§7 · Root · root: registry, redirects, synthesis · 2026-10-18T03:29:10Z

Remark 7.1 · Registry after three returns: the theorem in characteristic \(p\) reduces to (A3a); audits launched

open audit: correct

Second audit: correct. Checked each registry entry against the cited steps and the agents' records: Theorem 4.8 is stated modulo (I_Λ) (the [Sa08] condition surfaced only later, in Audit 5.5), A3's status matches Remark 3.11/Summary 3.12 (all but (A3a) and the analytic Statement 6.3.1), A2 had not yet returned (03:33 > 03:29) and A1's relaunch and A5's launch match the agent record. The Cartan decomposition K\X(F_q((t))) ≃ Λ_S^+ is indeed reached by two distinct arguments, Theorem 3.8 (Delorme–Sécherre applied in characteristic p plus the one-class lemma Proposition 3.4) and Proposition 4.5 (orbit counting from F'_e, with Delorme–Sécherre and Kneser used only in characteristic 0); they are independent with respect to the characteristic-p risk, although both rest on Delorme–Sécherre's theorem (at different fields) and on S4-13's realization of W_X in K. The Chen–Yi remark is confirmed in chen-yi-v4 (the proof of Prop. 17(i) cites [GN10, Thm 8.2.9] and [Nad04, Thm 4.2], results over C, while the paper works over F_p-bar). The decision (the transfer bypasses the analytic Statement 6.3.1 because it uses Sakellaridis's theorem only at p-adic fields; (A3a) is the input every route needs) is an accurate reading of Theorem 4.8, Dead end 4.10 and Remark 3.11 at that time.

Two agents proved the Cartan decomposition over \(\mathbb F_q(\!(t)\!)\) by different routes, and the close-field transfer gave Sakellaridis's theorem there modulo (A3a) over \(W(\mathbb F_q)\). The direct rerun was relaunched after a failed first attempt, and the first auditor was assigned.

Table 2. family and agents and state
family	agents	state
close fields: \(C_K\) carried from \(\mathbb Q_q(p^{1/e})\) to \(\mathbb F_q(\!(t)\!)\)	A4	theorem modulo (A3a) over \(W(\mathbb F_q)\) (Theorem 4.8, Dead end 4.10), unaudited
hypotheses over \(\mathbb F_q(\!(t)\!)\)	A3	all but (A3a) and the analytic Statement 6.3.1 (Remark 3.11), unaudited
integral models (A3a)	A2	running
direct rerun	A1	first attempt ended at the output limit with nothing saved; relaunched

Independent evidence. The Cartan decomposition \(K\backslash X(\mathbb F_q(\!(t)\!))\simeq\Lambda_S^+\) was reached twice: by Delorme–Sécherre with a direct one-class lemma (Theorem 3.8), and by counting orbits over close fields (Proposition 4.5). Both agents also found that Chen–Yi prove their Prop. 17(i), the pages' input D0, by citing results over \(\mathbb C\).

Decision. The open analytic Statement 6.3.1 is bypassed by the transfer, which uses Sakellaridis's theorem only over \(p\)-adic fields. The one input every route needs is (A3a). Launch the audit of A4 and A3 (agent A5).

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Theorem 4.8 · Theorem: Theorem 8.0.2 for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\); (SG) fails
Dead end 4.10 · Dead end: Axiom 2.2.1 over \(W(\mathbb F_q)\) is not transferred
Theorem 3.8 · (A3b) for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\): \(K\backslash X(F)\simeq\Lambda_X^+\)
Proposition 4.5 · Proposition: \(K\backslash X(\mathbb F_q((t)))\simeq\Lambda_S^+\) by counting orbits
Remark 3.11 · Not proved here: the analytic part of Statement 6.3.1 over \(\mathbb F_q((t))\)

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

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