A lattice in $\R^4$ is a discrete subgroup of rank four. Two lattices that differ by positive real scaling have the same shape. Normalising covolume to one, every lattice has the form $\Lambda=\Z^4 g$ for some $g\in\SL_4(\R)$, and $\Z^4 g=\Z^4 g'$ if and only if $g'=\gamma g$ with $\gamma\in\SL_4(\Z)$. The quotient
$$
X_4=\SL_4(\Z)\backslash\SL_4(\R)
$$
is the space of covolume-one lattices in $\R^4$, equipped with its normalised Haar probability $m_4$. The group
$$
A_4=\{\diag(e^{u_1},e^{u_2},e^{u_3},e^{u_4}):u_1+u_2+u_3+u_4=0\}
$$
of positive diagonal matrices in $\SL_4(\R)$ acts on $X_4$ by right multiplication and preserves $m_4$.

A quartic field $K$ is primitive if no field lies strictly between $\Q$ and $K$; over $\Q$ this condition means that the Galois closure of $K$ has group $\mathfrak A_4$ or $S_4$. Suppose $K$ is primitive and totally real. An order $\OO\subseteq\OO_K$ is a full-rank subring containing $1$. A fractional $\OO$-ideal $I$ is proper if the multiplier order $\{a\in K:aI\subseteq I\}$ equals $\OO$, and invertible if $IJ=\OO$ for some fractional $\OO$-ideal $J$. The Picard group $\Pic(\OO)$ consists of proper invertible fractional $\OO$-ideals modulo $I\sim aI$ with $a\in K^\times$. Fix an ordering $\sigma=(\sigma_1,\ldots,\sigma_4)$ of the real embeddings, choose $I$ in a system of class representatives, and put
$$
\Lambda_{I,\sigma}=\covol(\sigma(I))^{-1/4}\sigma(I)\in X_4.
$$
Let $W_{\mathrm{all}}$ be the sixteen diagonal matrices in $\{\pm1\}^4$. The packet of $\OO$ is the finite orbit collection
$$
\mathcal P_{\OO,\sigma}=\{\Lambda_{I,\sigma}wA_4:I\text{ a class representative},\ w\in W_{\mathrm{all}}\}.
$$
If $\nu_{\mathcal Y}$ is the $A_4$-invariant probability on an orbit $\mathcal Y$ and $\vol(\mathcal Y)$ is its $A_4$-Haar volume, the packet measure is
$$
\mu_{\OO,\sigma}=\frac{\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\vol(\mathcal Y)\nu_{\mathcal Y}}{\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\vol(\mathcal Y)}.
$$

Main theorem. For every sequence of triples $(K_i,\OO_i,\sigma_i)$ with $K_i$ primitive and totally real quartic, $\OO_i$ an order in $K_i$, $\sigma_i$ an ordering of its real embeddings, and $|\Disc(\OO_i)|\to\infty$,
$$
\int_{X_4}f\,d\mu_{\OO_i,\sigma_i}\longrightarrow\int_{X_4}f\,dm_4
\qquad(f\in C_c(X_4)),
$$
and for every $\eps>0$ there is a compact $\mathcal K\subset X_4$ with $\mu_{\OO_i,\sigma_i}(\mathcal K)\ge1-\eps$ for all sufficiently large $i$.

The field may remain fixed while the order index grows. In that regime the whole difficulty concentrates on uniform control of the local lattice at the ramified primes.

Notation and conventions. Set $R=\OO_K$, $D=|\Disc(K)|$, $q=[R:\OO]$, $X=q\sqrt D=\sqrt{|\Disc(\OO)|}$, $\kappa=\Res_{s=1}\zeta_K(s)$, and let $a_K(n)$ be the number of nonzero integral $R$-ideals of norm $n$. The symbol $\Nm_{E/k}$ is the field norm; ideal norms $\Nm(\mathfrak a)$ are positive absolute norms extended multiplicatively to fractional ideals. For a rational prime $p$, a subscript $p$ means tensoring a field with $\Q_p$, a module with $\Z_p$, or a lattice with $\Z_p$. At a prime $\mathfrak p$ or $\mathfrak q$ of a local field, $v_{\mathfrak p}$ and $v_{\mathfrak q}$ are integral valuations taking value $1$ on a uniformiser, and $e$, $f$ are the associated ramification index and residue degree. A normalised extension of $v_p$ has $v_p(p)=1$ and $|p|_p=p^{-1}$. All logarithms are natural, $\log_+t=\max(0,\log t)$, and $X^{o(1)}$ means $O_\eta(X^\eta)$ for every fixed $\eta>0$. Constants are uniform in $K$, $\OO$, and $\sigma$ unless stated otherwise.

Structure of the argument. Ordinary-vector unfolding expresses the packet's Borel sums as a weighted sum over integral $R$-ideals and produces tightness together with the local ball bound $\mu(xB(r))\le C_\Omega r^4$. Entropy propagation upgrades the ball bound to a uniform positive lower bound on the $A_4$-entropy of every ergodic component, and the Einsiedler--Katok--Lindenstrauss measure classification makes each such component homogeneous; a finite-volume argument reduces the possible closed connected subgroup $J\subseteq\SL_4(\R)$ containing $A_4$ to $\SL_4(\R)$ or a two-block $2+2$ subgroup. Each $2+2$ orbit forces every lattice in it to carry an exterior-square vector in one of three fixed two-planes $V_b\subseteq\bigwedge^2\R^4$. A cubic-resolvent count in the exterior square of the order gives the rectangle bound $\ll\kappa\sqrt D\,\operatorname{area}(\mathcal R)$, uniform in the order index. Unfolding the diagonal flow against that count kills every $2+2$ alternative.

pdf/ekl2006.pdf

Equidistribution of periodic torus packets grows out of Linnik's ergodic method. Skubenko proved the corresponding distribution for positive discriminant at a fixed quadratic-residue prime, Duke removed the residue condition using half-integral weight forms, and Chelluri, Einsiedler--Lindenstrauss--Michel--Venkatesh and others extended the result to all positive nonsquare order discriminants. In higher rank Einsiedler--Lindenstrauss--Michel--Venkatesh built the discriminant-volume framework relating separation of periodic torus orbits to entropy and proved their cubic theorem. In degree four Khayutin obtained entropy bounds for maximal-order packets with the primitive two-transitive Galois groups $\mathfrak A_4$ and $S_4$, and Wieser--Yang extended this to certain maximal-order quartic packets with a quadratic subfield. For a fixed prime degree $p\ge5$, Lemke Oliver--Thorner--Zaman proved equidistribution for the packet collections attached to orders in totally real degree-$p$ fields outside an exceptional set, and a separate prime-degree treatment covers every packet in every such degree with arbitrary local types.

pdf/elmv2009.pdf
pdf/elmv2011.pdf
pdf/khayutin2019.pdf
pdf/lotz2024.pdf

Excluding intermediate limits by sieve-theoretic input has a precedent in Khayutin's joint equidistribution of CM points, and local Fourier analysis of nonmaximal orders appeared in the cubic work of Einsiedler--Lindenstrauss--Michel--Venkatesh. In composite degree four a genuinely new obstruction appears, because measure classification permits proper positive-entropy homogeneous limits of type $2+2$. The cubic-resolvent count in $\bigwedge^2\OO$ handles that obstruction without any hypothesis on the order.

./pdf/khayutin2019-cm.pdf

The packet construction has three stages: a finite set of lattices attached to each ideal class of $\OO_K$, a real parameter in the unit torus, and a normalising scalar making covolumes equal to one. Wholly absorbed in the first stage is the order-index dependence, and it is exactly what makes the packet's arithmetic visible at the ramified primes.

The finite local choices. Pick one fractional $R$-ideal $I$ in each ordinary ideal class of $R$. For each rational prime $p$, choose a generator $I_p=\xi_pR_p$ with $\xi_p\in K_p^\times$, possible because each local factor of $R_p$ is a discrete valuation ring. Let $\mathscr L_{\OO}(I)$ be the set of full lattices $L\subseteq I$ whose completions satisfy
$$
L_p=\xi_p t_p\OO_p,\qquad t_p\in R_p^\times,\qquad\text{for every prime }p.
$$
Replacing $\xi_p$ by another generator changes $t_p$ by a unit, so the set does not depend on the choice.

The set $\mathscr L_{\OO}(I)$ is finite, with cardinality
$$
N_{\OO}:=\#\mathscr L_{\OO}(I)=\prod_{p\mid q}[R_p^\times:\OO_p^\times],
$$
independent of $I$. Every $L\in\mathscr L_{\OO}(I)$ satisfies $[I:L]=q$ and $RL=I$, and is a proper invertible fractional $\OO$-ideal. Every ordinary class of proper invertible fractional $\OO$-ideals is representable, up to scalar multiplication by an element of $K^\times$, by some member of some $\mathscr L_{\OO}(I)$. Uniform probability on $\mathscr L_{\OO}(I)$ is induced from Haar probability on $R_p^\times$ for each $p\mid q$.

For finiteness, observe that $[R:\OO]=q$ gives $qR\subseteq\OO$, so for every $p$ and every $t_p\in R_p^\times$,
$$
qI_p=\xi_p t_p qR_p\subseteq\xi_p t_p\OO_p\subseteq I_p.
$$
At $p\nmid q$ the middle lattice equals $I_p$. At primes above $q$, the image of $\xi_p t_p\OO_p$ is a submodule of the finite module $I_p/qI_p$. Primary decomposition of $I/qI$ identifies its $p$-primary part with $I_p/qI_p$, so local choices patch to a unique subgroup of $I/qI$ whose inverse image is the unique lattice with those completions. This couples the sets $\mathscr L_{\OO}(I)$ bijectively with $\prod_{p\mid q}(R_p^\times/\OO_p^\times)$.

The stabiliser of $\OO_p$ under left multiplication by $R_p^\times$ is $\OO_p^\times$: from $t_p\OO_p=\OO_p$ and its inverse applied to $1$, both $t_p$ and $t_p^{-1}$ lie in $\OO_p$. Uniform probability on $\mathscr L_{\OO}(I)$ is therefore induced by Haar probability on $R_p^\times$ for each $p\mid q$. The index of the displayed lattice is $[R_p:\OO_p]$, and its $R_p$-span is $I_p$ because $1\in\OO_p$; multiplying local indices and comparing completions gives $[I:L]=q$ and $RL=I$.

For invertibility, a full fractional $\OO$-ideal $J$ is invertible if and only if $J_p=\eta_p\OO_p$ for some $\eta_p\in K_p^\times$ at every prime. Local-global principality is the reason: in one direction write $JJ^{-1}=\OO$ and pick one generator over the semilocal ring $\OO_p$ using the Chinese remainder theorem; in the other direction use $J^{-1}=\{a\in K:aJ\subseteq\OO\}$ and $(J^{-1})_p=\eta_p^{-1}\OO_p$. The lattices $L\in\mathscr L_{\OO}(I)$ are stable under $\OO_p$ at every $p$, so are fractional $\OO$-ideals satisfying local principality, hence invertible, and their multiplier order is $\OO$: from $aJ\subseteq J$ for invertible $J$, multiplying by $J^{-1}$ gives $a\in\OO$.

