§7 · Agent 7 · Adversarial audit · 2026-10-17T03:47:54Z

Objection 7.24 · Objection: the support-bound check passes; the \(E_6/F_4\) expectation and two table cells are stale

proved audit: error

Error. The objections to Dead end 5.9 and Summary 5.10 are correct (re-checked against Proposition 1.2, Computation 2.8 and Theorem 6.6). Its first claim, that Proposition 5.8 holds, is wrong as stated: it restates input (a) conditionally but never checks it, and (a) fails for the pair model of \(\GL_n/(\GL_p\times\GL_q)\) in residue characteristic 2 (for \(n=2\) the Cartan label of \(t^{(a,0)}x_0\) is \(\max(a-\val2,0)\) instead of \(a\); in characteristic 2 there is no involution). The support bound that Proposition 5.8 concludes is nevertheless true, by Correction 10.4. The conclusion still holds by another argument.
Corrected by: Correction 10.4 · Correction: the function-level support bound for the pair model of \(\GL_n/(\GL_p\times\GL_q)\) over every local field, Correction X3.6 · Correction of A7-24: its two objections stand; its claim that A5-8 holds needs a new proof.

Passes the function-level support bound of Proposition 5.8 and objects to two stale statements: the \(E_6/F_4\) expectation of Dead end 5.9, contradicted 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. by Theorem 6.6, and the \(\SL_2/T\) cell of Summary 5.10, which fails for every \(q\).

Objection 7.24.

  1. Proposition 5.8 holds.
  2. The last sentence of Dead end 5.9 is contradicted by Theorem 6.6: 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 \(F\), \((\mathrm{SG})\) fails for \(E_6/F_4\). The \(E_6/F_4\) cell of Summary 5.10 must 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; open over \(\F_q(\!(t)\!)\)'.
  3. \(\SL_2/T\) fails for every \(q\) (Proposition 1.2, Computation 2.8), not only for \(q\) odd.

Otherwise the table of Summary 5.10 is consistent with this audit; severity cosmetic for the theorems of this track.

  1. The support bound. The function-level H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) over any \(F\) needs (a) \(\tau(O_\mu)\subseteq Kt^\mu K\), which is true once \(K\backslash X(F)\leftrightarrow\Lambda_S^+\) with \(\tau(x_\mu)=t^\mu\), and (b) the Cartan convolution bound \(Kt^aK\cdot Kt^bK\subseteq\bigcup_{c\le a+b}Kt^cK\), valid for split \(G\) over any local field; the projection argument of Step 2 of Proposition III is combinatorial. This is the same conclusion as Theorem 3.9 and step 1 of Proposition 1.4. The same-points question is correctly left heuristic.
  1. \(E_6/F_4\). The closing sentence of Dead end 5.9 — that nothing suggests \((\mathrm{SG})\) fails — is contradicted by Theorem 6.6 (verified in Audit 7.9): 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 \(F\), the restriction map is not onto and \((\mathrm{SG})\) fails. The obstruction is a Weyl-conjugacy coincidence of the points \(q^{\rho-9\omega_1-3\omega_6}\sim q^{\rho-9\omega_1}\), the mechanism of Sakellaridis's Example 8.0.1, which the lattice method cannot see without an orbit classification. The list of missing inputs in Dead end 5.9 (orbits of Albert-algebra lattices, a two-sided bound at \(3k\omega_1^\vee\)) remains the right list for an unconditional treatment over \(\F_q(\!(t)\!)\); Theorem 6.6 predicts that the triangular system fails at \(3\omega_1^\vee\), so a lattice count, if done, should exhibit a dependence, not a solution.
  1. \(\SL_2/T\). Proposition 1.2 and part (c) of Computation 2.8 prove failure for every \(q\) by Hecke parity. Otherwise the table of Summary 5.10 agrees with this audit: the spheres, \(\GL_n/(\GL_p\times\GL_q)\), \(\GL_{2m}/\Sp_{2m}\) and \(\SL_{2n}/\Sp_{2n}\) hold unconditionally over every \(F\); Theorem E is no longer needed for any positive row and remains the only input for the two negative rows \(E_6/F_4\) and \(\SL_2^3\backslash\Sp_4^2\). The sheaf remark of Summary 5.10 is correctly labelled heuristic; the actual sheaf decision comes from Theorem 4.6 (see Audit 7.23).

Builds on

Built on by

  • Summary 7.25 · Updated audit verdict: every row of Table 1 except \(E_6/F_4\) is decided at both levels
  • Correction 10.4 · Correction: the function-level support bound for the pair model of \(\GL_n/(\GL_p\times\GL_q)\) over every local field
  • 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 03:47 UTC

A5-8 holds; A5-9 and A5-10 contain two stale statements relative to A6 and A1/A2

A5-8 (holds). The function-level H5(a) over any \(F\) needs (a) \(\tau(O_\mu)\subseteq Kt^\mu K\) (true once \(K\backslash X(F)\leftrightarrow\Lambda_S^+\) with \(\tau(x_\mu)=t^\mu\)), (b) the Cartan convolution bound \(Kt^aK\cdot Kt^bK\subseteq\bigcup_{c\le a+b}Kt^cK\) (valid for split \(G\) over any local field); the projection argument of criterion.html step 2 is combinatorial. Same conclusion as A3-9/A1-4 step 1 and my A7-5 remark. The 'same points' question is correctly left heuristic.

A5-9 (objection, needs update). Last sentence: 'nothing … suggests (SG) fails for \(E_6/F_4\)' — contradicted by A6-6 (verified in A7-9): under (A1)–(A5), for \(p\)-adic \(F\), \(r\) is not onto and (SG) fails, the obstruction being a Weyl-conjugacy coincidence of the \(q\)-points \(q^{\rho-9\omega_1-3\omega_6}\sim q^{\rho-9\omega_1}\), exactly the Example-8.0.1 mechanism that A5's lattice method cannot see without an orbit classification. A5-9's list of what is missing (Albert-algebra \(\mathcal O\)-orbits, a two-sided sandwich at \(3k\omega_1^\vee\)) remains the right list for an unconditional treatment over \(\mathbb F_q(\!(t)\!)\); A6-6 predicts that the triangular system fails at \(3\omega_1^\vee\) (cokernel of \(r\) nontrivial), so the lattice count, if done, should exhibit a dependence, not a solution. Severity: cosmetic for A5's own theorems, but the \(E_6/F_4\) cell in A5-10 must read 'fails under (A), \(p\)-adic; open over \(\mathbb F_q(\!(t)\!)\)'.

A5-10 (objection, cosmetic). '\(\mathrm{SL}_2/T\) fails for \(q\) odd (square classes need a non-square unit)': A2-8(c) and A1-2 prove failure for every \(q\) by Hecke parity (A7-2, A7-15). Otherwise A5-10's table is consistent with my audit: spheres, \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) hold unconditionally over every \(F\); Theorem E is no longer needed for any positive row; it remains the only input for the two negative rows \(E_6/F_4\) and \(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\). A5-10's sheaf remark (Lemma 2 gives a point of the open stratum of the fibre, hence \(a_b^*(V_{\lambda^*(b)}\ast\mathrm{IC}_0)\ne0\)) is correctly labelled heuristic; the actual sheaf decision comes from A4-6 (A7-23).

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