Audit 6.6 · Audit of the special fibres of orbit closures over \(\mathcal O\) (claim A2-7)

1. The closure. \(Z_x\) is reduced and \(\mathcal O\)-flat (torsion-free), being the schematic closure of the reduced \(Y_{x,F}\). It is \(B\)-stable: \(B\times Z_x\) is \(\mathcal O\)-flat, hence the schematic closure of its generic fibre, which the action maps into \(Z_{x,F}\). It contains the section \(y_x(\operatorname{Spec}\mathcal O)\) (the closure of its generic point), hence the orbit \(B_\kappa y_{x,\kappa}=Y_{x,\kappa}\) (Lemma 2.4 item 3), hence \(\overline{Y_{x,\kappa}}\). Verified.
2. Item 2. \(\tilde\tau^{-1}(B)\) is closed; by Audit 6.3 step 3, its geometric points in either fibre are exactly those of \(Y_{w_0}\) (\(\tilde\tau(Y_v)\subseteq B\dot{(vw_0)}B\) and \(vw_0=1\iff v=w_0\)). A reduced locally closed subscheme with closed support is closed, so \(Y_{w_0}\) is closed and \(Z_{w_0}=Y_{w_0}\). Verified.
3. Item 1 (geometric fibres, induction on \(\dim Y_x\) from \(w_0\), with \(\alpha\) such that \(x\alpha>0\) and \(x'=s_\alpha xs_\alpha\)). \(P_\alpha\overline{Y_{x',k}}\) is closed (image of \(P_\alpha\times^B\overline{Y_{x',k}}\) under a proper map), irreducible, of dimension \(\le\dim Y_{x'}+1=\dim Y_x\), and contains \(Y_{x,k}\), so it equals \(\overline{Y_{x,k}}\). The decomposition \(P_\alpha Y_{v,k}=Y_{v,k}\) or \(Y_{v,k}\sqcup Y_{s_\alpha vs_\alpha,k}\) uses only the \(\bar k\)-part of Proposition 2.5. The resulting \(\Sigma_x\) was recomputed and is choice-independent (Audit 6.1 item 6).
4. Item 3, with the repair. Induction as in step 3. Closedness: \(P_\alpha\times^BX_{\mathcal O}\simeq P_\alpha/B\times_{\mathcal O}X_{\mathcal O}\) via \([p,z]\mapsto(pB,pz)\); the image of \(P_\alpha\times^BZ_{x'}\) is closed (closedness descends along the \(B\)-torsor \(P_\alpha\times X\to P_\alpha\times^BX\), and \(P_\alpha\times Z_{x'}\) is closed and \(B\)-stable), and the second projection is proper because \(P_\alpha/B\simeq\mathbb P^1_{\mathcal O}\); so \(P_\alpha Z_{x'}\) is closed in \(X_{\mathcal O}\). It contains \(P_\alpha Y_{x',F}\supseteq Y_{x,F}\), hence \(Z_x\). Its special fibre is \(P_\alpha(Z_{x'}\cap X_\kappa)=P_\alpha\overline{Y_{x',\kappa}}=\overline{Y_{x,\kappa}}\) by induction and item 1. With step 1, \(Z_x\cap X_\kappa=\overline{Y_{x,\kappa}}\).
5. Points. All sets are constructible unions of the locally closed \(Y_{v,\kappa}\), which partition \(X_{\bar\kappa}\) (Audit 6.3); equalities checked on geometric points are equalities of underlying sets.
6. Characteristic. Nothing uses \(\operatorname{char}F\) or \(\operatorname{char}\kappa\) beyond \(\ne2\); the same \(\Sigma_x\) governs \(\bar F\) and \(\bar\kappa\), which is what makes the special fibre of the closure equal to the closure in the special fibre.

Outcome: holds with the repair of step 4.