§7 · Agent 7 · Adversarial audit · 2026-10-17T03:47:54Z

Summary 7.25 · Updated audit verdict: every row of Table 1 except \(E_6/F_4\) is decided at both levels

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. Both columns and the divergence paragraph follow from Summary 7.19 and Audit 7.20...A7-24, with the hypotheses listed. Defects, none changing a claim's truth: (1) it repeats Summary 7.19's misattribution: Conjecture 6.8 does not point to sheaf equality (it says the function failure neither proves nor refutes anything at sheaf level); (2) 'the only row where a divergence can still exist is E6/F4, in the direction C^Sat = C but (SG) fails' needs a qualifier: the reverse direction at E6/F4 is excluded over F_q((t)) by the transfer theorem and p-adically only 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.; 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., a p-adic reverse case was not excluded while the E6/F4 sheaf level was open; (3) the list of carried-over corrections omits Audit 7.23's (Proposition B invoked with reversed order, replaced by a direct induction). 'Every non-archimedean F' for GL_n/(GL_p x GL_q) is correctly restricted to the pair model.

After the audit of the lattice-count results, (SG) holds without Sakellaridis's hypotheses over every local field in every positive row, and the transfer theorem turns the symplectic rows into sheaf-level equalities over \(\overline{\mathbb F}_q\). Wherever both levels are decided they agree; only \(E_6/F_4\) can still diverge, in the direction 'sheaves hold, functions fail'.

Summary 7.25.

Function level. Unconditionally (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., any characteristic of \(F\)), (SG) holds for the group (Proposition D); for \(\mathbb G_m\) and \(\mathrm{GL}_2/T\) (Proposition J); for \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), all \(n\ge2\) (Theorem 5.3; for \(q\) odd also Proposition 1.3, over \(\mathbb F_q(\!(t)\!)\) also Lemma 4.8); for \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) in the pair model, which is the \(J\)-model when \(q\) is odd (Theorem 5.4); and for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), all \(m\), and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), all \(n\) (Theorem 5.7). (SG) fails for \(\mathrm{SL}_2/T\), every \(q\) (Computation 2.8, Proposition 1.2); for \(\mathrm{GL}_n/\mathrm O_n\) and \(\mathrm{PGL}_2/\mathrm{PO}_2\); and for \(E_6/F_4\) 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. for \(p\)-adic \(F\) (Theorem 6.6), the last being open over \(\mathbb F_q(\!(t)\!)\).

Sheaf level. \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) holds for the group, the spheres, \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathbb G_m\) and \(\mathrm{SL}_4/\mathrm{Sp}_4\) (Proposition III, whose steps 3 and 6 are ℓ-adic, with Betti via the pages' open item 1); for \(\mathrm{GL}_4/\mathrm{Sp}_4\) (Proposition 3.3 with Theorem 3.6, or the transfer theorem); and, new, for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge3\), and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\), over \(\overline{\mathbb F}_q\), \(q\) odd, 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., by Theorem 5.7 and Theorem 4.6 (Audit 7.23), and over \(\mathbb C\) under 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).. It fails for \(\mathrm{SL}_2/\mathrm{SO}_2\) (Betti, unconditional, Theorem 2.4) and for \(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\) (support). It is open only for \(E_6/F_4\).

Divergence. Wherever both levels are decided they agree. The transfer theorem ((SG) over \(\mathbb F_q(\!(t)\!)\) implies \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) for connected \(Z_{G^\theta}(S)\), 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.) excludes "functions hold, sheaves fail" in every row with \(Z_{G^\theta}(S)\) connected; for \(\mathrm{SL}_2/\mathrm{SO}_2\) both fail. The only row where a divergence can still exist is \(E_6/F_4\), in the direction "\(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) but (SG) fails": the function side 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. for \(p\)-adic \(F\) (Theorem 6.6), the sheaf side is open, with the relative-Satake heuristic of Conjecture 3.8 and Conjecture 6.8 pointing to equality, and the dictionary cannot exclude the divergence (Proposition 4.7). Settling it needs (a) a decision of \(\mathrm{IC}_{3\omega_1^\vee}\in\mathcal C_K^{\mathrm{Sat}}\) (the maximal-defect lemma on a Bott–Samelson model, or Hecke compatibility of Chen–Yi, Thm. 87(ii)); and (b) either Sakellaridis's Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\), or an Albert-lattice count exhibiting the dependence predicted by Theorem 6.6, so that the function-level failure lives in the field of the dictionary, or acceptance of the cross-field comparison through the two bridges of Audit 7.16.

