Summary · Iwahori generation

1 · The statements

$$\text{sheaves:}\qquad\mathcal C^{\mathrm{Sat}}\overset{?}{=}\mathcal C=\mathcal D_c(K\backslash LX),\qquad\mathcal C_I^{\mathrm{Iwa}}:=\operatorname{Thick}_{\mathcal C_I}\{\mathcal G\ast_I\mathrm{IC}_0:\mathcal G\in\mathcal H_{c,I}\}\overset{?}{=}\mathcal C_I=\mathcal D_c(I\backslash LX);$$ $$\text{functions:}\qquad\text{(SG)}\quad\mathcal H_K\cdot\varphi_0\overset{?}{=}C_K,\qquad\text{(IG)}\quad\mathcal H_I\cdot\varphi_0\overset{?}{=}C_I,\qquad Q_X:=C_I\big/\mathcal H_I\cdot C_K .$$

Sheaves. Spherical generation is the spherical workspace (H1–H5; two root-datum conditions for symmetric $X$). Proposition I: $\mathcal C_I^{\mathrm{Iwa}}$ is the Iwahori span of the whole of $\mathcal C$ exactly under spherical generation, and then a relative tensor product. Proposition II: for the group it is a proper two-sided ideal, detected by Whittaker averaging. Proposition III: three level-wise conditions I1–I3 give the equality; Proposition IV checks them for $\mathbb A^n$, the group and $\mathbb G_m$.

Functions. On the Iwahori function pages: (IG) holds iff (SG) holds and $Q_X=0$ (Proposition B); an irreducible $\pi$ with $\pi^I\ne0=\pi^K$ and $\operatorname{Hom}_G(\mathcal S(X),\pi)\ne0$ forces $Q_X\ne0$ (Corollary D), and for split symmetric $X$ the converse holds by finite generation. (SG) at function level is decided on the spherical workspace's classical page: unconditionally for the group, $\mathbb A^n$, $\mathbb G_m$ and $\mathrm{GL}_2/T$, and for the $\mathrm{GL}$ families only under Sakellaridis's hypotheses (A1)–(A5), written “under (A)” below. No implication between the two levels is proved.

