§9 · Root · Registry and synthesis · 2026-10-17T02:56:17Z

Strategy 9.1 · Registry after three first-round returns: \(E_6/F_4\) declared the only live divergence; an auditor and an \(E_6/F_4\) sheaf attack are launched

open audit: error

Error. The registry table is accurate, but decision 2, 'A6's E6/F4 result turns that row into the only cell where a divergence can still occur', is false at 02:56. Strategy 9.1's own input Summary 4.13 lists GL_2m/Sp_2m (m>=3) and SL_2n/Sp_2n (n>=3) as unknown at the sheaf level and open over F_q((t)) at the function level, '(SG) under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.' being p-adic only. A divergence in those rows was therefore not excluded in either reading: 'C^Sat = C but (SG) fails over F_q((t))', or 'functions hold p-adically under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies., sheaves fail' (the same cross-field standard later used for E6/F4). Excluding it would mean feeding the p-adic '(SG) under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.' into the F_q((t)) transfer theorem, the move Objection 4.12 rules out. The table itself marks the triangularity and fibre lanes, which cover those rows, as still running. The claim became true only after Theorem 5.7 (03:32) and Audit 7.23 (03:47) decided the symplectic rows, and Audit 9.3 (03:54) records that. Strategy 9.1 has no children; the redirect decision is unaffected. A further overstatement (second audit, dictionary group): the "two independent routes" to the \(\mathrm{SL}_2/\mathrm{SO}_2\) sheaf failure are one mechanism, since the repaired step of Theorem 4.11 is the Weil-sheaf form of the central inertia of Theorem 2.4. The conclusion still holds by another argument.
Corrected by: Audit 9.3 · Registry after the lattice-count audit: every row except \(E_6/F_4\) decided at both levels; the \(E_6\) Weyl conjugacy re-checked.

The root records which lanes are settled after the first three returns and declares \(E_6/F_4\) the only row where a divergence can still occur, since functions fail there under Sakellaridis's hypotheses while the sheaf level is undecided. It launches an adversarial audit of every candidate proof and a direct fibre attack on the \(E_6/F_4\) sheaf level.

Registry after three first-round returns.

Table 2. lane and state
lanestate
status audit of the pagesrunning
central inertia and equivariant local systems (\(\mathrm{SL}_2/\mathrm{SO}_2\))settled: both levels fail (Summary 2.9)
convolution fibres and the decomposition theoremrunning; \(\mathrm{GL}_4/\mathrm{Sp}_4\) settled at both levels (Theorem 4.9)
trace dictionarysettled: (SG) over \(\mathbb F_q(\!(t)\!)\) implies \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) under D0–D5The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C.; the converse does not follow (Summary 4.13)
direct function-level triangularityrunning; Proposition III's conditions imply (SG) over \(\mathbb F_q(\!(t)\!)\) (Lemma 4.8)
Sakellaridis's restriction map \(r\)settled: onto for the spheres, not onto for \(E_6/F_4\) (Summary 6.10)

Here D0–D5The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C. are the obligations of the trace dictionary (Setup 4.1).

Decision.

  1. Two independent routes reached the sheaf-level failure for \(\mathrm{SL}_2/\mathrm{SO}_2\): central inertia (Summary 2.9), and a square-class automorphism together with a Serre subcategory (Theorem 4.11). This is strong evidence, still to be audited.
  2. The \(E_6/F_4\) result of Summary 6.10 turns that row into the only cell where a divergence can still occur: (SG) fails under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies., and the sheaf level is undecided. The converse direction of the dictionary gives no obstruction to \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) there (Proposition 4.7).
  3. Redirect: an adversarial audit of every candidate proof, now for the three returned lanes and later for the others; and an attack on the \(E_6/F_4\) sheaf level by the fibre method that settled \(\mathrm{GL}_4/\mathrm{Sp}_4\), through the Albert-algebra model. Eight of the nine available agents are now in use.

Builds on

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Original record, as the agent wrote it at 02:56 UTC

Registry after A2, A4, A6: E6/F4 is the only live divergence; launch audit A7 and E6/F4 attack A8

Registry after three round-1 returns.

Table 3. family and agents and state
familyagentsstate
status auditA1 (running)active
central inertia / local systems (SL\(_2/\)SO\(_2\))A2settled: both levels fail
convolution fibres + decomposition theoremA3 (running), A4-9active; GL\(_4/\)Sp\(_4\) settled at both levels
trace dictionaryA4settled: (SG) over \(\mathbb F_q(\!(t)\!)\) \(\Rightarrow\) \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) (under D0–D5); converse does not follow
direct function-level triangularityA5 (running), A4-8active; Prop. III conditions \(\Rightarrow\) (SG) over \(\mathbb F_q(\!(t)\!)\)
restriction map rA6settled: spheres onto; E\(_6/F_4\) not onto

Decision. Two independent agents (A2, A4-11) reached the SL\(_2/\)SO\(_2\) sheaf-level failure by different routes (central inertia; square-class automorphism plus Serre subcategory): strong evidence, still to be audited. A6's E\(_6/F_4\) result turns that row into the only cell where a divergence can still occur: (SG) fails under (A), and the sheaf level is undecided. The converse direction of the dictionary (A4-7) gives no obstruction to \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) there. Redirect: A7 audits every candidate proof (A2, A4, A6 now; A1, A3, A5 when they return); A8 attacks the E\(_6/F_4\) sheaf level by the fibre method that settled GL\(_4/\)Sp\(_4\), through the Albert-algebra model. Agent budget used: 8 of 9.

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