§7 · Agent 7 · Audit: criterion and bridge · 2026-10-18T19:29:01Z

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

proved ★ key step

Every audited step survives, some with minor repairs: the Iwahori summand criterion (both versions) over $\overline{\mathbb F}_q$ and $\mathbb C$; functions $\Rightarrow$ sheaves (both versions, plus disconnected stabilizers) over $\overline{\mathbb F}_q$ with $\mathbb F_q((t))$; no general sheaves $\Rightarrow$ functions implication.

Summary 7.11.

Claim I (Iwahori summand criterion) survives, version 1 (A2-1–A2-5) as stated and version 2 (A5-4) with (L) strengthened, over $k=\overline{\mathbb F}q$ ($\overline{\mathbb Q}\ell$) and $k=\mathbb C$ (Betti), under: (L1) affine-space transitions in $M$; (L2) Cartesian closed embeddings in $N$; (L3) $I/I_{N,M}$ connected, $I_{N,M}$ pro-unipotent (relative quotients unipotent), transitions fully faithful on $D^b$; (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; (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) connected stabilizers, or (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') with local systems counted at every level; every $G_w=\mathrm{IC}w\ast_I\mathrm{IC}_0\in\mathcal C_I$ (two-sided generators $\mathrm{IC}{w_1}\ast c\ast\mathrm{IC}{w_2}$ for the group). Then unconditionally $\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D{\mathcal M_I^+}$, and under (DT$_I$): $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}(\mathcal M_I)$, $S\in\mathcal C_I^{\mathrm{Iwa}}\iff S\in\mathcal M_I$, $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I$. (DT$_I$) holds over $\overline{\mathbb F}_q$ given a proper finite-type model (FM$_w$) with $r$ smooth (Sun, Thm 1.2), verified for $\mathbb A^n$ (lattice bundle) and the group (twisted product); over $\mathbb C$ it is the cited Bernstein–Lunts equivariant decomposition theorem.

Claim II (functions $\Rightarrow$ sheaves) survives over $\overline{\mathbb F}q$ with $F=\mathbb F_q((t))$, version 1 (A3-1–A3-6) as stated and version 2 (A5-7) with the equivariant splitting (Sun, Thm 3.12) and (Lin) over every $\mathbb F{q^r}$, under (L1) (including the topology condition (d) and $I_{N,M}$ connected), (L2) ($\mathrm{Fr}$-stable orbits, connected stabilizers), (D1) bounded convolution, (D2) Lang and Grothendieck–Lefschetz, (D3) the equivariant decomposition: (IG) $\Rightarrow\mathcal M_I=\mathrm{Irr}\,\mathcal P_I\Rightarrow\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$; pointwise, $\mathbf 1_{P(\mathbb F_q)}\in\mathcal H_I\ast\varphi_0\Rightarrow\mathrm{IC}(P)\in\mathcal C_I^{\mathrm{Iwa}}$. With disconnected stabilizers (A3-7): (IG)$_n$ for all $n$ $\Rightarrow\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$; (IG)$_1$ sees only $\mathrm{Fr}$-stable simple objects. Not proved over $\mathbb C$.

Converses (A3-10, A3-11) survive as stated: sheaf generation gives non-zero rows of some $m^{(n)}$, not (IG); (WT) is a common sufficient condition; generation is an IC-coordinate test over all $\mathbb F_{q^n}$; for the group, $\mathrm{IC}_\omega\notin\mathcal M_I$ for every $\omega\in\Omega$ by sign functionals, hence $\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}$ over $\overline{\mathbb F}_q$.

Repairs required (none changes a statement's substance).

  1. A5-7 step 3 and A5-4 (over $\mathbb C$): equivariant splitting from Sun, Thm 3.12/1.2 (over $\overline{\mathbb F}_q$) and Bernstein–Lunts (over $\mathbb C$), not BBD 5.4.5/6.2.5; necessity shown by $\mathbb G_m$ acting on a point (Counterexample 7.3).
  2. A5-4 (L): add $D^b$-full faithfulness of the transitions (affine-space fibres, unipotent kernels); necessity shown by $\mathbb G_m\to\mathrm{pt}$ (Audit 7.4). A5-4's "vector bundle $E_w$" is specific to $\mathbb A^n$.
  3. A2-4(e) under (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.'): local systems at every level (Audit 7.2).
  4. A2-3(c): proposition-I.tex Step 1 is an isomorphism up to $[-2d]$ (Audit 7.2).
  5. A5-7: (Lin) over each $\mathbb F_{q^r}$ for the "all $r$" clause (Audit 7.9).
  6. Citations: BBD 4.2.5–4.2.6.2 on pp. 109–111; t-exactness of $i_*$ is BBD 1.4.16; purity of $\mathrm{IC}_w$ is BBD Cor. 5.3.2 (Audit 7.1, Audit 7.6).
  7. Verification remark for the group: $I_{N,M}$ must be a congruence (pro-unipotent) subgroup, not the full kernel, for Lang (fails for $\mathrm{SL}_2$ since $H^1(\mathbb F_q,\mu_2)=\mathbb Z/2$) (Lemma 7.5).

