§4 · Agent 4 · Close fields · 2026-10-18T02:48:51Z

Theorem 4.8 · Theorem: Theorem 8.0.2 for $\mathcal M_\theta$ over $\mathbb F_q((t))$; (SG) fails

proved audit: correct ★ key step

Second audit: correct. Re-derived: {1_{Kx_kappa}} is a basis by Proposition 4.5; Phi_{F'e} sends {1{K'x'kappa}} to the e-independent basis {P_kappa} (Audit 4.7(c)), so Psi is bijective with P_0 = 1; for each (l, mu) the chain Psi(T_l*b_mu) = sum c(F)P_nu = sum c(F'_e)P_nu = Phi(T'_l*b'_mu) = r(S T'_l)P_mu = r(S T_l)Psi(b_mu) holds with e >= max(D(l,mu), D'(l)), and the Hecke and Satake conventions are the same formulas over both fields. Uniqueness re-derived: Delta = Psi' Psi^-1 is R-linear with Delta(1) = 1, the ideal {g : g(y1) = g(sigma y1) = 0} of the domain S is non-zero and lies in R = im r (S4-7), so a Delta(s) = as forces Delta = id; S4-7 and S4-8 then give (SG)-failure, C_K = H_K phi0 + C 1{Kx_{3w1v}} and exactly two generators, p = 3 included. The only open input is (I_Lambda) ([Sa08] holds over every characteristic-0 local field, checked). Remark, not a defect: S is built from the p-adic data A_X, delta(X), W_X = S3 fixed in the setting; reading Psi literally as Sakellaridis's theorem for X_F needs these to be the characteristic-p invariants (Proposition 3.6, Proposition 3.7), which the step signals by saying no statement over the algebraic closure is used, and (SG)-failure needs only R != S.

Over $\mathbb F_q(\!(t)\!)$ the map $\Psi(\mathbf 1_{Kx_\kappa})=P_\kappa$ is checked to be $\mathcal H_K$-linear for the Satake action, one pair $(\lambda,\mu)$ at a time, against $\mathbb Q_q(p^{1/e})$ with $e\ge D(\lambda,\mu)$, where Sakellaridis's theorem holds; it sends $\varphi_0$ to $1$ and is the only such isomorphism. With Proposition VII, (SG) fails over $\mathbb F_q(\!(t)\!)$ and $C_K$ needs exactly two generators. The only open input is the one the $p$-adic theorem over $\mathbb Q_q$ already needs.

Theorem 4.8.

Let $p$ be odd ($p=3$ included), $F=\mathbb F_q(\!(t)\!)$, $\mathcal O=\mathbb F_q[\![t]\!]$, $X=\mathcal M_\theta$, and assume $(I_\Lambda)$: Axiom 2.2.1 holds for $\mathcal M_\theta$ over $\Lambda=W(\mathbb F_q)$ as $\Lambda$-schemes (for the 45 maximal-rank Borel orbits and the 6 simple roots). Then the linear map $\Psi:C_c^\infty(X(F))^K\to S=\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}$, $\Psi(\mathbf 1_{Kx_\kappa})=P_\kappa$ ($\kappa\in\Lambda_S^+$, $P_\kappa$ as in Audit 4.7(c)), is an isomorphism with $$\Psi(f*\varphi)=r(\mathcal Sf)\cdot\Psi(\varphi)\quad(f\in\mathcal H_K),\qquad\Psi(\mathbf 1_{X(\mathcal O)})=1,$$ and it is the unique $\mathcal H_K$-module isomorphism $C_K\to S$ with $\mathbf 1_{X(\mathcal O)}\mapsto1$. Consequently, over $F$: $\mathcal H_K*\varphi_0=\Psi^{-1}(\operatorname{im}r)\ne C_K$, so (SG) fails; $C_K=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{Kx_{3\omega_1^\vee}}$, $C_K$ is not cyclic and needs exactly two generators; $C_K/\mathcal H_K\varphi_0$ is the character with Satake parameter $q^{\rho-\omega_4}$. Without $(I_\Lambda)$: the Cartan decomposition Proposition 4.5 and the equality of all structure constants with those over $F'_e$, $e\gg0$ (Proposition 4.6), still hold.

Proof.

Bases. By Proposition 4.5 the indicators $b_\kappa=\mathbf 1_{Kx_\kappa}$, $\kappa\in\Lambda_S^+$, form a basis of $C_K(F)$; the same holds over every $F'_e$ (S4-13). By Audit 4.7(b),(c), $(I_\Lambda)$ gives Theorem 8.0.2 over every $F'_e$, so $\{P_\kappa\}=\Phi_{F'_e}(\{b'_\kappa\})$ is a basis of $S$, the same set for all $e$. Hence $\Psi$ is a linear bijection.
Basic function. $X(\mathcal O)=Kx_0$ ((E2) of Proposition 4.5), $x_0=x_{\kappa=0}$, and $P_0=1$ ($P_0=1$ is stated in the proof on p. 64).
Equivariance. Fix $\lambda\in\Lambda_T^+$, $\mu\in\Lambda_S^+$ and choose $e\ge\max(D(\lambda,\mu),D'(\lambda))$ (Proposition 4.6); $F'_e$ is $e$-close to $F$ (Lemma 4.2(a)). Then

