§1 · Agent 1 · Direct rerun · 2026-10-18T03:51:57Z

Strategy 1.1 · Every external input of Theorem 8.0.2 and its status over \(\mathbb F_q((t))\)

proved audit: minor defect ★ key step

Minor defect (second audit). I checked all 27 rows against arXiv:0905.4244v3 (where each input sits, page, what it is used for) and against the replacing nodes: the inventory is accurate and complete for the proof of Theorem 8.0.2 (Lemma 7.1.4 is row 22), and psi(x)=psi_0(Tr Res(x dt)) has conductor exactly O with dx(O)=1 self-dual. One misfiling: row 25 lists Lemma 4.2.3 as characteristic-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, but its proof (p. 27) needs the image of Phi_K to be finite at delta^{1/2}, taken from (4.5), i.e. [CS80]; this also contradicts row 17 ('not needed'). Nothing on the E6/F4 route depends on it, because Computation 1.7 computes c directly.

Lists the 27 external inputs on which Sakellaridis's proof of Theorem 8.0.2 rests, with the step using each and its status in characteristic p; isolates the four that need real work.

Method. I read §§1–8 of the text extract line by line and listed every citation and every 'it is known', together with the step that uses it. Each input is classified as: char-free (the argument is valid over any non-archimedean local field), residue-characteristic only, a characteristic-0 citation that is replaced in a later node, avoided (the route below does not need it), or kept as a hypothesis.

Table 2. # and input (place in arXiv:0905.4244v3) and used for
#input (place in arXiv:0905.4244v3)used forsource / characteristicstatus over F_q((t)), p odd
1ψ of conductor O, dx(O)=1, 'k unramified over Q_p' (§1.1 p.2, §3.8 p.23, §5.3 p.32)normalizations of all Fourier transformsfor F_q((t)) take ψ(x)=ψ_0(Tr_{F_q/F_p} Res(x dt)), conductor O; dx(O)=1 is self-dual. No 'unramified' issuechar-free
2x_0 ∈ X̊(O) via [Sa08, Prop. 3.2.1] + Hensel (§2.1 p.8)choice of base pointLang's theorem for the connected solvable B_{F_q} on the homogeneous X̊_{F_q}, then smooth liftingresidue field only; char-free
3[BLV86, Thm 3.4]: [L(X),L(X)] ⊆ H (§2.1) and (B_y)_u = L(X)∩U (§3.4, eq. (3.5))definition of A_X; δ_{P(X)} ∈ A_X^*char 0 sourcereplaced for symmetric X, p odd: Lemma 1.3
4[Kn91, Cor. 1.8], [Kn91, §5]: Λ_X^+ ⊆ Λ_X, V = image of the antidominant chamber for symmetric X (§2.1)labels of K-orbits in (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits.char 0 sourceavoided: (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits. is restated with antidominant labels (A3b′); not used in the proof
5[Kn95] Knop's W-action on B_{F̄}-orbits; W(X)=W_X⋉W_{P(X)} (§2.2)definition of W_X; which T_w survivechar 0 source (not opened)avoided: W_X′ := Stab_W(δ(X)^{1/2}A_X^*) ∩ [W/W_{P(X)}] defined analytically (Proposition 1.6); for M_θ computed (Computation 1.7)
6[Kn94], [Kn96]: W_X a reflection group with fundamental domain V; root system (§2.2, §6.1)triangularity in §8; Statement 6.3.1char 0 sourceavoided as above; for M_θ, W_X′ = W(A_2) explicitly (Computation 1.7)
7[Sa06, Prop. 3.6.1]: max-rank B_{F̄}-orbits are defined over F with F-points (§2.2)intermediate orbits of Brion paths (§3.3, §6)p-adic source (not opened)not needed for the route of Proposition 1.5–Proposition 1.6; needed only to rerun §6 (Dead end 1.8)
8Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. consequences (p. 9): every max-rank orbit is reached from X̊ by type-U reflectionssmoothness of all max-rank Y over Ouses Knop's transitivity (char 0)not needed in the route below; needed for Dead end 1.8
9[Br01, Props 2, 4; Thm 4] (§2.3)Prop. 3.3.6 (path independence of measures), §6.2char 0 sourcenot needed for Thm 8.0.2 in the route below (only the open orbit is normalized); needed for Dead end 1.8
10[Sa08, Thm 5.2.1] in Lemma 3.3.5 (p. 15)a priori proportionality T_{w_α}S^Z ∼ S^Ychar 0 sourceunnecessary: Fubini over Z×U_α ≅ Y gives the equality directly; char-free
11convergence and rationality of the integrals (3.2) [Sa08] (p. 12)existence of S_χ, Δ_χ as rational familieschar 0 source (not opened)replaced: Proposition 1.5 (convergence by eigenfunction bounds; rationality by Bernstein's principle)
12[Sa08] Thm 3.2.2 (multiplicity g·a) (p. 13)spanning statements of Thm 4.2.2char 0 sourcenot needed for Thm 8.0.2; only generic uniqueness is used, proved in Proposition 1.5
13[Sa08] Thm 3.7.1 / Cor. 3.7.2 (pp. 21–22)functional equations F_wΔ_χ = b_wΔ^{wX̊}char 0 sourcereplaced: Proposition 1.6 (uniqueness + rank-one kill for α ∈ Δ(X))
14finiteness of B(F)-orbits on X(F) and of H^1 (implicit in [Sa08])Bruhat-filtration argumentsfails in general in char p (H^1(F,Z/p), H^1(F,μ_p) infinite)proved for symmetric X, p odd: Lemma 1.4
15[BK98] Schwartz space S(U\G), F_wF_{w′}=F_{ww′}, basic vector (§3.8, §4.3)cocycle (3.12), Prop. 4.2.1stated over p-adic fields (not opened)replaced: F_w := normalized intertwining operator on I(χ) via (4.9); cocycle from T_w (Proposition 1.6)
16[Ca80, Thm 3.1] Gindikin–Karpelevich (4.8) (p. 28)relation F_w vs T_wnot openedrank-one SL_2 integral + cocycle; char-free (Proposition 1.6)
17[CS80, Thm 5.4] (4.5), Cor. 4.3.1normalization of Ω vs φ_{K,χ}—not needed for Thm 8.0.2 (a scalar independent of λ cancels; Proposition 1.2)
18Tate's local functional equation (§5.3)rank-one constants bany local field (Tate's thesis; Weil, Basic Number Theory VII)char-free with ψ of conductor O
19[Lu01, §1] (type T split, p. 31); §5.6–5.8 types T, N (quadratic characters, F^×/F^{×2})rank-one constants for T, Nresidue char ≠ 2 onlynot needed for M_θ (types G, U only)
20[Was96], [Ak83]: rank ≤ 2 classification; Lemma 6.2.2, Prop. 6.2.1, Thm 6.11.1, §§6.4–6.21Thm 6.3.2 ((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.(i),(ii) ⇒ Statement 6.3.1)char 0 classification(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.(iii) kept as a hypothesis; for M_θ open (Dead end 1.8)
21[SV]: normalized spherical roots form a root datum (§6.1)Φ_X, βcombinatorics over F̄for M_θ explicit: Φ_X = A_2, Δ_X = {γ_1, γ_6}
22Lemmas 7.1.1–7.1.4 (case-by-case), colours and Θ (§7.1)definition of β, Statement 7.1.5char 0 case analysisnot needed for M_θ: β defined directly from Φ̌_X^+ (Computation 1.7)
23[Kn91, §6] affine ⟺ V and colour valuations separated (§7.2)ω constant; T ∩ V = {0}char 0 sourceM_θ: T = cone(Φ̌_X^+), T ∩ V = {0} elementary; constant c computed (Computation 1.7)
24Lemma 3.4.2, Lemma 5.1.1 (unfolding)Δ_χ as an integral over U\BG_ξneeds smooth orbit mapschar-free once H and B_y are smooth: Lemma 1.3
25Lemma 3.9.1, Lemma 4.2.3, Prop. 4.2.1, §8 proofsum formula and last step—char-free given (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., invariant measure, rank-one Fourier on F^2 (Proposition 1.2, Proposition 1.6)
26Axioms 2.4.1, 2.4.2 ((A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits., (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x.); [LV83], [Sa12], [BO07], [DS11]Cartan and Iwahori orbitshypotheses of the theoremassumed (verification is (Q3))
27'hold at almost every place' (§1.7 p.7, §2.2 p.9)justification of the axiomsglobal statementnot usable at a given place; the axioms stay hypotheses

