§7 · Agent 7 · Adversarial audit · 2026-10-17T02:55:57Z

Audit 7.2 · Audit: the \(\SL_2/\SO_2\) verdict holds at both levels; recursions and fibre counts re-derived

proved audit: correct

Second audit: correct.

Re-derives the non-generation argument, the fibre counts, the single-IC lemma and the Hecke recursions for \(\SL_2/\SO_2\). All hold; the sheaf proof is Betti over \(\C\) and needs no transfer.

Audit 7.2.

  1. Theorem 2.4 holds (severity none); its generalisation needs only a central \(z\in Z(G)\cap G^\theta\) representing a nontrivial class of some \(\pi_0\Stab(x_\mu)\), connected \(G^\theta\) (so \(\varepsilon_{\IC_0}=\mathrm{id}\)) and connected \(L^+G\)-stabilizers on \(\Gr\).
  2. Computation 2.5 holds: the fibres are the type-0 vertices of the line \(y_0A_T\) at distance \(\le2n\) from \(v_0\), and \(s_1=(3,2,1)\).
  3. Lemma 2.6 holds under the perverse t-structure and slice inputs (H) listed in Summary 2.9.
  4. Computation 2.8(c) holds, verified by hand, and is the same statement as \(\C[y^2]\subsetneq\C[y]\) in Proposition 1.2.
  5. Summary 2.9 holds: both levels fail, and the sheaf proof is Betti over \(\C\), stronger than Theorem 4.11 and Proposition 1.7.
  1. Theorem 2.4. \(\Delta(\mathcal L_1^-)\in\mathcal C_K^-\setminus\{0\}\), \(\Csat\subseteq\mathcal C_K^+\) and \(\mathcal C_K^+\cap\mathcal C_K^-=0\). Closure under retracts is correctly handled (Lemma 2.2(a)).
  2. Computation 2.5. The fibre description and \(s_1=(3,2,1)\) were re-derived. The vertex \(u(x)v_0\) with \(\val x=-1\) is adjacent to the apartment vertex \(\cO\oplus t\cO\), so \(d=1\): the labelling agrees with the pages' orbit \(\mathcal O_{\alpha^\vee}\).
  3. Lemma 2.6. The lemma "\(\nu\) finite and an isomorphism over a dense open \(\Rightarrow\nu_{*}\IC_Z=\IC_Y\)" is correct: for finite \(\nu'\), \(\nu'^!\) is left t-exact, so the \(\Hom\)-vanishing follows from the characterisation of \(j_{!*}\). The isomorphism over \(\mathcal O_{2n}\) uses: finite, injective on points and characteristic \(0\) give birationality, and \(\mathcal O_{2n}\) is smooth, hence normal, so Zariski's main theorem applies.
  4. Computation 2.8(c). With \(T=A^2-(q+1)\) and the tree counts \(Af_0=2f_0+f_1\), \(Af_1=(q-1)f_0+f_2\), \(Af_d=qf_{d-1}+f_{d+1}\), the four recursions of Computation 2.8 follow and agree with direct neighbour counts. \(T^nf_0=f_{2n}+(\text{lower})\), so \(f_1\notin\mathcal H_K\varphi_0\) for every \(q\); this is literally \(\C[y^2]\subsetneq\C[y]\) of Proposition 1.2 (see Objection 7.15).
  5. Summary 2.9. Both levels fail; the sheaf argument needs no transfer from \(\overline{\F}_q\) (Audit 7.7).

Builds on

  • Theorem 2.4 · \(\Csat\ne\mathcal C_K\) for \(\SL_2/\SO_2\): the sign local system on the first orbit is not generated
  • Computation 2.5 · Fibres of the convolution map are finite sets of tree vertices; the map is finite and an isomorphism over the top orbit
  • Lemma 2.6 · Each Satake object is a single IC sheaf: \(V_{n\alpha^\vee}\ast\IC_0\simeq\IC(\overline{\mathcal O_{2n}})\)
  • Proposition 2.7 · The Satake span is exactly the even-label, constant-coefficient part, strictly inside the even part
  • Computation 2.8 · Function level for \(\SL_2/T\): Lang count of \(K\)-orbits, the sign-class dictionary, and a Hecke dimension count for every \(q\)
  • Summary 2.9 · Verdict for \(\SL_2/\SO_2\): both levels fail, and the sheaf argument is the geometric form of Proposition J(2)

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 02:55 UTC

A2-4 to A2-8: the SL2/SO2 verdict holds at both levels; recursions and fibre counts re-derived

A2-4 (holds, severity none). \(\Delta(\mathcal L_1^-)\in\mathcal C^-\setminus0\), \(\mathcal C^{\mathrm{Sat}}\subseteq\mathcal C^+\), \(\mathcal C^+\cap\mathcal C^-=0\). Retract-closure is correctly handled (A2-2(a)). The generalisation sentence is correct: it needs only a central \(z\in Z(G)\cap G^\theta\) representing a nontrivial class in some \(\pi_0\mathrm{Stab}(x_\mu)\), connected \(G^\theta\) (so \(\varepsilon_{\mathrm{IC}_0}=\mathrm{id}\)) and connected \(L^+G\)-stabilizers on \(\mathrm{Gr}\).

A2-5 (holds). Fibre \(F_y=\{\text{type-0 vertices of the line }y_0A_T\text{ at distance}\le2n\text{ from }v_0\}\); \(s_1=(3,2,1)\) re-derived; the vertex \(u(x)v_0\) with \(\mathrm{val}\,x=-1\) is adjacent to \(\mathcal O\oplus t\mathcal O\in A_T\), so \(d=1\): labelling consistent with the pages' \(\mathcal O_{\alpha^\vee}\).

A2-6 (holds under H). The lemma '\(\nu\) finite, iso over dense open \(\Rightarrow\nu_*\mathrm{IC}_Z=\mathrm{IC}_Y\)' is correct (finite \(\nu'\) has \(\nu'^!\) left t-exact, so the \(\mathrm{Hom}\)-vanishing follows from BBD's characterisation of \(j_{!*}\)). 'Isomorphism over \(\mathcal O_{2n}\)' uses: finite + injective on points + char 0 \(\Rightarrow\) birational, and \(\mathcal O_{2n}\) smooth hence normal, so ZMT applies. H = perverse t-structure on \(\mathcal C\) with Chen–Yi's slices/finite-type models (A2-9 (ii)), as A2 states.

A2-8(c) (holds, verified by hand). With \(T=A^2-(q+1)\) and the tree counts \(A f_0=2f_0+f_1\), \(Af_1=(q-1)f_0+f_2\), \(Af_d=qf_{d-1}+f_{d+1}\): \(Tf_0=2f_0+2f_1+f_2\), \(Tf_1=2(q-1)f_0+(q-2)f_1+f_3\), \(Tf_2=q(q-1)f_0+(q-1)f_2+f_4\), \(Tf_d=q^2f_{d-2}+(q-1)f_d+f_{d+2}\): all four agree with A2-8 and with the direct neighbour counts. \(T^nf_0=f_{2n}+\)lower, so \(f_1\notin\mathcal H_K\varphi_0\) for every \(q\). This is literally the same statement as A1-2's \(\mathbb C[y^2]\subsetneq\mathbb C[y]\) (A7-15).

A2-9 (holds). Both levels fail; the sheaf proof is Betti over \(\mathbb C\) and needs no \(\overline{\mathbb F}_q\) transfer — stronger than A4-11 and A1-7 (A7-7).

Sources