§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:24:47Z

Theorem 6.6 · $E_6/F_4$: two points of the shifted torus that are not $W_X$-conjugate are $W(E_6)$-conjugate, so $r$ is not onto and (SG) fails under (A1)–(A5)

proved audit: minor defect ★ key step

Minor defect (second audit). Main claim verified independently, in a Bourbaki R^8 model of E6 with theta=-w_L built from scratch: lambda2* lies in the W-orbit of lambda1* (orbit of size 25920, unique dominant element rho-omega4); both reflection words end at rho-omega4; the explicit w is an element of W(E6) with <lambda1*,mu>=<lambda2*,w mu>, and exactly two elements of W have this property; the W_X-orbits, sigma(y1), and the values of M_(3,0), M_(0,3), M_(1,1) recomputed. Wording defect: the gloss 'the Hecke module H_K phi_0 misses the orbit label 3omega1^vee' conflates the orbit sum M_(3,0) with the image of the orbit indicator (Sakellaridis's P_lambda, not M_lambda), and no argument is given for the stronger reading 1_{O_{3omega1^vee}} not in H_K phi_0. That reading is true under (A): in the proof of Sakellaridis Thm 8.0.2, 1_{x_lambda K} maps to P_lambda whose other weights lie in lambda+T, T the cone of colour valuations; T is sigma-stable, strictly convex and meets the antidominant cone only in 0, so T lies in {a+b>0} union {0}; hence P_(3,0)=c M_(3,0)+c1 M_(1,1)+c0 with c!=0, and l(g)=g(y1)-g(sigma y1) gives l(P_(3,0))=c l(M_(3,0))!=0. No downstream node uses the strong reading as a proof step.

The points $y_1=(q^{-5},q)$ and $y_2=(q^{-5},q^4)$ of the dual torus $A_X^*$ are not related by the little Weyl group $S_3$, but after the shift by $q^{\rho_L}$ they become conjugate under $W(E_6)$, and the orbit sum $M_{(3,0)}$ takes different values at them. So $M_{(3,0)}\notin\operatorname{im}r$ and, by Theorem E, spherical generation fails for $E_6/F_4$ over $p$-adic fields under Sakellaridis's hypotheses: the function-level half of the only candidate divergence.

Theorem 6.6.

Let $X=E_6/F_4$, with $r$, $u,v$, $M_\kappa$, $m_\mu$ and $\sigma$ as in Computation 6.4, and let $y_1,y_2\in A_X^*$ be the positive real points with $(u,v)$-coordinates $$y_1=(q^{-5},\,q),\qquad y_2=(q^{-5},\,q^{4}).$$ Then:

$q^{\rho_L}y_1$ and $q^{\rho_L}y_2$ are $W$-conjugate in $A^*$;
$y_1$ and $y_2$ are not $W_X$-conjugate;
$M_{(3,0)}(y_1)\ne M_{(3,0)}(y_2)$.

Hence $M_{(3,0)}\notin\operatorname{im}r$, $r$ is not onto, and by Theorem E(2) (SG) fails for $E_6/F_4$ under (A1)–(A5), for every $q$. The conclusion does not depend on the sign of the shift or on left/right conventions (Setup 6.1). More precisely, $$\operatorname{im}r\subseteq\mathcal K:=\{g\in\C[\Lambda_S]^{S_3}:g(y_1)=g(\sigma y_1)\}\subsetneq\C[\Lambda_S]^{S_3},$$ and the missing generator is $M_{(3,0)}$: the Hecke module $\cH_K\varphi_0$ misses the orbit label $3\omega_1^\vee$ (and, symmetrically, $3\omega_6^\vee$).

A point $x\in A^*$ with $x^{\alpha_j^\vee}=q^{n_j}$, $n_j\in\Z$, is written $x=q^\lambda$, where $\lambda$ is the weight with $\langle\lambda,\alpha_j^\vee\rangle=n_j$; then $x^\mu=q^{\langle\lambda,\mu\rangle}$, and $w\cdot q^\lambda=q^{w\lambda}$ for $w\in W$.

Proof of 1.

Write $(u_i,v_i)$ for the coordinates of $y_i$. By Computation 6.4, $x_i:=q^{\rho_L}y_i$ has $x_i^{\alpha_1^\vee}=q^{-3}u_i$, $x_i^{\alpha_6^\vee}=q^{-3}v_i$ and $x_i^{\alpha_j^\vee}=q$ for $j=2,\dots,5$. So $x_1^{\alpha_1^\vee}=x_2^{\alpha_1^\vee}=q^{-8}$, $x_1^{\alpha_6^\vee}=q^{-2}$, $x_2^{\alpha_6^\vee}=q$, and $x_i=q^{\lambda_i^*}$ with, in fundamental-weight coordinates,

$$\lambda_1^*=(-8,1,1,1,1,-2)=\rho-9\omega_1-3\omega_6,\qquad\lambda_2^*=(-8,1,1,1,1,1)=\rho-9\omega_1 .$$

It suffices to show that $\lambda_1^*$ and $\lambda_2^*$ are $W$-conjugate.

