\). That open meets the dense set of the hypothesis, a contradiction.
4. Group. Proposition III(a) (props/summand.html; nodes A4-5, A8-5): for every odd \(q\), over \(\overline{\mathbb F}_q\), no \(\Delta_\omega\) (\(\omega\in\Omega\)) is a perverse constituent of any \(\mathrm{IC}_x\ast\mathrm{IC}_0\ast\mathrm{IC}_y\). Hence the hypothesis holds for all geometric closed points of \(\operatorname{Spec}\mathbb Z[1/2,1/\ell]\), a dense set.
5. (Sp) for the group holds over \(\mathbb Z\). The Chevalley group scheme, the Iwahori, and the Bruhat closures in \(LH\) (images of Demazure resolutions) are all defined over \(\mathbb Z\). The congruence-quotient levels of (Lev) are finite-type \(\mathbb Z\)-schemes, and the double-coset orbits are uniform in \(p\). A1-2 holds in the two-sided form. So \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}(\mathbb C)\).
6. Remark: this is the 'failure' half of transfer. It needs only one fixed Betti witness, which would be transferred to almost all \(p\). That is why no uniformity issue arises here.

1. Orbits. \(\{0\}\) (stabilizer \(\mathbb G_m\)) and \(\mathbb G_m\) (trivial stabilizer), both connected. The simple equivariant perverse sheaves are \(\delta_0=i_{0*}\underline k\) and \(\underline k_{\mathbb A^1}[1]\). The heart has finite length (BBD 4.3.1), so \(D^b_{\mathbb G_m}(\mathbb A^1)=\operatorname{Thick}\{\delta_0,\underline k[1]\}\). Invariance: \((tx)v=n(tu)\iff xv=nu\).
2. *\(n\) invertible in \(k\).* If \(v=0\) then \(nu=0\), so \(u=0\), which is impossible. So \(v\neq0\), and \(x\mapsto(x,[x:n])\) is an equivariant isomorphism \(\mathbb A^1\simeq Z_n\). Hence \(G_n=\underline k[1]\).
3. *\(p\mid n\), \(k=\overline{\mathbb F}_p\).* \(Z_n=\{xv=0\}=(\mathbb A^1\times[1:0])\cup(\{0\}\times\mathbb P^1)\): two smooth components meeting transversally at \((0,[1:0])\) (chart \(u=1\): \(xv=0\)). For a curve, \(\mathrm{IC}=\nu_*\underline k[1]\) with \(\nu\) the normalization, so

$$G_n=\underline k_{\mathbb A^1}[1]\ \oplus\ i_{0*}R\Gamma(\mathbb P^1)[1]=\underline k_{\mathbb A^1}[1]\oplus\delta_0[1]\oplus\delta_0[-1].$$

The splitting is equivariant: \([(\mathbb A^1\sqcup\mathbb P^1)/\mathbb G_m]\to[\mathbb A^1/\mathbb G_m]\) is proper, representable, with pure kernel, so Sun's Theorem 1.2 applies. Hence \(\delta_0\in\mathcal C^{\mathrm{gen}}\), and with step 1, \(\mathcal C^{\mathrm{gen}}\) is everything (take \(n=p\)).

1. *\(\mathbb C\).* By step 2, all \(G_n=\underline k[1]\). The constituent subcategory \(\mathcal D_{\{\underline k[1]\}}\) is thick (Proposition I(a), one level) and contains \(\underline k[1]\), but not \(\delta_0\) (\({}^pH^0\delta_0=\delta_0\)). So \(\delta_0\notin\mathcal C^{\mathrm{gen}}(\mathbb C)\).
2. Diagnosis. For each fixed \(n\), the transfer of A1-2 holds for \(p\nmid n\), consistent with BBD 6.1.10. The set of \(p\) where a fixed witness works is \(\{p\mid n\}\), finite and not dense, so the pigeonhole of A1-3 has nothing to act on. A multiplicity can jump up at bad primes: a component appears in a special fibre.
3. Scope. This is a toy, not a loop space: the generators are not \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\). It certifies that no proof of ℓ-adic ⇒ Betti can use only spreading out, (DT) and the summand criterion. Such a proof must use (W) or extra structure of the Iwahori generators.

1. At \(\bar s_p\), \(Z_{\bar s_p}=\{xv=0\}\), so \(G=\underline k[1]\oplus\delta_0[1]\oplus\delta_0[-1]\) equivariantly, by A1-5 step 3.
2. Over \(\mathbb C\), \(\pi\neq0\), so \(Z_{\mathbb C}\simeq\mathbb A^1\) and \(G=\underline k[1]\). Then \(\delta_0\notin\operatorname{Thick}\), by A1-5 steps 2 and 4.
3. The same holds at every geometric point with \(\pi\mapsto a\neq0\). The good locus for transfer is \(D(\pi)\), and \(\mathcal P\cap D(\pi)=\varnothing\), so A1-3 step 4 cannot start.
4. If \(\dim R=1\), then \(R\) is finite over \(\mathbb Z[1/N]\) and proper closed subsets of \(\operatorname{Spec}R\) are finite, so any infinite set of closed points is dense. This applies to \(R=\mathbb Z[1/N]\), the case of \(\mathbb A^n\), the group and the split forms in Table 1.

1. (a) \(H^j(S[-i])_y=H^{j-i}(S)_y\) gives \(P_{S[-i],o}=v^iP_{S,o}\). Stalk cohomology is additive on direct sums, which gives the first identity. The support of \(\mathrm{IC}(\overline{o'})\) is \(\overline{o'}\), and its restriction to \(o'\) is \(\underline k[\delta(o')]\), which gives the other two.
2. Order \(O\) by decreasing dimension (a linear extension of the closure order). Descending induction: for \(o\) maximal among those not yet treated, \(P_{M,o}=m_{S_o}v^{-\delta(o)}+\sum_{o'\neq o,\ o\subseteq\overline{o'}}m_{S_{o'}}P_{S_{o'},o}\), where the sum involves only treated \(o'\). So \(m_{S_o}\) is determined. The splitting exists at every geometric point: \(G_w=\pi^*f_{w*}K_w\) with \(K_w\) pure, so BBD 5.4.5 applies over \(\overline{\mathbb F}_q\) (non-equivariant), and 6.2.5 over \(\mathbb C\).
3. (b) Fix \((S,w)\) and a level carrying both. \(U\cap U_{S,w}\) is a nonempty open of a Jacobson scheme, so it has a closed point \(\bar s_0\). A1-2 gives \(e(\mathbb C)=e(\bar s_0)\).
4. For any \(\bar s\in U\), step 2 computes \(e(\bar s)\) from the same polynomials as \(e(\bar s_0)\), by (U-stalk). So \(e(\bar s)=e(\bar s_0)=e(\mathbb C)\).
5. By I2 on \(U\), \(\beta_{\bar s}\) is a bijection \(\mathcal S_I(\mathbb C)\to\mathcal S_I(\bar s)\) (A1-1, uniform on \(U\)). By Proposition I(c) on both sides, generation at \(\bar s\) \(\iff\) \(\forall S\,\exists w,i:\ e_{S_{\bar s},i}(w)\neq0\) \(\iff\) \(\forall S\,\exists w,i:\ e_{S,i}(w)_{\mathbb C}\neq0\) \(\iff\) Betti generation.
6. (c) heuristic. If \(H^j(\cdot)_y\) has all Frobenius eigenvalues equal to \(q^{j/2}\), then \(m\mapsto\operatorname{tr}(\mathrm{Fr}^m)\) determines all \(\dim H^j\). If these traces are universal polynomials in \(q\) (e.g. computed in a Hecke module over \(\mathbb Z[v^{\pm1}]\), as for Kazhdan–Lusztig polynomials), (U-stalk) follows. Whether the Chen–Yi / Mars–Springer module gives this for the symmetric rows of Table 1 is not checked here.

1. Which category. The glossary entry 'paper-coherent' (iwahori-generation/assets/glossary.js) is the paper's Definition 3.27: coherent, not a priori holonomic. Objects on a quotient stack are strongly equivariant complexes. On the Betti side, \(\mathcal C_I\) is the Bernstein–Lunts equivariant derived category, i.e. \(D^b_c\) of the stack, so the two sides match.
2. Coherent ⇒ regular holonomic (cited). For a smooth \(H\)-variety with finitely many orbits, every \(H\)-equivariant coherent \(\mathcal D\)-module is regular holonomic. The characteristic variety lies in \(\mu^{-1}(0)=\bigcup_oT^*_oY\) (holonomic), and regularity is HTT, Theorem 11.6.1 (cited from memory, not in the local sources). Singular levels (gap, minor). \(Y_\lambda\) affine admits an \(H\)-equivariant closed embedding into a representation \(V\). Kashiwara's equivalence identifies modules on \(Y_\lambda\) with modules on \(V\) supported on \(Y_\lambda\). The characteristic-variety argument needs finitely many orbits only on the support. That the cited regularity proof also needs nothing more is not re-checked here. The cohomology objects of a complex on \([Y/H]\) are equivariant modules, and they are finitely many ('finite complex'). Hence \(\mathcal D_c=\mathcal D_{rh}\).
3. RH (cited). Kashiwara–Mebkhout: \(\mathrm{DR}:D^b_{rh}\simeq D^b_c\), compatible with proper \(f_*\), smooth \(f^*\) (up to shift), \(\boxtimes\), and \(j_{!*}\) (minimal extension ↦ IC). The equivariant/stack version follows by smooth descent along \(Y\to[Y/H]\) (Bernstein–Lunts). It is compatible with the affine-space transitions \(Y_{\lambda'}\to Y_\lambda\) and the closed embeddings \(Y_N\subseteq Y_{N+1}\), so it glues over the level system.
4. \(\mathrm{IC}^{dR}_w\mapsto\mathrm{IC}_w\). \(\mathrm{IC}^{dR}_0\) (the \(\mathcal O\)-module or \(\omega\) on \(L^+X\)) maps to \(\underline{\mathbb C}_{L^+X}\) up to shift. Convolution is \(m_*\) of a twisted external product along smooth maps, so \(\mathrm{DR}(G_w^{dR})\simeq G_w[c]\).
5. An equivalence preserves shifts, cones and summands, so \(\mathrm{DR}(\operatorname{Thick}\{G_w^{dR}\})=\operatorname{Thick}\{G_w\}\), which proves 3.

1. *Levels over \(\mathbb Z\).* \(Y_{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n\simeq\mathbb A^{n(N+M)}_{\mathbb Z}\). \(H_{N,M}=I/(1+t^{N+M}M_n(\mathcal O))\) is the image of \(I\) in \(\mathrm{GL}_n(\mathbb Z[t]/t^{N+M})\): a split unipotent group extended by \(B\), smooth with connected fibres over \(\mathbb Z\). (Lev) and I1 are as in props/summand.html (A5-1, A8-1), and their proofs are characteristic-free.
2. Orbits. With \(\tau=t^{\varepsilon_n}\sigma\) and \(\Lambda_{m,j}=\tau^{nm+n-j}\mathcal O^n\), the orbits are the images of \(\tau^k\mathcal O^n\setminus\tau^{k+1}\mathcal O^n\) and \(\{0\}\). They are defined by lattice conditions over \(\mathbb Z\) and are single orbits over every field (A5-1, which uses only linear algebra over \(k\)). So \(U_\lambda=\operatorname{Spec}\mathbb Z[1/2]\) for every \(\lambda\) in A1-1, and I2 holds everywhere.
3. (Dict)(d) uniform. \(\overline{IwI}\cdot t^{-N}\mathcal O^n\subseteq t^{-N-c(w)}\mathcal O^n\), with \(c(w)\) the largest exponent of \(t^{-1}\) in a representative of \(w\), independent of the field.
4. Witness. \(\tau^k\in\Omega\) has length 0, so \(\mathrm{IC}_{\tau^k}=\Delta_{\tau^k}\), and \(\Delta_{\tau^k}\ast_I\) is push-forward by \(\tau^k\) (A5-3). Hence \(G_{\tau^k}(k)=S_k[k]\) for every \(k\in\{\mathbb C,\overline{\mathbb F}_p\}\). So \(e_{S_k,-k}(\tau^k)=1\) at every point, and (W1) holds with \(\mathcal P_S=\) all points.
5. (DT) is needed only at \(\bar\eta\), where it is the equivariant decomposition theorem. Here it is not even needed: \(G_{\tau^k}\) is already simple up to shift. A1-3 (2 ⇒ 1) gives Betti generation; then A1-8 gives the \(\mathcal D\)-module version (I1 holds for \(\mathbb A^n\)). This matches III(b), proved directly over \(\mathbb C\) and \(\overline{\mathbb F}_q\).
6. Group. A1-4 step 4–5.
7. *\(\mathrm{GL}_2/T\).* \(G=\mathrm{GL}_2\), \(T\), and \(X=G/T\) are defined over \(\mathbb Z[1/2]\). The orbit description of Proposition IV (geodesics in the tree) is uniform in odd \(p\). (Sp) further requires I1 for \(N\ge1\), which is open (A9-5) and is also needed for IV itself. Given (Sp), the failure transfers by A1-4.

1. Support. \(S=\mathrm{IC}(P)\mid G_w\) forces \(P\subseteq\operatorname{supp}G_w=\overline{IwI\cdot L^+X}\). This is a lower bound on \(w\). Nothing bounds \(w\) from above.
2. Multiplicativity. If \(\ell(sw)=\ell(w)+1\), then \(\mathrm{IC}_s\ast\mathrm{IC}_w\simeq\mathrm{IC}_{sw}\oplus\bigoplus_{v<sw}\mathrm{IC}_v^{\oplus}\), with Kazhdan–Lusztig multiplicities that are the same in every characteristic. So \(G_{sw}\) is a summand of \(\mathrm{IC}_s\ast G_w\). From \(S\mid G_{sw}\) one gets only \(S\mid\mathrm{IC}_s\ast S'\) for some \(S'\mid G_w\), which is a statement about \(\mathrm{IC}_s\ast S'\), not about a smaller generator. Witnesses propagate upward, never downward.
3. Length-zero witnesses. For \(\mathbb A^n\) and \(\mathbb G_m\), \(\mathcal S_I=\Omega\cdot\mathrm{IC}_0\) gives canonical bounded witnesses (A1-9). For the other symmetric rows \(\mathcal S_I\neq\Omega\cdot\mathrm{IC}_0\) (verdict, routes ruled out), so this is unavailable.
4. Ultraproduct. A non-principal ultraproduct of the \(\overline{\mathbb F}_p\) is isomorphic to \(\mathbb C\). An unbounded family \(w_p\) defines a nonstandard element of \(\widetilde W\), not a witness over \(\mathbb C\), so the obstruction of A1-5 reappears.
5. Effective bad primes. BBD 6.1.10 shrinks \(\operatorname{Spec}A\) non-effectively. A bound on \(U_{S,w}\) uniform in \(w\) would be (U-stalk) of A1-7, which needs stalk-polynomial input (e.g. Kazhdan–Lusztig-type polynomials for the \(I\)-orbits on \(LX\), independent of \(p\)). This is not available here beyond \(\mathbb A^n\) and \(\mathbb G_m\).
6. The toy A1-5 shows the gap cannot be closed by formal arguments: some property specific to \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\) is needed.

1. (a) ⇐ and ⇒: A1-3 (pigeonhole over the finite \(W_S\), then the transfer A1-2 of one multiplicity via BBD 6.1.2, 6.1.9, 6.1.10, and the summand criterion, Proposition I(c), on both sides). \(\mathcal D\)-modules: A1-8.
2. (b): A1-7.
3. (c): A1-4, which uses only Proposition I(a),(b) over \(\mathbb C\) and A1-2.
4. (d): A1-5 (witness \(n=p\) unbounded, generation at every \(p\), failure over \(\mathbb C\)) and A1-6 (\(R=\mathbb Z[\pi]\), points over every \(p\) inside \(V(\pi)\)).
5. Check: A1-9.
6. Open: whether (W) follows from ℓ-adic generation at all \(p\gg0\) for a genuine \(LX\) (A1-10).
7. Answer to 'one / infinitely many / all \(p\gg0\)'. One \(p\) suffices only under (U-stalk). 'All \(p\gg0\)' is not enough by itself (A1-5). What is needed is a dense set of \(\bar s\) together with bounded witnesses (W).

Lemma 1.

