Strategy 4.1 · Strategy: move Theorem 8.0.2 from \(\mathbb Q_q(p^{1/e})\) to \(\mathbb F_q((t))\)

Notation (fixed for all nodes). \(p\) is odd, \(q=p^f\), \(\Lambda:=W(\mathbb F_q)\), \(\mathbb Q_q:=\Lambda[1/p]\). A test field \(E\) is a non-archimedean local field with residue field \(\mathbb F_q\) that is a \(\Lambda\)-algebra: either \(F=\mathbb F_q(\!(t)\!)\) (through \(\Lambda\to\mathbb F_q\)) or a finite extension \(F'\) of \(\mathbb Q_q\) with residue field \(\mathbb F_q\). Write \(\mathcal O_E,\varpi_E\) (\(\varpi_F=t\)), \(K=G(\mathcal O_E)\), \(K(m)=\ker(K\to G(\mathcal O_E/\varpi^m))\), \(t^\mu:=\mu(\varpi_E)\). \(G\) is the split simply connected \(E_6\) over \(\Lambda\), \(\theta=\theta_0\) (defined over \(\mathbb Z\), S4-11), \(\theta(T)=T\), \(\theta|_{X_*(T)}=-w_L\); \(H=G^\theta\) (smooth with connected fibres of type \(F_4\), S4-4); \(X=G/H\), \(x_0=eH\), \(\tau(gx_0)=g\theta(g)^{-1}\). \(\Lambda_T=X_*(T)\), \(\Lambda_T^+\) its dominant cone, \(\varphi=1-\theta\), \(\Lambda_S=\varphi(\Lambda_T)=\mathbb Ze_1\oplus\mathbb Ze_6\), \(\Lambda_S^+=\Lambda_S\cap\Lambda_T^+\). For \(\kappa\in\Lambda_S^+\), \(x_\kappa:=t^\mu x_0\) with \(\varphi(\mu)=\kappa\) (Computation 4.4). \(T_\lambda=\mathbf 1_{Kt^\lambda K}\), \(\varphi_0=\mathbf 1_{X(\mathcal O)}\), \(C_K=C_c^\infty(X(E))^K\), \(\mathcal S\) the Satake isomorphism, \(r\) the restriction to \(\delta(X)^{1/2}A_X^*\), \(S=\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\). \((\text{H-Cartan})_E\): \(X(E)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa\). Two test fields are \(D\)-close if there is a \(\Lambda\)-algebra isomorphism \(\mathcal O_E/\varpi_E^D\cong\mathcal O_{E'}/\varpi_{E'}^D\) with \(\bar\varpi_E\mapsto\bar\varpi_{E'}\) (Deligne's truncated rings). \(F'_e:=\mathbb Q_q(p^{1/e})\) (Eisenstein, totally ramified of degree \(e\), residue field \(\mathbb F_q\)).

Why a transfer. Sakellaridis's proof of Theorem 8.0.2 rests on characteristic-\(0\) inputs (Knop's action, Brion's paths, BLV, Luna, [Sa08], Braverman–Kazhdan, Casselman–Shalika, Tate's thesis as written in §5.3). A transfer uses all of them only over \(F'_e\), where they hold, and needs in characteristic \(p\) only facts about \(G(F)\), \(H(F)\), \(K\).

Steps.

1. Truncation: \(F\) and \(F'_e\) are \(e\)-close; conditions on elements of \(K\), \(H(\mathcal O)\) read through valuations of matrix coefficients are determined modulo \(\varpi^D\), so their solution counts agree over \(D\)-close fields (Lemma 4.2).
2. The Cartan decomposition over \(F'_e\) (Delorme–Sécherre, S4-12, S4-13) transfers to \(F\) by counting the \(K\)-orbits in each \(Kt^\lambda X(\mathcal O)\) (Proposition 4.5).
3. With the Cartan decomposition on both sides, every structure constant \((T_\lambda*\mathbf 1_{Kx_\mu})(x_\nu)\) and every Satake coefficient of \(T_\lambda\) agrees over \(D\)-close fields (Proposition 4.6).
4. Theorem 8.0.2 holds over every \(F'_e\) once Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. holds over \(\Lambda\); the paper's normalizations are checked for ramified fields, and the image \(P_\kappa\) of each orbit indicator depends only on \(q\) (Audit 4.7).
5. Over \(F\) define \(\Psi(\mathbf 1_{Kx_\kappa})=P_\kappa\) and check \(\Psi(T_\lambda*\mathbf 1_{Kx_\mu})=r(\mathcal ST_\lambda)P_\mu\) one pair \((\lambda,\mu)\) at a time, each time against an \(F'_e\) with \(e\) large enough (Theorem 4.8).

Relation to Lemma 1.7 of the earlier run (satake-converse, node A1-7, audited correct in A4-17). That lemma transfers the counts \((T_\lambda*\varphi_0)(x_\mu)\) and identities inside \(\mathcal H_K*\varphi_0\), under an embedding hypothesis (H-int) and (H-Cartan). Here: its hypotheses are checked for \(\mathcal M_\theta\) (Audit 4.3); it is extended to all structure constants and the Satake counts with labels read through \(\tau\) instead of an embedding (Proposition 4.6); and (H-Cartan) is supplied on the \(\mathbb F_q(\!(t)\!)\) side (Proposition 4.5), where the pages' earlier route (IIb step 1, input D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers.) rests on Chen–Yi, Prop. 17(i), whose proof cites results over \(\mathbb C\).