§13 · Agent X3 · Pair model at 2 · 2026-10-17T20:13:33Z

Lemma X3.1 · The pair model as rank-$(n-p)$ projectors; its orbit label is a polar Smith part

proved audit: correct

Second audit: correct. I re-derived (a), the isomorphism (b) π_V(L0)/(L0∩V) ≅ (L0+P L0)/L0, the minors formula, the partial-sum form of dominance on Λ_S^+ (no condition on the totals, finite lower sets) and the identification φ(λ)=(λ_i−λ_{n+1−i}). I checked the formula against the label computed from Theorem 5.4's lattice definition (a saturated basis of L0∩V, without using P_x minors) at 80 random points per field over Q_2, Q_3, Q_2(√2) and F_2((t)), for n≤6 (V3/pair_v3.py). I found no defect.

Writing a point $x=(U,V)$ of the pair model as its projector $P_x$ makes $X(\mathcal O)$ the integral projectors and the label of Theorem 5.4 the polar part of the Smith form of $P_x$; the labels are $\Lambda_S^+$, and dominance on them is a partial-sum order on $p$-tuples.

Lemma X3.1.

Notation. $F$ is any non-archimedean local field (either characteristic, any residue characteristic), with $\mathcal O$, uniformizer $\varpi$, $\operatorname{val}\varpi=1$, residue field $\mathbb F_q$; $K=\mathrm{GL}_n(\mathcal O)$, $L_0=\mathcal O^n$, $\operatorname{vol}K=1$. Fix $n\ge2$ and $1\le p\le n-p$; the second block has size $n-p$ (the $q$ of $\mathrm{GL}_q$ in Theorem 5.4, renamed to avoid the residue cardinality). Put $e:=\operatorname{val}(2)\in\{0,1,2,\dots\}\cup\{\infty\}$: $e=0$ iff the residue characteristic is odd, $e=\infty$ iff $\operatorname{char}F=2$.

The pair model. $X(F)=\{(U,V):F^n=U\oplus V,\ \dim V=p\}$, and $x=(U,V)\mapsto P_x:=$ the projection onto $U$ along $V$ identifies $X(F)$ with $\{P\in M_n(F):P^2=P,\ \operatorname{rk}P=n-p\}$, $g\cdot P=gPg^{-1}$. The base point of Theorem 5.4 is $x_0=(U_0,V_0)$, $U_0=\langle e_i+e_{n+1-i}\ (i\le p),\ e_j\ (p<j\le n-p)\rangle$, $V_0=\langle e_{n+1-i}\ (i\le p)\rangle$, so $P_0:=P_{x_0}$ has $P_0e_i=e_i+e_{n+1-i}$, $P_0e_j=e_j$, $P_0e_{n+1-i}=0$. $X(\mathcal O):=\{x:L_0=(L_0\cap U)\oplus(L_0\cap V)\}$, and $\ell(x):=$ the type of $\pi_V(L_0)/(L_0\cap V)$ ($\pi_V=1-P_x$), as in Theorem 5.4.

(a) $X(\mathcal O)=\{x:P_x\in M_n(\mathcal O)\}$.

(b) $\pi_V(L_0)/(L_0\cap V)\cong(L_0+P_xL_0)/L_0$. Hence, if $P_x=k_1\operatorname{diag}(\varpi^{c_1},\dots,\varpi^{c_n})k_2$ with $k_i\in K$, $c_1\le\dots\le c_n$ in $\mathbb Z\cup\{\infty\}$, then $\ell(x)=(-c_i)_{c_i<0}$, with at most $p$ parts; with $\delta_j(M):=\min\{\operatorname{val}\det M_{I,J}:|I|=|J|=j\}$, $\delta_0=0$, $$\ell_1(x)+\dots+\ell_k(x)=-\min_{0\le j\le k}\delta_j(P_x)\qquad(k\le p).$$

(c) $b\mapsto\mu(b):=(b_1,\dots,b_p,0^{\,n-2p},-b_p,\dots,-b_1)$ is a bijection from $\{b_1\ge\dots\ge b_p\ge0\}$ onto $\Lambda_S^+$ (families page gl-pq, sections 2 and 5), under which $\varphi(\lambda)=\lambda-\theta\lambda$ becomes $(\lambda_1-\lambda_n,\dots,\lambda_p-\lambda_{n+1-p})$. The dominance order of $\Lambda_T$, restricted to $\Lambda_S^+$, is $$\mu(b')\le\mu(b)\iff\textstyle\sum_{i\le k}b'_i\le\sum_{i\le k}b_i\quad(1\le k\le p),$$ with no condition on the totals; every lower set is finite.