Logical skeleton of Theorem 8.0.2 (p. 64), as the route used here.

  1. A rational family of \((B,\chi)\)-equivariant functionals \(S_\chi\) on \(C_c^\infty(X(F))\), \(\chi\in\delta(X)^{1/2}A_X^*\), with generic uniqueness (Lemma 1.4, Proposition 1.5).
  2. Functional equations \(F_wS_\chi=b_w(\chi)S_{w\chi}\) for \(w\in W_X'\) and the sum formula (4.2)/(6.2) for \(\Omega_\chi(x_\lambda)\) (Proposition 1.6).
  3. The cocycle values \(B_{w_\gamma}\) (Statement 6.3.1) and the coboundary \(\beta\) (Theorem 7.2.1) (Computation 1.7, Dead end 1.8).
  4. The formal passage from (7.4) and the Cartan decomposition to the module isomorphism (Proposition 1.2).

Finding. Every characteristic-0 citation of the paper enters either through the geometry of Borel orbits over \(\bar F\) (Knop, Brion, Luna, Wasserman, Akhiezer, [BLV86], [Kn91]) or through [Sa08] (rationality, multiplicity, composition with intertwining operators). The harmonic analysis on \(F\) itself (Tate, Fourier transform on \(F^2\), Gindikin–Karpelevich, Iwahori decomposition) is characteristic-free once \(\psi\) has conductor \(O\).

Builds on

  • —

Built on by

  • Proposition 1.2 · Triangular spherical transforms give \(C_K\cong\mathbb C[D]^{W_X}\), any characteristic
  • Lemma 1.3 · Local structure and smooth Borel stabilizers for \(G/G^\theta\) in characteristic \(\neq2\)
  • Audit 5.19 · Audit of A1-1 (inventory of inputs) and A1-8 (the open item (S6.3.1)_p)

Sources