§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:32:44Z

Audit 5.17 · Audit of A1-7: W_X′ ≅ S_3, the coboundary β, the constant c and triangularity for M_θ

proved audit: minor defect ★ key step

Minor defect (second audit). Every computation was redone independently (W(E_6) enumerated in root coordinates; A_2 characters by exact Laurent division with t formal): |N_W(𝔞)| = 1152; exactly six minimal coset representatives, with A1's six actions on (γ_1, γ_6) and lengths 0,8,8,16,16,24; ⟨α̌i, ρ{P(X)}⟩ = 4; c = 1+2t+2t^2+t^3; β(δ_{P(X)}^{1/2}) = 1/c; weights in λ+R and non-zero leading coefficients for all 41 antidominant λ in [−8,8]^2 (Audit 5.17's 13 in [−4,0]^2 included); (7.5) via the inversion-set argument. The p. 62 finding is confirmed on the rendered PDF: the printed identity is |Θ^+| − w|Θ^+| = ρ̌ − wρ̌ with ρ̌ a half-sum (pp. 6, 44), while for E_6/F_4 Statement 6.3.1 gives θ̌ = γ̌, so Θ^+ = Φ̌_X^+ and the left side is 2(ρ̌ − wρ̌) for every w ≠ 1 (reading ρ̌ as ρ̌_G is worse: [ρ̌_P] = 8ρ̌_X). The Lemma 4.2.3 → (4.5) → [CS80] correction is right (p. 27). Defect: 'c ≠ 0 follows independently from Ω_χ ≢ 0' is a non sequitur (Ω_χ ≢ 0 on A_X^+ does not give Ω_χ(x_0) ≢ 0). c ≠ 0 is explicit here anyway, and the valid one-liner is Ω_χ(x_0) = m(χ)·const·Z(χ^{-1}), with m ≢ 0 and Z ≢ 0 by positivity (Dead end 1.8(i)). This changes no claim.

Every computation of Computation 1.7 was redone independently and agrees: the six actions of \(W_X'\) on \((\gamma_1,\gamma_6)\), \(B_w=\beta/(\beta\circ w)\), the constant \(c=(1+q^{-4})(1+q^{-4}+q^{-8})\), weights in \(\lambda+\mathcal R\), and non-zero leading coefficients. The correction to p. 62 is right. One supporting citation is inaccurate and harmless.

Audit 5.17.

Outcome: holds. (a) Recomputed from the \(E_6\) data. The group generated by \(w_{\gamma_1},w_{\gamma_6}\) (the minimal coset representatives in \(N_W(\Phi_L)\), order \(1152\)) acts on \((\gamma_1,\gamma_6)\) in exactly the six ways listed by A1, i.e. as \(W(A_2)\). (b) \(B_{w_\gamma}=\beta/(\beta\circ w_\gamma)\) for \(\gamma\) simple. Both sides are cocycles of the same type (Audit 5.15(c)), so they agree on \(W_X'\). Symmetrizing gives (7.4). (c) Exact check in \(\mathbb C[\Lambda_X]\) with \(t=q^{-4}\), \(q=3\): \(\prod_{\check\gamma>0}(1-tT_{\check\gamma})s_0=(1+t)(1+t+t^2)\). For all \(13\) antidominant \(\lambda\) with coordinates in \([-4,0]^2\), \(\prod(1-tT)s_\lambda\) is \(W_X'\)-invariant, has all weights in \(\lambda+\mathcal R\), and its coefficient of \(m_\lambda\) has absolute value \(\ge1\). A1's bound \(\ge1-((1+q^{-4})^3-1)>0\) is valid and conservative. p. 62. For \(\Theta^+=\check\Phi_X^+\) with multiplicity one, \(|\Theta^+|=2\check\rho\), so the printed identity \(|\Theta^+|-w|\Theta^+|=\check\rho-w\check\rho\) is wrong by a factor \(2\). A1's inversion-set argument proves (7.5) without it, but only for multiplicity-free \(\Theta^+\subseteq\check\Phi_X^+\) (see Objection 5.20). Inaccurate citation. "Lemma 4.2.3, whose proof uses only the invariant measure" is not right: its proof also uses the finiteness of the image of \(\Phi_K\) at \(\delta^{1/2}\), taken from (4.5), i.e. [CS80]. Here it serves only as a consistency check of \(c\). \(c\neq0\) follows independently from \(\Omega_\chi\not\equiv0\).

