§3 · Agent 3 · Hypotheses in char p · 2026-10-18T02:55:41Z

Audit 3.1 · Where the \(p\)-adic verification (S4, S4b) used characteristic 0, and what replaces it

proved audit: minor defect

Minor defect (second audit). I re-read S4-3 to S4-13 against the table: every row is accurate (the input is as cited there and each replacement node exists), and the side claims of step 2 hold: |F^x/F^x2| = 4 for F_q((t)), p odd; N_H(S)/Z_H(S) is S3, constant; X^(T)/(1-theta)X^*(T) is torsion-free (I recomputed the elementary divisors (1,1)). Defect: the claim that the table lists every characteristic-0 or unchecked input is overstated. It omits four inputs: S4-12 step 2 (S is maximal because of the 'rank 2 of E IV', a classification input); S4-3 step 8 (type-G colours are defined over F, from Sakellaridis's general remark on p. 44); S4-11 step 5 (the classification of outer involutions, with Steinberg 8.1 also used there and in S4-12 step 1); and the use in S4-5(3)(a) of the transitivity of Knop's action to count 45 orbits. Each is superseded by Proposition 3.4(a)(b), Proposition 3.10 step 3, Proposition 3.2 step 3 and Proposition 3.6(2), so no conclusion changes. The closing sentence holds once Proposition 3.6 and Proposition 3.7 receive their one-line repairs.

Fixes the setting \(F=\mathbb F_q(\!(t)\!)\), \(p\) odd, and the model \(\mathcal M_\theta\) over \(\mathcal O=\mathbb F_q[[t]]\), and lists every step of the earlier \(p\)-adic verification of (A1), (A2), (A3b), (A3c), (A4), (A5) that rested on a characteristic-0 theorem, with the characteristic-\(p\) argument of this run that replaces it.

Audit 3.1.

Setting. \(p\) an odd prime, \(q=p^f\), \(F=\mathbb F_q(\!(t)\!)\), \(\mathcal O=\mathbb F_q[[t]]\), \(\varpi=t\), \(\bar F\) an algebraic closure and \(F^{s}\) the separable closure. \(G=G_{\mathbb Z}\otimes\mathcal O\) is the split simply connected group of type \(E_6\) with pinning \((T,B,\{x_{\alpha_i}\})\), Bourbaki labels; \(\theta=\theta_0=\mathrm{Int}(\sigma_L)\circ C\), an involution defined over \(\mathbb Z\) with \(\theta|_T=-w_L\) (S4-11). \(H:=G^{\theta}\) (scheme-theoretic fixed points), \(X:=G/H\) with the left action, \(x_0=eH\), \(K=G(\mathcal O)\), \(J\) the Iwahori subgroup, \(\tau(gH)=g\theta(g)^{-1}\). \(P=U_P\rtimes L\) is the standard parabolic of the black nodes \(\{2,3,4,5\}\), \(L=Z_G(S)\), \(S=(T^{-\theta})^\circ\), \(\Phi_L\) the 24 roots of the black \(D_4\), \(T_8\) the maximal torus of \([L,L]=\mathrm{Spin}_8\) (cocharacters \(\alpha_2^\vee,\dots,\alpha_5^\vee\)), \(\beta_i:=w_L\alpha_i=-\theta\alpha_i\), \(\gamma_1=\alpha_1+\beta_1=2\omega_1-\omega_6\), \(\gamma_6=\alpha_6+\beta_6=-\omega_1+2\omega_6\). \(A_X=T/T^\theta\), \(\Lambda_X=X_*(A_X)\), \(e_i\in\Lambda_X\) the image of \(\alpha_i^\vee\) (\(i=1,6\)); \(\varphi=1-\theta\) on \(\Lambda_T\), \(\Lambda_S=\varphi(\Lambda_T)\). \(B\)-eigenfunctions are normalized by \(f(b^{-1}x)=\chi(b)f(x)\), which is Sakellaridis's right-action convention under \(x\cdot g\leftrightarrow g^{-1}x\).

Claim (audit). The table lists every characteristic-0 or unchecked input that S4 and S4b (nodes S4-3 to S4-13) used for the hypotheses of (Q3), and its replacement. After the replacements, the only obligation of (Q3) not proved in characteristic \(p\) is the analytic content of Statement 6.3.1 (Remark 3.11).

