§7 · Agent 7 · Adversarial audit · 2026-10-17T03:21:52Z

Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)

proved audit: minor defect

Minor defect (second audit). Repeats the flawed repair of Objection 7.11 (corrected in Correction 10.2); its own conclusions are unaffected. Conclusions (i)-(v) and the team-question paragraph follow from Audit 7.1...A7-18 with the fields and hypotheses stated (Betti for the SL2/SO2 sheaf failure; l-adic over F_q-bar 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 (ii)-(iii); F_q((t)) for (iii)-(iv); p-adic under (A1)-(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit. for (v)). The E6 facts used in (v) were recomputed independently (e6_g7.py: both weights reach rho-omega_4 = (1,1,1,0,1,1) by words of length 18 and 13; the S_3-orbits of (-5,1) and (-5,4) are disjoint). Defect: Conjecture 6.8 is cited as one of 'two heuristics predicting C^Sat = C', but Conjecture 6.8's sheaf-side reading says the function-level failure 'neither proves nor refutes anything at sheaf level'; only the extension of Conjecture 3.8's Peter-Weyl heuristic to PGL_3 -> E6^ad made in Audit 7.13 predicts equality. Wrong cross-reference; no verdict depends on it. 'The one candidate' is right as a soft claim: the symplectic rows with m>=3 were still undecided at the sheaf level, but nothing pointed to a divergence there.

The adversarial audit of the first-round results confirms five team-level conclusions, each in its own field and under stated hypotheses, and finds only cosmetic defects. The one row where the evidence points to a divergence is \(E_6/F_4\): functions fail there under Sakellaridis's hypotheses, while the sheaf level is still open.

Summary 7.19.

(i) \(\mathrm{SL}_2/\mathrm{SO}_2\): both levels fail. At sheaf level, \(\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K\) in the Betti category over \(\mathbb C\), using only the standing realisation of \(\mathcal C_K\) by equivariant sheaves whose local systems on an orbit \(\mathcal O_\mu\) are the representations of \(\pi_0\operatorname{Stab}(x_\mu)\) (Theorem 2.4); independently ℓ-adically (Theorem 4.11, Proposition 1.7), using Weil structures over all \(\mathbb F_{q^r}\), the decomposition theorem and purity. At function level, (SG) fails for the split model \(\mathrm{SL}_2/T\) for every \(q\) and every \(F\) by Hecke parity (Computation 2.8, Proposition 1.2), and for \(q\) odd also by square classes (Proposition J(2)).

(ii) \(\mathrm{GL}_4/\mathrm{Sp}_4\): both levels hold. (SG) holds over every non-archimedean local field (Theorem 3.9, Proposition 1.4). \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) holds over \(\overline{\mathbb F}_q\), \(q\) odd, by the decomposition of \(V_{\omega_2}\ast\mathrm{IC}_0\) (Proposition 3.3) with shift stability (Theorem 3.6), and independently by (iii); both routes need Chen–Yi's placid formalism and the decomposition theorem on finite-type equivariant models (D0–D2The 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.). Over \(\mathbb C\) it follows by 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)., which is standard for large \(p\) by spreading out (BBD §6.1) and suffices because (SG) holds for every \(q\).

(iii) (SG) over \(\mathbb F_q(\!(t)\!)\) implies \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) over \(\overline{\mathbb F}_q\), for \(G,\theta\) defined over \(\mathbb F_q\) with \(T\) split, \(q\) odd, \(G^\theta\) and \(Z_{G^\theta}(S)\) connected, under 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., 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., 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). (only the semisimplicity of Satake objects, no weights) and D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q. (only the rationality of \(x_\mu\) and the compact-openness of \(\mathcal O_\mu(\mathbb F_q)\)); D4D4 · Unitriangularity of the Satake traces tr V_λ in the basis T_λ = 1_{Kt^λK} (Zhu (5.6.2), Lemma 5.6.3; Lusztig, Kato). is elementary; 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).. The converse is not proved and is not a formal consequence of the dictionary (Proposition 4.7).

(iv) Proposition III's two conditions imply (SG) over \(\mathbb F_q(\!(t)\!)\), under 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. and D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q., for \(q\) odd and \(T\) split (Lemma 4.8). This gives the spheres (all \(n\)) and \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) 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..

(v) \(E_6/F_4\): (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.; spheres: (SG) holds 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 \(F\), with (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit., (A3)(A3) Sakellaridis's three axioms over O, relabelled: (A3a) rank-one models over O (Axiom 2.2.1); (A3b) the Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2). For E6/F4 all three hold in both characteristics (VI run). at the given place and (A4)(A4) Spherical roots and colours: no spherical root of type N, every Levi L_γ of a simple spherical root classical, and his Statements 6.3.1 and 7.1.5 on the spherical roots and the virtual coloured weights. beyond "\(L_\gamma\) classical" unverified. The \(E_6\) computation (the \(W(E_6)\)-conjugacy of \(\rho-9\omega_1-3\omega_6\) and \(\rho-9\omega_1\), and the non-conjugacy of \(y_1,y_2\) under \(S_3\)) is independently confirmed. The spheres hold moreover unconditionally for \(q\) odd over every \(F\) (Proposition 1.3) and over \(\mathbb F_q(\!(t)\!)\) (Lemma 4.8).

