§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:23:37Z

Audit 8.10 · Audit of (IG) for $\mathbb A^n$ and the reverse route (A5-6, A5-8)

proved

T_{τ^k} ∗ φ_0 = 1_{τ^k O^n} (vol I = 1) and these indicator functions form a basis of C_I, so (IG) holds over every non-archimedean field; the T_{s_0}-formula was recomputed pointwise. The reverse route A5-8 is correct given A5-7, which is not in the audited file.

Audit 8.10.

Holds: for every non-archimedean \(F\) with \(\operatorname{vol}(I)=1\), \(T_{\tau^k}\ast\varphi_0=b_k:=\mathbf 1_{\tau^k\mathcal O^n}\), \(\{b_k\}_{k\in\mathbb Z}\) is a basis of \(C_I\), and \(h\mapsto h\ast\varphi_0\) restricts to an isomorphism \(\mathbb C[\Omega]\xrightarrow{\sim}C_I\); so \(\mathcal H_I\ast\varphi_0=C_I\). Also \(T_{s_0}\ast\varphi_0=b_{-1}+q\,b_1-b_0\), \(T_{t^{\lambda_{m,j}}}\ast\varphi_0=q^{j(n-j)}b_k\), and the traces of Proposition 5.6(e).

Holds conditionally on A5-7 (not in the file): Corollary 5.8's inputs (L\(_q\)), (W), (Lin), (P), (Tr) are verified correctly for \(\mathbb A^n\). The route needs purity and the decomposition theorem, and only over \(\overline{\mathbb F}_q\).

  1. Translates. \((T_{\tau^k}\ast f)(x)=\int_{\tau^kI}f(g^{-1}x)\,dg=\operatorname{vol}(I)f(\tau^{-k}x)\), so \(T_{\tau^k}\ast\mathbf 1_L=\mathbf 1_{\tau^kL}\).
  2. Basis. \(\varphi\in C_I\) is supported in some \(t^{-N}\mathcal O^n=\tau^{-nN}\mathcal O^n\), constant on some \(t^M\mathcal O^n=\tau^{nM}\mathcal O^n\) (local constancy at \(0\), the function shadow of Audit 8.8 step 6), and constant on each \(o_k\); so \(\varphi=\sum_k\varphi(o_k)(b_k-b_{k+1})+\varphi(0)b_{nM}\). Independence comes from the strictly decreasing lattices. Hence (IG), even for the commutative subalgebra \(\mathbb C[\Omega]\simeq\mathbb C[z^{\pm1}]\).
  3. \(T_{s_i}\)-action. For \(i\not\equiv-k\), \(P_{s_i}\) fixes \(\tau^k\mathcal O^n\), so \(T_{s_i}b_k=q\,b_k\). For \(i\equiv-k\), \(T_{s_i}T_{\tau^k}=T_{\tau^k}T_{s_0}\) (lengths add with \(\ell(\tau)=0\)), and \(T_{s_0}\varphi_0=\sum_{L\ne\mathcal O^n}\mathbf 1_L\) over the \(q\) other lattices of the \(\mathbb P^1(\mathbb F_q)\). A vector of \(\tau^{-1}\mathcal O^n\setminus\tau\mathcal O^n\) lies in exactly one of the \(q+1\) lattices, giving \(b_{-1}+qb_1-b_0\). The quadratic relation \(T_{s_0}^2=(q-1)T_{s_0}+q\) on \(\varphi_0\) checks: both sides equal \((q-1)b_{-1}+q(q-1)b_1+b_0\).
  4. Pointwise recomputation. \((T_{s_0}\ast\varphi_0)(x)=\#\{c\in\mathbb F_q:x\in u_cs_0\mathcal O^n\}\) was evaluated, with the coset representatives \(u_cs_0\), \(u_c=1+ctE_{n1}\), of \(Is_0I/I\) and \(s_0=t^{-1}E_{1n}-tE_{n1}+\sum_{1<i<n}E_{ii}\), at every point of \(t^{-1}\mathcal O^n/t^2\mathcal O^n\) over \(\mathbb F_q=\mathbb F_p\), and compared with \(b_{-1}+qb_1-b_0\). The precision suffices because all lattices involved lie between \(t^{-1}\mathcal O^n\) and \(t\mathcal O^n\). For \((n,q)=(2,2),(2,3),(3,2),(3,3),(4,2)\) (64, 729, 512, 19683, 4096 points) there was no mismatch.
  5. Traces. \(\operatorname{tr}(S_k)=(-1)^kb_k\); \(\operatorname{tr}(\mathrm{IC}_{s_0}\ast\mathrm{IC}_0)=-(1+T_{s_0})\varphi_0=-b_{-1}-qb_1\) agrees with \(\operatorname{tr}(S_{-1}\oplus S_1(-1))\). For \(\delta_{\tau^k}\), trace of convolution is Frobenius-equivariance of \(x\mapsto\tau^kx\).
  6. Reverse route. (P): \(E_w\subseteq\mathrm{Fl}_{\le w}\times V_{N,M}\) is a vector bundle of constant rank \(nM-\operatorname{val}\det g\) (\(\operatorname{val}\det\) is constant on the connected \(\overline{IwI}\); Proposition 2.6 step 5 makes this explicit), \(m_w\) is proper, and \(\mathrm{IC}_w\ast\mathrm{IC}_0=\pi^*m_{w!}\mathrm{pr}_1^*\mathrm{IC}_w\) is pure. (Lin): the \(b_k\) are independent. (Tr): Lang for the connected pro-algebraic \(I\). The authors are right that traces alone give only constituents, so the summand property needs the decomposition theorem. The direct route Audit 8.9 does not need it and also works over \(\mathbb C\). The statement of A5-7 (general (II\(_I\)) with these inputs) is outside this audit, so Corollary 5.8 is correct relative to it.