Hypotheses. Function column: lattice algebra only, except \(E_6/F_4\) 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.. Sheaf column: Chen–Yi's placid formalism with Prop. 17 and Thm. 18 over \(\overline{\mathbb F}_q\), \(p\) odd (D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers.); proper base change, the trace formula and the perverse t-structure (D1D1 · Traces and t-structure: proper base change and the Grothendieck–Lefschetz trace formula on convolution fibres, with a Frobenius-compatible perverse t-structure whose simple objects are the IC sheaves.); the decomposition theorem on finite-type equivariant models (Sun, Thm. 1.2; D2D2 · Decomposition: each Satake object V_λ ∗ IC_0 is pure and splits into shifted IC sheaves with finite-dimensional multiplicity spaces (Sun, Thm 1.2; Chen–Yi, Prop. 70).); Lang's theorem (D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q.); Betti over \(\mathbb C\) by spreading out (BBD §6.1; 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).), except \(\mathrm{SL}_2/\mathrm{SO}_2\), which is decided in Betti directly. The corrections carried over (the remark in Proposition 1.2, the hypothesis of Chen–Yi's Prop. 64, unneeded inputs, stale cells in Summary 1.9, Dead end 5.9, Summary 5.10) are all cosmetic.

  1. Spheres in every residue characteristic. The model \(Q=\{q(x)=1\}\), with \(q\) hyperbolic of even rank, is regular over \(\mathbb Z\); the relevant orthogonal and spin groups are smooth; and the classification of \(K\)-orbits by pole order uses only Hensel's lemma, Witt cancellation over the residue field and the spinor norm over \(\mathcal O\). So residue characteristic 2 is covered (Audit 7.20).
  2. The pair model. A lattice \(L\) in \(U\oplus V\) has a normal form determined by the elementary divisors of \(\pi_V L/(L\cap V)\), which gives \(K\backslash X(F)=\Lambda_S^+\) for every \(F\); the type of a quotient of \(L'/L_0\) is bounded componentwise by that of \(L'/L_0\), which is the support bound needed for Corollary C (Audit 7.21).
  3. Symplectic families. The three lattice lemmas of Lemma 5.6 — the count for a single row, a unimodular point on the open stratum, and reverse dominance from the perfect pairing \(\Lambda_0^\#/\Lambda''\times\Lambda''/\Lambda_0\to F/\mathcal O\) — were checked line by line, and the table for \(m=3\) was recomputed over \(\mathbb F_2[\![t]\!]\) and \(\mathbb F_3[\![t]\!]\) (Audit 7.22). The induction runs downward along dominance inside the finite sets \(\{b:\ b_m\ge2c,\ \sum_ib_i=d\}\), not through Proposition B; the degree-0 reduction gives \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\); and the transfer theorem applies, since \(G^\theta=\mathrm{Sp}_{2m}\) and \(Z_{G^\theta}(S)=\mathrm{SL}_2^m\) are connected (Audit 7.23).
  4. Stale statements. The remark of Dead end 5.9 that nothing suggests a failure at \(E_6/F_4\), and the restriction "\(q\) odd" for \(\mathrm{SL}_2/T\) in Summary 5.10, predate the results that contradict them (Objection 7.24).
  5. Assembly. Steps 1–4 and the first audit verdict Summary 7.19 give both columns; the divergence paragraph follows by inspecting the rows.

