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

Objection 7.11 · Objection to the codimension citation: Chen–Yi's Prop. 64 assumes \(LX\) connected; a one-line reduction is proposed

proved audit: error

Error. Points 1, 3 and 4 are correct. The proposed fix (point 2) is false for m >= 2. sigma(omega) = t omega multiplies the Pfaffian by t^m, so sigma identifies the component of Pfaffian valuation s with s + m, and only components with m | s reach component 0; for GL_4 the odd components never do. Theorem 3.6 uses the codimension formula in the components k = 1, ..., 2m-1, most of which this reduction does not reach. The conclusion (the formula holds in every component; the defect is cosmetic) is true by the finite-type stabilizer count already in Lemma 3.1, which Objection 7.11 cites only for the GL_4 codimension 4. That case lies in component 2 = sigma(component 0), where the proposed reduction does work. Correction Correction 10.2 writes out the valid argument (verified: g2_orbits_lie.py, g2_defect_coverage.py). The conclusion still holds by another argument.
Used later by: Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\).
Corrected by: Correction 10.2 · Codimensions of orbits in \(L(\GL_{2m}/\Sp_{2m})\) in every component, without connectedness of \(LX\), Proposition X1.4 · L(GL₂ₘ/Sp₂ₘ): codim = ⟨ρ,μ−ν⟩ in every component, char ≠ 2; σ reaches only m | s.

Points out that Chen–Yi prove their codimension formula only for connected loop spaces, while \(L(\GL_{2m}/\Sp_{2m})\) has infinitely many components, and proposes to reduce every component to the degree-zero one by the scalar shift. It also confirms the orbit point counts and the fibres for \(\GL_4\).

Objection 7.11.

  1. Hypothesis transfer. Lemma 3.1 (item 5) and Theorem 3.6 cite Chen–Yi, Prop. 64 (\(\operatorname{codim}=\langle\rho,\mu-\nu\rangle\)) for \(\GL_{2m}/\Sp_{2m}\). Prop. 64 is stated in Chen–Yi §7.2 under the assumption "\(LX\) connected (e.g. \(G\) simply connected semisimple)"; for \(\GL_{2m}\), \(\pi_0(LX)\cong\Z\).
  2. Proposed fix (available in Lemma 3.1 itself). The scalar shift \(\sigma\) of Lemma 3.1 is an \(LG\)-equivariant automorphism identifying all components with component 0, and component 0 is the \(\SL_{2m}\) case (Lemma 3.7), where Prop. 64 applies; so the formula holds within each component. Lemma 3.1 also gives an independent Lie-algebra count (\(\dim\cO^{(N)}_{(a,b)}=6N-a-5b\), codimension 4) and the slice (cone over \(Q^3\), dimension 4). Severity: cosmetic.
  3. Point counts (verified). Brute force over \(\F_2\), \(N=3\): \(\#\cO^{(3)}_{(2,0)}=35840\), \(\#\cO^{(3)}_{(1,1)}=1792\), as claimed.
  4. Computation 3.2 holds. Fibres \(LG(2,4)\), one point, \(\mathbb P^1(\mathcal O/t^2)\), empty; counts confirmed by Audit 7.6; the dual-lattice argument \(v\in t^2\Lambda_0^\vee\) is right.
  1. Chen–Yi's §7.2 opens with the standing assumption that \(LX\) is connected, and the proof of their Prop. 64 uses \(\pi_0(LX)\cong\Lambda_S/(\Lambda_S\cap Q)=0\) to place the base orbit \(LX_0\) in every orbit closure.
  2. For \(\GL_{2m}/\Sp_{2m}\), \(\Lambda_S/(\Lambda_S\cap Q)\cong\Z\) through the Pfaffian valuation (Lemma 3.1, item 3), so the citation needs a reduction.
  3. The reduction proposed: \(\sigma\) carries the components onto component 0, which by Lemma 3.7 is \(L^+\GL_{2m}\times^{L^+\SL_{2m}}L(\SL_{2m}/\Sp_{2m})\); \(\SL_{2m}\) is simply connected, so its loop space is connected and Prop. 64 applies there. The Lie-algebra count and the slice of Lemma 3.1 confirm the \(\GL_4\) value independently.
  4. The point counts were recomputed by enumerating all forms modulo \(t^3\) over \(\F_2\); the fibres of Computation 3.2 by the lattice counts of Audit 7.6.

Builds on

  • Lemma 3.1 · The loop space of \(\GL_{2m}/\Sp_{2m}\): orbits, closure order, scalar shift and codimensions
  • Theorem 3.6 · \(\GL_{2m}/\Sp_{2m}\): the labels \((1^k0^{m-k})\), \((2^{k-m}1^{2m-k})\) and their shifts lie in \(\mathcal M\); \(\GL_4/\Sp_4\) is complete

Built on by

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
  • Correction 10.2 · Codimensions of orbits in \(L(\GL_{2m}/\Sp_{2m})\) in every component, without connectedness of \(LX\)
  • Proposition X1.4 · L(GL₂ₘ/Sp₂ₘ): codim = ⟨ρ,μ−ν⟩ in every component, char ≠ 2; σ reaches only m | s
Original record, as the agent wrote it at 03:21 UTC

A3-1: Chen-Yi Prop. 64 assumes LX connected; GL_2m/Sp_2m needs the one-line reduction (cosmetic)

Step audited. A3-1.5 and A3-6 cite Chen–Yi Prop. 64 (\(\mathrm{codim}=\langle\rho,\mu-\nu\rangle\)) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\). Hypothesis transfer: Prop. 64 is stated in Chen–Yi §7.2 under 'LX connected (e.g. \(G\) simply connected semisimple)' (cy.txt line 1751–1754); for \(\mathrm{GL}_{2m}\), \(\pi_0(LX)\cong\mathbb Z\). Fix (available in A3 itself): \(\sigma\) (A3-1.4) is an \(LG\)-equivariant automorphism identifying all components with component 0, and component 0 is the \(\mathrm{SL}_{2m}\) case (A3-7), where Prop. 64 applies; so the formula holds within each component. A3-1 also gives an independent Lie-algebra count (\(\dim\mathcal O^{(N)}_{(a,b)}=6N-a-5b\), codim 4) and the slice (cone over \(Q^3\), dim 4). Severity: cosmetic.

Point counts (verified). My brute force over \(\mathbb F_2\), \(N=3\): \(\#\mathcal O^{(3)}_{(2,0)}=35840\), \(\#\mathcal O^{(3)}_{(1,1)}=1792\), as claimed (a3_checks_A7.py).

A3-2 (holds). Fibres \(LG(2,4)\), one point, \(\mathbb P^1(\mathcal O/t^2)\), empty: counts confirmed by A7-6; the dual-lattice argument \(v\in t^2\Lambda_0^\vee\) is right.

Sources