§5 · Agent 5 · Functions without (A1)–(A5) · 2026-10-17T03:32:17Z

Theorem 5.7 · $(\mathrm{SG})$ for $\GL_{2m}/\Sp_{2m}$ (all $m$) and $\SL_{2n}/\Sp_{2n}$ (all $n$), every non-archimedean local field

proved audit: minor defect ★ key step

Minor defect (second audit). The theorem and corollary are correct; every step re-derived (finiteness of the slices, support inside the slice via $\omega\mapsto\varpi^{2c}\omega$ and Lemma 3, diagonal coefficient $\ge1$ by Lemma 2, the $\SL$ reduction). For the slice $S_{4,0}$, $m=3$, the computed matrix is lower triangular with diagonal $q^3(q+1)$, $q^3(q+1)^2$, $q^3(q+1)(q^2+1)$, $q^3(q+1)^2(q^2+1)$. One citation is imprecise: Proposition B as stated needs the chosen orbits to span $C_K$, which a single slice does not; what is used is the induction in its proof, applied inside a finite slice that the translates do not leave — valid. The coset map in the corollary is written in the reverse direction (bijective either way).

A downward induction in finite slices of fixed degree, using the triangular rows of Lemma 5.6 and central shifts, puts every orbit indicator in the Hecke span of the basic function; restricting to Pfaffian valuation zero gives the $\SL$ case. This replaces 'holds under (A)' by 'holds' for both rows of Table 1, with no dominant lifts, no Theorem E and no (A1)–(A5).

Theorem 5.7.

Theorem. $\cH_K\cdot\varphi_0=C_K$ for $X=\GL_{2m}/\Sp_{2m}$, every $m\ge1$, and every non-archimedean local field $F$.

Corollary. $(\mathrm{SG})$ holds for $\SL_{2n}/\Sp_{2n}$, every $n$, every $F$.

Proof of the theorem.

Orbits. By the symplectic Smith form, $K\backslash X(F)\leftrightarrow\{b\in\Z^m:b_1\ge\dots\ge b_m\}$; the orbits are compact open and their indicators span $C_K$.
Slices. For $d,c\in\Z$ the set $S_{d,c}:=\{b:\ b_m\ge2c,\ |b|=d\}$ is finite, and every label $b$ lies in $S_{|b|,\lfloor b_m/2\rfloor}$.
Triangular translates. For $b\in S_{d,c}$ put $f_b:=T_{(c,\dots,c)}\ast T_{\lambda^*(b-2c\mathbf 1)}$. By the central shift of Computation 5.5 and Lemmas 2 and 3 of Lemma 5.6,

$$f_b\ast\varphi_0=\sum_{b'\in S_{d,c},\ b'\ge_{\mathrm{dom}}b}N\big(\lambda^*(b-2c),\,b'-2c\big)\,\one_{O_{b'}},\qquad N\big(\lambda^*(b-2c),\,b-2c\big)\ge1 .$$

Induction. Descending induction on $\ge_{\mathrm{dom}}$ in the finite poset $S_{d,c}$ — Proposition B with the order $b'\preceq b\iff b'\ge_{\mathrm{dom}}b$, restricted to $S_{d,c}$ — gives $\one_{O_b}\in\cH_K\cdot\varphi_0$ for every $b$. $\square$

Proof of the corollary. The degree-zero reduction in the last part of Proposition F is re-verified without hypotheses. $X_{\SL}(F)=\{\Pf=1\}$ is a single $\SL_{2n}(F)$-orbit (a symplectic basis and $\Pf(gJg^{\mathsf T})=\det g$); $O_b\cap X_{\SL}(F)$ is a single $K_{\SL}$-orbit; the coset spaces $Kt^\lambda K/K$ and $K_{\SL}t^\lambda K_{\SL}/K_{\SL}$ correspond bijectively when $|\lambda|=0$ (adjust by $\operatorname{diag}(u,1,\dots,1)$); and $T_\lambda\ast\varphi_0$ is supported on $\val\Pf=|\lambda|$. So the degree-zero part of $(\mathrm{SG})$ for $\GL_{2n}$ is $(\mathrm{SG})$ for $\SL_{2n}$. $\square$

Consistency. $m=1$ is Proposition J(3); $m=2$ is Computation 5.5; $\SL_4/\Sp_4\cong\Spin_6/\Spin_5$ agrees with Theorem 5.3.

Mechanism versus coverage. For an uncovered label $b$ (some $b_i=b_{i+1}$ odd) the row used is $\lambda^*(b)$, whose top orbit $\varphi(\lambda^*(b))=(b_1+b_2,b_1+b_2,b_3+b_4,\dots)$ lies strictly above $b$: the induction runs downward from the top of $S_{d,c}$ instead of upward from dominant lifts. Nothing from Theorem E or (A1)–(A5) is used, and $\operatorname{char}F$ and $q$ are arbitrary.

