The function-level half of the \(E_6/F_4\) divergence rests on Sakellaridis's description of the unramified Hecke module, which the pages had cited under five hypotheses, three of them unverified. The user asked for that theorem to be made self-contained, and for a further question: is \(C_K\) generated by any single function at all, not only by the basic function?
The agent stated Theorem 8.0.2 and Corollary 8.0.4 with every assumption in Sakellaridis's own terms and checked the pages' summary against them (Theorem (cited) X4.1), (Audit X4.2). For \(E_6/F_4\) it computed the spherical roots, colours and Levis and verified the hypotheses on them in full (Proposition X4.3), fixed an integral model and verified the conditions on it for odd \(p\) (Proposition X4.4), and proved one of the two axioms on orbits and half of the other (Proposition X4.5). Exactly two hypotheses remain open (Gap X4.6).
It then determined the image of the restriction map exactly: the invariants that agree at the two points glued by \(W(E_6)\), a subring of codimension one, in a ring with only constant units (Proposition X4.7). Under Sakellaridis's theorem this means that \(C_K\) is not cyclic and needs exactly two generators, the basic function and one orbit indicator (Corollary X4.8).
Result as the agent reported it partial
For \(X=E_6/F_4\) over a \(p\)-adic \(F\) with \(p\) odd and the integral model \(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0}\), \(\theta_0=\operatorname{Int}(\sigma_L)\circ C\) an involution of type E IV defined over \(\mathbb Z\): (i) unconditionally, \(\operatorname{im}r\) is the codimension-one subring \(\{g:g(y_1)=g(\sigma y_1)\}\) of \(S=\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), and \(S\) has only constant units; (ii) under Sakellaridis's Theorem 8.0.2, \(C_K\) is not cyclic, \(C_K=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{O_{3\omega_1^\vee}}\) needs exactly two generators, and \(C_K/\mathcal H_K\varphi_0\) is the character \(q^{\rho-\omega_4}\); (iii) every hypothesis of Theorem 8.0.2 holds for this model except Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). as \(\mathcal O\)-schemes. These hypotheses are: the conditions of §1.1/§1.7 on \(F\), \(G\), \(X\) and \(\mathring X\); the Part 2 assumption of no type-\(N\) reflections; the §7 conditions and Statements 6.3.1 and 7.1.5; the §8 conditions; Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. (via Delorme–Sécherre's split Cartan decomposition, whose index set has one element because \(H^1(F,\mathrm{Spin}_8)=1\), and \(W_X\) realized in \(H(\mathcal O)\)); Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points.; and the types in Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run)..
Gap
(I) Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). as \(\mathcal O\)-schemes, for the 45 maximal-rank Borel orbits and the 6 simple roots (types \(U\) and \(G\) only). Cited, not re-checked here: Delorme–Sécherre Theorem 4.8 (statement read from arXiv:math/0612545v1) and Helminck–Wang Prop. 10.3 inside it; conjugacy of maximal \(\theta\)-split tori over \(\bar F\) (Vust, Richardson); Kneser's \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\); Bruhat–Tits for \(K=\operatorname{Stab}(x)\); smoothness and connectedness of \(G^{\theta_0}_{\mathcal O}\) for \(p\ne2\) (Edixhoven, Steinberg); quasi-affineness of \(G_{\mathcal O}/G_{\mathcal O}^{\theta_0}\) (Alper); SGA 3 XXIII for the Chevalley involution over \(\mathbb Z\); Knop's transport of weight lattices (types in Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run)., no type-\(N\) reflections); the paper's claim that removing '\(F/\mathbb Q_p\) unramified' is trivial. Over \(\mathbb F_q(\!(t)\!)\) nothing is decided.
Blocked routes
Brief as sent
You are agent S4 in a bounded salvage team repairing a finished, audited search on one mathematical question. Work independently; you will not see the other agents.
SETTING. First read /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/setting.md. It holds the problem, the setting, the completion criterion, the references with absolute paths, the run's record and the failure checklist, and every agent gets the same text. To read references: sed -e 's/<[^>]*>/ /g' FILE | tr -s ' \n' for HTML; pdftotext -layout FILE - for PDFs. Sakellaridis's paper: /Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/two-definitions-iwahori/sources/sakellaridis-v3.pdf.
QUESTION (team-level). The run's verdict says that E6/F4 is the one divergence: categorical generation holds, while function-level generation (SG) fails over p-adic fields. The function half rests on Sakellaridis's theorem (Theorem 8.0.2 and Corollary 8.0.4 of the paper above), applied under hypotheses that the pages summarize as (A1)–(A5). Those are defined on /Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/classical.html, sections #s-hypotheses (anchors #hyp-a1 … #hyp-a5) and #sakellaridis-theorem, where the pages call the result "Theorem E". The verdict records that (A5) is verified for E6/F4 (step A6-9), while (A2), (A3) at the given place, and (A4) beyond "L_γ classical", are not. Your team must make this half of the answer self-contained and say exactly what holds.
YOUR TARGET, in three parts. (1) Sakellaridis's theorem, self-contained. From the paper itself, not the pages' summary, state:
- Theorem 8.0.2 and Corollary 8.0.4 exactly (and the role of Example 8.0.1), with page numbers;
- every assumption they use, in the paper's own terms: §1.7, Axioms 2.2.1, 2.4.1 and 2.4.2, Statements 6.3.1 and 7.1.5, and the assumptions of Theorem 7.2.1;
- the invariants A_X, Λ_X, W_X, δ(X) and the restriction map r, each defined.
