§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:33:42Z

Audit 5.19 · Audit of A1-1 (inventory of inputs) and A1-8 (the open item (S6.3.1)_p)

proved audit: correct ★ key step

Second audit: correct. I checked the inventory against every bracketed citation of §§1–8: the only other unlisted item, [Sa08, §3.8] (p. 7, eigenmeasures), is part of hypothesis (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7)., and [Ca80, Thm 4.2], [Ga99], [Kn94], [GN10] occur only in remarks or in §§5–6, which the route avoids. The row-25 correction is right (p. 27: Lemma 4.2.3 uses that the image of Φ_K is finite at δ^{1/2}, from (4.5), proved with [CS80, Thm 5.4]), and Dead end 1.8(i), (ii) re-derive (Gindikin–Karpelevich product c'_w; Weil volumes q^{−N dim X}; O_E/ϖ^N ≅ F_q[t]/t^N for e ≥ N). I opened [Sa08] (arXiv:math/0606130v2): §1.7 and §4.1 assume only a locally compact non-archimedean field of characteristic 0 and a split connected group, with no unramified hypothesis, so the coordinator's report Audit 5.19 relied on is accurate and H_Sa08 is discharged. The chain (I_Λ) ⇒ Theorem 7.2.1 and (4.5) over every F'_e (Audit 4.7, graded elsewhere) ⇒ M4 ⇒ Z_F = R_q ⇒ (S6.3.1)_p is valid (Laurent expansions on the same cone R; constant normalizations cancel in b_w), and for M_θ the constancy ω = c comes from (7.4) at λ = 0, so it does not use the faulty p. 62 step.

The inventory of external inputs is accurate up to one misfiled row (Lemma 4.2.3 uses (4.5), hence [CS80]). The dead-end node proves its two reductions correctly. Its route B coincides with the close-fields transfer of A4, so \((\mathrm{S6.3.1})_p\) follows from \((I_\Lambda)\); with the coordinator's information on [Sa08], nothing else remains.

Audit 5.19.

Outcome: Strategy 1.1 holds, with one correction. Dead end 1.8 holds (its "blocked" status is accurate). (i) Correction to Strategy 1.1 row 25: Lemma 4.2.3 is not "char-free given (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x. and the invariant measure". Its proof uses that the image of \(\Phi_K\) is finite at \(\delta^{1/2}\), taken from (4.5), i.e. [CS80] (p. 27). The lemma fixes only the value of \(c\), which the theorem does not need, so the route is unaffected. All other rows match the text, including the replacements of [BLV86] (Lemma 1.3), [Sa08] (Proposition 1.5, Proposition 1.6), [BK98] and [Ca80] (Audit 5.15 step 2). (ii) Dead end 1.8(i) \(b_w=c'_w\,Z_F(\chi)/Z_F(w\chi)\) and (ii) the level lemma are correct. (iii) Link to the first audit. Route B's missing lemma M4 (\(Z_E=R_q\) for all totally ramified \(E\)) follows from Theorem 8.0.2 over every \(F'_e\). By Audit 4.7 that theorem holds under \((I_\Lambda)\), and the residual \(H_{\mathrm{Sa08}}\) of Audit 5.5 is discharged by the coordinator's information that [Sa08] (arXiv:math/0606130v2, §§1.7, 4.1) assumes only characteristic \(0\), with no unramified assumption. I did not open [Sa08]. Hence \((I_\Lambda)\Rightarrow(\mathrm{S6.3.1})_p\), and the two routes A4 and A1 rest on the same open item.

  1. Inventory spot checks. \(\psi(x)=\psi_0(\operatorname{Tr}_{\mathbb F_q/\mathbb F_p}\operatorname{Res}(x\,dt))\) is trivial on \(\mathcal O\) (no \(t^{-1}\) term) and non-trivial on \(t^{-1}\mathcal O\): conductor \(\mathcal O\). Lemma 3.3.5 and Proposition 3.3.6 concern non-open orbits and are not used in the open-orbit route (Audit 5.15 Scope). (A3b′) with antidominant \(W_X'\)-labels sidesteps [Kn91, §5]. For \(\mathcal M_\theta\) it follows from Theorem 3.8 or Proposition 4.5 through \(\kappa\mapsto-\bar\mu\), with no valuation cone. [Sa06, Prop. 3.6.1] and Brion's paths enter only §6, i.e. the proof of Statement 6.3.1, which is kept as a hypothesis.
  2. A1-8 (i). \(\mathbf 1_{X(\mathcal O)}\) is \(K\)-invariant, so \(S_\chi(\mathbf 1_{X(\mathcal O)})=Z_F(\chi)\varphi_{K,\chi}\). \(F_w\varphi_{K,\chi}=c'_w(\chi)\varphi_{K,w\chi}\) with \(c'_w=\prod\frac{1-q^{-1}e^{\check\alpha}}{1-q^{-1}e^{-\check\alpha}}\) (Audit 5.15 step 2). Applying \(F_wS_\chi=b_wS_{w\chi}\) gives the formula. \(Z_F\not\equiv0\) because on real points of \(R\) it is the integral of a positive function.
  3. A1-8 (ii). On \(X(\mathcal O)\cap\mathring X(F)=\{f_1f_6\neq0\}\), the label \(\lambda(x)\) is determined by \((\operatorname{ord}f_1,\operatorname{ord}f_6)\), because \(\omega_1,\omega_6\) is a basis of \(\mathcal X(X)\). With the residually non-vanishing invariant form (A3-2 item 5), the fibres of \(X(\mathcal O)\to X(\mathcal O/\varpi^N)\) have volume \(q^{-N\dim X}\). So \(z_\mu\) is a point count over \(\mathcal O/\varpi^N\), \(N>\max m_i\), with \(f_i\) and \(X\) defined over \(\Lambda\). It agrees for \(\mathbb F_q(\!(t)\!)\) and \(F'_e\), \(e\ge N\) (Lemma 4.2(a)). This is a special case of A4's determinacy.
  4. Route B closes under \((I_\Lambda)\). For each \(\mu\), choose \(e>\max m_i\). Then \(z_\mu(F)=z_\mu(F'_e)\), and \(z_\mu(F'_e)\) is the \(\mu\)-coefficient of \(R_q\) by Theorem 8.0.2 over \(F'_e\): (1.4) together with (4.5) and (7.3). So the Laurent expansions of \(Z_F\) and \(R_q\) on \(R\) coincide, \(Z_F=R_q\), and (i) gives \((\mathrm{S6.3.1})_p\).
  5. Route A and Theorem 6.3.2. Route A (rerunning §6.19 in characteristic \(p\)) correctly lists M1 (orbit-by-orbit convergence, uniqueness, Fubini) and M2 (orbit geometry on the \(D_5\) quadric) as missing. The general Theorem 6.3.2 rests on the classification of [Was96] and [Ak83], available only in characteristic \(0\). Both statements are accurate.

Builds on

  • Strategy 1.1 · Every external input of Theorem 8.0.2 and its status over \(\mathbb F_q((t))\)
  • Dead end 1.8 · \(B_{w_\gamma}\) for \(M_\theta\) over \(\mathbb F_q((t))\): two routes, missing lemmas

Built on by

  • Summary 5.21 · Verdict of the second audit: A1-1 … A1-9 (direct rerun over F_q((t)))

Sources