§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:30:06Z

Conjecture 6.8 · Conjecture: the image of \(r\) is exactly the codimension-one subalgebra of invariants that agree at \(y_1\) and \(\sigma y_1\)

heuristic audit: minor defect

Minor defect (second audit). Correctly labelled heuristic and never promoted. All reported counts reproduced by an independent rerun: rank 0: 1152; rank 2: 36864, of which 876 consistent (one positive real solution each), giving 144 non-W_X gluings, 72 in each direction, all between W_X y1 and W_X y2 = W_X sigma(y1); rank 1: 13824, of which 13698 inconsistent and 126 curves. Imprecision: of the 1152 rank-0 elements only the 6 rho_L-fixing ones (= W_X) preserve q^{rho_L}A_X^; the other 1146 map it to disjoint translates q^{w rho_L}A_X^ and identify nothing, so 'the W_X.W_L identifications' is loose. Strengthening found in this audit: along every component of the 126 curves the image is identically W_X-conjugate (exact monomial check, e6_curves_exact.py), so the point-level statement 'the only gluing is {y1, sigma y1}' is exact; only the unramifiedness gap remains.

Window computations and an exhaustive search over \(W(E_6)\) indicate that the pair of Theorem 6.6 is the only gluing of the shifted torus. If the induced map is moreover unramified, the cokernel of \(r\), which is \(C_K/\cH_K\varphi_0\) under Theorem E, is one-dimensional: the Hecke character of \(q^{\rho-\omega_4}\). Heuristic, and not needed for the verdict.

Conjecture 6.8.

Heuristic. With \(y_1\) as in Theorem 6.6, let \(\mathcal K:=\{g\in\C[\Lambda_S]^{S_3}:g(y_1)=g(\sigma y_1)\}\).

Conjecture. \(\operatorname{im}r=\mathcal K\). Equivalently, under Theorem E, \(C_K/\cH_K\varphi_0\) is one-dimensional and \(\cH_K\) acts on it by the character of the point \(q^{\rho-\omega_4}\in A^*/W\); the missing function has leading orbit \(3\omega_1^\vee\).

Evidence. (a) In every window of Computation 6.5 the image of \(r\) has codimension exactly one in the span of the occurring \(M_\kappa\), and it is contained in \(\mathcal K\), which also has codimension one there; so \(\operatorname{im}r\) and \(\mathcal K\) have the same intersection with that span. (b) An exhaustive search over \(W\) finds that the only identification of points of \(q^{\rho_L}A_X^*\) not induced by \(W_X\) is \(W_Xy_1\leftrightarrow W_X\sigma y_1\), the curve cases being tested at generic points only.

Gap. It is not shown that \(\delta(X)^{1/2}A_X^*/W_X\to A^*/W\) is unramified (injective on Zariski tangent spaces) at every point, and the curve cases were tested only at generic points. Not needed for the verdict.

Windows. In each window computed in Computation 6.5, \((\mu,\mu)\le12,16,20,24\), the image has codimension exactly one in the span of the occurring \(M_\kappa\). Since \(\operatorname{im}r\subseteq\mathcal K\) by Theorem 6.6, and \(\mathcal K\) has codimension one in that span (it does not contain \(M_{(3,0)}\)), the image and \(\mathcal K\) have the same intersection with the span.

Exhaustive point search. For each of the 51840 elements \(w\in W\), the condition \(w\cdot(q^{\rho_L}y)\in q^{\rho_L}A_X^*\) is a system of four monomial equations in \((u,v)\), one for each black simple coroot \(\alpha_j^\vee\): \((w\cdot x)^{\alpha_j^\vee}=q\). The search gives:

So the only point-gluing is \(\{y_1,\sigma y_1\}\); the curve test is generic-point only, hence of heuristic strength.

