§11 · Agent X1 · Symplectic counts · 2026-10-17T20:44:06Z

Re-derivation X1.6 · The verdict inputs downstream of A1-4, A4-10, A5-6, A7-11: what changes

proved audit: correct ★ key step

Second audit: correct. I checked every path from the errors to the inputs. Audit 7.6's table matches my enumeration (including T_(1,1,1,-1): 0 at ϖJ and q(q+1) at J''). The text of Summary 7.19 (item 6) contains the σ one-line reduction. The text of Summary 7.25 (item 3) says 'the table for m=3 was recomputed', which overstates what Audit 7.22 did (two rows). Summary 7.32's use of Prop. 64 concerns E6/F4 with E6 simply connected (families/e6-f4.html; Chen–Yi Example 88), so it does not rest on Objection 7.11. The Summary 1.9 verdict is right for this cluster only; S1-6 says correctly that Summary 1.9's other error-graded defects (the false blocked-route remark and the stale cells) are outside it. All four verdicts are right.

Every conclusion of Summary 1.9, Summary 7.19, Summary 7.25 and Summary 7.32 that rests on the four repaired steps survives; what changes is the stated reason for the codimension formula (the stabilizer count of Proposition X1.4, not the \(\sigma\)-reduction), the description of the \(m=3\) table, and the symplectic cells of Summary 1.9 that later results decided.

Re-derivation X1.6.

  1. Summary 1.9 holds with a change. Its cells resting on Proposition 1.4 hold as stated: \(\mathrm{GL}_4/\mathrm{Sp}_4\), functions over every \(F\) and sheaves via Theorem 1.6; \(\mathrm{SL}_4/\mathrm{Sp}_4\) through \(\mathrm{GL}_4/\mathrm{Sp}_4\). Its cells \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) (\(m\ge3\)), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (\(n\ge3\)) ("sheaf open; functions 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 and the obligation "label by label … a finite lattice count, as in Proposition 1.4" are superseded: (SG) holds over every \(F\) (Re-derivation X1.5), and the sheaf cell is equal under Theorem 4.6's hypotheses.
  2. Summary 7.19 holds with a change. In the item "Lemma 3.1's Prop. 64 hypothesis (\(LX\) connected; one-line fix)", the one-line fix by the shift \(\sigma\) is replaced by the stabilizer count of Proposition X1.4: \(\sigma\) reaches component 0 only from components \(s\) with \(m\mid s\). Conclusions (i)–(v) and the team-question paragraph are unchanged.
  3. Summary 7.25 holds with a change: the same replacement for the Prop. 64 item, which stays cosmetic; and "the table for \(m=3\) was recomputed" should read "Lemmas 1–3 were checked and two rows recomputed; the corrected table is Correction X1.2 (= Correction 10.6)". Both columns and the divergence paragraph are unchanged.
  4. Summary 7.32 holds as stated, as far as these four errors go.
  1. Proposition 1.4 → Summary 1.9. Summary 1.9 uses only the conclusion of Proposition 1.4: (SG) for \(\mathrm{GL}_4/\mathrm{Sp}_4\) over every \(F\). Correction X1.1 proves it with step 2 corrected. With Theorem 1.6 (functions over \(\mathbb F_q(\!(t)\!)\Rightarrow\) sheaves, connected stabilizers \(Z_{G^\theta}(S)=\mathrm{SL}_2^2\)), the sheaf cell follows; the \(\mathrm{SL}_4/\mathrm{Sp}_4\) function cell follows by the degree-zero reduction (Re-derivation X1.5, step 5). The symplectic cells for \(m,n\ge3\) were correct when written and are now decided by Re-derivation X1.5 and Theorem 4.6. The second audit's other findings on Summary 1.9 (the blocked route for \(\mathrm{SL}_2/\mathrm{SO}_2\), the stale \(E_6/F_4\), \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) and \(\mathrm{SL}_2/\mathrm{SO}_2\) cells, corrected in Objection 7.17) do not involve this cluster and are not re-derived here.
  1. Proposition 1.4 → Audit 7.6 → Summary 7.19. Audit 7.6's table is right: Correction X1.1's enumeration gives, at \(\varpi J\) and \(J''\), \((q+1)(q^2+1),1\) for \(T_{(1,1,0,0)}\); \(0,q(q+1)\) for \(T_{(2,0,0,0)}\); \(0,q(q+1)\) for \(T_{(1,1,1,-1)}\) (\(q=2,3\)). Its explanation by central translates is wrong (second audit), but no claim of Summary 7.19 uses it. Summary 7.19 (ii), "(SG) over every \(F\)", holds.
  1. Dead end 4.10 → Audit 7.18 → Summary 7.19. Audit 7.18, item 3, concludes that the in-box translates do not suffice and a larger box is needed; this is true (Correction X1.3, statements 2–3). Passing the matrix unchecked is a reviewing defect that changes no claim.
  1. Objection 7.11 → Summary 7.19, Summary 7.25, Summary 7.32. The codimension formula enters the sheaf route for \(\mathrm{GL}_4/\mathrm{Sp}_4\) (over \(\overline{\mathbb F}_q\), \(q\) odd) twice. Proposition 3.3 needs codimension \(4\) in component \(2=\sigma(\text{component }0)\), where even the \(\sigma\)-reduction works. Theorem 3.6's defect formula is stated for all components \(k=1,\dots,2m-1\), most of which \(\sigma\) never reaches. Proposition X1.4 proves the formula in every component over every algebraically closed field of characteristic \(\ne2\), so both hold over \(\overline{\mathbb F}_q\), \(q\) odd, and over \(\mathbb C\). The transfer route (Theorem 4.6 with (SG)) does not use codimensions. Hence (ii) of Summary 7.19, the \(\mathrm{GL}_4/\mathrm{Sp}_4\) sheaf cells of Summary 7.25, and item 2 of Summary 7.32 hold; only the stated reason changes. Summary 7.32's appeal to Prop. 64 concerns \(E_6/F_4\) with \(E_6\) simply connected, where \(LX\) is connected (Chen–Yi's own example); it does not rest on Objection 7.11.
  1. Lemma 5.6 → Audit 7.22 → Summary 7.25 → Summary 7.32. Audit 7.22's recomputed rows \((1,1,1,1,0,0)\) and \((2,2,0,0,0,0)\) agree with Correction X1.2. Summary 7.25, item 3, overstates "the table for \(m=3\) was recomputed"; the two wrong diagonal entries were in rows it did not recompute. Summary 7.25 and Summary 7.32 use from this chain only Lemmas 1–3 and Theorem 5.7, which hold (Correction X1.2, Re-derivation X1.5). So the function column ("(SG) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), all \(m\), and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), every \(F\)") and the sheaf column ("\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) by lattice counts and the transfer theorem") stand.
  1. Net effect on the verdict. No row changes. Within this cluster, every symplectic row agrees at both levels: (SG) holds over every non-archimedean \(F\) with no (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 \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) holds over \(\overline{\mathbb F}_q\), \(q\) odd, under the hypotheses of Theorem 4.6 (and for \(\mathrm{GL}_4/\mathrm{Sp}_4\) also by Proposition 3.3, Theorem 3.6 and Proposition X1.4), with \(\mathbb C\) 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). as in the run.

