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

Audit 7.13 · Check of the summand criterion, the maximal-defect lemma, the minuscule fibre lemma and the \(\SL\)/\(\GL\) reduction

proved audit: minor defect

Minor defect (second audit). Each verdict is right (re-verified independently, see the audits of Proposition 3.4 to Conjecture 3.8). Defects that change no claim: (i) the codimension input of Theorem 3.6 is delegated to Objection 7.11, whose scalar-shift reduction fails in the components k not divisible by m that Theorem 3.6 uses; the valid input is Lemma 3.1's dimension count (correction Correction 10.2). Its own defect check for m <= 4 used the formula of Prop. 64, so it confirms the algebra but not the codimension input. (ii) The point-count remark of Conjecture 3.8, which holds only for top-degree terms, was not examined.

Audits the summand criterion, the maximal-defect lemma, the minuscule fibre lemma with its defect formula and shift stability, and the reduction of \(\SL_{2n}/\Sp_{2n}\) to a component of \(\GL_{2n}/\Sp_{2n}\); all hold, with the codimension input deferred to the objection on Chen–Yi's Prop. 64. It also confirms that the relative-Satake node stays heuristic.

Audit 7.13.

  1. Proposition 3.4 holds. Proposition I′: (1) by the definition of \(\mathcal M\) and (DT); (2), \(\Leftarrow\), as in Lemma 4.5 (needs H1–H3Hypotheses of the pages' Proposition I (H1–H5 imply categorical generation): H1 finite orbit filtration; H2 equivariant recollement; H3 constant coefficients on orbits; H4 the Hecke action preserves the category; H5 every orbit carries a Satake object with nonzero top splitting.); (2), \(\Rightarrow\), by a Serre-subcategory argument; (3) needs associativity and Satake semisimplicity, both standing. The obstruction example (\(\operatorname{Cone}(u)\) on \(B\mathbb G_m\)) is correctly used: a nonvanishing restriction does not put the constant sheaf in the thick span; purity is what splits \(a_\mu^*P\). The fibre-proposition step forcing \(\varphi(\lambda)=\mu\) is correct.
  2. Lemma 3.5 holds, and "ties allowed" is right.
  3. Theorem 3.6 holds: fibre lemma re-derived; \(\operatorname{def}(\nu)=n_1(\nu)\) confirmed for all \(\nu\in\{0,1,2\}^m\), \(|\nu|=k\), \(m\le4\); shift stability; \(\mathcal M=\Lambda_S^+\) for \(\GL_4\). Codimension input: see Objection 7.11 (cosmetic).
  4. Lemma 3.7 holds.
  5. Conjecture 3.8 is labelled heuristic and Summary 3.11 does not promote it. The same Peter–Weyl heuristic applied to \(E_6/F_4\) (\(\check G_X=\PGL_3\hookrightarrow E_6^{ad}\)) would predict \(\Csat=\cC_K\) there.
  1. Summand criterion. (1) is immediate from the definition of \(\mathcal M\) and (DT). For (2), \(\Leftarrow\), the boundary step uses standard generation, hence H1–H3Hypotheses of the pages' Proposition I (H1–H5 imply categorical generation): H1 finite orbit filtration; H2 equivariant recollement; H3 constant coefficients on orbits; H4 the Hecke action preserves the category; H5 every orbit carries a Satake object with nonzero top splitting.. For (2), \(\Rightarrow\), the subcategory of objects whose perverse constituents lie in \(\{\IC_\nu:\nu\in\mathcal M\}\) is thick and generated by a Serre subcategory of the heart. (3) uses associativity of the action and semisimplicity of geometric Satake.
  1. Maximal-defect lemma. Step 1: \(\dim\supp\mathcal H^i(K)\le\max\{D-c_\nu:2f_\nu\ge i+D\}\le r-i\), so \(K\in{}^pD^{\le r}\), and \(\mathbb DK\simeq K\) gives \(K\in{}^pD^{\ge-r}\). Step 3: for \(\nu\ne\mu\) with \(x_\mu\in\overline{\cO_\nu}\), \(\IC_\nu[-j]\) lives at \(x_\mu\) in degrees \(\le-\dim\cO_\mu-1+r=-D+2f_\mu-1\), whether or not \(\operatorname{def}(\nu)=r\); \(\IC_\mu[-j]\) reaches \(-D+2f_\mu\) exactly when \(j=r\); and \(H^{2f_\mu}(m^{-1}x_\mu)\ne0\) for a proper scheme of dimension \(f_\mu\).
  1. Fibre lemma and defects. \(W\subseteq\ker\bar\omega_0=V_1\oplus R\), \(\bar\omega_1\)-isotropic of dimension \(n_1+2n_2\), forces \(W\supseteq R\) with \(W/R\) Lagrangian; \(\nu_j\le2\) follows from \(M=M^*\subseteq t^{-1}\Lambda_0\). The defect identity was checked by script for \(m\le4\), using \(\langle\rho,e_{2j-1}+e_{2j}\rangle=2m+2-4j\) and \(D=\dim\cO_{top}+\epsilon\). Shift stability: \(\sigma\) is \(LG\)-equivariant and \(\sigma_*^2=V_{(1^{2m})}\ast(-)\).
  1. SL/GL reduction. \([g,\omega]\mapsto g\omega g^T\) is bijective (\(\det h=1\) from the Pfaffian), and the component-0 identification of Satake objects is right.
  1. Heuristic. Conjecture 3.8 remains a heuristic in Summary 3.11, where \((1,1,-2)\) stays open.

