Proposition IV as a paper · statement and complete proof (PDF) →

Proof

Steps 1–2 set up the forms model and show where H5 fails. Steps 3–7 prove (a) by a triangular system of lattice counts. Step 8 gives (b) through Proposition II. Steps 9–12 compute the \(\mathrm{GL}_4/\mathrm{Sp}_4\) decomposition directly, and step 13 assembles the statement.

1. Purpose Pass to alternating forms

   Orbits are elementary-divisor classes of alternating matrices, with the Pfaffian valuation as degree:

   $$\mathcal O_a=G(\mathcal O)\cdot D_a=\mathcal O_{\mu_a},\qquad\operatorname{val}\operatorname{Pf}=|a|\ \text{on}\ \mathcal O_a,\qquad\mu_b\le\mu_a\iff b\preceq a .$$
   Proof

   • Model. \(\tau(gG^\theta)J=gJg^TJ^{-1}J=gJg^T\). Over a field all nondegenerate alternating forms of rank \(2m\) are congruent, so \(X(R)\) is all of them, with \(g\cdot\omega=g\omega g^T\).
   • Orbits. By the elementary-divisor theorem for alternating matrices over the principal ideal domain \(\mathcal O\), \(\omega=kD_ak^T\) with \(k\in G(\mathcal O)\) and \(a\) unique. As \(t^{\mu_a}J=D_a\), this is Chen–Yi's parametrization \(\tau(x_\mu)=t^\mu\).
   • Degree and order. \(\operatorname{Pf}(g\omega g^T)=\det g\,\operatorname{Pf}\omega\). The partial sums of \(\mu_a-\mu_b\) are \(2S_i\) at position \(2i\) and \(S_{i-1}+S_i\) at position \(2i-1\), with \(S_i=\sum_{j\le i}(a_j-b_j)\): $$\mu_a-\mu_b\in\textstyle\sum\mathbb Z_{\ge0}(e_i-e_j)\iff S_1,\dots,S_{m-1}\ge0,\ S_m=0\iff b\preceq a .$$
   • Shifts. \(t\) is central with \(\operatorname{Pf}(t\omega)=t^m\operatorname{Pf}\omega\), so \(\sigma\) moves the degree by \(m\); \(t^c\cdot1\) acts by \(\omega\mapsto t^{2c}\omega\).
2. Result H5 fails at the wall labels

   A label is covered iff no two adjacent entries are equal and odd; so H5 fails for \(m\ge2\) and for \(\mathrm{SL}_{2n}\), \(n\ge3\):

   $$\mu_{(1,1,0,\dots,0)}\notin\varphi(\Lambda_T^+)\ \ (\mathrm{GL}_{2m},\ m\ge2),\qquad\mu_{(1,1,0,\dots,0,-2)}\notin\varphi(\Lambda_T^+)\ \ (\mathrm{SL}_{2n},\ n\ge3).$$
   Proof

   • Coverage (symplectic family page, §5). If \(a_i=a_{i+1}=a\) is odd, a lift has \(\lambda_{2i}\le\frac{a-1}2\lt\frac{a+1}2\le\lambda_{2i+1}\), not dominant. Otherwise split each pair as \((\lceil a_i/2\rceil,\lfloor a_i/2\rfloor)\).
   • H5. \(Z_{G^\theta}(S)=\mathrm{SL}_2^m\) (resp. \(\mathrm{SL}_2^n\)) is connected, and Proposition III of the pages (criterion.html, step 7) says H1–H5 hold iff \(\varphi(\Lambda_T^+)=\Lambda_S^+\).
   • Mechanism. At an uncovered \(\mu\), \(\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_\mu}\) forces \(\varphi(\lambda)\le\mu\), and \(a_\mu^*(V_\lambda\ast\mathrm{IC}_0)\ne0\) forces \(\mu\le\varphi(\lambda)\) (nodes/fibre.html); no Satake object qualifies.
   • \(\mathrm{GL}_4\). The uncovered labels are exactly \((c,c)\), \(c\) odd; \((1,1)\) is the least label of degree \(2\), so \(\mathcal O_{(1,1,1,1)}\) is closed in its component.
3. Method Hecke values as lattice counts

