§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:34:18Z

Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses

proved audit: error ★ key step

Error. What Objection 5.20 states is right: the M_θ corollary holds; 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). is needed (p. 62 identity off by 2, verified on the PDF for all six w; s_{2γ̌} = −(e^{−γ̌}+1+e^{γ̌}) recomputed; by W-invariance a doubled coroot forces a doubled simple coroot); 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. is a real gap with a short fix (H^1(Z/2, U''(F)) = 0); H3H3 · No spherical root of type N. is real (Statement 6.3.1, p. 45, has no type-N clause). But it misses a fourth gap of the general theorem: Theorem 1.9 step 3 invokes Proposition 1.2(c), which needs non-zero leading coefficients, and Objection 5.20 step 1 marks these as checked by Audit 5.16, Audit 5.17, which treat only M_θ. Under 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). the coefficient of m_λ is a_J = CT(∏{θ̌∈Θ^+}(1−σq^{−r}e^{θ̌}) ∏{γ̌∈Φ̌_J^+}(1−e^{−γ̌})) with W_J = Stab(λ) (I proved this and checked a_J = 1, 1+t, c for E_6/F_4), and for a general Θ (r = 1/2, σ = −, repeated or non-coroot weights) nothing in the run or in Sakellaridis shows a_J ≠ 0: his Remark after Thm 7.2.1 uses the non-strict order ≻_2 and says nothing about this coefficient, and the strict reading is false at walls. So 'the general theorem needs H = 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).+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.+H3H3 · No spherical root of type N.' leaves out a needed, unproved condition; its 'Minor' repair 'c ≠ 0 follows from Ω_χ ≢ 0' is also a non sequitur (fix: positivity of Z, Dead end 1.8(i)). The conclusion still holds by another argument.
Used later by: Summary 5.21 · Verdict of the second audit: A1-1 … A1-9 (direct rerun over F_q((t))), 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: Lemma S1.1 · (7.5) for every \(\Theta\): 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). is equivalent to one colour set per simple root, Lemma S1.3 · Positivity of \(\hat S\) forces \(c\neq0\) and \(a_J/c>0\) over every local field, Theorem S1.5 · Theorem 8.0.2 over \(\mathbb F_q((t))\) for symmetric \(X\): the complete hypothesis list, Summary S1.6 · Verdict inputs A5-20 and A5-21 re-derived; how R-4(b) and its open items read.

For \(\mathcal M_\theta\) the theorem and corollary of Theorem 1.9 are correct: everything reduces to \((\mathrm{S6.3.1})_p\), which by Audit 5.19 follows from \((I_\Lambda)\). For a general symmetric \(X\) the stated list of hypotheses is incomplete. The proof also uses the inequality (7.5) for a general \(\Theta\) (Sakellaridis's own argument for it is wrong by a factor 2, and A1's fix covers only multiplicity-free \(\Theta^+\subseteq\check\Phi_X^+\)), the \(F\)-rationality of the adapted data, and the absence of type-\(N\) roots.

Objection 5.20.

Outcome: the \(\mathcal M_\theta\) corollary holds. The general theorem needs hypothesis H = 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). + 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. + H3H3 · No spherical root of type N..

Minor. Step 2's "\(c\neq0\) by Lemma 4.2.3, whose proof needs only the invariant measure" misdescribes the lemma (it uses (4.5)); \(c\neq0\) follows from \(\Omega_\chi\not\equiv0\) (Audit 5.15 step 7).

  1. What the proof uses, step by step. Lemma 1.3, Lemma 1.4, Proposition 1.5 hold with 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. and the cited characteristic-\(\ne2\) inputs (Springer's finiteness of \(B_{\bar F}\)-orbits, Steinberg 7.5, CGP 2.1.10, BZ, Bernstein's principle): Audit 5.14, Audit 5.18. Proposition 1.6 holds with (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x. and the invariant measure (Audit 5.15). The passage to (7.4) needs Statements 6.3.1 and 7.1.5 in characteristic \(p\) (so H3H3 · No spherical root of type N.) 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).. Proposition 1.2 needs (A3b′), triangularity (H1 plus the cone condition (E)) and non-zero leading coefficients (Audit 5.16, Audit 5.17).
  2. Why (E) suffices once 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). holds. Under Statement 7.1.5, each simple coroot lies in \(\operatorname{cone}(\Theta^+)\): it is the virtual colour of a type-\(G\) root, or the sum of the two colour valuations of a type-\(T\) root. So \(\mathcal R\subseteq\operatorname{cone}(\Theta^+)\), and (E) is the condition \(\mathcal T\cap C^-=\{0\}\) for \(\mathcal T=\operatorname{cone}(\Theta^+)\).
  3. The counterexample to multiplicity two. In rank one with \(\check\rho=\check\gamma/2\): \(s_\nu=\big(e^{\nu}-e^{\check\gamma-\nu}\big)/(1-e^{\check\gamma})\). Then \(s_{2\check\gamma}=(e^{2\check\gamma}-e^{-\check\gamma})/(1-e^{\check\gamma})=-(e^{-\check\gamma}+1+e^{\check\gamma})\). Its weight \(-\check\gamma\) violates "all weights in \(0+\mathcal R\)". It contributes to \(\prod_{\Theta^+}(1-q^{-r}T)s_0\) unless cancelled. So the shape of \(\Theta\) matters, and the printed argument cannot be cited.
  4. \(\mathcal M_\theta\) corollary. \(\Theta^+=\check\Phi_X^+\) is multiplicity-free, and Computation 1.7 does not need Statement 7.1.5 at all (it builds \(\beta\) from \(\check\Phi_X^+\)). 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. is explicit (S4-11). The only simple spherical roots are \(\gamma_1,\gamma_6\), of type \(G\) (A3-6, A3-7). So the hypotheses reduce to (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7)., (A3b′), (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., \((\mathrm{S6.3.1})_p\). The first three hold over \(\mathbb F_q(\!(t)\!)\): A3-2, Theorem 3.8 or Proposition 4.5 read with antidominant labels, and Audit 3.9 (with the repairs of Audit 5.10, Objection 5.11, which do not touch these items). \((\mathrm{S6.3.1})_p\) follows from \((I_\Lambda)\) (Audit 5.19). Then Proposition VII (\(r\) not onto, every \(q\)) gives the failure of (SG) over \(\mathbb F_q(\!(t)\!)\).
  5. (SG) \(\iff\) \(r\) onto. Under the isomorphism with \(\mathbf 1_{x_0K}\mapsto1\), \(\mathcal H_K*\mathbf 1_{x_0K}\mapsto r(\mathbb C[A^*]^W)\). This is correct and insensitive to \(f\mapsto f^\vee\) (Audit 5.16).
  6. What A1-9 achieves for (Q1). A characteristic-\(p\) proof of the analytic half of Theorem 8.0.2 for symmetric \(X\), with characteristic-\(\ne2\) inputs only. For \(\mathcal M_\theta\) the whole theorem holds modulo the single generating-function identity \(Z_F=R_q\) (Dead end 1.8(i)). That identity is a point-count statement over \(\mathbb F_q[t]/t^N\) by Dead end 1.8(ii). It is implied by \((I_\Lambda)\), and it might also be attacked directly, without \((I_\Lambda)\).

Builds on

  • Theorem 1.9 · Theorem 8.0.2 over \(\mathbb F_q((t))\), \(p\) odd, symmetric \(X\): hypotheses

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