$$\Psi(T_\lambda*b_\mu)=\sum_{\nu\in N(\lambda,\mu)}c^\nu_{\lambda\mu}(F)P_\nu=\sum_{\nu\in N(\lambda,\mu)}c^\nu_{\lambda\mu}(F'_e)P_\nu=\Phi_{F'_e}(T'_\lambda*b'_\mu)=r(\mathcal ST'_\lambda)P_\mu=r(\mathcal ST_\lambda)\Psi(b_\mu),$$

using Proposition 4.6(a),(b) for the first two equalities, Theorem 8.0.2 over $F'_e$ for the fourth, and Proposition 4.6(c) for the last. The $T_\lambda$ span $\mathcal H_K$ and the $b_\mu$ span $C_K$.

Uniqueness. Let $\Psi'$ be another such isomorphism and $\Delta=\Psi'\Psi^{-1}$: an automorphism of $S$, linear over $R=\operatorname{im}r$, with $\Delta(1)=1$, hence $\Delta|_R=\mathrm{id}$. By S4-7 (unconditional, every $q$), $R=\{g\in S:g(y_1)=g(\sigma y_1)\}$; so the ideal $\mathfrak f=\{g\in S:g(y_1)=g(\sigma y_1)=0\}$ of $S$ is non-zero and $\mathfrak f\subseteq R$. For $s\in S$ and $0\ne a\in\mathfrak f$: $as\in\mathfrak f\subseteq R$, so $a\Delta(s)=\Delta(as)=as$; $S$ is a domain, so $\Delta(s)=s$.
Consequences. $\Psi(\mathcal H_K*\varphi_0)=r(\mathbb C[A^*]^W)\cdot1=R$, and $R\ne S$ with $\dim S/R=1$ (S4-7; Proposition VII). So (SG) fails over $F$. S4-8 derives the remaining statements from an isomorphism $C_K\simeq S$ with $\varphi_0\mapsto1$ together with $P_{3\omega_1^\vee}\notin R$; $\Psi$ is such an isomorphism and $\Psi(\mathbf 1_{Kx_{3\omega_1^\vee}})$ is the same $P_{3\omega_1^\vee}$ (it depends only on $q$), so S4-8 applies verbatim over $F$.
Inputs. Characteristic $p$: only (E1)–(E4) of Proposition 4.5 and the Iwasawa decomposition. Characteristic $0$: Sakellaridis's Theorem 8.0.2 over $F'_e$ for all $e$, whose hypotheses are (b) of Audit 4.7 plus $(I_\Lambda)$; Delorme–Sécherre over $F'_e$. No hypothesis (A2)–(A5) over $\mathbb F_q[\![t]\!]$, and no statement over an algebraic closure of $\mathbb F_q(\!(t)\!)$, is used. In particular Axiom 2.2.1 over $\mathbb F_q[\![t]\!]$ (part of (Q2)) is not needed for (Q1) by this route. $\square$

Comparison with Lemma 1.7(c). Lemma 1.7(c) with S4-8 over $F'_e$ already gives $\mathbf 1_{Kx_{3\omega_1^\vee}}\notin\mathcal H_K\varphi_0$ over $F$ under $(I_\Lambda)$ (its (H-Cartan) on the $p$-adic side is S4-13). The theorem above adds the whole module structure, uniqueness, and the Cartan decomposition over $F$.

Builds on

Proposition 4.5 · Proposition: $K\backslash X(\mathbb F_q((t)))\simeq\Lambda_S^+$ by counting orbits
Proposition 4.6 · Proposition: all structure constants and Satake coefficients agree over close fields
Audit 4.7 · Check: Theorem 8.0.2 over ramified $F'$ needs only Axiom 2.2.1 over $W(\mathbb F_q)$

Built on by

Proposition 4.9 · Proposition: what the characteristic-$p$ theorem needs for symmetric $X$
Dead end 4.10 · Dead end: Axiom 2.2.1 over $W(\mathbb F_q)$ is not transferred
Summary 4.11 · Verdict: Cartan proved; Theorem 8.0.2 over $\mathbb F_q((t))$ given $(I_\Lambda)$
Audit 5.6 · Audit of A4-8: Theorem 8.0.2 for M_θ over F_q((t)); uniqueness; (SG)
Remark 7.1 · Registry after three returns: the theorem in characteristic $p$ reduces to (A3a); audits launched
Theorem 7.4 · Verdict: Sakellaridis's theorem over $\mathbb F_q(\!(t)\!)$; for $E_6/F_4$ in both characteristics, (A3a) included

Sources

sakellaridis-v3.pdf, Theorem 8.0.2 and proof p. 64, (8.1), Corollary 8.0.4 p. 65
satake-two-levels/audit/S4.json S4-7 (im r), S4-8 (two generators); S4b.json S4-13
satake-two-levels/props/e6-f4-module.html (Proposition VII)
satake-converse/nodes/A1-7.html (Lemma 1.7(c))

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

Theorem 4.8 · Theorem: Theorem 8.0.2 for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\); (SG) fails

Builds on

Built on by

Sources