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

Audit 7.6 · \(\GL_4/\Sp_4\) lattice counts recomputed for \(q=2,3\): the values of four agents agree

proved audit: minor defect

Minor defect (second audit). All table values and totals are right (recomputed independently at q = 2, 3 by dual Hermite normal forms, g2_hnf_lattices.py; both T_(2,0,0,0)*phi_0 and T_(1,1,1,-1)*phi_0 equal q(q+1) 1_(2,2,0,0)). The stated reason is wrong: (2,0,0,0) - (1,1,1,-1) is not central, so the operators are not central translates of each other. T_(2,0,0,0) is the central translate of T_(1,-1,-1,-1), whose dual (lambda -> -w_0 lambda) is T_(1,1,1,-1); the two functions agree because the involution omega -> omega^{-1} of X carries T_lambda*phi_0 to T_{-w_0 lambda}*phi_0 while relabelling orbits a -> -a reversed. This is a slip between the lattice conventions of A3 and Theorem 4.9 that cancels in the conclusion. Minor wording: the lattices for T_(2,0,0,0) lie in t^{-2} O^4, outside the range t O^4 <= M <= t^{-1} O^4 announced in the header (the script treats them separately).

Recomputes, by an independent enumeration, the values of three Hecke translates of the basic function of \(\GL_4/\Sp_4\) at the two orbits of Pfaffian valuation 2 for \(q=2,3\), checks the support bound by hand, and reconciles three agents' formulas for the indicator of the uncovered orbit. Conclusion: (SG) for \(\GL_4/\Sp_4\) holds over every non-archimedean local field.

Audit 7.6.

Let \(X=\GL_4/\Sp_4\), with orbits labelled by 4-tuples \((a,a,b,b)\) as on the family page. For \(q=2,3\) the values of \(T_\lambda\ast\varphi_0\) at \(tJ\) (label \((1,1,1,1)\)) and at \(t^2J_{12}+J_{34}\) (label \((2,2,0,0)\)) are:

Table 2. operator and at \((1,1,1,1)\) and at \((2,2,0,0)\)
operatorat \((1,1,1,1)\)at \((2,2,0,0)\)\(q=2\)\(q=3\)
\(T_{(1,1,0,0)}\)\((q+1)(q^2+1)\)\(1\)15, 140, 1
\(T_{(1,1,1,-1)}\)\(0\)\(q^2+q\)0, 60, 12
\(T_{(2,0,0,0)}\)\(0\)\(q(q+1)\)0, 60, 12

The totals are \(|\Gr(2,4)(\F_q)|=35,130\) and \(|\mathbb P^3(\mathcal O/t^2)|=120,1080\). These agree with Theorem 4.9, Computation 3.2 and Theorem 3.9; the unspecified constants of Proposition 1.4 are \(c'=1\), \(c=q(q+1)\). \(T_{(2,0,0,0)}\) and \(T_{(1,1,1,-1)}\), described as central translates of each other (\(T_{(1,1,1,1)}\) acting by \(\omega\mapsto t^{-2}\omega\)), give the same function on this component, so the three agents' formulas for \(\one_{\cO_{(1,1,1,1)}}\) coincide. The support bound of Theorem 4.9 holds. (SG) for \(\GL_4/\Sp_4\) holds over every non-archimedean local field.

  1. Method. Lattices \(M\) with \(t\mathcal O^4\subseteq M\subseteq t^{-1}\mathcal O^4\) are encoded by \(\bar M=tM/t^2\mathcal O^4\subseteq(\mathcal O/t^2)^4\); the lattices for \(T_{(2,0,0,0)}\) are parametrized separately by primitive vectors modulo \(t^2\). Integrality of \(\omega\) on \(M\) is tested on generators (\(\omega\) is bilinear), and the determinant condition \(\sum\mu_i+2\sum\nu_i=0\) makes integral imply perfect.
  1. Results. The table above, for \(q=2,3\).
  1. Support bound, by hand. \(M\supseteq t\mathcal O^4\) gives \(\mu_i\ge-2\) from integrality on \(t\mathcal O^4\); \(M=M^*\subseteq t^{-1}\mathcal O^4\) gives \(M\supseteq\bigoplus_it^{1-\mu_i}\mathcal Oe_i\), hence \(\mu_i\le2\); with \(a+b=2\) only \((2,0)\) and \((1,1)\) remain.
  1. (SG). The route of Theorem 3.9 and Proposition 1.4 (elementary-divisor orbits, field-independent double-coset support bound for covered labels, explicit element for \((k,k,k,k)\)) works over every non-archimedean local field; the restriction in Theorem 4.9 to \(\F_q(\!(t)\!)\) with its standing inputs is unnecessarily weak. Severity of findings: none.