1. On the Hecke side, \(\mathrm{BS}_{\underline w}:=\pi_{\underline w*}\underline k[r]\) for the Demazure resolution \(\pi_{\underline w}\) of \(\mathrm{Fl}_{\le w}\) is \(I\)-equivariantly a direct sum \(\mathrm{IC}_w\oplus\bigoplus_{v<w}\mathrm{IC}_v\otimes V_v\) (decomposition theorem on the finite-type quotient stack: over \(\overline{\mathbb F}_x\) Sun Thm 1.2 + purity; over \(\mathbb C\) Bernstein–Lunts §5; the multiplicity of \(\mathrm{IC}_w\) is 1 because \(\pi_{\underline w}\) is an isomorphism over \(IwI/I\)).
2. \(-\ast_I\mathrm{IC}_0\) is additive and \(G^{\mathrm{BS}}_{\underline w}=\mathrm{BS}_{\underline w}\ast_I\mathrm{IC}_0\) (associativity of the twisted product). So \(G^{\mathrm{BS}}_{\underline w}\simeq G_w\oplus\bigoplus_{v<w}G_v\otimes V_v\).
3. (⊆) \(G_w\) is a summand of \(G^{\mathrm{BS}}_{\underline w}\), so every summand of \(G_w\) is one of \(G^{\mathrm{BS}}_{\underline w}\).
4. (⊇) By (DT) each \(G_v\) is a finite sum of shifted simples; \(\mathcal C_I^{(L)}(k)\) is Hom-finite and idempotent complete (summand.html step 3), hence Krull–Schmidt, and a shifted simple has local endomorphism ring \(k\). So a shifted simple summand of \(G^{\mathrm{BS}}_{\underline w}\) is isomorphic to a summand of some \(G_v\).

Lemma 2.

1. \(H_L\) is connected ((Lev)), so the forgetful functor \(\operatorname{Perv}_{H_L}(Y_L)\to\operatorname{Perv}(Y_L)\) is fully faithful with image closed under subquotients (BBD 4.2.5, 4.2.6.2); simple objects stay simple and non-isomorphic simples stay non-isomorphic.
2. Forgetting is triangulated and commutes with \({}^pH^i\); an equivariant splitting \(K\simeq\bigoplus S[-i]\otimes E_{S,i}\) forgets to a non-equivariant one, and by Krull–Schmidt in \(D^b_c(Y_L)\) the multiplicity of \(S[-i]\) is \(\dim E_{S,i}\) in both.
3. Hence comparisons of multiplicities between \(\mathbb C\) and \(\overline{\mathbb F}_x\) may be made in the non-equivariant categories \(D^b_c(Y_{L,k})\), where BBD §6.1 and nearby cycles apply; (DT) itself is assumed equivariantly on both sides (trap: a non-equivariant splitting does not give (DT), summand.html step 10).

Why Bott–Samelson. \(E_{\underline w}\) can be taken smooth over \(R_0\) (an iterated \(\mathbb P^1\)-bundle times a jet-type fibre), whereas the model of \(G_w\) involves \(\mathrm{Fl}_{\le w}\) and \(\mathrm{IC}_w\); smoothness of the source is what makes the specialization of the generators uniform in \(w\) (A2-4).

1. (a) \(K=\pi_L^*\bar K\) with \(\bar K\) \(H_L\)-equivariant on \(Y_L\). Each \(\mathcal H^j(\bar K)|_o\) is an \(H_L\)-equivariant local system on the orbit \(o=H_L/S_y\), i.e. a representation of \(\pi_0(S_y)\), trivial by I2; so it is constant on \(o\) and its pull-back is constant on \(P=\pi_L^{-1}(o)\).
2. (b) \(\mathrm{IC}(\overline{P'})\) is supported on \(\overline{P'}\) and restricts to \(\underline k[\delta(P')]\) on \(P'\) (trace.html, definitions; I2 makes the local system trivial). The stalk Poincaré polynomial of \(\underline k[\delta]\) is \(v^{-\delta}\). Closure of pieces is a partial order (pieces are preimages of orbits; summand.html step 2), so any refinement to a total order makes \(H\) triangular with unit diagonal entries \(v^{-\delta(P)}\in\mathbb Z[v^{\pm1}]^\times\).
3. (c) Stalks are additive and \(\dim\mathcal H^j(A[-i])_x=\dim\mathcal H^{j-i}(A)_x\) multiplies Poincaré polynomials by \(v^i\). Summing over the decomposition gives \(h_K(P)=\sum_{P'}m_{P'}(v)H_{P,P'}\).
4. (d) is (c) applied in both settings with the same matrix and vector.
5. No weights, traces or Frobenius are used: the argument is valid verbatim for Betti sheaves over \(\mathbb C\) and \(\ell\)-adic sheaves over \(\overline{\mathbb F}_x\).

**Example (\(\mathbb A^n\)).** Piece closures are lattices, smooth, so \(\mathrm{IC}(\overline{P'})=\underline k_{\Lambda(P')}[\delta(P')]\) and \(H_{P,P'}=v^{-\delta(P')}[P\le P']\). The pieces form a chain \(P_1<\dots<P_r\), so \(m_{P_i}=v^{\delta(P_i)}\big(h_K(P_i)-h_K(P_{i+1})\big)\): a difference of consecutive stalks (A2-8, script).

Use. (d) reduces equality of multiplicities to equality of finitely many stalk polynomials per \((L,K)\); A2-3 explains why this alone does not close the transfer, and A2-6 gives a numerical variant.

1. (a) BBD 6.1.7: given finitely many schemes, morphisms and constructible sheaves over \(\mathbb C\), they descend to \(S=\operatorname{Spec}A\), \(A\subset\mathbb C\) of finite type, and after shrinking \(S\) the usual operations commute with every base change \(S'\to S\) (generic base change, SGA 4½ [Th. finitude] 1.9). BBD 6.1.9–6.1.10: for \(S\) large enough, the categories \(D^b_{\mathcal T,\mathcal L}\) over \(\mathbb C\) and over a closed point \(s\) are equivalent, compatibly with perversity and finitely many chosen operations, and stalk ranks along strata are preserved. Applied to the data \((Y_L,\text{orbits},p_{\underline w},\underline k_{E_{\underline w}})\) this gives \(U(L,\underline w)\); its complement is a proper closed subset whose size BBD does not control in terms of \(\underline w\).
2. (b) The set of words is infinite and \(\underline w(x,S)\) is produced by the hypothesis at \(x\). If \(\dim R_0=1\), each \(\operatorname{Spec}R_0\setminus U(L,\underline w)\) is finite, but the union over \(\underline w\) can be all closed points. Even 'generation for all \(p\gg0\)' only supplies \(x\mapsto\underline w(x,S)\), and the needed relation \(x\in U(L,\underline w(x,S))\) is not implied. When \(\dim R_0>1\) the complement of \(U\) can lie over every prime (e.g. \(V(f)\subset\operatorname{Spec}\mathbb Z[f]\)), so even '\(p\gg0\)' is the wrong notion: goodness is a property of the closed point \(x\).
3. (c) Proof of the inequality. By A2-4 \(\psi(G^{\mathrm{BS}}_{\bar\eta})\simeq G^{\mathrm{BS}}_s\). Decompose \(G^{\mathrm{BS}}_{\bar\eta}\simeq\bigoplus S'_{\bar\eta}[-i]^{m}\) (DT over \(\mathbb C\), carried to \(\bar\eta\) by BBD 6.1.2 and the base change \(\bar\eta\subset\mathbb C\)); \(\psi\) is additive, so \(G^{\mathrm{BS}}_s\simeq\bigoplus\psi(S'_{\bar\eta})[-i]^m\). Each \(\psi(S'_{\bar\eta})\) is a summand of the semisimple \(G^{\mathrm{BS}}_s\) (DT at \(x\)), hence a sum of shifted simples (Krull–Schmidt). The coefficient of \(S_s\) in \(\psi(S_{\bar\eta})\) is 1 (A2-5, step 4). Counting summands \(S_s[-i]\) gives the displayed formula, all terms \(\ge0\).
4. The extra terms \([\psi(S'_{\bar\eta}):S_s]\), \(S'\neq S\), are the obstruction to the converse: \(S_s\) can be a summand of \(G^{\mathrm{BS}}_s\) only because some other \(S'_{\bar\eta}\) degenerates. They vanish under (ψIC\(_x\)) (A2-6); the toy A2-10 shows the analogous failure when stabilizers are not smooth.

Status: the obstruction to the argument is proved; no certified Iwahori example shows that the conclusion fails when only (UP) holds (see verdict).

1. Model. By (BS), \(G^{\mathrm{BS}}_{\underline w}(k)=\pi_L^*\,p_{k*}\underline k_{E_k}[d]\) for \(p=p_{\underline w}:E\to Y_L\) proper over \(R_0\) (base-changed to \(V\)) and \(k=\bar\eta,s\) (and \(\mathbb C\)). It suffices to treat \(p_*\underline k_E\) on \(Y_L\); the level pull-back is compatible with the transitions, which are smooth over \(R_0\).
2. Proper base change for nearby cycles: for \(p\) proper over \(V\), \(\psi_{Y}\circ Rp_{\bar\eta*}\simeq Rp_{s*}\circ\psi_E\) (SGA 7 II, Exp. XIII, 2.1.7).
3. Smooth case: for \(E\) smooth over \(V\), \(\psi_E(\underline{\overline{\mathbb Q}}_{\ell})\simeq\underline{\overline{\mathbb Q}}_{\ell}\) (SGA 7 II, Exp. XIII, 2.1.5; more generally \(\psi\) commutes with smooth pull-back). Hence \(\psi(p_{\bar\eta*}\underline k)\simeq p_{s*}\underline k\).
4. Shifts: the shift \([d]\) in (BS) is one integer for all \(k\) (it is \(r\) plus the level normalization \(e_M\), neither depending on \(k\)), so the normalizations over \(\bar\eta\) and \(s\) agree.
5. \(\bar\eta\) vs \(\mathbb C\): \(\overline{\operatorname{Frac}V}\subset\mathbb C\); for finite-type schemes over an algebraically closed field of characteristic 0, extension to a larger algebraically closed field commutes with \(Rp_*\) and preserves stalks, and Artin's comparison identifies \(\ell\)-adic and Betti constructible complexes compatibly with \(Rp_*\), \(p^*\), \(\otimes\) (BBD 6.1.2; the chain \(\alpha,\beta,\gamma,\delta\) of Bezrukavnikov–Finkelberg Prop. 5, whose step \(\beta\) is this base change and \(\gamma\) is BBD 6.1.2(B″)). Coefficients: \(G^{\mathrm{BS}}\) and the \(\mathrm{IC}(\overline P)\) (constant coefficients, by I2) are Betti complexes defined over \(\mathbb Q\), and the \(\mathrm{IC}(\overline P)\) are absolutely simple; Hom spaces commute with extension of the coefficient field, so multiplicities are the same with \(\mathbb Q\), \(\mathbb C\) or (via Artin comparison) \(\overline{\mathbb Q}_\ell\) coefficients; no abstract isomorphism \(\overline{\mathbb Q}_\ell\simeq\mathbb C\) is needed for dimensions.
6. Uniformity: steps 2–4 hold at every closed point \(x\) of \(\operatorname{Spec}R_0\), because \(E_{\underline w}\) is smooth and \(p_{\underline w}\) proper over all of \(R_0\). No localization depending on \(\underline w\) occurs; this is exactly what BBD 6.1.10 alone does not give (A2-3(a)).

Remark. The same argument with \(\mathrm{IC}_w\) in place of \(\underline k_E\) would need \(\psi(\mathrm{IC}_{w,\bar\eta})\simeq\mathrm{IC}_{w,s}\) on Schubert varieties, which is true but requires Kazhdan–Lusztig input (A2-9); Lemma 1 of A2-1 avoids it.

1. Bijection. By (UP\(_x\)) the orbits at each level over \(\mathbb C\) and \(\overline{\mathbb F}_x\) are the fibres of the same \(o_R\), with the same closure relations, compatibly with transitions; with I2 on both sides, \(\mathcal S_I(k)=\{\mathrm{IC}(\overline P)\}\) (summand.html step 5), so \(\beta_x\) is a bijection.
2. Let \(S=\mathrm{IC}(\overline P)\), \(P=\pi_L^{-1}(o)\), with \(S[-i]\) a summand of \(G^{\mathrm{BS}}_{\underline w}(\mathbb C)\). Work at a common level containing both, non-equivariantly (Lemma 2 of A2-1). By A2-4 step 5, \(S_{\bar\eta}[-i]\) is a summand of \(G^{\mathrm{BS}}_{\bar\eta}\).
3. Apply \(\psi\) (additive): \(\psi(S_{\bar\eta})[-i]\) is a summand of \(\psi(G^{\mathrm{BS}}_{\bar\eta})\simeq G^{\mathrm{BS}}_s\) (A2-4). By (DT) at \(x\), \(G^{\mathrm{BS}}_s\) is a finite sum of shifted simples; \(D^b_c(Y_{L,s})\) is Krull–Schmidt, so \(\psi(S_{\bar\eta})\simeq\bigoplus_b S_b[n_b]\) with \(S_b\) simple.
4. Identify one summand. Let \(U=Y_L\setminus(\overline{o_R}\setminus o_R)\), open. \(S_{\bar\eta}|_{U}=i_{o*}\underline k_o[\delta]\) with \(o\) closed in \(U\). \(\psi\) commutes with restriction to opens (étale base change), with \(i_*\) for closed immersions (2.1.7), and \(\psi(\underline k_{o})=\underline k_{o_s}\) since \(o_R\) is smooth over \(V\) (SGA 7 XIII 2.1.5). So \(\psi(S_{\bar\eta})|_{U_s}\simeq i_*\underline k_{o_s}[\delta]\). Also \(\operatorname{supp}\psi(S_{\bar\eta})\subseteq(\overline{o_R})_s=\overline{o_s}\) (proper base change for the closed immersion, and (UP\(_x\))).
5. Among simple objects supported in \(\overline{o_s}\), only \(\mathrm{IC}(\overline{o_s})=\beta_xS\) is nonzero on \(o_s\), where it is \(\underline k[\delta]\) (I2: no other local systems). Restricting the decomposition of step 3 to \(U_s\) gives \(\underline k_{o_s}[\delta]=\bigoplus_{b:S_b=\beta_xS}\underline k_{o_s}[\delta+n_b]\); so exactly one \(b\) has \(S_b=\beta_xS\), with \(n_b=0\).
6. Therefore \(\beta_xS[-i]\) is a summand of \(G^{\mathrm{BS}}_s\). Doing this for every summand of the decomposition over \(\mathbb C\) (Krull–Schmidt, distinct summands of \(G^{\mathrm{BS}}_{\bar\eta}\) map to distinct summands \(\psi(\cdot)\) of \(G^{\mathrm{BS}}_s\)) gives the coefficientwise inequality.
7. Conclusion. If Betti generation holds, \(\mathcal M_I(\mathbb C)=\mathcal S_I(\mathbb C)\) (Prop. III, (⇒) uses (DT) over \(\mathbb C\)); by Lemma 1 of A2-1 and steps 2–6, \(\mathcal M_I(\overline{\mathbb F}_x)\supseteq\beta_x\mathcal S_I(\mathbb C)=\mathcal S_I(\overline{\mathbb F}_x)\); Prop. III (⇐, no (DT) needed) gives \(\ell\)-adic generation at \(x\).

Uniformity. Only (UP\(_x\)) at the level of \(S\) and (BS) enter; the witness \(\underline w\) is the one found over \(\mathbb C\) and imposes no condition on \(x\). This direction is therefore free of the circularity of A2-3.

Proof of the theorem.

1. Work at a level \(L\) containing \(G^{\mathrm{BS}}_{\underline w}\), non-equivariantly (Lemma 2 of A2-1). By (DT) over \(\mathbb C\), transported to \(\bar\eta\) (A2-4 step 5): \(G^{\mathrm{BS}}_{\bar\eta}\simeq\bigoplus_{S'\in\mathcal S^{(L)},\,i}S'_{\bar\eta}[-i]^{\oplus m_{S',i}}\), \(m_{S',i}\) the Betti multiplicities.
2. By A2-4, \(G^{\mathrm{BS}}_s\simeq\psi(G^{\mathrm{BS}}_{\bar\eta})\simeq\bigoplus\psi(S'_{\bar\eta})[-i]^{\oplus m_{S',i}}\).
3. By (ψIC\(_x\)) at level \(L\), \(\psi(S'_{\bar\eta})\simeq\mathrm{IC}(\overline{o'_s})=\beta_xS'\) (the level pull-back commutes with \(\psi\) since the transitions are smooth and closed over \(R_0\)). So \(G^{\mathrm{BS}}_s\simeq\bigoplus\beta_xS'[-i]^{\oplus m_{S',i}}\).
4. \(\beta_x\) is injective ((UP\(_x\))), and multiplicities of shifted simples in a semisimple object are unique (Krull–Schmidt, or \({}^pH^i\)). Hence \(m_x(\beta_xS',\cdot)=m_{\mathbb C}(S',\cdot)\) for \(G^{\mathrm{BS}}_{\underline w}\).
5. For \(G_w\): \(G^{\mathrm{BS}}_{\underline w}\simeq G_w\oplus\bigoplus_{v<w}G_v\otimes V_v\) with the same graded \(V_v\) over \(\mathbb C\) and \(\overline{\mathbb F}_x\) (Kazhdan–Lusztig: \(V_v\) are given by the expansion of \(\underline H_{s_1}\cdots\underline H_{s_r}\) in the KL basis, independently of the field; Kazhdan–Lusztig 1980). Induction on \(\ell(w)\) then gives \(m_x(\beta_xS,G_w)=m_{\mathbb C}(S,G_w)\). (Only this sentence uses KL; the boxed equivalence needs only step 4 and Lemma 1.)
6. Equivalence: by Lemma 1 of A2-1 and step 4, \(\beta_x\mathcal M_I(\mathbb C)=\mathcal M_I(\overline{\mathbb F}_x)\); \(\beta_x\mathcal S_I(\mathbb C)=\mathcal S_I(\overline{\mathbb F}_x)\); Prop. III over each field.

