§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:07:08Z

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

proved audit: correct ★ key step

Second audit: correct. The summary table matches my grades of Audit 5.1-Audit 5.12: no audited claim fails and the four repairs are valid. The transfer theorem over F_q((t)) is conditional on (I_Lambda) and H_Sa08, and H_Sa08 holds because Sa08 assumes only characteristic 0 (see Audit 7.3). (A1), (A2), (A3b), (A3c), (A4)(i),(ii),(iv) and (A5) hold for every odd p, including 3. Its only inaccuracy, that [Sa08] was not available locally, is immaterial. Correct.

None of the 18 audited claims fails. Four need repairs, all supplied here. The characteristic-\(p\) Theorem 8.0.2 for \(\mathcal M_\theta\) is proved by the transfer modulo \((I_\Lambda)\) and one unchecked citation of [Sa08]. The hypotheses (A1), (A2), (A3b), (A3c), (A4)(i),(ii),(iv), (A5) hold over \(\mathbb F_q(\!(t)\!)\) for every odd \(p\).

Summary 5.13.

Table 2. claim and outcome and node
claim	outcome	node
A4-2 truncation and determinacy	holds	Audit 5.1
A4-4 \(\Lambda_S\), labels	holds (recomputed)	Audit 5.2
A4-5 Cartan transfer	holds; repair: "Axiom 2.4.1" needs Proposition 3.7(3)	Audit 5.3
A4-6 structure constants	holds with repair: uniform \(\nu\)-box for \(\mathcal S(T_\lambda)\)	Objection 5.4
A4-7 ramified Theorem 8.0.2	holds; needs \(H_{\mathrm{Sa08}}\)	Audit 5.5
A4-8 Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\)	holds under \((I_\Lambda)\) and \(H_{\mathrm{Sa08}}\)	Audit 5.6
A4-3 \(\tau\) closed immersion	holds (item 2 cites \(G_\Lambda\hookrightarrow\mathrm{GL}(W)\))	Audit 5.7
A4-9 general symmetric \(X\)	holds as a conditional theorem	Audit 5.7
A3-1 audit table	holds, with the repairs of A3-6, A3-7	Audit 5.8
A3-2 (A1), (A2)	holds	Audit 5.8
A3-3 \(H^1\), (A5)	holds (Bruhat–Tits III cited)	Audit 5.8
A3-4 \(\theta\)-split tori	holds	Audit 5.9
A3-5 Delorme–Sécherre	holds (building references cited)	Audit 5.9
A3-6 orbits, types	holds with repair in (2)	Audit 5.10
A3-7 colours, cone	step 3 fails as written; holds after the \(\delta\)-repair	Objection 5.11
A3-8 (A3b)	holds	Audit 5.9
A3-9 (A3c)	holds	Audit 5.12
A3-10 (A4)	holds (combinatorial; analytic 6.3.1 open)	Audit 5.12

\(H_{\mathrm{Sa08}}\): [Sa08, Thm 5.2.1] and [Sa08, §3] hold over every finite extension of \(\mathbb Q_p\), \(p\) odd. \((I_\Lambda)\): Axiom 2.2.1 for \(\mathcal M_\theta\) over \(W(\mathbb F_q)\) as \(\Lambda\)-schemes.

