§13 · Agent X3 · Pair model at 2 · 2026-10-17T20:22:32Z

Summary X3.7 · A7-25, A7-32 re-derived; the \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) row

proved audit: minor defect ★ key step

Minor defect (second audit). Both verdicts are right (see inputs). Two cosmetic defects. (1) Summary 7.25 is labelled 'holds as stated' while S3-7 asks for an addition to its corrections list. (2) The 'further path Proposition 5.8→Computation 5.5→Theorem 5.7→Summary 7.25' overstates the dependence: Theorem 5.7 uses Computation 5.5 only for the central shift (Computation 5.5 step 3) and Lemma 5.6's Lemmas 2–3, not Computation 5.5 step 5's support bound. The conclusion is unaffected because the alternating-forms case of Proposition 5.8 is valid in every characteristic. The proposed row text is correct, and the pair-model convention matches classical.html §1 (a smooth affine O-model); X_J's non-affine special fibre is cited (Richardson), not verified by me.

Every cell of Summary 7.25 and Summary 7.32 reached from Proposition 5.8 or Objection 7.24 survives: the pair-model cell rests on Theorem 5.4, the symplectic cells use only the alternating-forms case of Proposition 5.8, which is valid; Summary 7.32 must name the pair model, because at residue characteristic 2 the involution model's basic function does not generate.

Summary X3.7.

Paths. \(A5\text{-}8\to A7\text{-}24\to A7\text{-}25\to A7\text{-}32\) carries one sentence, "Proposition 5.8 holds"; \(A7\text{-}24\to A7\text{-}25\) also carries the two objections. A further path, \(A5\text{-}8\to A5\text{-}5\to A5\text{-}7\to A7\text{-}25\), uses the alternating-forms case of Proposition 5.8 for covered labels.

Summary 7.25 holds as stated. Its \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) function cell says "in the pair model, which is the \(J\)-model when \(q\) is odd" (\(q\) the residue cardinality) and rests on Theorem 5.4 and Audit 7.21, which use neither \(\tau\) nor input (a) (Theorem X3.4(2)); the endorsement of Proposition 5.8 is true as a statement (Theorem X3.4(1)); the symplectic cells use only the alternating-forms case, valid in every characteristic (Theorem X3.4, steps 2–3); the objections it carries from Objection 7.24 are correct (Correction X3.6). Its list of carried-over corrections should add the correction of Proposition 5.8 (cosmetic).

Summary 7.32 holds with a change. "Function column: every row is decided unconditionally over every \(F\)" is true for \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) only for the pair model; it must say so. For the involution model the row holds iff the residue characteristic is odd, and for \(1\le\operatorname{val}2<\infty\) its basic function \(\mathbf 1_{Kx_0^J}\) does not generate (Proposition X3.5). That model has bad reduction there (non-affine special fibre, cited), whereas the pair model is smooth and affine over \(\mathcal O\) with reductive stabilizer at every place; with the pair model fixed, as Summary 7.25 already does, no new divergence arises. The answer — \(E_6/F_4\) the unique divergence, conditional and cross-field — and "no row has (SG) holding and \(\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K\)" are unchanged.

The row should read. \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(p\le q\). Sheaves: \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) (Proposition III). Functions: (SG) holds for every non-archimedean \(F\) (\(\operatorname{char}F\) arbitrary, residue characteristic \(2\) included) for the pair model \(X=\{P\in M_n:P^2=P,\ \operatorname{rk}P=q\}\cong\mathrm{GL}_n/(\mathrm{GL}_q\times\mathrm{GL}_p)\) over \(\mathcal O\), \(\varphi_0=\mathbf 1_{Kx_0}\): Theorem 5.4 with Proposition B, or Corollary C with the support bound for every dominant \(\lambda\) (Proposition X3.2); no \(\tau\), 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.. This is the involution (\(J\)-)model iff the residue characteristic is odd; for \(1\le\operatorname{val}2<\infty\) the involution base point lies in \(O_{(\operatorname{val}2,\dots,\operatorname{val}2)}\) and does not generate; for \(\operatorname{char}F=2\) there is no involution model. Levels agree.

  1. What Summary 7.25 takes from the erroneous steps. From Objection 7.24: the stale-cell corrections (\(E_6/F_4\) 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.; \(\mathrm{SL}_2/T\) for every \(q\)), both correct (Correction X3.6, steps 3–4), and the endorsement of Proposition 5.8, which enters only the sentence "the corrections carried over … are all cosmetic". From Proposition 5.8 directly: nothing for the pair model. Step 2 of the argument of Summary 7.25 cites Audit 7.21 (normal form and componentwise bound), and Audit 7.21 cites Theorem 5.4; both were graded correct and use only lattice algebra. The quotient argument of Theorem 5.4(iii) bounds the label of \(gx_0\), \(g\in Kt^{\lambda(a)}K\), by a quotient of \(L'/L_0\); no involution enters.
  1. The symplectic cells. Computation 5.5, step 5, uses the support bound of Proposition 5.8 for covered labels of \(\mathrm{GL}_4/\mathrm{Sp}_4\), and Theorem 5.7 builds on Computation 5.5. That is the alternating-forms case, proved in every characteristic in Theorem X3.4, steps 2–3. So the \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) cells stand.
  1. The two models. Theorem X3.4(3): for odd residue characteristic the pair model and the involution model have the same \(X(F)\), \(K\)-orbits and basic function. Proposition X3.5(iii): for \(1\le e<\infty\), \(\mathcal H_K\cdot\mathbf 1_{Kx_0^J}\ne C_K\). Both statements are consistent with Summary 7.25's wording, which restricts to the pair model and identifies it with the \(J\)-model only for odd residue characteristic.
  1. Why Summary 7.32 needs the change. Summary 7.32 states the function column "over every \(F\)" without a model. Read for the involution model at a \(2\)-adic \(F\), the \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) cell would be false, and the row would look like a cross-field mismatch of the kind used for \(E_6/F_4\) (sheaves hold over \(\mathbb C\); functions fail \(p\)-adically). It is an artefact of the integral model: the pair model is a closed subscheme of \(M_n\), smooth and affine over \(\mathcal O\), with reductive stabilizer \(\mathrm{GL}(L_0\cap U_0)\times\mathrm{GL}(L_0\cap V_0)\) at every place, while \(X_J=\mathrm{GL}_n/Z(J)\) has special fibre \(\mathrm{GL}_n/Z(\bar J)\) with \(Z(\bar J)\) non-reductive, which is not affine (cited: Richardson, a homogeneous space of a reductive group is affine only if the stabilizer is reductive). The function level of the verdict is meaningful only for a fixed model; the run's own wording (Summary 7.25) fixes the pair model, and Summary 7.32 must state this convention, otherwise the cell is false for the involution model at \(2\)-adic \(F\).
  1. Divergence analysis unchanged. The sheaf side lives over \(\mathbb C\) and \(\overline{\mathbb F}_q\) (\(p\) odd), and the trace dictionary over \(\mathbb F_q(\!(t)\!)\), \(q\) odd; there \(e=0\) and the models coincide. The \(E_6/F_4\) analysis of Summary 7.32 does not involve the pair model.

Builds on

Built on by

  • Re-check V3 · Re-check by V3
  • Summary 9.6 · Verdict after the repairs: every input held or repaired; \(E_6/F_4\) still the one divergence

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