A simple reflection $s_j$ changes fundamental-weight coordinates by $c\mapsto c-c_j\cdot(\text{row }j\text{ of the Cartan matrix})$. Applying it repeatedly at a negative coordinate, the words (applied from left to right)

$$\lambda_1^*:\ s_1s_3s_4s_2s_5s_4s_3s_1s_6s_5s_4s_2s_3s_1s_4s_3s_5s_6,\qquad\lambda_2^*:\ s_1s_3s_4s_2s_5s_4s_3s_1s_6s_5s_4s_2s_3$$

take both weights to the dominant weight $(1,1,1,0,1,1)=\rho-\omega_4$. Hence $\lambda_1^*$ and $\lambda_2^*$ are $W$-conjugate.

Norm check: $(\lambda_1^*,\lambda_1^*)-(\lambda_2^*,\lambda_2^*)=-6(\rho,\omega_6)+54(\omega_1,\omega_6)+9(\omega_6,\omega_6)=-48+36+12=0$.
An explicit $w\in W$ with $x_1^\mu=x_2^{w\mu}$ for all $\mu$, found by exhaustive search over the 51840 elements of $W$, is

$$w(\alpha_1^\vee)=\alpha_1^\vee,\quad w(\alpha_2^\vee)=\alpha_4^\vee,\quad w(\alpha_3^\vee)=-(\alpha_1^\vee+\alpha_2^\vee+\alpha_3^\vee+2\alpha_4^\vee+2\alpha_5^\vee+\alpha_6^\vee),$$ $$w(\alpha_4^\vee)=\alpha_5^\vee,\quad w(\alpha_5^\vee)=\alpha_6^\vee,\quad w(\alpha_6^\vee)=\alpha_1^\vee+\alpha_2^\vee+2\alpha_3^\vee+2\alpha_4^\vee+\alpha_5^\vee .$$

Independently of $w$: $m_\mu(x_1)=m_\mu(x_2)$ as exact Laurent polynomials in $q^{1/2}$ for all 23 dominant $\mu$ of norm at most 20.

Proof of 2. In logarithmic coordinates $(\log_qu,\log_qv)$ the generators of $W_X$ act by $s_1(\alpha,\beta)=(-\alpha,\alpha+\beta)$ and $s_6(\alpha,\beta)=(\alpha+\beta,-\beta)$, so $$W_Xy_1=\{(-5,1),(-4,-1),(-1,5),(1,4),(4,-5),(5,-4)\},\qquad W_Xy_2=\{(-5,4),(-4,5),(-1,-4),(1,-5),(4,1),(5,-1)\}.$$ These are disjoint, and since $q>1$ a positive real point is determined by its logarithms. Moreover $\sigma(u,v)=(v,u)$, so $\sigma y_1=(q,q^{-5})\in W_Xy_2$: the gluing identifies $y_1$ with its $\sigma$-image, matching the $\sigma$-odd functional of Computation 6.5.

Proof of 3. $W_X(3,0)=\{(3,0),(-3,3),(0,-3)\}$. Writing $(a,b)=c_1e_1+c_6e_6$ with $c_1=\tfrac{2a+b}3$ and $c_6=\tfrac{a+2b}3$, one has $y^{(a,b)}=u^{c_1}v^{c_6}$, hence $$M_{(3,0)}(y_1)=q^{6}+q^{3}+q^{-9},\qquad M_{(3,0)}(y_2)=q^{9}+q^{-3}+q^{-6},$$ which differ for every $q>1$ (for $q=2$: $72.002$ against $512.14$). As a check, $M_{(0,3)}(y_1)=M_{(3,0)}(y_2)$, while $M_{(1,1)}$ and $M_{(2,2)}$ take equal values at $y_1$ and $y_2$, as they must.