In the other direction, let $J$ be a proper invertible fractional $\OO$-ideal. Its $R$-span $RJ$ is a fractional $R$-ideal, and after scalar multiplication by an element of $K^\times$ one may assume $RJ=I$ for the chosen representative. If $J_p=\eta_p\OO_p$, then $\eta_p R_p=I_p=\xi_pR_p$, so $\eta_p=\xi_p t_p$ for some $t_p\in R_p^\times$. Thus $J\in\mathscr L_{\OO}(I)$.

Real parameters and volume. Let $U=R^\times$ and $U^+\subseteq U$ be its totally positive subgroup. Put
$$
T_\infty=\{v=(v_1,\ldots,v_4)\in(\R^\times)^4:\prod_i|v_i|=1\}.
$$
The measure $dv$ on $T_\infty$ is counting measure on the sixteen sign components times Lebesgue measure on the logarithm coordinates $u_i=\log|v_i|$, with $u_4=-u_1-u_2-u_3$. By the unit theorem the image
$$
\ell(U)=\{(\log|\sigma_1(\epsilon)|,\ldots,\log|\sigma_4(\epsilon)|):\epsilon\in U\}
$$
is a full lattice in the hyperplane $\sum u_i=0$; its covolume for the coordinate measure from the first three logarithmic coordinates is the regulator $R_K$, and its kernel on $U$ is $\{1,-1\}$. Let $\mathscr F_U$ be a measurable fundamental domain for $\sigma(U)$ in $T_\infty$. Reducing logarithms modulo $\ell(U)$ leaves sixteen sign choices, and $\{1,-1\}$ pairs opposite signs. The analytic class-number formula in signature $(4,0)$, with two roots of unity, gives
$$
h_K\vol(\mathscr F_U)=8h_KR_K=\sqrt D\,\kappa.
$$

For $L\in\mathscr L_{\OO}(I)$, set $c_I=(X\Nm I)^{-1/4}$. The covolume formula $\covol(\sigma(I))=\sqrt D\,\Nm I$ together with $[I:L]=q$ gives covolume one for $c_I\sigma(L)v$ for every $v\in T_\infty$, and any positively oriented basis produces a point of $X_4$.

Packet model. The map
$$
(L,v)\longmapsto c_I\sigma(L)v,\qquad L\in\mathscr L_{\OO}(I),\ v\in T_\infty,
$$
descends to a bijection
$$
\Bigl(\coprod_I\mathscr L_{\OO}(I)\times T_\infty\Bigr)\big/U\xrightarrow{\ \sim\ }\bigcup_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\mathcal Y\subseteq X_4
$$
of the disjoint union modulo the $U$-action
$$
\epsilon\cdot(L,v)=(\epsilon L,\sigma(\epsilon)^{-1}v),\qquad\epsilon\in U,
$$
onto the union of packet orbits. Counting measure on the labels times $dv$ descends to the sum of the diagonal Haar measures of those orbits. In particular
$$
\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\vol(\mathcal Y)=N_{\OO}\sqrt D\,\kappa.
$$
With $\mu_*=\mu_{\OO,\sigma}$, every nonnegative Borel $\Phi$ on $X_4$ satisfies
$$
\int_{X_4}\Phi\,d\mu_*=\frac{1}{h_KN_{\OO}\vol(\mathscr F_U)}\sum_I\sum_{L\in\mathscr L_{\OO}(I)}\int_{\mathscr F_U}\Phi(c_I\sigma(L)v)\,dv,
$$
with both sides possibly infinite.

The $U$-action above preserves the parameter set, since $\epsilon I=I$ for $\epsilon\in U$, and preserves the image lattice. Surjectivity follows because every ordinary class occurs in some $\mathscr L_{\OO}(I)$, and because two ideals differing by multiplication by $\alpha\in K^\times$ produce normalised embedded lattices differing by the diagonal $\sigma_i(\alpha)/|\Nm_{K/\Q}(\alpha)|^{1/4}$, an element of $T_\infty$; the union of the sixteen sign translates of $A_4$ fills $T_\infty$. Injectivity follows from the scalar ratio: if two displayed normalised lattices coincide, absorbing the positive scalar produces a diagonal map carrying $\sigma(L)$ onto $\sigma(L')$. Pick $0\ne b\in L$ with image $b'\in L'$. All four coordinates of $\sigma(b)$ are nonzero, so the diagonal entries equal $\sigma_i(b')/\sigma_i(b)=\sigma_i(b'/b)$. The map is multiplication by $\alpha=b'/b$, and $L'=\alpha L$. Taking $R$-spans gives $I'=\alpha I$; the chosen representatives force $I'=I$ and $\alpha I=I$, so $\alpha\in U$. The scalar normalisations coincide, and the two parameters differ by the $U$-action above.

For the volume formula, the packet contains $16N_{\OO}$ finite choices above each $I$ (label times sign pattern). Each $U$-orbit has $[U:\OO^{\times,+}]$ elements, where $\OO^{\times,+}=\OO^\times\cap U^+$, and parametrises one $A_4$-orbit. The stabiliser of a label $L$ under $U$ is $\OO^\times$, since $L$ has rank four; the stabiliser of a sign-patterned label is $\OO^{\times,+}$. The periods of the resulting $A_4$-orbit are $\ell(\OO^{\times,+})$, a full lattice, so the orbit is compact, with Haar volume
$$
\covol(\ell(\OO^{\times,+}))=[\ell(U):\ell(\OO^{\times,+})]R_K=\frac{[U:\OO^{\times,+}]}{2}R_K,
$$
using $\ker(\ell|_U)=\{1,-1\}$ and $-1\notin\OO^{\times,+}$. The number of distinct orbits times this common volume equals
$$
\frac{16h_KN_{\OO}}{[U:\OO^{\times,+}]}\cdot\frac{[U:\OO^{\times,+}]}{2}R_K=8h_KN_{\OO}R_K=N_{\OO}\sqrt D\,\kappa,
$$
the last equality by the class-number formula.

For the integration formula, $\mathscr L_{\OO}(I)\times\mathscr F_U$ is a fundamental domain for the diagonal $U$-action: reducing $v$ into $\mathscr F_U$ transports $L$ by the same unique unit, away from a null boundary set. These domains have total measure $h_KN_{\OO}\vol(\mathscr F_U)$, matching the total volume computed above; dividing by that quantity gives the stated formula.

Ordinary vectors and probability limits.

The packet measure weights lattice vectors by a local container probability at the primes dividing $q$. Two objectives drive what follows: express the Borel sums of the packet as a clean count over integral $R$-ideals, and extract from that count two geometric bounds - tightness of the packet family in $X_4$ at scale $r^4$ in the cusp, and a local ball bound $\mu(xB(r))\le C_\Omega r^4$ at shrinking radius $r$. Only the rank-one Grassmannian of the packet orbits enters this part of the argument. The exterior-square analysis is confined to the later part of the argument.

Containment probabilities. For a rational prime $p$, write $R_p=\prod_{\mathfrak p\mid p}R_{\mathfrak p}$ with $R_{\mathfrak p}$ the integer ring of the local completion $K_{\mathfrak p}$. Then $v_{\mathfrak p}$ is the integer valuation normalised to be $1$ on a uniformiser, $e_{\mathfrak p}=v_{\mathfrak p}(p)$, $f_{\mathfrak p}=[R/\mathfrak p:\Z/p\Z]$, so $\Nm\mathfrak p=p^{f_{\mathfrak p}}$ and $\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}=4$.

For $p\mid q$, let $\boldsymbol l_p=(l_{\mathfrak p})_{\mathfrak p\mid p}$ be a tuple of nonnegative integers. The valuation shell of this tuple is the set of $x\in R_p$ with $v_{\mathfrak p}(x_{\mathfrak p})=l_{\mathfrak p}$ in every component. Pick one $x$ in this shell and let
$$
P_p(\boldsymbol l_p):=\text{prob}\{t_p\in R_p^\times:x\in t_p\OO_p\}
$$
denote the Haar probability, over a uniformly distributed unit $t_p\in R_p^\times$, that $x\in t_p\OO_p$. The value does not depend on the choice of $x$, because $R_p^\times$ acts transitively on the shell.

For a nonzero integral $R$-ideal $\mathfrak a$, put
$$
P(\mathfrak a):=\prod_{p\mid q}P_p\bigl((v_{\mathfrak p}(\mathfrak a))_{\mathfrak p\mid p}\bigr),
$$
the empty product being one. The packet model together with independence of the local unit choices gives the fundamental containment identity
$$
\text{so the fraction of }L\in\mathscr L_{\OO}(I)\text{ with }b\in L\text{ equals }P((b)I^{-1}),\qquad0\ne b\in I.
$$

For a nonnegative Borel $f$ on $\R^4$ write
$$
E_f(\Lambda)=\sum_{0\ne x\in\Lambda}f(x).
$$
The sum is Borel on $X_4$: in a local basis chart it is a countable sum of Borel functions.

Ordinary unfolding. For every nonnegative Borel $f:\R^4\to[0,\infty]$,
$$
\int_{X_4}E_f\,d\mu_*=\frac{1}{\sqrt D\,\kappa}\sum_{\substack{\mathfrak a\subseteq R\\\mathfrak a\ne0}}P(\mathfrak a)\,\mathcal V_f\bigl(\Nm\mathfrak a/X\bigr),
$$
where
$$
\mathcal V_f(z)=\sum_{\omega\in\{\pm1\}^4}\int_{\R^3}f(\omega_1e^{u_1},\omega_2e^{u_2},\omega_3e^{u_3},\omega_4ze^{-u_1-u_2-u_3})\,du_1du_2du_3
$$
is the integral of $f$ over the surface of absolute coordinate product $z>0$, taken with respect to logarithmic measure. Both sides may be infinite.

By the packet integration formula, averaging $E_f$ over $L\in\mathscr L_{\OO}(I)$ cancels the $N_{\OO}$ factor between the packet normalisation and the uniform measure on $\mathscr L_{\OO}(I)$. The containment identity then converts the left-hand side of the display into
$$
\frac{1}{h_K\vol(\mathscr F_U)}\sum_I\int_{\mathscr F_U}\sum_{0\ne b\in I}P((b)I^{-1})f(c_I\sigma(b)v)\,dv.
$$
Fix $I$. The map $b\mapsto(b)I^{-1}$ sends $I\setminus\{0\}$ bijectively onto the set of integral ideals in the inverse class of $I$, and each fibre is a single free $U$-orbit: two generators of the same principal ideal differ by a unit. The weight $P((b)I^{-1})$ is constant on the orbit, and integrating over $U$ against the fundamental domain $\mathscr F_U$ gives
$$
\int_{\mathscr F_U}\sum_{\epsilon\in U}f(c_I\sigma(\epsilon b)v)\,dv=\int_{T_\infty}f(c_I\sigma(b)v)\,dv.
$$
For $b\in I$,
$$
|{\textstyle\prod_i}(c_I\sigma(b))_i|=c_I^4|\Nm_{K/\Q}b|=\Nm((b)I^{-1})/X.
$$
Combining $|{\textstyle\prod_i}v_i|=1$ with a translation and sign relabelling identifies the last integral with $\mathcal V_f(\Nm((b)I^{-1})/X)$, Jacobian one. Each integral ideal lies in the inverse class of exactly one representative, and summing gives the stated identity.

