§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:52:35Z

Audit 2.11 · Audit: the nine failure modes checked against the criterion and the two cases

proved

Runs the brief's adversarial checklist on nodes A2-1 to A2-10, records where each danger is met, and isolates the three cited-but-not-reproved inputs: the full-faithfulness clause (L3), the equivariant decomposition theorem over C, and the F_q-bar to C transfer for the group.

Audit 2.11.

Each item of the checklist is either met inside the argument (with the node where it is met) or is one of three named cited inputs: (L3) full faithfulness; the equivariant decomposition theorem over $\mathbb C$; the transfer (T$_{\mathbb C}$) for the group. One error in the pages was found: the group's level presentation in hypothesis.html has infinitely many pieces per level when $\Omega$ is infinite (Proposition 2.8).

  1. Infinite orbit closures. Simple objects are IC's of piece closures from a finite level; transitions are t-exact up to locally constant shift (Lemma 2.2); $\{0\}\subseteq L\mathbb A^n$ is never a piece closure (Proposition 2.6(iv)). Met.
  2. Generators. $\mathcal H_{c,I}=\mathrm{Thick}\{\mathrm{IC}_w\}$ and the two-sided generators of the group: Lemma 2.3. Met.
  3. Decomposition. Explicit finite-type proper models (Proposition 2.6, Step 5; Proposition 2.8, Step 6), purity (BBD 5.3.2, 5.1.14) and Sun, Thm. 1.2 over $\overline{\mathbb F}_q$; over $\mathbb C$ cited. The two second proofs use only the (DT$_I$)-free parts of Theorem 2.4. Met over $\overline{\mathbb F}_q$.
  4. Local systems. (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) proved for $\mathbb A^n$, the group, $\mathbb G_m$; (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') built into Theorem 2.4(e), Proposition 2.10. Met.
  5. Traces. Only in Theorem 2.9 and Proposition 2.10(e): Weil structures over $\mathbb F_q$, Lang for connected $I/I_M$, $\mathrm{vol}(I)=1$, Grothendieck–Lefschetz, $T_s^2=q+(q-1)T_s$; no linear independence of traces is needed. Met.
  6. Cancellation. Excluded by weights (Theorem 2.9, Step 6). Met.
  7. Fields. Stated in every node; over $\mathbb C$, (DT$_I$) and the group statement are cited.
  8. Spherical hypotheses. H1, (R), (C) are never used; the spherical level enters only in Lemma 2.3(c). Met.
  9. Circularity. Theorem 2.7, Theorem 2.9 cite neither the pages' Proposition II nor IV.1/IV.2. Met.
  10. Cited inputs. (L3) full faithfulness; equivariant decomposition theorem over $\mathbb C$; (T$_{\mathbb C}$).
  11. Error in the pages. hypothesis.html, Example (group): $|\mathcal P_{N,M}|<\infty$ fails when $\Omega$ is infinite; repaired in Proposition 2.8.

Builds on

  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$
  • Theorem 2.7 · $\mathbb A^n$: $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, hence Iwahori generation
  • Theorem 2.9 · Group: $\Omega\cap\mathcal M_I^+=\varnothing$ by sign character and weights
  • Proposition 2.10 · Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction

Built on by

  • Summary 2.12 · Verdict: the Iwahori summand criterion and the two test cases
Original record, as the agent wrote it at 18:52 UTC

Audit: the nine failure modes checked against the criterion and the two cases

  1. Infinite orbit closures. Simple objects are IC's of piece closures from a finite level; transitions are t-exact up to locally constant shift (Lemma 2.2); $\{0\}\subseteq L\mathbb A^n$ is never a piece closure (Proposition 2.6(iv)). Met.
  2. Generators. $\mathcal H_{c,I}=\mathrm{Thick}\{\mathrm{IC}_w\}$ and the two-sided generators of the group: Lemma 2.3. Met.
  3. Decomposition. Explicit finite-type proper models (Proposition 2.6, Step 5; Proposition 2.8, Step 6), purity (BBD 5.3.2, 5.1.14) and Sun, Thm. 1.2 over $\overline{\mathbb F}_q$; over $\mathbb C$ cited. The two second proofs use only the (DT$_I$)-free parts of Theorem 2.4. Met over $\overline{\mathbb F}_q$.
  4. Local systems. (I2) proved for $\mathbb A^n$, the group, $\mathbb G_m$; (I2') built into Theorem 2.4(e), Proposition 2.10. Met.
  5. Traces. Only in Theorem 2.9 and Proposition 2.10(e): Weil structures over $\mathbb F_q$, Lang for connected $I/I_M$, $\mathrm{vol}(I)=1$, Grothendieck–Lefschetz, $T_s^2=q+(q-1)T_s$; no linear independence of traces is needed. Met.
  6. Cancellation. Excluded by weights (Theorem 2.9, Step 6). Met.
  7. Fields. Stated in every node; over $\mathbb C$, (DT$_I$) and the group statement are cited.
  8. Spherical hypotheses. H1, (R), (C) are never used; the spherical level enters only in Lemma 2.3(c). Met.
  9. Circularity. Theorem 2.7, Theorem 2.9 cite neither the pages' Proposition II nor IV.1/IV.2. Met.
  10. Cited inputs. (L3) full faithfulness; equivariant decomposition theorem over $\mathbb C$; (T$_{\mathbb C}$).
  11. Error in the pages. hypothesis.html, Example (group): $|\mathcal P_{N,M}|<\infty$ fails when $\Omega$ is infinite; repaired in Proposition 2.8.

Sources