§7 · Agent 7 · Adversarial audit · 2026-10-17T03:43:09Z

Audit 7.20 · Audit of the sphere results: the residue-characteristic-2 case is genuinely covered

proved audit: minor defect

Minor defect (second audit). Agrees with my audit of Lemma 5.2 and Theorem 5.3, and its characteristic-2 analysis is right. It repeats the slip of Lemma 5.2: after the Hensel step \(B(y,z')=1+2a\varpi^{2m}\), so the displayed Gram matrix holds only after dividing \(z'\) by this unit (harmless). Calling \(\bar y^\perp/\bar y\) 'hyperbolic' is meaningful only for the bilinear form in characteristic 2, which is all that is used.

An independent check of Lemma 5.2 and Theorem 5.3: the hyperbolic model is smooth over \(\Z\), the Hensel and lifting steps work in characteristic 2, and the spinor-norm step holds, so \((\mathrm{SG})\) for every sphere over every non-archimedean local field, residue characteristic 2 included, is confirmed.

Audit 7.20.

Lemma 5.2 and Theorem 5.3 hold: \((\mathrm{SG})\) for \(\Spin_{2n}/\Spin_{2n-1}\), every \(n\ge2\), every non-archimedean \(F\), including residue characteristic 2. Severity of the defects found: none.

  1. The model. \(Q=\{N=1\}\), \(N=\sum_{i\le n}x_ix_{2n+1-i}\), is a regular quadratic form of even rank over \(\Z\) whose polar form is perfect in every characteristic; \(\SO_{2n}\), \(\Spin_{2n}\) and \(\Spin_{2n-1}\) are smooth over \(\Z\), and \(\SO_{2n}/\SO_{2n-1}\simeq Q\) (transitivity on geometric points by Witt's theorem). So \(X(F)=Q(F)\) and \(X(\cO)=Q(\cO)\) for every \(F\). The characteristic-2 pitfalls — the Arf invariant and the non-smooth odd-rank orthogonal groups — do not arise: the lattice has even rank and is hyperbolic, and only \(\mathrm O(H^{n-1})\), of even rank, is used for lifting.
  1. The transitivity step. \(f(a)=N(z)+a+\varpi^{2m}a^2\). For \(m\ge1\), \(f'\equiv1\) and Hensel's lemma applies. For \(m=0\) in characteristic 2, \(B(y,y)=2N(y)=0\), so \(\bar y\in\bar y^\perp\) and \(\bar y^\perp/\bar y\) is nondegenerate of dimension \(2n-2\ge2\). Lifts \(s_0,t_0\in\bar y^\perp\) of a pair with \(B(s_0,t_0)=1\) give \(N(z+ss_0+tt_0)=Q_0(s,t)+\gamma s+\delta t+N(z)\) with \(Q_0\) a nondegenerate binary form, and \(Q_0(s,t)+\gamma s+\delta t=Q_0(s+\delta,t+\gamma)-Q_0(\delta,\gamma)\) takes every value of \(\F_q\) (a nondegenerate binary form over a finite field represents every element). So \(N(z+w)\) can be moved into the image of \(a\mapsto a^2+a\) — for \(q\) odd one chooses a value with a simple root, for example \(0\) — and Hensel's lemma applies. The plane \(P=\cO y\oplus\cO z'\) has Gram matrix \(\begin{pmatrix}2\varpi^{2m}&1\\1&0\end{pmatrix}\), is unimodular and \(P\simeq H\); \(\overline{P^\perp}\simeq H^{n-1}\) by Witt cancellation over \(\F_q\), valid for regular forms in characteristic 2; the isometry lifts because the isometry scheme is a torsor under the smooth \(\mathrm O(H^{n-1})\) (Conrad, Appendix C) and \(\cO\) is complete. The determinant or Dickson invariant is corrected by an orthogonal transvection in a vector of unit norm in \(P^\perp\).
  1. The spinor-norm step. \(1\to\mu_2\to\Spin\to\SO\to1\) is exact in the fppf topology; \(H^1_{\mathrm{fppf}}(\cO,\mu_2)=\cO^\times/\cO^{\times2}\) by Kummer theory and \(\operatorname{Pic}\cO=0\); the connecting map is the spinor norm; and \(r_{e_2+e_{2n-1}}r_{e_2+ue_{2n-1}}\in\Stab(y_m)\) has spinor norm \(u\).
  1. Generation. The lift \((m,m,0,\dots,0)\), the pole-order support bound, Lemma A(2) and Proposition B (order \(m'\le m\)) work as in step 3 of the odd-residue proof Proposition 1.3.