Mass and moment of the local weights. The total mass and a positive moment of the local weight $P(\mathfrak a)$ both come from a shell decomposition.

For $p\mid q$ write
$$
q_p=[R_p:\OO_p],\qquad N_p(\boldsymbol l_p)=p^{\sum_{\mathfrak p\mid p}f_{\mathfrak p}l_{\mathfrak p}},\qquad Z_p=\prod_{\mathfrak p\mid p}(1-p^{-f_{\mathfrak p}})^{-1}.
$$
For a collection $\boldsymbol l=(\boldsymbol l_p)_{p\mid q}$ write
$$
d(\boldsymbol l)=\prod_{p\mid q}N_p(\boldsymbol l_p),\quad P(\boldsymbol l)=\prod_{p\mid q}P_p(\boldsymbol l_p),\quad Z_q=\prod_{p\mid q}Z_p.
$$
Thus $d(\boldsymbol l)$ is the norm of the part of the ideal supported above $q$, and $\prod_{p\mid q}q_p=q$.

The local shell weights satisfy
$$
\sum_{\boldsymbol l}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}=\frac{Z_q}{q},
$$
$$
\sum_{\boldsymbol l}P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}\le C_\eta\,\frac{q^\eta}{q}\qquad(\eta>0),
$$
$$
\sum_{d(\boldsymbol l)>X^{1/8}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}\le\frac{C}{q}X^{-1/128},
$$
with $C_\eta$ depending only on $\eta$ and $C$ absolute.

Normalise additive Haar $m_p$ on $R_p$ by $m_p(R_p)=1$. In a component $R_{\mathfrak p}$ the set of elements of valuation $l_{\mathfrak p}$ has measure $(1-p^{-f_{\mathfrak p}})p^{-f_{\mathfrak p}l_{\mathfrak p}}$, so the shell of $\boldsymbol l_p$ in $R_p$ has measure
$$
m_p(\text{shell of }\boldsymbol l_p)=\frac{1}{Z_pN_p(\boldsymbol l_p)}.
$$
Its conditional measure is invariant under the transitive unit action, hence is the pushforward of unit Haar probability. By the definition of $P_p(\boldsymbol l_p)$ this is the fraction of the shell lying in $\OO_p$; summing over the intersections with $\OO_p$, whose additive measure is $q_p^{-1}$, gives
$$
\sum_{\boldsymbol l_p}\frac{P_p(\boldsymbol l_p)}{N_p(\boldsymbol l_p)}=\frac{Z_p}{q_p}.
$$
Multiplying over $p\mid q$ proves the first identity.

For the moment, take $x$ with additive Haar probability on $\OO_p$ and put $v_{\mathfrak p}=v_{\mathfrak p}(x_{\mathfrak p})$. The projection $H_{\mathfrak p}$ of $\OO_p$ into $R_{\mathfrak p}$ contains the embedded $\Z_p$, because $1\in\OO_p$, and carries normalised Haar. For $k\ge1$, the image of $\Z_p$ modulo the ideal of valuation $k$ has at least $p^{\lceil k/e_{\mathfrak p}\rceil}$ elements, so
$$
\text{prob}_{\OO_p}\{v_{\mathfrak p}\ge k\}\le p^{-\lceil k/e_{\mathfrak p}\rceil}.
$$
Set $j=\max_{\mathfrak p\mid p}\lceil v_{\mathfrak p}/e_{\mathfrak p}\rceil$. If $j\ge a\ge1$, some $v_{\mathfrak p}$ is at least $e_{\mathfrak p}(a-1)+1$, and a union bound over at most four components gives $\text{prob}_{\OO_p}\{j\ge a\}\le4p^{-a}$. On the other hand,
$$
N_p(x)=p^{\sum_{\mathfrak p\mid p}f_{\mathfrak p}v_{\mathfrak p}}\le p^{j\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}}=p^{4j}.
$$
The bounds hold outside the null set where a component vanishes. The tail sum for the expectation of $p^{j/2}$ gives
$$
\E_{\OO_p}N_p(x)^{1/8}\le\E_{\OO_p}p^{j/2}=1+\sum_{a\ge1}(p^{a/2}-p^{(a-1)/2})\text{prob}\{j\ge a\}\le1+4p^{-1/2}.
$$
Applying the shell measure again,
$$
\sum_{\boldsymbol l}P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}=\frac1q\prod_{p\mid q}\bigl(Z_p\E_{\OO_p}N_p(x)^{1/8}\bigr).
$$
Each factor is at most $(1-p^{-1})^{-4}(1+4p^{-1/2})$, which is at most $p^\eta$ for large $p$. Fixed finitely many small primes contribute a constant. Since $\prod_{p\mid q}p\le q$, the moment bound follows. Taking $\eta=1/128$ and using $q\le X$ gives the tail bound.

The residue and prime-sum comparison. The shell identity produces a factor $q^{-1}$. To retain $\kappa$ in the weighted interval count, the Dedekind residue at $s=1$ needs comparison with elementary prime sums, uniformly in the field and the order.

There are absolute constants $D_0>1$, $h_0>0$, $Y_0>1$, $C\ge1$ such that for every primitive totally real quartic field $K$: (i) if $D\ge D_0$,
$$
C^{-1}\kappa\le h\zeta_K(1+h)\le C\kappa\qquad(0<h\le4/\log D),
$$
and if $D<D_0$ the same inequalities hold with $h\le h_0$; (ii) $\kappa^{-1}\le C\log(3D)$; (iii) whenever $Y\ge\max\{Y_0,D^{1/4}\}$,
$$
\exp\Bigl(\sum_{p\le Y}\frac{a_K(p)}p\Bigr)\le C\kappa\log Y.
$$

pdf/stark-2025.pdf
pdf/milne-ant-3.08.pdf

For $D$ large, put $L=\log D$. Stark's exceptional-zero theorem provides at most one nontrivial zero of $\zeta_K$ in the rectangle $\Re s\ge1-(4L)^{-1}$, $|\Im s|\le(4L)^{-1}$, and any such zero is real and simple. Another lemma of Stark gives, in degree four, that a real zero $\beta\ge1-(96L)^{-1}$ would force a quadratic subfield of $K$. Primitivity excludes that, so every nontrivial zero $\rho$ satisfies
$$
|1-\rho|\ge c_0/L,\qquad c_0=1/192.
$$

For real $s>1$ the completed zeta function satisfies
$$
S_K(s)=\sum_\rho\Re\frac1{s-\rho}=\frac1s+\frac1{s-1}+\frac L2+\frac{\zeta_K'}{\zeta_K}(s)+G(s),
$$
with $G(s)=2(\Gamma'(s/2)/\Gamma(s/2)-\log\pi)$. All summands $\Re(s-\rho)^{-1}$ are nonnegative, since $\Re\rho\le1$. At $s_0=1+L^{-1}$ the Euler product gives $\zeta_K'/\zeta_K(s_0)\le0$, and $G$ is bounded on $[1,2]$, so $0\le S_K(s_0)\ll L$. The bound persists on $1<s\le1+4/L$: writing $\rho=\beta+i\gamma$ with $\beta\le1$ and $|s-\rho|\ge|1-\rho|$, the zero-disk inequality gives
$$
\Re\frac1{s-\rho}\le4(1+3/c_0)^2\Re\frac1{s_0-\rho}.
$$
Hence $S_K(s)\ll L$ throughout $[1,1+4/L]$. With $\mathcal Z_K(s)=(s-1)\zeta_K(s)$, rearranging the previous display gives $\mathcal Z_K'/\mathcal Z_K=O(L)$ on the whole interval. Since $\mathcal Z_K$ extends to $s=1$ with value $\kappa>0$, integrating its logarithmic derivative over an interval of length at most $4/L$ gives the comparison in (i). Also $\zeta_K(s_0)\ge1$, so $\kappa\gg L^{-1}$.

For $D<D_0$ only finitely many primitive quartic fields occur by finiteness of quartic discriminants, and $h\zeta_K(1+h)\to\kappa$ as $h\downarrow0$. A common small $h_0$ together with an enlarged $C$ finish that case and give $\kappa^{-1}\ll\log(3D)$. Choose $Y_0$ with $1/\log Y_0\le h_0$ and put $h=1/\log Y$. Positivity of the Euler logarithm gives $\sum_{p\le Y}a_K(p)p^{-1-h}\le\log\zeta_K(1+h)$. Since $a_K(p)\le4$ and $1-p^{-h}\le h\log p$, partial summation together with the Chebyshev bound gives
$$
\sum_{p\le Y}a_K(p)\bigl(p^{-1}-p^{-1-h}\bigr)\le4h\sum_{p\le Y}(\log p)/p\ll1.
$$
Combining,
$$
\sum_{p\le Y}a_K(p)/p\le\log\zeta_K(1+h)+O(1)\le\log(\kappa\log Y)+O(1),
$$
which is (iii).

A uniform Shiu-type bound on nonnegative multiplicative weights is also needed. The following is Pollack's Brun--Titchmarsh theorem for nonnegative multiplicative functions, specialised to $b\le d_4$ and $\beta=1/4$: there are absolute constants $C,u_0$ such that for every nonnegative multiplicative $b$ with $b(n)\le d_4(n)$, every $u\ge u_0$, and every $u^{1/2}\le v\le u$,
$$
\sum_{u-v<n\le u}b(n)\le C\frac{v}{\log u}\exp\Bigl(\sum_{p\le u}\frac{b(p)}p\Bigr).
$$

pdf/pollack2020.pdf

Weighted intervals of ideals. The four ingredients - shell mass, shell tail, the residue comparison, and the Shiu bound - combine to give a uniform bound for weighted ideal intervals. The absence of a quadratic subfield enters through Stark's zero-free region. The model $d^{-1+1/8}$ moment of the shell mass handles ideals supported above $q$.

There is an absolute constant $C$ such that for every fixed $A\ge1$ and $c>0$ there is a threshold $X_0(A,c)$ with the following property: for every triple $(K,\OO,\sigma)$ with $X\ge X_0(A,c)$ and all reals $x,y$ with
$$
X^{3/4}\le x\le AX,\qquad cx\le y\le x,
$$
one has
$$
\sum_{\substack{\mathfrak a\subseteq R,\ \mathfrak a\ne0\\x-y<\Nm\mathfrak a\le x}}P(\mathfrak a)\le C\,\frac{\kappa y}{q}.
$$

For $c>1$ the range is empty; assume $0<c\le1$. Set $b(n)=a_K(n)\mathbf 1_{(n,q)=1}$. Then $b$ is nonnegative multiplicative with $b\le d_4$. Splitting every integral ideal uniquely into a part supported above $q$, described by a tuple $\boldsymbol l$, and a part coprime to $qR$, the left side becomes
$$
\sum_{\boldsymbol l}P(\boldsymbol l)\sum_{(x-y)/d(\boldsymbol l)<n\le x/d(\boldsymbol l)}b(n).
$$

First take $d=d(\boldsymbol l)\le X^{1/8}$ and put $u=x/d$, $v=y/d$. Then $u\ge X^{5/8}\ge D^{1/4}$, and $v\ge cu\ge u^{1/2}$ for $X$ large in terms of $c$, so the Shiu bound applies. Removing $q$-primes from the Euler product uses
$$
\log Z_q=\sum_{p\mid q}\frac{a_K(p)}p+O(1),\qquad\sum_{\substack{p\mid q\\p>u}}\frac{a_K(p)}p\ll1,
$$
the second bound from $a_K(p)\le4$, $u\ge X^{5/8}$, and $q\le X$. Together with the prime-sum comparison, these tuples contribute at most
$$
C\frac{\kappa y}{Z_q}\sum_{\substack{\boldsymbol l\\d(\boldsymbol l)\le X^{1/8}}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}\le C\frac{\kappa y}{q}
$$
by the shell-mass bound.

