§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:05:46Z

Objection 5.11 · Audit of A3-7: colour weights (gap repaired by δ) and the valuation cone

proved audit: correct ★ key step

Second audit: correct. The gap is real: Proposition 3.7 step 3 only shows that the colour weights form a Z-basis of Z omega1 + Z omega6 made of dominant weights, and {omega1, omega1+omega6} is another such basis, while step 5 needs regular eigenfunctions of weights exactly omega1 and omega6. The delta-repair is valid: delta C delta^{-1} = C and delta fixes the Tits lift sigma_L, so delta commutes with theta_0, preserves H and permutes the colours, and a delta-stable basis of Z^2 inside N^2 must be {(1,0),(0,1)}. I re-checked steps 4-5 (arc valuations with mu(t)x0 = kappa(s)x0, and the D5 localization with the Spin10-invariant form). My script confirms that the weights of V(omega1) in X(X) are omega1, omega6-omega1 and -omega6, and that only the first two lie in the D5 vector module. Correct.

Proposition 3.7 step 3 infers the colour weights \(\{\omega_1,\omega_6\}\) from 'a basis of dominant weights', which does not follow. The diagram automorphism commuting with \(\theta_0\) forces it. With that repair, the colours, the valuation cone \(\mathcal V=C^-\) and the spherical roots hold in every characteristic \(\ne2\).

Objection 5.11.

Outcome: step 3 fails as written; the conclusion holds after a repair. (i) Gap. Step 3 shows that the colour weights \(\{\lambda_{D}\}\) form a \(\mathbb Z\)-basis of \(\mathcal X(X)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\) consisting of dominant weights, and concludes "i.e. \(\{\omega_1,\omega_6\}\)". But \(\{\omega_1,\omega_1+\omega_6\}\), and more generally \(\{\omega_1,n\omega_1+\omega_6\}\), are bases of dominant weights. Step 5 needs regular \(B\)-eigenfunctions of weights exactly \(\omega_1\) and \(\omega_6\) (\(\ell=f_1|_{X_1}\) must generate a finite-dimensional \(M_1\)-module). So does Proposition 3.10(iv) (\(v_{D_i}=e_i\), \(\Theta^+\) = cone of colour valuations). (ii) Repair (Lemma). Let \(\delta\) be the pinned automorphism of \(G_{\mathbb Z}\) with \(1\leftrightarrow6\), \(3\leftrightarrow5\). It preserves \(T\), \(B\) and \(H\), induces an automorphism of \(X\) fixing \(x_0\), permutes the colours, and acts on \(\mathcal X(X)\) by \(\omega_1\leftrightarrow\omega_6\). A \(\delta\)-stable \(\mathbb Z\)-basis of \(\mathbb Z^2\) contained in \(\mathbb N^2\) equals \(\{(1,0),(0,1)\}\). Hence the colours are \(D_1,D_6\) with weights \(\omega_1,\omega_6\), as claimed. (iii) The remaining parts hold: (1), (3) \(\mathcal V=C^-\) and \(\Lambda_X^+=\Lambda_X\cap C^-\), and (4).

Proof of the repair.

  1. \(\delta\) commutes with \(\theta_0=\operatorname{Int}(\sigma_L)\circ C\). \(\delta C\delta^{-1}\) inverts \(T\) and sends the pinning to the opposite pinning exactly as \(C\) does, so \(\delta C\delta^{-1}=C\) by the isomorphism theorem. \(\delta(\dot s_i)=\dot s_{\delta i}\) and \(\delta(w_L)=w_L\), since \(\{2,3,4,5\}\) is \(\delta\)-stable. Tits lifts do not depend on the reduced word, so \(\delta(\sigma_L)=\sigma_L\). Hence \(\delta\theta_0\delta^{-1}=\theta_0\) and \(\delta(H)=H\). This same fact is used without proof in Audit 3.9 steps 7–12.
  2. \(\delta_X(gx_0):=\delta(g)x_0\) is an automorphism of \(X\) with \(\delta_X(bx)=\delta(b)\delta_X(x)\). It maps \(B\)-stable prime divisors to \(B\)-stable prime divisors, and a \(B\)-eigenfunction of weight \(\chi\) to one of weight \(\delta\chi\). So \(\{\lambda_{D_1},\lambda_{D_2}\}\) is \(\delta\)-stable.
  3. In the basis \(\omega_1,\omega_6\), \(\delta\) swaps coordinates. Two vectors on the diagonal cannot form a basis. So the basis is \(\{(a,b),(b,a)\}\) with \(a,b\ge0\) and \(a^2-b^2=\pm1\), which forces \(a+b=1\), i.e. \(\{(1,0),(0,1)\}\). \(\square\)

