§7 · Agent 7 · Adversarial audit · 2026-10-17T04:00:32Z

Audit 7.26 · Audit of the orbit-label lemma and the Hecke label: both hold

proved audit: correct

Second audit: correct.

An independent check of the cone, the dimension formula, the two-element interval and the Hecke label \(\lambda_k\), including the hypotheses of Chen–Yi's codimension formula. No defect found.

Audit 7.26.

Lemma 8.2 and Lemma 8.3 hold as stated. In particular Chen–Yi, Prop. 64 applies for every \(p\ne2\), and \(\Gr^{\le\lambda_1}=\Gr^{\lambda_1}\sqcup\Gr^{\omega_2^\vee}\sqcup\Gr^0\), with cells of dimensions \(32,22,0\), is singular.

  1. Cone. \(p=\tfrac12(1-\theta)\) kills the black coroots and sends \(\alpha_1^\vee,\alpha_6^\vee\) to \(\tfrac12e_1,\tfrac12e_6\). Both \(\Lambda_S\) and \(\Z e_1\oplus\Z e_6\) have index \(3\) in \(\Z\omega_1^\vee\oplus\Z\omega_6^\vee\), and the second is contained in the first; so they are equal, and \(\Lambda_S\cap Q^+=\Z_{\ge0}e_1\oplus\Z_{\ge0}e_6\).
  2. Codimension formula. Chen–Yi's Prop. 64 assumes that \(LX\) is connected. Here \(\Lambda_T=Q^\vee\) because \(G\) is simply connected, so \(\pi_0(LX)=\Lambda_S/(\Lambda_S\cap Q^\vee)=0\). Its proof (Lemma 65) spreads out over a discrete valuation ring between \(\Z[1/2]\) and \(\C\) with smooth fixed-point schemes, so it holds for every \(p\ne2\), not only for large \(p\). Since \(2\rho\) has coefficient \(16\) at \(\alpha_1\) and at \(\alpha_6\), \(\delta(a,b)=8(a+b)\).
  3. Interval. \(e_6\) generates an edge of the cone, so \([\mu_k,\mu_k']=\{\mu_k,\mu_k'\}\), of codimension \(8\).
  4. Hecke label. \(\lambda_k\in\Lambda_T\) (its class is \(c_k+2=\tfrac{3(k+1)}2\equiv0\bmod3\)), it is dominant, and \(\varphi(\lambda_k)=\mu_k'\). An independent computation gives \(\langle2\rho,\lambda_k\rangle=32,80,128=\delta(\mu_k')\) for \(k=1,3,5\). The identity \(\langle2\rho,\lambda\rangle-\langle\rho,\varphi\lambda\rangle=\langle2\rho_L,\lambda\rangle\) follows again from \(\theta\rho=2\rho_L-\rho\). The stratification of \(\Gr^{\le\lambda_1}\) is as stated.

Severity: none.

Verified by computation: e6_fibre_A7.py

Builds on

  • Lemma 8.2 · Orbit labels, the dimension function, and the two-element interval \([\mu_k,\mu_k']\)
  • Lemma 8.3 · The Hecke label \(\lambda_k\): a dominant lift of \(\mu_k'\) whose cell has dimension \(\delta(\mu_k')\)

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Original record, as the agent wrote it at 04:00 UTC

A8-2/A8-3: labels, two-element interval, codimension 8 and the Hecke label verified

A8-2 (holds). (i) \(p=\tfrac12(1-\theta)\) kills the black coroots and sends \(\alpha_1^\vee,\alpha_6^\vee\) to \(\tfrac12e_1,\tfrac12e_6\); since \(\Lambda_S=\mathbb Ze_1\oplus\mathbb Ze_6\) (both of index 3 in \(\mathbb Z\omega_1^\vee\oplus\mathbb Z\omega_6^\vee\)), \(\Lambda_S\cap Q^+=\mathbb Z_{\ge0}e_1\oplus\mathbb Z_{\ge0}e_6\). (ii) Hypotheses of Chen–Yi Prop. 64: \(LX\) connected because \(\Lambda_T=Q\) (\(G\) simply connected), so \(\pi_0(LX)=\Lambda_S/(\Lambda_S\cap Q)=0\); its proof (Lemma 65) spreads out over a DVR between \(\mathbb Z[1/2]\) and \(\mathbb C\) with smooth fixed-point schemes, so it holds for every \(p\ne2\), not only \(p\gg0\). \(2\rho\) has coefficient 16 at \(\alpha_1,\alpha_6\), so \(\delta(a,b)=8(a+b)\). (iii) \(e_6\) is a generator of the cone, so \([\mu_k,\mu_k']=\{\mu_k,\mu_k'\}\), codimension \(8\).

A8-3 (holds). \(\lambda_k\in\Lambda_T\) (class \(c_k+2=\tfrac{3(k+1)}2\equiv0\bmod3\)), dominant, \(\varphi(\lambda_k)=\mu_k'\); \(\langle2\rho,\lambda_k\rangle=32,80,128=\delta(\mu_k')\) for \(k=1,3,5\) (my script); the identity \(\langle2\rho,\lambda\rangle-\langle\rho,\varphi\lambda\rangle=\langle2\rho_L,\lambda\rangle\) re-derived from \(\theta\rho=2\rho_L-\rho\). \(\mathrm{Gr}^{\le\lambda_1}=\mathrm{Gr}^{\lambda_1}\sqcup\mathrm{Gr}^{\omega_2^\vee}\sqcup\mathrm{Gr}^0\) (dims 32, 22, 0) is correct and singular. Severity: none.

Sources