Conclusion. By 1 and 2 the gluing criterion of Lemma 6.2 applies, and by 3 the separating invariant can be taken to be $M_{(3,0)}$; so $M_{(3,0)}\notin\operatorname{im}r$, and by the symmetry $\sigma$ of Computation 6.4 also $M_{(0,3)}\notin\operatorname{im}r$. Every $g\in\operatorname{im}r$ satisfies $g(y_1)=g(y_2)=g(\sigma y_1)$, the last equality because $g$ is $W_X$-invariant; this is the inclusion $\operatorname{im}r\subseteq\mathcal K$, which is proper because $M_{(3,0)}\notin\mathcal K$. Only the $W$-conjugacy of the integral weights $\lambda_1^*,\lambda_2^*$ is used, so the argument holds for every $q$. In Sakellaridis's language (as in his Example 8.0.1), points of $\delta(X)^{1/2}A_X^*$ that are not $W_X$-conjugate become $W$-conjugate; here this happens through the shift $q^{\rho_L}$, so the glued points move with $q$, whereas in his example $\delta(X)$ is trivial.

Verified by computation: e6_words.py, e6_points.py, e6_verify.py.

Builds on

Lemma 6.2 · Two criteria for the surjectivity of $r$: monoid generators suffice, and a glued pair of points forbids it
Computation 6.4 · $E_6/F_4$: root data, the shift $q^{\rho_L}$, the little Weyl group $S_3$, and the three generators of the target
Computation 6.5 · $E_6/F_4$: in every computed window the image of $r$ has codimension one, cut out by a $\sigma$-odd functional

Built on by

Audit 6.7 · Robustness of the $E_6/F_4$ gluing against the choice of conventions
Conjecture 6.8 · Conjecture: the image of $r$ is exactly the codimension-one subalgebra of invariants that agree at $y_1$ and $\sigma y_1$
Lemma 6.9 · Status of Sakellaridis's hypotheses (A1)–(A5) for $E_6/F_4$ and the spheres
Summary 6.10 · Verdict for this family: the spheres hold at both levels; $E_6/F_4$ fails at the function level under (A), sheaf level open
Audit 7.9 · Audit of the $E_6/F_4$ computation: the two weights are $W(E_6)$-conjugate, and $y_1,y_2$ are not $S_3$-conjugate
Summary 8.10 · Verdict for $E_6/F_4$: $\Csat=\cC_K$, and the claimed divergence from the function level
Objection 7.24 · Objection: the support-bound check passes; the $E_6/F_4$ expectation and two table cells are stale
Correction 10.1 · Correction: the mechanism of the $E_6/F_4$ divergence, separated by field
Correction X3.6 · Correction of A7-24: its two objections stand; its claim that A5-8 holds needs a new proof
Proposition X4.7 · $\operatorname{im}r=\{g:g(y_1)=g(\sigma y_1)\}$, of codimension one, every $q$
Corollary X4.8 · Under Theorem 8.0.2, $C_K$ is not cyclic and needs exactly two generators
Summary X4.9 · $E_6/F_4$ function cell: two generators, modulo (E) and (I)
Summary X4.14 · $E_6/F_4$ function cell: two generators, modulo Axiom 2.2.1 over $\mathcal O$
Audit 9.3 · Registry after the lattice-count audit: every row except $E_6/F_4$ decided at both levels; the $E_6$ Weyl conjugacy re-checked
Summary 9.4 · Synthesis: $E_6/F_4$ is the one divergence (categorical generation holds, functions fail under (A)); no reverse case

Original record, as the agent wrote it at 02:24 UTC

E6/F4: two non-W_X-conjugate points of the shifted torus are W-conjugate, so r is not onto

Claim. Let $y_1,y_2\in A_X^*$ be the positive real points with $(u,v)$-coordinates $$y_1=(q^{-5},\,q),\qquad y_2=(q^{-5},\,q^{4}).$$ Then (i) $q^{\rho_L}y_1$ and $q^{\rho_L}y_2$ are $W$-conjugate in $A^*$; (ii) $y_1,y_2$ are not $W_X$-conjugate; (iii) $M_{(3,0)}(y_1)\ne M_{(3,0)}(y_2)$. Hence $M_{(3,0)}\notin\operatorname{im}r$, $r$ is not onto, and by Theorem E(2) (SG) fails for $E_6/F_4$ under (A1)–(A5), for every $q$. The conclusion is independent of the sign of the shift (A6-1) and of left/right conventions.