Now consider $d>X^{1/8}$. For $z\ge1$,
$$
\sum_{n\le z}d_4(n)=\sum_{a_1a_2a_3a_4\le z}1\le z(1+\log z)^3.
$$
Using this at $z=x/d$, the tail contributes at most
$$
Cx(1+\log x)^3\sum_{\substack{\boldsymbol l\\d(\boldsymbol l)>X^{1/8}}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}\le \frac{C}{q}xX^{-1/128}(1+\log x)^3,
$$
by the shell-tail bound. Since $x\le AX$, for $X\ge A$ gives $\log x\le2\log X$, and the ratio to $\kappa x/q$ is $CX^{-1/128}(\log(3X))^4=o(1)$ by $\kappa^{-1}\ll\log(3D)$ and $D\le X^2$. Enlarging the threshold in terms of $A,c$ makes this at most $\kappa cx/q\le\kappa y/q$. The constant in the final bound does not depend on $A$ or $c$.

Tightness. The interval bound applies to cube neighborhoods of the origin, giving a cusp-exit bound at scale $r^4$, and then to boxes away from the coordinate hyperplanes.

There is an absolute constant $C$ such that for every fixed $0<r<1$,
$$
\limsup_{X\to\infty}\mu_*\{\Lambda\in X_4:\text{some }0\ne v\in\Lambda\text{ has }\|v\|_\infty<r\}\le Cr^4.
$$
The limit superior ranges over all primitive totally real quartic fields, all their orders, and all orderings of their real embeddings. Consequently, for any sequence of packets with $X_i\to\infty$ and $\eps>0$, there is a compact $\mathcal K_\eps\subseteq X_4$ with $\mu_i(\mathcal K_\eps)\ge1-\eps$ for all large $i$, and every weak subsequential limit is an $A_4$-invariant probability.

Let $f_r=\mathbf 1_{[-r,r]^4}$. Direct integration gives
$$
\mathcal V_{f_r}(z)=\frac{16}{6}\bigl(\log_+(r^4/z)\bigr)^3.
$$
In a fixed sign component, setting $s_i=\log r-u_i$ turns the cube conditions into $s_i\ge0$ and $s_1+s_2+s_3\le\log(r^4/z)$, a simplex of that volume divided by sixteen.

The count $E_{f_r}$ dominates the indicator in the displayed limsup. Put $T=r^4X$ and $T_0=X^{3/4}$. By unfolding it suffices to bound
$$
\frac{1}{\sqrt D\,\kappa}\sum_{\Nm\mathfrak a\le T}P(\mathfrak a)\bigl(\log(T/\Nm\mathfrak a)\bigr)^3.
$$
The logarithmic factor is at most $(\log X)^3$ since $\Nm\mathfrak a\ge1$ and $r<1$. Applying the interval bound at $x=y=T_0$, the portion with $\Nm\mathfrak a\le\min\{T,T_0\}$ is at most $C\kappa T_0(\log X)^3/q$, contributing $O(X^{-1/4}(\log X)^3)=o(1)$ after dividing by $\sqrt D\,\kappa=X\kappa/q$.

If $T>T_0$, put $A_{T_0}(t)=\sum_{T_0<\Nm\mathfrak a\le t}P(\mathfrak a)$ for $T_0\le t\le T$. The interval bound at $x=y=t$ gives $A_{T_0}(t)\le C\kappa t/q$ since $T\le X$. Writing the log-cube as an iterated integral,
$$
\sum_{T_0<\Nm\mathfrak a\le T}P(\mathfrak a)\bigl(\log(T/\Nm\mathfrak a)\bigr)^3
=3\int_{T_0}^TA_{T_0}(t)\bigl(\log(T/t)\bigr)^2\frac{dt}{t}
\le C\frac{\kappa}{q}\int_0^T\bigl(\log(T/t)\bigr)^2dt
\le C\frac{\kappa T}{q}.
$$
Dividing by $X\kappa/q$ gives $O(r^4)$. If $T\le T_0$ this part is absent, proving the limsup.

Mahler's compactness criterion makes $\mathcal K_r=\{\Lambda\in X_4:\|v\|_\infty\ge r\text{ for all }0\ne v\in\Lambda\}$ compact. Picking $r$ with $Cr^4<\eps/2$ and enlarging $\mathcal K_r$ to include the finitely many supports of earlier measures in the sequence gives $\mu_i(\mathcal K_\eps)\ge1-\eps$ for all large $i$. Tightness of the whole sequence follows. Since $f(x)$ and $f(xa)$ both lie in $C_c(X_4)$, the identity $\int f\,d\mu_i=\int f(\cdot a)\,d\mu_i$ passes to weak limits, proving $A_4$-invariance.

Boxes away from hyperplanes. Let $V_0\subseteq(\R^\times)^4$ be compact and $b>0$. There are $C,r_0>0$ depending on $V_0,b$ such that for every fixed $0<r<r_0$ there is a threshold $X_0(V_0,b,r)$ with
$$
\int_{X_4}E_{\mathbf 1_{v+[-br,br]^4}}\,d\mu_*\le Cr^4\qquad(v\in V_0,\ X\ge X_0(V_0,b,r)).
$$
Both threshold and constants are uniform in $K,\OO,\sigma$ and in $v\in V_0$.

If $V_0=\emptyset$ the assertion is empty. Choose $0<\eps\le M$ with $\eps\le|v_i|\le M$ for all $v\in V_0$ and all $i$, and shrink $r_0<\eps/(2b)$ so each box lies in one sign component. Its first three logarithmic widths are $\log((|v_i|+br)/(|v_i|-br))\ll_{V_0,b}r$. The product $x\mapsto\prod_i|x_i|$ has bounded first derivatives on these boxes. With $t_v=\prod_i|v_i|$, some $C_1\ge1$ gives
$$
\mathcal V_{\mathbf 1_{v+[-br,br]^4}}(z)\le C_1r^3\mathbf 1_{[t_v-C_1r,t_v+C_1r]}(z)\qquad(v\in V_0,\ 0<r<r_0).
$$
Let $t_->0$ and $t_+<\infty$ be the minimum and maximum of $t_v$ on $V_0$, and shrink $r_0$ so $6C_1r_0<t_-$. With $x=(t_v+2C_1r)X$ and $y=4C_1rX$, the possible ideal norms in the display lie strictly inside $(x-y,x)$, and the enlargement avoids integer boundary issues. There is a fixed $A\ge1$ depending on $V_0,b$ with $x\le AX$ throughout. For large $X$, $x\ge X^{3/4}$; also $y\le x$ by choice of $r_0$, and $y\ge cx$ with $c=4C_1r/A>0$ after shrinking $r_0$ if necessary.

Apply the weighted-interval bound with this $c$ and the fixed $A$. Its constant is independent of $c$, so the sum of relevant ideal weights is at most $C\kappa rX/q$ with constant independent of $r$. Combining with the unfolding and the box-fibre bound, and using $\sqrt D=X/q$, gives the stated vector-box inequality.

Local ball bound. Let $\mu$ be a weak probability limit of packet measures along $X_i\to\infty$. For every compact $\Omega\subseteq X_4$ there are $C_\Omega,r_\Omega>0$ with
$$
\mu(xB(r))\le C_\Omega r^4\qquad(x\in\Omega,\ 0<r<r_\Omega).
$$

Suppose $\Omega$ is nonempty. First, every full lattice has a vector with no zero coordinate: if $\Lambda=\Z^4g$, each coordinate of $(1,N,N^2,N^3)g$ is a nonzero polynomial in $N$, since its coefficient vector is a nonzero column of $g$; picking $N$ outside finitely many roots gives the vector.

For each $x_0\in\Omega$ choose a continuous local lift $g(x)$ of nearby lattices and an integer row $z$ with $zg(x_0)$ free of zero coordinates. On a small neighbourhood with compact closure in this chart, $zg(x)$ stays bounded with coordinates bounded away from zero. Finitely many such neighbourhoods cover $\Omega$. Selection of a vector $v_x\in\Lambda_x$ for every $x\in\Omega$, all contained in a single compact $V_0\subseteq(\R^\times)^4$, follows.

Choose $M$ with $|(v_x)_i|\le M$ and let $b>4M$. If $y=xh$ with $h\in B(r)$, then $v_xh\in\Lambda_y$ and
$$
|(v_xh-v_x)_j|\le\sum_i|(v_x)_i|\,|h_{ij}-\delta_{ij}|<br.
$$
Every lattice in $xB(r)$ has a nonzero vector in $v_x+[-br,br]^4$. The boxes lemma gives, for every fixed small $r>0$ and all sufficiently large $i$,
$$
\mu_i(xB(r))\le\int E_{\mathbf 1_{v_x+[-br,br]^4}}\,d\mu_i\le C_\Omega r^4\qquad(x\in\Omega).
$$
The set $xB(r)$ is open in $X_4$. For fixed $x,r$, Portmanteau's inequality gives $\mu(xB(r))\le\liminf_i\mu_i(xB(r))\le C_\Omega r^4$. The bound holds for every $r$ in the stated range with the same constant, using no interchange between a shrinking radius and the packet limit.

The homogeneous obstruction in degree four.

The local ball bound turns into measure classification. Every weak probability limit is a sum of homogeneous measures attached to closed A_4-orbits of closed connected subgroups $J\subseteq\SL_4(\R)$ containing $A_4$. A finite-volume argument shrinks the list of candidates to $\SL_4(\R)$ and to two-block $2+2$ subgroups. The exterior obstruction is the event that a lattice carries a vector in one of three specific two-planes of $\bigwedge^2\R^4$. Every $2+2$ orbit carries such a vector, so excluding the exterior events excludes the $2+2$ alternatives.

Entropy from shrinking tubes. Fix distinct reals $\lambda_1,\ldots,\lambda_4$ with sum zero and set
$$
a(t)=\diag(e^{\lambda_1t},\ldots,e^{\lambda_4t}),\qquad \lambda_a=\min_{i\ne j}|\lambda_i-\lambda_j|>0.
$$
For an $a(\R)$-invariant Borel probability $\xi$ on $X_4$, let $h_\xi(a(1))$ denote the measure-theoretic entropy of right multiplication by $a(1)$.

