§10 · Root · root: registry, redirects, synthesis · 2026-10-18T20:38:00Z

Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels

proved ★ key step

The root combines the surviving claims into five propositions: the Iwahori summand criterion, functions ⇒ sheaves, a second proof of rows 1–2 (and of \(\mathbb G_m\)), the row \(\mathrm{GL}_2/T\), and what remains possible for a divergence.

Verdict after the three audits. Every audited claim survives; the repairs are minor and are built into the propositions.

  1. Iwahori summand criterion (Proposition I). Over \(\overline{\mathbb F}_q\) and \(\mathbb C\), under the level axioms, 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). and I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.: \(\operatorname{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\operatorname{Thick}(\mathcal M_I^+)\), and under the decomposition \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I\) (Theorem 2.4, Summary 7.11). The decomposition must be the equivariant one (Counterexample 7.3); it holds over \(\overline{\mathbb F}_q\) from a finite-type proper model (Lemma 2.5).
  2. Functions ⇒ sheaves (Proposition II). Over \(\overline{\mathbb F}_q\), (IG) over \(\mathbb F_q(\!(t)\!)\) puts every simple object among the summands (Theorem 3.6, Audit 7.7); with disconnected stabilizers, (IG) over every \(\mathbb F_{q^n}(\!(t)\!)\) (Proposition 3.7). Not over \(\mathbb C\). The converse fails in general (Proposition 3.10).
  3. Rows 1–2 again (Proposition III). The group: no \(\Delta_\omega\) is reached, so \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) over \(\overline{\mathbb F}_q\) (Theorem 4.5, Theorem 2.9, Audit 8.5); levels truncated in \(\Omega\) (Audit 8.1); (IG) fails (Corollary 4.6). \(\mathbb A^n\): \(\mathcal S_I=\{S_k\}=\Omega\cdot\mathrm{IC}_0\), both levels hold (Theorem 5.5, Proposition 5.6, Summary 8.13); \(\mathbb G_m\) likewise (Proposition 2.10).
  4. \(\mathrm{GL}_2/T\) (Proposition IV). Functions fail with codimension two over every \(F\) (Proposition 6.6); sheaves fail exactly at \(\underline k_{A_\pm}\), 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). (Proposition 6.5, Objection 9.5, Summary 9.12). Not a divergence.
  5. Divergences (Proposition V). By 2, over \(\mathbb F_q(\!(t)\!)\) only "sheaves generate, functions do not" can occur. A Hecke functional that kills the generators and is nonnegative on every IC class excludes summands (Lemma 6.7, Audit 9.8); the function half of the closed-orbit test needs a nonzero specialization (Objection 9.10).

No decided row of Table 1 diverges. \(E_6/F_4\) fails at function level over \(\mathbb F_q(\!(t)\!)\) and \(p\)-adic \(F\), \(p\) odd, since (IG) implies (SG) and (SG) fails there (Summary 3.12); its sheaf level, and that of the other symmetric rows, is open.