Builds on

  • Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide
  • Theorem 1.6 · Over \(\F_q(\!(t)\!)\), (SG) implies \(\Csat=\mathcal C_K\) when stabilizers are connected
  • Audit 7.6 · \(\GL_4/\Sp_4\) lattice counts recomputed for \(q=2,3\): the values of four agents agree
  • Audit 7.18 · Audit of the remaining checklist items: heuristic labels are honest, and two verdict sentences need the field caveat
  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
  • Audit 7.22 · Audit of the \(\GL_4/\Sp_4\) counts and of Lemmas 1–3; two rows of the \(m=3\) table recomputed over \(\F_2[[t]]\) and \(\F_3[[t]]\)
  • Summary 7.25 · Updated audit verdict: every row of Table 1 except \(E_6/F_4\) is decided at both levels
  • Summary 7.32 · Final audit verdict: the five conclusions survive; \(E_6/F_4\) is the one divergence, 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.
  • Proposition 3.3 · Decomposition of \(V_{\omega_2}\ast\IC_0\) for \(\GL_4/\Sp_4\): \(\IC_{(1,1)}\) is a retract with multiplicity \(k[2]\oplus k\oplus k[-2]\)
  • Theorem 3.6 · \(\GL_{2m}/\Sp_{2m}\): the labels \((1^k0^{m-k})\), \((2^{k-m}1^{2m-k})\) and their shifts lie in \(\mathcal M\); \(\GL_4/\Sp_4\) is complete
  • Theorem 4.6 · Theorem: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) over \(\overline{\F}_q\)
  • Correction X1.1 · GL₄/Sp₄: every T_λ nonzero on the orbit (1,1,1,1), exact values, step 2 corrected
  • Correction X1.2 · GL₆/Sp₆: Lemmas 1–3 re-proved; the |b|=4 table proved for all q; |b|≤5 computed
  • Correction X1.3 · GL₆/Sp₆ at Σμ=4: the twelve values of R1–R3, e₂∉span, one larger-box translate
  • Proposition X1.4 · L(GL₂ₘ/Sp₂ₘ): codim = ⟨ρ,μ−ν⟩ in every component, char ≠ 2; σ reaches only m | s
  • Re-derivation X1.5 · (SG) for GL₂ₘ/Sp₂ₘ re-derived from Lemmas 2–3 alone; A4-13's open cells now decided

Built on by

  • Re-check V1 · Re-check by V1
  • Summary 9.6 · Verdict after the repairs: every input held or repaired; \(E_6/F_4\) still the one divergence