§10 · Audit · Second audit, then re-checks · 2026-10-17T06:59:16Z

Correction 10.2 · Codimensions of orbits in $L(\GL_{2m}/\Sp_{2m})$ in every component, without connectedness of $LX$

proved correction ★ key step

Correction. Corrects Objection 7.11 · Objection to the codimension citation: Chen–Yi's Prop. 64 assumes $LX$ connected; a one-line reduction is proposed.

The scalar shift $\omega\mapsto t\omega$ moves the Pfaffian valuation by $m$, so it reduces only the components whose valuation is divisible by $m$ to the degree-zero one, where Chen–Yi's codimension formula applies. A direct count in the truncated loop space gives the formula $\operatorname{codim}=\langle\rho,\mu-\nu\rangle$ in every component. This justifies the defect computation of the minuscule fibres for all $k$, and with it the $\GL_4/\Sp_4$ result.

Correction 10.2.

Let $X=\GL_{2m}/\Sp_{2m}$ over $k=\C$, with the notation of Lemma 3.1: labels $a=(a_1\ge\dots\ge a_m)\in\Z^m$, $\mu_a=(a_1,a_1,\dots,a_m,a_m)\in\Lambda_S^+$, $\cO_a$ the $L^+G$-orbit of $D_a:=\bigoplus_jt^{a_j}J_2$, and $\sigma(\omega)=t\omega$.

$\sigma$ maps the component $\{\val\Pf=s\}$ of $LX$ onto $\{\val\Pf=s+m\}$. Hence some power of $\sigma$ carries the component $s$ to the component $0$ if and only if $m\mid s$.
For labels $b\le a$ in one component,

$$\operatorname{codim}_{\overline{\cO_a}}\cO_b=\langle\rho,\mu_a-\mu_b\rangle=\sum_{j=1}^m(2m+2-4j)(a_j-b_j).$$

Consequently the codimensions used in Theorem 3.6, $c_\nu=\epsilon+\langle\rho,\nu^{top}-\nu\rangle$ for $\nu\in\{0,1,2\}^m$ with $|\nu|=k$, hold for every $1\le k\le2m-1$, and $\operatorname{def}(\nu)=n_1(\nu)$ stands, without appeal to Chen–Yi, Prop. 64.

Components. For a $2m\times2m$ alternating $\omega$, $\Pf(t\omega)=t^m\Pf(\omega)$. So $\sigma^j$ carries the component $s$ onto $s+jm$, and the component $s$ reaches the component $0$ exactly when $m\mid s$. For $\GL_4$ the odd components are never reached, so the reduction proposed in Objection 7.11 covers only the components with $m\mid s$. There it is valid: $\sigma^{-s/m}$ lands in component $0$, which by Lemma 3.7 is the $\SL_{2m}/\Sp_{2m}$ case, where Chen–Yi, Prop. 64 applies.

Reduction to nonnegative labels. $\sigma$ is an $L^+G$-equivariant automorphism of $LX$ with $\sigma(\cO_e)=\cO_{e+(1^m)}$, so $\sigma^c$ preserves orbit closures and codimensions and leaves $\mu_a-\mu_b$ unchanged. If $b\le a$ in one component, the partial sums satisfy $\sum_{j\le i}b_j\le\sum_{j\le i}a_j$ with equality at $i=m$, so $b_m\ge a_m$; after a shift we may assume $a_m\ge0$. By Chen–Yi, Thm. 18 (which needs no connectedness of $LX$), $\overline{\cO_a}=\bigsqcup_{e\le a}\cO_e$, and every such $e$ has $a_m\le e_m\le e_1\le a_1$. Put $s=\sum_ja_j$. Then $\overline{\cO_a}$ lies in the set $Z_s$ of integral forms with $\val\Pf=s$, and every form in $Z_s$ has all elementary divisors in $[0,s]$.

Finite-type model. Fix $N>s$ and let $\pi:\Lambda^2\mathcal O^{2m}\to\Lambda^2(\mathcal O/t^N)^{2m}$ be reduction modulo $t^N$: a trivial bundle with fibre $t^N\Lambda^2\mathcal O^{2m}$, equivariant for $L^+G\to G_N:=\GL_{2m}(\mathcal O/t^N)$. Since $N>s$, $Z_s=\pi^{-1}(Z_s^{(N)})$ with $Z_s^{(N)}=\{\bar\omega:\val\Pf\bar\omega=s\}$ open. Let $\omega'\in Z_s$ with $\omega'\equiv D_e\pmod{t^N}$, where $e$ is a label with $\sum e_j=s$, $e_j\ge0$. Then

$$\omega'=D_e\big(1+D_e^{-1}(\omega'-D_e)\big),$$

and the entries of $D_e^{-1}(\omega'-D_e)$ have valuation $\ge N-e_1\ge1$. So $\omega'$ and $D_e$ have the same Smith invariants, and alternating matrices over $\mathcal O$ with the same Smith invariants are congruent. Hence each $\cO_e\subseteq Z_s$ is the full preimage of the $G_N$-orbit $\cO_e^{(N)}$ of $D_e\bmod t^N$. As $\pi$ is open, $\overline{\cO_a}=\pi^{-1}\big(\overline{\cO_a^{(N)}}\big)\cap Z_s$. The codimension of $\cO_b$ in $\overline{\cO_a}$ is therefore computed in the finite-type model, where the orbits are irreducible: $$\operatorname{codim}_{\overline{\cO_a}}\cO_b=\dim\cO_a^{(N)}-\dim\cO_b^{(N)} .$$

