§7 · Agent 7 · Adversarial audit · 2026-10-17T03:21:52Z

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

proved audit: correct

Second audit: correct. No defect; every item re-derived independently (Bourbaki R^8 model; W(E6) enumerated; W_X = the 6 rho_L-fixing elements; lambda2* in the W-orbit of lambda1*).

An independent recomputation, from a re-entered Cartan matrix, of the shift, the two weights, their common dominant representative \(\rho-\omega_4\), the \(S_3\)-orbits and the values of \(M_{(3,0)}\) confirms Computation 6.4 and Theorem 6.6: (SG) fails for \(E_6/F_4\) under (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit. over \(p\)-adic fields, for every residue field. The (A5)(A5) X = H\G is homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points. verification of Lemma 6.9 is right; (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit., (A3)(A3) Sakellaridis's three axioms over O, relabelled: (A3a) rank-one models over O (Axiom 2.2.1); (A3b) the Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2). For E6/F4 all three hold in both characteristics (VI run). and most of (A4)(A4) Spherical roots and colours: no spherical root of type N, every Levi L_γ of a simple spherical root classical, and his Statements 6.3.1 and 7.1.5 on the spherical roots and the virtual coloured weights. stay open.

Audit 7.9.

An independent computation confirms:

  1. \(2\rho_L=6\alpha_2+6\alpha_3+10\alpha_4+6\alpha_5\) (the 12 positive roots of the black \(D_4\)), and \(\langle\rho_L,\alpha_j^\vee\rangle_{j=1,\dots,6}=(-3,1,1,1,1,-3)\).
  2. \(\varphi\alpha_1^\vee=(2,-1)\), \(\varphi\alpha_6^\vee=(-1,2)\) in \((a,b)\)-coordinates, and \(\theta^2=1\). Hence \(x=q^{\rho_L}y\) has \(x^{\alpha_1^\vee}=q^{-3}u\), \(x^{\alpha_6^\vee}=q^{-3}v\) and black coordinates \(q\), and \(y_1=(q^{-5},q)\mapsto\lambda_1^*=\rho-9\omega_1-3\omega_6\), \(y_2=(q^{-5},q^4)\mapsto\lambda_2^*=\rho-9\omega_1\).
  3. Reflecting negative coordinates, both weights reach \((1,1,1,0,1,1)=\rho-\omega_4\), with words of lengths 18 and 13 identical to those of Theorem 6.6: they are \(W\)-conjugate.
  4. \(W_X=S_3\) acts by \(s_1(a,b)=(-a,a+b)\) and \(s_6(a,b)=(a+b,-b)\), the reflections of the restricted \(A_2\) with simple coroots \(e_1,e_6\); dually \((u,v)\mapsto(u^{-1},uv)\) and \((u,v)\mapsto(uv,v^{-1})\). The orbits \(W_Xy_1\) and \(W_Xy_2\) are disjoint, and \(\sigma y_1=(1,-5)\) (logarithmic coordinates) lies in \(W_Xy_2\).
  5. The exponents of \(q\) in \(M_{(3,0)}\) are \(\{-9,3,6\}\) at \(y_1\) and \(\{-6,-3,9\}\) at \(y_2\); \(M_{(1,1)}\) and \(M_{(2,2)}\) agree at the two points.
  6. \(w_1=w_0(D_5^{\{1,\dots,5\}})w_0(D_4)\) and \(w_6\) fix \(2\rho_L\), induce \(s_1,s_6\) on \((a,b)\) and commute with \(\varphi\).

Outcome. Theorem 6.6 holds given Theorem E: (SG) fails for \(E_6/F_4\) under (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit., for \(p\)-adic \(F\) and every residue field. Only the \(W\)-conjugacy of the integral weights \(\lambda_1^*,\lambda_2^*\) is used, so the independence of \(q\) is genuine. The (A5)(A5) X = H\G is homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points. verification of Lemma 6.9 is right: \(B\cap H=B_L\cap L^\theta\) is a Borel subgroup of \(\Spin_8\), and \(\dim BH=42+52-16=78\), so \(BH\) is open. (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit., (A3)(A3) Sakellaridis's three axioms over O, relabelled: (A3a) rank-one models over O (Axiom 2.2.1); (A3b) the Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2). For E6/F4 all three hold in both characteristics (VI run). at the given place, and (A4)(A4) Spherical roots and colours: no spherical root of type N, every Levi L_γ of a simple spherical root classical, and his Statements 6.3.1 and 7.1.5 on the spherical roots and the virtual coloured weights. beyond "\(L_\gamma\) classical" remain unverified, as Lemma 6.9 says. Severity: none.