Why this resolves A2-3. The bad set for comparing \((S,\underline w)\) is contained in the set where (UP) or (ψIC) fail at some level, which does not involve \(\underline w\): the generator side is uniform by A2-4. The extra terms \([\psi(S'_{\bar\eta}):S_s]\) of A2-3(c) vanish by step 3.

Proof of the numerical variant.

1. Fix \(S\) at level \(L_0\) and \(\underline w\); let \(L\ge L_0\) contain \(G^{\mathrm{BS}}_{\underline w}\). BBD 6.1.7 and 6.1.10 give a dense open \(U(L,\underline w)\) on which stalk ranks of \(G^{\mathrm{BS}}_{\underline w}\) and all \(\mathrm{IC}(\overline P)\), \(P\in\mathcal P_L\), agree with \(\mathbb C\).
2. \(\mathcal X\) is Zariski-dense, so pick \(x'\in\mathcal X\cap U(L,\underline w)\). For any \(x\in\mathcal X\): \(h^{x}=h^{x'}=h^{\mathbb C}\) for these objects by (US) and step 1.
3. A2-2(d) gives \(m_x(\beta_xS,G^{\mathrm{BS}}_{\underline w})=m_{\mathbb C}(S,G^{\mathrm{BS}}_{\underline w})\); conclude as in step 6.

The witness \(\underline w\) enters only through the auxiliary point \(x'\), never through \(x\): this is how \(p\)-independence on \(\mathcal X\) breaks the circularity (the same mechanism as Kazhdan–Lusztig's transfer of \(P_{y,w}\) from \(\overline{\mathbb F}_p\) to \(\mathbb C\)).

**Sufficient conditions for (ψIC\(_x\)).** (i) Every orbit closure \(\overline{o_R}\) is smooth over \(R_0\) near \(x\): then \(\mathrm{IC}=\underline k[\dim]\) and \(\psi(\underline k)=\underline k\). (ii) Every \(\overline{o_R}\) has an equivariant resolution \(\rho:\widetilde Z\to\overline{o_R}\), proper with \(\widetilde Z\) smooth over \(R_0\), whose decomposition multiplicities over \(\bar\eta\) and \(s\) have equal total counts (e.g. given by a field-independent formula): then \(\psi(\rho_*\underline k)=\rho_*\underline k\), each \(\psi(\mathrm{IC}_{\bar\eta})\) contains \(\mathrm{IC}_s\) once (as in A2-5 steps 3–5), and equality of total counts leaves no room for extra summands; this applies to every IC occurring in \(\rho_*\underline k\), in particular \(\mathrm{IC}(\overline o)\).

1. (a) \(H_L\) is a connected algebraic group with finitely many orbits on \(Y_L\) (I1). A strongly \(H_L\)-equivariant coherent \(\mathcal D_{Y_L}\)-module is regular holonomic (Hotta–Takeuchi–Tanisaki, D-modules, perverse sheaves and representation theory, Thm 11.6.1): its characteristic variety lies in the union of conormal bundles to the orbits, which is Lagrangian, and regularity holds along each orbit. Cohomology objects of the equivariant derived category are equivariant modules, so (a) follows.
2. Trap. The statement fails for weaker notions: \(\mathcal D_{Y}\) itself is a weakly equivariant coherent module that is not holonomic, and coherent-but-not-holonomic objects have no constructible counterpart. 'Coherent \(\mathcal D\)-modules' on the Iwahori pages must be read as strongly equivariant (or as holonomic), otherwise the category is not the one RH matches.
3. (b) Equivariant Riemann–Hilbert at a finite level: \(\mathrm{DR}:D^b_{H}(\mathcal D_Y)_{\mathrm{rh}}\xrightarrow{\sim}D^b_{H,c}(Y,\mathbb C)\), compatible with \(f_*,f_!,f^*,f^!,\boxtimes\) for \(H\)-equivariant maps (Kashiwara–Mebkhout RH, equivariant form: Bernstein–Lunts LNM 1578; Kashiwara, Equivariant derived category and representation of real semisimple Lie groups, LNM 1931 — cited, not re-proved; result number not checked locally).
4. Compatibility with levels: the transitions are smooth affine-space fibrations (pull-back) and closed immersions (push-forward); DR commutes with both up to the relative-dimension shift, so the \(\mathrm{DR}_L\) glue to a functor on the union, fully faithful and essentially surjective level by level.
5. Generators: \(\mathrm{IC}^{\mathcal D}_w\) (the minimal extension of \(\mathcal O\) on \(IwI/I\)) maps to \(\mathrm{IC}_w\), \(\mathrm{IC}^{\mathcal D}_0\) (structure sheaf of \(L^+X\) at finite level) to the constant sheaf, and the convolution \(m_!(-\widetilde\boxtimes-)\) is computed at finite level by \(\boxtimes\), smooth descent along the twisted product, and proper push-forward, all compatible with DR. So \(\mathrm{DR}(G^{\mathcal D}_w)\simeq G_w(\mathbb C)\).
6. (c) An equivalence of triangulated categories preserves thick closures and identifies the generating sets, so the two generation statements are equivalent; combine with A2-6.

Gap recorded. The local sources do not contain the equivariant RH statement with a result number; it is used as cited background, at finite type only.

1. (a) Orbits: Iwahori row reduction puts \(v\) with \(\min_i\operatorname{val}v_i=m\), attained last at \(j\), into the form \(t^me_j\); the argument uses only that \(\mathcal O^\times\) acts transitively on units, valid over \(\mathcal O=k[[t]]\) for every field \(k\) (case-affine.html, step 1). The stabilizer in \(H_{N,M}\) of \(t^me_j\bmod t^M\) is cut out by linear conditions on column \(j\) inside the smooth connected group \(H_{N,M}\) (unit diagonal entries, free off-diagonal entries subject to Iwahori conditions): an affine space times a connected group, over \(\mathbb Z\) (case-affine.html, step 2). Closures \(\overline{O_{m,j}}=\Lambda_{m,j}\) are \(\mathbb Z\)-linear subspaces of \(Y_{N,M}\simeq\mathbb A^{(N+M)n}_{\mathbb Z}\) and their fibres are the closures over every field.
2. (b) \(g_b\mathcal O^n\) depends only on \(g_bI\), and for \(N,M\) large (depending on \(\underline w\)) \(t^M\mathcal O^n\subseteq g_b\mathcal O^n\subseteq t^{-N}\mathcal O^n\); the condition \(\bar y\in g_b\mathcal O^n/t^M\) defines a subbundle of constant rank of the trivial bundle, so \(E_{\underline w}\) is smooth over \(\mathbb Z\); \(p_{\underline w}\) is the restriction of the projection \(\mathrm{BS}\times Y\to Y\), proper. This is the page's \(E_w\) (summand.html, (FM\(_w\)); audit A7-4) pulled back along \(\mathrm{BS}(\underline w)\to\mathrm{Fl}_{\le w}\).
3. (c) Smooth closures give constant IC sheaves; \(\psi\) of a constant sheaf on a scheme smooth over \(V\) is constant (SGA 7 XIII 2.1.5).
4. (H\(_k\)): (Lev), I1, I2 hold over every field by 1; (DT) holds over \(\overline{\mathbb F}_p\) by Prop. IV (Sun) and over \(\mathbb C\) by Bernstein–Lunts, as on the pages. So A2-5 and A2-6 apply with \(x\) any prime, including \(p=2\) (no trace argument, hence no '\(q\) odd' condition, is used).
5. (d) \(I=\mathcal O^\times\), \(\mathrm{Fl}\) is discrete, \(G_k=\Delta_{t^k}\ast\mathrm{IC}_0=\underline k_{t^k\mathcal O}\), and \(t^k\mathcal O\) is the closure of the piece \(t^k\mathcal O^\times\).
6. (e) \(G_\tau\): \(I\tau I=\tau I\), so \(G_\tau=\underline k_{\tau\mathcal O^2}\), \(\tau\mathcal O^2=\Lambda_{0,1}\). \(G_{s_1}\): \(\mathrm{Fl}_{\le s_1}=K/I\simeq\mathbb P^1\) and \(K\mathcal O^2=\mathcal O^2\), so \(G_{s_1}=\underline k_{\mathcal O^2}\otimes H^*(\mathbb P^1)[1]\). \(G_{s_0}\): \(\mathrm{Fl}_{\le s_0}\cdot\mathcal O^2=\{L:\Lambda_{0,1}\subset L\subset t^{-1}\Lambda_{0,1}=\Lambda_{-1,1},\ \dim L/\Lambda_{0,1}=1\}\simeq\mathbb P^1\). The fibre over \(y\) is \(\{L\ni y\}\): all of \(\mathbb P^1\) if \(y\in\Lambda_{0,1}\), one point if \(y\in\Lambda_{-1,1}\setminus\Lambda_{0,1}\), empty otherwise. Stalks \(k[1]\oplus k[-1]\), \(k[1]\), \(0\) match the displayed sum, which is semisimple, and by A2-2 the decomposition is unique. Nothing depends on the field.
7. (f) Script `a2_counts.py` enumerates \(\mathbb F_p\)-galleries in the lattice-chain model of \(\mathrm{Fl}\) for \(\mathrm{GL}_2\) (\(s_1\) moves \(L_1\) between \(tL_0\) and \(L_0\); \(s_0\) moves \(L_0\) between \(L_1\) and \(t^{-1}L_1\)), counts those with \(y\in L_0\) for \(y=t^me_j\) and \(y=0\), and inverts the chain as in A2-2. Results (multiplicity of \(\underline k_\Lambda\) as a polynomial in \(q\), exact fit at 5 primes with one degree of over-determination):

| word | \(\Lambda\): multiplicity |

|---|---|

| \(\tau\) | \(\Lambda_{0,1}\): 1 |

| \(s_1\) | \(\mathcal O^2\): \(1+q\) |

| \(s_0\) | \(\Lambda_{-1,1}\): 1; \(\Lambda_{0,1}\): \(q\) |

| \(s_0s_1\) | \(\Lambda_{-1,1}\): \(1+q\); \(\Lambda_{0,1}\): \(q+q^2\) |

| \(s_1s_0\) | \(t^{-1}\mathcal O^2\): 1; \(\mathcal O^2\): \(2q\); \(t\mathcal O^2\): \(q^2\) |

| \(s_0s_1s_0\) | \(\Lambda_{-2,1}\): 1; \(\Lambda_{-1,1}\): \(3q\); \(\Lambda_{0,1}\): \(3q^2\); \(\Lambda_{1,1}\): \(q^3\) |

| \(s_1s_0s_1\) | \(t^{-1}\mathcal O^2\): \(1+q\); \(\mathcal O^2\): \(2q+2q^2\); \(t\mathcal O^2\): \(q^2+q^3\) |

These are Frobenius-trace shadows (a check, not a proof); the proof of \(p\)-independence is (a)–(c) with A2-6.

1. Consistency: generation holds over every field directly, since every \(\Lambda_{m,j}=\tau^k\mathcal O^n\) and \(G_{\tau^k}=\underline k_{\tau^k\mathcal O^n}\) is the simple object of \(\overline{O_{m,j}}\) up to shift (summand.html, remarks; A2-7 of the earlier run). The theorem's prediction (equivalence of the two sides) is consistent.

1. (a) \(I_H\backslash LH/I_H=\widetilde W\) over every field (Bruhat–Tits/Iwahori decomposition, valid for split \(H\) over any \(k\)); the cells \(I_HvI_H/I_H\simeq\mathbb A^{\ell(v)}\) and Schubert closures are defined over \(\mathbb Z\) with field-independent closure order. The stabilizer of \(vI_H\) in \(I_H\) is \(I_H\cap vI_Hv^{-1}\), a product of root subgroups and the torus part, smooth with connected fibres over \(\mathbb Z\) (summand.html, I2 for the group; Chen–Yi Lemma 71(i)). Truncation in \(\Omega\) makes each level have finitely many pieces (audit A8-1).
2. (b) Demazure varieties are iterated \(\mathbb P^1\)-bundles over \(\mathbb Z\); twisted products of them with the finite-type quotient \(Y_{N,M}\) are smooth and the multiplication map is proper (summand.html step 11, group).
3. (c) Contrapositive of A2-5: if Betti generation held, \(\ell\)-adic generation would hold at every \(x\) satisfying (UP\(_x\)), in particular at \(x=(p)\), \(p\) odd, contradicting the earlier run. This is an independent proof of Betti failure from the \(\ell\)-adic one, using no Whittaker averaging.
4. (d) Let \(\rho=\pi_{\underline v}:\mathrm{BS}(\underline v)\to\overline{I_HvI_H}/I_H\). \(\psi(\rho_*\underline k)=\rho_*\underline k\) (A2-4). The decomposition multiplicities of \(\rho_*\underline k\) are given by the expansion of \(\underline H_{s_1}\cdots\underline H_{s_r}\) in the Kazhdan–Lusztig basis, over \(\overline{\mathbb F}_p\) for every \(p\) (Kazhdan–Lusztig 1980, via purity and point counts; affine flag case by the same argument) and over \(\mathbb C\) (by BBD 6.1.10 at one auxiliary large prime and \(p\)-independence, the mechanism of A2-6, numerical variant). Equal totals and the sufficient condition (ii) of A2-6 give \(\psi(\mathrm{IC}_{v,\bar\eta})\simeq\mathrm{IC}_{v,s}\). Level pull-backs are compatible.
5. Consistency: A2-6 says ℓ-adic and Betti generation are equivalent at every \(x\); both fail. The prediction matches Table 1 of the earlier run.

Caveat. The Kazhdan–Lusztig input for affine Schubert varieties is cited background, not re-proved here; (c) does not use it.

1. Orbits: \(t\mapsto t^2\) is surjective on \(\overline k^\times\) for every algebraically closed \(k\), so the orbits are \(\{0\}\) and \(\mathbb G_m\). Stabilizer of \(1\): \(\{t:t^2=1\}=\mu_2\); over \(\mathbb Z[1/2]\) it is étale with two points, over \(\mathbb F_2\) it is \(\operatorname{Spec}\mathbb F_2[t]/(t-1)^2\), connected and non-reduced.
2. Simple equivariant perverse sheaves are \(\mathrm{IC}(\bar o,\xi)\) with \(\xi\) an irreducible representation of \(\pi_0\) of the (reduced) stabilizer (BBD 4.3.1 and summand.html step 2; the étale site does not see nilpotents, cf. trace.html step 3 'inseparability'). (a): \(\pi_0=1\), so \(\mathcal S=\{\delta_0,\underline k_Y[1]\}\). (b): \(\pi_0=\mathbb Z/2\), so in addition \(\mathrm{IC}(\mathcal L)=j_{!*}\mathcal L[1]\) with \(\mathcal L\) the sign representation.
3. (a) Proposition I(a) (summand.html; finite type, finitely many orbits, connected group, no weights used) gives \(\mathcal C=\operatorname{Thick}(\mathcal S)=\operatorname{Thick}(\mathcal G)\).
4. (b) The generators are simple, so the summand set is \(\mathcal M=\{\delta_0,\underline k_Y[1]\}=\mathcal M^+\). Proposition I(b) gives \(\mathcal C^{\mathcal G}\subseteq\operatorname{Thick}(\mathcal M^+)=\mathcal D_{\mathcal M^+}\), the objects all of whose perverse constituents lie in \(\mathcal M^+\); \(\mathrm{IC}(\mathcal L)\) is simple and not in \(\mathcal M^+\), so it is not in \(\mathcal C^{\mathcal G}\).
5. Point counts confirm the asymmetry: \(\#\big(\mathbb F_q^\times/(\mathbb F_q^\times)^2\big)=2\) for \(q\) odd (two rational orbits in the geometric orbit, the shadow of \(\pi_0\mu_2=\mathbb Z/2\)), and \(=1\) for \(q=2^r\) (squaring is bijective).
6. What it certifies. (i) The single prime in A2-6 cannot be arbitrary: generation at \(x=(2)\) holds while Betti generation fails. (ii) I2 at \(x\) does not imply I2 over \(\mathbb C\): the simple sets have different sizes, so no bijection \(\beta_x\) exists — the smoothness of stabilizers in (UP\(_x\)) is what makes \(\beta_x\) exist. (iii) It realizes the failure mode of A2-3(c) in the other direction: \(\psi\) kills \(\mathrm{IC}(\mathcal L)\) (wild Kummer cover \(y=x^2\) at 2; \(\psi(f_*\underline k)=\underline k\) for the radicial \(f\)), so generic summands are lost at \(s\).
7. Limitation. This is a finite-type toy, not a loop-space example: for \(X=\mathbb A^1\) under \(\mathbb G_m\) with weight 2, the Iwahori levels have infinitely many orbits in characteristic 2 (squares of \(k[[t]]^\times\) are series in \(t^2\)), so I1 fails at \(p=2\) there, and I2 fails over \(\mathbb C\). An Iwahori-level example with (UP) at \(x\) but failure of (ψIC\(_x\)) was not found.