Team question. No decided row diverges. The one candidate is \(E_6/F_4\): its function level 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. in characteristic 0 (Theorem 6.6); its sheaf level is open, with two heuristics predicting \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\), the Peter–Weyl argument of Conjecture 3.8 transported along \(\mathrm{PGL}_3\hookrightarrow E_6^{\mathrm{ad}}\) and Conjecture 6.8; and the dictionary cannot exclude this divergence (Proposition 4.7). To make it a theorem one needs (a) the sheaf statement at the labels \(3k\omega_1^\vee\), for instance the maximal-defect lemma Lemma 3.5 applied to a Bott–Samelson model for some \(\lambda\) with \(\varphi(\lambda)\ge3\omega_1^\vee\), and (b) either Sakellaridis's Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\) or acceptance of a comparison of \(p\)-adic functions with Betti sheaves (Objection 1.8).

Notation. 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 fixed in Setup 4.1: 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. the orbit geometry of Chen–Yi over \(\overline{\mathbb F}_q\), \(p\ne2\); 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. proper base change, the trace formula and the perverse t-structure; 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). the decomposition of Satake objects; D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q. Lang's theorem; D4D4 · Unitriangularity of the Satake traces tr V_λ in the basis T_λ = 1_{Kt^λK} (Zhu (5.6.2), Lemma 5.6.3; Lusztig, Kato). the Satake transform; 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). the comparison of decomposition multiplicities between \(\overline{\mathbb F}_q\) and \(\mathbb C\). "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." means under Sakellaridis's hypotheses (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit. of Theorem E on the classical page.

  1. (i). The central element \(-I\in L^+\mathrm{SL}_2\) acts trivially on \(LX\) and on \(\mathrm{Gr}\), so it defines a natural involution of the identity functor of \(\mathcal C_K\); convolution multiplies its eigenvalues, so the Satake span lies in the \(+1\) part, while the sign local system on the first orbit lies in the \(-1\) part (Audit 7.1, Audit 7.2). This uses neither purity nor a transfer from \(\overline{\mathbb F}_q\). The two ℓ-adic proofs need purity to exclude cancellation of Frobenius eigenvalues between degrees, and agree on the witness (Audit 7.7). The function-level statement \(\mathcal H_K\varphi_0=\mathbb C[y^2]\subsetneq\mathbb C[y]\) was re-derived from the recursions on the tree (Audit 7.2, Objection 7.15).
  2. (ii). The values of \(T_{(1,1,0,0)}\ast\varphi_0\) and \(T_{(2,0,0,0)}\ast\varphi_0\) on the orbits \((1,1,1,1)\) and \((2,2,0,0)\), namely \((q+1)(q^2+1),\,1\) and \(0,\,q(q+1)\), were recomputed for \(q=2,3\) (Audit 7.6). The decomposition of \(V_{\omega_2}\ast\mathrm{IC}_0\) was re-derived from the stalk at the closed orbit (the cohomology of \(LG(2,4)\cong Q^3\)), the support condition and Verdier self-duality; Sun's decomposition theorem is proved over finite fields and only stated over \(\mathbb C\), so the Betti version rests on 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). (Audit 7.12). The summand criterion, the maximal-defect lemma, the fibre lemma and the reduction from \(\mathrm{SL}_{2n}\) to \(\mathrm{GL}_{2n}\) hold (Audit 7.13).
  3. (iii). The three write-ups of the transfer theorem are one proof: the trace dictionary, the decomposition of Satake objects with constant coefficients, the expansion of their traces in the basis \(\operatorname{tr}\mathrm{IC}_\nu\), the extraction of each label from a maximal nonzero coefficient, and the summand criterion (Audit 7.14, Audit 7.3, Audit 7.4). The extraction needs only that \(x_\mu\) is rational and that \(\operatorname{tr}\mathrm{IC}_\mu\) is nonzero on \(\mathcal O_\mu(\mathbb F_q)\).
  4. (iv). Corollary C applies to a dominant lift \(\lambda\) of each label, with \(t^\lambda x_0\in\mathcal O_{\varphi(\lambda)}\) and the support bound \(Kt^\lambda X(\mathcal O)\subseteq\bigcup_{\nu\le\varphi(\lambda)}\mathcal O_\nu(\mathbb F_q)\) (Audit 7.5).
  5. (v). The convention \(\delta(X)^{1/2}=q^{\pm\rho_L}\), the irrelevance of its sign and the transport of the \(W_X\)-action were checked against Proposition F (Audit 7.8); the \(E_6\) computation was redone with an independent encoding of the root datum (Audit 7.9); the three sphere arguments are consistent (Audit 7.10). The status of (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit. is the one recorded in Lemma 6.9.
  6. Defects found. The remark in Proposition 1.2 that \(\mathrm{IC}_0\) is a summand of \(V_{(1,0)}\ast\mathrm{IC}_0\) is false and unused (Objection 7.15). Lemma 3.1 cites Chen–Yi, Prop. 64, which assumes \(LX\) connected; a one-line reduction to the degree-0 component is needed (Objection 7.11). Theorem 1.6 and Theorem 4.6 list inputs (Chen–Yi, Thm. 81; Lusztig–Kato) that the proof does not use (Audit 7.14). Three cells of Summary 1.9 are stale (Objection 7.17); one sentence each in Summary 3.11 and Summary 4.13 needs the field caveat (Audit 7.18).

