§7 · Agent 7 · Adversarial audit · 2026-10-17T04:11:59Z

Audit 7.29 · Audit of the fixed-point computation: 8 fixed points, \(\dim Z_k=36\), stabilizer algebra of dimension \(29\), so \(\dim F_k\ge7\)

proved audit: correct

Second audit: correct.

An independent recomputation of the fixed points of \(F_k\), the Lie algebra of the centralizer and the stabilizer dimension, with a justification of each counting step. No defect found.

Audit 7.29.

Computation 8.6 holds. For \(k=1,3,5\): \(|W\lambda_k|=270\); eight elements of \(W\lambda_k\) map to \(\mu_k\) under \(\varphi\) and one maps to \(\mu_k'\); \(\dim Z_k=36\); and \(\dim(\operatorname{Lie}Z_k\cap\operatorname{Lie}P(-\nu))=29\) at each of the eight fixed points. Hence \(\dim F_k\ge7\). The mirror case gives identical numbers.

An independent script re-enters the root data, takes \(\theta\) from the family page, and checks that it is an isometric involution.

For \(k=1,3,5\): \(|W\lambda_k|=270\), \(\#\{\nu\in W\lambda_k:\varphi(\nu)=\mu_k\}=8\) and \(\#\{\nu\in W\lambda_k:\varphi(\nu)=\mu_k'\}=1\).
The Levi of \(\mu_k\) is the \(D_5\) on nodes \(2,\dots,6\) (40 roots); \(\theta\) fixes the 24 black roots and pairs the other 16 into 8 orbits. So \(\dim\operatorname{Lie}Z_k=4+24+8=36\) (type \(B_4\)), and \(\dim(\operatorname{Lie}Z_k\cap\operatorname{Lie}P(-\nu))=4+18+7=29\) at each of the 8 fixed points.
\(\langle\rho_L,\nu\rangle\) takes the values \(-3,-2,-1,0,0,1,2,3\), the eight terms of \(c_8(q)\) in Computation 6.4.
Mirror case (Levi \(D_5\) on nodes \(1,\dots,5\), \(\lambda=\omega_1^\vee+c_k\omega_6^\vee\)): identical numbers.

Justification of the counting steps.

\(Z_k\subseteq\Stab(x_{\mu_k})\), since \(h\in Z_G(\mu_k)^\theta\) commutes with \(t^{\mu_k}\).
The intersection can be computed weight by weight: both subspaces are \(T'\)-stable, and the \(T'\)-weights of the \(B_4\) lines are distinct and nonzero (the long roots are the black roots, the short roots are the projections of the pairs).
A line \(X_\alpha+\theta X_\alpha\) lies in \(\operatorname{Lie}P(-\nu)\) iff \(\langle\alpha,\nu\rangle\le0\) and \(\langle\theta\alpha,\nu\rangle\le0\). The two numbers are equal: roots of the \(D_5\)-Levi have no \(\alpha_1\)-component, so \(\langle\gamma,\nu\rangle=\langle\gamma,m\rangle\) with \(m=\nu^\theta\), and \(\langle\theta\alpha,m\rangle=\langle\alpha,m\rangle\).
\(\operatorname{Lie}(H^\theta)=(\operatorname{Lie}H)^\theta\) for \(p\ne2\), and \(\dim\Stab\le\dim\operatorname{Lie}(\Stab)\) in every characteristic.

So \(\dim Z_k\cdot t^\nu L^+G\ge7\) and \(\dim F_k\ge7\). Outcome: holds. Severity: none.

Verified by computation: e6_fibre_A7.py

Builds on

Computation 8.6 · Torus-fixed points of \(F_k\) and a seven-dimensional orbit of \(Z_{G^\theta}(\mu_k)^\circ\)

Built on by

Summary 7.32 · Final audit verdict: the five conclusions survive; \(E_6/F_4\) is the one divergence, under (A)

Original record, as the agent wrote it at 04:11 UTC

A8-6 recomputed: 8 fixed points, dim Z = 36, stabilizer Lie algebra 29, so dim F_k >= 7

Independent script (e6_fibre_A7.py; root data re-entered, \(\theta\) from families/e6-f4.html, isometry checked). For \(k=1,3,5\): \(|W\lambda_k|=270\); \(\#\{\nu\in W\lambda_k:\varphi(\nu)=\mu_k\}=8\), \(\#\{\varphi(\nu)=\mu_k'\}=1\). The Levi of \(\mu_k\) is the \(D_5\) on nodes 2–6 (40 roots); \(\theta\) fixes the 24 black roots and pairs the other 16 into 8 \(\theta\)-orbits. So \(\dim\mathrm{Lie}\,Z_k=4+24+8=36\) (\(B_4\)), and \(\dim(\mathrm{Lie}\,Z_k\cap\mathrm{Lie}\,P(-\nu))=4+18+7=29\) at each of the 8 fixed points. \(\langle\rho_L,\nu\rangle\) takes the values \(-3,-2,-1,0,0,1,2,3\), the 8 terms of \(c_8(q)\) in A6-4. Mirror case (Levi \(D_5\) on nodes 1–5, \(\lambda=\omega_1^\vee+c_k\omega_6^\vee\)): identical numbers.

Justification checks.

\(Z_k\subseteq\mathrm{Stab}(x_{\mu_k})\), since \(h\in Z_G(\mu_k)^\theta\) commutes with \(t^{\mu_k}\).
The intersection can be computed weight by weight: both subspaces are \(T'\)-stable, and the \(T'\)-weights of the \(B_4\) lines are distinct and nonzero (long roots = black roots; short roots = projections of the pairs).
A line \(X_\alpha+\theta X_\alpha\) lies in \(\mathrm{Lie}\,P(-\nu)\) iff \(\langle\alpha,\nu\rangle\le0\) and \(\langle\theta\alpha,\nu\rangle\le0\). The two numbers are equal: \(D_5\) roots have no \(\alpha_1\)-component, so \(\langle\gamma,\nu\rangle=\langle\gamma,m\rangle\) with \(m=\nu^\theta\), and \(\langle\theta\alpha,m\rangle=\langle\alpha,m\rangle\).
\(\mathrm{Lie}(H^\theta)=(\mathrm{Lie}\,H)^\theta\) for \(p\ne2\), and \(\dim\mathrm{Stab}\le\dim\mathrm{Lie}(\mathrm{Stab})\) in every characteristic.

So \(\dim Z_k\cdot t^\nu L^+G\ge7\) and \(\dim F_k\ge7\). Outcome: holds. Severity: none.

Sources

A8.json Computation 8.6
scratchpad/solve/A7/e6_fibre_A7.py
families/e6-f4.html §2–4

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