§5 · Agent 5 · Functions without (A1)–(A5) · 2026-10-17T03:15:58Z

Theorem 5.3 · $(\mathrm{SG})$ for $\Spin_{2n}/\Spin_{2n-1}$, every $n\ge2$, every non-archimedean local field

proved audit: correct ★ key step

Second audit: correct. Re-derived: the dominance order on $\Lambda_S^+$, the lift and $t^\lambda x_0=x_m$, the pole-order bound (for dominant $\lambda$ with $\lambda_1=m$ every coordinate satisfies $|\lambda_i|\le m$), Lemma A(2) and Proposition B. No hypothesis on $F$ is used.

With the orbit classification of Lemma 5.2, the dominant lifts $(m,m,0,\dots,0)$ and a pole-order support bound make Proposition B apply directly: the basic function generates for every sphere over every local field, without Theorem E or (A1)–(A5). The Table 1 cell 'open for $n\ge4$' becomes 'holds'.

Theorem 5.3.

For $X=\Spin_{2n}/\Spin_{2n-1}$ with the model $Q=\{N=1\}$ of Lemma 5.2, every $n\ge2$ and every non-archimedean local field $F$: $\cH_K\cdot\varphi_0=C_K$.

Labels. $\Lambda_S^+=\{2m\varepsilon_1:m\ge0\}$ (families page for the spheres, sections 2 and 5), identified with $m\ge0$ and with the shells $O_m$ of Lemma 5.2. In the dominance order $2m'\varepsilon_1\le2m\varepsilon_1$ iff $m'\le m$, because $2\varepsilon_1=(\varepsilon_1-\varepsilon_2)^\vee+(\varepsilon_1+\varepsilon_2)^\vee$; lower sets are finite.

Spanning. By Lemma 5.2 the $O_m$ are compact open. A $K$-invariant $\varphi\in C_c^\infty(Q(F))$ is constant on each of them, and its compact support meets only finitely many. So $\{\one_{O_m}\}_{m\ge0}$ is a basis of $C_K$, and $\varphi_0=\one_{O_0}$.

Lift. $\lambda=(m,m,0,\dots,0)$ lies in $\Lambda_T^+$ (the coroot lattice of type $D_n$ consists of the vectors with even coordinate sum). Its image in $\SO_{2n}$ is $\bar t^\lambda=\operatorname{diag}(\varpi^m,\varpi^m,1,\dots,1,\varpi^{-m},\varpi^{-m})$, and $t^\lambda x_0=\bar t^\lambda(e_1+e_{2n})=x_m\in O_m$.

Support bound. $\bar t^\lambda\cO^{2n}\subseteq\varpi^{-m}\cO^{2n}$ and $K$ preserves $\varpi^{-m}\cO^{2n}$, so

$$K\,t^\lambda Q(\cO)\ \subseteq\ Q(F)\cap\varpi^{-m}\cO^{2n}\ =\ \bigcup_{m'\le m}O_{m'} .$$

The same inclusion holds for every dominant $\lambda$ with $\lambda_1=m$, that is, for every dominant lift of $2m\varepsilon_1$.

Generation. Lemma A(2) gives $(T_\lambda\ast\varphi_0)(x_m)\ge1$, and Proposition B with $f_m=T_{(m,m,0,\dots,0)}$ gives $(\mathrm{SG})$. $\square$

Inputs. Used: Lemma 5.2, Lemma A(2), Proposition B. Not used: Theorem E, (A1)–(A5), the Cartan decomposition, $\operatorname{char}F=0$, finiteness of the set of $G(F)$-orbits. The result is consistent with $n=2$ (the group $\SL_2$, Proposition D) and $n=3$ ($\SL_4/\Sp_4$). In Table 1 the row $\Spin_{2n}/\Spin_{2n-1}$, '$n=3$ under (A); $n\ge4$ open', becomes 'holds for all $n\ge2$ and all $F$'.

Builds on

Lemma 5.2 · Spheres: the $\Spin_{2n}(\cO)$-orbits on $Q(F)$ are the pole-order shells, for every local field

Built on by

Summary 5.10 · Table changes at function level; no row disagrees between the levels; nothing here depends on (A)
Audit 7.20 · Audit of the sphere results: the residue-characteristic-2 case is genuinely covered
Theorem X3.4 · Corrected A5-8 for the three families, and (SG) for the pair model without $\tau$

Original record, as the agent wrote it at 03:15 UTC

(SG) for Spin_{2n}/Spin_{2n-1}, every n>=2, every local field F

Labels: $\Lambda_S^+=\{2m\varepsilon_1\}\leftrightarrow m\ge0$ (families/spin.html §2,§5); dominance on $\Lambda_S^+$ is $m'\le m$ ($2\varepsilon_1=(\varepsilon_1-\varepsilon_2)^\vee+(\varepsilon_1+\varepsilon_2)^\vee$); lower sets finite.

Spanning: by node 2 the $O_m$ are compact open, a $K$-invariant $\varphi\in C_c^\infty$ is constant on them and its (bounded) support meets finitely many; so $\{\mathbf 1_{O_m}\}$ is a basis of $C_K$. $\varphi_0=\mathbf 1_{O_0}$.
Lift: $\lambda=(m,m,0,\dots,0)\in\Lambda_T^+$ (coroot lattice of $D_n$), $\bar t^{\lambda}=\mathrm{diag}(\varpi^m,\varpi^m,1,\dots,\varpi^{-m},\varpi^{-m})$, $t^\lambda x_0=\bar t^\lambda(e_1+e_{2n})=x_m\in O_m$.
Support bound: $\bar t^\lambda\mathcal O^{2n}\subseteq\varpi^{-m}\mathcal O^{2n}$ and $K$ preserves $\varpi^{-m}\mathcal O^{2n}$, so

$$K\,t^\lambda Q(\mathcal O)\subseteq Q(F)\cap\varpi^{-m}\mathcal O^{2n}=\textstyle\bigcup_{m'\le m}O_{m'}.$$

(Holds for every dominant $\lambda$ with $\lambda_1=m$, i.e. every dominant lift.)

Lemma A(2): $(T_\lambda\ast\varphi_0)(x_m)\ge1$. Proposition B with $f_m=T_{(m,m,0,\dots)}$ gives (SG). $\square$

Used: node 2, Lemma A(2), Proposition B. Not used: Theorem E, (A1)–(A5), Cartan decomposition, $\operatorname{char}F=0$, finiteness of $G(F)$-orbits. Consistent with $n=2$ (group $\mathrm{SL}_2$, Prop. D) and $n=3$ ($\mathrm{SL}_4/\mathrm{Sp}_4$). Table 1 row $\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$: "$n=3$ under (A); $n\ge4$ open" becomes "holds, all $n\ge2$, all $F$".

Sources

classical.html#Corollary C
families/spin.html

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

Theorem 5.3 · \((\mathrm{SG})\) for \(\Spin_{2n}/\Spin_{2n-1}\), every \(n\ge2\), every non-archimedean local field

Builds on

Built on by

(SG) for Spin_{2n}/Spin_{2n-1}, every n>=2, every local field F

Sources