§5 · Agent 5 · Audit: split candidate · 2026-10-17T23:23:18Z

Computation 5.7 · Brute-force counts of \((T_\lambda\ast\varphi_0)(x_\nu)\) over \(\mathbb F_q(\!(t)\!)\)

proved

Lattice positions computed directly from valuations of minors, without the position formula, reproduce A3-7's counts. The volume identity \(\sum_\nu(T_\lambda\ast\varphi_0)(x_\nu)\operatorname{vol}(Kx_\nu)/\operatorname{vol}(Kx_0)=|Kt^\lambda K/K|\) holds exactly for every \(\lambda\) tested.

Computation 5.7.

Model: \(W_i=\langle e,f\rangle\), \(L_i(d)=\langle e_1,e_2,t^{-d}e_1+f_2,f_1+t^{-d}e_2\rangle\) (self-dual, \(W\cap L=\langle e_1,t^df_1\rangle\)), and \(\varphi(e)=t^{-k}e'\), \(\varphi(f)=t^kf'\) with \(e=|d_1-d_2+2k|\). For all \(\lambda=(\lambda^1,\lambda^2)\) with \(a_i\le2\) at \(q=3,5,7\), and with \(a_i\le3\) at \(q=3\):

  1. \(\#\{L'\in\mathrm{Gr}^\lambda(\mathbb F_q):x_\nu\ \text{integral for}\ L'\}\), computed by brute force, equals the count given by A3-7's position formula and tree counts. It was checked on all 4068 (resp. 31900) pairs \((\nu,\lambda)\), zeros included.
  2. The volume (mass) identity holds exactly for all 36 (resp. 100) \(\lambda\), with volumes \([(q+1)q^{e-1}]_{e\ge1}\prod_{d_i\ge1}(q^{3d_i}-q^{3d_i-2})\) and \(|Kt^{(a,b)}K/K|=q^{4a+2b}W(q^{-1})/W_{(a,b)}(q^{-1})\) for \(\mathrm{Sp}_4\).
  3. The number of \(Q\) with \(\operatorname{pos}(C\oplus Q)=\mu\) depends only on \(\operatorname{dist}(C,[N])\): no violation in a complete enumeration for \(d\le2\).

Method (independent of A3's scripts).

  1. Enumeration. Unimodular lattices in a symplectic plane were enumerated in Hermite normal form \(\langle t^ae,\ ye+t^{-a}f\rangle\), with \(y\) a Laurent polynomial of degree \(<a\).
  2. Integral points. By A3-7(a), which is re-proved in Audit 5.8, integrality of \(x_\nu\) for \(L'\) means \(L_1'=C\oplus Q_1\) and \(L_2'=\varphi C\oplus Q_2\) with all three lattices unimodular.
  3. Positions. For each pair \((C,Q)\), the matrix \(M=B_{L(d)}^{-1}B_{C\oplus Q}\) was formed over \(\mathbb F_q[t,t^{-1}]\). Then \(\lambda_1=-\min v(M_{ij})\) and \(\lambda_1+\lambda_2=-\min v(\text{2×2 minors})\). These are the elementary divisors \(t^{\mp\lambda_1},t^{\mp\lambda_2}\) of a self-dual lattice relative to \(L\).
  4. Summation. The \(Q\)-counts were tabulated by \(D=\operatorname{dist}(C,[N])\), using diagonal representatives \(C=\langle t^\alpha e,t^{-\alpha}f\rangle\). This is legitimate because \(\operatorname{Stab}_K(W)\) acts on \(W\) through all of \(\mathrm{SL}(N)\), which is transitive on spheres around \([N]\); it was also verified exhaustively for \(d\le2\). Finally, \(C\) was summed over all unimodular lattices within distance \(2a_1\) of \(\langle e,f\rangle\).

Results.

Why it suffices. The mass identity tests labels, volumes and counts together. The exact agreement with the closed formula, at three values of \(q\), tests the position formula of A3-7(b) and the tree counts of A3-7(c) independently of A3's own derivation and scripts. Combined with the algebraic re-derivation in Audit 5.8, this makes the polynomials used for semismallness reliable.

Builds on

  • Computation 3.7 · Convolution fibres and the counts \((T_\lambda\ast\varphi_0)(x_\nu)\)
  • Proposition 3.6 · Orbits \((d_1,d_2,e)\), connected stabilizers, \(\delta=3d_1+3d_2+e\)

Built on by

  • Audit 5.8 · Audit of A3-7, A3-8: position formula, semismallness, dual branching
  • Summary 5.12 · Verdict: (SG) fails, proved; \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) modulo (O1)–(O3)

Sources