§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:24:32Z

Objection 8.12 · Objection to the sign-obstruction lemma (A2-10(e))

proved

The proof of A2-10(e) expands the generator's trace as Σ(−1)^s Tr(F^r|V_{S,s}) t_S over all simple S, which is meaningless for constituents not fixed by Frobenius, the typical case once non-constant L_{ξ,v} occur. The lemma survives with a repaired expansion over Frobenius orbits and a purity hypothesis on S_0.

Objection 8.12.

Holds with a minor repair for Proposition 2.10(e). Read the hypothesis as: for every \(r\ge1\) and every \(F^r\)-stable simple \(S\not\simeq S_0\) at the level, \(\lambda_r(t^{(r)}_S)=0\) for one (equivalently every) \(F^r\)-Weil structure on \(S\). Add: \(S_0\) carries a Weil structure pure of some weight \(w_{S_0}\), which holds for \(\mathrm{IC}(\overline{O_v},L_{\xi,v})\) with \(\xi\) Frobenius-stable, by finite monodromy and Gabber. Then, if every generator \(G_w\) has a finite-type proper model with pure input, \(S_0\) is not a perverse constituent of any \(G_w\), so \(S_0\notin\mathcal C_I^{\rm Iwa}\). As written, the step “\(t_{G_w}=\sum_{S,s}(-1)^s\operatorname{Tr}(F^r|V_{S,s})t_S\)” fails when some constituent \(S\) has \(F^*S\not\simeq S\): then \(V_{S,s}\) carries no Frobenius and \(t_S\) is undefined.

  1. Where the gap is. In Theorem 2.9 every \(\mathrm{IC}_v\) is Frobenius-stable, so \(V_{v,s}\) carries a Frobenius. For symmetric \(X\) the simple objects \(\mathrm{IC}(\overline{O_v},L_{\xi,v})\) are indexed by \(\xi\in X(\pi_0(T_v))\), and Frobenius acts on these characters (non-split \(T_v\)); a constituent \(S\) may satisfy \(F^*S\simeq S'\ne S\). Frobenius then permutes isotypic components, and the authors' displayed expansion has no meaning for such \(S\) (checklist 5).
  2. Repair: orbit decomposition. Purity of \(G_w\) (proper model, pure \(\mathrm{IC}_w\) and \(\mathrm{IC}_0\)) makes \(P_s={}^pH^s(G_w)\) geometrically semisimple (BBD 5.3.8). Its isotypic decomposition is canonical, so the Weil structure permutes components along Frobenius orbits: \(P_s=(S_0\otimes V_{S_0,s})\oplus\bigoplus_{\mathcal O}R_{\mathcal O}\), each summand Frobenius-stable, \(S_0\) being stable.
  3. Orbits of length \(d\). \(R_{\mathcal O}=\bigoplus_{i\in\mathbb Z/d}S_i\otimes V_i\) with \(\varphi\) mapping the \(i\)-th summand of \(F^*R_{\mathcal O}\) to the \((i+1)\)-th. At \(x\in LX(\mathbb F_{q^r})\), \(\varphi^r\) moves the \(i\)-th stalk summand to the \((i+r)\)-th, so \(t^{(r)}_{R_{\mathcal O}}=0\) if \(d\nmid r\). If \(d\mid r\), each \(S_i\otimes V_i\) is \(F^r\)-stable, and since \(\operatorname{End}S_i=\overline{\mathbb Q}_\ell\) it is \(S_i\) (any \(F^r\)-Weil structure) \(\otimes(V_i,B_i)\). So \(t^{(r)}_{R_{\mathcal O}}=\sum_i\operatorname{tr}(B_i)\,t^{(r)}_{S_i}\) and \(\lambda_r(t^{(r)}_{R_{\mathcal O}})=0\) by the repaired hypothesis (independent of the Weil structure chosen, which changes \(t_{S_i}\) by a scalar).
  4. Rest of the proof. \(\lambda_r(\varphi_0)=0\) from \(\mathbf 1_K\ast\varphi_0=[K:I]\varphi_0\), \(\varepsilon_r(\mathbf 1_K)=0\), \([K:I]\ne0\). Then \(\lambda_r(t_{G_w})=\varepsilon_r(t_{\mathrm{IC}_w})\lambda_r(\varphi_0)=0\) (trace of convolution, \(\operatorname{vol}(I)=1\)). By steps 2–3, \(0=c\beta^r\sum_s(-1)^s\operatorname{Tr}(F^r|V_{S_0,s})\). \(V_{S_0,s}\) is pure of weight \(w_G+s-w_{S_0}\) (the argument of Audit 8.4 step 3, which needs the pure Weil structure on \(S_0\) added above), so the eigenvalues of \(\beta F\) on different \(V_{S_0,s}\) have different absolute values and Audit 8.5 steps 3–4 give \(V_{S_0,s}=0\). Then Audit 8.2(i) gives \(S_0\notin\mathcal C_I^{\rm Iwa}\).
  5. Scope. The lemma is a criterion; the file applies it to no symmetric row other than the group, where it is Audit 8.5. So the repair changes no conclusion in the file.