2 · Every variety, both levels

Table 1. · Each cell reads “spherical → Iwahori”. Sheaves: H1–H5 (or the level form), then $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$. Functions: (SG), then (IG). Letters refer to the classical page and F1–F3.
$X$	sheaves: SG → IG	functions: SG → IG	witness or reason
$H$ under $H\times H$, $W\ne1$	holds → false (II; IV.2)	holds → fails (E)	The sign character $\varepsilon$ of $\mathcal H_{I_H}$ kills $\mathcal H_{I_H}e_{K_H}\mathcal H_{I_H}$: the witness is $\mathrm{St}_H\boxtimes\widetilde{\mathrm{St}}_H$. At sheaf level I3 fails at the closed cell $I\omega I$, and the affine Whittaker averaging $\Phi$ detects the ideal.
$\mathbb A^n$ under $\mathrm{GL}_n$	holds (level form) → true, (IIa)	holds → holds (𝔸ⁿ pages)	The $I$-stable lattices $\Lambda_{m,j}$ form a chain and their indicators are translates of $\varphi_0$; the only quotients with Iwahori-fixed vectors come from the open orbit and do not extend over $0$.
$\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m$	holds → true (IV.3)	holds → holds	$I\backslash X(F)=K\backslash X(F)=F^\times/\mathcal O^\times$.
$\mathrm{GL}_2/T=\mathrm{PGL}_2/T$	holds → open	holds → fails (F)	$\mathrm{St}_2$ is $T$-distinguished (the difference of values at the two fixed points); Theorem A applies since $X$ is quasi-split.
$\mathrm{GL}_{2p}/(\mathrm{GL}_p\times\mathrm{GL}_p)$; $\mathrm{GL}_3/(\mathrm{GL}_1\times\mathrm{GL}_2)$	holds → open	under (A) → fails (F2, B)	$\mathrm{St}_{2p}$ has a Shalika model, hence is $M_{p,p}$-distinguished; also $\mathrm{St}_{2p}\otimes\chi_0$ by the Steinberg sweep. $\mathrm{St}_2\times\mathbf 1$ is $M_{2,1}$-distinguished (Matringe).
$\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$, $q>p$, $n\ge4$	holds → open	under (A) → open	No Steinberg twist and, for $q\ge p+2$, no generic representation is a witness; the candidate is $\mathrm{St}_{2p}\times\mathbf 1_{\mathrm{GL}_{q-p}}$ (F2, B(iii)).
$\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$, $n\ge2$	H5 fails → open	under (A) → fails, $\operatorname{char}F=0$ (G)	The Speh representation $U(\mathrm{St}_2,2)\times\mathbf 1_{\mathrm{GL}_2}^{\times(n-2)}$ is Iwahori-spherical, has no $K$-fixed vector and is $\mathrm{Sp}_{2n}$-distinguished (Offen–Sayag). At sheaf level $(1,1,1,1,0,\dots)$ has no dominant lift; the function level holds under (A) by the restriction map.
$\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$, $n\ge2$	$n=2$ holds; $n\ge3$ H5 fails → open	under (A) for $\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$ → fails, $\operatorname{char}F=0$ (F2, C)	The Speh witness restricted to $\mathrm{SL}_{2n}(F)$. (SG) for $\mathrm{SL}$ is the degree-$0$ part of (SG) for $\mathrm{GL}$.
$\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$	holds → open	$n=2$ holds, $n=3$ under (A), $n\ge4$ open → $n=2,3$ fail; $n\ge4$ open	$n=2$ is the group $\mathrm{SL}_2$, $n=3$ is $\mathrm{SL}_4/\mathrm{Sp}_4$. Not quasi-split for $n\ge3$, so the Steinberg sweep is silent; candidates are the constituents of $C^\infty(Q(F))/\mathbb C$ for the quadric $Q$ (F2, 2).
$E_6/F_4$	H5 fails → open	open → open	Not quasi-split ($L=S\cdot\mathrm{Spin}_8$); Steinberg silent; $r$ not computed; not in Sakellaridis's table.
$\mathrm{SL}_2/T$	H1–H5 fail → —	fails → fails (F2, D)	Square classes split the $K$-orbits (the parametrization of the orbits as $(d,k\bmod2,\epsilon)$ on F2 needs $k\equiv d\bmod2$, since $\mathrm{SL}_2(F)/K$ is the set of even vertices); the sign character is a direct summand of $\mathcal S(X)^I$ (Shtotland, Proposition 9.1).
$M_n$ under $\mathrm{GL}_n\times\mathrm{GL}_n$, $n\ge2$	not on these pages	$n=2$ holds; $n\ge3$ open → fails (F2, E)	$\mathrm{St}_n\boxtimes\mathrm{St}_n$ through the Godement–Jacquet zeta integral. The (SG) argument of F2, E, inducts along infinite lower sets; the spherical workspace's Proposition H settles $n=2$ and leaves $n\ge3$ open.
$\mathrm{PGL}_2/T_u$, $T_u$ the unramified torus	not split; not on these pages	holds → holds (F2, 3)	$T_u\backslash G/I=K\backslash G/I$: $C_c(T_u\backslash G)^I\simeq\mathbf 1_K\mathcal H_I$ with $\mathbf 1_{T_uK}\mapsto\mathbf 1_K$.

Where F2's table writes “holds (paper)” in an SG column, it means the sheaf level; the function-level (SG) of the $\mathrm{GL}$ families is now Proposition F of the spherical workspace, under (A). Pattern: among the homogeneous families, (IG) fails wherever it is decided, by a Steinberg (quasi-split $X$), a Speh (symplectic $X$) or an induced-Steinberg witness, and the only positive homogeneous case is the torus-like $\mathbb G_m$; $\mathbb A^n$ is positive because its Iwahori-spherical quotients do not extend over the origin, while $M_n$ keeps the group's Steinberg quotient.

