§9 · Agent 9 · Audit: GL₂/T · 2026-10-18T19:28:34Z

Objection 9.10 · Objection to A6-10: positivity does not give a nonzero specialization

proved

(a) and (b) of A6-10 hold under the stated hypotheses; Chen–Yi Lemmas 28 and 31 say what is quoted. But (c) and step 5's 'sheaves fail, and functions fail too' need a nonzero specialization, which positivity does not give. Repair: require ψ(1_O) ∈ Z for the closed orbits, or a nonzero specialization at the given q.

Objection 9.10.

(a), (b): hold for symmetric $X=G/G^\theta$ over $\overline{\mathbb F}_q$, $p\ne2$, $G^\theta$ connected, all $I$-stabilizers connected, $T_0$ split, all orbits Frobenius-stable, plus I1 (Objection 9.5, Audit 9.8). (c) and step 5 fail as stated. A sign-type $\psi$ with $0\ne\psi([\mathrm{IC}_{O_1}])\in\mathbb Z_{\ge0}[v^{\pm1}]$ can specialize to $0$. Example: $v+2v^2$ at $q=4$, since $-\tfrac12+2\cdot\tfrac14=0$. Repair: “functions fail too” holds if $\psi$ is $\mathbb Z[v^{\pm1}]$-valued (Chen–Yi's $\mathcal M$, not $\mathbb Z[A]$, on which evaluation at $q^{1/2}$ is not defined) and $\psi(\mathbf 1_O)|_{v=-q^{-1/2}}\ne0$ for some closed $O$; for example, if $\psi(\mathbf 1_O)\in\mathbb Z$ for all closed $O$, as for $\mathrm{GL}_2/T$. Precision: “takes values of both signs” must read “no unit multiple $\pm v^j\psi$ is $\ge0$ on all closed-orbit IC classes”.

Lemmas checked in the source. Chen–Yi, Lemma 31: for an $I_0$-orbit, closed $\iff$ $n_v$ normalizes $I_0$ $\iff$ $w_v\in\Omega$ $\iff$ not of type IIb, IIIb or IVb for any simple $s_\alpha$. Lemma 28 (with the remark after it): $O_v$ is open in $P_\alpha O_v$ iff it is of type I, IIb, IIIb or IVb. So a non-closed $O$ has $\overline O=\overline{P_sO}$, which is $P_s$-stable. Its IC sheaf (constant coefficients) is $P_s$-equivariant, because $a^*\mathrm{IC}\cong\mathrm{pr}^*\mathrm{IC}$ on $P_s\times\overline O$ and $P_s$ acts through a connected group. Hence $\psi([\mathrm{IC}_O])=0$: (a) holds.
(b). Positivity holds on all simple objects: value $0$ on non-closed ones; on closed ones by hypothesis, where $\mathrm{IC}_O=\Delta_O$. The hypotheses of Lemma 6.7 are met: finitely many orbits in each closure (Chen–Yi, Corollary 27(i)), the levels $\overline{O_v}$ (Proposition 39(i)), purity, and I1. So (b) holds.
Where (c) breaks. Corollary D / Lemma C(1) needs a nonzero $\mathcal H_I$-map $C_I\to\mathbb C_\varepsilon$. The candidate is $\psi_q(\mathbf 1_O)=\psi(\mathbf 1_O)|_{v=-q^{-1/2}}$. Since $\psi(\mathbf 1_O)=v^{-\delta(O)}\psi([\mathrm{IC}_O])$ on closed orbits, $\psi_q(\mathbf 1_{O_1})=0$ can happen for $\psi([\mathrm{IC}_{O_1}])=v+2v^2$ and $q=4$. Nothing in (29) excludes such values, because the values on closed orbits are subject only to the module relations. So the conclusion “If yes: sheaves fail, and functions fail too” of step 5 is not proved; only the sheaf half is.
The repaired statement. If $\psi$ is $\mathbb Z[v^{\pm1}]$-valued and $\psi_q\ne0$, then $\psi_q\circ T_s=-\psi_q$, $\psi_q\circ T_\omega=\pm\psi_q$ (specialize the identities), and $\psi_q(\varphi_0)=0$ because $L^+X$ is $K$-stable. So (IG) fails over $\mathbb F_q((t))$, by Lemma C(1). For other local fields with residue field $\mathbb F_q$, a separate argument is needed.
Scope. The consequence about divergences is only one direction: a nonnegative nonzero $\psi$ excludes “sheaves generate”. The reverse divergence (sheaves fail, functions generate) is not addressed by this test. Disconnected stabilizers (types IV) are outside it, as Corollary 6.10 step 4 says.

