§4 · Agent 4 · Close fields · 2026-10-18T02:50:26Z

Dead end 4.10 · Dead end: Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(W(\mathbb F_q)\) is not transferred

blocked audit: minor defect

Minor defect (second audit). Checked: Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over O enters at §3.3.4 ('as o-schemes Z x U_alpha = Y'), Lemma 3.3.5, Prop. 3.3.6 and §5.1/§5.4 ('up to an integral change of coordinates'); Thm 3.7.1 gives the functional equations only up to non-zero rational functions; D(l, mu) is unbounded, so infinitely many F'e are needed; (I_Lambda) is exactly S4-13's open input (I) for Q_q and implies Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over every O'_e by Audit 4.7 step 8 ('are one statement' is loose: sufficient, not equivalent). Defects in the blocked alternative (item 4): Lemma 1.7(c) needs one fixed mu with 1{Kx_mu} not in H_K phi0 over F'_e for arbitrarily large e, whereas non-cyclicity of C_K(F'_e) for each e only gives an e-dependent mu; and Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points. enters the normalization mu(x0 J) = 1 of Delta_chi (§3.4), not its existence. The dead-end conclusion itself (the transfer needs (I_Lambda); nothing over F_q[[t]] replaces it on this route) is correct, and (I_Lambda) was later supplied by Theorem 2.8.

The transfer uses Sakellaridis's theorem at infinitely many ramified places, and his proof there uses Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(\mathcal O'_e\); by flat base change this is one statement over \(W(\mathbb F_q)\), the same open input (I) as for the unramified \(p\)-adic field. Neither a finite set of \(p\)-adic fields nor any statement over \(\mathbb F_q[\![t]\!]\) replaces it. A uniform check over \(\mathbb Z[1/2]\) of the \(45\times6\) pairs would discharge it for every \(p\); an eigenfunctional route to (SG)-failure alone is blocked at the special parameters.

Dead end 4.10.

Missing lemma \((I_\Lambda)\): for each of the \(45\) Borel orbits \(Y_{\mathbb Q_q}\subseteq X_{\mathbb Q_q}\) of maximal rank and each simple root \(\alpha\), the \(\Lambda\)-scheme \(YP_\alpha/R(P_\alpha)\), with Sakellaridis's scheme structure on \(Y\) (p. 9), is isomorphic to \(SU\backslash\mathrm{PGL}_2\) (type \(U\)) or \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\) (type \(G\)), and the complement of \(YP_\alpha\) is the closure of the complement of \((YP_\alpha)_{\mathbb Q_q}\).

  1. Why the transfer needs it. Theorem 4.8 step 3 checks \(T_\lambda*\mathbf 1_{Kx_\mu}\) against \(F'_e\) with \(e\ge D(\lambda,\mu)\), unbounded in \((\lambda,\mu)\); so Theorem 8.0.2 is needed over infinitely many \(F'_e\). In Sakellaridis's proof, Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(\mathcal O'_e\) enters the compatible normalization of eigenmeasures on the orbits of maximal rank (§3.3.4, Lemma 3.3.5, Proposition 3.3.6) and the reduction to \(\mathrm{SL}_2\) (§5.1); the constants \(b_w\), hence \(P_\kappa\), depend on it. By Audit 4.7 step 8 these infinitely many statements are one statement over \(\Lambda\), which is exactly the open item (I) of the pages for the unramified field \(\mathbb Q_q\).
  2. What does not replace it. Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(\mathbb F_q[\![t]\!]\) (part of (Q2)) is not used by the transfer and would not help: Sakellaridis's theorem is not available over \(\mathbb F_q(\!(t)\!)\). Finitely many \(p\)-adic fields give only finitely many structure constants.
  3. Route that would discharge it (not done). S4-13's route: for each pair, a point \(y\in Y(\mathbb Z[1/2])\) with \(B_y\) smooth over \(\mathbb Z[1/2]\) and \(Y=B\cdot y\) on every fibre; then, as in S4-4 steps 3–4, \(B/B_y\to X\) is a fibrewise open immersion, hence an open immersion with affine image, purity of the complement identifies it with Sakellaridis's \(Y\), and \(YP_\alpha/R(P_\alpha)\) is computed in \(\mathrm{PGL}_2\). Done over \(\mathbb Z[1/2]\), this gives \((I_\Lambda)\) for every odd \(p\), hence Theorem 8.0.2 over every \(\mathbb F_q(\!(t)\!)\) by Theorem 4.8. For \(p=3\) (bad for \(E_6\), \(F_4\)) the smoothness of the \(B_y\) is the point to watch; nothing in Lemma 4.2–Theorem 4.8 excludes \(p=3\).
  4. Blocked alternative for (SG)-failure only. Without \((I_\Lambda)\), over \(F'_e\) the open-orbit morphisms \(\Delta_\chi\) exist (they need Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points., proved in S4-5) and Theorem 3.7.1 gives the functional equations only up to non-zero rational factors. Two eigenfunctionals \(\varphi\mapsto\langle\varphi,\Omega_{y_1}\rangle\), \(\langle\varphi,\Omega_{\sigma y_1}\rangle\) with the same Hecke character \(q^{\rho-\omega_4}\), if regular and linearly independent, would show \(C_K\ne\mathcal H_K\varphi_0\) over \(F'_e\) (criterion (c) of S4-8), and Lemma 1.7(c) would transfer the failure to \(\mathbb F_q(\!(t)\!)\) without \((I_\Lambda)\). Missing: regularity and independence of \(\Omega_{y_1},\Omega_{\sigma y_1}\) at these special (non-generic) parameters, which needs the exact cocycles that \((I_\Lambda)\) provides.

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