3 · Cited theorems and their hypotheses

Table 2. · What each citation is used for, the hypotheses it carries, and whether they are verified or assumed on these pages.
result	used for	hypotheses there	here
Shtotland, Distinction of the Steinberg representation and the dual group of a symmetric space, arXiv:2501.01701v2: Theorem 1.5, Remark 1.6, Proposition 2.7	Theorem A, the Steinberg sweep (Tool 2)	$G$ split, $\sigma$ an involution over $\mathcal O$, $H=G^\sigma$ split, residue characteristic $\ne2$, $\operatorname{char}F=0$; $X$ quasi-split ($L=Z_G(S)$ a torus); no simple factor $\mathrm{PGL}_{2n+1}/P(\mathrm{GL}_n\times\mathrm{GL}_{n+1})$	Assumed as stated; quasi-splitness and the exceptional factor are checked family by family on F1 and F2.
Shtotland, Iwahori Matsumoto presentation for modules of Iwahori fixed functions on symmetric spaces, arXiv:2406.16070v3 (IMRN 2026): Theorem 5.9, Propositions 9.1–9.3	Finite generation of $C_I$ (converse in Tool 1, Corollary D); the $T\backslash\mathrm{SL}_2$ and $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$ modules	$G$ split, $H=G^\sigma$, residue characteristic $\ne2$; Proposition 9.2 stated for $q\equiv3\bmod4$	Assumed as stated; numbering checked on F1.
Offen–Sayag (J. Number Theory 2007, Theorem 1; IMRN 2008, Theorem 8, Corollary 2; J. Funct. Anal. 2008, Theorem 1), Heumos–Rallis (Theorems 2.4.2, 11.1), Tadić (Theorem D, Theorem A.8)	Speh witnesses for $\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$ (Proposition G, Tool 4, F2 Theorem C)	$\operatorname{char}F=0$; Heumos–Rallis's Theorems 3.1 and 3.2.2 rest on a false proposition of Klyachko, corrected by Offen–Sayag	Assumed: every symplectic failure carries “$\operatorname{char}F=0$”.
Matringe, arXiv:1210.4307 (J. Number Theory 2014): Theorems 3.1, 3.2, 6.1, Proposition 6.1; Prasad, Venketasubramanian for $\mathrm{St}_2\times\mathbf 1$	Tool 3; F2, Proposition B	$G=\mathrm{GL}_n$ over a $p$-adic field	Assumed as stated.
Godement–Jacquet, LNM 260 (zeta integrals); Borel, Invent. Math. 35 (1976) (Theorem 4.10, Corollary 4.11, 3.5, 5.7); Casselman (Propositions 2.6–2.7); Iwahori–Matsumoto; Haines–Kottwitz–Prasad §4.1	F2 Proposition E; Iwahori-fixed vectors as $\mathcal H_I$-modules; the Steinberg sign line; the spherical idempotent	Split (or connected reductive) $G$ over a non-archimedean local field; no characteristic restriction	Verified as standing setting.
Sakellaridis, Spherical functions on spherical varieties (2013), Theorem 8.0.2, Corollary 8.0.4	(SG) at function level for the $\mathrm{GL}$ families (spherical workspace, Theorem E, Proposition F); every “under (A)” above	(A1)–(A5), listed there: $\operatorname{char}F=0$, $G$ split with simply connected derived group, Axioms 2.2.1, 2.4.1, 2.4.2, no type-$N$ roots, Statements 6.3.1 and 7.1.5, $X$ affine homogeneous with one open $B(F)$-orbit	Assumed, never verified. He asserts the hypotheses of his Theorem 7.2.1 for the rows of his table; (A5) follows from connected generic stabilizers and fails for $\mathrm{SL}_2/T$.
Sakellaridis–Venkatesh, Astérisque 396 (2017), §2.1; Ben-Zvi–Sakellaridis–Venkatesh, Relative Langlands duality (2024), Conjecture 7.5.1; Devalapurkar, Conjecture 3.4.14	The invariants $A_X$, $W_X$ (function level); the conjectures whose automorphic sides are $\mathcal C^{\mathrm{Sat}}$ and $\mathcal C_I^{\mathrm{Iwa}}$, the origin of both questions	SV: a spherical variety over an algebraically closed field of characteristic $0$. BZSV: $T^*X$ polarized hyperspherical ($X$ smooth affine spherical with connected generic $B$-stabilizers)	SV §2.1: verified (every $X$ here is spherical over $\mathbb C$). The conjectures are not used in any proof; $\mathrm{SL}_2/T$ and $\mathrm{PGL}_2/T_u$ lie outside the paper's class.
Chen–Yi, arXiv:2310.20006v4 (Proposition 17, Proposition 41, Theorem 87); the companion paper's level presentation (I1)	Orbits of $LX$ and spherical generation (via the spherical workspace); the finite levels of Proposition III	$X=G/G^\theta$ with $G^\theta$ connected over $\mathbb C$; the colimit description of $\mathcal D$-modules on placid ind-schemes	$G^\theta$ connected verified per family; I1 is assumed (Proposition III).

