Audit 7.7 · Audit of A3-4, A3-5, A3-6: unitriangular IC-trace basis, $m_{S,w}=\sum_i(-1)^i\mathrm{tr}\,A_{S,i}(w)$, and (IG) $\Rightarrow\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$

Audit of A3-4, A3-5, A3-6: unitriangular IC-trace basis, $m_{S,w}=\sum_i(-1)^i\mathrm{tr}\,A_{S,i}(w)$, and (IG) $\Rightarrow\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$

1. A3-4(i). $\mathrm{tr}\,S$ is supported on $\overline P(\mathbb F_q)$, constant on fibres and $I(\mathcal O)$-invariant (normalized Weil structure, Audit 7.6 step 2), hence constant on each $P'(\mathbb F_q)$ (one $I(\mathcal O)$-orbit, A3-2(c)); value $(-1)^{\delta(P)}$ on $P$. Triangular for the closure order with diagonal $\pm1$. Across levels: finitely many simple objects live at a common level, where they are IC's of distinct pieces, and the normalized trace is level-independent (Audit 7.6 step 2). Coordinates are finitely supported, so the coordinate map $\kappa\colon C_I\otimes\overline{\mathbb Q}\ell\to\bigoplus_S\overline{\mathbb Q}\ell$ is an isomorphism. Holds (checklist item 5: unitriangularity at a common level).
2. A3-4(ii). The identity $\mathrm{tr}(K)=\sum_i(-1)^i\mathrm{tr}\,{}^pH^i(K)$ needs only the $\mathrm{Fr}$-compatible perverse truncation triangles (Laumon (1.1.1.1)); no splitting. The isotypic decomposition $P_i\simeq\bigoplus_SS\otimes E_{S,i}$ needs semisimplicity of $P_i$ over $k$ (from (D3), or Sun Thm 3.11 since $P_i$ is pure), and $\Phi_i=\bigoplus\Phi_S\otimes A_{S,i}$ by functoriality of evaluation. $E_{S,i}(w)\neq0\iff S[-i]$ is a summand of $\mathrm{IC}_w\ast\mathrm{IC}_0$ in $\mathcal C_I$ uses the equivariant splitting in (D3) (cf. Counterexample 7.3). Non-$\mathrm{Fr}$-stable constituents: $\Phi_i$ maps the $S$-isotypic stalk to the $(\mathrm{Fr}^{-1})^*S$-isotypic one, a block permutation without fixed blocks, trace $0$. Holds.
3. A3-4(iii). $\kappa$ sends $\mathrm{tr}(\mathrm{IC}w\ast\mathrm{IC}_0)$ to column $w$; with A3-3(c), equality of spans. Field independence: (IG) is a family of rank conditions on finite integral matrices ($\mathbb Z$-valued $T_w\ast\varphi_0$, $\mathbb Z$-basis of piece indicators), the same over $\mathbb C$, $\mathbb Q$, $\overline{\mathbb Q}\ell$. Holds.
4. A3-5. Step 0: $\mathcal H_{c,I}=\mathrm{Thick}\{\Delta_w\}=\mathrm{Thick}\{\mathrm{IC}w\}$ by induction on Bruhat intervals with $j!j^*\mathrm{IC}w\to\mathrm{IC}_w\to ii^\mathrm{IC}_w$; steps 1–4 as in Audit 7.2. Holds.
5. A3-6(a), the chain. (IG) $\Rightarrow$ columns span $\Rightarrow$ each coordinate form $\pi_S$ is non-zero on the span $\Rightarrow\exists w$: $m_{S,w}\neq0\Rightarrow\exists i$: $E_{S,i}(w)\neq0\Rightarrow S\in\mathcal M_I$; (L2) puts every simple object in the basis; A3-5(b). Only "$E_{S,i}(w)=0\ \forall i\Rightarrow m_{S,w}=0$" is used, so cancellation inside $m_{S,w}$ can only lose information, never create a false summand (checklist item 6). Holds.
6. A3-6(b), pointwise. Write $\mathbf 1_{P(\mathbb F_q)}=\sum_Qc_Q\,\mathrm{tr}\,\mathrm{IC}(Q)$ at a common finer level; uniqueness of coordinates gives $c_Q\neq0\Rightarrow\mathrm{IC}(Q)\in\mathcal M_I$. A maximal $Q_0$ with $c_{Q_0}\ne0$: evaluating at $Q_0(\mathbb F_q)$ only $Q_0$ contributes, so $Q_0\subseteq P$. Evaluating at the open piece $P^\circ$ gives $1$, so some $Q\ge P^\circ$ has $c_Q\neq0$; a maximal $Q_1\ge Q$ with $c_{Q_1}\ne0$ is maximal overall, lies in $P$ and has $P^\circ\subseteq\overline{Q_1}$; a piece of $P$ whose closure contains the open piece $P^\circ$ of the irreducible $P$ is $P^\circ$ (otherwise it lies in the closed set $P\setminus P^\circ$). So $\mathrm{IC}(P)=\mathrm{IC}(P^\circ)\in\mathcal M_I$. Holds.
7. A3-6(c). $\kappa_S$ kills the span of the $\mathrm{tr}(\mathrm{IC}_w\ast\mathrm{IC}_0)$ and not $\mathrm{tr}\,S$; integrality transports the failure to $\mathbb C$-valued functions. Holds.
8. Replacement table and fields. The satake-run inputs (R), (C) concern symmetric $X$ and H1 is false for $\mathbb A^n$ at Iwahori level; A3-6 uses neither, only (L2) (checklist item 8). Sheaves over $\overline{\mathbb F}_q$ only (item 7).