Stabilizer Lie algebra. Let $X\in\mathfrak{gl}_{2m}(\mathcal O/t^N)$ satisfy $XD_a+D_aX^T\equiv0\pmod{t^N}$. Write $X=(X_{ij})$ in $2\times2$ blocks and use $YJ_2+J_2Y^T=\tr(Y)\,J_2$.
- Diagonal block $j$: the condition is $t^{a_j}\tr(X_{jj})\equiv0$, i.e. $\tr X_{jj}\in t^{N-a_j}\mathcal O$, leaving $3N+a_j$ dimensions.
- Pair $i<j$ (so $a_i\ge a_j$): $X_{ij}t^{a_j}J_2+t^{a_i}J_2X_{ji}^T\equiv0$ determines $X_{ij}\bmod t^{N-a_j}$ from the free block $X_{ji}$, leaving $4N+4a_j$ dimensions; the $(j,i)$ equation is the transpose of the $(i,j)$ one.
The total is $\sum_j(3N+a_j)+\sum_{i<j}(4N+4a_j)=(2m^2+m)N+\sum_j(4j-3)a_j$.

Orbit dimension. In characteristic $0$ the stabilizer group scheme is smooth, and $G_N$ is connected of dimension $4m^2N$. Therefore

$$\dim\cO_a^{(N)}=4m^2N-(2m^2+m)N-\sum_j(4j-3)a_j=m(2m-1)N-\sum_j(4j-3)a_j .$$

The formula. By steps 3 and 5,

$$\operatorname{codim}_{\overline{\cO_a}}\cO_b=\sum_j(4j-3)(b_j-a_j)=\sum_j4j\,(b_j-a_j)=\sum_j(2m+2-4j)(a_j-b_j),$$

using $\sum_j(a_j-b_j)=0$ twice. Since $\rho=\big(\tfrac{2m+1}2-i\big)_{i=1}^{2m}$ gives $\langle\rho,e_{2j-1}+e_{2j}\rangle=2m+2-4j$, this is $\langle\rho,\mu_a-\mu_b\rangle$.

Application to Theorem 3.6. For $\lambda=\omega_k$ all strata $\nu\in\{0,1,2\}^m$ with $|\nu|=k$ are nonnegative labels in the component $k$. With $D=\dim\cO_{\nu^{top}}+\epsilon$ (generic fibre of dimension $\epsilon=k\bmod2$), step 6 gives $c_\nu=D-\dim\cO_\nu=\epsilon+\langle\rho,\nu^{top}-\nu\rangle$ for every $k$, and the computation of Theorem 3.6 gives $\operatorname{def}(\nu)=n_1(\nu)$. The maximal-defect lemma Lemma 3.5 then applies as stated there, so the conclusion $\mathcal M=\Lambda_S^+$ for $\GL_4/\Sp_4$ and the list of labels in $\mathcal M$ for $\GL_{2m}/\Sp_{2m}$ are unaffected.

The argument is written over $k=\C$, the setting of the sheaf statement. The count of step 4 is characteristic-free, and the same codimensions hold over $\overline{\F}_p$ for all but finitely many $p$ by spreading out, which is the form in which the run passes ℓ-adic statements to $\C$.

Verified by computation: g2_orbits_lie.py, g2_defect_coverage.py

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Objection 7.11 · Objection to the codimension citation: Chen–Yi's Prop. 64 assumes $LX$ connected; a one-line reduction is proposed
Lemma 3.1 · The loop space of $\GL_{2m}/\Sp_{2m}$: orbits, closure order, scalar shift and codimensions
Lemma 3.7 · $\SL_{2n}/\Sp_{2n}$ is the degree-zero component of $\GL_{2n}/\Sp_{2n}$; its smallest uncovered label stays open

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Proposition X1.4 · L(GL₂ₘ/Sp₂ₘ): codim = ⟨ρ,μ−ν⟩ in every component, char ≠ 2; σ reaches only m | s
Summary 9.5 · Verdict after the second audit: $E_6/F_4$ is still the one divergence

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

Correction 10.2 · Codimensions of orbits in \(L(\GL_{2m}/\Sp_{2m})\) in every component, without connectedness of \(LX\)

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