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

Audit 7.14 · Audit: the three proofs that (SG) implies \(\Csat=\cC_K\) are one proof in three notations; inputs listed and compared

proved audit: minor defect

Minor defect (second audit). The comparison is correct (checked against Theorem 4.6, Proposition 3.10, Theorem 1.6, Lemma 1.5: the maximal-ν0 argument needs only support and the nonzero value at x_ν; Lemma 1.5 needs no H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf.; Proposition 3.10 does not use its one-orbit hypothesis). One wrong citation number, repeated from Theorem 1.6: the fact that a Weil-simple perverse sheaf whose geometric constituent IC_ν is Frobenius-stable is IC_ν ⊗ α is BBD 5.3.9(ii) with the discussion before it (checked in the scanned source), not 5.3.8, which is the geometric semisimplicity of pure perverse sheaves.

An in-run comparison showing that the three independent proofs of the transfer direction share one skeleton, that none is weaker than another, and listing the exact obligations of the transfer theorem.

Audit 7.14.

Common skeleton.

Exact obligations of the transfer theorem: 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). (semisimplicity form), D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q. (only \(x_\mu\) rational and \(\cO_\mu(\F_q)\) compact open), \(G,\theta\) over \(\F_q\) with \(T\) split, \(q\) odd, \(G^\theta\) and \(Z_{G^\theta}(S)\) connected; conclusion over \(\overline{\F}_q\); 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).. With \(Z_{G^\theta}(S)\) disconnected the argument yields only "for each \(\mu\) some \(\IC_{\chi,\mu}\) is a summand", not \(\Csat=\cC_K\).

  1. Differences. Theorem 1.6 lists Chen–Yi's Theorem 81 (pointwise purity) as an assumption; it is not needed for this direction (cosmetic). Theorem 4.6 lists Lusztig–Kato (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).); it is elementary (cosmetic). Proposition 3.10 is stated for \(\GL_{2m}/\Sp_{2m}\), "more generally H1–H4Hypotheses of the pages' Proposition I (H1–H5 imply categorical generation): H1 finite orbit filtration; H2 equivariant recollement; H3 constant coefficients on orbits; H4 the Hecke action preserves the category; H5 every orbit carries a Satake object with nonzero top splitting., connected stabilizers and (DT)": the same hypotheses. Proposition 3.10 states the hypothesis "one \(K\)-orbit per label" but does not use it; Theorem 1.6 uses Lang's theorem only for the basis, which is likewise dispensable. None of the three proves the converse, and all three say so.
  2. The maximal-\(\nu_0\) argument. If \(\one_{\cO_\mu(\F_q)}=\sum_{\nu\in\mathcal M}c_\nu\tr\IC_\nu\) and \(\nu_0\) is maximal with \(c_{\nu_0}\ne0\), then at \(x_{\nu_0}\) every other term vanishes (its support misses \(\cO_{\nu_0}\), or its coefficient is zero by maximality), so the left side is nonzero at \(x_{\nu_0}\) and \(\nu_0=\mu\in\mathcal M\).

Builds on

  • Theorem 4.6 · Theorem: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) over \(\overline{\F}_q\)
  • Proposition 3.10 · Function level implies sheaf level: (SG) over \(\F_q(\!(t)\!)\) forces \(\mathcal M=\Lambda_S^+\)
  • Theorem 1.6 · Over \(\F_q(\!(t)\!)\), (SG) implies \(\Csat=\mathcal C_K\) when stabilizers are connected

Built on by

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
Original record, as the agent wrote it at 03:21 UTC

A4-6, A3-10, A1-6 are one proof in three notations; inputs listed and compared

Common skeleton. (a) dictionary \(\mathrm{tr}(V_\lambda\ast\mathrm{IC}_0)=\mathrm{tr}V_\lambda\ast\varphi_0\) and \(\{\mathrm{tr}V_\lambda\}\) spans \(\mathcal H_K\) (proper base change + Grothendieck–Lefschetz on finite-type fibres; unitriangularity of \(\mathrm{tr}V_\lambda\), elementary); (b) Satake objects are direct sums of shifted \(\mathrm{IC}_\nu\) with constant coefficients (semisimplicity + connected stabilizers); (c) \(\mathrm{tr}\) of such an object lies in \(\mathrm{span}\{\mathrm{tr}\,\mathrm{IC}_\nu:\nu\in\mathcal M\}\) — A4-4.3 (Frobenius on multiplicity spaces), A1-6 step 4 (BBD 5.3.8: Weil-simple constituents are scalar twists \(\mathrm{IC}_\nu\otimes\alpha\)), A3-10 step 2 (same, less explicit): equivalent, none weaker; (d) extraction of \(\mu\in\mathcal M\) from \(\mathbf 1_{\mathcal O_\mu}\in\) that span — A4-6 via the basis/coordinate functional, A3-10/A1-6 via a maximal \(\nu_0\): the latter needs only 'support in \(\overline{\mathcal O_\nu}\) and nonzero on \(\mathcal O_\nu(\mathbb F_q)\)', not unitriangularity nor one \(K\)-orbit per label (A3-10 states that hypothesis but does not use it; A1-6 uses Lang only for the basis, likewise dispensable); (e) \(\mathcal M=\Lambda_S^+\Rightarrow\mathcal C^{\mathrm{Sat}}=\mathcal C\): A4-5(iii)/A3-4(2) through Prop. I's standard generation (needs H3), A1-5(\(\Leftarrow\)) through finite length + boundedness (needs no H3).

Differences. A1-6 lists Chen–Yi Thm 81 (pointwise purity) as an assumption: not needed for this direction (cosmetic). A4-6 lists Lusztig–Kato (D4): elementary (cosmetic). A3-10 is restricted to \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) 'more generally H1–H4 + connected stabilizers + (DT)': same hypotheses. None of the three proves the converse, and all say so. Exact obligations of (iii): D0, D1, D2 (semisimplicity form), D3 (only \(x_\mu\) rational and \(\mathcal O_\mu(\mathbb F_q)\) compact open), \(G,\theta\) over \(\mathbb F_q\) with \(T\) split, \(q\) odd, \(G^\theta\) and \(Z_{G^\theta}(S)\) connected; conclusion over \(\overline{\mathbb F}_q\); Betti via D5. With disconnected \(Z\) the argument only yields 'for each \(\mu\) some \(\mathrm{IC}_{\chi,\mu}\) is a summand', not \(\mathcal C^{\mathrm{Sat}}=\mathcal C\).

Sources