§9 · Root · Registry and synthesis · 2026-10-17T03:32:49Z

Summary 9.2 · Registry after the first audit round: the first-round results survive with their hypotheses, three cosmetic corrections, and \(E_6/F_4\) the only live cell

proved audit: minor defect

Minor defect (second audit). Repeats the flawed repair of Objection 7.11, passed on by Summary 7.19 (corrected in Correction 10.2); conclusions unaffected. Items 1-5 and the three corrections follow from Summary 7.19 and the audit nodes it cites (Objection 7.15, Objection 7.11, Objection 7.17). Defects: (1) the title's 'E6/F4 the only live cell' overstates Summary 7.19's 'only candidate divergence': at 03:32 the symplectic rows with m>=3 were still undecided at the sheaf level, as Summary 9.2's own parents Summary 1.9 and Summary 3.11 and the audit note Audit 7.18 say, pending the lattice-count lane. Unlike Strategy 9.1, the claim appears only as a headline, and the body (wait for the lattice-count lane and the E6/F4 attack) is consistent with the true state, so I grade it as wording; (2) 'GL4/Sp4 holds at both levels' omits that the sheaf half is l-adic over F_q-bar, Betti via D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p).; (3) 'sheaves unconditionally over C' means modulo the standing realisation of C by equivariant sheaves (Objection 7.17). No later node uses Summary 9.2.

The root accepts the first audit verdict: \(\mathrm{SL}_2/\mathrm{SO}_2\) fails and \(\mathrm{GL}_4/\mathrm{Sp}_4\) holds at both levels, the one-way transfer theorem and the Proposition III route to (SG) stand, and the \(E_6/F_4\) computation of \(r\) is confirmed. It lists the corrections to carry into the verdict and keeps the \(E_6/F_4\) sheaf attack running.

Summary 9.2.

  1. \(\mathrm{SL}_2/\mathrm{SO}_2\) fails at both levels: at sheaf level unconditionally over \(\mathbb C\), by central inertia (Theorem 2.4); at function level for every \(q\) (Computation 2.8, Proposition 1.2).
  2. \(\mathrm{GL}_4/\mathrm{Sp}_4\) holds at both levels (Proposition 3.3, Theorem 3.6, Theorem 3.9, Theorem 4.9, Proposition 1.4; the coefficients were recomputed at \(q=2,3\) in Audit 7.6).
  3. The one-way transfer theorem (Theorem 4.6, Proposition 3.10, Theorem 1.6: one proof in three notations, Audit 7.14) holds under D0–D3The 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., with D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p). for the Betti category.
  4. Proposition III's two conditions give (SG) over \(\mathbb F_q(\!(t)\!)\) without (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. (Lemma 4.8, Audit 7.5).
  5. For \(E_6/F_4\) the restriction map \(r\) is not onto, independently confirmed (Audit 7.9); the sphere results of three lanes are consistent (Audit 7.10).
  6. \(E_6/F_4\) is the only live cell.

Source. The registry follows the first audit verdict Summary 7.19; the classifications being corrected are Summary 1.9 and Summary 3.11. 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 Setup 4.1.

Corrections to carry into the verdict.

  1. The remark in step 4 of Proposition 1.2 that \(\mathrm{IC}_0\) is a summand of \(V_{(1,0)}\ast\mathrm{IC}_0\) is false, since that object is simple (Objection 7.15). The claim that the parity obstruction does not lift to sheaves "because of retracts" is withdrawn; its conclusion is unaffected.
  2. Lemma 3.1 and Theorem 3.6 cite Chen–Yi, Prop. 64, which assumes \(LX\) connected. For \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) the scalar shift identifies every component with the degree-0 component, where the proposition applies; this is a one-line reduction (Objection 7.11).
  3. Three cells of Summary 1.9 must be updated: the \(E_6/F_4\) function cell (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.); the \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) function cell over \(\mathbb F_q(\!(t)\!)\) (holds by Lemma 4.8); and the \(\mathrm{SL}_2/\mathrm{SO}_2\) sheaf cell (decided over \(\mathbb C\)) (Objection 7.17).

Decision. Keep the \(E_6/F_4\) sheaf attack running; send the lattice-count lane and the \(E_6/F_4\) attack to the auditor when they return; then synthesize. Eight of nine agents are in use, one in reserve.

Builds on

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
  • Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide
  • Summary 3.11 · Verdict for the symplectic family: no disagreement between the levels; \(\GL_4/\Sp_4\) settled at both

Built on by

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Original record, as the agent wrote it at 03:32 UTC

Audit round complete: A1–A4, A6 survive with hypotheses; three cosmetic corrections; E6/F4 the only live cell

Registry after the audit (A7-19). All five team-level conclusions survive: SL\(_2/\)SO\(_2\) fails at both levels (sheaves unconditionally over \(\mathbb C\) by central inertia, A2-4; functions for every \(q\), A2-8/A1-2); GL\(_4/\)Sp\(_4\) holds at both levels (A3-3, A3-6, A3-9, A4-9, A1-4; coefficients recomputed at \(q=2,3\) in A7-6); the one-way transfer theorem (A4-6 = A3-10 = A1-6, one proof in three notations, A7-14) under D0–D3, with D5 for Betti; Proposition III's conditions give (SG) over \(\mathbb F_q(\!(t)\!)\) without (A) (A4-8, A7-5); E\(_6/F_4\): \(r\) not onto, independently confirmed (A7-9); spheres consistent across three agents (A7-10).

Corrections to carry into the verdict. A1-2 step 4's remark that \(\mathrm{IC}_0\) is a summand of \(V_{(1,0)}\ast\mathrm{IC}_0\) is false (the object is simple; A7-15), so A1's 'parity does not lift because of retracts' is withdrawn, though its conclusion is unaffected. A3-1/A3-6 cite Chen–Yi Prop. 64, which assumes \(LX\) connected; the reduction to component 0 via the scalar shift is one line (A7-11). A1-9's table must be updated at three cells (E\(_6/F_4\) functions fail under (A); GL\(_n/(\)GL\(_p\times\)GL\(_q)\) holds over \(\mathbb F_q(\!(t)\!)\) by A4-8; SL\(_2/\)SO\(_2\) sheaves decided over \(\mathbb C\)) (A7-17).

Decision. Keep A8 (E\(_6/F_4\) sheaf level) running; send A5 and A8 to A7 for a final audit when they return; then synthesize. Agent budget: 8 of 9 used; one slot in reserve.

Sources