4 · Open

Sheaves. $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ for every symmetric family other than the group: deciding I3 needs the orbit graph of Chen–Yi's module, and the function-level failures suggest that no Hecke translate of $\mathrm{IC}_0$ is a nonzero constant sheaf on a closed $I$-orbit.
Functions. $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ with $q>p$, $n\ge4$: is $\mathrm{St}_{2p}\times\mathbf 1_{\mathrm{GL}_{q-p}}$ distinguished? The spheres with $n\ge4$ and $E_6/F_4$: compute $Q_X$ from Shtotland's presentation. The ramified torus $\mathrm{PGL}_2/T_r$.
Hypotheses. Every “under (A)” is conditional on Sakellaridis's (A1)–(A5); every symplectic failure on $\operatorname{char}F=0$.

\(X\)	sheaves: SG → IG	functions: SG → IG	witness or reason
\(H\) under \(H\times H\), \(W\ne1\)	holds → false (II; IV.2)	holds → fails (E)	The sign character \(\varepsilon\) of \(\mathcal H_{I_H}\) kills \(\mathcal H_{I_H}e_{K_H}\mathcal H_{I_H}\): the witness is \(\mathrm{St}_H\boxtimes\widetilde{\mathrm{St}}_H\). At sheaf level I3 fails at the closed cell \(I\omega I\), and the affine Whittaker averaging \(\Phi\) detects the ideal.
\(\mathbb A^n\) under \(\mathrm{GL}_n\)	holds (level form) → true, (IIa)	holds → holds (𝔸ⁿ pages)	The \(I\)-stable lattices \(\Lambda_{m,j}\) form a chain and their indicators are translates of \(\varphi_0\); the only quotients with Iwahori-fixed vectors come from the open orbit and do not extend over \(0\).
\(\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\)	holds → true (IV.3)	holds → holds	\(I\backslash X(F)=K\backslash X(F)=F^\times/\mathcal O^\times\).
\(\mathrm{GL}_2/T=\mathrm{PGL}_2/T\)	holds → open	holds → fails (F)	\(\mathrm{St}_2\) is \(T\)-distinguished (the difference of values at the two fixed points); Theorem A applies since \(X\) is quasi-split.
\(\mathrm{GL}_{2p}/(\mathrm{GL}_p\times\mathrm{GL}_p)\); \(\mathrm{GL}_3/(\mathrm{GL}_1\times\mathrm{GL}_2)\)	holds → open	under (A) → fails (F2, B)	\(\mathrm{St}_{2p}\) has a Shalika model, hence is \(M_{p,p}\)-distinguished; also \(\mathrm{St}_{2p}\otimes\chi_0\) by the Steinberg sweep. \(\mathrm{St}_2\times\mathbf 1\) is \(M_{2,1}\)-distinguished (Matringe).
\(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(q>p\), \(n\ge4\)	holds → open	under (A) → open	No Steinberg twist and, for \(q\ge p+2\), no generic representation is a witness; the candidate is \(\mathrm{St}_{2p}\times\mathbf 1_{\mathrm{GL}_{q-p}}\) (F2, B(iii)).
