§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:31:44Z

Audit 5.15 · Audit of A1-6: F_w S_χ = b_w S_{wχ}, the sum over W_X′, and the open-orbit route

proved audit: correct ★ key step

Second audit: correct. I recomputed the SL_2 intertwiner on the Iwahori basis with the bottom-row criterion (g ∈ BJ iff val c > val d) and got exactly Audit 5.15's formulas: Tφ_e = (1−q^{-1})ζ/(1−ζ)φ'e + q^{-1}φ'_w, Tφ_w = φ'_e + (1−q^{-1})/(1−ζ)φ'_w, Tφ_K = (1−q^{-1}ζ)/(1−ζ)φ'_K, and φ_e − ζF_sφ_e = q^{-1}(1−ζ)/(1−q^{-1}ζ)(φ_e+φ_w). By hand: F_s^2 = 1 with the (4.9) normalization; the normalizers of F_w and F{w^{-1}} match under α̌ = −w^{-1}β̌; the four cocycles c, b^D, B, β/(β∘w) satisfy u_{w1w2} = (u_{w1}∘w2)u_{w2}; the pairing w = w's_β gives the factor −e^{β̌}(w'^{-1}χ); and the reindexed coefficient c(v)b_v(χ^{-1}) is Sakellaridis's (4.2) coefficient σ(w)e^{ρ̌−w^{-1}ρ̌}b_w. Part (a) (wD = D iff w ∈ N_W(𝔞) and wρ_L = ρ_L, since QΦ_L ∩ 𝒳(X)_Q = 0) is right, as is the scope remark. The only thing not mentioned is a sign slip in Proposition 1.6 step 2 (the value for |x| = q^n is e^{+nα̌}(χδ^{1/2}), not e^{−nα̌}), which does not reach the stated c_α; immaterial.

The functional equations from generic uniqueness, the normalized intertwiners on $I(\chi)$ replacing [BK98] and [Ca80], and the sum formula over $W_X'$ are correct over every non-archimedean local field. I recomputed the rank-one $K$-invariance and checked that all cocycles involved satisfy the same cocycle rule. Knop's action and Axiom 2.2.1 are indeed not used in this route.

Audit 5.15.

Outcome: holds. (a) $W_X'=\{w:wD=D\}=N_W(\mathfrak a)\cap[W/W_L]$ is correct, by the argument via $w\rho_L=\rho_L$. (b) Rank-one check. With $\zeta=e^{\check\alpha}(\chi)$ on $\mathrm{SL}_2$, $\psi$ of conductor $\mathcal O$ and $dx(\mathcal O)=1$: $$T\varphi_e=\tfrac{(1-q^{-1})\zeta}{1-\zeta}\varphi'_e+q^{-1}\varphi'_w,\qquad T\varphi_w=\varphi'_e+\tfrac{1-q^{-1}}{1-\zeta}\varphi'_w .$$ Hence $T\varphi_K=\frac{1-q^{-1}\zeta}{1-\zeta}\varphi'_K$ (that is (4.8)), and $$\varphi_{J,\chi}-\zeta\,F_s\varphi_{J,s\chi}=\frac{q^{-1}(1-\zeta)}{1-q^{-1}\zeta}\,(\varphi_e+\varphi_w)$$ is $K$-invariant. So Proposition 4.2.1 holds on $I(\chi)$ with A1's coefficients $\prod_{\alpha>0,\,w^{-1}\alpha<0}(-e^{\check\alpha}(\chi))$. (c) Cocycle rules. $c(w)=\sigma(w)e^{\check\rho-w^{-1}\check\rho}$, the transported $b^D_v(\chi):=b_v(\chi^{-1})$, the product $B_v=c(v)b^D_v$, and $\beta/(\beta\circ w)$ all satisfy the same rule $u_{w_1w_2}=(u_{w_1}\circ w_2)\,u_{w_2}$. So "two cocycles agreeing on generators agree" (Computation 1.7 step 2) is legitimate. (d) In the $w$-term of $\Delta_\chi(\Phi_{K,\chi})(x)$ the character is ${}^{w^{-1}}\tilde\chi$. The formula is correct after reindexing $v=w^{-1}$, which matches A1's display. Scope. Axiom 2.2.1 and the orbits $w\mathring X\ne\mathring X$ are not needed for (6.2) with unknown $b_w$. Sakellaridis normalizes eigenmeasures on all maximal-rank orbits only to compute the $b_{w_\gamma}$ along Brion paths (Statement 6.3.1), which A1 keeps as a hypothesis. To state that hypothesis for type-$T$ roots one needs Brion paths in characteristic $p$; for $\mathcal M_\theta$ (type $G$ only) the formula is explicit.

