§7 · Root · root: registry, redirects, synthesis · 2026-10-18T07:22:08Z

Summary 7.5 · Verdict after the audit: unchanged for \(E_6/F_4\); the general direct form completed

proved ★ key step

A fresh audit graded all 76 steps: 52 correct, 21 minor, 3 errors, all three in the hypothesis list of the direct route for a general symmetric variety. One salvage agent repaired them, and the reviewers who had found the errors re-checked the repairs. Of the 16 inputs of the verdict, 14 held and 2 were salvaged; none was lost. The answer for \(E_6/F_4\) is unchanged.

Summary 7.5.

Answer (status proved, unchanged by the audit).

  1. \(E_6/F_4\), \(p\) odd. Sakellaridis's Theorem 8.0.2 holds for the model \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\) (Theorem 4.8) and over every \(p\)-adic field of any ramification. Over a \(p\)-adic field it follows from (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. (Theorem 2.8), the other hypotheses (S4, S4b, with the characteristic-free proofs of Summary 3.12), the normalizations (Audit 4.7) and [Sa08] (Audit 7.3). The isomorphism \(C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) with \(\varphi_0\mapsto1\) is unique. With Proposition VII, (SG) fails: \(C_K=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{Kx_{3\omega_1^\vee}}\).
  2. Symmetric \(X\), by the transfer. Under (T1)–(T5)Hypotheses of the characteristic-p transfer for symmetric X over W(F_q): (T1) H split with connected fibres; (T2) X(E) = G(E)x0; (T3) τ detects integral points; (T4) the Cartan decomposition over the ramified fields F'_e; (T5) Sakellaridis's theorem over F'_e = Q_q(p^{1/e}), e ≫ 0. the theorem holds over \(\mathbb F_q(\!(t)\!)\) (Proposition 4.9); the target is built from the invariants of \(X\) over \(F'_e\), which agree with those over \(\mathbb F_q(\!(t)\!)\) for \(\mathcal M_\theta\) (Proposition 3.6, Proposition 3.7).
  3. Symmetric \(X\), directly. The theorem holds over \(\mathbb F_q(\!(t)\!)\) under the hypotheses of Theorem S1.5: (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 \(p\) with the simple spherical roots those of \(W_X'\), the cone condition, (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.. The leading coefficients and \(c\) are non-zero by positivity (Lemma S1.3). For \(E_6/F_4\) the one open input, Statement 6.3.1 in characteristic \(p\), comes from (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. through the transfer (Audit 5.19). So this route confirms item 1 but does not prove it independently.
  4. Characteristic 0. Sakellaridis's printed proof has two gaps: the identity on p. 62 is off by a factor 2, and the non-vanishing of the leading coefficients is not proved. Both are closed by characteristic-free arguments (Lemma S1.1, Lemma S1.3, Remark S1.4), so Theorem 8.0.2 holds as stated.

The inputs after the audit: 16 inputs; 14 held, 2 salvaged, 0 lost.

Table 2. input and what the verdict uses it for and after the audit
inputwhat the verdict uses it forafter the audit
Theorem 2.8(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. for \(E_6/F_4\) over every DVR with \(2\) invertibleheld
Summary 2.10the answer to (Q2)held
Theorem 3.8Cartan decomposition over \(\mathbb F_q(\!(t)\!)\), first proofheld
Remark 3.11the analytic Statement 6.3.1 left openheld
Summary 3.12the hypotheses over \(\mathbb F_q[\![t]\!]\)held
Proposition 4.5Cartan decomposition over \(\mathbb F_q(\!(t)\!)\), by countingheld; "Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits." also needs the valuation cone of Proposition 3.7
Theorem 4.8Theorem 8.0.2 for \(E_6/F_4\) over \(\mathbb F_q(\!(t)\!)\)held
Proposition 4.9the transfer for symmetric \(X\) under (T1)–(T5)Hypotheses of the characteristic-p transfer for symmetric X over W(F_q): (T1) H split with connected fibres; (T2) X(E) = G(E)x0; (T3) τ detects integral points; (T4) the Cartan decomposition over the ramified fields F'_e; (T5) Sakellaridis's theorem over F'_e = Q_q(p^{1/e}), e ≫ 0.held; for general \(X\), choose the \(e\) along which the data of \(\Theta\) are constant
Dead end 4.10the transfer needs (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. over \(W(\mathbb F_q)\)held
Audit 5.5normalizations over ramified fieldsheld
Audit 5.6audit of Theorem 4.8held
Audit 5.7audit of Audit 4.3, Proposition 4.9held
Summary 5.13audit of the transfer and the hypothesesheld
Objection 5.20the general list of the direct routesalvaged: holds with change, 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). and \(c\neq0\) by positivity (Summary S1.6)
Summary 5.21verdict on the direct routesalvaged: holds as stated, the list is now proved (Theorem S1.5, Summary S1.6)
Summary 6.9audit of the proof of (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.held

What was done. Eight fresh reviewers graded all 76 steps of the run, the in-run audits and the root's steps included (Review G1 to Review G8). They found three errors, all in one place. The direct rerun's theorem for a general symmetric variety (Theorem 1.9) used, without proof, that the leading coefficients of the images \(P_\lambda\) are non-zero. The in-run audit missed this (Objection 5.20, Summary 5.21). Every input that rests on these steps does so only through that general list, so the budget was one salvage agent. Its repairs (Lemma S1.1, Lemma S1.3, Theorem S1.5, Summary S1.6) were re-checked by the two reviewers who had found the errors (Re-check V1, Re-check V2), and every one holds.

The root's own corrections (review Review G8, all minor).

What remains 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., the cone condition, and Statements 6.3.1 and 7.1.5 in characteristic \(p\), with Sakellaridis's Theorem 6.3.2, which rests on the classification of spherical varieties of rank at most two in characteristic \(0\). Citations used within their standing hypotheses but not opened locally: Steinberg, Mem. AMS 80, Thms 7.5, 8.1; Conrad–Gabber–Prasad A.8.10; Edixhoven 3.4; EGA IV; Bruhat–Tits III, Thm 4.7; Macdonald's product formula (checked numerically).

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