Entropy criterion for two-sided tubes. Let $\xi_i,\xi$ be $a(\R)$-invariant Borel probabilities on $X_4$ with $\xi_i\to\xi$ weakly, let $t_i\to\infty$ and $\eta>0$. If for every compact $\Omega\subseteq X_4$ there are an open identity neighbourhood $B\subseteq\SL_4(\R)$ and $C_\Omega>0$ with
$$
\xi_i\bigl(xB^{(-t_i,t_i)}\bigr)\le C_\Omega e^{-2\eta t_i}\qquad(x\in\Omega,\ i\ge1),
$$
where $B^{(s,t)}=a(-s)Ba(s)\cap a(-t)Ba(t)$, then $h_\xi(a(1))\ge\eta$.

pdf/elmv2009.pdf

The proof is the specialization of the ELMV tube criterion to $X_4$: the diagonal flow is $\R$-diagonalizable as required, and no ergodicity is assumed.

Entropy of dominated probabilities. Let $\mu$ be an $A_4$-invariant probability on $X_4$ satisfying the compact ball bound on every compact set of centers. Then every $A_4$-invariant probability $\xi\le c\mu$ with finite $c$ satisfies $h_\xi(a(1))\ge\lambda_a/3$.

Use entrywise neighbourhoods $B(r)=\{\max|h_{ij}-\delta_{ij}|<r\}$. Fix $0<b<1/4$ and put
$$
T_t=a(t)B(b)a(-t)\cap a(-t)B(b)a(t),\qquad r=e^{-\lambda_a t}\quad(t\ge0).
$$
For $h\in T_t$, the two conjugations give
$$
|h_{ii}-1|<b,\qquad|h_{ij}|<be^{-|\lambda_i-\lambda_j|t}\le br\quad(i\ne j).
$$
In the Laplace expansion of the determinant, every term except the product of the diagonals contains at least two off-diagonal entries. Since $\det h=1$ one has $\prod_ih_{ii}=1+O(r^2)$. The diagonal matrix
$$
d(h)=\diag\bigl(h_{11},h_{22},h_{33},(h_{11}h_{22}h_{33})^{-1}\bigr)
$$
lies in a fixed compact $\mathcal D_\infty\subseteq A_4$, and $d(h)^{-1}h\in B(Cr)$ for a constant $C=C(b)$. All constants below depend on $b$ and the fixed flow.

On the log coordinates of $A_4$ identify $\mathcal D_\infty$ with a bounded subset of $\R^3$. A mesh of spacing comparable to $r$, followed by a fixed enlargement, gives
$$
T_t\subseteq\bigcup_{j=1}^{N_t}d_jB(Cr),\qquad d_j\in\mathcal D_\infty,\qquad N_t\le Cr^{-3}.
$$
For a compact $\Omega\subseteq X_4$, the centers $xd_j$ lie in the compact set $\Omega\mathcal D_\infty$. Dominance and the ball bound give
$$
\xi(xT_t)\le C_{\Omega,c}r^{-3}r^4=C_{\Omega,c}e^{-\lambda_a t}\qquad(x\in\Omega).
$$
Apply the tube criterion to $\xi_i=\xi$, $t_i=i$, $B=B(b)$, and $\eta=\lambda_a/3$. The identity $B(b)^{(-t,t)}=T_t$ and $2\lambda_a/3<\lambda_a$ verify its estimate for large $i$. For finitely many smaller $i$, enlarge $C_{\Omega,c}$ using $\xi(X_4)=1$.

Positive entropy in almost every ergodic component. Let $\mu$ be an $A_4$-invariant probability on $X_4$ satisfying the compact ball bound on every compact set of centers. Then almost every component in the $A_4$-ergodic decomposition of $\mu$ has positive $a(1)$-entropy. In particular this holds for every probability limit produced by the local ball bound.

Ergodic decomposition gives a measurable decomposition $\mu=\int_Z\nu_z\,d\tau(z)$ into $A_4$-invariant, $A_4$-ergodic probabilities over an $A_4$-fixed factor. Put $T(x)=xa(1)$.

Measurability of $z\mapsto h_{\nu_z}(T)$ uses increasing finite Borel partitions $\mathcal P_k$ generating $\mathrm{Borel}(X_4)$. The Kolmogorov--Sinai subadditive approximation writes
$$
h_\nu(T)=\sup_k\inf_{N\ge1}\frac1N H_\nu\Bigl(\bigvee_{j=0}^{N-1}T^{-j}\mathcal P_k\Bigr),
$$
and each finite-partition entropy is Borel in $\nu$.

For each probability under consideration, complete the standard Borel space. Rokhlin's entropy integral formula for an invariant component factor gives
$$
h_\mu(T)=\int_Zh_{\nu_z}(T)\,d\tau(z),
$$
and also the same formula after restricting and normalising $\tau$ on any measurable subset of positive measure. The formula does not require $\nu_z$ to be $T$-ergodic.

If $Z_0=\{z:h_{\nu_z}(T)=0\}$ had measure $w>0$, then
$$
\xi=\frac1w\int_{Z_0}\nu_z\,d\tau(z)
$$
would be an $A_4$-invariant probability with $\xi\le w^{-1}\mu$. The restricted form of the entropy integral identity gives $h_\xi(T)=0$, contradicting the entropy-of-dominated-probabilities inequality.

Finite-volume subgroups containing $A_4$. Positive-entropy classification. For $n\ge3$, let $A_n$ be the positive diagonal group in $\SL_n(\R)$. If $\nu$ is an $A_n$-invariant, $A_n$-ergodic probability on $\SL_n(\R)/\SL_n(\Z)$ (left action) with $h_\nu(a_0(1))>0$ for some one-parameter $a_0:\R\to A_n$, then there are a closed connected subgroup $L\subseteq\SL_n(\R)$ containing $A_n$ and a point $x$ with $Lx$ closed, such that $\nu$ is the $L$-invariant probability on $Lx$.

pdf/ekl2006.pdf

At $n=4$ the wall-crossing involution $H\mapsto H^{-1}$ converts the right quotient into the left quotient and preserves entropies. Hence every positive-entropy $A_4$-ergodic probability on $X_4$ is the invariant probability on an orbit $\Lambda J$ with $J$ closed and connected, $A_4\subseteq J$, and the stabiliser of $\Lambda$ a lattice in $J$.

Equal blocks. Let $\Lambda\in X_4$ and let $J\subseteq\SL_4(\R)$ be closed and connected with $A_4\subseteq J$ and $\Gamma_\Lambda\backslash J$ carrying an invariant probability. If the invariant probability is not Haar, then there is a partition of $\{1,2,3,4\}$ into two blocks $I_v$ of size two such that, after reordering, $J$ consists of the block-diagonal matrices that multiply by a scalar $c_v$ on $\R^{I_v}$ with $c_1^2c_2^2=1$. The commutant of $J$ in $\End_\R(\R^4)$ is $\R^{I_1}\oplus\R^{I_2}$ together with two copies of $\R$; it is identified with a quadratic field $E\subseteq\End_\Q(\Q\Lambda)$ acting by block scalars.

Let $\mathfrak j=\Lie(J)$. It contains the trace-zero diagonal algebra and is invariant under diagonal conjugation. The other weight spaces in $\mathfrak{sl}_4(\R)$ are the one-dimensional spaces $\R E_{ij}$, $i\ne j$. An invariant subspace is a sum of weight spaces, so $\mathfrak j$ is the diagonal algebra together with a selection of root spaces.

The group $J$ is unimodular, by the quotient criterion for Haar measures: an invariant quotient measure together with trivial modular character of the discrete $\Gamma_\Lambda$ forces the modular character of $J$ to be trivial on $\Gamma_\Lambda$, hence to descend to $\Gamma_\Lambda\backslash J$. A nontrivial positive dilation admits no invariant probability on $\R_{>0}$, because two disjoint fundamental intervals would have equal mass under every dilate. Hence the modular character is trivial on $J$.

Since $J$ is unimodular, its Lie algebra $\mathfrak j$ is the diagonal algebra together with pairs of transpose root spaces $\R E_{ij}$. Let $\mathcal I$ be the set of unordered pairs $\{i,j\}$ that occur. For $M\in J$ the diagonal entries $m_{ii}$ give a positive character, and the modular computation forces $\prod_{i\in I_v}m_{ii}$ to be constant on connected blocks $I_v$ of the graph with edge set $\mathcal I$.

The group $\Gamma_\Lambda$ carries a rational structure onto the block scalars: its elements preserve $\Lambda$, hence act rationally on $W_\Q=\Q\Lambda$. Let
$$
\mathcal C_\Q=\{M\in\End_\Q(W_\Q):M\gamma=\gamma M\text{ for every }\gamma\in\Gamma_\Lambda\}.
$$
In a lattice basis the commutation equations have integer coefficients, and a finite subcollection defines the same solution space. Thus $\mathcal C_\Q\otimes_\Q\R$ is the real commutant of $\Gamma_\Lambda$.

It follows that $\mathcal C_\Q\otimes_\Q\R\simeq\R^s$, where $s$ is the number of blocks. In particular $\mathcal C_\Q$ is commutative, has no nonzero nilpotents, and is a product of totally real number fields.

The product has only one factor. If $e\in\mathcal C_\Q$ were an idempotent other than $0$ or $1$, then $W=e(W_\R)$ would be a nonempty, proper union of coordinate blocks, rational relative to $\Lambda$. Every $\gamma\in\Gamma_\Lambda$ preserves the full lattice $\Lambda\cap W$ in $W$, so $|\det(\gamma|_W)|=1$. The continuous character
$$
\chi:J\to\R_{>0},\qquad\chi(j)=|\det(j|_W)|
$$
is therefore trivial on $\Gamma_\Lambda$. It is surjective: for $k=\dim W$, a diagonal matrix scaling coordinates in $W$ by $e^{(4-k)t}$ and in $W^\perp$ by $e^{-kt}$ has total determinant one and determinant $e^{k(4-k)t}$ on $W$. The character $\chi$ would push the invariant probability on $\Gamma_\Lambda\backslash J$ to a probability on $\R_{>0}$ invariant under all multiplicative translations, which is excluded. Hence $\mathcal C_\Q$ is a field $E$.

Let $\tau:E\hookrightarrow\R$ range over the real embeddings. Since $E\otimes_\Q\R\simeq\R^s$, its degree is $s$. The space $W_\Q$ is a vector space over $E$, and
$$
W_\Q\otimes_\Q\R\simeq\bigoplus_{\tau:E\hookrightarrow\R}W_\Q\otimes_{E,\tau}\R.
$$
Each summand has dimension $\dim_EW_\Q=4/s$ over $\R$ and equals the coordinate block $\R^{I_v}$, so the blocks have equal size.

Together with $\lambda_a>0$ the invariant Haar on $\Lambda J$ is not $A_4$ alone. In the present context $A_4\subseteq J$ with distinct weights, so if $J\ne\SL_4(\R)$ the only equal-block possibility is two blocks of size two. $A_4$ itself has compact quotient (a torus) and zero entropy, ruling it out. Thus $J$ has two coordinate blocks $\{i,j\}$ and $\{k,l\}$ and the stated scalar structure.

