§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:42:18Z

Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$

proved ★ key step

The Iwahori-level bridge: under level-wise finiteness, connected stabilizers and rationality of the finite-level orbits, Iwahori generation of functions over $\mathbb F_q(\!(t)\!)$ forces every simple object to be a shifted summand of some $\mathrm{IC}_w*_I\mathrm{IC}_0$, hence sheaf generation over $\overline{\mathbb F}_q$; contrapositively each missing simple object gives an explicit linear form killing $\mathcal H_I*\varphi_0$.

Theorem 3.6.

Assume (L1), (L2), (D1)–(D3) of Setup 3.1.

(a) If $\mathcal H_I*\varphi_0=C_I$ over $F=\mathbb F_q(\!(t)\!)$, then $\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$, and therefore $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ over $k=\overline{\mathbb F}q$ with $\overline{\mathbb Q}\ell$-coefficients.

(b) (Pointwise form.) If $\mathbf 1_{P(\mathbb F_q)}\in\mathcal H_I*\varphi_0$ for one piece $P$, then $\mathrm{IC}(P)\in\mathcal M_I$, hence $\mathrm{IC}(P)\in\mathcal C_I^{\mathrm{Iwa}}$.

(c) (Witness.) If $S\in\mathrm{Irr}\,\mathcal P_I\setminus\mathcal M_I$, the coordinate form $\kappa_S\colon C_I\otimes\overline{\mathbb Q}\ell\to\overline{\mathbb Q}\ell$ (coefficient of $\mathrm{tr}\,S$ in the basis of Lemma 3.4(i)) vanishes on $\mathcal H_I*\varphi_0$ and $\kappa_S(\mathrm{tr}\,S)=1$; so (IG) fails over $F$.

Consequently, under these inputs the divergence "(IG) holds over $\mathbb F_q(\!(t)\!)$ but $\mathcal C_I^{\mathrm{Iwa}}\neq\mathcal C_I$ over $\overline{\mathbb F}_q$" does not occur.

Chain. $$\text{(IG)}\ \xRightarrow{[[A3-4]](iii)}\ \text{columns of }m\text{ span}\ \xRightarrow{\text{rows}}\ \forall S\ \exists w:\ m_{S,w}\neq0\ \xRightarrow{[[A3-4]](ii)}\ \mathcal M_I=\mathrm{Irr}\,\mathcal P_I\ \xRightarrow{[[A3-5]](b)}\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I .$$

  1. Rows. If the columns span $\bigoplus_S\overline{\mathbb Q}_\ell$, the coordinate form $\pi_S$ is non-zero on their span, so row $S$ is non-zero.
  2. Extraction. $m_{S,w}\neq0$ forces some $E_{S,i}(w)\neq0$, i.e. $S[-i]$ is a direct summand of $\mathrm{IC}_w*_I\mathrm{IC}_0$ by (D3). Only "zero multiplicity space $\Rightarrow$ zero coefficient" is used: no purity beyond (D3), no positivity.
  3. Every simple object is reached. Step 2 for every $S$; by (L2) every simple object is $\mathrm{Fr}$-stable and appears in the basis of Lemma 3.4(i). Lemma 3.5(b) concludes (a).
  4. (b). By step 2 in contrapositive, $m_{S,w}=0$ for $S\notin\mathcal M_I$, so by Lemma 3.3(c) and Lemma 3.4(ii) $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}\ell\subseteq\mathrm{span}\{\mathrm{tr}\,S:S\in\mathcal M_I\}$. Write $\mathbf 1{P(\mathbb F_q)}=\sum_Qc_Q\,\mathrm{tr}\,\mathrm{IC}(Q)$ over the pieces $Q$ of a common finer level; by uniqueness of coordinates, $c_Q\neq0\Rightarrow\mathrm{IC}(Q)\in\mathcal M_I$. Let $P^\circ$ be the open piece of $P$ at that level, so $\mathrm{IC}(P)=\mathrm{IC}(P^\circ)$ (Setup 3.1(ii)). (i) A maximal $Q_0$ with $c_{Q_0}\neq0$ lies in $P$: evaluating at $Q_0(\mathbb F_q)$, only $Q=Q_0$ contributes (triangularity), giving $\pm c_{Q_0}\neq0$. (ii) Evaluating at $P^\circ(\mathbb F_q)$ gives $1$, so some $Q\ge P^\circ$ has $c_Q\neq0$; a maximal $Q_1\ge Q$ lies in $P$ by (i) and has $P^\circ\subseteq\overline{Q_1}$, so $Q_1$ is open in the irreducible $P$, i.e. $Q_1=P^\circ$. Hence $\mathrm{IC}(P)\in\mathcal M_I$, and $\in\mathcal C_I^{\mathrm{Iwa}}$ by Lemma 3.5(a).
  5. (c). $\kappa_S$ kills every $\mathrm{tr}(\mathrm{IC}w*_I\mathrm{IC}_0)$ (step 4), hence $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}\ell$; $\mathrm{tr}\,S\in C_I\otimes\overline{\mathbb Q}_\ell$ is not killed; the integrality argument of Lemma 3.4(iii) transports the failure to $\mathbb C$-valued functions. $\square$