1. Logic: Prop. III reduces generation to \(\mathcal M_I=\mathcal S_I\) on each side; Lemma 1 (A2-1) computes \(\mathcal M_I\) with Bott–Samelson generators; A2-4 specializes these generators uniformly in \(w\); A2-5 gives special ≥ generic with only (UP); A2-6 gives equality under (ψIC); A2-7 passes to \(\mathcal D\)-modules.
2. Answer to 'one / infinitely many / all \(p\gg0\)': one closed point \(x\) where (UP\(_x\)) and (ψIC\(_x\)) hold at all levels. '\(p\gg0\)' must be read as '\(x\) outside the bad-reduction locus of the level stratifications', which is independent of \((S,w)\); for \(\dim R_0>1\) goodness is a property of the closed point, not of \(p\).
3. Hypothesis roles: (UP\(_x\)) makes \(\beta_x:\mathcal S_I(\mathbb C)\to\mathcal S_I(\overline{\mathbb F}_x)\) exist and makes \(\psi(S_{\bar\eta})\supseteq\beta_xS\); (BS) removes the \(w\)-dependence of bad primes; (ψIC\(_x\)) kills the cross terms \([\psi(S'_{\bar\eta}):S_s]\), \(S'\ne S\); (DT) on both sides is needed for Prop. III (⇒) and for Krull–Schmidt counting.

Gaps (explicit).

• Whether (ψIC\(_x\)) can be dropped when (UP\(_x\)) holds is open: no proof of \(\ell\)-adic ⇒ Betti without it, and no certified Iwahori counterexample.
• (UP), (BS), (ψIC) are hypotheses; verified for \(\mathbb A^n\) and the group only.
• Cited, not re-proved: SGA 7 XIII properties of \(\psi\) (2.1.5, 2.1.7); equivariant Riemann–Hilbert and HTT 11.6.1; Kazhdan–Lusztig field-independence for affine Schubert varieties (used only in A2-6 step 5 and A2-9(d)); (DT) over \(\mathbb C\) via Bernstein–Lunts as on the pages.

1. What the pages use. iwahori-generation/index.html §1: "\(k=\mathbb C\) … categories are coherent \(\mathcal D\)-modules", \(\mathcal C_I:=\mathcal D_c(I\backslash LX)\); \(\mathrm{IC}_0\) is "the \(!\)-extension of the constant \(\mathcal D\)-module on \(L^+X\)" (main-draft p. 16). The words holonomic and regular never occur. case-affine.html, remark "Assumed": that every object of \(\mathcal D_c(I\backslash L\mathbb A^n)\) comes from some \(Y_{N,M}\) is a reading of the paper's colimit description, not derived. So (I1\(^{\mathcal D}\)) is a hypothesis, the \(\mathcal D\)-module counterpart of I1.
2. Singular levels. \(Y\) need not be smooth. Choose an \(H\)-equivariant closed embedding \(Y\subset V\) into a finite-dimensional representation (an \(H\)-stable finite-dimensional space of generators of \(\mathcal O(Y)\); \(Y\) affine in all examples, otherwise use a quasi-projective equivariant embedding); \(\mathcal D\)-modules on \(Y\) := \(\mathcal D_V\)-modules supported on \(Y\) (Kashiwara's equivalence), independent of the embedding.
3. Strong vs weak. "Equivariant" means strongly equivariant (Kashiwara §3: the differentiated \(H\)-action equals the action of the vector fields \(\xi_Y\), \(\xi\in\mathrm{Lie}\,H\)). Weak equivariance is not enough for anything below (A3-2, step 5).
4. Derived equivariant vs derived of the heart. Use \(D^b_H\), not \(D^b(\mathrm{Mod}_{H,\mathrm{coh}})\): for \(H=\mathbb G_m\), \(Y=\mathrm{pt}\), \(\operatorname{End}^*_{D^b_H}(\mathbb C)=H^*(B\mathbb G_m)=\mathbb C[u]\), \(\deg u=2\), while \(\operatorname{End}^*_{D^b(\mathrm{Vect})}(\mathbb C)=\mathbb C\); the convolution \(m_*\) of equivariant objects is defined in \(D^b_H\) (Kashiwara (4.5.3)). The generation question only meets the heart through composition factors (A3-4), where the two agree: both hearts are \(\mathrm{Mod}_{H,\mathrm{coh}}(\mathcal D_Y)\) resp. \(\mathrm{Perv}_H(Y)\).
5. Coherent vs compact. Stabilizers contain \(T\) (e.g. \(\operatorname{Stab}(0)=H\) for \(\mathbb A^n\)), so \(Y/H\) is not a 'safe' stack and compact objects of the DG category \(\mathcal D\text{-mod}(Y/H)\) are strictly fewer than coherent ones (Drinfeld–Gaitsgory, background). We follow the pages' word coherent: bounded with coherent cohomology.
6. Hearts. With finitely many orbits and connected \(H\), \(\mathrm{Perv}_H(Y)\) is a full subcategory of \(\mathrm{Perv}(Y)\) closed under subquotients, of finite length, with simple objects \(\mathrm{IC}(\bar o,\xi)\) (BBD 4.3.1); by I2, \(\xi\) trivial, so \(\mathcal S^{(N,M)}=\{\mathrm{IC}(\bar o)\}_{o\in Y/H}\).

1. **Good \(H\)-stable filtration.** \(H\) acts locally finitely on \(M\); pick a finite-dimensional \(H\)-stable generating subspace \(M_0\) and put \(F_jM=F_j\mathcal D_V\cdot M_0\). The order filtration is \(H\)-stable, so each \(F_jM\) is \(H\)-stable.
2. Characteristic variety. For \(\xi\in\mathrm{Lie}\,H\), strong equivariance says the vector field \(\xi_V\in F_1\mathcal D_V\) acts on \(M\) as the derivative \(d\rho(\xi)\), which preserves every \(F_jM\). So the principal symbol \(\sigma_1(\xi_V)\) kills \(\operatorname{gr}^FM\) and

$$\operatorname{Ch}(M)\subseteq\mu^{-1}(0)\cap T^*V|_Y=\bigcup_{o\subseteq Y}T^*_oV,\qquad\mu^{-1}(0)=\{(x,\eta):\langle\eta,\xi_V(x)\rangle=0\ \forall\xi\},$$

since \(\{\xi_V(x)\}=T_x(Hx)\). Finitely many orbits: a finite union of conormal bundles, each of dimension \(\dim V\). So \(\dim\operatorname{Ch}(M)\le\dim V\): \(M\) is holonomic.

1. Reduction to simples. Regular holonomic modules form a Serre subcategory of holonomic modules (HTT, Ch. 6), so it suffices that each composition factor is regular. A simple strongly equivariant holonomic module is \(j_{!*}L\), \(j:o\hookrightarrow V\) an orbit, \(L\) a strongly equivariant integrable connection on \(o=H/S_x\), i.e. a representation of the finite group \(\pi_0(S_x)\) (Kashiwara notes §1.2; trivial under I2).
2. Regularity of simples. Pulled back along the finite étale cover \(H/S_x^\circ\to o\), \(L\) becomes a sum of copies of \(\mathcal O\), which is regular; regularity descends along finite étale maps, so \(L\) is regular. \(j_{!*}L=\operatorname{im}(j_!L\to j_*L)\), and \(j_*,j_!\) preserve regular holonomicity (HTT Thm. 6.1.5). So \(j_{!*}L\) is regular.
3. Cited form. Kashiwara, Equivariant derived category…, §1.3 ("finitely many \(K\)-orbits. Therefore any coherent \(K\)-equivariant \(\mathcal D_X\)-module is regular holonomic", citing Kashiwara, Astérisque 173–174) and §7.14; Hotta–Takeuchi–Tanisaki Thm. 11.6.1 (recalled, not in the local sources).
4. Counterexample to the weak version. \(M=\mathbb C[x]\) with \(\partial\) acting by \(f\mapsto f'+f\). Translation \(\tau_a\): \(\tau_a^*M\simeq M\), so \(M\) is weakly \(\mathbb G_a\)-equivariant; it is irregular at \(\infty\) (exponential type), and \(H^*_{\mathrm{dR}}(\mathbb A^1,M)=0\) since \(f\mapsto f'+f\) is bijective on \(\mathbb C[x]\). It is not strongly equivariant: the derivative of an algebraic \(\mathbb G_a\)-action is locally nilpotent, while \(\partial=d/dx+1\) is invertible on \(\mathbb C[x]\). (This is the \(\exp\) of group.html; it lives in the Whittaker category, outside \(\mathcal C_I\).)

1. One level. By A3-2, \(D^b_{H,\mathrm{coh}}(\mathcal D_Y)=D^b_{H,\mathrm{rh}}(\mathcal D_Y)\), and the equivariant Riemann–Hilbert theorem (Kashiwara notes, Thm. 4.6.2, deduced from Kashiwara's Thm. 4.6.1) gives \(\mathrm{DR}_Y:D^b_{H,\mathrm{rh}}(\mathcal D_Y)\xrightarrow{\sim}D^b_{H,c}(\mathbb C_{Y^{an}})\), t-exact for the perverse t-structure on the right (\(\mathrm{Perv}:=\mathrm{DR}(\mathrm{RH})\), notes §1.3); simple objects go to simple objects, so \(\mathrm{IC}^{\mathcal D}(\bar o)\mapsto\mathrm{IC}(\bar o)\).
2. **Change of \(M\).** \(\pi:Y_{N,M'}\to Y_{N,M}\) is smooth of relative dimension \(d\) and \(H'\to H\) has unipotent kernel. \(\mathrm{DR}\) commutes with smooth pullback up to the shift \([d]\) and with \(f^*,f^!\) in general (Brav et al., Thm. 2.18, local source; HTT Thm. 7.1.1), and with restriction/inflation of the group (Kashiwara (4.5.4)–(4.5.6)).
3. **Change of \(N\).** \(i:Y_N\hookrightarrow Y_{N+1}\) closed: \(\mathrm{DR}\circ i_*\simeq i_*\circ\mathrm{DR}\) (same references). Steps 1–3: \(\mathrm{DR}\) is compatible with the fully faithful transition functors defining both unions, hence induces an equivalence of the unions (an equivalence on each term of a filtered union of full subcategories, compatible with inclusions, is an equivalence of the unions).
4. Generators. By bounded convolution, at a level \(G_w=f_*(\mathrm{IC}_w\mathbin{\widetilde\boxtimes}\mathrm{IC}_0)\) for a proper map \(f\) of finite-type quotients; \(\widetilde\boxtimes\) is an external product descended along a smooth quotient map. \(\mathrm{DR}\) commutes with \(\boxtimes\), smooth pullback/descent and proper \(f_*=f_!\) (Brav et al., Thm. 2.18; HTT 7.1.1). \(\mathrm{DR}(\mathrm{IC}^{\mathcal D}_w)=\mathrm{IC}_w\) (step 1) and \(\mathrm{DR}(\mathcal O_{L^+X})=\mathbb C_{L^+X}[\dim]\) at each level. Shifts are irrelevant for thick closures.
5. Thick closures. For an exact equivalence \(\Phi\), (T2) applied to \(\Phi\) and \(\Phi^{-1}\) gives \(\Phi(\operatorname{Thick}S)=\operatorname{Thick}(\Phi S)\). Apply to \(\Phi=\mathrm{DR}\), \(S=\{G_w^{\mathcal D}\}\).
6. Group. group.html proves \(\Delta^{\mathcal D}_e\notin\operatorname{Thick}\{\mathcal A\ast c\ast\mathcal B\}\) in \(\mathcal C_I^{\mathcal D}\) using \(\Phi=\mathrm{Av}_\psi\) with the irregular \(\exp\); \(\exp\) is used only inside the functor \(\Phi\), not as an object of \(\mathcal C_I^{\mathcal D}\). Step 5 transports the conclusion: \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}(\mathbb C)\), with no Betti exponential sheaf needed.
7. What is not claimed. Without I1\(^{\mathcal D}\), i.e. if the paper's \(\mathcal D_c(I\backslash LX)\) (a DG colimit on a placid ind-scheme) is not the union of the levels, nothing here identifies it with \(\mathcal C_I(\mathbb C)\).

1. A thick subcategory. Let \(\mathcal D_\Sigma\subseteq\mathcal C_I(k)\) be the objects all of whose perverse cohomology objects have all composition factors in \(\Sigma\) (hearts have finite length, A3-1 step 6). It is closed under shifts and isomorphisms; under summands since \({}^pH^i(A\oplus B)={}^pH^i A\oplus{}^pH^iB\); under cones since for \(A\to B\to C\) the exact \({}^pH^iA\to{}^pH^iB\to{}^pH^iC\) makes \({}^pH^iB\) an extension of a subobject of \({}^pH^iC\) by a quotient of \({}^pH^iA\), and \(\Sigma\)-composition factors form a Serre subcategory.
2. **(a) \(\Rightarrow\).** \(G_w\in\mathcal D_\Sigma\) by definition of \(\Sigma\), so \(\operatorname{Thick}\{G_w\}\subseteq\mathcal D_\Sigma\). A simple \(S\in\mathcal D_\Sigma\) has \({}^pH^0S=S\), so \(S\in\Sigma\).
3. **(a) \(\Leftarrow\) under (DT).** (DT): \(G_w\simeq\bigoplus_i{}^pH^i(G_w)[-i]\) with semisimple \({}^pH^i\), so a composition factor is a summand of \({}^pH^i(G_w)\), hence of \(G_w[i]\), hence in the thick closure.
4. (b). If \(\Sigma=\mathcal S_I\), every simple is in \(\operatorname{Thick}\); an object of a level category has finitely many nonzero \({}^pH^i\) of finite length, so truncation triangles and (T1) put it in \(\operatorname{Thick}(\mathcal S_I)\subseteq\operatorname{Thick}\{G_w\}\). Conversely use (a).
5. (c) Non-equivariant splitting. At a level, \(G_w=f_*K\), \(f\) proper, \(K=\mathrm{IC}_w\mathbin{\widetilde\boxtimes}\mathrm{IC}_0\) semisimple of geometric origin (BBD 6.2.4: constituents of \(\mathrm{IC}\)'s under \(f^*,\boxtimes\)). BBD Thm. 6.2.5: \(f_*K\) is semisimple of geometric origin, i.e. \(\simeq\bigoplus{}^pH^i[-i]\) with semisimple \({}^pH^i\).
6. (c) Equivariant splitting. Use Bernstein–Lunts approximation (Kashiwara notes Def. 4.2.3): for \(k\gg0\), \(D^{[a,b]}_H(Z)\simeq D^{[a,b]}((V_k\times Z)/H)\) fully faithfully, compatibly with \(f_*\) and perverse truncations up to the shift \(\dim V_k\). Apply step 5 to the proper map \((V_k\times Z)/H\to(V_k\times Y)/H\) of varieties (choose \(V_k\) with quasi-projective quotient) and the semisimple kernel \(K_k\). The splitting isomorphism lies in \(D^{[a,b]}\) with \([a,b]\) containing the amplitude, so it lifts to \(D_H\). Semisimplicity of the equivariant \({}^pH^i\): \(\mathrm{Perv}_H\subseteq\mathrm{Perv}\) is full and closed under subquotients for connected \(H\).
7. Comparison with the pages. This is Proposition III/summand criterion (props/summand.html), with (DT) proved rather than assumed at \(k=\mathbb C\). At \(k=\overline{\mathbb F}_p\), (DT) needs Sun Thm. 1.2 and a pure model (props/trace.html), but part (a) \(\Rightarrow\) needs nothing.

1. Finite-type data. For fixed \((N,M,w)\) only finitely many finite-type objects enter: \(H\), \(Y=Y_{N,M}\), the target level \(Y'=Y_{N',M''}\) of \(G_w\), the proper map \(f_w:Z_w\to Y'\) and kernel \(\mathrm{IC}_w\mathbin{\widetilde\boxtimes}\mathrm{IC}_0\), and one approximation \((V_k\times-)/H\) with \(k\) large for the bounded amplitude of \(G_w\) (A3-4 step 6); \(V_k\) is a Stiefel-type open of a representation, defined over \(\mathbb Z\).
2. Orbits. \(\operatorname{Gal}(\mathbb C/\operatorname{Frac}R)\) permutes the finitely many orbits; adjoining finitely many elements to \(R\) makes each orbit \(o\) an \(R\)-subscheme \(o_R\). By constructibility (EGA IV §9: surjectivity of \(H_R\to o_R\), smoothness, disjointness and covering, connectedness of geometric fibres of the stabilizer schemes (9.7.7), and formation of closures are constructible conditions holding at the generic point), they hold on a dense open. This is (i).
3. Sheaf equivalences. BBD Lemma 6.1.9 and (6.1.10.1) (p. 157): for the orbit stratification \(\mathcal T\) and constant \(\mathcal L\) (by I2), for \(\bar s\) in a dense open, \(D^b_{\mathcal T,\mathcal L}(X_{\mathbb C})\leftrightarrow D^b_{\mathcal T,\mathcal L}(X_{\bar s})\) are equivalences. BBD p. 158: after refining \((\mathcal T,\mathcal L)\) and shrinking, they commute with \(f_*,f_!,f^*,f^!,\otimes\) (via 6.1.7) and respect the perverse t-structures. The \(\overline{\mathbb Q}_\ell\) and étale/classical variants are BBD p. 158 and the remark after Thm. 6.2.5 (p. 163).
4. Matching generators. \(\mathrm{IC}\)'s are built from \(j_*\), \(i^*\) and perverse truncations, so they correspond; hence \(\mathrm{IC}_w\mathbin{\widetilde\boxtimes}\mathrm{IC}_0\) and \(G_w=f_{w*}(\cdots)\) correspond, on the approximation and so equivariantly.
5. Multiplicities. A t-exact equivalence matches perverse cohomology and simple objects, so the composition multiplicities agree. This is (ii).
6. Non-uniformity. BBD p. 158: each \(K\) defines only a germ of objects over \(S\); "on ne peut pas passer à la limite". \(U(N,M,w)\) shrinks with \((N,M,w)\), and \(\mathcal S_I\), \(\widetilde W\) are infinite.

