§7 · Agent 7 · Audit: criterion and bridge · 2026-10-18T19:25:52Z

Audit 7.6 · Audit of A3-1, A3-2, A3-3: glued Weil heart, piece-indicator bases by Lang, $\mathrm{tr}(\mathcal G\ast_IM)=\mathrm{tr}\,\mathcal G\ast\mathrm{tr}\,M$ with $\mathrm{vol}(I)=1$

proved

The $\mathbb F_q$ set-up of Claim II, version 1, the finite-level function bases and the trace of convolution are correct under (L1), (L2), (D1), (D2); one citation repair and one filled detail on inseparable orbit maps.

Audit 7.6.

Holds with a minor repair. Over $k=\overline{\mathbb F}_q$, functions over $F=\mathbb F_q((t))$:

  1. A3-1(i)–(iii) hold; the gluing shifts $e_M=e_{M'}+d(M,M')$ need $d(M,M')$ independent of $N$ (true in Examples A, B: $d=n(M'-M)$, resp. $\dim I_M/I_{M'}$), which (L1a) implicitly asserts.
  2. A3-2(a)–(c) hold; in characteristic $p$ the orbit map $H/S_{y,\mathrm{red}}\to o$ may be purely inseparable, but a radicial surjective morphism of $\mathbb F_q$-varieties is bijective on $\mathbb F_q$-points ($\mathbb F_q$ perfect), so the Lang count is unaffected.
  3. A3-3(a)–(c) hold with the stated normalization $\mathrm{vol}(I)=1$, and $\mathrm{tr}\,\mathrm{IC}w=(-1)^{\ell(w)}T_w+\sum{v<w}c_{vw}T_v$.

Repair: in (D3) the purity of $\mathrm{IC}_w$ is BBD Cor. 5.3.2 (checked on the scan, journal p. 135); the citation "Cor. 5.4.3" was not verified.