   For \(\lambda_{2m}\ge0\), a Hecke value is a count of self-dual lattices, which vanishes off degree \(|\lambda|\):

   $$N(\lambda,b)=\#\Big\{\Lambda''\supseteq\Lambda_0:\ \Lambda''/\Lambda_0\simeq\textstyle\bigoplus_i\mathcal O/t^{\lambda_i},\ D_b\ \text{unimodular on}\ \Lambda''\Big\},\qquad N(\lambda,b)\ne0\Rightarrow b_m\ge0,\ |b|=|\lambda| .$$
   Proof

   • Lemma A(2). \((T_\lambda\ast\varphi_0)(x)=\#\{gK\subseteq Kt^\lambda K:\ g^{-1}x\in X(\mathcal O)\}\) for \(\operatorname{vol}K=1\).
   • Lattices. \(g^{-1}\cdot\omega=g^{-1}\omega g^{-T}\) is unimodular iff \(\omega\) is unimodular on \(\Lambda''=g^{-T}\Lambda_0\). Since \(g^{-T}\in Kt^{-\lambda}K\), the map \(gK\mapsto g^{-T}\Lambda_0\) is a bijection $$Kt^\lambda K/K\xrightarrow{\ \sim\ }\{\Lambda''\supseteq\Lambda_0:\ \Lambda''/\Lambda_0\simeq\textstyle\bigoplus_i\mathcal O/t^{\lambda_i}\}.$$
   • Constraints. Integrality on \(\Lambda_0\subseteq\Lambda''\) gives \(b_m\ge0\). On a basis of \(\Lambda''\) the Pfaffian valuation of \(D_b\) is \(|b|-|\lambda|\); it must vanish, and an integral form of unit Pfaffian is unimodular.
   • Central translates. \((T_{c\cdot1}\ast f)(x)=f(t^{-c}x)\) and \(t^{-c}\cdot\omega=t^{-2c}\omega\), so \(T_{c\cdot1}\ast\mathbf 1_{\mathcal O_b}=\mathbf 1_{\mathcal O_{b+2c(1^m)}}\); and \(T_{c\cdot1}\ast T_\lambda=T_{\lambda+c\cdot1}\).
4. Result GL₄/Sp₄: four lattice counts

   For \(m=2\) both translates live on the two orbits of degree \(2\), and a cancellation isolates the uncovered closed orbit, over every \(F\):

   $$\mathbf 1_{\mathcal O_{(1,1,1,1)}}=\frac{1}{(1+q)(1+q^2)}\Big(T_{(1,1,0,0)}-\frac{1}{q(q+1)}T_{(2,0,0,0)}\Big)\ast\varphi_0 .$$
   Proof

   • Support. By step 3, \(b\ge0\) and \(|b|=2\): \(b\in\{(2,0),(1,1)\}\), i.e. \(\mathcal O_{(2,2,0,0)}\cup\mathcal O_{(1,1,1,1)}\).
   • \(\lambda=(1,1,0,0)\). \(\Lambda''=\Lambda_0+t^{-1}\widetilde W\), \(W\in\operatorname{Gr}(2,\mathbb F_q^4)\). At \(tJ\): \(tJ(t^{-1}w,t^{-1}w')=t^{-1}J(w,w')\), so \(W\) is Lagrangian: \((1+q)(1+q^2)\). At \(D_{(2,0)}\): pairing \(t^{-1}w\) with \(\Lambda_0\) kills the \(e_3,e_4\)-coordinates, so \(W=\langle\bar e_1,\bar e_2\rangle\): \(1\).
   • \(\lambda=(2,0,0,0)\). \(\Lambda''=\Lambda_0+\mathcal Ot^{-2}v\), \(v\) primitive mod \(t^2\). At \(tJ\): \(t^{-1}J(v,\Lambda_0)\subseteq\mathcal O\) forces \(v\in t\Lambda_0\): \(0\). At \(D_{(2,0)}\): the \(e_3,e_4\)-coordinates of \(v\) lie in \(t^2\mathcal O\): \(|\mathbb P^1(\mathcal O/t^2)|=q(q+1)\).
   • Result. $$T_{(1,1,0,0)}\ast\varphi_0=\mathbf 1_{\mathcal O_{(2,2,0,0)}}+(1+q)(1+q^2)\,\mathbf 1_{\mathcal O_{(1,1,1,1)}},\qquad T_{(2,0,0,0)}\ast\varphi_0=q(q+1)\,\mathbf 1_{\mathcal O_{(2,2,0,0)}} .$$ The counts are configurations over \(\mathbb F_q\) and \(\mathcal O/t^2\), independent of \(F\).
   • Audit corrections. The translates nonzero on \(\mathcal O_{(1,1,1,1)}\) are exactly \(T_{(1+a,1+b,-b,-a)}\), \(a\ge b\ge0\) (C-3); \(T_{(2,0,0,0)}\) is used only through its value on \(\mathcal O_{(2,2,0,0)}\). \(T_{(1,1,1,-1)}\ast\varphi_0\) equals \(T_{(2,0,0,0)}\ast\varphi_0\) because \(\omega\mapsto\omega^{-1}\) exchanges \(\lambda\) and \(-w_0\lambda\), not because the two are central translates (audit of A7-6).
5. Result The row λ*(b) is triangular

