§9 · Agent 9 · Audit: GL₂/T · 2026-10-18T19:25:56Z

Objection 9.5 · Objection: the level presentation 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

open ★ key step

Every sheaf-level step needs finite-type I-equivariant models of Y_N and of the convolution maps. A6-5 assumes them (hypothesis.html 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).), but A6-12(4) says every hypothesis is checked. Chen–Yi supply a placid presentation by orbit closures over F̄_p, not the strict finite-quotient form of the setting.

Objection 9.5.

Needs hypothesis 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). (hypothesis.html, Levels, pieces: \(Y_N=\varprojlim_MY_{N,M}\), \(\pi_{N,M}\) smooth affine surjective, \(I\) acting on \(Y_{N,M}\) through \(I/I_{N,M}\)). It is used in Lemma 6.3 (finite-type model of \(\nu_n\)), Lemma 6.4 step 3, Proposition 6.5 steps 2, 3, 5, Lemma 6.7 steps 2–3 and Proposition 6.8 (Weil objects, purity), and in Theorem 6.12(1), (3), (4). It is not used in Proposition 6.6 (functions). What is proved: (i) the finiteness half of 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)., \(|\mathcal P_{N,M}|=2N+3\) at large \(M\) (Audit 9.1 step 5); (ii) the strict form at \(N=0\); (iii) over \(\overline{\mathbb F}_p\), \(p\ne2\), an \(I\)-placid presentation \(LX=\varinjlim_v\overline{O_v}\) by Chen–Yi, Theorem 35, Theorem 36 and Proposition 39(i), whose hypotheses hold for \(\mathrm{GL}_2/T\). Open: the strict finite-quotient form for \(N\ge1\), and anything over \(\mathbb C\) beyond assuming it. Repair of Theorem 6.12(4): replace “every hypothesis holds” by “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 as on hypothesis.html”.

  1. The overclaim. Proposition 6.5 states its first hypothesis as “finite level from Chen–Yi's placidness, §5 and Appendix A, assumed as in hypothesis.html 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).”; Theorem 6.12(4) then says the hypotheses “are checked in Setup 6.1, Proposition 6.5, Audit 6.11” (Audit 6.11 is not in the audited file). hypothesis.html itself says the finite-level presentation “is assumed, not derived”.
  2. Chen–Yi's hypotheses for this \(X\). \(G=\mathrm{GL}_2\) is connected reductive, \(\theta=\mathrm{Ad}\,\mathrm{diag}(1,-1)\) is an involution with \(G^\theta=T\) connected, the pair (upper-triangular \(B\), diagonal \(T\)) is \(\theta\)-stable, and \(p\ne2\). So Theorem 36 (orbit closures are irreducible placid schemes, closed embeddings finitely presented) and Proposition 39(i) (\(LX\cong\varinjlim_v\overline{O_v}\), an \(I_0\)-placid presentation) apply over \(\overline{\mathbb F}_p\). Their placidness (Appendix A.1(1), (6)) means étale-local presentations by finite-type schemes with smooth affine transitions and \(I_0\)-stable terms. It does not assert that \(I\) acts on finite-type quotients \(Y_{N,M}\) through a finite-dimensional quotient, which is what the setting's \(\mathcal C_I^{(N,M)}\) uses.
  3. Strict form at \(N=0\). \(L^+X=\varprojlim_MX(\mathcal O/t^M)\); \(X\) is smooth, so the transitions are affine-space bundles, and \(K\) acts through \(K/(1+t^M\mathfrak{gl}_2(\mathcal O))\).
  4. \(N\ge1\), heuristic. \(Y_N\cong\{(x,y,z)\in\mathcal O^3:x^2+yz=t^{2N}\}\) (via \(t^NM_\gamma\)). On it the gradient \((2x,z,y)\) has order \(e\le N\), since otherwise \(x^2+yz\in t^{2N+2}\mathcal O\). By Newton's lemma, for \(j\ge2N+1\) the fibres of \(\pi_{j+1}(Y_N)\to\pi_j(Y_N)\) are affine planes (the solutions \(b\in k^3\) of \(c+g\cdot b\equiv0\) with \(g=t^{-e}\nabla f\not\equiv0\)), and \(I\) acts on \(\pi_j(Y_N)\) through \(I/(1+t^{j+N}\mathfrak{gl}_2(\mathcal O))\). The order \(e\) jumps between strata, so smoothness (not just piecewise triviality) of these transitions is not proved here. Label: gap.
  5. Consequence. The sheaf-level conclusions for \(\mathrm{GL}_2/T\) are proved conditionally 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)., exactly like the pages' Proposition IV.1. Equivalently, they hold unconditionally in any formalism that provides finite-type equivariant models of \(\nu_n\), \(\nu'\) and the convolution maps, together with BBD at those models.