1. (a). \(\mathcal S=\{S_0\}\), \(\widetilde W\supseteq\{w_1,w_2,\dots\}\) (infinitely many elements, as in any \(\widetilde W\) with \(X_*(T)\neq0\)), \(p_n\) the \(n\)-th prime. Put \(m_0\equiv0\), \(m_p(S_0,w_n)=1\) if \(p=p_n\) and \(0\) otherwise, \(m_p(S_0,w)=0\) for the other \(w\). (C) holds with \(B(S_0,w_n)=\{p_n\}\), \(B=\varnothing\) otherwise. At each \(p=p_n\) the witness is \(w_n\); at \(0\) there is none.
2. (b). \(\mathcal S=\{S_1,S_2,\dots\}\), \(m_0(S_n,w)=\delta_{w,w_n}\), and \(m_p=m_0\) except \(m_{p_n}(S_n,w_n)=0\). (C) holds with \(B(S_n,w_n)=\{p_n\}\). At \(0\) every \(S_n\) has witness \(w_n\); at \(p=p_n\), \(S_n\) has none.
3. Which hypothesis this isolates. In (a) the witness \(w_p(S_0)=w_n\) escapes every finite subset of \(\widetilde W\) as \(p\) varies, and the bad sets \(B(S_0,w)\) are unbounded in \(w\). Either control blocks the example: a \(p\)-independent finite set of candidate witnesses (A3-7), or a finite bad set uniform in \((N,M,w)\) (A3-7(c)).
4. Scope. This refutes an inference schema, not the geometric implication: no \((G,X)\) realizing (a) is constructed. It certifies that the "standard spreading-out argument" is incomplete exactly at the uniformity step.

1. Reduce to one orbit. By I2 the simples of \(\mathcal C_I(\mathbb C)\) are the \(\mathrm{IC}(\bar o)\), pulled back along the levels; by A3-4(b),(c) Betti generation \(\iff\) each is a composition factor of some \({}^pH^i(G_w(\mathbb C))\). Fix \(o\) at level \((N,M)\); enlarging the level, assume \(S=\mathrm{IC}(\bar o)\) and all \(G_w\), \(w\in F(o)\), live at one level (finitely many \(w\); bounded convolution).
2. Pigeonhole. \(F(o)\) finite and infinitely many \(p\in\mathcal P\) have a witness in \(F(o)\), so one \(w\in F(o)\) is a witness for an infinite \(\mathcal P'\subseteq\mathcal P\).
3. Avoid the bad set. \(B(N,M,w)\) is finite (A3-5); choose \(p\in\mathcal P'\setminus B(N,M,w)\). Then \(o_p\) is the orbit matched to \(o\) (A3-5(i)) and \([{}^pH^i(G_w(\overline{\mathbb F}_p)):\mathrm{IC}(\bar o_p)]\neq0\) for some \(i\).
4. Transfer. A3-5(ii): \([{}^pH^i(G_w(\mathbb C)):\mathrm{IC}(\bar o)]\ne0\). A3-4(c): \(\mathrm{IC}(\bar o)\) is a summand of \(G_w(\mathbb C)[i]\), so it lies in \(\operatorname{Thick}\{G_w\}\). All \(o\): Betti generation by A3-4(b). \(\mathcal D\)-modules: A3-3.
5. The implied form of (BW). If \(\mathrm{IC}(\bar o_p)\in\operatorname{Thick}\{G_w:w\in F(o)\}\), then A3-4(a), run with the generator set \(\{G_w\}_{w\in F(o)}\) (the Serre argument works for any set of generators), makes it a composition factor of some \(G_w\), \(w\in F(o)\).
6. (b). Betti generation and A3-4(a) give \(w(o)\) with \([{}^pH^i(G_{w(o)}(\mathbb C)):\mathrm{IC}(\bar o)]\neq0\); by A3-5(ii) the same holds at every \(p\notin B(N,M,w(o))\). No (DT) at \(p\) is used.
7. (c). Steps 3–4 with the fixed \(p\notin B\) work for every \(o\) at once. Conversely, step 6 gives composition factors at every \(p\notin B\), and (DT) at \(p\) with A3-4(b) gives generation.
8. Sharpness. A3-6(a) satisfies everything except (BW) and (UGR): its witnesses \(w_n\) escape every finite set and its bad sets are unbounded. So neither "one \(p\)" nor "all \(p\gg0\)" suffices without one of the two uniformity hypotheses, at the level of the comparison mechanism.

1. Trace of a simple. props/trace.html, IC-trace basis: \(\operatorname{tr}\mathrm{IC}(P)=(-1)^{\delta(P)}\mathbf 1_{P(\mathbb F_p)}+\sum_{P'<P}c_{P'P}\mathbf 1_{P'(\mathbb F_p)}\), a finite sum over pieces of one level.
2. Fixed support. By (IG\(_F\)), \(\operatorname{tr}\mathrm{IC}(P)\in\operatorname{span}\{T_w\ast\varphi_0: w\in F_1\}\), \(F_1=\bigcup_{P'\le P}F(P')\), finite and independent of \(p\) (the \(c_{P'P}\) depend on \(p\); the support does not).
3. **From \(T_w\) to \(\mathrm{IC}_w\).** \(\operatorname{tr}\mathrm{IC}_w=\sum_{v\le w}\epsilon_{v,w}(q)\,T_v\) with \(\epsilon_{w,w}=\pm q^{-\ell(w)/2}\neq0\) (Kazhdan–Lusztig), triangular in the Bruhat order; so \(\operatorname{span}\{T_v:v\in F\}=\operatorname{span}\{\operatorname{tr}\mathrm{IC}_v:v\in F\}\) for the finite lower set \(F=F(o)\) of the statement. With trace–convolution compatibility (props/trace.html, step "Trace converts the sheaf action"), \(\operatorname{tr}\mathrm{IC}(P)=\sum_{w\in F}a_w\operatorname{tr}G_w\).
4. Coefficients. In \(K_0\) of Weil perverse sheaves, \([G_w]=\sum_i(-1)^i[{}^pH^iG_w]\), and each Weil composition factor of \({}^pH^iG_w\) is geometrically \(S'\otimes V\) with \(S'\) Frobenius-stable ((Dict)(c)) and \(V\) a Frobenius module. So \(\operatorname{tr}G_w=\sum_{S'}e_{S',w}\operatorname{tr}S'\), with \(e_{S',w}\neq0\) only if \(S'\) is a composition factor of some \({}^pH^iG_w\). This uses no splitting, so no (DT).
5. Extract a witness. \(\{\operatorname{tr}S'\}\) is a basis (props/trace.html). Comparing coefficients of \(\operatorname{tr}\mathrm{IC}(P)\): \(1=\sum_{w\in F}a_we_{\mathrm{IC}(P),w}\), so \(e_{\mathrm{IC}(P),w}\neq0\) for some \(w\in F(o)\). This holds for every \(p\in\mathcal P\): (BW).
6. Conclude by A3-7(a).
7. Remark (generic form). If, at a level, the entries of the finite spanning matrix of (IG\(_F\)) are polynomials in \(q\) (as for \(\mathbb A^n\)), full rank at infinitely many \(q\) is equivalent to full rank over \(\mathbb Q(q)\), hence at all but finitely many \(q\). Polynomiality is not proved in general; this remark is heuristic outside the examples.

1. **\(\mathbb A^n\), geometry.** case-affine.html: \(Y_{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n\), \(I_{N,M}=1+t^{N+M}M_n(\mathcal O)\), all defined over \(\mathbb Z\). The pieces form a chain of \((N+M)n+1\), and the stabilizers \(\{g:ge_j\equiv e_j\bmod t^{M-m}\}\) are an affine space times a connected group. These proofs use no property of \(k\), so I1, I2 hold over \(\mathbb C\) and every \(\overline{\mathbb F}_p\), with orbit bijection at every level and every \(p\). (Dict) for \(\mathbb A^n\): props/trace.html, "Where it holds".
2. **\(\mathbb A^n\), functions.** case-affine.html step 3: \(\Delta^{\mathrm{Fl}}_{t^\lambda}\ast_I\mathrm{IC}_0\simeq\underline k_{\Lambda_{m,j}}[-2\ell]\) via fibres \(\mathbb A^{\ell}\), \(\ell=\ell(t^\lambda)\); with the Weil structure, \(H^*_c(\mathbb A^\ell)\) has trace \(q^\ell\), so \(T_{t^\lambda}\ast\varphi_0=q^{\ell}\mathbf 1_{\Lambda_{m,j}}\). Subtract consecutive lattices: \(\mathbf 1_{O_{m,j}}=\mathbf 1_{\Lambda_{m,j}}-\mathbf 1_{\Lambda_{m,j-1}}\). The supports are independent of \(p\).
3. **\(\mathbb A^n\), direct route.** IV.1's triangles \(\Delta_{O_{m,j}}\to\underline k_{\Lambda_{m,j}}\to\underline k_{\Lambda_{m,j-1}}\) hold verbatim in \(\mathcal C_I(\mathbb C)\) and \(\mathcal C_I^{\mathcal D}\) (the pages work over \(\mathbb C\) with \(\mathcal D\)-modules). So \(\mathbb A^n\) needs no transfer, and the transfer theorem agrees with it.
4. Group, functions. group.html step 1: \(\varepsilon(T_w)=(-1)^{\ell(w)}\) kills \(\mathcal H_Ie_K\mathcal H_I\) and \(\varepsilon(T_e)=1\), for every \(q\). So \(\mathbf 1_{IeI}\notin\mathcal H_I\ast\varphi_0\ast\mathcal H_I\) at every \(p\).
5. **Group, \(\mathbb C\).** \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}(\mathbb C)\) by A3-3 step 6. By A3-7(a) (contrapositive), (BW) fails for the point orbit \(IeI\).
6. Characteristic-free reason (remark, recalled). By A3-4(a), \(\Sigma\) consists of the constituents of \(\mathrm{IC}_x\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_y\). Their multiplicities are the positive Kazhdan–Lusztig structure constants, the same over \(\mathbb C\) and \(\overline{\mathbb Q}_\ell\). \(\mathrm{IC}_e\) occurs only if \(e\le_{LR}w_0\), which is impossible for \(W\neq1\): Lusztig's \(a\)-function is monotone along \(\le_{LR}\), with \(a(e)=0<\ell(w_0)=a(w_0)\) (Lusztig, Cells in affine Weyl groups). This gives the group failure uniformly in \(k\), with no exponential sheaf.

1. Basis. \(\chi(\mathrm{IC}(P))|_P=[\mathbb Q(k)]^{\pm1}\) for some Tate twist, a unit of \(A\). Its support is \(\bar P\), so \(\{\chi(\mathrm{IC}(P))\}_P\) is unitriangular against the indicators, hence an \(A\)-basis of \(\operatorname{Fun}(\mathcal P_{N,M},A)\).
2. Decomposition. Saito: a pure complex splits as \(\bigoplus{}^pH^i[-i]\) in \(D^b\mathrm{MHM}\) with semisimple pure \({}^pH^i\). An \(H\)-equivariant polarizable VHS on an orbit with connected stabilizer is constant (I2), so \({}^pH^i(G_w)\simeq\bigoplus_{S}S\otimes E_{S,i}(w)\) with Hodge structures \(E_{S,i}(w)\), and \(\chi(G_w)=\sum_Se_{S,w}\chi(S)\) with \(e_{S,w}=\sum_i(-1)^i[E_{S,i}(w)]\in A\).
3. Extraction. (MH2) and step 1: \(\chi(\mathrm{IC}(P))=\sum_wa_w\chi(G_w)\). Taking the \(\mathrm{IC}(P)\)-coefficient gives \(1=\sum_wa_we_{\mathrm{IC}(P),w}\) in the nonzero ring \(A\), so some \(e_{\mathrm{IC}(P),w}\ne0\), hence some \(E_{\mathrm{IC}(P),i}(w)\neq0\). Its underlying vector space is the multiplicity space, so \(\mathrm{IC}(P)\) is a summand of \(G_w[i]\) (A3-4(c)).
4. Comparison with II(a). This is II(a)'s argument with \(\operatorname{tr}\mapsto\chi\), Weil structures replaced by Hodge structures, and Sun/BBD 5.4.5 replaced by Saito. It needs no spreading out.
5. Gap: the input. Over \(\mathbb F_q(\!(t)\!)\), (IG) is a statement about a \(p\)-adic group, accessible by representation theory (sign character, Steinberg, Speh). \(A\)-valued functions on pieces have no such group. In mixed-Tate situations \(\chi\) lands in \(\mathbb Q[\mathbb L^{\pm1}]\) and (MH2) becomes a statement over the generic Hecke algebra. Relating it to (IG) at specific \(q\) needs polynomial count and \(q\)-independent supports: the same uniformity as (BW) (A3-8 step 7). So this is an alternative route for II(b), not a way around the uniformity issue.
6. Inputs recalled, not locally checked. Saito, Mixed Hodge modules (1990): purity under proper pushforward and the decomposition theorem; equivariant MHM via Bernstein–Lunts approximation (Achar's notes, background).

1. Complex side. Strongly equivariant coherent \(=\) regular holonomic at each level (A3-2; weak equivariance fails, \(e^x\)). Equivariant RH (Kashiwara Thm. 4.6.2) matches \(\mathcal C_I^{\mathcal D}\) with \(\mathcal C_I(\mathbb C)\) and \(G_w^{\mathcal D}\) with \(G_w\), so the two complex generation statements are equivalent, given I1\(^{\mathcal D}\) (A3-3).
2. Criterion. Over any \(k\): a simple is in \(\operatorname{Thick}\{G_w\}\) iff it is a perverse composition factor of some \(G_w\). Over \(\mathbb C\), (DT) is BBD 6.2.5 plus approximation, so "summand" can replace "composition factor" (A3-4).
3. Comparison. For fixed (level, \(w\)), multiplicities agree for \(p\notin B(N,M,w)\), a finite set (BBD 6.1.9–6.1.10). It is not uniform (A3-5).
4. Uniformity. Toy systems show that generation for every \(p\), together with pairwise agreement, implies nothing, in either direction (A3-6). With (BW): pigeonhole, then avoid one finite bad set (A3-7). Answer to "one / infinitely many / all \(p\gg0\)": infinitely many \(p\) with \(p\)-independent finite witness sets. One \(p\) suffices only under a uniform bad set (UGR). "All \(p\gg0\)" without witness control is not enough for this mechanism.
5. Checks. \(\mathbb A^n\): (IG\(_F\)) with \(|F|\le2\) at every odd \(p\), so the theorem applies and agrees with the characteristic-free IV.1. Group: the hypotheses fail at every \(p\) and the conclusion fails over \(\mathbb C\) (A3-9).
6. Alternative route. Hodge stalk classes replace Frobenius traces in the II(a) extraction. That implication is proved; its input is not (A3-10).
7. Open. Whether ℓ-adic generation at all \(p\gg0\) forces (BW) for a genuine \((G,X)\): no geometric counterexample is known. (UGR) is not verified even for \(\mathbb A^n\). I1\(^{\mathcal D}\) (the paper's DG category is the union of levels) is assumed, as on the pages.