   The translate of \(T_{\lambda^*(b)}\) is nonzero at \(b\) and supported on labels above \(b\):

   $$N(\lambda^*(b),b)\ge1,\qquad N(\lambda^*(b),b')\ne0\ \Longrightarrow\ b'_m\ge0,\ |b'|=|b|,\ b'\succeq b .$$
   Proof

   • Nonvanishing. \(\Lambda''=\bigoplus_i(t^{-b_i}\mathcal Oe_{2i-1}\oplus\mathcal Oe_{2i})\) has \(\Lambda''/\Lambda_0\simeq\bigoplus_i\mathcal O/t^{b_i}\), and \(D_b(t^{-b_i}e_{2i-1},e_{2i})=1\).
   • Exact sequence. Let \(\omega\) be unimodular on \(\Lambda''\supseteq\Lambda_0\) of type \(\lambda^*(b)\), with \(\omega|_{\Lambda_0}\simeq D_{b'}\), and \(\Lambda_0^\#=\{v:\omega(v,\Lambda_0)\subseteq\mathcal O\}\). Then \(\Lambda''=\Lambda''^\#\subseteq\Lambda_0^\#\) and $$0\to M=\Lambda''/\Lambda_0\to M'=\Lambda_0^\#/\Lambda_0\simeq\textstyle\bigoplus_i(\mathcal O/t^{b'_i})^2\to N=\Lambda_0^\#/\Lambda''\simeq\operatorname{Hom}(M,F/\mathcal O)\to0,$$ with \(M\) and \(N\) of type \(b\) (the pairing \(N\times M\to F/\mathcal O\) induced by \(\omega\) is perfect).
   • Submodule bound. For a finite \(\mathcal O\)-module of type \(t_1\ge t_2\ge\cdots\), a \(k\)-generated \(S\) has \(\log_q|S|\le t_1+\dots+t_k\), with equality for the top \(k\) summands: multiplication by \(t^{j-1}\) embeds \(S[t^j]/S[t^{j-1}]\) in \(S[t]\cap t^{j-1}M'\), of dimension \(\le\min(k,\#\{i:t_i\ge j\})\).
   • Comparison. Take optimal \(k\)-generated \(S_1\subseteq M\), \(S_2\subseteq N\); \(S_1\) plus lifts of generators of \(S_2\) is \(2k\)-generated of order \(\ge|S_1||S_2|\): $$2(b'_1+\dots+b'_k)\ge2(b_1+\dots+b_k)\qquad(1\le k\le m).$$ With \(|b'|=|b|\) (step 3) this is \(b'\succeq b\).
6. Result (SG) for GL₂ₘ/Sp₂ₘ, every F

   Inside each finite slice the translates \(f_b\ast\varphi_0\) form a triangular system with positive diagonal, so every orbit indicator is a Hecke translate:

   $$f_b\ast\varphi_0=\sum_{b'\in S_{d,c},\ b'\succeq b}N\big(\lambda^*(b-2c),b'-2c\big)\,\mathbf 1_{\mathcal O_{b'}},\qquad\mathcal H_K\cdot\varphi_0=C_K .$$
   Proof