Builds on

Lemma 5.6 · $\GL_{2m}/\Sp_{2m}$: the rows $\lambda^*(b)=(b,0)$ are triangular for reverse dominance
Computation 5.5 · $\GL_4/\Sp_4$: lattice counts and a Hecke element for the uncovered label $(1,1)$

Built on by

Summary 5.10 · Table changes at function level; no row disagrees between the levels; nothing here depends on (A)
Dead end 5.9 · $E_6/F_4$ at function level: what the lattice method would need
Audit 7.23 · Audit of the $\GL_{2m}/\Sp_{2m}$ theorem: the induction holds (directly, not through Proposition B as stated); the $\SL$ corollary and the sheaf consequence hold
Re-derivation X1.5 · (SG) for GL₂ₘ/Sp₂ₘ re-derived from Lemmas 2–3 alone; A4-13's open cells now decided
Summary X3.7 · A7-25, A7-32 re-derived; the $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ row
Audit 9.3 · Registry after the lattice-count audit: every row except $E_6/F_4$ decided at both levels; the $E_6$ Weyl conjugacy re-checked
Summary 9.4 · Synthesis: $E_6/F_4$ is the one divergence (categorical generation holds, functions fail under (A)); no reverse case

Original record, as the agent wrote it at 03:32 UTC

(SG) for GL_{2m}/Sp_{2m} (all m) and SL_{2n}/Sp_{2n} (all n), every local field F

Theorem. $\mathcal H_K\cdot\varphi_0=C_K$ for $X=\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}$, every $m\ge1$, every non-archimedean local field $F$. Proof. 1. Orbits: $K\backslash X(F)\leftrightarrow\{b\in\mathbb Z^m:b_1\ge\dots\ge b_m\}$ (symplectic Smith form), compact open, indicators span $C_K$.

Fix $d\in\mathbb Z$, $c\in\mathbb Z$; $S_{d,c}:=\{b:\ b_m\ge2c,\ |b|=d\}$ is finite. Every label lies in some $S_{|b|,\lfloor b_m/2\rfloor}$.
For $b\in S_{d,c}$ put $f_b:=T_{(c,\dots,c)}\ast T_{\lambda^*(b-2c\mathbf 1)}$. By node 5 (central shift) and node 6, Lemmas 2–3,

$$f_b\ast\varphi_0=\sum_{b'\in S_{d,c},\ b'\ge_{\mathrm{dom}}b}N(\lambda^*(b-2c),b'-2c)\,\mathbf 1_{O_{b'}},\qquad N(\lambda^*(b-2c),b-2c)\ge1 .$$

Descending induction on $\ge_{\mathrm{dom}}$ in the finite poset $S_{d,c}$ (Proposition B with the order "$b'\preceq b\iff b'\ge_{\mathrm{dom}}b$" restricted to $S_{d,c}$): $\mathbf 1_{O_b}\in\mathcal H_K\cdot\varphi_0$ for all $b$. $\square$

Corollary. (SG) for $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$, every $n$, every $F$: the degree-$0$ reduction of classical.html (Prop. F, last part) is re-verified without hypotheses: $X_{\mathrm{SL}}(F)=\{\mathrm{Pf}=1\}$ is one $\mathrm{SL}_{2n}(F)$-orbit (symplectic basis + $\mathrm{Pf}(gJg^{\mathsf T})=\det g$), $O_b\cap X_{\mathrm{SL}}(F)$ is one $K_{\mathrm{SL}}$-orbit, $Kt^\lambda K/K\to K_{\mathrm{SL}}t^\lambda K_{\mathrm{SL}}/K_{\mathrm{SL}}$ is bijective for $|\lambda|=0$ (adjust by $\mathrm{diag}(u,1,\dots)$), $T_\lambda\ast\varphi_0$ lives on $\operatorname{val}\mathrm{Pf}=|\lambda|$, so the degree-$0$ part of (SG) for $\mathrm{GL}_{2n}$ is (SG) for $\mathrm{SL}_{2n}$.

Consistency: $m=1$ is Prop. J(3); $m=2$ is node 5; $\mathrm{SL}_4/\mathrm{Sp}_4\cong\mathrm{Spin}_6/\mathrm{Spin}_5$ agrees with node 3.

Mechanism vs. coverage: for an uncovered $b$ (some $b_i=b_{i+1}$ odd) the row used is $\lambda^*(b)$, whose top orbit $\varphi(\lambda^*(b))=(b_1+b_2,b_1+b_2,b_3+b_4,\dots)$ lies strictly above $b$; the induction runs downward from the top of $S_{d,c}$ instead of upward from dominant lifts. Nothing from Theorem E or (A1)–(A5) is used; $\operatorname{char}F$ and $q$ are arbitrary.

Sources

classical.html#Proposition F
classical.html#Proposition B

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 5.7 · \((\mathrm{SG})\) for \(\GL_{2m}/\Sp_{2m}\) (all \(m\)) and \(\SL_{2n}/\Sp_{2n}\) (all \(n\)), every non-archimedean local field

Builds on

Built on by

(SG) for GL_{2m}/Sp_{2m} (all m) and SL_{2n}/Sp_{2n} (all n), every local field F

Sources