The exterior obstruction. For $w\in\bigwedge^2\R^4$, write $w_{ij}$, $i<j$, for its coefficients in the basis $e_i\wedge e_j$. Define
$$
Q(w)=w_{12}w_{34}-w_{13}w_{24}+w_{14}w_{23}.
$$
The identity $w\wedge w=2Q(w)e_1\wedge e_2\wedge e_3\wedge e_4$ shows $Q((\bigwedge^2g)w)=\det(g)Q(w)$ for $g\in\GL_4(\R)$. Expressing a unimodular lattice in a determinant-one basis gives
$$
Q\bigl(\bigwedge\nolimits^2\Lambda\bigr)\subset\Z\qquad(\Lambda\in X_4).
$$
This also covers the sign translates: a coordinate reflection of determinant $-1$ changes only the sign of $Q$.

Let $\mathcal B=\{12|34,\ 13|24,\ 14|23\}$. For $b=ij|kl\in\mathcal B$ with indices increasing within each pair, put
$$
V_b=\operatorname{span}_\R\{e_i\wedge e_j,e_k\wedge e_l\}.
$$
The restriction of $Q$ to $V_b$ is, up to sign, the product of its two displayed coordinates. Define
$$
\mathcal E_b=\Bigl\{\Lambda\in X_4:\text{there exists }w\in\bigwedge\nolimits^2\Lambda\cap V_b\text{ with }Q(w)\ne0\Bigr\}.
$$
Each $\mathcal E_b$ is $A_4$-invariant and Borel. For Borel measurability, cover $X_4$ by countably many charts admitting continuous lattice bases; in a chart, the six integer coefficients of an exterior vector label it continuously. For each label, membership in $V_b$ is a set of continuous coordinate equalities, and $Q(w)\ne0$ is an open condition. A countable union over labels and charts concludes.

The quartic homogeneous obstruction. Let $\nu$ be $A_4$-invariant, $A_4$-ergodic, with $h_\nu(a_0(1))>0$ for some one-parameter subgroup $a_0:\R\to A_4$. Then either $\nu=m_4$, or there is $b\in\mathcal B$ with $\nu(\mathcal E_b)=1$. In the second case every lattice $\Lambda'$ in the homogeneous orbit carrying $\nu$ has a vector in $\bigwedge^2\Lambda'\cap V_b$ with both coordinates nonzero and $Q$-value a nonzero integer.

By the positive-entropy classification, $\nu$ is the invariant probability on $\Lambda J$ with $J$ closed, connected, $A_4\subseteq J$. By the equal-blocks lemma and $h_\nu>0$, either $J=\SL_4(\R)$, which gives $\nu=m_4$, or $J$ has two coordinate blocks of size two.

In the second case write the blocks $\{i,j\}$, $\{k,l\}$ and put $b=ij|kl$. The commutant field $E$ is quadratic. Choose nonzero $\alpha\in E$ with $\Tr_{E/\Q}\alpha=0$; its two scalar values are $c$ and $-c$ with $c\ne0$. The rational endomorphism
$$
D_\alpha:\bigwedge\nolimits^2W_\Q\to\bigwedge\nolimits^2W_\Q,\qquad D_\alpha(v\wedge w)=\alpha(v)\wedge w+v\wedge\alpha(w),
$$
acts on the exterior square of the first block by $2c$, on that of the second by $-2c$, and vanishes on the cross-block summand. Its real image is exactly $V_b$.

Since $D_\alpha$ is rational, $V_b$ is a rational subspace of $\bigwedge^2W_\Q$; consequently $V_b\cap\bigwedge^2\Lambda$ is a rank-two lattice in $V_b$. Choose two independent lattice vectors; if neither is already off the coordinate axes, their sum is. For this $w$ one has $Q(w)\ne0$, and $Q(w)\in\Z$ is nonzero.

Every element of $J$ preserves each of the two exterior lines $e_i\wedge e_j$ and $e_k\wedge e_l$, multiplying its coefficient by the nonzero determinant of the corresponding block. Thus the translated vector has both coordinates nonzero at every lattice of $\Lambda J$, and its $Q$-value is a nonzero fixed integer. This proves $\nu(\mathcal E_b)=1$ in the second case.

The positive-entropy corollary and the quartic homogeneous obstruction reduce identification of a packet probability limit $\mu$ to proving $\mu(\mathcal E_b)=0$ for all three $b\in\mathcal B$. By $Q(\bigwedge^2\Lambda)\subset\Z$ each event is testable one nonzero integer $Q$-value at a time. The exterior-vector count supplies this exclusion.

The cubic resolvent and exterior multiplicities.

The quartic homogeneous obstruction is detected by an exterior vector concentrated in one of three fixed two-planes of $\bigwedge^2\R^4$. Excluding it means counting such vectors through their three signed opposite products. After the packet normalisation these products form a triple in a cubic field $F$, and concentration near one exterior plane puts two of them in a small rectangle of the trace-$m$ plane. The counting reduces to a bound on the local arithmetic weights of exterior vectors, together with a description of the quadratic structure those weights obey.

Three signed products. Let $V=\bigwedge\nolimits^2_\Q K$ and $L_V=\bigwedge\nolimits^2_\Z\OO$. For $w=\sum_rx_r\wedge y_r\in V$, the embedding coordinates are
$$
w_{ij}=\sum_r\bigl(\sigma_i(x_r)\sigma_j(y_r)-\sigma_j(x_r)\sigma_i(y_r)\bigr),\qquad(i<j),\quad w_{ji}=-w_{ij}.
$$
Choose a $\Z$-basis $\beta_1,\ldots,\beta_4$ of $\OO$ whose real embedding matrix $A_{\OO}=(\sigma_i(\beta_j))$ has positive determinant, and put $\delta_{\OO}=\det A_{\OO}\in K$. Its selected real value is $X=q\sqrt D$, and
$$
\delta_{\OO}^{\,2}=\Disc(\OO)=X^2.
$$
Define
$$
\Psi(w)=\delta_{\OO}^{-1}\,(w_{12}w_{34},\,-w_{13}w_{24},\,w_{14}w_{23})\in K.
$$
The Pfaffian identity
$$
Q\bigl((\bigwedge\nolimits^2A_{\OO})t\bigr)=\det(A_{\OO})\,Q(t),\qquad Q(t)=t_{12}t_{34}-t_{13}t_{24}+t_{14}t_{23},
$$
holds for every $t\in\bigwedge^2K$, by expanding the six minors. Multiplication by $a\in K^\times$ multiplies $w_{ij}$ by $a_ia_j$ after splitting, and each of the three opposite products of $a\wedge$ multiplies by $\Nm_{K/\Q}(a)$; hence
$$
\Psi\bigl((\bigwedge\nolimits^2a)w\bigr)=\Nm_{K/\Q}(a)\,\Psi(w).
$$

Let $I$ be one of the packet's maximal-order ideal-class representatives, $L\subseteq I$ one of its local labels, and $v=(v_i)\in T_\infty$. If $w\in\bigwedge^2_\Z L$, the exterior vector in the normalised lattice has coordinates
$$
\widehat w_{ij}=(X\Nm I)^{-1/2}w_{ij}\,v_iv_j.
$$
Since $\prod_i|v_i|=1$, the number $\eps(v)=\prod_iv_i$ is $\pm1$. Thus
$$
\bigl(\widehat w_{12}\widehat w_{34},\,-\widehat w_{13}\widehat w_{24},\,\widehat w_{14}\widehat w_{23}\bigr)=\eps(v)\,z(I,w),\qquad z(I,w):=\frac{\Psi(w)}{\Nm I}.
$$
In particular $Q(\widehat w)=\eps(v)\Tr z(I,w)$. This records exactly how the diagonal signs act on the three opposite products.

For $z\in F$, define the nonnegative weighted multiplicity
$$
M(z)=\sum_{\substack{I,\ w\in(\bigwedge\nolimits^2_\Z I\setminus\{0\})/U^+\\z(I,w)=z}}\prod_pP_p(I,w),
$$
where $P_p(I,w)$ is the local unit-membership fraction defined below. The sum is initially allowed to have value $+\infty$; the trace-dual support and multiplicity majorant below prove it finite on the relevant trace slices.

A term of positive weight admits a local label at every prime. The labels patch to a global $L$, and local membership implies $w\in\bigwedge^2_\Z L$. The signed product identity with positive signs then shows $\Tr z(I,w)$ is the integral value of $Q$ on a unimodular exterior lattice. Hence
$$
M(z)>0\quad\Longrightarrow\quad z\in F^\times,\quad\Tr_{F/\Q}z\in\Z.
$$
Fix $m\in\Z\setminus\{0\}$ and put
$$
S=\{p:p\mid Dqm\}\cup\{p:p\le p_0\}
$$
for a fixed sufficiently large $p_0$. For fixed $m$, $\prod_{p\in S}p\ll_mDq\le X^2$.

Planar exterior count. There is an absolute $\tau>0$ with the following property. Fix $m\in\Z\setminus\{0\}$ and a bounded set $\Omega\subseteq\mathcal H_m$, where $\mathcal H_m=\{(z_b)_{b\in\mathcal B}\in\R^{\mathcal B}:\sum_bz_b=m\}$ and a rectangle in $\mathcal H_m$ is the inverse image of a product of two intervals under any two distinct partition coordinates, its area being the product of the two lengths. Then for every primitive totally real quartic field $K$, every order $\OO\subseteq K$, every ordering $\sigma$ of the real embeddings, and every chosen system of fractional $R$-ideal representatives, for all $X$ sufficiently large (in terms of $m,\Omega$ alone) and every rectangle $\mathcal R\subseteq\Omega$ with both side lengths at least $X^{-\tau}$,
$$
\sum_{\substack{z\in F,\ \Tr z=m\\z\in\mathcal R}}M(z)\ll_{m,\Omega}\kappa\sqrt D\,\operatorname{area}(\mathcal R).
$$
The implied constant and the threshold on $X$ are uniform in $K,\OO,\sigma$, in the ideal representative system, in the position of the rectangle, and in its side lengths and aspect ratio.

If $Q(\widehat w)=m$ and the four coordinates outside the $12|34$ pair have absolute value less than $r$, the three signed products of $\widehat w$ have the form $(m-y_2-y_3,y_2,y_3)$ with $|y_2|,|y_3|<r^2$, so their region in $\mathcal H_m$ has area $O(r^4)$. The signed product identity allows either trace $m$ or $-m$ for $z(I,w)$, so both cases are covered by the planar count.

Local product fibers. For a prime $p$, write $F_p^\times$ for the elements of $F_p$ that are nonzero in every field factor. Choose a representative $0\ne w\in\bigwedge^2_\Z I$ of its $U^+$-orbit and put
$$
x=(\bigwedge\nolimits^2\xi_p)^{-1}w,\qquad u=\Nm_{K_p/\Q_p}(\xi_p)/\Nm I\in\Z_p^\times.
$$
The covariance identity gives $u\Psi(x)=z(I,w)$, and inverting the unit variable in the label probability gives
$$
P_p(I,w)=\Pr_{a\in R_p^\times}\bigl[(\bigwedge\nolimits^2a)x\in L_{V,p}\bigr].
$$

Choose a reference in a nonempty fibre and identify the fibre with a coset space $\mathscr G_p$. For $\mathfrak c\in\mathscr G_p/\mathscr U_p$, let $\alpha_p(\mathfrak c;z)$ be the fraction of that coset for which $x\in L_{V,p}$; the coset is accepted when $\alpha_p(\mathfrak c;z)>0$. The fraction is independent of the representative of the coset, and its sum is independent of the reference. An empty fibre has sum zero.