Which satake-run inputs are replaced. D0 (Chen–Yi orbit geometry, H1) $\to$ level-wise finiteness (L1), (L2); (R) $\to$ split $G$ and $\mathrm{Fr}$-stable finite-level orbits (L2); (C) $\to$ connected stabilizers at every level (L2); D1 (sheaf theory, $\mathrm{Fr}$-compatible t-structure; base change on convolution fibres) $\to$ the glued t-structure of Setup 3.1 and (D1) with the shearing isomorphism of Lemma 3.3; D2 (decomposition of $V_\lambda*\mathrm{IC}0$) $\to$ (D3) for $\mathrm{IC}_w*_I\mathrm{IC}_0$; D3 (Lang for $L^+G$, stabilizers) $\to$ (D2) for $I$, $I{N,M}$ and finite-level stabilizers. D5 (comparison with $\mathbb C$) is not used.

Fields. Functions over $\mathbb F_q(\!(t)\!)$, sheaves over $\overline{\mathbb F}_q$. Nothing here is claimed over $\mathbb C$: the transfer needs, for each $S$, a $w$ realizing $S$ over $\overline{\mathbb F}_p$ for some $p$ outside the finite bad set of BBD §6.1 spreading out for that $w$; uniformity in $w$ is not proved (gap for the Betti version).

Builds on

  • Lemma 3.3 · Trace of $*_I$ is convolution (vol $I=1$); IC traces span $\mathcal H_I$
  • Lemma 3.4 · Unitriangular IC-trace basis of $C_I$ and the decomposition matrix $m_{S,w}$
  • Lemma 3.5 · Iwahori summand criterion: $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\,\mathcal M_I$

Built on by

  • Proposition 3.7 · Disconnected stabilizers: (IG)$_n$ for all $n$ gives $\mathcal M_I=\mathrm{Irr}$
  • Corollary 3.8 · $\mathbb A^n$: inputs checked; (IG) gives sheaf generation over $\overline{\mathbb F}_q$
  • Proposition 3.9 · Group: no $\mathrm{IC}_\omega$ is a summand, so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Proposition 3.10 · Sheaves to functions: non-zero rows only; (WT) gives (IG)
  • Theorem 3.11 · Sheaf generation as a test on IC-coordinates over all $\mathbb F_{q^n}$
  • Summary 3.12 · Verdict: (II$_I$) proved; rows 1–3 agree; one divergence type excluded
  • Audit 7.7 · Audit of A3-4, A3-5, A3-6: unitriangular IC-trace basis, $m_{S,w}=\sum_i(-1)^i\mathrm{tr}\,A_{S,i}(w)$, and (IG) $\Rightarrow\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
  • 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 18:42 UTC

(II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$

Chain. $$\text{(IG)}\ \xRightarrow{[[A3-4]](iii)}\ \text{columns of }m\text{ span}\ \xRightarrow{\text{rows}}\ \forall S\ \exists w:\ m_{S,w}\neq0\ \xRightarrow{[[A3-4]](ii)}\ \mathcal M_I=\mathrm{Irr}\,\mathcal P_I\ \xRightarrow{[[A3-5]](b)}\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I .$$