\(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge2\)	H5 fails → open	under (A) → fails, \(\operatorname{char}F=0\) (G)	The Speh representation \(U(\mathrm{St}_2,2)\times\mathbf 1_{\mathrm{GL}_2}^{\times(n-2)}\) is Iwahori-spherical, has no \(K\)-fixed vector and is \(\mathrm{Sp}_{2n}\)-distinguished (Offen–Sayag). At sheaf level \((1,1,1,1,0,\dots)\) has no dominant lift; the function level holds under (A) by the restriction map.
\(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge2\)	\(n=2\) holds; \(n\ge3\) H5 fails → open	under (A) for \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\) → fails, \(\operatorname{char}F=0\) (F2, C)	The Speh witness restricted to \(\mathrm{SL}_{2n}(F)\). (SG) for \(\mathrm{SL}\) is the degree-\(0\) part of (SG) for \(\mathrm{GL}\).
\(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\)	holds → open	\(n=2\) holds, \(n=3\) under (A), \(n\ge4\) open → \(n=2,3\) fail; \(n\ge4\) open	\(n=2\) is the group \(\mathrm{SL}_2\), \(n=3\) is \(\mathrm{SL}_4/\mathrm{Sp}_4\). Not quasi-split for \(n\ge3\), so the Steinberg sweep is silent; candidates are the constituents of \(C^\infty(Q(F))/\mathbb C\) for the quadric \(Q\) (F2, 2).
\(E_6/F_4\)	H5 fails → open	open → open	Not quasi-split (\(L=S\cdot\mathrm{Spin}_8\)); Steinberg silent; \(r\) not computed; not in Sakellaridis's table.
\(\mathrm{SL}_2/T\)	H1–H5 fail → —	fails → fails (F2, D)	Square classes split the \(K\)-orbits (the parametrization of the orbits as \((d,k\bmod2,\epsilon)\) on F2 needs \(k\equiv d\bmod2\), since \(\mathrm{SL}_2(F)/K\) is the set of even vertices); the sign character is a direct summand of \(\mathcal S(X)^I\) (Shtotland, Proposition 9.1).
\(M_n\) under \(\mathrm{GL}_n\times\mathrm{GL}_n\), \(n\ge2\)	not on these pages	\(n=2\) holds; \(n\ge3\) open → fails (F2, E)	\(\mathrm{St}_n\boxtimes\mathrm{St}_n\) through the Godement–Jacquet zeta integral. The (SG) argument of F2, E, inducts along infinite lower sets; the spherical workspace's Proposition H settles \(n=2\) and leaves \(n\ge3\) open.
\(\mathrm{PGL}_2/T_u\), \(T_u\) the unramified torus	not split; not on these pages	holds → holds (F2, 3)	\(T_u\backslash G/I=K\backslash G/I\): \(C_c(T_u\backslash G)^I\simeq\mathbf 1_K\mathcal H_I\) with \(\mathbf 1_{T_uK}\mapsto\mathbf 1_K\).