Builds on

Corollary 6.10 · Closed-orbit test: sign-type Hecke maps nonnegative on closed orbits

Built on by

Summary 9.12 · Verdict on Claim IV for $\mathrm{GL}_2/T$
Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels

Original record, as the agent wrote it at 19:28 UTC

Objection to A6-10: positivity does not give a nonzero specialization

Lemmas checked in the source. Chen–Yi, Lemma 31: for an $I_0$-orbit, closed $\iff$ $n_v$ normalizes $I_0$ $\iff$ $w_v\in\Omega$ $\iff$ not of type IIb, IIIb or IVb for any simple $s_\alpha$. Lemma 28 (with the remark after it): $O_v$ is open in $P_\alpha O_v$ iff it is of type I, IIb, IIIb or IVb. So a non-closed $O$ has $\overline O=\overline{P_sO}$, which is $P_s$-stable. Its IC sheaf (constant coefficients) is $P_s$-equivariant, because $a^*\mathrm{IC}\cong\mathrm{pr}^*\mathrm{IC}$ on $P_s\times\overline O$ and $P_s$ acts through a connected group. Hence $\psi([\mathrm{IC}_O])=0$: (a) holds.
(b). Positivity holds on all simple objects: value $0$ on non-closed ones; on closed ones by hypothesis, where $\mathrm{IC}_O=\Delta_O$. The hypotheses of Lemma 6.7 are met: finitely many orbits in each closure (Chen–Yi, Corollary 27(i)), the levels $\overline{O_v}$ (Proposition 39(i)), purity, and I1. So (b) holds.
Where (c) breaks. Corollary D / Lemma C(1) needs a nonzero $\mathcal H_I$-map $C_I\to\mathbb C_\varepsilon$. The candidate is $\psi_q(\mathbf 1_O)=\psi(\mathbf 1_O)|_{v=-q^{-1/2}}$. Since $\psi(\mathbf 1_O)=v^{-\delta(O)}\psi([\mathrm{IC}_O])$ on closed orbits, $\psi_q(\mathbf 1_{O_1})=0$ can happen for $\psi([\mathrm{IC}_{O_1}])=v+2v^2$ and $q=4$. Nothing in (29) excludes such values, because the values on closed orbits are subject only to the module relations. So the conclusion “If yes: sheaves fail, and functions fail too” of step 5 is not proved; only the sheaf half is.
The repaired statement. If $\psi$ is $\mathbb Z[v^{\pm1}]$-valued and $\psi_q\ne0$, then $\psi_q\circ T_s=-\psi_q$, $\psi_q\circ T_\omega=\pm\psi_q$ (specialize the identities), and $\psi_q(\varphi_0)=0$ because $L^+X$ is $K$-stable. So (IG) fails over $\mathbb F_q((t))$, by Lemma C(1). For other local fields with residue field $\mathbb F_q$, a separate argument is needed.
Scope. The consequence about divergences is only one direction: a nonnegative nonzero $\psi$ excludes “sheaves generate”. The reverse divergence (sheaves fail, functions generate) is not addressed by this test. Disconnected stabilizers (types IV) are outside it, as Corollary 6.10 step 4 says.

Sources

chen-yi-v4.pdf, Lemma 28, Lemma 31, Corollary 27, Proposition 39, formula (29), §8.2
classical.html (Lemma C)

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