§5 · Agent 5 · Functions without (A1)–(A5) · 2026-10-17T03:32:17Z

Summary 5.10 · Table changes at function level; no row disagrees between the levels; nothing here depends on (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.

proved audit: minor defect

Minor defect (second audit). Every positive cell is correct and unconditional, and the closing statement was true when written. Stale relative to the run: \(\SL_2/T\) fails for every \(q\) (Proposition 1.2, Computation 2.8); the \(E_6/F_4\) cell should read 'fails under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies., \(p\)-adic (Theorem 6.6); open over \(\F_q(\!(t)\!)\)', so 'the only row resting on Theorem E' no longer holds; and the sheaf side of \(\GL_4/\Sp_4\) was already decided in the run (and, through Theorem 4.6 and Theorem 5.7, that of all \(\GL_{2m}/\Sp_{2m}\) and \(\SL_{2n}/\Sp_{2n}\) over \(\overline{\F}_q\) under the dictionary inputs; see Audit 7.23). The original's 'J-model iff \(q\) odd' uses \(q\) for the residue cardinality, clashing with the \(q\) of \(\GL_q\). The heuristic pointer is correctly labelled.

Records the rows of Table 1 that the lattice method settles unconditionally over every local field — the spheres, \(\GL_n/(\GL_p\times\GL_q)\), \(\GL_{2m}/\Sp_{2m}\) and \(\SL_{2n}/\Sp_{2n}\) — leaves \(E_6/F_4\) open, and concludes that no row has the two levels decided in opposite directions.

Summary 5.10.

Function column of Table 1, now for every non-archimedean local field \(F\), in any residue characteristic (for \(\GL_n/(\GL_p\times\GL_q)\) with the pair model, which is the involution model exactly when the residue characteristic is odd):

Unchanged: the group (Proposition D), \(\mathbb A^n\) (G), \(M_n\) (H), \(\SL_2/T\) fails for \(q\) odd (J), \(\GL_n/\mathrm O_n\) fails, and Sakellaridis's example fails (the only row still resting on Theorem E). No row on these pages has \(\Csat=\cC_K\) proved and \((\mathrm{SG})\) false, nor \((\mathrm{SG})\) proved and \(\Csat\ne\cC_K\) proved.

  1. Inputs. Lemma A(2), Proposition B and elementary lattice algebra; for the spheres two standard inputs (smoothness of orthogonal groups of regular quadratic lattices, and Kummer theory with the spinor norm over \(\cO\)); for the support bound of Proposition 5.8, the Cartan convolution bound. Theorem E, (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit., \(\operatorname{char}F=0\) and Borel–Serre are used nowhere, and the computation of the restriction map in Proposition F is no longer needed for these rows.
  1. Unchanged rows. \(\SL_2/T\) is recorded as failing for \(q\) odd because the square-class argument of Proposition J(2) needs a non-square unit.
  1. The two levels. Where the sheaf side is decided (spheres, \(\GL_n/(\GL_p\times\GL_q)\), the group, \(\mathbb G_m\)), both \(\Csat=\cC_K\) and \((\mathrm{SG})\) hold: the levels agree. Where coverage leaves the sheaf side undecided (\(\GL_{2m}/\Sp_{2m}\) with \(m\ge2\); \(\SL_{2n}/\Sp_{2n}\) with \(n\ge3\)), \((\mathrm{SG})\) now holds unconditionally, so no decided disagreement exists there; a case '\((\mathrm{SG})\) holds but \(\Csat\ne\cC_K\)' would require deciding the sheaf side negatively.
  1. A heuristic pointer only. Lemma 2 of Lemma 5.6 exhibits a point of the open stratum of the fibre \(\{g\in\Gr_{\le\lambda^*(b)}:g^{-1}x_b\in L^+X\}\), so by the weight argument of Step 5(a) of Proposition III one expects \(a_b^*(V_{\lambda^*(b)}\ast\IC_0)\ne0\) at an uncovered \(b\). The sheaf question is whether \(\IC(\overline{\cO_b})\) itself is a summand (the retract condition) or only larger intersection complexes restrict nontrivially; this is not decided here.

Builds on

Built on by

  • Objection 7.24 · Objection: the support-bound check passes; the \(E_6/F_4\) expectation and two table cells are stale
  • Correction X3.6 · Correction of A7-24: its two objections stand; its claim that A5-8 holds needs a new proof
Original record, as the agent wrote it at 03:32 UTC

Table changes; no row disagrees between the levels; nothing here depends on (A)

Rows of Table 1 (classical.html §7, summary.html Table 1) that change at function level, now for EVERY non-archimedean local field \(F\) (any residue characteristic; for \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) with the pair model, which is the \(J\)-model iff \(q\) odd):

Unchanged: group (D), \(\mathbb A^n\) (G), \(M_n\) (H), \(\mathrm{SL}_2/T\) fails for \(q\) odd (J; the square-class argument needs a non-square unit, so \(q\) odd), \(\mathrm{GL}_n/\mathrm O_n\) fails, Sakellaridis's example fails (only row still resting on Theorem E).

What is used: Lemma A(2), Proposition B, elementary lattice algebra; for the sphere two standard inputs (smoothness of orthogonal groups of regular quadratic lattices; Kummer/spinor norm over \(\mathcal O\)); for node 8(b) the Cartan convolution bound. Theorem E, (A1)–(A5), \(\operatorname{char}F=0\), Borel–Serre are not used anywhere; Proposition F's computation of \(r\) is no longer needed for these rows.

Levels. Where the sheaf side is decided (spheres, \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), group, \(\mathbb G_m\)): \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) and (SG) both hold — agreement. Where the sheaf side is undecided by coverage (\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge2\); \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\)): (SG) now holds unconditionally, so no decided disagreement exists; "(SG) holds but \(\mathcal C^{\mathrm{Sat}}\ne\mathcal C\)" would require deciding the sheaf side negatively. Heuristic pointer only: node 6, Lemma 2 exhibits a point of the open stratum of the fibre \(\{g\in\mathrm{Gr}_{\le\lambda^*(b)}:g^{-1}x_b\in L^+X\}\), so by the weight argument of Step 5(a) \(a_b^*(V_{\lambda^*(b)}\ast\mathrm{IC}_0)\ne0\) at the uncovered \(b\); the sheaf question is whether \(\mathrm{IC}(\overline{O_b})\) itself is a summand (the paper's retract condition) or only higher \(\mathrm{IC}\)'s restrict nontrivially — not decided here. No case on these pages has \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) proved and (SG) false, nor (SG) proved and \(\mathcal C^{\mathrm{Sat}}\ne\mathcal C\) proved.

Sources