Builds on

Built on by

  • 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

A3-4 to A3-7: summand criterion, maximal-defect lemma, fibre lemma and SL/GL reduction hold

A3-4 (holds). Prop. I′: (1) by definition of \(\mathcal M\) and (DT); (2)\(\Leftarrow\) as A4-5(iii) (needs H1–H3); (2)\(\Rightarrow\) Serre subcategory; (3) needs associativity and Satake semisimplicity, both standing. The notes-obstruction example (\(\mathrm{Cone}(u)\) on \(B\mathbb G_m\)) is correctly used: nonvanishing restriction does not give \(\underline k\in\mathrm{Thick}\); purity is what splits \(a_\mu^*P\). fibre.html (a),(b) \(\Rightarrow\varphi(\lambda)=\mu\): correct.

A3-5 (holds, 'ties allowed' is right). Step 1: \(\dim\mathrm{supp}\,\mathcal H^i(K)\le\max\{D-c_\nu:2f_\nu\ge i+D\}\le r-i\), so \(K\in{}^pD^{\le r}\); \(\mathbb DK\simeq K\) gives \(\ge-r\). Step 3: for \(\nu\ne\mu\) with \(x_\mu\in\overline{\mathcal O_\nu}\), \(\mathrm{IC}_\nu[-j]\) lives at \(x_\mu\) in degrees \(\le-\dim\mathcal O_\mu-1+r=-D+2f_\mu-1\) regardless of whether \(\mathrm{def}(\nu)=r\); \(\mathrm{IC}_\mu[-j]\) reaches \(-D+2f_\mu\) iff \(j=r\); \(H^{2f_\mu}(m^{-1}x_\mu)\ne0\) for a proper scheme of dimension \(f_\mu\). Correct.

A3-6 (holds). Fibre lemma re-derived: \(W\subseteq\ker\bar\omega_0=V_1\oplus R\), \(\bar\omega_1\)-isotropic of dimension \(n_1+2n_2\) forces \(W\supseteq R\), \(W/R\) Lagrangian; \(\nu_j\le2\) from \(M=M^*\subseteq t^{-1}\Lambda_0\). Defect: my script confirms \(\mathrm{def}(\nu)=n_1(\nu)\) for all \(\nu\in\{0,1,2\}^m\), \(|\nu|=k\), \(m\le4\) (a3_checks_A7.py), using \(\langle\rho,e_{2j-1}+e_{2j}\rangle=2m+2-4j\) and \(D=\dim\mathcal O_{top}+\epsilon\). Shift stability: \(\sigma\) is \(LG\)-equivariant, \(\sigma_*^2=V_{(1^{2m})}\ast\); \(\mathcal M+\mathbb Z(1^m)=\mathcal M\). GL\(_4\): \(\mathcal M=\Lambda_S^+\). Codimension input: see A7-11 (cosmetic).

A3-7 (holds). \([g,\omega]\mapsto g\omega g^{\mathsf T}\) is bijective (\(\det h=1\) from \(\mathrm{Pf}\)) and the component-0 identification of Satake objects is right.

A3-8 is labelled heuristic and A3-11 does not promote it (SL\(_6\) label \((1,1,-2)\) stays open). Peter–Weyl remark: the same heuristic applied to \(E_6/F_4\) (\(\check G_X=\mathrm{PGL}_3\hookrightarrow E_6^{ad}\), \(\mathbb C[E_6^{ad}]\twoheadrightarrow\mathbb C[\mathrm{PGL}_3]\)) would predict \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) there — see A7-19.

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