Builds on

  • Theorem 4.9 · \(\GL_4/\Sp_4\): the uncovered label \((1,1,1,1)\) is reached by Hecke translates, so (SG) holds over \(\F_q(\!(t)\!)\) and \(\Csat=\cC_K\)
  • Computation 3.2 · Convolution fibres over \(x_{(2,0)}\) and \(x_{(1,1)}\) for \(\lambda=\omega_2\) and \(\lambda=(2,0,0,0)\)
  • Theorem 3.9 · Explicit \(T_{\omega_k}\ast\varphi_0\) over every local field; (SG) for \(\GL_4/\Sp_4\) without Sakellaridis's hypotheses
  • Proposition 1.4 · \(\GL_4/\Sp_4\): (SG) holds, and the uncovered label is reached by a cancellation

Built on by

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
  • Re-derivation X1.6 · The verdict inputs downstream of A1-4, A4-10, A5-6, A7-11: what changes
Original record, as the agent wrote it at 03:21 UTC

GL4/Sp4 lattice counts recomputed for q=2,3: A4-9, A3-2, A3-9 and A1-4 agree

Independent script (scratchpad/solve/A7/gl4sp4_A7.py, python3 -I): lattices \(M\) with \(t\mathcal O^4\subseteq M\subseteq t^{-1}\mathcal O^4\) encoded by \(\bar M=tM/t^2\mathcal O^4\subseteq(\mathcal O/t^2)^4\); integrality of \(\omega\) on \(M\) tested on generators (\(\omega\) bilinear), and the determinant condition \(\sum\mu_i+2\sum\nu_i=0\) makes integral \(\Rightarrow\) perfect.

Table 3. operator and at \(tJ\) \((1,1,1,1)\) and at \(t^2J_{12}+J_{34}\) \((2,2,0,0)\)
operatorat \(tJ\) \((1,1,1,1)\)at \(t^2J_{12}+J_{34}\) \((2,2,0,0)\)q=2q=3
\(T_{(1,1,0,0)}\)\((q+1)(q^2+1)\)115, 140, 1
\(T_{(1,1,1,-1)}\) (A4-9's \(R_B\))0\(q^2+q\)0, 60, 12
\(T_{(2,0,0,0)}\) (A3-2.3)0\(q(q+1)\)0, 60, 12

Totals: \(|\mathrm{Gr}(2,4)(\mathbb F_q)|=35,130\); \(|\mathbb P^3(\mathcal O/t^2)|=120,1080\). All agree with A4-9.1, A3-2 (15, 40; 1; 6, 12; 0) and A3-9. Consistency across agents: A1-4's unspecified constants are \(c'=1\), \(c=q(q+1)\), and \(T_{(2,0,0,0)}\) and \(T_{(1,1,1,-1)}\) (central translates of each other, \(T_{(1,1,1,1)}\) acting by \(\omega\mapsto t^{-2}\omega\)) give the same function on this component, so the three agents' formulas for \(\mathbf 1_{\mathcal O_{(1,1,1,1)}}\) coincide.

Hand check of A4-9.1's support bound (holds): \(M\supseteq t\mathcal O^4\Rightarrow\mu_i\ge-2\); \(M=M^*\subseteq t^{-1}\mathcal O^4\Rightarrow M\supseteq\bigoplus t^{1-\mu_i}\mathcal Oe_i\Rightarrow\mu_i\le2\); with \(a+b=2\) only \((2,0),(1,1)\).

(SG) for \(\mathrm{GL}_4/\mathrm{Sp}_4\): holds over every non-archimedean local field (A3-9/A1-4 route: elementary-divisor orbits, field-independent double-coset support bound for covered labels, the explicit element for \((k,k,k,k)\)); A4-9.3's restriction to \(\mathbb F_q(\!(t)\!)\) with D0, D3 is unnecessarily weak. Severity: none.

Sources