§7 · Agent 7 · Adversarial audit · 2026-10-17T03:21:52Z

Objection 7.17 · Objection: three cells of the status classification disagree with other results and must be updated

proved audit: minor defect

Minor defect (second audit). Cell updates correct. Slip: the 'substantive conflict' is attributed to Setup 1.1 not having seen Theorem 6.6; the stale cell is in the classification Summary 1.9 (the status table Setup 1.1 correctly reproduces the pages' 'open' entry at its time).

Compares the status classification cell by cell with the other lines of attack. Three cells are stale — one genuine conflict (\(E_6/F_4\) functions) and two strengthenings — and all other cells agree.

Objection 7.17.

The classification of Summary 1.9 must be updated at three cells:

  1. \(E_6/F_4\), functions: "open" against "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." (Theorem 6.6, verified). New entry: fails under (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. for \(p\)-adic \(F\), every residue field; open over \(\F_q(\!(t)\!)\). The sheaf entry stays open (two heuristics point to equality).
  2. \(\GL_n/(\GL_p\times\GL_q)\), functions over \(\F_q(\!(t)\!)\): "open" against "holds" (Lemma 4.8); the Lang argument there supplies the orbit classification that Summary 1.9 asks for.
  3. \(\SL_2/\SO_2\), sheaves: "fails (ℓ-adic, Betti through open item 1)" against "fails over \(\C\) directly" (Theorem 2.4). Not a contradiction; new entry: fails (Betti, unconditional modulo the standing realisation of \(\mathcal C_K\)).

All other cells agree with the other results.

Agreeing cells. The group case; the spheres (the routes of Proposition 1.3, Lemma 4.8 and Proposition 6.3 are consistent); \(\GL_2/T\) and \(\GL_2/\Sp_2\); \(\GL_4/\Sp_4\), both hold (functions by Proposition 1.4, Theorem 3.9, Theorem 4.9; sheaves by Theorem 1.6, Proposition 3.3 with Theorem 3.6, and Theorem 4.6); \(\SL_4/\Sp_4\), also unconditional over every \(F\) through Theorem 3.9 and the degree-zero reduction; \(\GL_{2m}/\Sp_{2m}\), \(m\ge3\), sheaf open (the partial result of Theorem 3.6 is compatible); \(\GL_n/\mathrm O_n\), both fail.

Severity. The only substantive conflict is the \(E_6/F_4\) function cell, which reflects that Setup 1.1 did not see Theorem 6.6; the other two updates are strengthenings.

Builds on

Built on by

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
Original record, as the agent wrote it at 03:21 UTC

A1-9 table: three cells disagree with other agents' results and must be updated

Cell-by-cell against A2/A3/A4/A6 (as audited above):

  1. \(E_6/F_4\), function column: A1 'open' vs A6-6 'fails under (A), char 0'. A6-6 is verified (A7-9). Update to 'fails under (A1)–(A5) for \(p\)-adic \(F\), every residue field; open over \(\mathbb F_q(\!(t)\!)\)'. Sheaf column: 'open' agreed by all (A3-8/A6-8 heuristics point to equal).
  2. \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), function over \(\mathbb F_q(\!(t)\!)\): A1 'open' vs A4-8 'holds (D0, D3)'. A4-2's Lang argument supplies the orbit classification A1-9 item 3 asks for. Update.
  3. \(\mathrm{SL}_2/\mathrm{SO}_2\), sheaf column: A1 'fails (\(\ell\)-adic, Betti via open item 1)' vs A2-4 'fails over \(\mathbb C\) directly'. Not a contradiction; A1's cell is weaker than available. Update to 'fails (Betti, unconditional modulo the standing realisation of \(\mathcal C\))'.

Agreeing cells: group, spheres (A1-3/A4-8/A6-3 consistent), \(\mathrm{GL}_2/T\), \(\mathrm{GL}_2/\mathrm{Sp}_2\), \(\mathrm{GL}_4/\mathrm{Sp}_4\) both-hold (A1-4/A3-9/A4-9 functions; A1-6/A3-3+A3-6/A4-6 sheaves), \(\mathrm{SL}_4/\mathrm{Sp}_4\) (also unconditional over every \(F\) via A3-9 + degree-0 reduction), \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) \(m\ge3\) sheaf-open (A3-6 'partial' is compatible), \(\mathrm{GL}_n/\mathrm O_n\) both-fail. Minor: A1-1's 'A1 did not see A6' is the only substantive conflict; the other two are strengthenings.

Sources