  1. Rows. If the columns span $\bigoplus_S\overline{\mathbb Q}_\ell$, the coordinate form $\pi_S$ is non-zero on their span, so row $S$ is non-zero.
  2. Extraction. $m_{S,w}\neq0$ forces some $E_{S,i}(w)\neq0$, i.e. $S[-i]$ is a direct summand of $\mathrm{IC}_w*_I\mathrm{IC}_0$ by (D3). Only "zero multiplicity space $\Rightarrow$ zero coefficient" is used: no purity beyond (D3), no positivity.
  3. Every simple object is reached. Step 2 for every $S$; by (L2) every simple object is $\mathrm{Fr}$-stable and appears in the basis of Lemma 3.4(i). Lemma 3.5(b) concludes (a).
  4. (b). By step 2 in contrapositive, $m_{S,w}=0$ for $S\notin\mathcal M_I$, so by Lemma 3.3(c) and Lemma 3.4(ii) $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}\ell\subseteq\mathrm{span}\{\mathrm{tr}\,S:S\in\mathcal M_I\}$. Write $\mathbf 1{P(\mathbb F_q)}=\sum_Qc_Q\,\mathrm{tr}\,\mathrm{IC}(Q)$ over the pieces $Q$ of a common finer level; by uniqueness of coordinates, $c_Q\neq0\Rightarrow\mathrm{IC}(Q)\in\mathcal M_I$. Let $P^\circ$ be the open piece of $P$ at that level, so $\mathrm{IC}(P)=\mathrm{IC}(P^\circ)$ (Setup 3.1(ii)). (i) A maximal $Q_0$ with $c_{Q_0}\neq0$ lies in $P$: evaluating at $Q_0(\mathbb F_q)$, only $Q=Q_0$ contributes (triangularity), giving $\pm c_{Q_0}\neq0$. (ii) Evaluating at $P^\circ(\mathbb F_q)$ gives $1$, so some $Q\ge P^\circ$ has $c_Q\neq0$; a maximal $Q_1\ge Q$ lies in $P$ by (i) and has $P^\circ\subseteq\overline{Q_1}$, so $Q_1$ is open in the irreducible $P$, i.e. $Q_1=P^\circ$. Hence $\mathrm{IC}(P)\in\mathcal M_I$, and $\in\mathcal C_I^{\mathrm{Iwa}}$ by Lemma 3.5(a).
  5. (c). $\kappa_S$ kills every $\mathrm{tr}(\mathrm{IC}w*_I\mathrm{IC}_0)$ (step 4), hence $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}\ell$; $\mathrm{tr}\,S\in C_I\otimes\overline{\mathbb Q}_\ell$ is not killed; the integrality argument of Lemma 3.4(iii) transports the failure to $\mathbb C$-valued functions. $\square$

Which satake-run inputs are replaced. D0 (Chen–Yi orbit geometry, H1) $\to$ level-wise finiteness (L1), (L2); (R) $\to$ split $G$ and $\mathrm{Fr}$-stable finite-level orbits (L2); (C) $\to$ connected stabilizers at every level (L2); D1 (sheaf theory, $\mathrm{Fr}$-compatible t-structure; base change on convolution fibres) $\to$ the glued t-structure of Setup 3.1 and (D1) with the shearing isomorphism of Lemma 3.3; D2 (decomposition of $V_\lambda*\mathrm{IC}0$) $\to$ (D3) for $\mathrm{IC}_w*_I\mathrm{IC}_0$; D3 (Lang for $L^+G$, stabilizers) $\to$ (D2) for $I$, $I{N,M}$ and finite-level stabilizers. D5 (comparison with $\mathbb C$) is not used.

Fields. Functions over $\mathbb F_q(\!(t)\!)$, sheaves over $\overline{\mathbb F}_q$. Nothing here is claimed over $\mathbb C$: the transfer needs, for each $S$, a $w$ realizing $S$ over $\overline{\mathbb F}_p$ for some $p$ outside the finite bad set of BBD §6.1 spreading out for that $w$; uniformity in $w$ is not proved (gap for the Betti version).

Sources