Re-check of the other steps.

  1. (1) The elementary divisors of \(1-\theta^*\) on \(P\) are \((1,1)\) (recomputed, Audit 5.2 step 7), so the weights of rational \(B\)-eigenfunctions on \(\mathring X=B/B_8\) are \(\mathbb Z\omega_1\oplus\mathbb Z\omega_6\).
  2. (2), other parts. \(k[G]^\times=k^\times\), \(\mathrm{Pic}\,G=0\) for simply connected \(G\) and \(X^*(F_4)=0\) give \(\mathrm{Pic}\,X=0\); \(X\) is smooth affine, hence factorial. The boundary \(X\smallsetminus\mathring X\) has pure codimension \(1\) (EGA IV 21.12.7), and units of \(\mathring X\) give the isomorphism \(\mathbb Z^{\text{colours}}\cong\mathcal X(X)\). Frobenius reciprocity uses that the minuscule Weyl modules are irreducible in every characteristic. The \(P_\alpha\)-stability criterion uses that the simple head of an \(\mathrm{SL}_2\) Weyl module \(V(n)\), \(n\ge1\), has dimension \(\ge2\).
  3. (3) Lower bound. For dominant \(\kappa\in\Lambda_S\), the arc valuation \(v_\kappa\) is \(G\)-invariant. The computation over \(k(\!(s)\!)\), \(s^2=t\), with \(\kappa(s)^{-1}u^-\kappa(s)\to u_0\in U^-\cap Z_G(\kappa)\), Hensel for the smooth \(U_P\times T\to\mathring X\), and \(\langle\varphi(\mu),\chi\rangle=2\langle\mu,\chi\rangle\) on \(\mathcal X(X)\), gives \(v_\kappa(f_\chi)=-\langle\mu,\chi\rangle\). No dominant lift is needed.
  4. (3) Upper bound. The \(D_5\)-localization is correct for \(p\ne2\). \(E\) is a quotient of the \(\mathrm{Spin}_{10}\) Weyl module of the vector weight (Jantzen II.2.13), which is irreducible. \(Q^*\) pairs the highest weight \(\omega_1\) with the lowest weight \(\omega_1-\gamma_1=\omega_6-\omega_1\); indeed \(\gamma_1=2\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5=2\varepsilon_1\) in \(D_5\). The weights of \(V(\omega_1)\) in \(\mathbb Z\omega_1\oplus\mathbb Z\omega_6\) are exactly \(\omega_1,\ \omega_6-\omega_1,\ -\omega_6\) (recomputed). \(\dot n_1\in M_1^\theta\) swaps the two weight lines, so \(F_1(x_0)\ne0\), and \(v'(F_1)\ge2v'(\ell)\) gives \(\langle v,\gamma_1\rangle\le0\).
  5. (4) \((\alpha_1,\beta_1)=0\), and \(\alpha_1\pm\beta_1\) are not roots (recomputed). \(\gamma_1=2\omega_1-\omega_6\) is primitive, and \(\langle e_i,\gamma_j\rangle\) is the \(A_2\) Cartan matrix (with \(e_i\in\Lambda_X\); see the notation warning in Audit 5.2).

Builds on

  • Proposition 3.7 · Char \(\ne2\): two colours, the valuation cone, spherical roots \(\gamma_1,\gamma_6\)

Built on by

  • Summary 5.13 · Verdict of audit A5: transfer route A4-2…A4-9 and char-p hypotheses A3-1…A3-10

Sources