§9 · Salvage 1 · Leading coefficients · 2026-10-18T06:51:04Z

Summary S1.6 · Verdict inputs A5-20 and A5-21 re-derived; how R-4(b) and its open items read

proved correction

Correction. Corrects Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses, Summary 5.21 · Verdict of the second audit: A1-1 … A1-9 (direct rerun over F_q((t))).

With Lemma S1.1 (H1 is (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1).) and Lemma S1.3 (\(c\neq0\), \(a_J\neq0\) by positivity), the general list 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. of Objection 5.20 is complete. A5-20 holds with two changes: 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 characterized as (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., not by the shape of \(\Theta^+\), and its \(c\neq0\) argument is replaced. A5-21 holds as stated, because its formulation 'Theorem 1.9 + 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.' is now proved (Theorem S1.5). The \(E_6/F_4\) conclusions and the final verdict are untouched.

Summary S1.6.

Objection 5.20 (A1-9 general X: three tacit hypotheses). Holds with change. The \(\mathcal M_\theta\) corollary and the need for 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. stand, 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. is the complete list (no H4): \(c\neq0\) and \(a_J\neq0\) follow (Lemma S1.3). Two statements are replaced. (i) 'A1's inversion-set argument proves 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). exactly when \(\Theta^+\subseteq\check\Phi_X^+\) without repetitions … half-coroots in type \(T\) need a separate check' becomes: 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). \(\iff\) (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., the virtual colours of each simple spherical root sum to its coroot, and then (7.5) holds for every \(\Theta\) with the cone \(\operatorname{cone}(\Theta^+)\) (Lemma S1.1). Half-coroots of type \(T\), e.g. \(T\backslash PGL_2\), satisfy 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 doubled-coroot example is the case \(m_\gamma=2\). (ii) '\(c\neq0\) follows from \(\Omega_\chi\not\equiv0\)' becomes: \(\Omega_\chi(x_0)=m(\chi)\hat S_{\mathbf 1_{x_0K}}(\chi^{-1})/\operatorname{vol}(x_0K)\), with \(m\not\equiv0\) and \(\hat S\) positive on the convergence region (Lemma S1.3(iii)).

Summary 5.21 (Verdict 2). Holds as stated. Item 4 says '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'. This was unproven only because the leading coefficients were unchecked for general \(\Theta\). With Lemma S1.3 it is exactly the theorem of Theorem S1.5: 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 (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1). by Lemma S1.1, and non-vanishing is automatic. The table rows and the \(E_6/F_4\) conclusions are unchanged.

R-4. Part (b), second sentence, and its open items should read: 'A second, direct form reruns the analytic half of the proof in characteristic \(p\); it needs (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)., (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8)., (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits. with antidominant labels, (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., Statements 6.3.1 and 7.1.5 in characteristic \(p\), the cone condition, 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. of Objection 5.20, 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). equivalent to the uniqueness (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1). of the colours of each simple spherical root (Lemma S1.1); the non-vanishing of \(c\) and of the leading coefficients is automatic (Lemma S1.3); and (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 not used (Theorem S1.5); open for a general symmetric \(X\) without the transfer: (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (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., Statements 6.3.1 and 7.1.5 and the cone condition in characteristic \(p\), and Theorem 6.3.2 in characteristic \(p\).' The \(E_6/F_4\) verdict, parts (a), (c), (d), and the transfer form via Proposition 4.9 do not change. Part (a) needs no '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). caveat' in characteristic 0 (Remark S1.4).

  1. Path. Theorem 1.9 → Objection 5.20 → Summary 5.21 → R-4, and Objection 5.20 → R-4 (target §3). Both reviews (G1, G6) graded the \(E_6/F_4\) conclusions of these steps correct. The errors were confined to the general list. That list lacked (a) a proof of (7.5) beyond multiplicity-free \(\Theta^+\subseteq\check\Phi_X^+\), (b) the non-vanishing of the leading coefficients, and (c) a valid argument for \(c\neq0\).
  2. (a) is Lemma S1.1: under Statement 7.1.5, 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). \(\iff\) (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., and (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1). gives (7.5) with \(\mathcal T=\operatorname{cone}(\Theta^+)\) for every \(\Theta\), including repeated weights, half-coroots and shared type-\(T\) colours.
  3. (b), (c) are Lemma S1.3. Comparing coefficients of \(e^\lambda\) in \(\Pi_\lambda\hat S_0=c\,C_\lambda\hat S_\lambda\) gives \(a_J=c\,C_\lambda V_\lambda(\lambda)/V_0(0)\) with \(C_\lambda,V_\lambda(\lambda),V_0(0)>0\), and \(c\neq0\) from the case \(\lambda=0\). It uses only steps of the run that hold over \(\mathbb F_q((t))\) (Proposition 1.5, Proposition 1.6, Audit 5.15), together with (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). and (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x..
  4. The corrected statement is Theorem S1.5. It 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). = (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (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. and H3H3 · No spherical root of type N. added, which is Objection 5.20's list. So Summary 5.21's item 4 is now a proved statement, and A5-21 holds as stated. Objection 5.20 holds with the two replacements listed, which concern its arguments and its description of 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)., not its outcome line.
  5. Effect on R-4. Only the general remark in (b) and the open items change: the list stays 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. 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). read as (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., and an explicit sentence records that no non-vanishing hypothesis is needed. The \(E_6/F_4\) results (Theorem 8.0.2 over \(\mathbb F_q((t))\) and over \(p\)-adic fields; failure of (SG) through Proposition VII) use only the \(\mathcal M_\theta\) corollary, where \(a_J\in\{1,1+t,c\}\) explicitly (Proposition S1.2(b)); they are unaffected. The published pages (VIb sub-tab part (e); the remark 'the direct form of (b)' on the Proposition VI page) should be read the same way.

Builds on

Built on by

  • Re-check V2 · Re-check by G6
  • Summary 7.5 · Verdict after the audit: unchanged for \(E_6/F_4\); the general direct form completed

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