   • Orbits. \(\mathcal O_b\) is compact (image of \(K\)) and open (a form congruent to \(D_b\) modulo a high power of \(t\) has the same elementary divisors, after a central translate making \(b_m\ge0\)); so the \(\mathbf 1_{\mathcal O_b}\) are a basis of \(C_K\).
   • Slices. \(S_{d,c}\) is finite (entries in \([2c,d-2c(m-1)]\)), and \(b\in S_{|b|,\lfloor b_m/2\rfloor}\). For \(b\in S_{d,c}\), steps 3 and 5 put \(T_{\lambda^*(b-2c)}\ast\varphi_0\) on labels \(b''\succeq b-2c\) with \(b''\ge0\), \(|b''|=d-2cm\); the central shift moves \(b''\) to \(b''+2c\in S_{d,c}\). The diagonal coefficient is \(\ge1\).
   • Induction. On \(\#\{b'\in S_{d,c}:b'\succ b\}\): the terms above \(b\) are in \(\mathcal H_K\varphi_0\) by induction, so \(\mathbf 1_{\mathcal O_b}\) is.
   • Correction (A7-23). A5-7 cited Proposition B for the reversed order, whose lower sets are infinite (obstruction); the induction above runs in the finite slices, which the translates do not leave.
7. Result (SG) for SL₂ₙ/Sp₂ₙ, every F

   The Pfaffian-one forms carry the degree-zero part of step 6, and the elements used there lie in the Hecke algebra of \(\mathrm{SL}_{2n}\):

   $$T^{\mathrm{SL}}_\lambda\ast\varphi^{\mathrm{SL}}_0=(T_\lambda\ast\varphi_0)\big|_{X^{\mathrm{SL}}(F)}\quad(|\lambda|=0),\qquad f_b=T_{\lambda^*(b-2c)+c\cdot1}\quad(|b|=0).$$
   Proof

   • Orbits. \(X^{\mathrm{SL}}(F)=\{\operatorname{Pf}=1\}\), and \(\mathcal O_b\cap X^{\mathrm{SL}}(F)\), \(|b|=0\), is one \(K_{\mathrm{SL}}\)-orbit: \(k\omega k^T=\omega'\) with equal Pfaffians forces \(\det k=1\); \(\operatorname{diag}(u,1,\dots,1)\) gives surjectivity.
   • Cosets. For \(|\lambda|=0\), \(K_{\mathrm{SL}}t^\lambda K_{\mathrm{SL}}/K_{\mathrm{SL}}\to Kt^\lambda K/K\) is bijective (\(K=K_{\mathrm{SL}}\cdot\operatorname{diag}(u,1,\dots,1)\), diagonal matrices commute with \(t^\lambda\), \(K\cap\mathrm{SL}_{2n}(F)=K_{\mathrm{SL}}\)), and \(x\in gX(\mathcal O)\iff x\in gX^{\mathrm{SL}}(\mathcal O)\) for \(g\in\mathrm{SL}_{2n}(F)\), \(\operatorname{Pf}x=1\).
   • Restriction. \(T_{c\cdot1}\ast T_{\lambda^*(b-2c)}=T_{\lambda'}\) with \(|\lambda'|=|b|=0\); restricting the identities of step 6 in \(S_{0,c}\) to \(X^{\mathrm{SL}}(F)\) gives every \(K_{\mathrm{SL}}\)-orbit indicator.
8. Result Sheaves: \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) over \(\overline{\mathbb F}_q\), all m, n

   Proposition II turns (SG) over \(\mathbb F_q(\!(t)\!)\) into categorical generation; both families satisfy its hypotheses:

   $$\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\ \ \text{over}\ \overline{\mathbb F}_q\ (\text{D0–D3}),\qquad\text{Betti over}\ \mathbb C\ (\text{D5}).$$
   Proof

   • Proposition II (node) assumes \(q\) odd, (R) \(T\) split in a θ-stable Borel over \(\mathbb F_q\), (C) \(G^\theta\) and \(Z_{G^\theta}(S)\) connected, D0–D3, and (SG) over \(\mathbb F_{q^n}(\!(t)\!)\) for one \(n\).
   • (R). In the basis \(e_1,e_3,\dots,e_{2m-1},e_{2m},\dots,e_4,e_2\) the matrix \(J\) is antidiagonal, so the upper-triangular Borel is θ-stable and contains the diagonal torus.
   • (C). \(G^\theta=\mathrm{Sp}\) and \(Z_{G^\theta}(S)=\mathrm{SL}_2^m\) (resp. \(\mathrm{SL}_2^n\)) are connected.
   • (SG). Steps 6–7, for every odd \(q\).
   • Mechanism. The traces of \(V_\lambda\ast\mathrm{IC}_0\) span \(\mathcal H_K\varphi_0=C_K\), so every \(\operatorname{tr}\mathrm{IC}_\mu\)-coordinate is reached, every \(\mathrm{IC}_\mu\) is a summand, and Proposition I concludes. D5 applies for \(p\) outside a finite set, available since every odd \(p\) works. This closes the \(\mathrm{SL}_6\) label \((1,1,-2)\) left open by A3-7.
9. Result Codimensions in every component