1. Lemma 1 step 1. pi_w: BS(w) -> Fl_{<=w} is proper with smooth source; over F̄_x the I-equivariant splitting is Sun Thm 1.2 (checked in sun-decomposition.pdf: f proper of finite diagonal between F_q-stacks of finite type with affine stabilizers, K_0 iota-pure => Rf_*K = ⊕ pR^if_*K[-i], each semisimple); here f = [I_L\BS] -> [I_L\Fl_{<=w}] after passing to a finite-type quotient, K_0 = constant sheaf (pure). Summands are IC_v with trivial local systems because I-stabilizers on Fl are connected. Multiplicity of IC_w is 1 since pi_w is an isomorphism over IwI/I. Over C: Bernstein-Lunts 5.3. Holds.
2. Step 2 needs m_w proper (it factors through Fl_{<=w} x~ L^+X -> LX, ind-proper), so m_* = m_! and G^BS = BS_w *_I IC_0 by associativity of twisted products. Holds.
3. Steps 3-4: G^BS ≅ G_w ⊕ ⊕_{v<w} G_v⊗V_v, each G_v a finite sum of shifted simples by (DT); C_I^(L) is Hom-finite (each Hom(A,B[n]) finite-dimensional) and idempotent complete (bounded t-structure, summand.html step 3), so Krull-Schmidt; a shifted simple summand of G^BS is a summand of some G_v. Holds; (DT) is used for every v<=w, not only for w.
4. Lemma 2: for connected H_L the forgetful functor Perv_{H_L}(Y_L) -> Perv(Y_L) is fully faithful with image stable under subquotients (BBD 4.2.5 applied to the action/projection; the exact sub-number 4.2.6.2 could not be checked: the local BBD PDF is a scan without text layer). Multiplicities read off pH^i agree. Holds.
5. (BS) for general X. To descend m_w to a proper map E_w -> Y_L of finite type one needs, for N,M large depending on w, that every translate g_b·ωL^+X (b in BS(w)) is pi_L-saturated (a union of fibres of pi_L) and that the family {pi_L(g_b ωL^+X)} is smooth (e.g. a vector subbundle) over BS(w)_{R_0}. For A^n both hold (g_bO^n ⊇ t^MO^n, lattice subbundle, A2-8(b)); for the group they hold by Demazure twisted products (A2-9(b)). For a general smooth affine X neither is automatic (pi_L-saturation fails when I_L is not normalized by the P_{s_i}). Hence the sentence 'E_w can be taken smooth over R_0' is heuristic; (BS) must stay a hypothesis, as A2-11 in fact states. Also: the word's g_b must include the Ω-factor ω (minor).

1. (a): H^j(K̄)|_o is an H_L-equivariant constructible sheaf on the homogeneous o, hence a local system; o is connected (H_L connected), so the rank is constant on o and on P = pi_L^{-1}(o). I2 (triviality of the local system) is not needed for (a); it is needed in (b).
2. (b): IC(P̄')|_{P'} = k[δ(P')] uses I2 (trivial local system); its Poincaré polynomial is v^{-δ}, a unit; support in P̄' gives triangularity for any linear refinement of the closure order. Holds.
3. (c): stalks are additive and A[-i] multiplies the Poincaré polynomial by v^{i}; h_K(P) = Σ_{P'} m_{P'}(v) H_{P,P'}. Holds.
4. (d): immediate from (c) given the closure-preserving bijection, H^k = H^{k'}, and semisimplicity of K, K' (needed so that K is a sum of shifted IC's: (DT) on both sides). Holds as stated in (c)'s hypothesis.
5. Example: for A^n the closures Λ_{m,j} form a chain ... ⊃ Λ_{m,j+1} ⊃ Λ_{m,j} ⊃ ... ⊃ Λ_{m,1} ⊃ Λ_{m+1,n} = t^{m+1}O^n, so h_K(P_i) = Σ_{i'>=i} m_{P_{i'}} v^{-δ(P_{i'})} and m_{P_i} = v^{δ(P_i)}(h_K(P_i) - h_K(P_{i+1})) with P_1 the smallest piece and h(P_{r+1}) = 0. Holds.

1. BBD 6.1.8 (scan p.156): for A ⊂ C of finite type over Z, V with A ⊂ V ⊂ C a strictly henselian DVR with residue field an algebraic closure of a finite field; 'pour tout point fermé s de Spec A, il existe un tel V' with closed point over s (take an analytic arc in Spec  not contained in the preimage of any proper closed subset, normalize, strictly henselize). So V exists at every closed x, also when dim R_0 > 1, with R_0 → V injective. Confirmed.
2. Nearby cycles of an object on Y_η̄ are defined (RΨF = ī^*Rj̄_*F over the normalization of V in η̄), so no descent of G^BS to η is needed (in any case p_*Λ is defined over R_0).
3. Step 2 (Rp_{s*}RΨ_E ≅ RΨ_Y Rp_{η*} for p proper) and step 3 (RΨ_E Λ = Λ for E smooth over V) are the standard properness and smoothness properties of RΨ. The local sources folder has no SGA 7, so the exact numbers XIII 2.1.5/2.1.7 were not checked; the facts are correct.
4. Step 1: the closed embeddings Y_N ⊂ Y_{N+1} are not smooth, contrary to the wording 'transitions smooth'; but ψ commutes with closed push-forward (proper) and with the smooth affine-space transitions Y_{N,M'} -> Y_{N,M}, which is all that is used. Holds.
5. Step 5 (η̄ vs C, coefficient fields): base change between algebraically closed fields of characteristic 0 is fully faithful on D^b_c and commutes with p_*, so a splitting over C descends to η̄; G^BS and IC(P̄) are defined with Q-coefficients and the IC's are absolutely simple, so multiplicities (dimensions of Hom into pH^i) are the same for Q, C, Q̄_ℓ (Artin comparison). Holds.
6. Uniformity: nothing in steps 1-5 depends on x beyond (BS), which holds over all of R_0. Holds. The Remark (ψ(IC_w) = IC_w via KL) is background, not used.

1. Step 4. U = Y_L ∖ (ō_R ∖ o_R) is open; S_η̄|_U = i_{o*}k_o[δ] (I2 over η̄ and (ō_R)_η̄ = closure of o_η̄); ψ commutes with open restriction and with i_* (proper), and ψ(k_o) = k_{o_s} because o_R is smooth over V near x; supp ψ(S_η̄) ⊆ (ō_R)_s = ō_s. Holds.
2. Step 5, try to break: if S_b were an arbitrary simple perverse sheaf on Y_{L,s} supported in ō_s, there could be several S_b nonzero on o_s (IC with nontrivial local systems on o_s, or IC's of non-orbit subvarieties meeting o_s), and the counting fails. Repair: ψ(S_η̄) is a summand of G^BS_s; by (DT) at x and I2, G^BS_s ≅ ⊕ IC(ō')[n] equivariantly, hence after forgetting (Lemma 2 of A2-1); IC(ō') stays indecomposable after forgetting (H_L connected), so by Krull-Schmidt every S_b is some IC(ō'), ō' ⊆ ō_s. Only IC(ō_s) is nonzero on U_s; k_{o_s}[δ] is indecomposable (o_s connected), so exactly one b with S_b = β_xS and n_b = 0. Holds with the repair.
3. Shifts: ψ (normalized fibrewise) is perverse t-exact, so ψ(S_η̄) is perverse and all n_b = 0 anyway; step 5 does not need this.
4. Step 6: G^BS_s ≅ ⊕ ψ(S'_η̄)[-i]^{m_{S',i}} and each ψ(S_η̄) contains β_xS once, so m_x(β_xS) ≥ m_C(S) coefficientwise. Holds. The parenthesis 'distinct summands map to distinct summands' is just additivity of ψ.
5. Step 7: Prop. III (⇒) over C uses (DT) over C; (⇐) at x uses no (DT) (checked in summand.html: '(⇐) ... (DT) is not used'); Lemma 1 at x uses (DT) at x. β_x is onto S_I(F̄_x) only if (UP_x) holds at every level (orbits over F̄_x are exactly fibres of the o_R). Holds.
6. Uniformity: the witness w comes from C and enters only through (BS), valid at every x. Correct: this direction has no circularity.

1. BBD 6.1.8-6.1.10 (scan pp.156-157): for (X,T,L) over C with strata smooth with geometrically connected fibres over S = Spec A and Ext^q(F,G) locally constant and compatible with base change (achieved after shrinking S depending only on (T,L)), Lemma 6.1.9 gives equivalences D^b_{T,L}(X) ←u^— D^b_{T,L}(X_V) —i^→ D^b_{T,L}(X_s); 6.1.10: 'pour S assez grand et pour tout V' an equivalence D^b_{T,L}(X) ↔ D^b_{T,L}(X_s). The size of S depends only on (T,L).
2. Apply with X = Y_L, T = level-L orbits, L = trivial local systems (I2). Shrink further (still depending on level-L data only) so that: (i) the finitely many objects j_{T!}k_T are universally locally acyclic over S (Deligne, SGA 4½ [Th. finitude] 2.13, generic ULA), hence so is every object of D^b_{T,L}(Y_{L,S}) (triangulated closure); (ii) j_T^*, j_T^! of the finitely many IC_S commute with base change (6.1.7 'Conséquence'); (iii) orbit maps and stabilizer group schemes are smooth with connected fibres and closures commute with base change ((UP) at level L, generic smoothness in characteristic 0). Call the result U(L).
3. For closed x ∈ U(L) and V at x: for F_V ∈ D^b_{T,L}(Y_{L,V}), ULA gives i^*F_V ≅ ψ(F_η̄). Taking F_V = the model of IC(ō) gives ψ(IC_η̄) ≅ i^*IC_V ≅ IC_s by (ii): (ψIC_x) at level L holds on U(L).
4. For any word w with G^BS_w at level L: G^BS_η̄ ∈ D^b_{T,L}(Y_{L,η̄}) (stalk sheaves constant along orbits), so G^BS_η̄ ≅ F_η̄ for some F_V ∈ D^b_{T,L}(Y_{L,V}), and by A2-4 G^BS_s ≅ ψ(G^BS_η̄) ≅ i^*F_V, i.e. G^BS_s corresponds to G^BS_C under the BBD equivalence, which preserves IC's and direct sums. Hence m_x(β_xS, G^BS_w) = m_C(S, G^BS_w) for all S at level L and all such w, for every x ∈ U(L). No dependence on w.
5. Consequence for A2-3(b): generation at x produces w(x,S) whose generator lives at some level L(x,S) ≥ level of S; the transfer needs x ∈ U(L(x,S)). That is the genuine residual coupling, and it is between x and the level, not the word. Without (BS) the word dependence is real (BBD 6.1.7 on each p_w).
6. A2-3(c): additivity of ψ, (DT) at x and step 2 of A2-4 give G^BS_s ≅ ⊕ψ(S'_η̄)[-i]^m, and [ψ(S_η̄):S_s] = 1 by A2-5 steps 4-5 (with the repair of A4-4); ψ is perverse t-exact so the cross-term coefficients are integers, not Laurent polynomials. Holds.

Label: proved modulo the cited generic ULA theorem (not in the local sources).

1. By A4-5 each level L has a proper closed bad set B_L ⊂ Spec R_0, depending only on level-L data, and the theorem needs x ∉ B_L for every L (generation at x involves simple objects at all levels, and their witnesses can sit at arbitrarily high levels).
2. The levels L = (N,M) are countably many. A finitely generated Z-algebra has countably many closed points. So ⋃_L B_L can contain every closed point (e.g. B_{L_k} ∋ the k-th prime, for R_0 = Z). Nothing in (Lev), I1, I2, (DT), (BS) prevents this: the smooth transitions Y_{N,M'} → Y_{N,M} create new orbits at each M', whose closures can acquire new bad primes.
3. So 'p ≫ 0' (a cofinite or Zariski-open set) is not implied; 'independent of w and S' is true but does not give existence. This is the same uniformity gap that A2-3 identifies, moved from words to levels.
4. Repaired statement (UL): assume there is a closed x with x ∉ B_L for all L (for instance: the level stratifications are base changes of finitely many stratified R_0-schemes up to smooth affine-space bundles, so ⋃B_L is a finite union). Then A2-6 holds at x. For A^n (R_0 = Z) and the group (R_0 = Z[1/N_H]) B_L = ∅ for all L by A2-8(a)(c) and A2-9(a)(d), so every prime works there.
5. No Iwahori example with ⋃B_L = all closed points is known to me: the claim is unproved, not refuted. Status open.

1. Steps 1-4: G^BS_s ≅ ⊕ψ(S'_η̄)[-i]^m ≅ ⊕β_xS'[-i]^m by (ψIC_x); β_x injective; multiplicities in a semisimple object are unique. Holds (level pull-back: ψ commutes with smooth pull-back, the transitions Y_{N,M'}→Y_{N,M} are smooth; the closed embeddings Y_N⊂Y_{N+1} commute with ψ by properness).
2. Step 5: V_v given by the KL expansion of C_{s_1}...C_{s_r}C_ω is the KL theorem for the (extended) affine Weyl group with characteristic-0 coefficients; over F̄_p it follows from pointwise purity of affine Schubert IC's, over C by transfer. Background; not used in the boxed equivalence. Holds as cited.
3. Step 6: β_xM_I(C) = M_I(F̄_x), β_xS_I(C) = S_I(F̄_x) (needs (UP_x) at all levels), Prop. III on both sides with (DT). Holds.
4. Numerical variant: X Zariski-dense and U(L,w) dense open, so X ∩ U(L,w) ≠ ∅; at x' there, BBD 6.1.10 gives stalk ranks equal to C; (US) transfers to every x ∈ X; A2-2(d) (with (DT) at x) gives equal multiplicities. The witness enters only via the auxiliary x'. Holds. Note (US) is itself an assumption about all words; by A4-5 it can be weakened to: stalk polynomials of the IC(P̄) only, at all levels (the G^BS stalks then follow).
5. (i): smooth closures over R_0 near x: IC = k[dim], ψ(k) = k. Holds. (ii): Σ_{S'} m_η̄(S')·#(simple summands of ψ(S'_η̄)) = total count at s, each ψ(S'_η̄) contains β_xS' once (A4-4), so equal totals force ψ(S'_η̄) = β_xS' for every S' with m_η̄(S') ≠ 0, in particular for IC(ō) (multiplicity 1 in ρ_*k). Holds; it requires (UP_x) for every orbit occurring in ρ_*k.

1. (a): Iwahori row reduction (upper-triangular mod t) brings v with min val m attained last at j to t^m e_j; closure Λ_{m,j} = ⊕_{i<=j} t^mOe_i ⊕ ⊕_{i>j} t^{m+1}Oe_i, a Z-linear subspace of Y_{N,M}; stabilizers are cut out by linear conditions in the smooth connected H_{N,M}; valid over every field. I_{N,M} = 1+t^{N+M}M_n(O) acts trivially on t^{-N}O^n/t^MO^n. Holds.
2. (b): t^MO^n ⊆ g_bO^n ⊆ t^{-N}O^n for N,M large, so {(b,ȳ): ȳ ∈ g_bO^n/t^M} is a constant-rank subbundle (rank = nM + val det g_b) over the smooth projective BS(w)_Z, and m-source = E_w ×_{Y_L} Y_N, so G^BS = pi_L^*p_*k by proper base change. Holds (g_b = p_1...p_r ω).
3. (c): closures are linear, IC constant, ψ(k) = k. Holds; so B_L = ∅ for every level (A4-6).
4. (e) rechecked: Fl_{<=s_0}·O^2 = {Λ_{0,1} ⊂ L ⊂ Λ_{-1,1}, index 1} ≅ P^1 (P_{s_0} stabilizes Λ_{0,1} and t^{-1}Λ_{0,1} = Λ_{-1,1}); fibres P^1 / point / empty give stalks k[1]⊕k[-1], k[1], 0, i.e. G_{s_0} = k_{Λ_{-1,1}}[1] ⊕ k_{Λ_{0,1}}[-1]. Holds over every field.
5. (f): lattice chain ... ⊃ t^mO^2 ⊃ Λ_{m,1} ⊃ t^{m+1}O^2 ⊃ ...; s_1 fixes t^mO^2 (factor 1+q) and sends 1_{Λ_{m,1}} to 1_{t^mO^2} + q·1_{t^{m+1}O^2}; s_0 fixes Λ_{m,1} and sends 1_{t^mO^2} to 1_{Λ_{m-1,1}} + q·1_{Λ_{m,1}}. Applying words right-to-left to 1_{O^2} reproduces every row, e.g. s_0s_1s_0 ↦ Λ_{-2,1}:1, Λ_{-1,1}:3q, Λ_{0,1}:3q², Λ_{1,1}:q³. Match at all eight q tested. These are trace identities in the Hecke module, i.e. consistency checks, as the node itself says.
6. Step 4 (p = 2 allowed): no trace or square-root argument is used; Sun's theorem needs no parity condition. Holds.

1. (a): stabilizer of v ∈ LH in I_H×I_H is {(a,b): avb^{-1} = v} ≅ I_H ∩ vI_Hv^{-1}, smooth with connected fibres over Z (root subgroups and torus part); cells and Bruhat order are field-independent. Holds.
2. (b): Demazure twisted products are iterated P^1-bundles over Z, the multiplication map is proper. Holds.
3. (c): contrapositive of A2-5 (needs (DT) over C and at x). Holds.
4. Independent check of the conclusion: G_{(w_1,w_2)} = IC_{w_1} * k_{L^+H} * IC_{w_2} and k_{L^+H} = IC_{w_0} (w_0 longest in the finite W, ℓ(w_0) > 0). In the Hecke algebra H C_x H ⊆ span{C_z : z ≤_LR x}, and the two-sided cell of length-0 elements is the unique top cell: C_ω occurs in a product of KL basis elements only if every factor has length 0. With positivity of the structure constants (decomposition theorem, characteristic-0 coefficients), IC_ω is never a constituent of any generator, over C and over every F̄_p alike. So Betti failure holds directly, consistent with (c) and with Table 1; the 'q odd' restriction of the earlier trace argument is not needed for this conclusion (heuristic-level confirmation, standard cell theory).
5. (d): the equal-totals argument (sufficient condition (ii) of A2-6) is correct given KL multiplicities for ρ_*k at every p (pointwise purity of affine Schubert IC's, field-independent) and over C. Holds as cited; (c) does not use it.