  1. A3-1, gluing. $\Phi_{N,M}=\pi_{N,M}^*[e_M]$; compatibility with $Y_{N,M'}\to Y_{N,M}$ is $e_M=e_{M'}+d(M,M')$, and with the closed $N$-transitions it needs the same $e_M$ for all $N$, i.e. $d(M,M')$ independent of $N$. Transitions are t-exact and fully faithful (affine-space fibres, unipotent kernels, as in Audit 7.1 steps 2–3), so the t-structures glue to a bounded one with heart the union of finite-length hearts. Holds.
  2. A3-1(iii), Weil structures. $\mathrm{Fr}^*\mathrm{IC}(P)\simeq\mathrm{IC}(\mathrm{Fr}^{-1}P)=\mathrm{IC}(P)$ for $\mathrm{Fr}$-stable orbits; $\mathrm{Hom}(\mathrm{Fr}^*S,S)$ is one-dimensional and restriction to the open piece is injective on maps between IC's, so the structure normalized to $1$ on $S|P=\overline{\mathbb Q}\ell[\delta(P)]$ is unique; it is equivariant because $\mathrm{Perv}H\to\mathrm{Perv}$ is fully faithful for connected $H$. Level-independence: $\delta$ computed at level $(N,M')$ on the open piece $P^\circ$ is $\dim o^\circ+e{M'}=\dim o+d+e_{M'}=\delta(P)$. Holds.
  3. A3-2(a). A compact support lies in one $Y_N(\mathbb F_q)$; local constancy on the compact $Y_N(\mathbb F_q)$ plus the nested fibre basis gives constancy on the fibres of one $\pi_{N,M}$ (finitely many basic sets; a level-$M$ fibre through $y$ lies in the chosen coarser fibre through $y$). Holds.
  4. A3-2(b),(c). Surjectivity on points: affine-space bundles over $\mathbb F_q$ have rational points in rational fibres, and an inverse limit of surjections of finite sets surjects; $I(\mathcal O)\twoheadrightarrow(I/I_{N,M})(\mathbb F_q)$ from $H^1(\mathbb F_q,I_{N,M})=1$ (Milne, Cor. 27.56, extract trace-milne-lang.txt; needs $I_{N,M}$ connected, see Lemma 7.5(d)). Rational orbits in $o(\mathbb F_q)$ $\leftrightarrow\ker(H^1(\mathbb F_q,S_y)\to H^1(\mathbb F_q,H))=H^1(\mathbb F_q,\pi_0S_y)=1$. Statement 2 above fills the inseparability detail. Holds.
  5. A3-3(a). Shearing $[g,y]\mapsto(gI,gy)$ with inverse $(gI,x)\mapsto[g,g^{-1}x]$ (well defined by $I$-equivariance); stalks $\mathcal G_g\otimes M_{g^{-1}x}$ compatibly with Frobenius for a rational lift $g\in G(F)$ (exists by Lang for $I$); reduction to the finite-type $\mathrm{Fl}{\le w}\times Y{N',M''}$ by (D1); Grothendieck–Lefschetz and $\mathrm{tr}(A\otimes B)=\mathrm{tr}A\cdot\mathrm{tr}B$ (Laumon (1.1.1.3), (1.1.1.2), extract trace-laumon-dict.txt); $\sum_{gI\in G(F)/I}f(g)\varphi(g^{-1}x)=\int_{G(F)}f(g)\varphi(g^{-1}x)\,dg$ for $\mathrm{vol}(I)=1$, the summand being right $I$-invariant in $g$ because $\varphi$ is $I$-invariant. $\mathrm{tr}\,\mathrm{IC}0=\mathbf 1{X(\mathcal O)}$ since $\mathrm{IC}0$ is unshifted with trivial Weil structure. Group case: the right factor acts through $g_2^{-1}$, giving $f_1\ast\varphi\ast\check f_2$, consistent with $\mathcal H{I_G}=\mathcal H_{I_H}\otimes\mathcal H_{I_H}$. Holds.
  6. A3-3(b),(c). $\mathrm{IC}w|{IwI/I}=\overline{\mathbb Q}\ell[\ell(w)]$ with Frobenius $1$; $\mathrm{Fl}{\le w}(\mathbb F_q)=\bigsqcup_{v\le w}I vI/I$ (Lang for the connected $I\cap vIv^{-1}$); Bruhat intervals in $W_{\mathrm{aff}}\rtimes\Omega$ are finite; unitriangularity gives a basis; (c) follows from (a). $T_w\ast\varphi_0(x)=\#\{gI\in IwI/I:\ g^{-1}x\in X(\mathcal O)\}\in\mathbb Z_{\ge0}$, so the integrality used later is right. Holds.

Builds on

  • Setup 3.1 · Set-up over $\mathbb F_q$: levels, the perverse heart of $\mathcal C_I$, Weil structures
  • Lemma 3.2 · Functions at a finite level: bases $\{\mathbf 1_{P(\mathbb F_q)}\}$ by Lang
  • Lemma 3.3 · Trace of $*_I$ is convolution (vol $I=1$); IC traces span $\mathcal H_I$

Built on by

  • Summary 7.11 · Verdict of the audit: (I$_I$) over $\overline{\mathbb F}_q$ and $\mathbb C$, (II$_I$) over $\overline{\mathbb F}_q$ only, converses as stated
Original record, as the agent wrote it at 19:25 UTC

Audit of A3-1, A3-2, A3-3: glued Weil heart, piece-indicator bases by Lang, $\mathrm{tr}(\mathcal G\ast_IM)=\mathrm{tr}\,\mathcal G\ast\mathrm{tr}\,M$ with $\mathrm{vol}(I)=1$

