§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:35:33Z

Summary 5.21 · Verdict of the second audit: A1-1 … A1-9 (direct rerun over F_q((t)))

proved audit: error ★ key step

Error. The table entries and the E_6/F_4 conclusions agree with my grades of Audit 5.14–Audit 5.19, and the [Sa08] discharge is correct (I opened [Sa08] myself). Item 4, however, asserts that 'the correct characteristic-p statement for symmetric X is Theorem 1.9's theorem with H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1).–H3H3 · No spherical root of type N. added'. The non-vanishing of the leading coefficients a_J, one for each stabilizer W_J of an antidominant label, is needed by Proposition 1.2(c) and is proved only for M_θ (Computation 1.7 step 6), so this formulation is not established; the error is inherited from Objection 5.20. Its stated conclusions stand: the A1 rerun is sound for M_θ, the analytic half holds with characteristic-≠2 inputs, the E_6/F_4 theorem and the failure of (SG) reduce to (S6.3.1)_p, which follows from (I_Λ) via Audit 4.7, and H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1).–H3H3 · No spherical root of type N. are indeed needed. The conclusion still holds by another argument.
Used later by: Theorem 7.4 · Verdict: Sakellaridis's theorem over \(\mathbb F_q(\!(t)\!)\); for \(E_6/F_4\) in both characteristics, (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. included.
Corrected by: Summary S1.6 · Verdict inputs A5-20 and A5-21 re-derived; how R-4(b) and its open items read.

None of the nine claims fails for \(\mathcal M_\theta\). The analytic half of Sakellaridis's proof (open-orbit functionals, functional equations, sum formula, last step) does hold over \(\mathbb F_q(\!(t)\!)\), \(p\) odd, with characteristic-\(\ne2\) inputs only. For \(E_6/F_4\), Theorem 8.0.2 and the failure of (SG) reduce to \((\mathrm{S6.3.1})_p\), which follows from \((I_\Lambda)\). The general symmetric statement needs three tacit hypotheses.

Summary 5.21.

Table 2. claim and outcome and node
claimoutcomenode
A1-1 inventoryholds; row 25 misfiled (Lemma 4.2.3 uses (4.5), i.e. [CS80])Audit 5.19
A1-2 formal last stepholds; Remark (i) = involution \(\iota(\psi)=w_L\psi^{-1}\); differs from Sakellaridis by \(f\mapsto f^\vee\)Audit 5.16
A1-3 local structureholds; "Springer's lemma" replaced by Steinberg 7.5Audit 5.18
A1-4 finitely many \(B(F)\)-orbitsholds (Springer finiteness, CGP 2.1.10 cited)Audit 5.18
A1-5 \(S_\chi\) without [Sa08]holds; step 7 refined for a central torusAudit 5.14
A1-6 functional equations, (6.2)holds; rank-one \(K\)-invariance and cocycle rules recomputedAudit 5.15
A1-7 \(W_X'\), \(\beta\), \(c\), triangularityholds; all recomputed exactly; p. 62 correction confirmedAudit 5.17
A1-8 dead endholds; route B closes under \((I_\Lambda)\)Audit 5.19
A1-9 theorem\(\mathcal M_\theta\) corollary holds; the general statement needs H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1). (7.5), H2H2 · F-rational adapted data: a θ-stable split maximal torus T ⊆ B over F containing a maximal θ-split torus, with B adapted and x0 in the open Borel orbit. (data over \(F\)), H3H3 · No spherical root of type N. (no type \(N\))Objection 5.20

Answers to the priority questions. (1) A1-5 is valid without [Sa08] in characteristic \(p\). (2) A1-6: the sum over \(W_X'=N_W(\mathfrak a)\cap[W/W_L]\) is correct, and neither Knop's action nor (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. is needed for the formula. They are needed only to compute \(b_{w_\gamma}\), i.e. for Statement 6.3.1, which A1 keeps as a hypothesis. (3) A1-2's last step is formal. (4) A1-7's numbers are right.

  1. Main finding. Every characteristic-\(0\) input of Sakellaridis's analytic machinery for the open orbit is replaced by a correct characteristic-\(\ne2\) argument:
    • convergence by eigenfunction bounds;
    • generic uniqueness by the Bernstein–Zelevinsky filtration over the finitely many \(B(F)\)-orbits, where only \(\mu_2\) torsion occurs and the twisted unipotent radicals are split;
    • rationality by Bernstein's principle;
    • functional equations by uniqueness;
    • [BK98] and [Ca80] by normalized intertwiners on \(I(\chi)\), with the rank-one \(\mathrm{SL}_2\) computation checked;
    • [CS80] absorbed into a \(\lambda\)-independent scalar.
  2. The remaining analytic content is Statement 6.3.1 in characteristic \(p\). For \(\mathcal M_\theta\) this is one identity, \(Z_F=R_q\) for \(Z_F(\chi)=S_\chi(\mathbf 1_{X(\mathcal O)})(1)\). By Dead end 1.8(ii) it is a family of point counts over \(\mathbb F_q[t]/t^N\). It follows from \((I_\Lambda)\) through Theorem 8.0.2 at the totally ramified \(F'_e\) (Audit 5.19). With the coordinator's report that [Sa08] assumes only characteristic \(0\), the hypothesis \(H_{\mathrm{Sa08}}\) of the first audit is discharged. Both routes, A4's transfer and A1's rerun, now rest on \((I_\Lambda)\) alone.
  3. Error in the source found by A1 and confirmed here. On p. 62, \(|\Theta^+|-w|\Theta^+|=\check\rho-w\check\rho\) is false for \(\Theta^+=\check\Phi_X^+\) (factor \(2\)). (7.5) survives for multiplicity-free \(\Theta^+\subseteq\check\Phi_X^+\) (which includes \(\mathcal M_\theta\)) by the inversion-set argument. In general it is an open point, H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1)..
  4. Formulation of (Q1). The correct characteristic-\(p\) statement for symmetric \(X\) is Theorem 1.9's theorem with H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1).–H3H3 · No spherical root of type N. added. For \(\mathcal M_\theta\), Theorem 8.0.2 holds over \(\mathbb F_q(\!(t)\!)\) modulo \((\mathrm{S6.3.1})_p\), and that in turn holds modulo \((I_\Lambda)\). Under either, (SG) fails, by Proposition VII.

Builds on

  • Audit 5.14 · Audit of A1-5: convergence, uniqueness and rationality of S_χ over F_q((t))
  • Audit 5.15 · Audit of A1-6: F_w S_χ = b_w S_{wχ}, the sum over W_X′, and the open-orbit route
  • Audit 5.16 · Audit of A1-2: triangular spherical transforms give C_K ≅ C[D]^{W_X}
  • Audit 5.17 · Audit of A1-7: W_X′ ≅ S_3, the coboundary β, the constant c and triangularity for M_θ
  • Audit 5.18 · Audit of A1-3 (smooth Borel stabilizers) and A1-4 (finitely many B(F)-orbits)
  • Audit 5.19 · Audit of A1-1 (inventory of inputs) and A1-8 (the open item (S6.3.1)_p)
  • Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses

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