Say how each assumption corresponds to the pages' (A1)–(A5), and whether the pages' summary is accurate or loses something. Give the structure of the proof (formula for the spherical function → isomorphism C_c^∞(X)^K ≅ C[δ(X)^{1/2}A_X^*]^{W_X} → the criterion) in a few numbered steps. (2) The hypotheses for X = E6/F4 (split E6, θ with fixed group F4, over a p-adic F). For each assumption, decide: verified (with a proof or an exact citation), false, or open (with the exact missing statement). Where it applies, use the run's steps A6-1 … A6-10 (in /Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/solve-maths/satake-two-levels/data.json; A6-9 is the (A5) check) and the families page families/e6-f4.html. Spherical roots, their types, the Levis L_γ and the colours of E6/F4 are computable from its restricted root system (type A2, root multiplicity 8). State which residue characteristics, and which integral model of X over O, each verdict needs. (3) Cyclicity at the function level for E6/F4. Decide whether C_K = C_c^∞(X(F))^K is a cyclic H_K-module, i.e. generated by some single function, not necessarily the basic function φ_0.
- Under Theorem 8.0.2, C_K ≅ S := C[δ(X)^{1/2}A_X^]^{W_X} with H_K = C[A^]^W acting through r. Examine whether "S is cyclic over C[A^*]^W" reduces to "r is onto": a generator s would satisfy 1 = r(a)s, so s would be a unit of S; determine the units of S for E6/F4.
- Determine the minimal number of generators of C_K (for instance from the fibre dimensions of S over C[A^*]^W at the points y_1, y_2 that W(E6) glues, step A6-6).
- Say what, if anything, can be said without Sakellaridis's hypotheses.
Then re-derive the verdict's E6/F4 function cell: what exactly is proved, under which named hypotheses, and over which fields.
FILES. The pages and PDFs in setting.md; classical.html and families/e6-f4.html; Sakellaridis's PDF; the run's data.json, nodes/ and scripts/A6/ (read-only); the run's E6/F4 proposition page /Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/solve-maths/satake-two-levels/props/e6-f4.html.
RULES.
- Return concrete lemmas, computations or counterexamples, each with a proof or an explicit gap; no status reports. Label every claim: proved (here, or cited with an exact result number and page) · heuristic · gap. An analogue at one level does not certify an inference at the other.
- Check the exact meaning of every term: p-adic versus F_q((t)); split; simply connected; quasi-affine versus affine; colours versus spherical roots; δ(X)^{1/2} as a point of A^*; the W_X-action on the translate.
- Never write "(A1)–(A5)" or a single "(Ak)" without saying in the same sentence what it requires, at least in a short phrase (for example "(A3), Sakellaridis's Axioms 2.2.1, 2.4.1, 2.4.2 on rank-one quotients and K-orbit representatives").
- Do not search the web for a solution of this question; ordinary background may be looked up. Prefer the local sources.
- Short Python checks: run them with
python3 -I, keep scripts in /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/ (create it), and keep each run short and single-threaded. Do not modify anything under /Users/miltonlin/Projects/Trees. - Timestamps: at the start of every node, run
date -u +%Y-%m-%dT%H:%M:%SZ and record the result. - Work in nodes and produce 5–10 of them. Node ids are S4-1, S4-2, …. A node that repairs or sharpens a run step carries "corrects": ["<id>"] (for example the E6/F4 function cell of R-4 or A7-32 if you sharpen it). Concepts: "e6f4-functions" for parts (2)–(3), "bridge" for part (1), "synthesis" for the re-derived verdict cell.
- Write every node as a stage of a mathematics paper: a plain-words name, a one-sentence summary, the precise statement, and the argument as numbered steps. Cite nodes as Theorem 6.6, never by bare code in prose. State computations as mathematics, not as program output.
- FINAL ANSWER: one JSON object inside a ```json fence, with nothing outside the fence. Escape JSON strings properly (double every backslash, escape quotes and newlines). Schema:
{"agent":"S4","family":"≤12 words","nodes":[{"id":"S4-1","t":"<date -u>","kind":"idea|lemma|computation|check|objection|blocked|verdict","status":"proved|refuted|blocked|open|heuristic|superseded","label":"Theorem|Proposition|Lemma|Computation|Audit|Summary|…","name":"≤44 chars, plain text","title":"≤90 chars, inline TeX allowed","summary":"1–2 sentences","statement":"Markdown+TeX","parents":["A6-6"],"corrects":[],"body":"argument, numbered steps, \\( \\) and $$ $$","concept":"e6f4-functions","key":false,"sources":["sakellaridis-v3.pdf, Theorem 8.0.2, p. …"],"inputs":{"A6-6":"holds as stated | holds with change: … | fails"}}],"result":{"status":"proved|partial|refuted|blocked","statement":"…","gap":"…or none"},"blocked_routes":[],"scripts":["paths"]} "inputs" is optional; use it on the node that re-derives the verdict's E6/F4 function cell, for the inputs A6-6 and A7-32.