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Original record, as the agent wrote it at 03:21 UTC

Verdict: (i)-(v) survive with stated hypotheses; the only candidate divergence is E6/F4

(i) \(\mathrm{SL}_2/\mathrm{SO}_2\), both levels fail — survives. Sheaves: A2-4 over \(\mathbb C\) (Betti, needs only the standing realisation of \(\mathcal C\) as equivariant sheaves with \(\mathrm{Rep}(\pi_0\mathrm{Stab})\) local systems; no purity, no transfer); independently \(\ell\)-adically A4-11/A1-7 (need Weil structures over all \(\mathbb F_{q^r}\), decomposition, purity). Functions: fails for every \(q\) and every \(F\) for the split model (A2-8(c) = A1-2 Hecke parity; verified), and by square classes for \(q\) odd (J(2)).

(ii) \(\mathrm{GL}_4/\mathrm{Sp}_4\), both levels hold — survives. Functions: (SG) over every non-archimedean local field (A3-9/A1-4; coefficients \(1,(q+1)(q^2+1)\) and \(0,q(q+1)\) recomputed for \(q=2,3\)). Sheaves: over \(\overline{\mathbb F}_q\), \(q\) odd, by the direct decomposition A3-3 (\(\mathrm{IC}_{(1,1)}\otimes(k[2]\oplus k\oplus k[-2])\) summand of \(V_{\omega_2}\ast\mathrm{IC}_0\)) + A3-6 shift stability, and independently by the transfer theorem (A4-6/A3-10/A1-6); both need Chen–Yi's placid formalism and Sun's decomposition theorem on finite-type equivariant models (D0–D2). Over \(\mathbb C\): D5, which is standard for large \(p\) by BBD §6.1 spreading out and suffices here since (SG) holds for all \(q\).

(iii) (SG) over \(\mathbb F_q(\!(t)\!)\Rightarrow\mathcal C^{\mathrm{Sat}}=\mathcal C\) — survives for \(G,\theta\) over \(\mathbb F_q\), \(T\) split, \(q\) odd, \(G^\theta\) and \(Z_{G^\theta}(S)\) connected; obligations: D0 (orbits/closures/support bound/stabilizer lemma over \(\overline{\mathbb F}_q\)), D1 (proper base change + trace formula on fibres; Fr-compatible perverse t-structure, finite-length heart), D2 in the semisimplicity form (weights not needed), D3 only for rationality of \(x_\mu\) and compact-openness of \(\mathcal O_\mu(\mathbb F_q)\); D4 elementary; conclusion over \(\overline{\mathbb F}_q\), Betti via D5. Converse not proved and formally false as a dictionary consequence (A4-7).

(iv) Prop. III conditions \(\Rightarrow\) (SG) over \(\mathbb F_q(\!(t)\!)\) — survives under D0, D3, \(q\) odd, \(T\) split (A4-8); gives spheres (all \(n\)) and \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) without (A).

(v) \(E_6/F_4\): (SG) fails under (A); spheres: (SG) holds under (A) — survives, \(p\)-adic \(F\), with (A2), (A3) at the given place and (A4) beyond '\(L_\gamma\) classical' unverified; the \(E_6\) computation (W-conjugacy of \(\rho-9\omega_1-3\omega_6\) and \(\rho-9\omega_1\), non-conjugacy of \(y_1,y_2\) under \(S_3\)) is independently confirmed. Spheres are moreover unconditional for \(q\) odd over any \(F\) (A1-3) and over \(\mathbb F_q(\!(t)\!)\) (A4-8).

Team question. No decided row diverges. The one candidate is \(E_6/F_4\): function level fails under (A) in characteristic 0 (A6-6), sheaf level open with two heuristics (A3-8 Peter–Weyl via \(\mathrm{PGL}_3\hookrightarrow E_6^{ad}\); A6-8) predicting \(\mathcal C^{\mathrm{Sat}}=\mathcal C\); the dictionary cannot exclude this divergence (A4-7). To make it a theorem one needs (a) the sheaf statement for the labels \(3k\omega_1^\vee\), e.g. A3-5 applied to a Bott–Samelson model for \(\lambda\) with \(\varphi(\lambda)\ge3\omega_1^\vee\), and (b) either Thm. 8.0.2 over \(\mathbb F_q(\!(t)\!)\) or acceptance of comparing \(p\)-adic functions with Betti sheaves (A1-8). Errors found: A1-2's summand remark (false, cosmetic); A3-1's Prop. 64 hypothesis (LX connected; one-line fix); A1-6/A4-6 list inputs (Thm 81, Lusztig–Kato) that are not needed; A1-9's three stale cells.

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