Builds on

  • Lemma 2.2 · Lemma (PH$_I$): perverse hearts glue; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$
  • Lemma 2.3 · Lemma: generators $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; spherical summands persist
  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$
  • Lemma 2.5 · Lemma: a finite-type proper model gives (DT$_I$) over $\overline{\mathbb F}_q$
  • Lemma 3.4 · Unitriangular IC-trace basis of $C_I$ and the decomposition matrix $m_{S,w}$
  • Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$
  • Proposition 3.7 · Disconnected stabilizers: (IG)$_n$ for all $n$ gives $\mathcal M_I=\mathrm{Irr}$
  • Proposition 3.10 · Sheaves to functions: non-zero rows only; (WT) gives (IG)
  • Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Corollary 4.6 · (IG) fails by the same character; (II$_I$) holds for the group
  • Proposition 4.7 · $\mathcal M_I$ for the group: top cells in, $\Omega$ out, exact in rank one
  • Theorem 2.9 · Group: $\Omega\cap\mathcal M_I^+=\varnothing$ by sign character and weights
  • Lemma 5.2 · Simple objects of $\mathcal C_I$: $S_k=\underline k_{\tau^k\mathcal O^n}[-k]$
  • Computation 5.3 · $\mathrm{IC}_{\tau^k}\ast_I\mathrm{IC}_0=S_k[k]$ and the simple reflections
  • Theorem 5.5 · $\mathbb A^n$: $\mathcal M_I=\{S_k\}$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$
  • Proposition 5.6 · (IG) for $\mathbb A^n$: $T_{\tau^k}\ast\varphi_0=\mathbf 1_{\tau^k\mathcal O^n}$
  • Proposition 2.10 · Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction
  • Proposition 6.5 · Sheaves fail for $\mathrm{GL}2/T$: $\mathrm{IC}{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}$
  • Proposition 6.6 · Functions: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two
  • Lemma 6.7 · Sheaf-level Corollary D: positive Steinberg certificates exclude summands
  • Proposition 6.8 · Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not
  • Corollary 6.10 · Closed-orbit test: sign-type Hecke maps nonnegative on closed orbits
  • Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument
  • Summary 3.12 · Verdict: (II$_I$) proved; rows 1–3 agree; one divergence type excluded
  • Counterexample 7.3 · Objection to A5-7 step 3 and A5-4 (DT over $\mathbb C$): BBD 5.4.5 splits only non-equivariantly
  • Summary 7.11 · Verdict of the audit: (I$_I$) over $\overline{\mathbb F}_q$ and $\mathbb C$, (II$_I$) over $\overline{\mathbb F}_q$ only, converses as stated
  • Audit 8.1 · Audit of the group set-up (A4-1, A2-8): pieces, stabilizers, generators, levels
  • Audit 8.5 · Audit of the main group theorem (A4-5, A2-9): no $\Delta_\omega$ constituent
  • Summary 8.13 · Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$
  • 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
  • Objection 9.10 · Objection to A6-10: positivity does not give a nonzero specialization
  • Summary 9.12 · Verdict on Claim IV for $\mathrm{GL}_2/T$

Built on by

  • —
Original record, as the agent wrote it at 20:38 UTC

Verdict after the audits

Verdict after the three audits. Every audited claim survives; the repairs are minor and are built into the propositions.

  1. Iwahori summand criterion (Proposition I). Over \(\overline{\mathbb F}_q\) and \(\mathbb C\), under the level axioms, I1 and I2: \(\operatorname{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\operatorname{Thick}(\mathcal M_I^+)\), and under the decomposition \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I\) (Theorem 2.4, Summary 7.11). The decomposition must be the equivariant one (Counterexample 7.3); it holds over \(\overline{\mathbb F}_q\) from a finite-type proper model (Lemma 2.5).
  2. Functions ⇒ sheaves (Proposition II). Over \(\overline{\mathbb F}_q\), (IG) over \(\mathbb F_q(\!(t)\!)\) puts every simple object among the summands (Theorem 3.6, Audit 7.7); with disconnected stabilizers, (IG) over every \(\mathbb F_{q^n}(\!(t)\!)\) (Proposition 3.7). Not over \(\mathbb C\). The converse fails in general (Proposition 3.10).
  3. Rows 1–2 again (Proposition III). The group: no \(\Delta_\omega\) is reached, so \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) over \(\overline{\mathbb F}_q\) (Theorem 4.5, Theorem 2.9, Audit 8.5); levels truncated in \(\Omega\) (Audit 8.1); (IG) fails (Corollary 4.6). \(\mathbb A^n\): \(\mathcal S_I=\{S_k\}=\Omega\cdot\mathrm{IC}_0\), both levels hold (Theorem 5.5, Proposition 5.6, Summary 8.13); \(\mathbb G_m\) likewise (Proposition 2.10).
  4. \(\mathrm{GL}_2/T\) (Proposition IV). Functions fail with codimension two over every \(F\) (Proposition 6.6); sheaves fail exactly at \(\underline k_{A_\pm}\), given I1 (Proposition 6.5, Objection 9.5, Summary 9.12). Not a divergence.
  5. Divergences (Proposition V). By 2, over \(\mathbb F_q(\!(t)\!)\) only "sheaves generate, functions do not" can occur. A Hecke functional that kills the generators and is nonnegative on every IC class excludes summands (Lemma 6.7, Audit 9.8); the function half of the closed-orbit test needs a nonzero specialization (Objection 9.10).

No decided row of Table 1 diverges. \(E_6/F_4\) fails at function level over \(\mathbb F_q(\!(t)\!)\) and \(p\)-adic \(F\), \(p\) odd, since (IG) implies (SG) and (SG) fails there (Summary 3.12); its sheaf level, and that of the other symmetric rows, is open.

Sources