§7 · Agent 7 · Audit: criterion and bridge · 2026-10-18T19:28:16Z

Audit 7.10 · Audit of A3-10, A3-11: sheaves $\Rightarrow$ functions only row-wise; (WT) $\Rightarrow$ (IG); generation as an IC-coordinate test over all $\mathbb F_{q^n}$

proved ★ key step

The converse statements claim exactly what is provable: non-zero rows via weights and a Vandermonde argument, no spanning; (WT) is a common sufficient condition; the sign-module exclusion is correct with the quadratic relation $(T_s-q)(T_s+1)=0$ and, for the group, isolates every $\mathrm{IC}_\omega$, $\omega\in\Omega$, giving $\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}$ without the Whittaker functor.

Audit 7.10.

Holds. Under the inputs of A3-6 plus purity of each $\mathrm{IC}w\ast_I\mathrm{IC}_0$ (weight $\ell(w)$), over $\overline{\mathbb F}_q$ / $\mathbb F{q^n}((t))$: (a) $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\Rightarrow$ every row of some $m^{(n)}$, $n\le\dim\bigoplus_iE_{S,i}(w)$, is non-zero (A3-10(a)); this does not give (IG), and the spherical instance cited (satake run, converse.tex, Proposition IX(a) and its Remark: $E_6/F_4$, rows non-zero, $C_K=\mathcal H_K\ast\varphi_0\oplus\mathbb C\mathbf 1_{O_{3\omega_1^\vee}}$) matches the source. (b) (WT) $\Rightarrow$ (IG) (A3-10(c)), and (I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages).) with Weil structures and (NC) $\Rightarrow$ (WT). (c) A3-11(a)–(c) are correct; with $\mathrm{vol}(I)=1$ the sign functional $T_s\mapsto-1$ is compatible with $(T_s-q)(T_s+1)=0$. For the group $X=H$, $W_H\ne1$: the component-restricted sign functionals $\varepsilon_\omega$ satisfy the hypotheses of A3-11(c) for $S_0=\mathrm{IC}\omega$, so every $\mathrm{IC}\omega\notin\mathcal M_I$; with (I$_I$), $\Delta_e=\mathrm{IC}_e\notin\mathcal C_I^{\mathrm{Iwa}}$ over $\overline{\mathbb F}_q$.

