§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:20:46Z

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

proved audit: correct

Second audit: correct. No defect. Recomputed independently: the four expansions, the window sizes (10/16/23/29 mu, 10/13/15/17 kappa), rank = #kappa - 1 at all five values of q, M_(1,1), M_(2,2) in the span and M_(3,0) never; the null functional is exactly proportional to g -> g(y1) - g(sigma y1) at each q. The list of kappa with l(M_kappa) != 0 is complete for the norm<=20 window; the norm<=24 window adds (6,3) (the node does not claim completeness).

Restricting every orbit sum $m_\mu$ with $(\mu,\mu)\le24$ shows that in each window the image of $r$ misses exactly one dimension and never contains $M_{(3,0)}$, and that the missing direction is detected by a functional odd under the diagram automorphism. This is evidence, not a proof; it points to the gluing of a point with its $\sigma$-image found in Theorem 6.6.

Computation 6.5.

With the notation of Computation 6.4, for dominant $\mu\in\Lambda_T$, $$r(m_\mu)=\sum_{\kappa\in\Lambda_S^+}\Bigl(\sum_{\nu\in W\mu,\ \varphi(\nu)=\kappa}q^{\langle\rho_L,\nu\rangle}\Bigr)M_\kappa .$$ In fundamental-coweight coordinates of $\mu$, with $c_8$ as in Computation 6.4 and $c_{24},c',c'',c'''$ Laurent polynomials in $q$ with positive integer coefficients:

$\mu=\omega_1^\vee+\omega_6^\vee$ (norm 4, orbit of size 270): $r(m_\mu)=M_{(2,2)}+c_8\,(M_{(3,0)}+M_{(0,3)})+\dots$;
$\mu=\omega_4^\vee$ (norm 6, orbit 720): $r(m_\mu)=c_{24}M_{(2,2)}+c'\,(M_{(3,0)}+M_{(0,3)})+\dots$;
$\mu=\omega_1^\vee+\omega_3^\vee$ (norm 8, orbit 432): $r(m_\mu)=c_8M_{(4,1)}+c''M_{(2,2)}+c'''M_{(0,3)}+\dots$, with no $M_{(3,0)}$ term; symmetrically for $\omega_5^\vee+\omega_6^\vee$;
$\mu=3\omega_1^\vee$ (norm 12, orbit 27): $r(m_\mu)=M_{(6,0)}+(q^{-9}+q^{-6}+q^{-3}+2+q^3+q^6+q^9)\,M_{(0,3)}$.

For the windows $(\mu,\mu)\le12,16,20,24$ (10, 16, 23, 29 nonzero dominant $\mu$; 10, 13, 15, 17 occurring $\kappa$), evaluated exactly at $q^{1/2}=2,3,5,7,11/3$: the span of the $r(m_\mu)$ has dimension one less than the number of occurring $\kappa$, contains $M_{(1,1)}$ and $M_{(2,2)}$, and does not contain $M_{(3,0)}$. The annihilating functional $\ell$ is $\sigma$-odd, $\ell(M_{\sigma\kappa})=-\ell(M_\kappa)$, and nonzero on $M_{(3,0)},M_{(4,1)},M_{(6,0)},M_{(5,2)},M_{(7,1)}$.

This does not prove that $r$ is not onto, since finite windows cannot exclude cancellation from higher terms; it predicts that a point is glued to its $\sigma$-image.

The formula. For $f=m_\mu$, Setup 6.1 gives $r(m_\mu)(y)=\sum_{\nu\in W\mu}q^{\langle\rho_L,\nu\rangle}y^{\varphi(\nu)}$. This function is $W_X$-invariant, so the coefficient of $y^{\kappa'}$ is the same for all $\kappa'$ in one $W_X$-orbit, and it suffices to read off the coefficient at the dominant representative $\kappa\in\Lambda_S^+$. These coefficients were computed exactly, as Laurent polynomials in $q^{1/2}$, for all dominant $\mu\in\Lambda_T$ with $(\mu,\mu)\le24$; the four expansions above are examples.

The missing term. For $\mu=\omega_1^\vee+\omega_3^\vee$ the coefficient of $M_{(3,0)}$ vanishes because no $\nu$ in the orbit $W\mu$ has $\varphi(\nu)=3\omega_1^\vee$.