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

Objection to the sign-obstruction lemma (A2-10(e))

  1. Where the gap is. In Theorem 2.9 every \(\mathrm{IC}_v\) is Frobenius-stable, so \(V_{v,s}\) carries a Frobenius. For symmetric \(X\) the simple objects \(\mathrm{IC}(\overline{O_v},L_{\xi,v})\) are indexed by \(\xi\in X(\pi_0(T_v))\), and Frobenius acts on these characters (non-split \(T_v\)); a constituent \(S\) may satisfy \(F^*S\simeq S'\ne S\). Frobenius then permutes isotypic components, and the authors' displayed expansion has no meaning for such \(S\) (checklist 5).
  2. Repair: orbit decomposition. Purity of \(G_w\) (proper model, pure \(\mathrm{IC}_w\) and \(\mathrm{IC}_0\)) makes \(P_s={}^pH^s(G_w)\) geometrically semisimple (BBD 5.3.8). Its isotypic decomposition is canonical, so the Weil structure permutes components along Frobenius orbits: \(P_s=(S_0\otimes V_{S_0,s})\oplus\bigoplus_{\mathcal O}R_{\mathcal O}\), each summand Frobenius-stable, \(S_0\) being stable.
  3. Orbits of length \(d\). \(R_{\mathcal O}=\bigoplus_{i\in\mathbb Z/d}S_i\otimes V_i\) with \(\varphi\) mapping the \(i\)-th summand of \(F^*R_{\mathcal O}\) to the \((i+1)\)-th. At \(x\in LX(\mathbb F_{q^r})\), \(\varphi^r\) moves the \(i\)-th stalk summand to the \((i+r)\)-th, so \(t^{(r)}_{R_{\mathcal O}}=0\) if \(d\nmid r\). If \(d\mid r\), each \(S_i\otimes V_i\) is \(F^r\)-stable, and since \(\operatorname{End}S_i=\overline{\mathbb Q}_\ell\) it is \(S_i\) (any \(F^r\)-Weil structure) \(\otimes(V_i,B_i)\). So \(t^{(r)}_{R_{\mathcal O}}=\sum_i\operatorname{tr}(B_i)\,t^{(r)}_{S_i}\) and \(\lambda_r(t^{(r)}_{R_{\mathcal O}})=0\) by the repaired hypothesis (independent of the Weil structure chosen, which changes \(t_{S_i}\) by a scalar).
  4. Rest of the proof. \(\lambda_r(\varphi_0)=0\) from \(\mathbf 1_K\ast\varphi_0=[K:I]\varphi_0\), \(\varepsilon_r(\mathbf 1_K)=0\), \([K:I]\ne0\). Then \(\lambda_r(t_{G_w})=\varepsilon_r(t_{\mathrm{IC}_w})\lambda_r(\varphi_0)=0\) (trace of convolution, \(\operatorname{vol}(I)=1\)). By steps 2–3, \(0=c\beta^r\sum_s(-1)^s\operatorname{Tr}(F^r|V_{S_0,s})\). \(V_{S_0,s}\) is pure of weight \(w_G+s-w_{S_0}\) (the argument of Audit 8.4 step 3, which needs the pure Weil structure on \(S_0\) added above), so the eigenvalues of \(\beta F\) on different \(V_{S_0,s}\) have different absolute values and Audit 8.5 steps 3–4 give \(V_{S_0,s}=0\). Then Audit 8.2(i) gives \(S_0\notin\mathcal C_I^{\rm Iwa}\).
  5. Scope. The lemma is a criterion; the file applies it to no symmetric row other than the group, where it is Audit 8.5. So the repair changes no conclusion in the file.

Sources