§12 · Agent X2 · SL₂/SO₂ traces · 2026-10-17T20:28:35Z

Summary X2.8 · Re-derivation of Summary 4.13, Summary 7.19, Summary 7.25, Summary 7.32 from the repaired steps

proved audit: correct ★ key step

Second audit: correct. Each input's SL2/SO2 content traces to Theorem 2.4 (Betti), to Computation 2.8/A1-2 (split-form functions, every q and every F), or to Theorem 4.11 as repaired in S2-4. None uses the Audit 7.16 remark. Its verdict 'holds as stated' is right for all four inputs. The second audit's unrelated minor defects in those inputs (the Objection 7.11 repair, the Conjecture 6.8 misattribution, Summary 4.13's wording) are outside this target and remain as graded.

After the repairs, every \(\mathrm{SL}_2/\mathrm{SO}_2\) entry of the four verdict inputs is re-derived. The sheaf cell comes from Theorem (repair of A4-11) X2.4 (ℓ-adic) or Theorem 2.4 (Betti); the function cell comes from the split form, and now also from the \(\mathrm{SO}_2\)-form (Proposition X2.6). All four inputs hold as stated; for Summary 7.32 the phrase “every \(q\)” refers to the split form, as Table 1 defines the row.

Summary X2.8.

Repaired steps. Theorem 4.11 holds after two named changes (Theorem (repair of A4-11) X2.4): σ is the \(\mathrm{PGL}_2(\mathcal O_n)\)-action of Lemma X2.1, and the invariance in step 2 is supplied by the Weil inertia lemma Lemma X2.2, \(\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\). This is Correction 10.8, now confirmed in both forms. The remark of Audit 7.16 is false for \(q\equiv3\bmod4\): Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\), which is exactly when that form is \(\mathbb F_q\)-isomorphic to the split form (Correction (repair of A7-16) X2.5, confirming Correction 10.5). Its conclusion is true, and nothing rests on it (Audit X2.7).

Inputs.

The verdict's \(\mathrm{SL}_2/\mathrm{SO}_2\) row, "both levels fail, no divergence", stands in both \(\mathbb F_q\)-forms.

  1. Summary 4.13 (path Theorem 4.11 → Summary 4.13). Its standing setting is: \(G^\theta\) connected, over \(\mathbb F_q\), \(q\) odd, \(T\) split, modulo D0–D4The 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., and 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 \(\mathbb C\). Its \(\mathrm{SL}_2/\mathrm{SO}_2\) sheaf entry cites Theorem 4.11 only, which Theorem (repair of A4-11) X2.4 re-proves under 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. for every odd \(q\) and both forms. The Betti statement is independent (Theorem 2.4). The function entry over \(\mathbb F_q(\!(t)\!)\) cites Proposition J (split form) and Lemma 4.2 (orbit count). The \(\mathrm{SO}_2\)-form satisfies the standing hypothesis "\(T\supseteq S\) split", since \(S=D\). For it, the count of Lemma 4.2 holds (Correction (repair of A7-16) X2.5(e)) but Proposition J's \(s\)-conjugation does not apply verbatim; Proposition X2.6 (with \(\sigma\) of Lemma X2.1, or the two \(\mathrm{SL}_2(F)\)-orbits) gives the failure. The entry is unchanged, and so is the answer paragraph ("where (SG) fails … the sheaf level fails too").
  1. Summary 7.19 (paths Theorem 4.11 → Audit 7.7 → Summary 7.19 and Audit 7.16 → Summary 7.19). Audit 7.7 said that Theorem 4.11 and Proposition 1.7 "use the same inputs". After the repair, Theorem 4.11 uses in addition the inertia lemma Lemma X2.2, derived from 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.. Audit 7.7's conclusions — purity is needed in the trace arguments, Theorem 2.4 needs none, all three proofs agree on the witnesses \(\mathrm{IC}_{\mathrm{sgn},m}\) — are unaffected. Summary 7.19(i) lists exactly "Weil structures over all \(\mathbb F_{q^r}\), the decomposition theorem and purity", which Theorem (repair of A4-11) X2.4 uses. Its function sentence names the split model and does not use the §8.2 remark (Audit X2.7). The rest of Summary 7.19 (GL₄/Sp₄, transfer theorem, \(E_6/F_4\)) does not involve \(\mathrm{SL}_2/\mathrm{SO}_2\).
  1. Summary 7.25 (paths through Summary 7.19). Function level: "(SG) fails for \(\mathrm{SL}_2/T\), every \(q\)" (Computation 2.8, Proposition 1.2). Sheaf level: "fails for \(\mathrm{SL}_2/\mathrm{SO}_2\) (Betti, unconditional, Theorem 2.4)". Divergence: "for \(\mathrm{SL}_2/\mathrm{SO}_2\) both fail". The hypotheses paragraph says \(\mathrm{SL}_2/\mathrm{SO}_2\) is decided in Betti directly. None of these uses the false remark, and the "two bridges of Audit 7.16" (spreading out \(\overline{\mathbb F}_p\leftrightarrow\mathbb C\); Theorem 8.0.2 in characteristic \(p\)) are the parts of Audit 7.16 the second audit found correct. Additionally, "both fail" is now true in both \(F\)-forms (Proposition X2.6).
  1. Summary 7.32 (paths through Summary 7.19). Conclusion 1 rests on Theorem 2.4 (sheaves) and on Computation 2.8(c)/Proposition 1.2 (functions, every \(q\) including even, every \(F\), for \(\mathrm{SL}_2/T\)). Table 1 writes the row as "\(\mathrm{SL}_2/\mathrm{SO}_2=\mathrm{SL}_2/T\)", so the statement holds as written. For the non-split form \(\mathrm{SO}(x^2+y^2)\), Proposition X2.6 proves failure for every odd \(q\); residue characteristic \(2\) is not covered there and is outside the symmetric-space setting (\(p\ne2\)). The sheaf column entry "\(\ne\) for \(\mathrm{SL}_2/\mathrm{SO}_2\)" and the answer ("no row has (SG) holding and \(\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K\)") are unaffected.
  1. Fields. Each conclusion stays in its field. ℓ-adic sheaves over \(\overline{\mathbb F}_q\) against functions over \(\mathbb F_{q^n}(\!(t)\!)\) give Theorem (repair of A4-11) X2.4; Betti over \(\mathbb C\) gives Theorem 2.4; functions over every local \(F\) give Computation 2.8, Proposition 1.2 and Proposition X2.6. No analogue at one level certifies an inference at the other. \(\square\)

Builds on

Built on by

  • Re-check V2 · Re-check by V2
  • Summary 9.6 · Verdict after the repairs: every input held or repaired; \(E_6/F_4\) still the one divergence

Sources