1. Orbits {0}, G_m over every algebraically closed field; stabilizer μ_2, étale for p ≠ 2, infinitesimal at p = 2. Equivariant local systems on the orbit correspond to representations of π_0 of the reduced stabilizer: trivial at p = 2, Z/2 otherwise. So S(F̄_2) = {δ_0, k[1]} and S(C) has the extra IC(L). Holds.
2. (a): finite type, finitely many orbits, so D^b_{G_m}(A^1) = Thick(S) = Thick(G). Holds. (b): Thick(G) ⊆ objects with all perverse constituents in {δ_0, k[1]} (a Serre subcategory, so a thick subcategory); IC(L) is not in it. Holds.
3. Independent check of (iii): with f: x ↦ x² over V, f_*Λ = Λ ⊕ L on G_{m,η̄}; f is finite, so ψ(f_*Λ) = f_{s*}Λ = Λ (f_s radicial at p = 2), hence ψ(L)|_{G_m} = 0. On P^1_V: RΓ(P^1_s, ψ(j_!L)) = RΓ(P^1_η̄, j_!L) = H_c^*(G_m, L) = 0, and ψ(j_!L) is supported on {0,∞}, so both stalks vanish. Thus ψ(IC(L)) = ψ(j_!L[1]) = 0 on A^1. Holds.
4. Scope: H = G_m on A^1 is not of the form 'Iwahori levels of a loop space', and the generators {k[1], δ_0} are not BS or G_w objects. Step 7 of A2-10 itself says the loop-space version breaks I1 at p = 2 and I2 over C. (UP_2) fails, so A2-5/A2-6 do not apply at 2 and nothing they claim is contradicted.
5. So A2-11's 'Necessity' only shows that the finite-type analogue of 'ℓ-adic at an arbitrary x ⇒ Betti' is false. An Iwahori-level example where generation holds at some x but fails over C is not given. Label heuristic.

1. A2-7 is cited for 'rh = strongly equivariant D-modules' but its statement and proof are not in this brief, so the third ⇔ is unaudited.
2. Coherent vs regular holonomic: on a smooth H-variety with finitely many H-orbits every strongly H-equivariant coherent D-module is regular holonomic (HTT Thm 11.6.1). Weak equivariance is not enough (D_Y itself is a weakly equivariant coherent D-module and is not holonomic). So the category on the pages must be read as strongly equivariant.
3. Singular levels: for general smooth affine X the levels Y_{N,M} need not be smooth. D-modules then need a Kashiwara-type definition via an H_L-equivariant closed embedding into a smooth affine variety (exists: H_L acts on the affine Y_{N,M} and the coordinate ring is a union of finite-dimensional representations). HTT 11.6.1 is applied to modules supported on the image, which has finitely many orbits; plausible, but not checked here.
4. Given 1-3, equivariant RH at each level is a t-exact equivalence onto D^b_{H_L,c} (Betti), compatible with smooth pull-back along transitions and with proper push-forward (hence with *_I and BS generators), so the D-module Thick-closure question is the Betti one. Heuristic until A2-7 is audited.

1. Step 1 (finitely many orbits by I1, models over a finitely generated \(R_\lambda\)): holds. Sharper: the orbits are permuted by \(\mathrm{Gal}(\mathbb C/K)\), \(K=\operatorname{Frac}R\) (the action is defined over \(K\)), so each orbit is stable under an open subgroup \(\mathrm{Gal}(\mathbb C/K')\), \([K':K]<\infty\), hence defined over \(K'\) (char 0). Take \(R_\lambda\) = a localization of the integral closure of \(R\) in \(K'\): finite over \(R[1/n]\).
2. Step 2 (partition spreads): holds. 'Disjoint and covering' are constructible conditions true on the generic fibre.
3. Step 3 (fibres are single orbits): holds. The image of \(a_j\) is constructible (Chevalley) and contains the generic fibre; its complement \(E\) is constructible and misses the generic fibre, so the image of \(E\) in \(\operatorname{Spec}R_\lambda\) is constructible without the generic point, hence nowhere dense.
4. Step 4: holds. Closure order via EGA IV 9.5; simple objects = IC's of orbits under I2 (BBD 4.3.1 plus equivariance, connected \(H\)). Connectedness of stabilizers also spreads (EGA IV 9.7.7 applied to the stabilizer group scheme over \(o^j_{R_\lambda}\), as A3-5 step 2 does). So I2 at \(\bar s\) holds automatically on a dense open: it need not be assumed.
5. Step 5 (lifting closed points): holds. Nonempty fibre of finite type over the finite field \(k(s)\) has a closed point \(s'\) with \(k(s')/k(s)\) finite, and \(k(s)\hookrightarrow\overline{\mathbb F}_p\) extends to \(k(s')\).
6. Lift dependence: two lifts of the same \(\bar s\) differ by a \(K\)-automorphism of \(K'\), which permutes the \(o^j\). So \(S\mapsto S_{\bar s}\) is defined up to this permutation. All later uses (A1-2, A1-3, A3-5) work with one fixed lift, and multiplicities in \(G_w\) are permutation-invariant because \(G_w\), \(\mathrm{Fl}_{\le w}\), \(Z_w\) are defined over \(R\). No gap, but the statements should say 'for the chosen lift'.
7. Step 6 (density survives finite shrinking): holds (irreducibility).

1. Citations (scan of bbd-perverse.pdf, printed pp. 155–159). 6.1.7: finitely many objects and operations descend to \(S=\operatorname{Spec}A\), \(A\subset\mathbb C\) of finite type; after shrinking, \(R^*f_*,Rf_!,f^*,Rf^!,\otimes,\mathrm{Tor},\mathrm{Ext}\) give constructible sheaves and commute with base change. 6.1.8: \(L(T)\) = finite families of irreducible lisse \(\mathbb Z/\ell\)-sheaves, and for every closed point \(a\) there is a strictly henselian DVR \(A\subset V\subset\mathbb C\) with residue field \(\overline{\mathbb F}_p\) through \(a\). Lemma 6.1.9: \(D^b_{T,L}(X,\mathbb Z_\ell)\xleftarrow{u^*}D^b_{T,L}(X_V,\mathbb Z_\ell)\xrightarrow{i^*}D^b_{T,L}(X_s,\mathbb Z_\ell)\) are equivalences. (6.1.10.1): for \(S\) large and every such \(V\). p. 158: after refining \((T,L)\) and shrinking \(S\), the equivalences commute with \(Rf_*,Rf_!,f^*,Rf^!\), \(R\mathcal Hom\), \(\otimes^L\) and respect perverse t-structures (conditions (a),(b)); 'on ne peut pas passer à la limite'. p. 159: \(\mathbb Q_\ell\), \(E_\lambda\), \(\overline{\mathbb Q}_\ell\) variants. Every citation in A1-2 step 4 and A3-5 step 3 matches.
2. Hypotheses of the comparison: finite type (yes: one level \(Y_{\lambda'}\), \(Z_w\), \(\mathrm{Fl}_{\le w}\), all of finite type by (Dict)(d)); dense open depending on data (acknowledged in both); coefficient field (\(\overline{\mathbb Q}_\ell\) via p. 159). No nearby-cycle identity \(\psi\,\mathrm{IC}=\mathrm{IC}\) is used: the DVR \(V\) enters only inside Lemma 6.1.9, and IC's are matched because the equivalence commutes with \(j_!,j_*\) and the t-structure, hence with \(j_{!*}\).
3. Equivariance. A1-2 step 3: for connected \(H\), \(\mathrm{For}:\mathrm{Perv}_H\to\mathrm{Perv}\) is fully faithful, t-exact (up to the fixed shift), with image closed under subquotients; so equivariant simples go to simples and Jordan–Hölder multiplicities are preserved. Holds. A3-5 step 1/4: one Bernstein–Lunts approximation \((V_k\times-)/H\) of bounded amplitude, on which the strata \((V_k\times o)/H\) are simply connected for \(k\gg0\) (connected stabilizers), so constant \(L\) suffice. Holds.
4. A1-2 steps 1–2: \(Z_w\) is closed in \(\mathrm{Fl}_{\le w}\times Y_{\lambda'}\) (\(L^+X\subset LX\) closed, \(X\) affine), so \(f_w\) is proper; set-theoretic images commute with base change (surjectivity is stable), so geometric fibres of \(\mathrm{Fl}_{\le w,R}\) are the Schubert varieties; IC depends only on the reduced structure. Holds.
5. Step 6 counter-check: on \(\mathbb G_m\) the \(\mathbb Q\)-local system with monodromy \(\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\) is simple over \(\mathbb Q\) and splits over \(\mathbb C\) into characters \(\pm i\); so 'multiplicities are unchanged by field extension' is not true for arbitrary simples. For \(S=\mathrm{IC}(\bar o,\mathbb Q)\): a composition factor \(T=\mathrm{IC}(\bar o',L)\) of \(M_0\) contributes to \(S\otimes K\) iff \(o'=o\) and \((L\otimes K)^{\pi_1}\ne0\) iff \(L^{\pi_1}\ne0\) iff \(L\) trivial of rank 1. So the repaired statement holds; the abstract-isomorphism alternative is also valid.
6. A3-5 last sentence ('in general \(\operatorname{Spec}R\setminus U\) lies over finitely many primes or every large \(p\) still has points in \(U\)'): the second clause is always true (Chevalley), but the points depend on \(U\); it does not make \(B(N,M,w)\) finite for a point \(\bar s_p\) fixed in advance. See A5-5.

1. Step 1 (1 ⇒ 3): Betti generation ⇒ \(S\in\operatorname{Thick}\{G_w\}\) ⇒ \(S\) a constituent of some \(G_{w_S}\) (Prop. I(a),(b), no (DT)) ⇒ summand by (DT) over \(\mathbb C\) (A3-4(c)). Transfer on \(U_{S,w_S}\) by A1-2. Summand at \(\bar s\) needs (DT) at \(\bar s\). Holds.
2. (DT) at \(\bar s\) is automatic: at a closed point \(s\) with finite residue field \(\mathbb F_q\), the spread-out data give a finite-type proper model \(f_w\) over \(\mathbb F_q\) with \(K_w\) pure (\(\mathrm{IC}_w\) and \(\underline k_{L^+X}\) pure, BBD 5.3.2), so \(f_{w*}K_w\) is pure; BBD 5.4.5 + 5.3.8 split it non-equivariantly with semisimple \({}^pH^i\), and the Bernstein–Lunts approximation argument of A3-4 step 6 (or Sun Thm. 1.2 on the quotient stack, checked: proper, finite diagonal, affine stabilizers, \(\iota\)-pure kernel) gives the equivariant splitting. This is Proposition IV of props/summand.html ('Given (FM\(_w\)), (DT) holds for \(w\) over \(\overline{\mathbb F}_q\) by purity and the decomposition theorem on stacks'). So '(DT) at the points of \(\mathcal P_S\)' is not an extra hypothesis on \(U_0\).
3. Step 3 (pigeonhole): \(\overline{\bigcup_{w\in W_S}\mathcal P_{S,w}}=\bigcup_w\overline{\mathcal P_{S,w}}=\operatorname{Spec}R\), irreducible ⇒ one \(\overline{\mathcal P_{S,w}}=\operatorname{Spec}R\). Holds.
4. Steps 4–5: a dense set meets the dense open \(U_{S,w}\); a summand gives \(e\ge1\); A1-2 gives \(e_{\mathbb C}\ge1\); (DT) over \(\mathbb C\) gives the summand. Holds. Only \(e\ge1\) is used, so (W\(^+\)) suffices.
5. Step 6: \(\bigcap_n\operatorname{Spec}\mathbb Z[1/n]\) has no closed point. Holds; the easy direction only gives a dense open per \(S\).
6. Implicit domain condition: \(S_{\bar s}\) is defined only for \(\bar s\) in the image of \(U_\lambda\) (with a chosen lift, A5-1); (W) should read '\(\mathcal P_S\subseteq\mathcal P\cap\operatorname{im}U_\lambda\)'. Harmless (finite shrinking preserves density).

1. Steps 1–3: \(S\in\operatorname{Thick}\{G_w\}\) ⇒ \(S\in\mathcal D_{\mathcal M^+_I}\) ⇒ \(S\in\mathcal M^+_I\) (\({}^pH^0S=S\)): Prop. I(a),(b), no (DT). A1-2 transfers \(e\ne0\) to the dense open \(U_{S,w}\), which meets the dense bad set: contradiction. Holds.
2. Step 4: verdict.md (iwahori-two-levels) Proposition III(a) reads 'Over \(\overline{\mathbb F}_q\), no \(\Delta_\omega\), \(\omega\in\Omega\), is a perverse constituent of any \(\mathrm{IC}_x\ast\mathrm{IC}_0\ast\mathrm{IC}_y\) … Over \(\mathbb C\), given the comparison of \(\overline{\mathbb F}_p\) with \(\mathbb C\) (BBD 6.1)', with \(p\) odd throughout. So the bad set contains every geometric closed point of \(\operatorname{Spec}\mathbb Z[1/2\ell]\); \(\dim=1\), so it is dense.
3. Matching: \(\Delta_e\) is the skyscraper on the closed cell \(I\cdot1\cdot I\); \(\Delta_{e,\bar s}\) is the same object at \(\bar s\), so no lift ambiguity.
4. Step 5 (Sp) for the group: I×I-orbits on the levels are the Bruhat cells (uniform in \(p\), defined over \(\mathbb Z\)), stabilizers \(I\cap wIw^{-1}\)-type pro-groups, connected (I2), closures are images of Demazure resolutions over \(\mathbb Z\). Bounded convolution (Dict)(d) for \(\mathrm{IC}_x\ast\mathrm{IC}_0\ast\mathrm{IC}_y\) with \(N',M''\) independent of \(p\) is plausible (the convolution is defined by the same group-theoretic formulas over \(\mathbb Z\)) but is not proved in A1-4; label: needs (Sp) written out.
5. Remark (step 6) is right: failure transfers from one Betti witness per \((S,w)\), so no uniformity is needed. The A3-3 route (Av\(_\psi\) with \(\exp\) on \(\mathcal D\)-modules, then RH) is independent; both conclusions agree.

1. A3-7 (S): 'for each prime \(p\) fix \(R\to\overline{\mathbb F}_p\) (through \(U(N,M,w)\) whenever \(p\notin B(N,M,w)\))'. So \(B(N,M,w)=\{p:\bar s_p\notin U(N,M,w)\}\). Step 3 needs it finite and cites A3-5, which proves this only for \(R=\mathbb Z[1/N]\) ('If \(R=\mathbb Z[1/N]\), \(U\) corresponds to all primes outside a finite set').
2. Counterexample to the mechanism: A1-6. \(R=\mathbb Z[\pi]\), \(Z=\{xv=\pi u\}\), \(\bar s_p:\pi\mapsto0\). At every \(p\) the single generator has \(\delta_0\) as a summand (so (BW) holds with \(|F|=1\) at all \(p\)), yet over \(\mathbb C\) the generator is \(\underline k[1]\). Here \(U=D(\pi)\), every \(\bar s_p\in V(\pi)\), \(B=\) all primes. (Checked numerically: \(|Z(\mathbb F_p)|=2p\) at \(\pi=0\), the nodal fibre; script toy_check.py.) The toy is not an Iwahori setting, but A3-7's proof uses only the abstract mechanism (spreading out, A3-4, A3-5), so the proof as written does not distinguish it.
3. Repair 1 (\(\dim R=1\)): proper closed subsets of \(\operatorname{Spec}R\) are finite, so \(\operatorname{Spec}R\setminus U(N,M,w)\) contains finitely many closed points and \(B(N,M,w)\) is finite whatever \(\bar s_p\) is. A3-7 then holds as stated. This covers every row of Table 1 (\(G\), \(X\), action over \(\mathbb Z[1/N]\); by A5-1 step 1 the orbit rings \(R_\lambda\) stay one-dimensional).
4. Repair 2 (dense): replace 'infinitely many \(p\in\mathcal P\)' by 'a Zariski-dense set \(\mathcal P_o\) of geometric closed points'. Pigeonhole over the finite \(F(o)\) in the irreducible \(\operatorname{Spec}R\) gives one \(w\) with dense witness set, which meets \(U(N,M,w)\); steps 4–7 go through. This is A1-3 (2 ⇒ 1) with constituent witnesses (A5-3 step 4).
5. Repair 3 (diagonal choice): enumerate the countably many opens \(U_k=U(N,M,w)\). Put \(m(p)=\max\{m:\ p\in\operatorname{im}(U_1\cap\dots\cap U_m\to\operatorname{Spec}\mathbb Z)\}\). By Chevalley each image contains all large \(p\), so \(m(p)\to\infty\); choose \(\bar s_p\in U_1\cap\dots\cap U_{m(p)}\) over \(p\). Then each \(U_k\) contains \(\bar s_p\) for all but finitely many \(p\), and A3-7 holds for these points. Caveat: the points depend on all \((N,M,w)\) and on the chosen \(U\)'s, so (BW) at 'some point over each \(p\)' cannot be checked without knowing them; an arbitrary choice can fail (step 2). Note also that the \(\overline{\mathbb F}_p\)-points of a positive-dimensional fibre are countable, so 'very general point' arguments do not apply.
6. (UGR) in (c) should be phrased with the canonical bad set \(B(N,M,w)=\{p:\ \)(i) or (ii) of A3-5 fails at \(\bar s_p\}\), not with a chosen \(U\); otherwise it depends on choices. With that reading (c) holds (steps 3–4 at one fixed \(p\notin B\) for all \(o\); (DT) at \(\mathbb C\) from A3-4(c)).
7. A3-8 inherits the same requirement; in addition its (IG\(_F\)) is stated over \(\mathbb F_p((t))\), which presumes \(k(s)=\mathbb F_p\); in general use \(\mathbb F_{q(s)}((t))\), \(q(s)=|k(s)|\).