Hypotheses checked here for rows 1–2 of Table 1 (the claims' own verification steps A2-6, A2-8, A2-9, A3-8, A3-9 are not in the audited file): $\mathbb A^n$ — (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).), (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) at every level (unit-group lemma), $\mathrm{Fr}$-stable pieces with one rational orbit each (orbit counts over $\mathbb F_3,\mathbb F_5$), (FM$w$) (Audit 7.4, Lemma 7.5); the group — connected stabilizers $(I\cap gIg^{-1})\times I_M$, (DT)(DT) · Decomposition: every generator IC_w ∗_I IC_0 is, equivariantly, a finite direct sum of shifted simple objects of C_I. Holds over F̄_q from a finite-type proper model of the convolution (purity and the stack decomposition theorem). by the twisted product, $\mathrm{IC}\omega\notin\mathcal M_I$ by A3-11(c) (Audit 7.10). Consequence, conditional on classical-affine.html Theorem 3 ((IG) for $\mathbb A^n$, not audited): A3-6 gives $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ for $\mathbb A^n$ over $\overline{\mathbb F}_q$ by an argument different from the I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages). dévissage.

Shared cited inputs (not re-proved by the claims or here): the identification of the Laszlo–Olsson category of $[Y/H]$ with the Bernstein–Lunts $D^b_{H,c}(Y)$; the equivariant decomposition theorem over $\mathbb C$; Karoubianness of triangulated categories with bounded t-structures.

Open (genuine gap, correctly flagged by A3-6 and A5-7): (II$_I$) for Betti sheaves over $\mathbb C$, which needs for each simple $S$ a witness $w$ valid at a prime outside the bad set of the spreading-out of $\mathrm{IC}_w\ast\mathrm{IC}_0$, i.e. some uniformity in $w$.

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  • 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 of the audit: (I$_I$) over $\overline{\mathbb F}_q$ and $\mathbb C$, (II$_I$) over $\overline{\mathbb F}_q$ only, converses as stated

Repairs required (none changes a statement's substance).

  1. A5-7 step 3 and A5-4 (over $\mathbb C$): equivariant splitting from Sun, Thm 3.12/1.2 (over $\overline{\mathbb F}_q$) and Bernstein–Lunts (over $\mathbb C$), not BBD 5.4.5/6.2.5; necessity shown by $\mathbb G_m$ acting on a point (Counterexample 7.3).
  2. A5-4 (L): add $D^b$-full faithfulness of the transitions (affine-space fibres, unipotent kernels); necessity shown by $\mathbb G_m\to\mathrm{pt}$ (Audit 7.4). A5-4's "vector bundle $E_w$" is specific to $\mathbb A^n$.
  3. A2-4(e) under (I2'): local systems at every level (Audit 7.2).
  4. A2-3(c): proposition-I.tex Step 1 is an isomorphism up to $[-2d]$ (Audit 7.2).
  5. A5-7: (Lin) over each $\mathbb F_{q^r}$ for the "all $r$" clause (Audit 7.9).
  6. Citations: BBD 4.2.5–4.2.6.2 on pp. 109–111; t-exactness of $i_*$ is BBD 1.4.16; purity of $\mathrm{IC}_w$ is BBD Cor. 5.3.2 (Audit 7.1, Audit 7.6).
  7. Verification remark for the group: $I_{N,M}$ must be a congruence (pro-unipotent) subgroup, not the full kernel, for Lang (fails for $\mathrm{SL}_2$ since $H^1(\mathbb F_q,\mu_2)=\mathbb Z/2$) (Lemma 7.5).

Hypotheses checked here for rows 1–2 of Table 1 (the claims' own verification steps A2-6, A2-8, A2-9, A3-8, A3-9 are not in the audited file): $\mathbb A^n$ — (I1), (I2) at every level (unit-group lemma), $\mathrm{Fr}$-stable pieces with one rational orbit each (orbit counts over $\mathbb F_3,\mathbb F_5$), (FM$w$) (Audit 7.4, Lemma 7.5); the group — connected stabilizers $(I\cap gIg^{-1})\times I_M$, (DT) by the twisted product, $\mathrm{IC}\omega\notin\mathcal M_I$ by A3-11(c) (Audit 7.10). Consequence, conditional on classical-affine.html Theorem 3 ((IG) for $\mathbb A^n$, not audited): A3-6 gives $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ for $\mathbb A^n$ over $\overline{\mathbb F}_q$ by an argument different from the I3 dévissage.

Shared cited inputs (not re-proved by the claims or here): the identification of the Laszlo–Olsson category of $[Y/H]$ with the Bernstein–Lunts $D^b_{H,c}(Y)$; the equivariant decomposition theorem over $\mathbb C$; Karoubianness of triangulated categories with bounded t-structures.

Open (genuine gap, correctly flagged by A3-6 and A5-7): (II$_I$) for Betti sheaves over $\mathbb C$, which needs for each simple $S$ a witness $w$ valid at a prime outside the bad set of the spreading-out of $\mathrm{IC}_w\ast\mathrm{IC}_0$, i.e. some uniformity in $w$.

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