§9 · Agent 9 · Audit: GL₂/T · 2026-10-18T19:29:04Z

Audit 9.11 · Audit of A6-12: both levels fail for $\mathrm{GL}_2/T$

proved ★ key step

Survives: (1) sheaves fail exactly at k_{A±} over F̄_q and C, given I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).; (2) the functions statement holds over every non-archimedean F with no further hypothesis; (3) the traces of C_I^Iwa are H_I·φ0; (4) is repaired, because I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III). is assumed, not checked.

Audit 9.11.

Holds with repairs. (1) holds over \(\overline{\mathbb F}_q\) (\(p\) odd) and over \(\mathbb C\), conditional on I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III). (Audit 9.6, Objection 9.5); the positive-certificate reproof holds over \(\overline{\mathbb F}_q\) only (Audit 9.9). (2) holds over every non-archimedean \(F\), unconditionally (Audit 9.7). (3) holds over \(\mathbb F_q((t))\) for Weil objects, conditional on I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).. (4) holds except for one point: “every hypothesis of (I_I) holds” must read “every hypothesis except I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III)., which is assumed”. (II_I) is indeed vacuous here, since (IG) fails. Corollary 6.9 and Audit 6.11 are cited but not in the audited file; the row verdict does not depend on them.

  1. (1). Steps of Proposition 6.5 audited in Audit 9.6, using Audit 9.1, Audit 9.3, Audit 9.4. The mechanism line (step 5 of Theorem 6.12) is correct: every generator is a sum of objects equivariant for \(K\) or \(K'\). \(A_\pm\) are the only orbits of type IIIa for both \(s_0\) and \(s_1\). The other orbits are of type IIb or IIIb for some \(s\), so their closures are \(P_s\)-stable (Computation 9.2, types from (29)).
  2. (2). Audit 9.7: \(\mathcal H_I\ast\varphi_0=C_K+C_{K'}\), the quotient is \(\varepsilon_-\oplus\varepsilon_+\), and the Steinberg map of Proposition F restricts to \(\mathrm{vol}(A_+)\lambda\). This refines classical.html, Proposition F, from “proper” to “codimension two”.
  3. (3). Every Weil object whose geometric object lies in \(\mathcal C_I^{\mathrm{Iwa}}\) has trace in \(\langle f_n,f^{(1)}_n\rangle=\mathcal H_I\ast\varphi_0\) (repair of Proposition 6.6(e) in Audit 9.7). Conversely \(\operatorname{tr}\mathrm{IC}_{i,n}=v^{n+1}f^{(i)}_n\), and \(\operatorname{tr}k_{A_\pm}=\mathbf 1_{A_\pm}\) spans the complement (pairing determinant \(1\)).
  4. Comparison with the pages. summary.html lists \(\mathrm{GL}_2/T\) as sheaves “holds → open”, functions “holds → fails (F)”. The run decides the open cell as fails, conditional on I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III)., the same condition as Proposition IV.1 for \(\mathbb A^n\). The row agrees at both levels.
  5. Checklist. 1: closures are finite unions of orbits, so pieces are orbits. 2: generators, one-sided action. 3: Bott–Samelson plus BBD. 4: connected stabilizers, no local systems. 5: orbits and ICs are defined over \(\mathbb F_q\); the function computation is direct and needs no Lang or independence argument. 6: handled by purity (Audit 9.8). 7: fields as stated above. 8: no transfer of spherical hypotheses is used. 9: the reproof does not use Lemma 6.3 or Lemma 6.4.

Builds on

  • Theorem 6.12 · Verdict for $\mathrm{GL}2/T$: both levels fail at the closed orbits $A\pm$

Built on by

Original record, as the agent wrote it at 19:29 UTC

Audit of A6-12: both levels fail for $\mathrm{GL}_2/T$

  1. (1). Steps of Proposition 6.5 audited in Audit 9.6, using Audit 9.1, Audit 9.3, Audit 9.4. The mechanism line (step 5 of Theorem 6.12) is correct: every generator is a sum of objects equivariant for \(K\) or \(K'\). \(A_\pm\) are the only orbits of type IIIa for both \(s_0\) and \(s_1\). The other orbits are of type IIb or IIIb for some \(s\), so their closures are \(P_s\)-stable (Computation 9.2, types from (29)).
  2. (2). Audit 9.7: \(\mathcal H_I\ast\varphi_0=C_K+C_{K'}\), the quotient is \(\varepsilon_-\oplus\varepsilon_+\), and the Steinberg map of Proposition F restricts to \(\mathrm{vol}(A_+)\lambda\). This refines classical.html, Proposition F, from “proper” to “codimension two”.
  3. (3). Every Weil object whose geometric object lies in \(\mathcal C_I^{\mathrm{Iwa}}\) has trace in \(\langle f_n,f^{(1)}_n\rangle=\mathcal H_I\ast\varphi_0\) (repair of Proposition 6.6(e) in Audit 9.7). Conversely \(\operatorname{tr}\mathrm{IC}_{i,n}=v^{n+1}f^{(i)}_n\), and \(\operatorname{tr}k_{A_\pm}=\mathbf 1_{A_\pm}\) spans the complement (pairing determinant \(1\)).
  4. Comparison with the pages. summary.html lists \(\mathrm{GL}_2/T\) as sheaves “holds → open”, functions “holds → fails (F)”. The run decides the open cell as fails, conditional on I1, the same condition as Proposition IV.1 for \(\mathbb A^n\). The row agrees at both levels.
  5. Checklist. 1: closures are finite unions of orbits, so pieces are orbits. 2: generators, one-sided action. 3: Bott–Samelson plus BBD. 4: connected stabilizers, no local systems. 5: orbits and ICs are defined over \(\mathbb F_q\); the function computation is direct and needs no Lang or independence argument. 6: handled by purity (Audit 9.8). 7: fields as stated above. 8: no transfer of spherical hypotheses is used. 9: the reproof does not use Lemma 6.3 or Lemma 6.4.

Sources