Part 1, main line. The close-fields transfer is sound. Truncation (Audit 5.1), orbit counting with a uniform \(D(\lambda)\) and the quantifier over \(H(\mathcal O)\) handled by the Cartan decomposition of \(H\) (Audit 5.3), structure constants and Satake coefficients (Objection 5.4, after the \(\nu\)-box repair), and the module transfer with uniqueness (Audit 5.6) contain no characteristic-\(0\) step applied in characteristic \(p\) (checklist items 1, 2, 10). The characteristic-\(0\) content sits entirely at the fields \(F'_e=\mathbb Q_q(p^{1/e})\), \(e\to\infty\). There (H-Cartan) holds for every ramification (S4-12/S4-13, Proposition 3.4), and Theorem 8.0.2 holds once \((I_\Lambda)\) holds, reading the paper's normalizations with \(\psi\) of conductor \(\mathcal O'\) as §1.7 prescribes (Audit 5.5). The one chain not verified is the ramification-independence of the results quoted from [Sa08] (Theorem 3.7.1 = [Sa08, Thm 5.2.1]); [Sa08] is not available locally.
Part 1, statements to correct. (a) Proposition 4.5's "Axiom 2.4.1 holds" needs the characteristic-\(p\) valuation cone (Proposition 3.7(3)); the parametrization \(K\backslash X(F)\simeq\Lambda_S^+\) does not. (b) Proposition 4.6(c) needs the two-sided bound \(\langle\lambda,w_0\omega_i\rangle\le\langle\nu,w_0\omega_i\rangle\le\langle\lambda,\omega_i\rangle\). (c) Theorem 4.8's "no statement over \(\bar F\)" applies to the proof of the isomorphism only; identifying \(S\), \(W_X\), \(\Lambda_X^+\) with Sakellaridis's invariants of \(X_{\bar F}\) in characteristic \(p\) uses Proposition 3.6, Proposition 3.7.
Part 2. Over \(\mathbb F_q(\!(t)\!)\), \(p\) odd (\(p=3\) included), the hypotheses (A1), (A2), (A3b), (A3c), (A4)(i),(ii),(iv), (A5) for \(\mathcal M_\theta\) are proved with characteristic-free inputs: Conrad 2.2.4 and 2.3.1 (checked), Prasad–Yu, Steinberg 7.5, Bruhat–Tits III Thm 4.7, Delorme–Sécherre (checked), Richardson. Two repairs are needed: the general converse in Proposition 3.6(2) (Audit 5.10) and the colour weights in Proposition 3.7 step 3 via the diagram automorphism \(\delta\) (Objection 5.11). Inseparability, non-reduced stabilizers and infinite \(H^1\) (checklist items 5, 6) do not arise: every stabilizer used is smooth, and every \(H^1\) is of a smooth connected group.
Numerical claims. Every numerical claim checked was recomputed in exact arithmetic (script a5_e6_checks.py) and agrees: \(e_1,e_6\), \(3\omega_1^\vee=2e_1+e_6\), elementary divisors \((1,1)\), \(\theta\)-fixed roots \(=\Phi_L\), \(|W\cdot\Phi_L|=45\), \(W_X\) of order \(6\), Levi sizes, \(\mu_1\)-bounds, Brion-path dimensions \(26,\dots,22\), edge types, rank-two vertex data.
For the team question. (Q1) for \(\mathcal M_\theta\): proved modulo \((I_\Lambda)\) and \(H_{\mathrm{Sa08}}\), with (SG) failing over \(\mathbb F_q(\!(t)\!)\) under these. (Q3): proved after the repairs, except Statement 6.3.1 in its analytic form, which the transfer route does not need. (Q2) was not audited here; \((I_\Lambda)\) is its unramified integral instance.

Builds on

Audit 5.1 · Audit of A4-2: truncated rings and finite determinacy
Audit 5.2 · Audit of A4-4: \(\Lambda_S=(1-\theta)X_*(T)\) and the labels \(x_\kappa\)
Audit 5.3 · Audit of A4-5: orbit count over \(\mathbb F_q((t))\), uniform \(D(\lambda)\)
Objection 5.4 · Audit of A4-6: structure constants transfer; the Satake support bound is supplied
Audit 5.5 · Audit of A4-7: Theorem 8.0.2 over ramified \(F'\) and the images \(P_\kappa\)
Audit 5.6 · Audit of A4-8: Theorem 8.0.2 for M_θ over F_q((t)); uniqueness; (SG)
Audit 5.7 · Audit of A4-3 (τ closed immersion) and A4-9 (general symmetric X)
Audit 5.8 · Audit of A3-1, A3-2, A3-3: integral model, \(H^1\) and (A5) over \(\mathbb F_q((t))\)
Audit 5.9 · Audit of A3-4, A3-5, A3-8: θ-split tori, Delorme–Sécherre, (A3b) in char p
Audit 5.10 · Audit of A3-6: the 45 maximal-rank Borel orbits and their types in char ≠ 2
Objection 5.11 · Audit of A3-7: colour weights (gap repaired by δ) and the valuation cone
Audit 5.12 · Audit of A3-9 (Iwahori orbits) and A3-10 ((A4) in characteristic p)

Built on by

Theorem 7.4 · Verdict: Sakellaridis's theorem over \(\mathbb F_q(\!(t)\!)\); for \(E_6/F_4\) in both characteristics, (A3a) included

Sources

A4.json and A3.json returns (claims audited)
nodes Audit 5.1 … Audit 5.12
script a5_e6_checks.py

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