(d) (cited, Theorem 5.4(i), re-derived in Audit 7.21 and by the second audit) $\ell$ is a bijection $K\backslash X(F)\to\{b_1\ge\dots\ge b_p\ge0\}$, the orbits $O_b$ are compact open, and $X(\mathcal O)=O_0=Kx_0$.

(a). If $P=P_x\in M_n(\mathcal O)$, each $l\in L_0$ is $Pl+(1-P)l$ with $Pl\in L_0\cap U$ and $(1-P)l\in L_0\cap V$, so $L_0$ splits. Conversely, if $L_0=(L_0\cap U)\oplus(L_0\cap V)$, then $P$ maps $L_0$ onto $L_0\cap U\subseteq L_0$, so $P\in M_n(\mathcal O)$.

(b), the cokernel. Since $l=Pl+(1-P)l$, $L_0+PL_0=L_0+(1-P)L_0$. The map $(1-P)L_0\to(L_0+(1-P)L_0)/L_0$ is onto with kernel $(1-P)L_0\cap L_0$. Now $(1-P)L_0=\pi_V(L_0)\subseteq V$, and $L_0\cap V=(1-P)(L_0\cap V)\subseteq(1-P)L_0$, so the kernel is $L_0\cap V$ and $(L_0+PL_0)/L_0\cong\pi_V(L_0)/(L_0\cap V)$. (This replaces the Goursat argument of step 1 of Correction 10.4, whose conclusion it confirms.)

(b), Smith form. With $P=k_1Dk_2$, $L_0+PL_0=k_1(L_0+DL_0)$ and $(L_0+DL_0)/L_0\cong\bigoplus_{c_i<0}\mathcal O/\varpi^{-c_i}$. At most $p$ of the $c_i$ are negative, because $(1-P)L_0$ has rank $p$. By Cauchy–Binet $\delta_j$ is invariant under $K\times K$, so $\delta_j(P)=c_1+\dots+c_j$. The partial sums of $-c_1\ge-c_2\ge\cdots$ increase while $-c_i>0$ and then do not increase, so $\sum_{i\le k}\max(-c_i,0)=\max_{0\le j\le k}(-\delta_j(P))$.

(c). On $\Lambda_T=\mathbb Z^n$, $\theta$ is the permutation matrix $J$ of $i\leftrightarrow n+1-i$ ($i\le p$); its $(-1)$-eigenlattice is $\{\mu(b)\}$, and $\mu(b)$ is dominant iff $b_1\ge\dots\ge b_p\ge0$. For $\lambda\in\Lambda_T$, $(\lambda-J\lambda)_i=\lambda_i-\lambda_{n+1-i}$ for $i\le p$. The partial sums of $\mu(b)$ are $b_1+\dots+b_k$ at position $k\le p$, $|b|$ for $p\le k\le n-p$, and $b_1+\dots+b_k$ at position $n-k$ ($k\le p$), since the total is $0$. An element of $\mathbb Z^n$ of total $0$ is a nonnegative combination of the simple coroots $\varepsilon_i-\varepsilon_{i+1}$ iff all its partial sums are $\ge0$; for $\mu(b)-\mu(b')$ these are the displayed inequalities. Lower sets are finite because $b'_i\le b'_1\le b_1$.

(d) is Theorem 5.4(i). Computation: the label is recomputed from minors in pair_tau.py.

Builds on

Theorem 5.4 · $\GL_n/(\GL_p\times\GL_q)$, pair model: orbits, lifts, a componentwise support bound and $(\mathrm{SG})$ over every $F$
Audit 7.21 · Audit of $\GL_n/(\GL_p\times\GL_q)$ in the pair model: normal form, componentwise bound and $(\mathrm{SG})$ over every $F$

Built on by

Proposition X3.2 · Support bound and top orbit for the pair model, every dominant $\lambda$, every $F$
Proposition X3.3 · Cartan label of $\tau(x)$ is $\mu((\ell(x)-e)^+)$, $e=\operatorname{val}2$
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
Re-check V3 · Re-check by V3

Sources

Theorem 5.4
families/gl-pq.html sections 1, 2, 5
Correction 10.4 step 1

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

Lemma X3.1 · The pair model as rank-\((n-p)\) projectors; its orbit label is a polar Smith part

Builds on

Built on by

Sources