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

Audit X2.7 · No \(\mathrm{SL}_2/\mathrm{SO}_2\) conclusion of the run rests on the false §8.2 remark

proved audit: correct

Second audit: correct. I searched every run node (and the root nodes) for §8.2, Audit 7.16 and the SO2-model. Setup 4.1 and Objection 4.12 misattribute 'T split' to §8.2, but they use only the rationality of S. Summary 7.25, Objection 7.31 and Summary 7.32 cite only the two bridges of Audit 7.16, which the second audit found correct. Proposition 1.7 uses Thm 81 in the split form, where §8.2 and the connectedness of K hold (θ=Ad diag(1,−1), upper triangular/diagonal, K=T). No conclusion rests on the false remark. The Summary 2.9 gap is now closed by S2-6.

Each \(\mathrm{SL}_2/\mathrm{SO}_2\) conclusion of the run is traced to its inputs. The Betti sheaf failure and the split-form function failure use neither the remark nor the gap. The ℓ-adic proofs use §8.2 only for the split form, where it holds. The one statement the run left open, the function level of the non-split \(\mathrm{SO}_2\)-form, is supplied by Proposition X2.6.

Audit X2.7.

For each conclusion of the run about \(\mathrm{SL}_2/\mathrm{SO}_2\), the field, the form, and whether it rests on the remark of Audit 7.16 ("§8.2 needs only \(T_0\) split, which the \(\mathrm{SO}_2\)-model satisfies") or on step 2 of Theorem 4.11:

Table 2. conclusion and field / form and rests on the remark?
conclusion	field / form	rests on the remark?	rests on A4-11 step 2?
Theorem 2.4 \(\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K\) (central inertia)	Betti over \(\mathbb C\)	no	no
Lemma 2.6, Proposition 2.7 exact Satake span	Betti over \(\mathbb C\)	no	no
Computation 2.8, Proposition 1.2, Prop. J(2): (SG) fails, every \(q\), every \(F\)	split form \(\mathrm{SL}_2/T\)	no	no
Proposition 1.7 ℓ-adic non-generation	split form over \(\mathbb F_q\); Chen–Yi Thm. 81	no (§8.2 holds for the split form, Correction (repair of A7-16) X2.5(b))	no
Theorem 4.11 ℓ-adic non-generation	split form	no	yes, repaired in Theorem (repair of A4-11) X2.4
Audit 7.16 "Lang's theorem applies to both forms"	both forms	its stated reason is the remark; the conclusion holds without it (Correction (repair of A7-16) X2.5(e), Lemma 4.2)	no

So no conclusion needs the false remark, and no replacement route is required. The run's only unsettled \(\mathrm{SL}_2/\mathrm{SO}_2\) statement, (SG) for the \(\mathrm{SO}_2\)-form with \(q\equiv3\bmod4\) (Summary 2.9, gap), is now decided: it fails (Proposition X2.6).

Betti. Theorem 2.4, Lemma 2.6 and Proposition 2.7 work over \(\mathbb C\), with the inertia of the central \(-I\) and the slices of Chen–Yi. There is no Frobenius, no \(\mathbb F_q\)-form and no Iwahori Weil structure, so §8.2 is not an input. Over \(\mathbb C\) the two forms coincide.

Functions. Computation 2.8(c) and Proposition 1.2 are tree and lattice computations for \(K=\mathrm{SL}_2(\mathcal O)\) on \(\mathrm{SL}_2/T\). Their only inputs are the Hecke relations and the parity of \(\varphi\), and the second audit rechecked them for \(q=2,3,4,5,7\). Proposition J(2) uses \(s=\operatorname{diag}(u,1)\) on the split form. None of these mentions Chen–Yi. Table 1 defines the row as \(\mathrm{SL}_2/\mathrm{SO}_2=\mathrm{SL}_2/T\).

ℓ-adic, Proposition 1.7. It works on \(\mathrm{SL}_2/T\) with \(\theta=\operatorname{Ad}\operatorname{diag}(1,-1)\). It uses Chen–Yi's Thm. 81, whose §8.2 hypotheses hold there for every odd \(q\) (upper triangular, diagonal; Correction (repair of A7-16) X2.5(b)). It avoids the gap of Theorem 4.11 by working with sign-isotypic parts on the orbit. Its input "the two \(K\)-orbits are exchanged by σ" is Lemma X2.1(c).

ℓ-adic, Theorem 4.11. As repaired in Theorem (repair of A4-11) X2.4, it uses no §8.2 input, and it holds for both forms.

Setup 4.1 and Objection 4.12. They attribute to §8.2 the assumption "\(T\) split", the torus of the θ-split pair. What they actually use is that \(S\), hence each \(x_\mu\), is \(\mathbb F_q\)-rational; that holds in both forms for \(\mathrm{SL}_2/\mathrm{SO}_2\), since \(S=D\) in the \(\mathrm{SO}_2\)-form. Item 3 of Objection 4.12 ("split forms exist") is correct if, for this row, the split form is chosen.

Conclusion. The false remark propagates nowhere. Its own conclusion is true by Correction (repair of A7-16) X2.5(e). The only new mathematics needed for completeness is the \(\mathrm{SO}_2\)-form at function level, which is Proposition X2.6; it is not an input of the verdict. \(\square\)

Builds on

Audit 7.16 · Audit: the comparability objection is correct — three fields, two bridges, and what each bridge needs
Theorem 4.11 · \(\SL_2/\SO_2\): the sign-twisted IC sheaves are not in the Satake span, so \(\Csat\ne\cC_K\) over \(\overline{\F}_q\)
Theorem 2.4 · \(\Csat\ne\mathcal C_K\) for \(\SL_2/\SO_2\): the sign local system on the first orbit is not generated
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 \(q\)
Proposition 1.2 · \(\SL_2/T\) versus \(\GL_2/T\): one variety, two Hecke algebras
Proposition 1.7 · \(\SL_2/\SO_2\): sign local systems never occur in Satake objects (ℓ-adic proof)
Lemma 4.2 · \(K\)-orbits on \(X(\F_q(\!(t)\!))\) via Lang's theorem: \(\Lambda_S^+\) exactly when \(Z_{G^\theta}(S)\) is connected
Objection 4.12 · Obstacles: \(\C\) versus \(\overline{\F}_q\), characteristic \(0\) in (A1) versus \(\F_q(\!(t)\!)\), Weil versus geometric, eigenvalue cancellation
Theorem (repair of A4-11) X2.4 · Repair of A4-11: \(\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}\) in both \(\mathbb F_q\)-forms, \(q\) odd
Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\); Lang’s count holds in both
Proposition X2.6 · (SG) fails for \(\mathrm{SL}_2/\mathrm{SO}(x^2+y^2)\) over every \(F\), \(q\) odd

Built on by

Summary X2.8 · Re-derivation of Summary 4.13, Summary 7.19, Summary 7.25, Summary 7.32 from the repaired steps
Re-check V2 · Re-check by V2

Sources

data.json nodes Proposition 1.2, Proposition 1.7, Theorem 2.4, Lemma 2.6, Proposition 2.7, Computation 2.8, Summary 2.9, Setup 4.1, Theorem 4.11, Objection 4.12, Audit 7.16, Audit 7.18
summary.html Table 1

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866