§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:28:15Z

Audit 6.7 · Robustness of the \(E_6/F_4\) gluing against the choice of conventions

proved audit: correct

Second audit: correct. No defect. The symmetry in item 1 holds in general: the longest element w_L of W_L negates <rho_L, . > and fixes phi, so each coefficient of r(m_mu) is invariant under q -> 1/q.

Each convention that could be read differently (the sign of \(\delta(X)^{1/2}\), the Levi and Borel, the \(W_X\)-action on the translate, the embedding \(A_X^*\hookrightarrow A^*\), left versus right modules, the residue field) leaves the conclusion of Theorem 6.6 intact, and an evaluation of 23 orbit sums confirms it without the conjugating element. The hypotheses (A1)–(A5) themselves are not checked here.

Audit 6.7.

The conclusion of Theorem 6.6, that \(r\) is not onto for \(E_6/F_4\), is unchanged under each of the following choices:

the sign of \(\delta(X)^{1/2}\) (\(q^{\rho_L}\) or \(q^{-\rho_L}\));
the choice of Levi and Borel;
transported versus restricted \(W_X\)-action on the translate;
the identification of \(A_X\) with \(S\) used to embed \(A_X^*\) in \(A^*\);
left versus right modules;
the residue field \(\mathbb F_q\).

Moreover (7) it is confirmed by evaluating 23 orbit sums at the two points, without using the conjugating element. The hypotheses (A1)–(A5) are not checked here (see Lemma 6.9).

Sign of \(\delta(X)^{1/2}\). By the sign lemma of Setup 6.1, \(r_+\) is onto if and only if \(r_-\) is; concretely, for the shift \(q^{-\rho_L}\) the glued pair is \(y_1^{-1},(\sigma y_1)^{-1}\). The coefficients computed in Computation 6.5 are symmetric under \(q^{1/2}\mapsto q^{-1/2}\).
Levi and Borel. \(L(X)=Z_G(S)\), and \(B\subseteq P\) with \(P\) minimal \(\theta\)-split; both are established on the \(E_6/F_4\) family page (sections 3 and 4). \(\delta(X)\) is the modular character of \(L\cap B\) (Sakellaridis §1.7), equal to \(\lvert e^{\pm2\rho_L}\rvert\).
The \(W_X\)-action on the translate. Transported from \(A_X^*\), or restricted from \(W\) through \(\rho_L\)-fixing representatives: the two actions are identical (Setup 6.1; for \(E_6\) the representatives \(w_1,w_6\) fix \(\rho_L\), Computation 6.4).
The embedding \(A_X^*\hookrightarrow A^*\). It is dual to \(\varphi:\Lambda_T\twoheadrightarrow\Lambda_S\), and \(\varphi(\Lambda_T)=\Lambda_S\) on the family page. The identification \(A_X=L/L^\theta=S/S[2]\simeq S\) by \(s\mapsto s^2\) is exactly \(t\mapsto t\theta(t)^{-1}\) on \(T\), so \(\Lambda_X=\varphi(\Lambda_T)\) with no rescaling ambiguity.
Left versus right modules (\(x\mapsto x^{-1}\)) do not change the ring map \(r\).
Dependence on \(q\). The glued points have coordinates that are integral powers of \(q\), and the conjugacy is a statement about integral weights; it holds for every \(q\), that is, for every residue field.
Independent verification. \(m_\mu(x_1)=m_\mu(x_2)\) for all 23 dominant \(\mu\) with \((\mu,\mu)\le20\), computed without using \(w\).

Not checked here: the hypotheses (A1)–(A5) themselves (Lemma 6.9).

Verified by computation: e6_verify.py.

Builds on

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)

Built on by

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

Original record, as the agent wrote it at 02:28 UTC

Robustness of A6-6 against convention choices

Each convention that could be read differently was checked not to affect the conclusion:

Sign of \(\delta(X)^{1/2}\) (\(q^{\rho_L}\) vs \(q^{-\rho_L}\)): Lemma of A6-1; concretely the pair \(y_1^{-1},(\sigma y_1)^{-1}\) is glued for the other sign. The data of A6-5 are symmetric under \(t\mapsto t^{-1}\).
Which Levi / which Borel: \(L(X)=Z_G(S)\) and \(B\subseteq P\) minimal \(\theta\)-split, both established on families/e6-f4.html §3–4; \(\delta(X)\) is the modular character of \(L\cap B\) (Sakellaridis §1.7), \(=|e^{\pm2\rho_L}|\).
\(W_X\)-action on the translate: transported from \(A_X^*\) or restricted from \(W\) through \(\rho_L\)-fixing representatives — identical (A6-1, A6-4: \(w_1,w_6\) fix \(\rho_L\)).
Embedding \(A_X^*\hookrightarrow A^*\): dual to \(\varphi:\Lambda_T\twoheadrightarrow\Lambda_S\) (\(\varphi(\Lambda_T)=\Lambda_S\) on the page); the identification \(A_X=L/L^\theta=S/S[2]\simeq S\) by \(s\mapsto s^2\) is exactly \(t\mapsto t\theta(t)^{-1}\), so \(\Lambda_X=\varphi(\Lambda_T)\) with no rescaling ambiguity.
Left vs right modules (\(x\mapsto x^{-1}\)): does not change the ring map \(r\).
Dependence on \(q\): the glued points have coordinates \(q^{\text{integer}}\) and the conjugacy is a statement about integral weights, valid for every \(q\) (every residue field).
Independent verification: \(m_\mu(x_1)=m_\mu(x_2)\) for all 23 dominant \(\mu\) with \((\mu,\mu)\le20\) computed in e6_verify.py without using \(w\).

What is not checked here: the hypotheses (A1)–(A5) themselves (A6-9).

Sources

classical.html#4
families/e6-f4.html
e6_verify.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