Reading of the conjecture. If \(\operatorname{im}r=\mathcal K\), the quotient \(\C[\Lambda_S]^{S_3}/\operatorname{im}r\) is one-dimensional, spanned by the class of \(M_{(3,0)}\). For \(f\in\C[A^*]^W\) the function \(r(f)\) takes the value \(f(q^{\rho_L}y_1)\) at both \(y_1\) and \(\sigma y_1\), so \(\cH_K\) acts on the quotient by \(f\mapsto f(q^{\rho_L}y_1)\); by Theorem 6.6, \(q^{\rho_L}y_1=q^{\lambda_1^*}\) is \(W\)-conjugate to \(q^{\rho-\omega_4}\). Under Theorem E this quotient is \(C_K/\cH_K\varphi_0\).

Gap. By the remark in Lemma 6.2, the conjecture amounts to the statement that the variety obtained from \(\delta(X)^{1/2}A_X^*/W_X\) by identifying the two glued points maps to \(A^*/W\) by a closed immersion. Besides injectivity on points (the search above), this needs the map to be unramified, i.e. injective on Zariski tangent spaces, at every point; that is not computed. The curve cases were tested only at generic points. None of this is needed for the verdict.

Sheaf-side reading (heuristic). The missing label is the same \(3\omega_1^\vee\) at which H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. fails (\(E_6/F_4\) family page). If \(\Csat=\cC_K\) were nevertheless true for \(E_6/F_4\), the \(E_6/F_4\) row would carry a divergence: sheaves equal, functions fail. The function-level failure is a Weyl-conjugacy coincidence of two \(q\)-points, with no obvious sheaf-theoretic counterpart; it neither proves nor refutes anything at sheaf level.

Verified by computation: e6_solve.py, e6_null.py, e6_points.py, e6_curves.py.

Builds on

Built on by

Original record, as the agent wrote it at 02:30 UTC

Image of r is probably exactly the codimension-one subalgebra K; cokernel one Hecke character

Evidence. (a) In every window computed (norm \(\le12,16,20,24\)) the image has codimension exactly \(1\) in the span of the occurring \(M_\kappa\), and \(\operatorname{im}r\subseteq K\) with \(K\) of codimension \(1\) in that span, so \(\operatorname{im}r\cap\operatorname{span}=K\cap\operatorname{span}\) there. (b) Exhaustive point search (e6_points.py, all 51840 \(w\in W\)): the condition \(w\cdot(q^{\rho_L}y)\in q^{\rho_L}A_X^*\) is a system of monomial equations in \((u,v)\); 1152 elements normalize the plane (the \(W_X\cdot W_L\) identifications), 36864 give rank-2 systems whose solutions (including roots of unity) glue exactly the two \(W_X\)-orbits \(W_Xy_1\leftrightarrow W_X\sigma y_1\) (144 solutions, 72 elements \(w\) each way) and nothing else; 13824 give rank-1 systems, of which 13698 are inconsistent and the remaining 126 define curves on which generic points (two random rational points per component, e6_curves.py) are mapped to \(W_X\)-conjugates. So the only point-gluing is \(\{y_1,\sigma y_1\}\) (the curve test is generic-point only: heuristic strength).

Conjecture (heuristic). \(\operatorname{im}r=K=\{g:g(y_1)=g(\sigma y_1)\}\); equivalently \(C_K/\mathcal H_K\varphi_0\) is one-dimensional, the \(\mathcal H_K\)-character of the point \(q^{\rho-\omega_4}\in A^*/W\); the missing function has leading orbit \(3\omega_1^\vee\). Gap: the finite-morphism criterion also needs \(\delta(X)^{1/2}A_X^*/W_X\to A^*/W\) unramified (injective on Zariski tangent spaces) at every point, not computed; and the curve cases were tested only at generic points. Not needed for the verdict.

Sheaf-side reading (heuristic). The missing label is the same \(3\omega_1^\vee\) where H5 fails (e6-f4.html); if \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) were nevertheless true for \(E_6/F_4\), the \(E_6/F_4\) row would carry a divergence (sheaves equal, functions fail). The function-level failure is a Weyl-conjugacy coincidence of two \(q\)-points, with no obvious sheaf-theoretic counterpart; it neither proves nor refutes anything at sheaf level.

Sources