  1. A3-1, gluing. $\Phi_{N,M}=\pi_{N,M}^*[e_M]$; compatibility with $Y_{N,M'}\to Y_{N,M}$ is $e_M=e_{M'}+d(M,M')$, and with the closed $N$-transitions it needs the same $e_M$ for all $N$, i.e. $d(M,M')$ independent of $N$. Transitions are t-exact and fully faithful (affine-space fibres, unipotent kernels, as in Audit 7.1 steps 2–3), so the t-structures glue to a bounded one with heart the union of finite-length hearts. Holds.
  2. A3-1(iii), Weil structures. $\mathrm{Fr}^*\mathrm{IC}(P)\simeq\mathrm{IC}(\mathrm{Fr}^{-1}P)=\mathrm{IC}(P)$ for $\mathrm{Fr}$-stable orbits; $\mathrm{Hom}(\mathrm{Fr}^*S,S)$ is one-dimensional and restriction to the open piece is injective on maps between IC's, so the structure normalized to $1$ on $S|P=\overline{\mathbb Q}\ell[\delta(P)]$ is unique; it is equivariant because $\mathrm{Perv}H\to\mathrm{Perv}$ is fully faithful for connected $H$. Level-independence: $\delta$ computed at level $(N,M')$ on the open piece $P^\circ$ is $\dim o^\circ+e{M'}=\dim o+d+e_{M'}=\delta(P)$. Holds.
  3. A3-2(a). A compact support lies in one $Y_N(\mathbb F_q)$; local constancy on the compact $Y_N(\mathbb F_q)$ plus the nested fibre basis gives constancy on the fibres of one $\pi_{N,M}$ (finitely many basic sets; a level-$M$ fibre through $y$ lies in the chosen coarser fibre through $y$). Holds.
  4. A3-2(b),(c). Surjectivity on points: affine-space bundles over $\mathbb F_q$ have rational points in rational fibres, and an inverse limit of surjections of finite sets surjects; $I(\mathcal O)\twoheadrightarrow(I/I_{N,M})(\mathbb F_q)$ from $H^1(\mathbb F_q,I_{N,M})=1$ (Milne, Cor. 27.56, extract trace-milne-lang.txt; needs $I_{N,M}$ connected, see Lemma 7.5(d)). Rational orbits in $o(\mathbb F_q)$ $\leftrightarrow\ker(H^1(\mathbb F_q,S_y)\to H^1(\mathbb F_q,H))=H^1(\mathbb F_q,\pi_0S_y)=1$. Statement 2 above fills the inseparability detail. Holds.
  5. A3-3(a). Shearing $[g,y]\mapsto(gI,gy)$ with inverse $(gI,x)\mapsto[g,g^{-1}x]$ (well defined by $I$-equivariance); stalks $\mathcal G_g\otimes M_{g^{-1}x}$ compatibly with Frobenius for a rational lift $g\in G(F)$ (exists by Lang for $I$); reduction to the finite-type $\mathrm{Fl}{\le w}\times Y{N',M''}$ by (D1); Grothendieck–Lefschetz and $\mathrm{tr}(A\otimes B)=\mathrm{tr}A\cdot\mathrm{tr}B$ (Laumon (1.1.1.3), (1.1.1.2), extract trace-laumon-dict.txt); $\sum_{gI\in G(F)/I}f(g)\varphi(g^{-1}x)=\int_{G(F)}f(g)\varphi(g^{-1}x)\,dg$ for $\mathrm{vol}(I)=1$, the summand being right $I$-invariant in $g$ because $\varphi$ is $I$-invariant. $\mathrm{tr}\,\mathrm{IC}0=\mathbf 1{X(\mathcal O)}$ since $\mathrm{IC}0$ is unshifted with trivial Weil structure. Group case: the right factor acts through $g_2^{-1}$, giving $f_1\ast\varphi\ast\check f_2$, consistent with $\mathcal H{I_G}=\mathcal H_{I_H}\otimes\mathcal H_{I_H}$. Holds.
  6. A3-3(b),(c). $\mathrm{IC}w|{IwI/I}=\overline{\mathbb Q}\ell[\ell(w)]$ with Frobenius $1$; $\mathrm{Fl}{\le w}(\mathbb F_q)=\bigsqcup_{v\le w}I vI/I$ (Lang for the connected $I\cap vIv^{-1}$); Bruhat intervals in $W_{\mathrm{aff}}\rtimes\Omega$ are finite; unitriangularity gives a basis; (c) follows from (a). $T_w\ast\varphi_0(x)=\#\{gI\in IwI/I:\ g^{-1}x\in X(\mathcal O)\}\in\mathbb Z_{\ge0}$, so the integrality used later is right. Holds.

Sources