Table 2. S4/S4b step and input used there and replacement over \(\mathbb F_q(\!(t)\!)\)
S4/S4b step	input used there	replacement over \(\mathbb F_q(\!(t)\!)\)	node
S4-4 (C1): \(H_{\mathcal O}\) smooth	Edixhoven, Prop. 3.4	Conrad, Lemma 2.2.4 (any base)	Proposition 3.2
S4-4 (C2): fibres of \(H\) connected	Steinberg 1968, Thm 8.1	direct proof: \(T^\theta=T_8\), \(Z_G(T_8)=T\), \(\mathrm{Aut}(F_4)\) inner	Proposition 3.2
S4-4: \(X_{\mathcal O}\) quasi-affine	Alper 2014 (number unchecked)	Conrad, Thm 2.3.1	Proposition 3.2
S4-4 step 7: \(X_F\) affine	Matsushima (char. 0)	Richardson 1977 / Haboush	Lemma 3.3
S4-12: \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\)	Kneser (char. 0)	Bruhat–Tits III, Thm 4.7	Lemma 3.3
S4-12 step 5: \(A'\) is \(G(F)\)-conjugate to \(S\)	Helminck–Wang, Prop. 10.3	not needed; direct proof	Proposition 3.4
S4-12 step 6: maximal \(\theta\)-split tori conjugate over \(\bar F\)	Vust (char. 0), Richardson	direct proof for char. \(\ne2\)	Proposition 3.4
S4-12: Delorme–Sécherre, Thm 4.8	proof not checked	proof checked for \(\mathbb F_q(\!(t)\!)\); one lemma replaced	Audit 3.5
S4-5(3), S4-10: only types \(G,U\)	Knop 1995 (lattice transport)	maximal-rank orbits via \(\theta\)-stable tori	Proposition 3.6
S4-3 step 3: valuation cone	Knop 1991, §5 (char. 0)	arc valuations + \(D_5\)-localization	Proposition 3.7
S4-3 step 6: colours	Helgason's theorem (char. 0)	\(\mathrm{Pic}\,X=0\), units, divisor map	Proposition 3.7
S4-3 step 9: colour of \(\gamma_1\)	Lemma 7.1.2 (case analysis)	first edge of every Brion path is \(\alpha_1\)	Proposition 3.10
S4-3 step 10: Statement 6.3.1	Thm 6.3.2 with Thm 6.11.1 (Wasserman, char. 0)	combinatorics checked; analytic part open	Proposition 3.10, Remark 3.11
S4-5(1): distinct labels	Cartan decomposition of \(G(F)\)	valid for every local field	Theorem 3.8
S4-5(2): Axiom 2.4.2	\(\sqrt\varpi\)	\(F(\sqrt t)/F\) separable, \(p\) odd	Audit 3.9

How the table was made. Every node S4-3, S4-4, S4-5, S4-10, S4-11, S4-12, S4-13 of the salvage returns was read and each cited input classified as characteristic-free, characteristic-0, or unchecked. Characteristic-free inputs are kept: existence of Chevalley group schemes over \(\mathbb Z\) and the isomorphism theorem (SGA 3, used in S4-11); EGA IV 17.9.1 and 21.12.7 (any locally noetherian scheme); Hensel plus Lang for \(H^1(\mathcal O,H)\) (Conrad, discussion after Remark 7.2.13, for every non-archimedean local field); Bruhat–Tits theory of the building (every non-archimedean local field); additive and multiplicative Hilbert 90 (every field).
What changes in characteristic \(p\) and how it is met. (a) \(\sqrt\varpi\): \(F(\sqrt t)/F\) is separable because \(p\ne2\) (Audit 3.9). (b) \(F^\times/F^{\times2}\) has order 4 because \(p\ne2\) (used only in Delorme–Sécherre, Lemma 3.6, which Theorem 4.8 does not need). (c) Inseparable orbit maps: every stabilizer used is computed scheme-theoretically and is smooth: \(H\), \(B\cap H\), \(P\cap H\), \(Q\cap H\), \(M^\theta\), the stabilizers of the rank-one and rank-two quotients, \(T^\theta\), \(N_H(S)\) (Proposition 3.2, Proposition 3.6). (d) Infinite Galois cohomology: \(H^1(F,\mathbb Z/p)\) and \(H^1(F,\mu_p)\) are infinite, but the only finite group schemes that occur are \(S[2]\simeq\mu_2^2\) (étale) and \(N_H(S)/Z_H(S)\simeq S_3\) (constant); the descent cocycle of Proposition 3.4 lands in \(Z_H(S)=\mathrm{Spin}_8\), never in \(N_H(S)\), so \(H^1(F,S_3)\), infinite for \(p=3\), does not enter. (e) The bad prime \(p=3\): the centre \(\mu_3\) of \(E_6\) is infinitesimal in characteristic 3 but is never used; \(X^*(T)/(1-\theta^*)X^*(T)\) is torsion-free, so \(T^\theta\) contains no \(\mu_3\).
Scope. Not covered here: (A3a) as \(\mathcal O\)-schemes (Q2), the proof of Theorem 8.0.2 in characteristic \(p\) (Q1), and the analytic part of Statement 6.3.1, which is part of (Q1) (Remark 3.11).

Builds on

Built on by

Proposition 3.2 · (A1), (A2) for \(\mathcal M_\theta\) over any complete DVR with 2 invertible
Summary 3.12 · (Q3) verdict: (A1)–(A3c), (A5), (A4)(i,ii,iv) hold over \(\mathbb F_q((t))\)
Audit 5.8 · Audit of A3-1, A3-2, A3-3: integral model, \(H^1\) and (A5) over \(\mathbb F_q((t))\)

Sources

S4.json and S4b.json (salvage returns, nodes S4-3 to S4-13)
sakellaridis-v3.pdf, §1.1 p. 2; §1.7 pp. 6–7; Axioms 2.2.1, 2.4.1, 2.4.2 pp. 9–11; opening of §6 p. 41; §7 p. 58; §8 p. 63
conrad-reductive-group-schemes.pdf, Lemma 2.2.4, Thm 2.3.1, discussion after Remark 7.2.13

Table 1. Page production by session
	tokens	US $
The run, its in-run audits and the pages (from 01:50 UTC)01:50–05:20 UTC	349 M	$144
This conversation: the audit pass (eight reviewers, one salvage agent, re-checks) and the simplified VIb page05:21–07:33 UTC	267 M	$106
Removing the detail switch from the published trees05:34–05:51 UTC	15 M	$7
Total	631 M	$257