Compared with Proposition 1.3 (\(q\) odd), Hensel's lemma and the smoothness of even-rank orthogonal groups replace the four odd-\(q\) lattice inputs; this is exactly what makes residue characteristic 2 work.

Builds on

  • Lemma 5.2 · Spheres: the \(\Spin_{2n}(\cO)\)-orbits on \(Q(F)\) are the pole-order shells, for every local field
  • Theorem 5.3 · \((\mathrm{SG})\) for \(\Spin_{2n}/\Spin_{2n-1}\), every \(n\ge2\), every non-archimedean local field
  • Proposition 1.3 · Spheres \(\Spin_{2n}/\Spin_{2n-1}\): (SG) for odd residue characteristic

Built on by

  • Summary 7.25 · Updated audit verdict: every row of Table 1 except \(E_6/F_4\) is decided at both levels
Original record, as the agent wrote it at 03:43 UTC

A5-2/A5-3 spheres over every F: the residue-characteristic-2 case is genuinely covered

Model. \(Q=\{q(x)=1\}\), \(q=\sum_{i\le n}x_ix_{2n+1-i}\): a regular (nondefective) quadratic form of even rank over \(\mathbb Z\); its polar form is perfect in every characteristic, \(\mathrm{SO}_{2n},\mathrm{Spin}_{2n},\mathrm{Spin}_{2n-1}\) are smooth over \(\mathbb Z\), and \(\mathrm{SO}_{2n}/\mathrm{SO}_{2n-1}\simeq Q\) (transitivity on geometric points by Witt). So \(X(F)=Q(F)\), \(X(\mathcal O)=Q(\mathcal O)\) for every \(F\). The char-2 pitfalls named by the coordinator (Arf invariant, non-smooth odd-rank orthogonal groups) do not arise: the lattice is even-rank hyperbolic and only \(\mathrm O(H^{n-1})\) (even rank) is used for lifting.

Step 2 (holds in all characteristics). \(f(a)=q(z)+a+\varpi^{2m}a^2\): for \(m\ge1\), \(f'(a)\equiv1\), Hensel. For \(m=0\): in char 2, \(B(y,y)=2q(y)=0\) so \(\bar y\in\bar y^\perp\) and \(\bar y^\perp/\bar y\) is hyperbolic of dimension \(2n-2\ge2\), hence contains a hyperbolic pair \((s_0,t_0)\); then \(q(z+ss_0+tt_0)=Q(s,t)+\gamma s+\delta t+q(z)\) with \(Q\) nondegenerate binary, and \(Q(s,t)+\gamma s+\delta t=Q(s+\delta,t+\gamma)-Q(\delta,\gamma)\) is onto \(\mathbb F_q\) (a nondegenerate binary form over a finite field represents every element, in char 2 as the norm form or \(st\)). So \(q(z+w)\) can be put in the image of \(a\mapsto a^2+a\) (for \(q\) odd choose a value with a simple root, e.g. \(0\)) and Hensel applies. \(P=\mathcal Oy\oplus\mathcal Oz'\) has Gram \(\begin{pmatrix}2\varpi^{2m}&1\\1&0\end{pmatrix}\), unimodular, \(P\simeq H\); \(\bar P^\perp\simeq H^{n-1}\) by Witt cancellation over \(\mathbb F_q\) (valid for regular forms in char 2); the isometry lifts because \(\mathrm{Isom}(P^\perp,H^{n-1})\) is a torsor under the smooth \(\mathrm O(H^{n-1})\) (Conrad, App. C) and \(\mathcal O\) is complete. Dickson/det correction by an orthogonal transvection \(r_v\), \(q(v)\in\mathcal O^\times\), \(v\in P^\perp\). ✓

Step 3 (holds). \(1\to\mu_2\to\mathrm{Spin}\to\mathrm{SO}\to1\) fppf; \(H^1_{\mathrm{fppf}}(\mathcal O,\mu_2)=\mathcal O^\times/\mathcal O^{\times2}\) (Kummer, \(\mathrm{Pic}\,\mathcal O=0\)); connecting map = spinor norm; \(r_{e_2+e_{2n-1}}r_{e_2+ue_{2n-1}}\in\mathrm{Stab}(y_m)\) has spinor norm \(u\). ✓

A5-3 (holds). Lift \((m,m,0,\dots)\), support bound by pole order, Lemma A(2), Proposition B (order \(m'\le m\)): same as A1-3 step 3. Outcome: (SG) for \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), all \(n\ge2\), every non-archimedean \(F\) including residue characteristic 2. Compared with A1-3 (\(q\) odd): A5 replaces the four odd-\(q\) lattice inputs (A7-10) by Hensel + smoothness of even-rank orthogonal groups, which is exactly what makes char 2 work. Severity: none.

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