   The codimension formula holds in every component, by a stabilizer count rather than Chen–Yi's Prop. 64; for \(\mathrm{GL}_4\) the closed orbit has codimension \(4\) in every characteristic:

   $$\operatorname{codim}_{\overline{\mathcal O_a}}\mathcal O_b=\langle\rho,\mu_a-\mu_b\rangle=\sum_{j=1}^m(2m+2-4j)(a_j-b_j),\qquad\operatorname{codim}_Z\mathcal O_{(1,1)}=4 .$$
   Proof

   • Why not cite Prop. 64 (obstruction). Chen–Yi assume \(LX\) connected; here \(\pi_0(LX)\simeq\mathbb Z\), and \(\sigma\) carries component \(s\) to \(s+m\), so it reaches component \(0\) only when \(m\mid s\).
   • Model. Shift so \(a_m\ge0\); put \(s=|a|\), \(N>s\). On \(Z_s=\{\omega\ \text{integral},\ \operatorname{val}\operatorname{Pf}=s\}\) reduction mod \(t^N\) is a trivial bundle onto an open \(Z_s^{(N)}\), and each orbit is a full preimage: \(\omega'\equiv D_e\) gives \(\omega'=D_e\big(1+D_e^{-1}(\omega'-D_e)\big)\).
   • Count. In \(2\times2\) blocks, with \(YJ_2+J_2Y^T=\operatorname{tr}(Y)J_2\), the stabilizer Lie algebra of \(D_a\) has dimension \((2m^2+m)N+\sum_j(4j-3)a_j\); so over \(\mathbb C\) $$\dim\mathcal O^{(N)}_a=m(2m-1)N-\textstyle\sum_j(4j-3)a_j,$$ and the difference is the formula since \(\sum_j(a_j-b_j)=0\).
   • \(\mathrm{GL}_4\) over \(\overline{\mathbb F}_q\). \(\#\mathcal O^{(3)}_{(2,0)}(\mathbb F_q)=q^{10}(q-1)(q^3-1)(q^2+1)\) and \(\#\mathcal O^{(3)}_{(1,1)}(\mathbb F_q)=q^8(q-1)(q^3-1)\) for every \(q\); the orbits are smooth and geometrically irreducible, so their dimensions are \(16\) and \(12\).
10. Result Convolution fibres for λ = (1,1,0,0)

    The minuscule convolution is birational onto \(Z\), with a Lagrangian Grassmannian over the closed orbit:

    $$m^{-1}(D_{(2,0)})=\mathrm{pt},\qquad m^{-1}(tJ)=\mathrm{LG}(2,4)\simeq Q^3,\qquad H^*(\mathrm{LG}(2,4))=k\oplus k[-2]\oplus k[-4]\oplus k[-6].$$
    Proof

    • \(\operatorname{Gr}^{\le\lambda}=\operatorname{Gr}^\lambda\simeq\operatorname{Gr}(2,4)\) since \(\lambda\) is minuscule, and \(gL^+G\in m^{-1}(\omega)\iff\omega\) is perfect on \(g^{-T}\Lambda_0\).
    • The two fibres are the computation of step 4 over \(k\).
    • \(\mathrm{LG}(2,4)\) is a smooth quadric threefold, paved by cells of dimensions \(0,1,2,3\); \(\#\mathrm{LG}(2,4)(\mathbb F_q)=(1+q)(1+q^2)\).
11. Result GL₄/Sp₄: the decomposition

    The decomposition theorem, the support condition and Verdier duality fix the multiplicity of the closed-orbit IC:

    $$P=m_!\underline k_Y[D]\simeq\mathrm{IC}_{(2,0)}\oplus\mathrm{IC}_{(1,1)}[2]\oplus\mathrm{IC}_{(1,1)}\oplus\mathrm{IC}_{(1,1)}[-2].$$
    Proof