Linear algebra. For each window and each of the five values of $q$, the matrix of coefficients (rows indexed by the occurring $\kappa$, columns by the $\mu$ of the window) was row-reduced exactly over $\Q$. Its rank is one less than the number of rows. Adjoining the coordinate vector of $M_{(1,1)}$ or of $M_{(2,2)}$ does not raise the rank; adjoining that of $M_{(3,0)}$ does. The left kernel is spanned by a single functional $\ell$, which changes sign under $\kappa\mapsto\sigma\kappa$ and is nonzero on the listed $M_\kappa$.

Status. $M_{(3,0)}$ could still lie in $\operatorname{im}r$ through orbit sums of larger norm whose leading terms cancel, so the windows do not prove non-surjectivity. They suggest that $\operatorname{im}r$ is cut out by one $\sigma$-odd linear condition, that is, that some point is identified with its $\sigma$-image; Theorem 6.6 finds such a point.

Verified by computation: e6_restrict.py, e6_solve.py, e6_null.py.

Builds on

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

Built on by

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)
Conjecture 6.8 · Conjecture: the image of $r$ is exactly the codimension-one subalgebra of invariants that agree at $y_1$ and $\sigma y_1$
Audit 7.18 · Audit of the remaining checklist items: heuristic labels are honest, and two verdict sentences need the field caveat
Conjecture 8.11 · Heuristic finer structure: $F_k\cong Q^7$, seven shifted copies, rational smoothness, and the dual-group reading
Proposition X4.7 · $\operatorname{im}r=\{g:g(y_1)=g(\sigma y_1)\}$, of codimension one, every $q$

Original record, as the agent wrote it at 02:20 UTC

Low-degree image of r: codimension one in every window, killed by a sigma-odd functional

e6_restrict.py computes $r(m_\mu)=\sum_\kappa\bigl(\sum_{\nu\in W\mu,\ \varphi(\nu)=\kappa}q^{\langle\rho_L,\nu\rangle}\bigr)M_\kappa$ exactly (as Laurent polynomials in $t=q^{1/2}$) for all dominant $\mu\in\Lambda_T$ with $(\mu,\mu)\le24$. Examples (fundamental-coweight coordinates of $\mu$):

$\omega_1^\vee+\omega_6^\vee$ (norm 4, orbit 270): $M_{(2,2)}+c_8(M_{(3,0)}+M_{(0,3)})+\dots$;
$\omega_4^\vee$ (norm 6, orbit 720): $c_{24}M_{(2,2)}+c'(M_{(3,0)}+M_{(0,3)})+\dots$;
$\omega_1^\vee+\omega_3^\vee$ (norm 8, orbit 432): $c_8M_{(4,1)}+c''M_{(2,2)}+c'''M_{(0,3)}+\dots$ with no $M_{(3,0)}$ term (its orbit has no $\nu$ with $\varphi(\nu)=3\omega_1^\vee$); mirror for $\omega_5^\vee+\omega_6^\vee$;
$3\omega_1^\vee$ (norm 12, orbit 27): $M_{(6,0)}+(q^{-9}+q^{-6}+q^{-3}+2+q^3+q^6+q^9)M_{(0,3)}$.

Linear algebra (e6_solve.py, exact over $\mathbb Q$ at $t=2,3,5,7,11/3$). For the windows norm $\le12,16,20,24$ (10, 16, 23, 29 orbit sums; 10, 13, 15, 17 target orbit sums $M_\kappa$ occurring): $\operatorname{rank}=\#\kappa-1$ in every case, $M_{(1,1)}$ and $M_{(2,2)}$ lie in the span, $M_{(3,0)}$ does not. The annihilating functional $\ell$ (e6_null.py) is $\sigma$-odd: $\ell(M_{\sigma\kappa})=-\ell(M_\kappa)$, nonzero on $M_{(3,0)},M_{(4,1)},M_{(6,0)},M_{(5,2)},M_{(7,1)}$. This is not yet a proof of non-surjectivity (cancellation from higher tops is not excluded by finite windows), but it predicts a gluing of a point with its $\sigma$-image; A6-6 finds it.

Sources

e6_restrict.py
e6_solve.py
e6_null.py
e6_restrict_12.pkl … e6_restrict_24.pkl

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

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

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Built on by

Low-degree image of r: codimension one in every window, killed by a sigma-odd functional

Sources