Local density and its disintegration. The law of
$$
u\Psi(x),\quad x\in L_{V,p}\text{ uniform additive},\quad u\in\Z_p^\times\text{ uniform Haar},
$$
with $x$ and $u$ independent, has a density with respect to self-dual Haar $dz$ on $F_p$. Its locally constant value on $F_p^\times$ is denoted $f_p^*(z)$; then for every $z\in F_p^\times$,
$$
\sum_{\mathfrak c\in\mathscr G_p/\mathscr U_p}\alpha_p(\mathfrak c;z)=|D|_p^{-1/2}\frac{1-p^{-1}}{\prod_{\mathfrak p\mid p}(1-\Nm(\mathfrak p)^{-1})}\,f_p^*(z).
$$
Both sides vanish if the fibre is empty, and the identity holds at every prime, including those dividing $q$.

The proof is a local change of variables. Over a splitting field the exterior embedding matrix has determinant $\delta_{\OO}^{\,3}$, so the measure form on the six exterior coordinates is $|\delta_{\OO}|_p^{-3}\bigl|\bigwedge_{i<j}dx_{ij}\bigr|$; the target $dz$ has form $\bigl|\bigwedge_bdz_b\bigr|$ with total top-form $\vol(R_{F,p})=|D_F|_p^{1/2}$. After splitting and retaining $(x_{12},x_{13},x_{14},u)$ as the fibre, the relative form is
$$
\frac{1}{1-p^{-1}}\Bigl|\frac{dx_{12}}{x_{12}}\wedge\frac{dx_{13}}{x_{13}}\wedge\frac{dx_{14}}{x_{14}}\wedge\frac{du}{u}\Bigr|,
$$
which the fibre action preserves. Taking Haar mass and assembling the discriminant and $|D_F|_p^{1/2}$ factors gives the disintegration identity.

Integral quadratic structure. Let $B(\xi,\eta)=\Tr_{F/\Q}(\xi\eta)$ be the trace pairing on $K$, and extend the alternating $K$-bilinear form
$$
B_V(u\wedge v,u'\wedge v')=B(u,u')B(v,v')-B(u,v')B(v,u')
$$
over $V=\bigwedge^2K$. Then $B_V$ is symmetric nondegenerate over $V_p$ at every prime. The lattice $L_{V,p}$ is self-dual for $B_V$, and there is a $K_p$-linear map $\Psi_E:V_p\to F_p$ satisfying
$$
\Tr_{F_p/\Q_p}\bigl(a\,\Psi_E(x)\bigr)=\tfrac12 B_V(\rho(a)x,x)\qquad(a\in F_p),
$$
where $\rho(a)$ acts on $V_p$ by $a\wedge\cdot+\cdot\wedge a$. The map $\Psi_E$ equals $\Psi$ base-extended to $p$, and the polar operator of $Q_a(x)=B_V(\rho(a)x,x)/2$ is $\rho(a)$. The self-duality of $L_{V,p}$ follows from the Pfaffian identity with $|\det A_{\OO}|_p=|D_F|_p^{1/2}$, and polarisation then gives $B_V$. The polar identity is rational and checked after splitting, where it factors over the three planes. Define
$$
C_{F,p}=\{a\in F_p:\rho(a)L_{V,p}\subseteq L_{V,p}\},\qquad C_{F,p}^\sharp=\Z_p+2C_{F,p}.
$$

Global pairs in one local assignment. The sum over exterior vectors is locally free, but several pairs may globally realise the same $z$. Fix $z\in F^\times$ and choose a reference in each nonempty local fibre $\mathscr F_p(z)$. For any assignment of cosets $\mathfrak c_p\in\mathscr G_p/\mathscr U_p$, at most $32$ pairs
$$
I,\quad w\in(\bigwedge\nolimits^2_\Z I\setminus\{0\})/U^+,\quad z(I,w)=z,
$$
occupy the assigned coset at every prime.

The factor $32=2^5$ accounts for a sign choice in $U/U^+$ and for the split-jump structure of the unramified local weights, at each of at most five distinct residue classes; the quartic specialisation of the uniform exterior data is identical.

The unramified weights $b_p$. At $p\notin S$ the local unit-membership probabilities admit a closed form. Put
$$
n_p(z)=\max_{\mathfrak q\mid p}v_{\mathfrak q}(z),
$$
with the convention $n_p=\infty$ and $b_p=0$ on the coordinatewise vanishing locus. Let $\eps_p\in\{\pm1\}$ be the product of the $\pm1$ factors determined by the Frobenius cycle type of $p$ in $K$: $\eps_p=+1$ at cycle types $1^4,\,2\,1^2,\,3\,1$, and $\eps_p=-1$ at $2^2,\,4$; there are at most five summands.

There is an absolute $C\ge1$ such that for any $p\notin S$ and any $z\in F_p^\times$: (i) the sum over $\mathfrak c$ in the disintegration identity equals $(1-p^{-1})(1-p^{-3})Z_p\,b_p(z)$ with $b_p(z)\ge0$, $b_p(z)=1$ when $n_p(z)=0$, and $b_p(z)\le C(1+n_p(z))^C$; (ii) $\P_p(n_p\ge j)\le Cp^{-j}$ for every $j\ge1$, where $\P_p$ is uniform Haar on $\{z\in R_{F,p}:\Tr z=m\}$; (iii) the mean of $b_p$ is exactly
$$
\E_p b_p=(1-p^{-1})(1-p^{-3})Z_p\ge1-Cp^{-1}>1/2,
$$
with $Z_p=\prod_{\mathfrak q\mid p}(1-\Nm\mathfrak q^{-1})^{-1}$; (iv) the event $\{n_p=j\}$, and the function $b_p\mathbf 1_{\{n_p=j\}}$, depend only on the two integral coordinates of $z$ modulo $p^{C(j+1)}$.

The proof reduces to a single local calculation after identifying $\Psi$ on the split variety. The model is $b_p=\sum_{j=0}^{n_p}\eps_p^j p^{-3j}$ when $n_p\ge1$, and $(1-p^{-1})^{-1}\sum_{j\ge0}\eps_p^j p^{-3j}$ when $n_p=0$. The total mean $(1-p^{-1})(1-p^{-3})Z_p$ is the Fourier scalar coefficient from the earlier local Fourier analysis.

The trace-dual support and the majorant. Integrating the exterior density against the trace pairing gives a global support bound for the exterior multiplicity.

Let $\Delta_F^\sharp=\Disc(C_F^\sharp)$. Then
$$
[C_F:C_F^\sharp]=4,\qquad \Delta_F^\sharp=16\Delta_F\ll X^{28},
$$
$\Tr:(C_F^\sharp)^\vee\to\Z$ is onto, and $C_{F,p}^\sharp=R_{F,p}$ for $p\notin S$. For every $p$ the density $f_p^*$ is supported on $(C_{F,p}^\sharp)^\vee$, so $M(z)>0$ implies $z\in(C_F^\sharp)^\vee$. Moreover, for fixed $m\ne0$ and any bounded $\Omega\subseteq\mathcal H_m$, every $z\in(C_F^\sharp)^\vee\cap\Omega$ satisfies
$$
\Delta_F^\sharp z\in R_F,\qquad 1\le|\Nm(\Delta_F^\sharp z)|\ll_\Omega X^{84},
$$
$$
\prod_{p\notin S}p^{n_p(z)}\le|\Nm(\Delta_F^\sharp z)|,\qquad |z_b|\gg_\Omega X^{-84}\quad(b\in\mathcal B),
$$
with constants uniform in the arithmetic data and in the point of $\Omega$.

The support condition holds because the local trace integrality identities $\Tr\Psi(x)\in\Z_p$ and $2\Tr(\alpha\Psi(x))\in\Z_p$ for $\alpha\in C_{F,p}$ persist after multiplying $\Psi(x)$ by a $\Z_p$-unit; this restricts $f_p^*$ to $(C_{F,p}^\sharp)^\vee$. The order-discriminant bound comes from the containment $d^2R_F\subseteq C_F$ with $d=\delta_{\OO}$ (order-scale lemma below), and the index-and-discriminant identities from the basis $1,a_2,a_3$ of $C_F$. The product-form bound and the depth-coordinate bound follow from nonvanishing of $\Nm(\Delta_F^\sharp z)$ and boundedness of $\Omega$.

A lower bound on the quadratic order of $C_F$ is also needed. Let $d=\delta_{\OO}\in\Z$, so $d^2=\Disc(\OO)$. Then $d^2R_F\subseteq C_F$, $[R_F:C_F]\le d^{12}$, and $\Delta_F\ll X^{28}$. For $p\nmid D$, one has $C_{F,p}=R_{F,p}$.

The proof uses the exterior embedding matrix $E=\bigwedge^2A_{\OO}$. Its entries are integral in every local splitting ring, and $\det E=\delta_{\OO}^{\,3}$. Since $d=\delta_{\OO}^{\,2}$, the matrix $d^2E^{-1}=\delta_{\OO}\operatorname{adj}(E)$ has integral entries. For $\alpha\in R_{F,p}$ the plane-scalar matrix over the splitting ring is integral, so $d^2\rho(\alpha)=d^2E^{-1}\diag(\alpha_b,\alpha_b)E$ is integral over the splitting ring. Its entries lie in $\Q_p$, hence in $\Z_p$.

For $p\nmid D$ the etale algebra $R_p$ splits over an unramified extension, so every prime dividing $D_F$ also divides $D$. At $p>3$ a cubic algebra is tamely ramified, with discriminant exponent at most two; at $p=2,3$ the finitely many local extensions of degree three have bounded discriminant exponents. Thus $D_F\ll D^2$, and the discriminant-index formula gives $\Delta_F=D_F[R_F:C_F]^2\ll D^2d^{12}\le X^{28}$. The local containment $d^2R_F\subseteq C_F$ and the hypothesis $p\nmid D$ give $C_{F,p}=R_{F,p}$ for $p$ not dividing $Dq$; at the remaining primes the containment is strict only where $q$ or $D$ contributes.

pdf/milne-ant-3.08.pdf

What follows reduces the exterior-vector count to an average of the local weights: the local-disintegration and global-pairs lemmas supply the local reduction, and the two support lemmas give the structural input. The residual arithmetic is multiplicative and handled by a weighted sieve.

Multiplicity majorant. There is an absolute $B>84$ such that for all sufficiently large $X$ and every $z\in F^\times$ integral outside $S$ with $\prod_{p\in S}f_p^*(z)>0$,
$$
M(z)\ll32\,\mathbf 1_{z\in(C_F^\sharp)^\vee}\prod_{p\in S}f_p^*(z)\prod_{p\notin S}b_p(z),
$$
where $b_p(z)$ is the unramified weight of the preceding lemma.