    • Setting. For \(N\ge3\), \(m\) comes from a proper map \(Y^{(N)}\to Z^{(N)}\) with smooth source, \(D=\dim Z^{(N)}\) (birational by step 10). Connected stabilizers (D0) make the simple objects \(\mathrm{IC}_{(2,0)}\) and \(\mathrm{IC}_{(1,1)}=\underline k_{\mathcal O_{(1,1)}}[D-4]\); D2 gives \(P\simeq\mathrm{IC}_{(2,0)}\otimes A\oplus\mathrm{IC}_{(1,1)}\otimes B\).
    • Open orbit. Point fibre: \(P=\underline k[D]\) there, so \(A=k\).
    • Closed orbit. Proper base change (D1): \(P_x=H^*(\mathrm{LG}(2,4))[D]\), rank one in degrees \(-D,-D+2,-D+4,-D+6\).
    • Degrees. \(\dim\mathcal O_{(1,1)}=D-4\) (step 9), so \(\mathcal H^i(\mathrm{IC}_{(2,0)})_x=0\) for \(i\ge-D+4\), and \(\mathrm{IC}_{(1,1)}[-j]\) sits in degree \(-D+4+j\): \(j=0,2\) occur.
    • Duality. \(P\) and \(\mathrm{IC}_{(2,0)}\) are Verdier self-dual, so \(B\) is symmetric: \(j=-2\) occurs and fills degree \(-D+2\); \(j=\pm4\) cannot occur. Hence \(B=k[2]\oplus k\oplus k[-2]\) and \((\mathrm{IC}_{(2,0)})_x=k[D]\). Trace check: \(1+(q+q^2+q^3)=(1+q)(1+q^2)\).
12. Result GL₄/Sp₄: \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) directly

    Every \(\mathrm{IC}_\mu\) of \(\mathrm{GL}_4/\mathrm{Sp}_4\) lies in \(\mathcal C_K^{\mathrm{Sat}}\), so Proposition I gives generation under D0–D2 alone:

    $$\mathcal M=\Lambda_S^+,\qquad\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K .$$
    Proof

    • Covered labels. For \(\mu=\varphi(\lambda)\), \(V_\lambda\ast\mathrm{IC}_0\) is supported on \(\overline{\mathcal O_\mu}\) and nonzero at \(t^\lambda x_0\in\mathcal O_\mu\); only \(\mathrm{IC}_\mu\) among its summands is nonzero there.
    • \((1,1,1,1)\). Step 11.
    • Scalar shift. \(\sigma_*\mathrm{IC}_0=\mathrm{IC}_{(1,1,1,1)}\) up to shift, a summand of \(V_{(1,1,0,0)}\ast\mathrm{IC}_0\). As \(\sigma\) is \(LG\)-equivariant, \(\sigma_*(V_\lambda\ast\mathrm{IC}_0)\simeq V_\lambda\ast\sigma_*\mathrm{IC}_0\) is a retract of \(V_\lambda\ast V_{(1,1,0,0)}\ast\mathrm{IC}_0\), a finite sum of shifted Satake objects.
    • All odd walls. \(\sigma_*^2=V_{(1,1,1,1)}\ast(-)\) is invertible, so \(\sigma_*^{\pm1}\) preserve \(\mathcal C_K^{\mathrm{Sat}}\), and \(\sigma_*^{c-1}\mathrm{IC}_{(1,1,1,1)}=\mathrm{IC}_{(c,c,c,c)}\) for odd \(c\).
13. Conclusion Both levels hold although H5 fails

    (a) is steps 6–7, (b) step 8, (c) steps 4 and 11 with \(V_{(1,1,0,0)}\ast\mathrm{IC}_0=P[4-D]\), (d) step 2:

    $$\text{(SG) over every }F,\qquad\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\ \text{over}\ \overline{\mathbb F}_q,\qquad\text{H5 fails for }m\ge2,\ n\ge3 .$$
    • On these rows of the verdict's table the two levels agree; the function side needs no (A1)–(A5).
    • The sheaf side is \(\ell\)-adic over \(\overline{\mathbb F}_q\), \(p\ne2\); the Betti statement of the pages needs D5.