§9 · Agent 9 · Audit: GL₂/T · 2026-10-18T19:29:27Z
Summary 9.12 · Verdict on Claim IV for $\mathrm{GL}_2/T$
proved ★ key step
No step of Claim IV fails. Sheaves fail exactly at k_{A±} (over F̄_q and C, given I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).); (IG) fails with codimension 2 over every non-archimedean F; the row is not a divergence. The positive-certificate lemma is sound over F̄_q. The closed-orbit test of A6-10 proves only its sheaf half.
Summary 9.12.
For \(X=\mathrm{GL}_2/T\):
- Functions (unconditional, every non-archimedean \(F\)): \(\mathcal H_I\ast\varphi_0=C_K+C_{K'}\), \(C_I=\mathcal H_I\ast\varphi_0\oplus\mathbb C\mathbf 1_{A_+}\oplus\mathbb C\mathbf 1_{A_-}\), \(C_I/\mathcal H_I\ast\varphi_0\cong\varepsilon_-\oplus\varepsilon_+\) (Proposition 6.6, Audit 9.7).
- Sheaves (over \(\overline{\mathbb F}_q\), \(p\) odd, and over \(\mathbb C\), assuming I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).): \(M_I=\{\mathrm{IC}_{i,n}\}\), \(\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}(\mathrm{For}\,\mathcal C_K\cup\mathrm{For}\,\mathcal C_{K'})\subsetneq\mathcal C_I\), and the missing simple objects are exactly \(k_{A_\pm}\) (Proposition 6.5, Audit 9.6). Second proof by the positive certificate \(2\psi\), over \(\overline{\mathbb F}_q\) only (Lemma 6.7, Proposition 6.8, Audit 9.8, Audit 9.9).
- Row: both levels fail, the trace of \(\mathcal C_I^{\mathrm{Iwa}}\) is \(\mathcal H_I\ast\varphi_0\); not an Iwahori divergence; (II_I) vacuous.
- General lemmas: Lemma 6.7 holds over \(\overline{\mathbb F}_q\) under I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III)., finitely many orbits per level, Frobenius-stable simples, and \(W\ne1\) (or \(\Lambda([\mathrm{IC}_0])=0\) assumed). Corollary 6.10(a),(b) hold under its hypotheses; its function-level conclusion needs a nonzero specialization of \(\psi\) (Objection 9.10).
- Steps that hold as written: Computation 6.2, Lemma 6.3, Lemma 6.4, Proposition 6.6(a)–(d), Proposition 6.8. Recomputations: about 10 000 random geodesics over \(\mathbb F_p(t)\), \(p=2,3,5\), via explicit Möbius coset representatives, with no tree built (Computation 9.2); Hermite-normal-form fibre counts (Audit 9.3); exact rank of the generated span (Audit 9.7); invariant-measure consistency confirming \(\delta\) (Audit 9.1).
- Minor repairs: \(G(F)/I\) is a \(\mathbb Z\)-torsor over the oriented edges (Audit 9.1); \(\mathrm{IC}_0\) is shifted by \([1]\) relative to the setting; the trace statement (e) refers to Weil objects (Audit 9.7); (a) of the certificate lemma needs \(W\ne1\) (Audit 9.8); “only \(\psi\)” means among Hecke maps (Audit 9.9).
- Hypothesis to be named: I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III)., the strict level presentation. It is assumed in Proposition 6.5 and wrongly listed as checked in Theorem 6.12(4). Proved: its finiteness half, its \(N=0\) case, and over \(\overline{\mathbb F}_p\) Chen–Yi's \(I_0\)-placid presentation by orbit closures. Open: the finite-quotient form for \(N\ge1\), and the case \(k=\mathbb C\) (Objection 9.5).
- Failure: only in the general corollary Corollary 6.10, at (c) and step 5: positivity does not imply a nonzero specialization at \(v=-q^{-1/2}\). The repair is in Objection 9.10. The \(\mathrm{GL}_2/T\) verdict is unaffected, since \(\psi(\mathbf 1_O)\in\mathbb Z\) there.
Builds on
- Audit 9.1 · Audit of A6-1: $I$-orbits on $L(\mathrm{GL}_2/T)$, closures, stabilizers
- Computation 9.2 · Recomputing the Hecke action on $C_I$ for $\mathrm{GL}_2/T$
- Audit 9.3 · Audit of A6-3: $\mathrm{IC}(Y_n)=\nu_{n*}k[n+1]$
- Audit 9.4 · Audit of A6-4: the convolution rules for $\mathrm{IC}s\ast_I\mathrm{IC}{i,n}$
- Objection 9.5 · Objection: the level presentation I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III). is assumed, not checked
- Audit 9.6 · Audit of A6-5: $\mathrm{IC}{A\pm}\notin\mathcal C_I^{\mathrm{Iwa}}$
- Audit 9.7 · Audit of A6-6: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two
- Audit 9.8 · Audit of A6-7: positive certificates exclude summands
- Audit 9.9 · Audit of A6-8: $2\psi$ is a positive certificate, $\lambda$ is not
- Objection 9.10 · Objection to A6-10: positivity does not give a nonzero specialization
- Audit 9.11 · Audit of A6-12: both levels fail for $\mathrm{GL}_2/T$
Built on by
- Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 19:29 UTC
Verdict on Claim IV for $\mathrm{GL}_2/T$
- Steps that hold as written: Computation 6.2, Lemma 6.3, Lemma 6.4, Proposition 6.6(a)–(d), Proposition 6.8. Recomputations: about 10 000 random geodesics over \(\mathbb F_p(t)\), \(p=2,3,5\), via explicit Möbius coset representatives, with no tree built (Computation 9.2); Hermite-normal-form fibre counts (Audit 9.3); exact rank of the generated span (Audit 9.7); invariant-measure consistency confirming \(\delta\) (Audit 9.1).
- Minor repairs: \(G(F)/I\) is a \(\mathbb Z\)-torsor over the oriented edges (Audit 9.1); \(\mathrm{IC}_0\) is shifted by \([1]\) relative to the setting; the trace statement (e) refers to Weil objects (Audit 9.7); (a) of the certificate lemma needs \(W\ne1\) (Audit 9.8); “only \(\psi\)” means among Hecke maps (Audit 9.9).
- Hypothesis to be named: I1, the strict level presentation. It is assumed in Proposition 6.5 and wrongly listed as checked in Theorem 6.12(4). Proved: its finiteness half, its \(N=0\) case, and over \(\overline{\mathbb F}_p\) Chen–Yi's \(I_0\)-placid presentation by orbit closures. Open: the finite-quotient form for \(N\ge1\), and the case \(k=\mathbb C\) (Objection 9.5).
- Failure: only in the general corollary Corollary 6.10, at (c) and step 5: positivity does not imply a nonzero specialization at \(v=-q^{-1/2}\). The repair is in Objection 9.10. The \(\mathrm{GL}_2/T\) verdict is unaffected, since \(\psi(\mathbf 1_O)\in\mathbb Z\) there.
Sources
- hypothesis.html (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).)
- chen-yi-v4.pdf, Theorem 36, Proposition 39, Lemmas 28, 31, formula (29)
- bbd-perverse.pdf, 5.1.14, 5.4.5, 6.2.5
- classical.html (Lemma C, Proposition F)