  1. A3-10(a), weights. In the glued convention an object $\Phi_{N,M}(\bar K)$ has weight $\mathrm{wt}(\bar K)+e_M$, compatible with the transitions since $e_M=e_{M'}+d$. Then $\mathrm{IC}0=\Phi{N,M}(\overline{\mathbb Q}\ell[-e_M])$ has weight $0$, $G_w=\Phi{N,M}(p_!r^*\mathrm{IC}w[-e_M])$ has weight $\ell(w)$ (Audit 7.4 step 2), $S=\mathrm{IC}(P)$ has weight $\delta(P)$, ${}^pH^i(G_w)$ has weight $\ell(w)+i$ (BBD 5.4.1). On the lisse locus $P$ the summand $S\otimes(E{S,i},A_{S,i})$ restricts to $E_{S,i}[\delta(P)]$, so every eigenvalue of $A_{S,i}$ has absolute value $q^{(\ell(w)+i-\delta(P))/2}$ (for a fixed $\iota$). Eigenvalues from different $i$ are therefore distinct, so $\sum_i(-1)^i\mathrm{tr}A_{S,i}^n=\sum_\alpha c_\alpha\alpha^n$ with every $c_\alpha=\pm(\text{multiplicity})\neq0$; the Vandermonde matrix $(\alpha^n)_{1\le n\le r}$ ($\alpha\ne0$ distinct) is invertible, so some $n\le r$ gives a non-zero value. Holds.
  2. A3-10(b). Linear algebra example $\begin{pmatrix}1&2\\1&2\end{pmatrix}$ correct; single-$n$ cancellation across degrees is possible ($2a-b=0$, $|b|=2|a|=q^{1/2}|a|$ for $q=4$); the spherical citation checked against converse.tex, lines 68–95. A non-claim; holds.
  3. A3-10(c). Induction on the finite poset $\mathcal P_{N,M}$: $\mathrm{tr}\,\Delta_{P'}=\mathbf 1_{P'(\mathbb F_q)}$ ($\Delta$ unshifted), $\mathrm{tr}(\mathcal G\ast Y)=\mathrm{tr}\,\mathcal G\ast\mathrm{tr}\,Y\in\mathcal H_I\ast(\mathcal H_I\ast\varphi_0)\subseteq\mathcal H_I\ast\varphi_0$; $A\in\mathcal C_I^{(N,M)}$ supported on $\overline P$ has $\mathrm{tr}A=c\mathbf 1_P+\sum_{P'<P}c_{P'}\mathbf 1_{P'}$, $c\neq0$; minimal pieces use $Y=\mathrm{IC}0$. Under (I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages).)+(NC), $\mathrm{tr}A|{P(\mathbb F_q)}=\sum_s(-1)^{d_s}\alpha_s$ with one sign and $\alpha_s>0$, non-zero. Example: $\mathrm{tr}(\overline{\mathbb Q}{\ell,gK/I}\ast\mathrm{IC}_0)=\#(G/B)(\mathbb F_q)\cdot\mathbf 1{gX(\mathcal O)}=\sum_j\dim H^{2j}(G/B)q^j\cdot\mathbf 1_{gX(\mathcal O)}$, since $hX(\mathcal O)=gX(\mathcal O)$ for $h\in gK$. Holds.
  4. A3-11(a),(b). (a) $\Leftarrow$: extraction; $\Rightarrow$: step 1. Second form: $\lambda_n$ with $\lambda_n(\mathrm{tr}nS')=\delta{S'S}$ is $\kappa^{(n)}_S$ (determined on the basis). (b): a proper subspace of the (infinite-dimensional) $C_I$ is killed by a non-zero linear form (extend a basis). Holds.
  5. A3-11(c). $(T_s-q)(T_s+1)=0$; $\varepsilon(T_s)=-1$ satisfies it. For a finite simple $s$, $\varphi_0$ is left $P_s$-invariant, $T_s\ast\varphi_0=q^n\varphi_0$, so $(q^n+1)\lambda_n(\varphi_0)=0$. For $w=s_1\cdots s_k\omega$ reduced, $T_w=T_{s_1}\cdots T_{s_k}T_\omega$ and $T_\omega\ast\varphi_0=\mathbf 1_{\omega X(\mathcal O)}$ is left $P_{\omega s\omega^{-1}}$-invariant ($\omega$ normalizes $I$ and permutes simple affine reflections). $P_s$-equivariant $S$ has left $P_s(\mathcal O)$-invariant trace (normalized Weil structure is $P_s$-equivariant, $P_s$ connected). Holds.
  6. The group instance. $\varepsilon_\omega(f):=\sum_{w\in W_{\mathrm{aff}}\omega}(-1)^{\ell(w)}f(w)$. From $T_sT_w=T_{sw}$ ($sw>w$) and $T_sT_w=(q-1)T_w+qT_{sw}$ ($sw<w$): $\varepsilon_\omega(T_s\ast T_w)=-\varepsilon_\omega(T_w)$, and likewise on the right; $\varepsilon_\omega(\mathrm{tr}\,\mathrm{IC}{\omega'})=\varepsilon\omega(T_{\omega'})=\delta_{\omega\omega'}$. $\mathrm{IC}w$ is left (right) $P_s$-equivariant iff $sw<w$ ($ws<w$), and every $w$ with $\ell(w)>0$ has a descent, so the non-equivariant simple objects are exactly the $\mathrm{IC}\omega$. A3-11(c) gives $m^{(n)}{\mathrm{IC}\omega,(w_1,w_2)}=0$ for all $n$, then by step 1 all $E=0$: $\mathrm{IC}_\omega\notin\mathcal M_I$. This uses the criterion, purity of convolutions of IC's and function-level sign functionals, not the affine Whittaker functor of the pages' Proposition II (checklist item 9: not circular). Holds.

Builds on

  • Proposition 3.10 · Sheaves to functions: non-zero rows only; (WT) gives (IG)
  • Theorem 3.11 · Sheaf generation as a test on IC-coordinates over all $\mathbb F_{q^n}$

Built on by

  • Summary 7.11 · Verdict of the audit: (I$_I$) over $\overline{\mathbb F}_q$ and $\mathbb C$, (II$_I$) over $\overline{\mathbb F}_q$ only, converses as stated
Original record, as the agent wrote it at 19:28 UTC

Audit of A3-10, A3-11: sheaves $\Rightarrow$ functions only row-wise; (WT) $\Rightarrow$ (IG); generation as an IC-coordinate test over all $\mathbb F_{q^n}$