(a). $wD=D$ forces $w$ to preserve $\mathcal X(X)_{\mathbb Q}$, hence $\mathfrak a=X_*(S)_{\mathbb Q}$, and $|e^{w\rho_L-\rho_L}|\in A_X^*$. Since $\mathbb Q\Phi_L\cap\mathcal X(X)_{\mathbb Q}=0$, the latter means $w\rho_L=\rho_L$, i.e. $w\Phi_L^+=\Phi_L^+$, i.e. $w$ is minimal in $wW_L$. Such $w$ commute with $w_L$ and fix $\delta(X)$, and $W_X'\cap W_L=1$. The simple reflections of $W_L$ move $\delta(X)^{1/2}$, so the type-$G$ vanishing of Theorem 3.7.1 is subsumed in (c), second half.
(b), rank one, my computation. Representatives of $B\backslash\mathrm{SL}_2/J$ are $1$ and $w=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. For $|x|\le1$, $wu(x)\in wJ$. For $|x|=q^n>1$, $wu(x)=\begin{pmatrix}-x^{-1}&1\\0&-x\end{pmatrix}\begin{pmatrix}1&0\\x^{-1}&1\end{pmatrix}\in BJ$. For the second representative, $wu(x)w=\begin{pmatrix}-1&0\\x&-1\end{pmatrix}$ lies in $J$ for $|x|<1$, and equals $\begin{pmatrix}-x^{-1}&1\\0&-x\end{pmatrix}w\,u(-x^{-1})\in BwJ$ for $|x|\ge1$. Summing with $\operatorname{vol}\{|x|=q^n\}=q^n(1-q^{-1})$ gives the two formulas in the statement. Then $F_s=\frac{1-\zeta^{-1}}{1-q^{-1}\zeta}T$ on $I(s\chi)$ gives the displayed $K$-invariant combination. The general case follows as in Proposition 4.2.1: for $w=w'w_\beta$ with $w'\in[W/W_\beta]$, the coefficients factor as $c(w'^{-1})(\chi)\cdot(-e^{\check\beta}(w'^{-1}\chi))$, and $F_{w'}$ preserves $K_\beta$-invariance. This is characteristic-free and uses neither [BK98] nor [Ca80].
Normalized operators. $F_s^2=1$ follows from $T_s^2=c(\chi)c(s\chi)$, using $\operatorname{End}_GI(\chi)=\mathbb C$ for generic $\chi$ (geometric lemma, valid for ℓ-groups) and the rank-one value. The braid relations follow from additivity of lengths. $F_w^*=F_{w^{-1}}$: the normalizing products match under $\check\beta=-w\check\alpha$.
(c). For $w\in W_X'$, $F_wS_\chi$ lies in the line of Proposition 1.5(b) at $w\chi\in D'$; rationality of $b_w$ comes from Proposition 1.5(c). For $w\notin W_X'$, the restriction of $\mathrm{ev}_1\circ F_wS_\chi$ to $C_c^\infty(\mathring X(F))$ is $(B,w\chi)$-equivariant, which forces $w\chi\in D'$; this fails for generic $\chi$.
(d), bookkeeping. $\Delta_\chi(F_w\varphi_{J,w^{-1}\chi})(x)=\langle\varphi_{J,w^{-1}\chi},F_{w^{-1}}S_{\chi^{-1}}(\mathbf 1_{xJ})\rangle/|\omega|(xJ)$. This vanishes unless $w^{-1}\in W_X'$. By Lemma 3.9.1 (only (A3c) used: $xJ\subseteq xB(\mathcal O)\subseteq\mathring X(F)$, where the character is constant), it then contributes $b_{w^{-1}}(\chi^{-1})\,\operatorname{vol}(J)\,({}^{w^{-1}}\tilde\chi)\delta^{-1/2}(x)$. With $v=w^{-1}$ the coefficient of ${}^v\tilde\chi$ is $c(v)(\chi)\,b_v(\chi^{-1})$, where $c(v)=\prod_{\check\alpha>0,\,v\check\alpha<0}(-e^{\check\alpha})$. This is A1's $B_v$.
Cocycle rule of $b^D$. $b^D_{v_1v_2}(\chi)=b_{v_1v_2}(\chi^{-1})=b_{v_1}(v_2\chi^{-1})\,b_{v_2}(\chi^{-1})=b^D_{v_1}(v_2\chi)\,b^D_{v_2}(\chi)$. The same rule holds for $c$ ($\sigma$ is a character and $e^\mu(w\chi)=e^{w^{-1}\mu}(\chi)$) and for $\beta/(\beta\circ w)$. So Computation 1.7 step 2 is valid.
Non-vanishing. The term $v=1$ gives $\tilde\chi\delta_{P(X)}^{-1/2}$, and the characters ${}^v\tilde\chi$ ($v\in W_X'$) are distinct for generic $\chi$, so $\Omega_\chi\not\equiv0$. This also gives $\Phi_{K,\chi}=m(\chi)\varphi_{K,\chi}$ with $m\not\equiv0$, which Theorem 1.9 step 3 needs and does not prove separately.

Builds on

Proposition 1.6 · $F_wS_\chi=b_wS_{w\chi}$ and the sum formula (6.2) over $\mathbb F_q((t))$

Built on by

Summary 5.21 · Verdict of the second audit: A1-1 … A1-9 (direct rerun over F_q((t)))
Lemma S1.3 · Positivity of $\hat S$ forces $c\neq0$ and $a_J/c>0$ over every local field

Sources

Proposition 1.6 (claim audited)
sakellaridis-v3.pdf Prop. 4.2.1, Thm 4.2.2 pp. 26–27; (4.8) p. 28; Lemma 3.9.1 p. 24; proof of Thm 7.2.1 p. 61 (cocycle relations)
Bernstein–Zelevinsky geometric lemma (cited)

Table 1. Page production by session
	tokens	US $
The run, its in-run audits and the pages (from 01:50 UTC)01:50–05:20 UTC	349 M	$144
This conversation: the audit pass (eight reviewers, one salvage agent, re-checks) and the simplified VIb page05:21–07:33 UTC	267 M	$106
Removing the detail switch from the published trees05:34–05:51 UTC	15 M	$7
Total	631 M	$257