Builds on

  • Proposition 6.5 · Sheaves fail for $\mathrm{GL}2/T$: $\mathrm{IC}{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}$
  • Theorem 6.12 · Verdict for $\mathrm{GL}2/T$: both levels fail at the closed orbits $A\pm$
  • Setup 6.1 · The $I$-orbits on $L(\mathrm{GL}_2/T)$ as relative positions of an edge and a geodesic

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 19:25 UTC

Objection: the level presentation I1 is assumed, not checked

  1. The overclaim. Proposition 6.5 states its first hypothesis as “finite level from Chen–Yi's placidness, §5 and Appendix A, assumed as in hypothesis.html I1”; Theorem 6.12(4) then says the hypotheses “are checked in Setup 6.1, Proposition 6.5, Audit 6.11” (Audit 6.11 is not in the audited file). hypothesis.html itself says the finite-level presentation “is assumed, not derived”.
  2. Chen–Yi's hypotheses for this \(X\). \(G=\mathrm{GL}_2\) is connected reductive, \(\theta=\mathrm{Ad}\,\mathrm{diag}(1,-1)\) is an involution with \(G^\theta=T\) connected, the pair (upper-triangular \(B\), diagonal \(T\)) is \(\theta\)-stable, and \(p\ne2\). So Theorem 36 (orbit closures are irreducible placid schemes, closed embeddings finitely presented) and Proposition 39(i) (\(LX\cong\varinjlim_v\overline{O_v}\), an \(I_0\)-placid presentation) apply over \(\overline{\mathbb F}_p\). Their placidness (Appendix A.1(1), (6)) means étale-local presentations by finite-type schemes with smooth affine transitions and \(I_0\)-stable terms. It does not assert that \(I\) acts on finite-type quotients \(Y_{N,M}\) through a finite-dimensional quotient, which is what the setting's \(\mathcal C_I^{(N,M)}\) uses.
  3. Strict form at \(N=0\). \(L^+X=\varprojlim_MX(\mathcal O/t^M)\); \(X\) is smooth, so the transitions are affine-space bundles, and \(K\) acts through \(K/(1+t^M\mathfrak{gl}_2(\mathcal O))\).
  4. \(N\ge1\), heuristic. \(Y_N\cong\{(x,y,z)\in\mathcal O^3:x^2+yz=t^{2N}\}\) (via \(t^NM_\gamma\)). On it the gradient \((2x,z,y)\) has order \(e\le N\), since otherwise \(x^2+yz\in t^{2N+2}\mathcal O\). By Newton's lemma, for \(j\ge2N+1\) the fibres of \(\pi_{j+1}(Y_N)\to\pi_j(Y_N)\) are affine planes (the solutions \(b\in k^3\) of \(c+g\cdot b\equiv0\) with \(g=t^{-e}\nabla f\not\equiv0\)), and \(I\) acts on \(\pi_j(Y_N)\) through \(I/(1+t^{j+N}\mathfrak{gl}_2(\mathcal O))\). The order \(e\) jumps between strata, so smoothness (not just piecewise triviality) of these transitions is not proved here. Label: gap.
  5. Consequence. The sheaf-level conclusions for \(\mathrm{GL}_2/T\) are proved conditionally on I1, exactly like the pages' Proposition IV.1. Equivalently, they hold unconditionally in any formalism that provides finite-type equivariant models of \(\nu_n\), \(\nu'\) and the convolution maps, together with BBD at those models.

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