§7 · Agent 7 · Adversarial audit · 2026-10-17T04:11:59Z

Objection 7.28 · Objection to the finiteness lemma: the \(K_N\) dimension count is wrong as written; \(F_k'\) is a point by a fixed-point argument

proved audit: correct ★ key step

Second audit: correct.

The congruence-quotient bookkeeping offered to prove that the covering fibre is finite is wrong: the quotients of all orbits have the same dimension, and the convolution does not descend. A torus-fixed-point argument proves the same conclusion directly, so the summand theorem stands.

Objection 7.28.

  1. Step 2 of Lemma 8.5 fails as written: for \(N\gg0\), \(K_N\backslash\cO_\lambda\) does not have dimension \(\dim X_N+\delta(\lambda)\), and the convolution does not descend to \(\Gr^{\le\lambda}\mathbin{\widetilde\times}(K_N\backslash L^+X)\to K_N\backslash LX\).
  2. Nevertheless \(F_k'=\{t^{\lambda_k}\}\). Hence \(E_{\mu_k',n}\) is \(k\) for one \(n\) and \(0\) otherwise, and the conclusion of Lemma 8.5 holds.

Why step 2 fails.

  1. Group case. Take \(X=H\), \(G=H\times H\) and \(K_N=K_N^H\times K_N^H\). Then \(\dim K_N\backslash\cO_\lambda=N\dim H=\dim X_N\), not \(\dim X_N+\langle2\rho,\lambda\rangle\). Indeed, the right quotient of \(\cO_\lambda=L^+Ht^\lambda L^+H\) by \(K_N^H\) fibres over \(\Gr^\lambda\) with fibre \(J_NH=H(k[t]/t^N)\), so it has dimension \(\langle2\rho,\lambda\rangle+N\dim H\); and the left \(K_N^H\)-orbits on it have dimension exactly \(\langle2\rho,\lambda\rangle\).
  2. Descent. \(m\) does not descend to \(\Gr^{\le\lambda}\mathbin{\widetilde\times}(K_N\backslash L^+X)\to K_N\backslash LX\): \([g,ky]\mapsto(gkg^{-1})gy\), and \(gK_Ng^{-1}\not\subseteq K_N\).
  3. The difference \(D_k-\delta(\mu_k')=0\) is right, and the bound is valid in a consistent renormalized dimension theory (Bezrukavnikov–Kazhdan–Varshavsky; Chen–Yi, App. A.4), but that is not what is written.

Repair, with no dimension theory.

  1. \(T':=(T^\theta)^\circ\), the torus of the black coroots, fixes every \(x_\mu\): for \(s\in T'\), \(\tau(sx_\mu)=st^\mu\theta(s)^{-1}=st^\mu s^{-1}=t^\mu\). So each fibre \(F_{x_\mu}\) is \(T'\)-stable.
  2. No root of \(E_6\) vanishes on \(T'\): a root orthogonal to the black \(D_4\) would be \(a\omega_1+b\omega_6\) with \(a,b\in\Z\) and \(2(a^2+ab+b^2)=3\), which is impossible (a computer check finds none of the \(72\) roots). Hence for a generic cocharacter \(\eta\) of \(T'\), \(Z_G(\eta)=T\), and the fixed points of \(\eta(\mathbb G_m)\) on \(\Gr_G\) are the points \(t^\nu\) of \(\Gr_T\).
  3. By Lemma 8.4, \(F_k'\subseteq\Gr^{\lambda_k}\), so its \(\eta\)-fixed points are the \(t^\nu\) with \(\nu\in W\lambda_k\) and \(t^{-\nu}x_{\mu_k'}\in L^+X\), i.e. \(\varphi(\nu)=\mu_k'\). Then \(\nu-\lambda_k\in\ker\varphi\), the span of the black coroots, which is orthogonal to \(\lambda_k\); together with \(|\nu|=|\lambda_k|\) this forces \(\nu=\lambda_k\). (A computer check finds exactly one such point for \(k=1,3,5\).)
  4. A positive-dimensional \(\eta\)-stable closed subvariety of a projective variety embedded \(T\)-equivariantly (with linear action) has at least two \(\eta\)-fixed points: the limits of a non-fixed point at \(0\) and at \(\infty\) are distinct fixed points. Hence \(F_k'=\{t^{\lambda_k}\}\), and \(E_{\mu_k',n}\) is \(k\) for one \(n\) and \(0\) otherwise.

Outcome. The conclusion of Lemma 8.5 holds; its step 2 needs this replacement. Severity: needs repair (supplied); not fatal.

Verified by computation: e6_fibre_A7.py

Builds on

  • Lemma 8.5 · The fibre over the covering orbit is finite, so \(\IC_{\mu_k'}\) occurs only unshifted

Built on by

Original record, as the agent wrote it at 04:11 UTC

A8-5: the K_N dimension count is wrong as written; F' is a point by a fixed-point argument

Step audited: A8-5 step 2, \(\dim(\text{source over }\mathrm{Gr}^{\lambda_k})=D_k+\dim X_N\) and \(\dim(\mathcal O_{\mu_k'}/K_N)=\dim X_N+\delta(\mu_k')\).

Fails as written. Group case \(X=H\), \(K_N=K_N^H\times K_N^H\): \(\dim K_N\backslash\mathcal O_\lambda=N\dim H=\dim X_N\), not \(\dim X_N+\langle2\rho,\lambda\rangle\) (the left \(K_N\)-orbits on \(\mathcal O_\lambda\) are larger by exactly \(\langle2\rho,\lambda\rangle\)). Also \(m\) does not descend to \(\mathrm{Gr}^{\le\lambda}\widetilde\times(K_N\backslash L^+X)\to K_N\backslash LX\): \([g,ky]\mapsto(gkg^{-1})gy\) and \(gK_Ng^{-1}\not\subseteq K_N\). Derivation: scratchpad/solve/A7/A8_audit_notes.txt §1. The difference \(D_k-\delta(\mu_k')=0\) is right, and the bound is valid in a consistent renormalized dimension theory (BKV, Chen–Yi A.4), but that is not what is written.

Repair (independent; no dimension theory). \(T':=(T^\theta)^\circ\), the black torus, fixes every \(x_\mu\), so each fibre is \(T'\)-stable. No \(E_6\) root vanishes on \(T'\) (script: 0 of 72; a root \(\perp D_4\) would be \(a\omega_1+b\omega_6\), \(a,b\in\mathbb Z\), with \(2(a^2+ab+b^2)=3\)). So for generic \(\eta\in X_*(T')\), \(\mathrm{Gr}^\eta=\mathrm{Gr}_T=\{t^\nu\}\). \(F_k'^{\,\eta}=\{t^\nu:\nu\in W\lambda_k,\ \varphi(\nu)=\mu_k'\}=\{t^{\lambda_k}\}\): \(\nu-\lambda_k\in\ker\varphi\) = black span, which is orthogonal to \(\lambda_k\), and \(|\nu|=|\lambda_k|\) (script: 1 point, \(k=1,3,5\)). A positive-dimensional \(\eta\)-stable closed subvariety of a \(T\)-equivariantly embedded projective variety has \(\ge2\) fixed points, so \(F_k'=\{t^{\lambda_k}\}\) and \(E_{\mu_k',n}\) is \(k\) for one \(n\), else \(0\).

Outcome: the conclusion of A8-5 holds; step 2 needs this replacement. Severity: needs repair (supplied); not fatal.

Sources