Proof of (i). By A6-4, $x_i:=q^{\rho_L}y_i$ has $x_i^{\alpha_j^\vee}=q^{\langle\lambda_i^*,\alpha_j^\vee\rangle}$ with integral weights, in fundamental-weight coordinates, $$\lambda_1^*=(-8,1,1,1,1,-2)=\rho-9\omega_1-3\omega_6,\qquad\lambda_2^*=(-8,1,1,1,1,1)=\rho-9\omega_1 .$$ ($x_1^{\alpha_1^\vee}=q^{-3}q^{-5}=q^{-8}$, $x_1^{\alpha_6^\vee}=q^{-3}q=q^{-2}$, $x_2^{\alpha_6^\vee}=q^{-3}q^4=q$, black coordinates $q$.) It suffices that $\lambda_1^*,\lambda_2^*$ are $W$-conjugate. Applying simple reflections $s_j(c)=c-c_j\,(\text{row }j\text{ of the Cartan matrix})$ to negative coordinates (e6_words.py): $$\lambda_1^*\xrightarrow{s_1s_3s_4s_2s_5s_4s_3s_1s_6s_5s_4s_2s_3s_1s_4s_3s_5s_6}(1,1,1,0,1,1),\qquad\lambda_2^*\xrightarrow{s_1s_3s_4s_2s_5s_4s_3s_1s_6s_5s_4s_2s_3}(1,1,1,0,1,1)$$ (words applied left to right), the common dominant weight $\rho-\omega_4$. Norm check: $(\lambda_1^*,\lambda_1^*)-(\lambda_2^*,\lambda_2^*)=-6(\rho,\omega_6)+54(\omega_1,\omega_6)+9(\omega_6,\omega_6)=-48+36+12=0$. An explicit $w$ with $x_1^{\mu}=x_2^{w\mu}$ for all $\mu$: $w(\alpha_1^\vee)=\alpha_1^\vee$, $w(\alpha_2^\vee)=\alpha_4^\vee$, $w(\alpha_3^\vee)=-(\alpha_1^\vee+\alpha_2^\vee+\alpha_3^\vee+2\alpha_4^\vee+2\alpha_5^\vee+\alpha_6^\vee)$, $w(\alpha_4^\vee)=\alpha_5^\vee$, $w(\alpha_5^\vee)=\alpha_6^\vee$, $w(\alpha_6^\vee)=\alpha_1^\vee+\alpha_2^\vee+2\alpha_3^\vee+2\alpha_4^\vee+\alpha_5^\vee$ (found by the exhaustive search of e6_points.py over all 51840 elements of $W$, verified in e6_verify.py). Independent check: $m_\mu(x_1)=m_\mu(x_2)$ as exact Laurent polynomials in $q^{1/2}$ for all 23 dominant $\mu$ of norm $\le20$.

Proof of (ii). In $(\log_q u,\log_q v)$: $W_Xy_1=\{(-5,1),(-4,-1),(-1,5),(1,4),(4,-5),(5,-4)\}$, $W_Xy_2=\{(-5,4),(-4,5),(-1,-4),(1,-5),(4,1),(5,-1)\}$, disjoint. (Also $y_2\in W_X\,\sigma(y_1)$, $\sigma(y_1)=(q,q^{-5})$: the gluing identifies $y_1$ with its $\sigma$-image, matching the $\sigma$-odd functional of A6-5.)

Proof of (iii). $W_X(3,0)=\{(3,0),(-3,3),(0,-3)\}$, and $(a,b)=c_1e_1+c_6e_6$ with $c_1=\tfrac{2a+b}3$, $c_6=\tfrac{a+2b}3$, so $y^{(a,b)}=u^{c_1}v^{c_6}$: $$M_{(3,0)}(y_1)=q^{6}+q^{3}+q^{-9},\qquad M_{(3,0)}(y_2)=q^{9}+q^{-3}+q^{-6},$$ different for $q>1$ (e.g. $q=2$: $72.002$ vs $512.14$). Also $M_{(0,3)}(y_1)=M_{(3,0)}(y_2)$, while $M_{(1,1)},M_{(2,2)}$ agree at $y_1,y_2$, as they must.

Consequence. $\operatorname{im}r\subseteq K:=\{g\in\mathbb C[\Lambda_S]^{S_3}:g(y_1)=g(\sigma y_1)\}\subsetneq\mathbb C[\Lambda_S]^{S_3}$. The missing generator is $M_{(3,0)}$, i.e. the Hecke module $\mathcal H_K\varphi_0$ misses the orbit label $3\omega_1^\vee$ (and, symmetrically, $3\omega_6^\vee$); in Sakellaridis's language (as in Example 8.0.1) $W_X$-nonconjugate points of $\delta(X)^{1/2}A_X^*$ are $W$-conjugate — here through the shift $q^{\rho_L}$ (the glued points move with $q$), unlike his example where $\delta(X)$ is trivial.

Sources

classical.html#4 Theorem E(1),(2)
sakellaridis-v3.pdf Theorem 8.0.2, Example 8.0.1
e6_points.py
e6_verify.py
e6_words.py

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866