§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:24:04Z

Audit 8.11 · Audit of $\mathbb G_m$ and the other rows (A2-10 (a)–(d), (f))

proved

For G_m = GL_2/Sp_2 both levels hold, by the same length-zero mechanism as A^n; correct over C and F̄_q. For the other symmetric rows the criterion (∗_X) is correctly reduced to Chen–Yi's inputs plus the unproved level-presentation obligation (L_X); nothing is decided there.

Audit 8.11.

(c) Holds over $\mathbb C$ and $\overline{\mathbb F}_q$: for $X=\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m$ (action through $\det$), the $I$-orbits are $t^m\mathcal O^\times$ (open and closed), $\mathcal S_I=\{\underline k_{t^m\mathcal O^\times}\}$, $\Delta_{\omega^m}\ast_I\mathrm{IC}_0=\underline k_{t^m\mathcal O^\times}$ for $\omega=P\operatorname{diag}(t,1)$ ($\det\omega=-t$), so $\mathcal C_I^{\rm Iwa}=\mathcal C_I$; and $T_{\omega^m}\ast\varphi_0=\mathbf 1_{t^m\mathcal O^\times}$ span $C_I$, so (IG) holds. The row agrees, consistent with Table 1 and IV.3.

(a), (f) Need hypothesis (L$_X$) (Chen–Yi's placid presentation satisfies the level axioms and full faithfulness), and Chen–Yi's standing hypotheses $p\ne2$, $K=G^\theta$ connected. The latter holds for every symmetric row of Table 1. Given these, “the row differs iff $(\ast_X)$” holds for the rows where (IG) is known to fail.

(b) depends on A2-3(c), not in the file; (d) is heuristic, correctly labelled.

$\mathbb G_m$ orbits and levels. $\det(I)=\mathcal O^\times$, so the orbits on $F^\times$ are $t^m\mathcal O^\times$. $I/I_{N,M}\simeq(\mathcal O/t^M)^\times$ (connected: $\mathbb G_m$ times a unipotent group) acts simply transitively on each $t^m(\mathcal O/t^M)^\times$, and $(\mathcal O/t^{M+1})^\times\to(\mathcal O/t^M)^\times$ is a $\mathbb G_a$-torsor. Stabilizers are trivial, so all local systems are constant; each piece is closed and smooth, and its IC is the shifted constant sheaf. The translate: $\omega\cdot x=\det(\omega)x=-tx$, so $(\omega^m)_*\underline k_{\mathcal O^\times}=\underline k_{t^m\mathcal O^\times}$. Functions: $I$-invariant compactly supported $\varphi$ on $F^\times$ are finite combinations of $\mathbf 1_{t^m\mathcal O^\times}$ (each orbit open). No weights, so the proof is valid over $\mathbb C$. This re-proves IV.3 by the route of Audit 8.9.
Chen–Yi inputs, checked in chen-yi-v4.pdf. §2.1: $G$ connected reductive over $\overline{\mathbb F}_p$, $p\ne2$, $K=G^\theta$. §1.2: $K$ is assumed connected from §4 to §10, which contains Cor. 27 (finitely many orbits in each closure), Thm. 36 (placid closures), Prop. 70(ii) (semisimple $\star$ semisimple is semisimple) and Lemma 71(ii) (equivariant local systems on $O_v$ are sums of $L_{\xi,v}$, $\xi\in X(\pi_0(T_v))$). A2-10 does not mention the connectedness hypothesis. It holds for every row ($\Delta H$, $\mathrm{Sp}_{2n}$, $T$, $\mathrm{GL}_p\times\mathrm{GL}_q$, $\mathrm{Spin}_{2n-1}$, $F_4$), so this is a missing statement, not a gap. Chen–Yi's $I_0$ comes from a $\theta$-stable Borel; conjugating by $g\in G(k)\subseteq L^+G$ moves it to the pages' $I$ and preserves $L^+X$, so $\mathrm{IC}_0$ is unchanged.
(L$_X$). The authors correctly say that Chen–Yi's placid presentation is not checked against the level axioms: affine-space transitions in $M$, Cartesian closed embeddings in $N$, and full faithfulness. This is the same unproved reading as the first half of the pages' I1. It is a genuine open obligation for all symmetric rows except $\mathbb G_m$ and the group (verified in Audit 8.1 and in step 1 above).
Local systems (checklist 4). $(\ast_X)$ counts the IC's with non-constant $L_{\xi,v}$, as it must. Functions see only traces, so a divergence “(IG) holds, sheaves fail” can only come from $\xi\ne1$ or from non-split extensions. With purity this reduces to $\xi\ne1$, as the authors say (heuristic as a classification, correct as a remark).
(f). For the rows of Table 1 where (IG) is known to fail (index.html, Table 1: $\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$, $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$ in characteristic $0$; $\mathrm{GL}_2/T$; $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ for $p=q$ and $(p,q)=(1,2)$; $\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$, $n=2,3$), divergence is equivalent to sheaf generation, which by the criterion is equivalent to $(\ast_X)$ given (L$_X$). The function failures themselves are cited from the pages and not audited here. For the characteristic-$0$ entries, the link with sheaves over $\overline{\mathbb F}_q$ needs the function statement over $\mathbb F_q(\!(t)\!)$, as stated.

