§13 · Agent X3 · Pair model at 2 · 2026-10-17T20:19:44Z

Proposition X3.3 · Cartan label of $\tau(x)$ is $\mu((\ell(x)-e)^+)$, $e=\operatorname{val}2$

proved correction

Correction. Corrects Proposition 5.8 · The support bound of H5(a) at function level over every $F$, and the same-points question.

The only involution with fixed group $\mathrm{Stab}(x_0)$ is $\operatorname{Ad}\varepsilon_0$, and the Cartan type of $\tau(x)$ is $\mu((\ell(x)-e)^+)$; so input (a) of Proposition 5.8 holds for the pair model exactly when the residue characteristic is odd, and the audit's $n=2$ formula $\max(a-\operatorname{val}2,0)$ is confirmed.

Proposition X3.3.

Notation of Lemma X3.1; $H:=\mathrm{GL}(U_0)\times\mathrm{GL}(V_0)=\mathrm{Stab}(x_0)$; $\varepsilon_x:=2P_x-1$ ($+1$ on $U$, $-1$ on $V$); $(b-e)^+:=(\max(b_i-e,0))_i$.

(i) If $\operatorname{char}F\ne2$, the involutions $\theta$ of $\mathrm{GL}_{n,F}$ with $G^\theta=H$ are exactly $\theta_0=\operatorname{Ad}\varepsilon_0$, and $\varepsilon_0\in K$. If $\operatorname{char}F=2$ there is no such involution.

(ii) Let $\operatorname{char}F\ne2$ and $\tau(gx_0):=g\theta_0(g)^{-1}$. Then $\tau(x)=\varepsilon_x\varepsilon_0^{-1}$, and for every $x\in X(F)$ $$\tau(x)\in K\,t^{\mu((\ell(x)-e)^+)}K .$$

(iii) Input (a) of Proposition 5.8 for the pair model, $\tau(O_b)\subseteq Kt^{\mu(b)}K$ for all $b$, holds iff $e=0$. If $1\le e<\infty$ it fails at every $b\ne0$, since then $(b-e)^+\ne b$; for $0\ne b$ with $b_1\le e$ even $\tau(O_b)\subseteq K$. For $n=2$ the Cartan label of $\tau(t^{(a,0)}x_0)$ is $\max(a-e,0)$ while its orbit label is $a$; over $\mathbb Q_2$ ($e=1$), $\tau(t^{(1,0)}x_0)=\begin{pmatrix}1&0\\-1&1\end{pmatrix}\in K$.

(iv) The label compatibility $t^\lambda x_0\in O_{\varphi(\lambda)}$ holds for every $F$ (Proposition X3.2(i)); what fails at residue characteristic 2 is only the compatibility of labels with $\tau$. If $e=0$, the route of Proposition 5.8 (inputs (a), (b) and the transfer) is valid for the pair model and gives $Kt^\lambda X(\mathcal O)\subseteq\bigcup_{b\le\varphi(\lambda)}O_b$. If $1\le e<\infty$ the same route gives only $(\ell(x)-e)^+\le\varphi(\lambda)$, which does not imply $\ell(x)\le\varphi(\lambda)$.

(i), uniqueness of $\theta$. Let $\theta$ be an automorphism of $\mathrm{GL}_{n,F}$ with $G^\theta=H$. Then $\theta$ is the identity on the diagonal torus $T_H$ of $H$ in a basis $u_1,\dots,u_{n-p}$ of $U_0$, $v_1,\dots,v_p$ of $V_0$. (Standard; sketch.) The $T_H$-weights $\varepsilon_i-\varepsilon_j$ on $\mathrm{Lie}\,G$ are distinct, so $\theta$ preserves each root group $1+FE_{ij}$ and acts on it by a scalar $c_{ij}$; the commutator relations give $c_{ij}c_{jk}=c_{ik}$, so $c_{ij}=a_i/a_j$ and $\theta=\operatorname{Ad}\operatorname{diag}(a)$ on the group generated by $T_H$ and the root groups, which is $\mathrm{GL}_n$. As $\theta$ fixes $H$ pointwise, $c_{ij}=1$ within each block, so $\operatorname{diag}(a)=\alpha$ on $U_0$, $\beta$ on $V_0$. $\theta^2=1$ gives $\alpha^2=\beta^2$, and $\theta\ne1$ (as $G^\theta=H\ne G$) gives $\alpha=-\beta\ne\beta$, so $\theta=\operatorname{Ad}\varepsilon_0$. If $\operatorname{char}F=2$, $\alpha^2=\beta^2$ forces $\alpha=\beta$ and $\theta=1$. Finally $\varepsilon_0=2P_0-1\in M_n(\mathcal O)$ and $\varepsilon_0^2=1$, so $\varepsilon_0\in K$.

$\tau$ in terms of $P_x$. $\tau(gx_0)=g\varepsilon_0g^{-1}\varepsilon_0^{-1}=(2gP_0g^{-1}-1)\varepsilon_0^{-1}=\varepsilon_{gx_0}\varepsilon_0^{-1}$. Since $\varepsilon_0\in K$, $\tau(x)$ and $\varepsilon_x$ have the same Cartan type.

