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

Audit 7.4 · Audit of the thick-span criterion and the one-way transfer theorem; the converse is correctly refused

proved audit: minor defect

Minor defect (second audit). Its verdicts on Lemma 4.5, Theorem 4.6 and Proposition 4.7 are right, and its list of obligations is accurate (Lemma 1.5 indeed avoids H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf.). Two loose statements: (1) 'Csat = C is "every row of (m_{νλ}) nonzero"' should refer to the multiplicity spaces E_{μ,•}(λ) — rows of M^(1) can vanish by cancellation, which is why Proposition 4.7 needs some n; (2) the aside that in Sakellaridis's Example 8.0.1 and in E6/F4 'the multiplicities, as polynomials in q^{1/2}, stay nonzero' was not known for either at the time (the E6/F4 sheaf level was open until Summary 8.10, and the sheaf level of Example 8.0.1 was never studied).

An in-run check confirming the Serre-subcategory criterion for the thick span, the transfer theorem "(SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\)" with its exact obligations, and the refusal of the converse.

Audit 7.4.

  1. Lemma 4.5 holds under H1–H3Hypotheses 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., 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).. Its part 3 (\(\Leftarrow\)) uses Proposition I's standard-generation lemma, which needs H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf. (every object of \(\cC_\mu\) lies in \(\Thick(\underline k_{\cO_\mu})\)); the finite-length-and-boundedness version of Lemma 1.5 avoids H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf.. "Every IC is a summand \(\Leftrightarrow\) the thick span is \(\cC_K\)" is correctly related through the Serre subcategory, not through the Grothendieck group.
  2. Theorem 4.6 holds. Exact obligations: 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. (orbits, closures, support bound and stabilizer lemma over \(\overline{\F}_q\), \(p\ne2\)); 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 and the trace formula on the fibre; a Frobenius-compatible perverse t-structure with finite-length heart); 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 in the form "Satake objects are finite direct sums of shifted \(\IC_\nu\) with constant coefficients" (semisimplicity; weights are not used here); D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q. (Lang; Theorem 4.6 uses the full "one \(K\)-orbit per label", while Proposition 3.10 and Theorem 1.6 need only that \(x_\mu\) is rational and \(\cO_\mu(\F_q)\) compact open); 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). elementary (Audit 7.3). Plus: \(G,\theta\) over \(\F_q\), \(T\) split, \(q\) odd, \(G^\theta\) and \(Z_{G^\theta}(S)\) connected. Betti over \(\C\): 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)..
  3. Proposition 4.7 holds; the refusal of the converse is right.
  1. Thickness. The retract step is right: \({}^pH^i\) of a retract is a subobject, and Serre subcategories are closed under subobjects.
  2. Transfer. In step 2 of Theorem 4.6, (SG) makes the coordinate functional of \(\tr\IC_\mu\) nonzero on some \(\tr(V_\lambda\ast\IC_0)\), i.e. \(m^{(1)}_{\mu\lambda}\ne0\). Step 3 uses only the trivial direction "\(E=0\Rightarrow m^{(n)}=0\)" (Lemma 4.4, part 4(i)): the direction is correct, there is no cancellation issue, and no purity is needed for this direction.
  3. Converse. \(\Csat=\cC_K\) is "every row of \((m_{\nu\lambda})\) is nonzero", while (SG) is "the columns span". Sakellaridis's Example 8.0.1 and now Theorem 6.6 (\(E_6/F_4\)) are exactly the shape in which a Weyl-conjugacy coincidence of \(q\)-points kills surjectivity while the multiplicities, as polynomials in \(q^{1/2}\), stay nonzero.

Builds on

  • Lemma 4.5 · \(\Csat=\cC_K\) if and only if every \(\IC_\mu\) is a direct summand of a Satake object
  • Theorem 4.6 · Theorem: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) over \(\overline{\F}_q\)
  • Proposition 4.7 · The converse fails in general: \(\Csat=\cC_K\) gives detection by Hecke translates, not spherical generation

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

A4-5 to A4-7: thick-span criterion and the one-way transfer theorem hold; the converse correctly refused

A4-5 (holds under H1–H3, D1, D2). \(\mathcal D_\Sigma\) thick: the retract step is right (\({}^p\mathcal H^i\) of a retract is a subobject, Serre subcategories are closed under subobjects). (iii)\(\Leftarrow\) uses Prop. I's standard-generation lemma, which needs H3 (every object of \(\mathcal C_\mu\) lies in \(\mathrm{Thick}(\underline k_{\mathcal O_\mu})\)); A1-5's version (finite length + boundedness) avoids H3. Checklist (6): 'every IC a summand \(\Leftrightarrow\) Thick \(=\mathcal C\)' is correctly related through the Serre subcategory, not through \(K_0\).

A4-6 (holds). Step 2: (SG) \(\Rightarrow\) the coordinate functional of \(\mathrm{tr}\,\mathrm{IC}_\mu\) is nonzero on some \(\mathrm{tr}(V_\lambda\ast\mathrm{IC}_0)\), i.e. \(m^{(1)}_{\mu\lambda}\ne0\). Step 3 uses only the trivial direction '\(E=0\Rightarrow m^{(n)}=0\)' (A4-4.4(i)); checklist (8): direction correct, no cancellation issue, no purity needed for this direction. Exact obligations: D0 (orbits, closures, support bound, stabilizer lemma over \(\overline{\mathbb F}_q\), \(p\ne2\)); D1 (proper base change + trace formula on the fibre; Fr-compatible perverse t-structure with finite-length heart); D2 only in the form 'Satake objects are finite direct sums of shifted \(\mathrm{IC}_\nu\) with constant coefficients' (semisimplicity; weights unused here); D3 (Lang; A4-6 uses the full 'one \(K\)-orbit per label', A3-10/A1-6 need only \(x_\mu\) rational and \(\mathcal O_\mu(\mathbb F_q)\) compact open); D4 elementary (A7-3). Plus: \(G,\theta\) defined over \(\mathbb F_q\), \(T\) split, \(q\) odd, \(G^\theta\) and \(Z_{G^\theta}(S)\) connected. Betti over \(\mathbb C\): D5.

A4-7 (holds). The refusal of the converse is right: \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) is 'every row of \((m_{\nu\lambda})\) nonzero', (SG) is 'columns span'; Example 8.0.1 and now A6-6 (\(E_6/F_4\)) are exactly the shape where a Weyl-conjugacy coincidence of \(q\)-points kills surjectivity while multiplicities as polynomials in \(q^{1/2}\) stay nonzero.

Sources