Builds on

Proposition 2.10 · Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction

Built on by

Summary 8.13 · Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$

Original record, as the agent wrote it at 19:24 UTC

Audit of $\mathbb G_m$ and the other rows (A2-10 (a)–(d), (f))

$\mathbb G_m$ orbits and levels. $\det(I)=\mathcal O^\times$, so the orbits on $F^\times$ are $t^m\mathcal O^\times$. $I/I_{N,M}\simeq(\mathcal O/t^M)^\times$ (connected: $\mathbb G_m$ times a unipotent group) acts simply transitively on each $t^m(\mathcal O/t^M)^\times$, and $(\mathcal O/t^{M+1})^\times\to(\mathcal O/t^M)^\times$ is a $\mathbb G_a$-torsor. Stabilizers are trivial, so all local systems are constant; each piece is closed and smooth, and its IC is the shifted constant sheaf. The translate: $\omega\cdot x=\det(\omega)x=-tx$, so $(\omega^m)_*\underline k_{\mathcal O^\times}=\underline k_{t^m\mathcal O^\times}$. Functions: $I$-invariant compactly supported $\varphi$ on $F^\times$ are finite combinations of $\mathbf 1_{t^m\mathcal O^\times}$ (each orbit open). No weights, so the proof is valid over $\mathbb C$. This re-proves IV.3 by the route of Audit 8.9.
Chen–Yi inputs, checked in chen-yi-v4.pdf. §2.1: $G$ connected reductive over $\overline{\mathbb F}_p$, $p\ne2$, $K=G^\theta$. §1.2: $K$ is assumed connected from §4 to §10, which contains Cor. 27 (finitely many orbits in each closure), Thm. 36 (placid closures), Prop. 70(ii) (semisimple $\star$ semisimple is semisimple) and Lemma 71(ii) (equivariant local systems on $O_v$ are sums of $L_{\xi,v}$, $\xi\in X(\pi_0(T_v))$). A2-10 does not mention the connectedness hypothesis. It holds for every row ($\Delta H$, $\mathrm{Sp}_{2n}$, $T$, $\mathrm{GL}_p\times\mathrm{GL}_q$, $\mathrm{Spin}_{2n-1}$, $F_4$), so this is a missing statement, not a gap. Chen–Yi's $I_0$ comes from a $\theta$-stable Borel; conjugating by $g\in G(k)\subseteq L^+G$ moves it to the pages' $I$ and preserves $L^+X$, so $\mathrm{IC}_0$ is unchanged.
(L$_X$). The authors correctly say that Chen–Yi's placid presentation is not checked against the level axioms: affine-space transitions in $M$, Cartesian closed embeddings in $N$, and full faithfulness. This is the same unproved reading as the first half of the pages' I1. It is a genuine open obligation for all symmetric rows except $\mathbb G_m$ and the group (verified in Audit 8.1 and in step 1 above).
Local systems (checklist 4). $(\ast_X)$ counts the IC's with non-constant $L_{\xi,v}$, as it must. Functions see only traces, so a divergence “(IG) holds, sheaves fail” can only come from $\xi\ne1$ or from non-split extensions. With purity this reduces to $\xi\ne1$, as the authors say (heuristic as a classification, correct as a remark).
(f). For the rows of Table 1 where (IG) is known to fail (index.html, Table 1: $\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$, $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$ in characteristic $0$; $\mathrm{GL}_2/T$; $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ for $p=q$ and $(p,q)=(1,2)$; $\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$, $n=2,3$), divergence is equivalent to sheaf generation, which by the criterion is equivalent to $(\ast_X)$ given (L$_X$). The function failures themselves are cited from the pages and not audited here. For the characteristic-$0$ entries, the link with sheaves over $\overline{\mathbb F}_q$ needs the function statement over $\mathbb F_q(\!(t)\!)$, as stated.

Sources

chen-yi-v4.pdf, §1.2, §2.1, Cor. 27, Thm. 36, Prop. 70(ii), Lemma 71(ii), Def. 72
case-gm.html, IV.3 (comparison)
index.html, Table 1

Table 1. Cumulative token usage and estimated cost.
Work	Tokens	US $
Original search, audits and page construction	384.00 M	≈ $171.00
Presentation edits and follow-up corrections	9.64 M	≈ $4.99
Presentation edits and follow-up corrections	9.82 M	≈ $4.29
Total	≈ 403.46 M	≈ $180.28