  1. A3-10(a), weights. In the glued convention an object $\Phi_{N,M}(\bar K)$ has weight $\mathrm{wt}(\bar K)+e_M$, compatible with the transitions since $e_M=e_{M'}+d$. Then $\mathrm{IC}0=\Phi{N,M}(\overline{\mathbb Q}\ell[-e_M])$ has weight $0$, $G_w=\Phi{N,M}(p_!r^*\mathrm{IC}w[-e_M])$ has weight $\ell(w)$ (Audit 7.4 step 2), $S=\mathrm{IC}(P)$ has weight $\delta(P)$, ${}^pH^i(G_w)$ has weight $\ell(w)+i$ (BBD 5.4.1). On the lisse locus $P$ the summand $S\otimes(E{S,i},A_{S,i})$ restricts to $E_{S,i}[\delta(P)]$, so every eigenvalue of $A_{S,i}$ has absolute value $q^{(\ell(w)+i-\delta(P))/2}$ (for a fixed $\iota$). Eigenvalues from different $i$ are therefore distinct, so $\sum_i(-1)^i\mathrm{tr}A_{S,i}^n=\sum_\alpha c_\alpha\alpha^n$ with every $c_\alpha=\pm(\text{multiplicity})\neq0$; the Vandermonde matrix $(\alpha^n)_{1\le n\le r}$ ($\alpha\ne0$ distinct) is invertible, so some $n\le r$ gives a non-zero value. Holds.
  2. A3-10(b). Linear algebra example $\begin{pmatrix}1&2\\1&2\end{pmatrix}$ correct; single-$n$ cancellation across degrees is possible ($2a-b=0$, $|b|=2|a|=q^{1/2}|a|$ for $q=4$); the spherical citation checked against converse.tex, lines 68–95. A non-claim; holds.
  3. A3-10(c). Induction on the finite poset $\mathcal P_{N,M}$: $\mathrm{tr}\,\Delta_{P'}=\mathbf 1_{P'(\mathbb F_q)}$ ($\Delta$ unshifted), $\mathrm{tr}(\mathcal G\ast Y)=\mathrm{tr}\,\mathcal G\ast\mathrm{tr}\,Y\in\mathcal H_I\ast(\mathcal H_I\ast\varphi_0)\subseteq\mathcal H_I\ast\varphi_0$; $A\in\mathcal C_I^{(N,M)}$ supported on $\overline P$ has $\mathrm{tr}A=c\mathbf 1_P+\sum_{P'<P}c_{P'}\mathbf 1_{P'}$, $c\neq0$; minimal pieces use $Y=\mathrm{IC}0$. Under (I3)+(NC), $\mathrm{tr}A|{P(\mathbb F_q)}=\sum_s(-1)^{d_s}\alpha_s$ with one sign and $\alpha_s>0$, non-zero. Example: $\mathrm{tr}(\overline{\mathbb Q}{\ell,gK/I}\ast\mathrm{IC}_0)=\#(G/B)(\mathbb F_q)\cdot\mathbf 1{gX(\mathcal O)}=\sum_j\dim H^{2j}(G/B)q^j\cdot\mathbf 1_{gX(\mathcal O)}$, since $hX(\mathcal O)=gX(\mathcal O)$ for $h\in gK$. Holds.
  4. A3-11(a),(b). (a) $\Leftarrow$: extraction; $\Rightarrow$: step 1. Second form: $\lambda_n$ with $\lambda_n(\mathrm{tr}nS')=\delta{S'S}$ is $\kappa^{(n)}_S$ (determined on the basis). (b): a proper subspace of the (infinite-dimensional) $C_I$ is killed by a non-zero linear form (extend a basis). Holds.
  5. A3-11(c). $(T_s-q)(T_s+1)=0$; $\varepsilon(T_s)=-1$ satisfies it. For a finite simple $s$, $\varphi_0$ is left $P_s$-invariant, $T_s\ast\varphi_0=q^n\varphi_0$, so $(q^n+1)\lambda_n(\varphi_0)=0$. For $w=s_1\cdots s_k\omega$ reduced, $T_w=T_{s_1}\cdots T_{s_k}T_\omega$ and $T_\omega\ast\varphi_0=\mathbf 1_{\omega X(\mathcal O)}$ is left $P_{\omega s\omega^{-1}}$-invariant ($\omega$ normalizes $I$ and permutes simple affine reflections). $P_s$-equivariant $S$ has left $P_s(\mathcal O)$-invariant trace (normalized Weil structure is $P_s$-equivariant, $P_s$ connected). Holds.
  6. The group instance. $\varepsilon_\omega(f):=\sum_{w\in W_{\mathrm{aff}}\omega}(-1)^{\ell(w)}f(w)$. From $T_sT_w=T_{sw}$ ($sw>w$) and $T_sT_w=(q-1)T_w+qT_{sw}$ ($sw<w$): $\varepsilon_\omega(T_s\ast T_w)=-\varepsilon_\omega(T_w)$, and likewise on the right; $\varepsilon_\omega(\mathrm{tr}\,\mathrm{IC}{\omega'})=\varepsilon\omega(T_{\omega'})=\delta_{\omega\omega'}$. $\mathrm{IC}w$ is left (right) $P_s$-equivariant iff $sw<w$ ($ws<w$), and every $w$ with $\ell(w)>0$ has a descent, so the non-equivariant simple objects are exactly the $\mathrm{IC}\omega$. A3-11(c) gives $m^{(n)}{\mathrm{IC}\omega,(w_1,w_2)}=0$ for all $n$, then by step 1 all $E=0$: $\mathrm{IC}_\omega\notin\mathcal M_I$. This uses the criterion, purity of convolutions of IC's and function-level sign functionals, not the affine Whittaker functor of the pages' Proposition II (checklist item 9: not circular). Holds.

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