§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:06:43Z

Audit 5.12 · Audit of A3-9 (Iwahori orbits) and A3-10 ((A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5. in characteristic p)

proved audit: correct

Second audit: correct. My scripts confirm phi(mu1) = 3omega1v, the bounds 0 and -1, <b,3omega1v> in {0,3}, and (by a dynamic search up to 40) that the labels without a P-dominant lift are exactly 3k omega1v and 3k omega6v with k odd. They also confirm the Brion-path dimensions 26,25,24,23,22,22, that all five edges have type U, that (s2s4s3s1)^{-1}s5s4s3s1 = w_g1, and the rank-two vertex data. The half-translation argument (zeta(-1) in S[2] inside T_8; F(sqrt t)/F separable) and the definitions used for relevant colours and Statement 7.1.5 match Sec. 6.2 and 7.1 of the text. Audit 5.12 correctly leaves the analytic part of Statement 6.3.1 open and correctly notes that the transfer does not need it. Correct.

Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points. over \(\mathbb F_q(\!(t)\!)\) and the combinatorial parts of (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5. are correct. All numerical claims (lifts, root pairings, Brion-path dimensions, edge types, rank-two vertex data) were recomputed and agree. The analytic part of Statement 6.3.1 remains open, as A3 says.

Audit 5.12.

Outcome: Audit 3.9 holds. Proposition 3.10 holds for (i), (ii), (iv) and for the combinatorial part of (iii). The colour-dependent statements of (iv) rest on Proposition 3.7(2), i.e. on the repair in Objection 5.11. The analytic identities of Prop. 6.5.1 over \(\mathbb F_q(\!(t)\!)\) (needed for Statement 6.3.1 itself) are not proved, and Proposition 3.10 does not claim them (Remark 3.11). This matters only for running Sakellaridis's own proof in characteristic \(p\), not for the transfer route of Theorem 4.8.

Audit 3.9.

  1. Recomputed. \(\varphi(\mu_1)=(4,3,5,6,4,2)=3\omega_1^\vee\) for \(\mu_1=(2,0,1,1,1,1)\). \(\min_{\Phi(U_Q)}\langle\beta,\mu_1\rangle=0\) over \(16\) roots, and \(\min_{\Phi(U_{P\cap M})}\langle\beta,\mu_1\rangle=-1\) over \(8\) roots. \(\langle\beta,3\omega_1^\vee\rangle\) is \(0\) on \(\Phi_M\) and \(3\) on \(\Phi(U_Q)\). So \(\mu_k=\mu_1+\tfrac{k-1}2\zeta\) satisfies the same bounds and \(\varphi(\mu_k)=3k\omega_1^\vee\).
  2. Iwahori factorization and open cells. Both are characteristic-free: an \(\mathcal O\)-point whose reduction lies in the open big cell lies in the cell, since \(\mathcal O\) is local. The reduction \(J\to B(\mathbb F_q)\) forces the \(U_P^-\)-component into \(U_P^-(\mathfrak p)\).
  3. Half-translation. \(F(\sqrt t)/F\) is separable for \(p\) odd, and \(z^{-1}\sigma(z)=\zeta(-1)\in S[2]\subseteq T_8\subseteq H\) (\(S[2]\subseteq T_8\) checked mod \(2\) in Audit 5.8 step 4). So \(R_z\) is well defined with values in \(X(F')^\sigma=X(F)\), and \(\tau(z^kx_0)=\varphi(\mu_k)(t)\) gives \(R_z^k(x_0)=\mu_k(t)x_0\).
  4. The labels \(3k\omega_6^\vee\) use \(\delta\theta_0=\theta_0\delta\), proved in Objection 5.11 step 1.

Proposition 3.10.

  1. (i), (ii). (i) is Proposition 3.6(3). \(\operatorname{supp}\gamma_1=\{1,\dots,5\}\) (type \(D_5\)) and \(\operatorname{supp}\gamma_6=\{2,\dots,6\}\).
  2. Brion path (recomputed). \(\dim Y_v=42-(4+|R_v\cap\Phi_L|+\tfrac12|R_v\smallsetminus\Phi_L|)\) with \(R_v=v^{-1}\Phi^+\cap\theta(v^{-1}\Phi^+)\) gives \(26,25,24,23,22,22\) for \(v=1,s_1,s_3s_1,s_4s_3s_1,s_2s_4s_3s_1,s_5s_4s_3s_1\). All five edges have \(v^{-1}\alpha\notin\Phi_L\), i.e. type \(U\). \((s_2s_4s_3s_1)^{-1}s_5s_4s_3s_1=w_{\gamma_1}\), of length \(8=2\cdot3+2\).
  3. Rank-two vertices (recomputed). At \((v,\delta,\epsilon)=(s_1;\alpha_1,\alpha_3)\), \((s_3s_1;\alpha_3,\alpha_4)\), \((s_4s_3s_1;\alpha_4,\alpha_2)\) and \((s_4s_3s_1;\alpha_4,\alpha_5)\), the Levi roots \(\{\pm\delta',\pm\epsilon',\pm(\delta'+\epsilon')\}\) form an \(A_2\), and \(\theta\) fixes exactly one pair, a black root (\(\alpha_3,\alpha_4,\alpha_2,\alpha_5\) respectively). At \((s_4s_3s_1;\alpha_2,\alpha_5)\), \(\theta\delta'=-\epsilon'\) and \(\theta\epsilon'=-\delta'\). The stabilizer arguments (maximal parabolic of \(\mathrm{PGL}_3\); \(\Delta\mathrm{PGL}_2\) via the decomposition \(\mathfrak{sl}_2\oplus\mathfrak{sl}_2=\Delta\oplus\Delta^-\), both irreducible for \(p\ne2\)) are correct.
  4. (iv). Every Brion path for \(w_{\gamma_1}\) spells a reduced word of \(w_{\gamma_1}\in W_{\{1,\dots,5\}}\), and its first edge lowers \(\mathring X\) with a type-\(U\) label, so the label is \(\alpha_1\) and the colour is \(D_1\). \(\rho_{P(X)}=4\omega_1+4\omega_6\) and \(r=4\). Against the definition on p. 60: \(\Theta^+=\{e_1,e_6,e_1+e_6\}\), every element of \(\Theta\) is \(>0\) or \(<0\), and \(\{\check\theta\in\Theta^+:s_{\gamma_1}\check\theta<0\}=\{e_1\}\), the virtual colour of \(D_1\). So Statement 7.1.5 holds.

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

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