  1. Method. The Cartan matrix of \(E_6\) was re-entered with Bourbaki labels and every quantity recomputed with exact integers, independently of the computations behind Computation 6.4 and Theorem 6.6.
  2. Shift and weights. Summing the 12 positive roots of the black \(D_4\) gives \(2\rho_L\); pairing with the simple coroots gives \((-3,1,1,1,1,-3)\). With \(\varphi\alpha_1^\vee=(2,-1)\) and \(\varphi\alpha_6^\vee=(-1,2)\), the shifted points \(q^{\rho_L}y_1\) and \(q^{\rho_L}y_2\) are \(q^{\lambda_1^*}\) and \(q^{\lambda_2^*}\).
  3. Conjugacy. Reflecting negative coordinates until the weight is dominant reproduces the two words of Theorem 6.6 and their common endpoint \(\rho-\omega_4\).
  4. Non-conjugacy and separation. \(s_1,s_6\) are the reflections of the restricted \(A_2\) with simple coroots \(e_1,e_6\). The six-element orbits of \(y_1\) and \(y_2\) in logarithmic coordinates are those listed in Theorem 6.6 and are disjoint. The exponent sets of \(M_{(3,0)}\) at the two points differ.
  5. Representatives. \(w_1\) and \(w_6\) fix \(2\rho_L\), induce \(s_1,s_6\) on \((a,b)\) and commute with \(\varphi\) on 300 random vectors, so the transported and restricted actions agree, as claimed in Computation 6.4.
  6. Openness of \(BH\). \(\dim B=42\), \(\dim F_4=52\), and \(\dim(B\cap H)=16\) for a Borel subgroup of \(\Spin_8\), so \(\dim BH=78=\dim E_6\).

Verified by computation: e6_A7.py.

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  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
Original record, as the agent wrote it at 03:21 UTC

A6-4/A6-6 verified independently: the two weights are W(E6)-conjugate, y1,y2 are not S3-conjugate

Independent script (scratchpad/solve/A7/e6_A7.py, Bourbaki labels, Cartan matrix re-entered).

  1. \(2\rho_L=6\alpha_2+6\alpha_3+10\alpha_4+6\alpha_5\) (12 positive roots of the black \(D_4\)); \(\langle\rho_L,\alpha_j^\vee\rangle=(-3,1,1,1,1,-3)\). ✓
  2. \(\varphi\alpha_1^\vee=(2,-1)\), \(\varphi\alpha_6^\vee=(-1,2)\) in \((a,b)\), \(\theta^2=1\). ✓ So \(x=q^{\rho_L}y\) has \(x^{\alpha_1^\vee}=q^{-3}u\), \(x^{\alpha_6^\vee}=q^{-3}v\), black \(=q\); \(y_1=(q^{-5},q)\mapsto\lambda_1^*=(-8,1,1,1,1,-2)=\rho-9\omega_1-3\omega_6\), \(y_2=(q^{-5},q^4)\mapsto\lambda_2^*=(-8,1,1,1,1,1)=\rho-9\omega_1\). ✓
  3. Reflecting negative coordinates to dominant: both reach \((1,1,1,0,1,1)=\rho-\omega_4\), with words of length 18 and 13 identical to A6-6's. \(W\)-conjugate. ✓
  4. \(W_X=S_3\) via \(s_1(a,b)=(-a,a+b)\), \(s_6(a,b)=(a+b,-b)\) (checked: these are the reflections of the restricted \(A_2\) with simple coroots \(e_1,e_6\)); dual action \((u,v)\mapsto(u^{-1},uv),(uv,v^{-1})\). Orbits in \((\log_qu,\log_qv)\): \(W_Xy_1=\{(-5,1),(-4,-1),(-1,5),(1,4),(4,-5),(5,-4)\}\), \(W_Xy_2=\{(-5,4),(-4,5),(-1,-4),(1,-5),(4,1),(5,-1)\}\): disjoint; \(\sigma y_1=(1,-5)\in W_Xy_2\). ✓
  5. \(M_{(3,0)}\): exponents \(\{-9,3,6\}\) at \(y_1\), \(\{-6,-3,9\}\) at \(y_2\): different; \(M_{(1,1)},M_{(2,2)}\) agree. ✓
  6. \(w_1=w_0(D_5^{1..5})w_0(D_4)\), \(w_6\) fix \(2\rho_L\), induce \(s_1,s_6\) on \((a,b)\) and commute with \(\varphi\) (300 random vectors). ✓ So A6-4's 'transported = restricted' holds.

Outcome: A6-6 holds given Theorem E, i.e. (SG) fails for \(E_6/F_4\) under (A1)–(A5) for \(p\)-adic \(F\), every residue field. Only \(\lambda_1^*\sim_W\lambda_2^*\) (integral weights) is used, so \(q\)-independence is genuine. The (A5) verification (A6-9) is right: \(B\cap H=B_L\cap L^\theta\) is a Borel of \(\mathrm{Spin}_8\), and \(\dim BH=42+52-16=78\) so \(BH\) is open. (A2), (A3) at the given place and (A4) beyond '\(L_\gamma\) classical' remain unverified, as A6-9 says. Severity: none.

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