result	used for	hypotheses there	here
Shtotland, Distinction of the Steinberg representation and the dual group of a symmetric space, arXiv:2501.01701v2: Theorem 1.5, Remark 1.6, Proposition 2.7	Theorem A, the Steinberg sweep (Tool 2)	\(G\) split, \(\sigma\) an involution over \(\mathcal O\), \(H=G^\sigma\) split, residue characteristic \(\ne2\), \(\operatorname{char}F=0\); \(X\) quasi-split (\(L=Z_G(S)\) a torus); no simple factor \(\mathrm{PGL}_{2n+1}/P(\mathrm{GL}_n\times\mathrm{GL}_{n+1})\)	Assumed as stated; quasi-splitness and the exceptional factor are checked family by family on F1 and F2.
Shtotland, Iwahori Matsumoto presentation for modules of Iwahori fixed functions on symmetric spaces, arXiv:2406.16070v3 (IMRN 2026): Theorem 5.9, Propositions 9.1–9.3	Finite generation of \(C_I\) (converse in Tool 1, Corollary D); the \(T\backslash\mathrm{SL}_2\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) modules	\(G\) split, \(H=G^\sigma\), residue characteristic \(\ne2\); Proposition 9.2 stated for \(q\equiv3\bmod4\)	Assumed as stated; numbering checked on F1.
Offen–Sayag (J. Number Theory 2007, Theorem 1; IMRN 2008, Theorem 8, Corollary 2; J. Funct. Anal. 2008, Theorem 1), Heumos–Rallis (Theorems 2.4.2, 11.1), Tadić (Theorem D, Theorem A.8)	Speh witnesses for \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\) (Proposition G, Tool 4, F2 Theorem C)	\(\operatorname{char}F=0\); Heumos–Rallis's Theorems 3.1 and 3.2.2 rest on a false proposition of Klyachko, corrected by Offen–Sayag	Assumed: every symplectic failure carries “\(\operatorname{char}F=0\)”.
Matringe, arXiv:1210.4307 (J. Number Theory 2014): Theorems 3.1, 3.2, 6.1, Proposition 6.1; Prasad, Venketasubramanian for \(\mathrm{St}_2\times\mathbf 1\)	Tool 3; F2, Proposition B	\(G=\mathrm{GL}_n\) over a \(p\)-adic field	Assumed as stated.
Godement–Jacquet, LNM 260 (zeta integrals); Borel, Invent. Math. 35 (1976) (Theorem 4.10, Corollary 4.11, 3.5, 5.7); Casselman (Propositions 2.6–2.7); Iwahori–Matsumoto; Haines–Kottwitz–Prasad §4.1	F2 Proposition E; Iwahori-fixed vectors as \(\mathcal H_I\)-modules; the Steinberg sign line; the spherical idempotent	Split (or connected reductive) \(G\) over a non-archimedean local field; no characteristic restriction	Verified as standing setting.
Sakellaridis, Spherical functions on spherical varieties (2013), Theorem 8.0.2, Corollary 8.0.4	(SG) at function level for the \(\mathrm{GL}\) families (spherical workspace, Theorem E, Proposition F); every “under (A)” above	(A1)–(A5), listed there: \(\operatorname{char}F=0\), \(G\) split with simply connected derived group, Axioms 2.2.1, 2.4.1, 2.4.2, no type-\(N\) roots, Statements 6.3.1 and 7.1.5, \(X\) affine homogeneous with one open \(B(F)\)-orbit	Assumed, never verified. He asserts the hypotheses of his Theorem 7.2.1 for the rows of his table; (A5) follows from connected generic stabilizers and fails for \(\mathrm{SL}_2/T\).
Sakellaridis–Venkatesh, Astérisque 396 (2017), §2.1; Ben-Zvi–Sakellaridis–Venkatesh, Relative Langlands duality (2024), Conjecture 7.5.1; Devalapurkar, Conjecture 3.4.14	The invariants \(A_X\), \(W_X\) (function level); the conjectures whose automorphic sides are \(\mathcal C^{\mathrm{Sat}}\) and \(\mathcal C_I^{\mathrm{Iwa}}\), the origin of both questions	SV: a spherical variety over an algebraically closed field of characteristic \(0\). BZSV: \(T^*X\) polarized hyperspherical (\(X\) smooth affine spherical with connected generic \(B\)-stabilizers)	SV §2.1: verified (every \(X\) here is spherical over \(\mathbb C\)). The conjectures are not used in any proof; \(\mathrm{SL}_2/T\) and \(\mathrm{PGL}_2/T_u\) lie outside the paper's class.
Chen–Yi, arXiv:2310.20006v4 (Proposition 17, Proposition 41, Theorem 87); the companion paper's level presentation (I1)	Orbits of \(LX\) and spherical generation (via the spherical workspace); the finite levels of Proposition III	\(X=G/G^\theta\) with \(G^\theta\) connected over \(\mathbb C\); the colimit description of \(\mathcal D\)-modules on placid ind-schemes	\(G^\theta\) connected verified per family; I1 is assumed (Proposition III).