Builds on

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
  • Audit 7.20 · Audit of the sphere results: the residue-characteristic-2 case is genuinely covered
  • Audit 7.21 · Audit of \(\GL_n/(\GL_p\times\GL_q)\) in the pair model: normal form, componentwise bound and \((\mathrm{SG})\) over every \(F\)
  • 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 \(\F_3[[t]]\)
  • Audit 7.23 · Audit of the \(\GL_{2m}/\Sp_{2m}\) theorem: the induction holds (directly, not through Proposition B as stated); the \(\SL\) corollary and the sheaf consequence hold
  • Objection 7.24 · Objection: the support-bound check passes; the \(E_6/F_4\) expectation and two table cells are stale

Built on by

Original record, as the agent wrote it at 03:47 UTC

Updated verdict: every Table 1 row but E6/F4 is decided at both levels; divergence possible only at E6/F4

Function level (SG), after A5 (all unconditional, every non-archimedean \(F\), no (A), no \(\operatorname{char}F=0\)): group (D); \(\mathbb G_m\), \(\mathrm{GL}_2/T\) (J); spheres all \(n\ge2\) (A5-3; \(q\) odd also A1-3; \(\mathbb F_q(\!(t)\!)\) also A4-8); \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), pair model (A5-4; = \(J\)-model for \(q\) odd); \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) all \(m\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) all \(n\) (A5-7, Lemma 3 verified, table recomputed). Fails: \(\mathrm{SL}_2/T\) every \(q\) (A2-8, A1-2); \(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\); \(E_6/F_4\) under (A), \(p\)-adic (A6-6), open over \(\mathbb F_q(\!(t)\!)\).

Sheaf level \(\mathcal C^{\mathrm{Sat}}=\mathcal C\): holds for group, spheres, \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathbb G_m\), \(\mathrm{SL}_4/\mathrm{Sp}_4\) (Prop. III; steps 3, 6 \(\ell\)-adic, Betti via open item 1); \(\mathrm{GL}_4/\mathrm{Sp}_4\) (A3-3 + A3-6 directly, or transfer); new: \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge3\), and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\), over \(\overline{\mathbb F}_q\) (\(q\) odd) under D0–D3 by A5-7 + A4-6 (A7-23), over \(\mathbb C\) under D5. Fails: \(\mathrm{SL}_2/\mathrm{SO}_2\) (A2-4, Betti, unconditional); \(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\) (support). Open: \(E_6/F_4\) only.

Divergence. Wherever both levels are decided they agree. The implication (SG) over \(\mathbb F_q(\!(t)\!)\Rightarrow\mathcal C^{\mathrm{Sat}}=\mathcal C\) (connected \(Z_{G^\theta}(S)\), D0–D3) excludes 'functions hold, sheaves fail' in every row with connected \(Z\); for \(\mathrm{SL}_2/\mathrm{SO}_2\) both fail. The only row where a divergence can still exist is \(E_6/F_4\), in the direction '\(\mathcal C^{\mathrm{Sat}}=\mathcal C\) but (SG) fails': function side fails under (A) for \(p\)-adic \(F\) (A6-6, verified), sheaf side open with the Peter–Weyl/relative-Satake heuristic (A3-8) and A6-8 pointing to equality, and the dictionary unable to exclude it (A4-7). To settle it: (a) decide \(\mathrm{IC}_{3\omega_1^\vee}\in\mathcal C^{\mathrm{Sat}}\) (A3-5 on a Bott–Samelson model, or Hecke-compatibility of Chen–Yi Thm 87(ii)); (b) either prove Sakellaridis Thm 8.0.2 over \(\mathbb F_q(\!(t)\!)\) (or an Albert-lattice count exhibiting the dependence predicted by A6-6) so that the function failure lives in the dictionary's field, or accept the cross-field comparison with the two bridges of A7-16.

Hypotheses summary. Function column: none beyond lattice algebra (A5), except \(E_6/F_4\) under (A). Sheaf column: Chen–Yi's placid formalism and Prop. 17/Thm 18 over \(\overline{\mathbb F}_q\), \(p\) odd (D0), proper base change/trace formula and perverse t-structure (D1), decomposition theorem on finite-type equivariant models (Sun Thm 1.2; D2), Lang (D3); Betti over \(\mathbb C\) via BBD §6.1 spreading out (D5), except \(\mathrm{SL}_2/\mathrm{SO}_2\) which is decided in Betti directly. Corrections carried over from A7-19 (A1-2 remark, Prop. 64 hypothesis, unneeded inputs, stale cells in A1-9, A5-9, A5-10) are all cosmetic.

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