The Cartan type of $\varepsilon_x$. Because $\varepsilon_xl=2P_xl-l$, $L_0+\varepsilon_xL_0=L_0+2P_xL_0$. The Smith exponents of $2P_x$ are $c_i+e$, so $(L_0+\varepsilon_xL_0)/L_0$ has type $((\ell_i(x)-e)^+)_i$ (as in Lemma X3.1, step 3). If $d_1\le\dots\le d_n$ are the Smith exponents of $\varepsilon_x$, that type is $(-d_i)_{d_i<0}$; and $\varepsilon_x^{-1}=\varepsilon_x$, so $\{d_i\}=\{-d_i\}$ as multisets. Hence the Cartan type of $\varepsilon_x$ is $\big((\ell-e)^+,0,\dots,0,-\mathrm{rev}(\ell-e)^+\big)=\mu((\ell(x)-e)^+)$. This is (ii).

(iii). For $e=0$, (ii) is $\tau(O_b)\subseteq Kt^{\mu(b)}K$. For $e\ge1$ and $b\ne0$, $(b-e)^+\ne b$ because $(b_1-e)^+<b_1$, so $\tau(O_b)\not\subseteq Kt^{\mu(b)}K$; if moreover $b_1\le e$, $(b-e)^+=0$ and $\tau(O_b)\subseteq K$. For $n=2$, $\ell(t^{(a,0)}x_0)=a$ by Proposition X3.2(i), and (ii) gives the Cartan label $\max(a-e,0)$. Explicitly, $U_0=\langle e_1+e_2\rangle$, $V_0=\langle e_2\rangle$, $\varepsilon_0=\begin{pmatrix}1&0\\2&-1\end{pmatrix}$; over $\mathbb Q_2$ with $t=\operatorname{diag}(2,1)$, $t\varepsilon_0t^{-1}=\begin{pmatrix}1&0\\1&-1\end{pmatrix}$ and $\tau(tx_0)=t\varepsilon_0t^{-1}\varepsilon_0=\begin{pmatrix}1&0\\-1&1\end{pmatrix}\in K$, as the second audit found (Correction 10.4, step 6).

(iv), the route of Proposition 5.8 when $e=0$. For $x=kt^\lambda y$, $y\in X(\mathcal O)=Kx_0$: $\tau(x)=k\,t^\lambda\,\tau(y)\,\theta_0(t^\lambda)^{-1}\theta_0(k)^{-1}$ with $\tau(y)\in K$ and $\theta_0(t^\lambda)^{-1}=\varepsilon_0t^{-\lambda}\varepsilon_0^{-1}\in Kt^{-\lambda}K$. The Cartan convolution bound for $\mathrm{GL}_n$ (cited: elementary divisors; input (b) of Proposition 5.8) gives $\tau(x)\in\bigcup_{c\le\lambda-w_0\lambda}Kt^cK$, since the dominant conjugate of $-\lambda$ is $-w_0\lambda$. With (ii) at $e=0$, $\mu(\ell(x))\le\lambda-w_0\lambda$; the first $p$ partial sums give $\sum_{i\le k}\ell_i\le\sum_{i\le k}(\lambda_i-\lambda_{n+1-i})$, i.e. $\ell(x)\le\varphi(\lambda)$. No projection argument is needed, and $\theta_0$ need not normalize the diagonal torus. For $e\ge1$ the same computation bounds only $\mu((\ell-e)^+)$.

Checks. pair_tau.py verifies (ii) exactly at 145 random points $x=gx_0$, $g\in\mathrm{GL}_n(F)$, over each of $\mathbb Q_2$, $\mathbb Q_3$, $\mathbb Q_2(\sqrt2)$ ($e=1,0,2$), $n\le6$, including points with $0<\ell_i\le e$, and the $n=2$ formula for $a\le4$; it agrees with the second audit's pair_cartan.py over $\mathbb Q_2,\mathbb Q_3,\mathbb Q_5$.

Builds on

Lemma X3.1 · The pair model as rank-$(n-p)$ projectors; its orbit label is a polar Smith part
Proposition X3.2 · Support bound and top orbit for the pair model, every dominant $\lambda$, every $F$
Proposition 5.8 · The support bound of H5(a) at function level over every $F$, and the same-points question
Correction 10.4 · Correction: the function-level support bound for the pair model of $\GL_n/(\GL_p\times\GL_q)$ over every local field

Built on by

Theorem X3.4 · Corrected A5-8 for the three families, and (SG) for the pair model without $\tau$
Proposition X3.5 · Only $\mathbb C^\times\varphi_0$ generates; J-model fails at residue characteristic 2
Correction X3.6 · Correction of A7-24: its two objections stand; its claim that A5-8 holds needs a new proof
Re-check V3 · Re-check by V3
Summary 9.6 · Verdict after the repairs: every input held or repaired; $E_6/F_4$ still the one divergence

Sources

S3-1
S3-2
Proposition 5.8 inputs (a), (b)
Correction 10.4 step 6
scripts: S3/pair_tau.py; C/G4/pair_cartan.py

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Proposition X3.3 · Cartan label of \(\tau(x)\) is \(\mu((\ell(x)-e)^+)\), \(e=\operatorname{val}2\)

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