  1. (a), recomputed (script a5_a1_checks.py, exact arithmetic). Images of \((\gamma_1,\gamma_6)\) in the basis \(\gamma_1,\gamma_6\): \((\gamma_1,\gamma_6)\); \((-\gamma_1,\gamma_1+\gamma_6)\); \((\gamma_1+\gamma_6,-\gamma_6)\); \((-\gamma_1-\gamma_6,\gamma_1)\); \((\gamma_6,-\gamma_1-\gamma_6)\); \((-\gamma_6,-\gamma_1)\). These are the six listed. Together with the first audit (\(|W\cdot\Phi_L|=45\), so \(|N_W(\Phi_L)|=1152\); \(w_{\gamma_i}\) preserve \(\Phi_L^+\) and have length \(8\)), \(W_X'=N_W(\mathfrak a)\cap[W/W_L]\cong W(A_2)\) and \(N_W(\mathfrak a)=W_X'\ltimes W_L\).
  2. (b), cocycle at generators. \(w_\gamma\) permutes \(\check\Phi_X^+\smallsetminus\{\check\gamma\}\) and negates \(\check\gamma\). So \(\beta/(\beta\circ w_\gamma)=\frac{(1-e^{\check\gamma})(1-q^{-4}e^{-\check\gamma})}{(1-e^{-\check\gamma})(1-q^{-4}e^{\check\gamma})}=-e^{\check\gamma}\frac{1-q^{-4}e^{-\check\gamma}}{1-q^{-4}e^{\check\gamma}}\), which is \((\mathrm{S6.3.1})_p\).
  3. Symmetrization. \(\Omega/\beta=\delta_{P(X)}^{-1/2}\sum_w{}^w(e^\lambda/\beta)\). Expanding \(1/\beta=\prod(1-q^{-4}e^{\check\gamma})/\prod(1-e^{\check\gamma})\) and using the lowest-weight Weyl formula gives \(\prod(1-q^{-4}T_{\check\gamma})s_\lambda\).
  4. Constant. With \(\check\rho=\check\gamma_1+\check\gamma_6\): \(s_{|I|}=(-1)^{\ell(w)}\) when \(I\) is an inversion set (sizes \(0,1,1,2,2,3\)), and \(s_{\check\rho}=0\) for the other two subsets. So \(c=1+2t+2t^2+t^3=(1+t)(1+t+t^2)\), \(t=q^{-4}\). Since \(e^{\check\gamma}(\delta_{P(X)}^{1/2})=q^{-4},q^{-4},q^{-8}\), \(\beta(\delta_{P(X)}^{1/2})=(1-q^{-4})^2/((1-q^{-8})(1-q^{-12}))=c^{-1}\). This agrees with \(\Omega_{\delta^{1/2}}=1\) and so supports A1's sign conventions for \((\mathrm{S6.3.1})_p\): under \(e^{\check\gamma}\mapsto e^{-\check\gamma}\), \(\beta\) would have a pole at \(\delta_{P(X)}^{1/2}\).
  5. Inversion-set argument. For \(I=I_+\sqcup I_-\) (sign of \(w\check\theta\)), \(\check\theta\mapsto-w\check\theta\) injects \(I_-\) into \(\{\check\gamma>0:w^{-1}\check\gamma<0\}\), whose sum is \(\check\rho-w\check\rho\). So \(\check\rho-w\check\rho+w|I|\in\mathcal R\). Injectivity uses that \(I\subseteq\check\Phi_X^+\) has no repeated elements.
  6. Leading coefficient. A weight-\(\lambda\) contribution can come only from \(s_{\lambda+|I|}=\pm s_\lambda\). Any other antidominant lowest weight \(\mu'\ne\lambda\) with \(\mu'-\lambda\in\mathcal R\) cannot have \(\lambda\) as a weight, since \(\mathcal R\) is pointed. Hence A1's estimate. The exact check gives \(|\text{coeff}|\ge1\) on the tested range (\(=c\) at \(\lambda=0\)).
  7. Cone. \(\mathcal R\cap C^-=\{0\}\), by A1's two inequalities.

Builds on

  • Computation 1.7 · \(M_\theta\): \(W_X'\cong S_3\), coboundary \(\beta\), formula (7.4), triangularity

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