1. Invariance: \((tx)v-n(tu)=t(xv-nu)\). Holds.
2. \(n\in k^\times\): \(v=0\) forces \(u=0\); so \(v=1\), \(u=x/n\), and \(x\mapsto(x,[x:n])\) is a \(\mathbb G_m\)-equivariant isomorphism \(\mathbb A^1\simeq Z_n\). \(G_n=\underline k[1]\). Holds.
3. \(p\mid n\): \(Z_n=\{xv=0\}=(\mathbb A^1\times[1:0])\cup(\{0\}\times\mathbb P^1)\), transversal at \((0,[1:0])\). \(\mathrm{IC}\) of a curve is \(\nu_*\underline k[1]\) (\(\nu\) normalization). \(f\circ\nu\) is \(\mathrm{id}\) on \(\mathbb A^1\) and \(\mathbb P^1\to0\), so \(G_n=\underline k[1]\oplus i_{0*}R\Gamma(\mathbb P^1)[1]\). Equivariant splitting of \(R\Gamma_{\mathbb G_m}(\mathbb P^1)\simeq k\oplus k[-2]\) in \(D_{\mathbb G_m}(\mathrm{pt})\): the obstruction lies in \(\operatorname{Hom}(k[-2],k[1])=H^3(B\mathbb G_m)=0\); Sun Thm. 1.2 (checked: proper, finite diagonal, affine stabilizers, pure kernel) also applies. So \(G_n=\underline k[1]\oplus\delta_0[1]\oplus\delta_0[-1]\). Holds.
4. Point counts (toy_check.py): \(|Z_n(\mathbb F_p)|=p\) for \(p\nmid n\) and \(2p\) for \(p\mid n\) (\(p+(p+1)-1\)); trace at \(0\) of \(f_*\mathrm{IC}\) is \(-1\) resp. \(-(p+2)\), matching \(\underline k[1]\oplus\delta_0[1]\oplus\delta_0[-1]\).
5. Hypotheses: two orbits \(\{0\},\mathbb G_m\), stabilizers \(\mathbb G_m\) and \(1\) connected; equivariant simples \(\delta_0,\underline k[1]\); finite-length heart, so \(D^b_{\mathbb G_m}(\mathbb A^1)=\operatorname{Thick}\{\delta_0,\underline k[1]\}\); models over \(\mathbb Z\); generators proper push-forwards of IC's, so (DT) at every field. All as claimed.
6. Over \(\mathbb C\): \(\operatorname{Thick}\{\underline k[1]\}\subseteq\mathcal D_{\{\underline k[1]\}}\not\ni\delta_0\). (W) fails: a witness at \(p\) lies in \(p\mathbb Z\), so a finite \(W_S\) serves finitely many \(p\), not dense in \(\operatorname{Spec}\mathbb Z\).
7. A1-6: at \(\pi\mapsto0\) the fibre is the \(n=0\) case (same as \(p\mid n\)); over \(\mathbb C\), \(\pi\ne0\). \(\{\bar s_p\}\subset V(\pi)\) is infinite but not dense in \(\operatorname{Spec}\mathbb Z[\pi]\). 'The two notions agree iff \(\dim R=1\)': holds (a 1-dimensional Noetherian scheme has finite proper closed subsets; in \(\dim\ge2\), \(V(f)\) contains infinitely many closed points).

1. (a) \(P_{S[-i],o}=v^iP_{S,o}\), additivity, \(\mathrm{IC}(\bar o')|_{o'}=\underline k[\dim o']\) so \(P_{S_o,o}=v^{-\delta(o)}\), support \(\bar o'\): correct. Descending induction over a linear extension of the closure order solves for \(m_{S_o}\); the diagonal \(v^{-\delta(o)}\) is a unit in \(\mathbb Z[v^{\pm1}]\). Holds.
2. Need for I2: with a nontrivial irreducible equivariant local system \(\xi\) on \(o\) of rank \(d\), \(P_{\mathrm{IC}(\bar o,\xi),o'}\) is \(d\) times (generically) the constant-coefficient polynomial near \(o\); e.g. on the open orbit the stalk polynomial is \(d\,v^{-\delta(o)}\), equal to that of \(\mathrm{IC}(\bar o)^{\oplus d}\). So stalk dimensions do not determine multiplicities. (b) assumes I2 on \(U\), so it is unaffected.
3. Splitting at \(\bar s\): \(G_w=\pi^*f_{w*}K_w\) with \(K_w\) pure over the finite field of definition, so BBD 5.4.5 gives \(\bigoplus{}^pH^i[-i]\) and 5.3.8 semisimplicity non-equivariantly; under connected \(H\) the non-equivariant multiplicities of \(\mathrm{IC}(\bar o)\) equal the equivariant ones (A5-2 step 3). Over \(\mathbb C\): BBD 6.2.5. Holds.
4. (b) steps 3–5: a closed point \(\bar s_0\in U\cap U_{S,w}\) exists (Jacobson); A1-2 gives \(e(\mathbb C)=e(\bar s_0)\); (U-stalk) and (a) give \(e(\bar s)=e(\bar s_0)\) for all \(\bar s\in U\). With I2 and the uniform bijection \(\beta\), Prop. I(c) on both sides (needs (DT) at \(\bar s\) for ⇐, automatic by A5-3 step 2) gives 'generation at one \(\bar s\in U\) ⇔ Betti generation'. Holds.
5. (c): if \(H^j\) of a stalk is pure Tate (Frobenius acts with all eigenvalues equal to \(q^{j/2}\)), then \(\operatorname{tr}(\mathrm{Fr}^m)=\sum_j(-1)^j\dim H^j\,q^{mj/2}\), and distinct \(j\) give distinct exponents, so traces for all \(m\) determine all \(\dim H^j\). If the traces are \(p\)-independent polynomials in \(q\), the dimensions are \(p\)-independent. Correct; hypotheses unverified for the symmetric rows, as stated.

1. Step 1: local finiteness of the \(H\)-action on \(\Gamma(V,M)\) needs \(V\) affine; \(Y_{N,M}\) is affine with an \(H\)-equivariant closed embedding into a representation \(V\), so this is available. Holds.
2. Step 2: \(F_jM=F_j\mathcal D_V\cdot M_0\) is \(H\)-stable, so \(d\rho(\xi)\) preserves it; strong equivariance identifies \(\xi_V\) with \(d\rho(\xi)\), so \(\sigma_1(\xi_V)\) kills \(\operatorname{gr}^FM\) and \(\operatorname{Ch}M\subseteq\mu^{-1}(0)\cap T^*V|_Y=\bigcup_{o\subseteq Y}T^*_oV\), a finite union of Lagrangians (only orbits in the support matter). Holonomic. Holds.
3. Steps 3–4: for connected \(H\), subquotients of strongly equivariant modules are strongly equivariant; simples are \(j_{!*}L\), \(L\) from \(\pi_0(S_x)\); \(L\) trivializes on the finite étale cover \(H/S_x^\circ\), and \(L\) is a summand of \(\pi_*\pi^*L\), so regular; \(j_*,j_!\) preserve RH (HTT Thm. 6.1.5, recalled), and RH is a Serre subcategory. Holds.
4. Step 5: kashiwara-equivariant.pdf §1.3 (verified): 'The flag manifold X has finitely many K-orbits. Therefore any coherent K-equivariant D_X-module is a regular holonomic D_X-module (see [15])'; the general form appears again in the twisted setting of a later section ('Assume that there are finitely many G-orbits in X. Then any coherent holonomic G-equivariant \(D_{X,\lambda}\)-module is regular holonomic').
5. Step 6: \(M=\mathbb C[x]\), \(\partial f=f'+f\); \(f\mapsto f'+f\) is unitriangular on degree \(\le d\) (toy_check.py), hence bijective, so \(H^*_{\mathrm{dR}}=0\); translation commutes with \(\partial\), giving an algebraic \(\mathbb G_a\)-action compatible with \(\mathcal D\) (weak equivariance); the Lie derivative is \(d/dx=\partial-1\), so strong equivariance fails, and no other strong structure exists since \(\partial\) is invertible while derivatives of rational \(\mathbb G_a\)-modules are locally nilpotent. Irregular at \(\infty\). Holds.
6. Relevance: the pages' \(\mathcal D_c\) (glossary 'dc', 'paper-coherent' = Def. 3.27) is \(\mathcal D\)-modules on prestacks, which on \([Y/H]\) are strongly equivariant by definition; so the word 'strongly' is automatic there.

1. Kashiwara notes Thm. 4.6.2 (verified in kashiwara-equivariant.pdf §4.6): for an affine algebraic group \(G\) and a quasi-projective \(G\)-manifold \(X\), \(\mathrm{DR}_X:D^b_{G,\mathrm{rh}}(\mathcal D_X)\xrightarrow{\sim}D^b_{G^{an},\mathbb C\text{-c}}(\mathbb C_{X^{an}})\). Smoothness is required; singular \(Y_{N,M}\) are handled by the \(H\)-equivariant closed embedding \(Y\subset V\) and Kashiwara's equivalence (objects supported on \(Y\)), as in A3-2. With A3-2, \(\mathrm{coh}=\mathrm{rh}\) at each level. Step 1 of A3-3 holds.
2. Steps 2–3 (transitions): affine-space smooth pullbacks (DR commutes up to \([d]\)), group inflation along a unipotent kernel, closed push-forwards; an equivalence on each term of a filtered union of full subcategories compatible with the inclusions is an equivalence of unions. Holds.
3. Step 4 (generators): \(G_w=f_*(\mathrm{IC}_w\widetilde\boxtimes\mathrm{IC}_0)\) with \(f\) proper at finite type (bounded convolution read over \(\mathbb C\)); DR commutes with \(\boxtimes\), smooth descent, proper \(f_*\), and sends \(j_{!*}\) to \(j_{!*}\) (t-exact equivalence). Holds up to a fixed shift.
4. Step 5: \(\Phi(\operatorname{Thick}S)=\operatorname{Thick}(\Phi S)\) for an exact equivalence. Holds. So \(\mathcal D\)-module generation ⇔ Betti generation.
5. A1-8 step 3: 'The equivariant/stack version follows by smooth descent along \(Y\to[Y/H]\)' is not a proof: triangulated derived categories do not satisfy descent (that is why Bernstein–Lunts or \(\infty\)-categories are needed). The conclusion is right with Thm. 4.6.2 as the citation. A1-8 step 2's singular-level gap is closed by A3-2 (A5-8).
6. Which category the pages use: glossary 'dc' = 'the small stable ∞-category of coherent D-modules on a prestack: objects whose pullback to any smooth test scheme is a bounded complex with coherent cohomology (paper, Definition 3.27)'; hypothesis.html: 'The finite-level presentation in I1 is assumed, not derived from the paper's definition of coherent D-modules'. So the transfer applies to the pages' category exactly under I1\(^{\mathcal D}\), as A3-3 step 7 says.
7. Group: A3-3 step 6 uses \(\exp\) only inside \(\Phi=\mathrm{Av}_\psi\) (group.html: '\(H^*_{\mathrm{dR}}(\mathbb A^1,\exp)=0\)'), not as an object of \(\mathcal C_I^{\mathcal D}\); consistent with A3-2 (the Whittaker category is outside \(\mathcal C_I\)). Agrees with A5-4.

1. A3-4 step 1: for \(A\to B\to C\), \({}^pH^iB\) is an extension of a subobject of \({}^pH^iC\) by a quotient of \({}^pH^iA\); \(\Sigma\)-constituent objects form a Serre subcategory. Thick. Holds. Step 2 (⇒, no (DT)): holds.
2. Step 6 (equivariant splitting): \(D^{[a,b]}_H(Y)\to D^{[a,b]}((V_k\times Y)/H)\) fully faithful for \(k\gg0\); perverse truncation commutes with smooth pullback (fixed shift), so \({}^pH^i(f_*K)\) on the approximation is the image of the equivariant \({}^pH^i\); the splitting isomorphism lifts by full faithfulness; semisimplicity lifts since \(\mathrm{Perv}_H\subseteq\mathrm{Perv}\) is full and closed under subquotients (a complement's projection is a morphism between equivariant objects). Holds. The same argument with BBD 5.4.5/5.3.8 works over \(\overline{\mathbb F}_q\) given a pure model (A5-3 step 2), so step 7's '(DT) at \(\overline{\mathbb F}_p\) needs Sun 1.2 and a pure model' is satisfied on the spread-out locus.
3. A3-8 steps 1–3: IC-trace basis unitriangular on pieces of one level; \(F(o)\) = Bruhat-lower closure of a finite set, finite; \(\operatorname{span}\{T_v\}=\operatorname{span}\{\operatorname{tr}\mathrm{IC}_v\}\) on lower sets (KL triangularity, nonzero diagonal); trace of convolution = convolution of traces (Grothendieck–Lefschetz with proper base change). Holds.
4. Step 4: a Weil simple constituent of \({}^pH^iG_w\) is geometrically a subquotient of an equivariant perverse sheaf, hence (connected \(H\), I2, Frobenius-stable orbits by (Dict)(c)) of the form \(\mathrm{IC}(\bar o)\otimes\overline{\mathbb Q}_\ell^{(b)}\) with a rank-one Weil twist; so \(\operatorname{tr}G_w=\sum_{S'}c_{S',w}\operatorname{tr}S'\) with \(c_{S',w}=\sum_i(-1)^i\sum b\), and \(c_{S',w}\ne0\) ⇒ \(S'\) a constituent (converse false by cancellation, not needed). Holds.
5. Step 5: comparing the coefficient of \(\operatorname{tr}\mathrm{IC}(P)\) gives \(1=\sum_wa_wc_{\mathrm{IC}(P),w}\), so some \(c\ne0\). Holds, no (DT).
6. Identification across \(p\): 'piece \(P\)' at different \(p\) means \(\pi^{-1}(o_p)\) for the matched orbit; this needs \(p\notin\) the orbit-matching bad set of the level (finite only in the setting of A5-5 repairs). Also \(\mathbb F_p((t))\) should be \(\mathbb F_{q(s)}((t))\) in general. Step 7 is correctly labelled heuristic.

1. Bounded witnesses (A1, A3, independently): ℓ-adic generation transfers to ℂ iff each simple object has witnesses in a finite subset of \(\widetilde W\) on a dense set of primes A1-3, A3-7. Conflict: A1 says Zariski-dense, A3 says infinitely many \(p\); toy A1-6 separates them when \(\dim R\ge2\).
2. One good prime (A2): Bott–Samelson generators are proper push-forwards of constant sheaves from schemes smooth over \(R_0\), so nearby cycles carry every generator to its counterpart uniformly in \(w\) A2-4; with level-only hypotheses at one closed point, multiplicities agree A2-6.
3. ℂ side (A1, A2, A3): strongly equivariant coherent \(\mathcal D\)-modules are regular holonomic and Riemann–Hilbert matches the two generation statements A3-2, A3-3, A1-8.
4. Blocked: per-\((S,w)\) spreading out with \(p\)-dependent witnesses A2-3, A1-10.
5. Round 2: A4 audits mechanism 2, A5 audits mechanism 1, the toys and RH.

1. One good prime: A2-6 with the repair A4-4 of A2-5 step 5; the per-level good open A4-5; the existence of one prime good for all levels is a hypothesis A4-6 (verified for 𝔸ⁿ and the group).
2. Bounded witnesses: A1-3 with the constituent strengthening A5-3; A3-7 repaired by A5-5 (dense, or \(\dim R=1\)).
3. Failures transfer: A1-4, A5-4; the group fails over ℂ.
4. ℂ side: A3-2, A3-3, A5-8, A5-9 (needs I1 for 𝒟-modules).
5. Checks: 𝔸ⁿ at every odd prime A2-8, A4-8, A1-9.
6. Sharpness of the abstract mechanism: A1-5, A1-6, A5-6.