Builds on

  • Proposition 5.6 · (IG) for $\mathbb A^n$: $T_{\tau^k}\ast\varphi_0=\mathbf 1_{\tau^k\mathcal O^n}$
  • Corollary 5.8 · Reverse route for $\mathbb A^n$: (IG) and (II$_I$) give sheaf generation

Built on by

  • Summary 8.13 · Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$
Original record, as the agent wrote it at 19:23 UTC

Audit of (IG) for $\mathbb A^n$ and the reverse route (A5-6, A5-8)

  1. Translates. \((T_{\tau^k}\ast f)(x)=\int_{\tau^kI}f(g^{-1}x)\,dg=\operatorname{vol}(I)f(\tau^{-k}x)\), so \(T_{\tau^k}\ast\mathbf 1_L=\mathbf 1_{\tau^kL}\).
  2. Basis. \(\varphi\in C_I\) is supported in some \(t^{-N}\mathcal O^n=\tau^{-nN}\mathcal O^n\), constant on some \(t^M\mathcal O^n=\tau^{nM}\mathcal O^n\) (local constancy at \(0\), the function shadow of Audit 8.8 step 6), and constant on each \(o_k\); so \(\varphi=\sum_k\varphi(o_k)(b_k-b_{k+1})+\varphi(0)b_{nM}\). Independence comes from the strictly decreasing lattices. Hence (IG), even for the commutative subalgebra \(\mathbb C[\Omega]\simeq\mathbb C[z^{\pm1}]\).
  3. \(T_{s_i}\)-action. For \(i\not\equiv-k\), \(P_{s_i}\) fixes \(\tau^k\mathcal O^n\), so \(T_{s_i}b_k=q\,b_k\). For \(i\equiv-k\), \(T_{s_i}T_{\tau^k}=T_{\tau^k}T_{s_0}\) (lengths add with \(\ell(\tau)=0\)), and \(T_{s_0}\varphi_0=\sum_{L\ne\mathcal O^n}\mathbf 1_L\) over the \(q\) other lattices of the \(\mathbb P^1(\mathbb F_q)\). A vector of \(\tau^{-1}\mathcal O^n\setminus\tau\mathcal O^n\) lies in exactly one of the \(q+1\) lattices, giving \(b_{-1}+qb_1-b_0\). The quadratic relation \(T_{s_0}^2=(q-1)T_{s_0}+q\) on \(\varphi_0\) checks: both sides equal \((q-1)b_{-1}+q(q-1)b_1+b_0\).
  4. Pointwise recomputation. \((T_{s_0}\ast\varphi_0)(x)=\#\{c\in\mathbb F_q:x\in u_cs_0\mathcal O^n\}\) was evaluated, with the coset representatives \(u_cs_0\), \(u_c=1+ctE_{n1}\), of \(Is_0I/I\) and \(s_0=t^{-1}E_{1n}-tE_{n1}+\sum_{1<i<n}E_{ii}\), at every point of \(t^{-1}\mathcal O^n/t^2\mathcal O^n\) over \(\mathbb F_q=\mathbb F_p\), and compared with \(b_{-1}+qb_1-b_0\). The precision suffices because all lattices involved lie between \(t^{-1}\mathcal O^n\) and \(t\mathcal O^n\). For \((n,q)=(2,2),(2,3),(3,2),(3,3),(4,2)\) (64, 729, 512, 19683, 4096 points) there was no mismatch.
  5. Traces. \(\operatorname{tr}(S_k)=(-1)^kb_k\); \(\operatorname{tr}(\mathrm{IC}_{s_0}\ast\mathrm{IC}_0)=-(1+T_{s_0})\varphi_0=-b_{-1}-qb_1\) agrees with \(\operatorname{tr}(S_{-1}\oplus S_1(-1))\). For \(\delta_{\tau^k}\), trace of convolution is Frobenius-equivariance of \(x\mapsto\tau^kx\).
  6. Reverse route. (P): \(E_w\subseteq\mathrm{Fl}_{\le w}\times V_{N,M}\) is a vector bundle of constant rank \(nM-\operatorname{val}\det g\) (\(\operatorname{val}\det\) is constant on the connected \(\overline{IwI}\); Proposition 2.6 step 5 makes this explicit), \(m_w\) is proper, and \(\mathrm{IC}_w\ast\mathrm{IC}_0=\pi^*m_{w!}\mathrm{pr}_1^*\mathrm{IC}_w\) is pure. (Lin): the \(b_k\) are independent. (Tr): Lang for the connected pro-algebraic \(I\). The authors are right that traces alone give only constituents, so the summand property needs the decomposition theorem. The direct route Audit 8.9 does not need it and also works over \(\mathbb C\). The statement of A5-7 (general (II\(_I\)) with these inputs) is outside this audit, so Corollary 5.8 is correct relative to it.

Sources