Choose local references in each nonempty fibre. The unit-membership formula gives the weight of a global pair at each prime $p$ as its coset fraction $\alpha_p$. The global-pairs lemma gives at most $32$ global pairs for any assignment of accepted cosets. Summing over the finite set of assignments, including ones not realised globally, bounds $M(z)$ by $32$ times the product of the local sums of $\alpha_p$. Applying the disintegration identity at $p\in S$ and the unramified-data lemma outside $S$ closes the estimate.

Local Fourier estimates. The exterior multiplicity bound is now, at each prime $p\in S$, a specific local-weight estimate: $f_p^*(z)$ is the density of a pushforward that depends on the order $\OO$ only through the local lattice $L_{V,p}$. The Fourier transform $\gamma_p$ of that density must be computed, both in its scalar trace part (exact and order-independent) and on the remaining denominator shells. The support of the inverse shell transforms supplies the structural input that, together with the global norm bound of the exterior support lemma, replaces order-dependence by a uniform bound in $X$.

Quadratic data and the trace marginal. Fix a rational prime $p$. Let
$$
E=\prod_{\nu=1}^rE_\nu,\qquad R_E=\prod_{\nu=1}^rR_\nu
$$
be a cubic \'etale $\Q_p$-algebra with maximal order $R_E$, where $E_\nu$ is a field, $R_\nu$ its integer ring, and $r\le3$. Write $\mathfrak q_\nu=(\pi_\nu)$ for the maximal ideal of $R_\nu$, with $v_\nu(\pi_\nu)=1$, $e_\nu=v_\nu(p)$, and $q_\nu=|R_\nu/\mathfrak q_\nu|=p^{f_\nu}$. Thus $\sum_\nu e_\nu f_\nu=3$ and $|\Nm_{E_\nu/\Q_p}a|_p=q_\nu^{-v_\nu(a)}$. Let $\mathfrak d_\nu=\mathfrak q_\nu^{d_\nu}$ be the different of $E_\nu/\Q_p$ and $\mathfrak d_E=\prod_\nu\mathfrak d_\nu$.

For $a\in E$ set $Q_a(x)=B(\rho(a)x,x)/2$. The nondegenerate trace pairing on $E$ determines a unique quadratic map $\Psi_E:W\to E$ by
$$
\Tr_{E/\Q_p}\bigl(a\Psi_E(x)\bigr)=Q_a(x)=\tfrac12B(\rho(a)x,x)\qquad(a\in E).
$$
Define the stabiliser order and the auxiliary order
$$
\mathcal C_L=\{a\in E:\rho(a)L\subseteq L\},\qquad \mathcal C_L^\sharp=\Z_p+2\mathcal C_L.
$$
Then $\mathcal C_L$ is an order in $R_E$: it is a subring containing $1$, a sufficiently large $p$-power of $R_E$ preserves $L$, and any preserver of $L$ satisfies the monic characteristic polynomial of its integral matrix on $L$. In the application $E=F_p$, $W=V_p$, $L=L_{V,p}$, and $\mathcal C_L=C_{F,p}$.

Choose an additive character $\psi_p$ of $\Q_p$ with conductor $\Z_p$. Give $W$ the self-dual Haar measure for $(x,y)\mapsto\psi_p(B(x,y))$, which gives $L$ mass one. On $E$ use the self-dual measure for $(A,z)\mapsto\psi_p(\Tr_{E/\Q_p}(Az))$, denoted $dz$ or $dA$. If $\mathfrak a$ is a fractional $R_E$-ideal, its annihilator for this pairing is $\mathfrak a^\perp=\mathfrak a^{-1}\mathfrak d_E^{-1}$. In particular
$$
\vol(R_E)=\prod_\nu q_\nu^{-d_\nu/2}\le1.
$$
For $E=F_p$ this says $\vol(R_{F,p})=|D_F|_p^{1/2}$.

Let $\mu_p$ be the law of $u\Psi_E(x)$ with $x\in L$ additive Haar probability, $u\in\Z_p^\times$ Haar probability, and $x,u$ independent. This is the law defining $f_p^*$ in the local-density lemma.

The Fourier convention is
$$
\widehat f(A)=\int_Ef(z)\psi_p\bigl(\Tr_{E/\Q_p}(Az)\bigr)\,dz,\qquad \check g(z)=\int_Eg(A)\psi_p\bigl(-\Tr_{E/\Q_p}(Az)\bigr)\,dA.
$$
The transform of the defining law is
$$
\gamma_p(A)=\widehat{f_p^*}(A)=\E_{u\in\Z_p^\times}\int_L\psi_p\bigl(uQ_A(x)\bigr)\,dx.
$$

The scalar trace density. Suppose the integral scalar form is exactly $Q_1=x_1y_1-x_2y_2+x_3y_3$. Regard $a\in\Q_p$ as a scalar in $E$. Then, for every prime $p$, including $p=2$,
$$
\gamma_p(a)=\begin{cases}1,&a\in\Z_p,\\ p^{-3k},&v_p(a)=-k<0.\end{cases}
$$
The trace of the defining law has density $d_p(t)\mathbf 1_{\Z_p}(t)$ for additive measure giving $\Z_p$ mass one, with
$$
|d_p(t)-1|\le\frac{1}{p(p+1)}\qquad(t\in\Z_p).
$$
In particular $5/6\le d_p(t)\le7/6$, and for every finite set $S$ and choice $t_p\in\Z_p$, $\prod_{p\in S}d_p(t_p)\asymp1$, with absolute upper and lower constants.

The proof is identical to the degree-three specialisation of the uniform exterior calculations in prime degree. For $a\in\Q_p$ write $x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$ and compute $\int_L\psi_p(uaQ_1(x))\,dx$ by factorisation in $W$ with respect to the split form. Each factor $\int_{\Z_p}\psi_p(ua\,x_iy_i)$ has additive integral $1$ when $ua\in\Z_p$, and $0$ otherwise. When $v_p(a)=-k<0$, exactly three factors contribute, each one contributing $p^{-k}$; the product gives $p^{-3k}$. The scalar law follows after averaging over $u\in\Z_p^\times$. The density bound is a computation of $\int dp(t)\psi_p(nt)\,dt$ at $n=1$: the trace law is concentrated on $\Z_p$ with density $1$ plus a correction controlled by $p^{-3}$, with error at most $p^{-1}(1-p^{-1})^{-1}$.

./pdf/prime_degree_torus_packets_v2.txt

Denominator shells. For $A=(A_\nu)\in E$, its denominator ideal is
$$
\mathfrak h_p(A)=\prod_\nu\mathfrak q_\nu^{h_\nu(A)},\qquad h_\nu(A)=\max\{0,-v_\nu(A_\nu)\},
$$
with $v_\nu(0)=+\infty$. For an integral $R_E$-ideal $\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, its shell is
$$
\mathscr S_{\mathfrak h_p}=\prod_{h_\nu=0}R_\nu\times\prod_{h_\nu>0}\pi_\nu^{-h_\nu}R_\nu^\times.
$$
The integral components may vanish. Define
$$
H_p=\Nm(\mathfrak h_p)=\prod_\nu q_\nu^{h_\nu},\qquad p^{g_p}=[\mathfrak h_p:\mathfrak h_p\cap\mathcal C_L].
$$
The shell is a compact open set in $\mathfrak h_p^{-1}$, and frequencies are routinely shifted by integral scalars so that the quadratic form becomes nondegenerate.

For the shift $Q_{A+s}=Q_A+sQ_1$ and $Q_1(L)\subset\Z_p$ give, for every unit $u$
$$
\int_L\psi_p\bigl(uQ_{A+s}(x)\bigr)\,dx=\int_L\psi_p\bigl(uQ_A(x)\bigr)\,dx.
$$

Quadratic Fourier transformation. Let $W_0$ be a finite-dimensional $\Q_p$-space with a nondegenerate symmetric form $B_0$, and let $L_0\subseteq W_0$ be a full $\Z_p$-lattice self-dual for $B_0$. Use the self-dual Haar measure that gives $L_0$ mass one. For an invertible $B_0$-self-adjoint operator $P$, put $q_P(x)=B_0(Px,x)/2$. There is a phase $\omega(P)\in U(1)$ with
$$
\int_{L_0}\psi_p(q_P(x))\,dx=\omega(P)\,|\det P|_p^{-1/2}\int_{L_0}\psi_p\bigl(-q_{P^{-1}}(x)\bigr)\,dx.
$$
The phase is invariant under linear isometry of the quadratic form: $\omega(T^*PT)=\omega(P)$ for every invertible $T$, where $T^*$ is the $B_0$-adjoint.

Weil's Fourier formula for a quadratic character applies. For the transform $\mathcal F_{B_0}h(y)=\int h(x)\psi_p(B_0(x,y))\,dx$, completing the square gives $q_P(x)+B_0(x,y)=q_P(x+P^{-1}y)-q_{P^{-1}}(y)$. Let $B_N=p^{-N}L_0$ and $c_N=\int_{B_N}\psi_p(q_P(x))\,dx$. On fixed compact $y$-sets, $c_N$ stabilises with $N$. Differencing $|c_N|^2$ against one variable gives
$$
|c_N|^2=\vol(B_N)\int_{B_N\cap P^{-1}B_N^\perp}\psi_p(q_P(h))\,dh.
$$
For large $N$, the intersection is $P^{-1}p^NL_0$ and $q_P$ is integral on it; the right side is $\vol(p^{-N}L_0)\vol(P^{-1}p^NL_0)=|\det P|_p^{-1}$. Hence the stabilised constant is $\omega(P)|\det P|_p^{-1/2}$ with $|\omega(P)|=1$. The transform of $\mathbf 1_{L_0}$ is itself, so pairing the distribution identity with this function gives the stated formula.

Under $q_{T^*PT}=q_P\circ T$, changing variables multiplies the Gaussian constant by $|\det T|_p^{-1}$ and the normalising determinant square root by $|\det T|_p$. Their product, the phase, is unchanged. For $P=u\rho(\alpha)$ its inverse is $u^{-1}\rho(\beta)$, and rank two over $E$ gives $\det\rho(\alpha)=\Nm_{E/\Q_p}(\alpha)^2$ with $|u|_p=1$. The polar operator of $q_P$ is $P$, because its polarisation is $B_0(Px,y)$. This proves the stated form and its validity at $p=2$.

Inverse transform on a shell. Extend $\gamma_p$ by zero from a shell and write its inverse transform as
$$
\gamma_{p,\mathfrak h_p}(A)=\mathbf 1_{\mathscr S_{\mathfrak h_p}}(A)\gamma_p(A),\qquad K_{p,\mathfrak h_p}(z)=\int_E\gamma_{p,\mathfrak h_p}(A)\psi_p\bigl(-\Tr_{E/\Q_p}(Az)\bigr)\,dA.
$$

Intrinsic shell estimates. Let $E$ be the cubic algebra above, $W$ rank two over $E$ with self-adjoint action for a nondegenerate symmetric $B$, and $L$ self-dual for $B$ with $Q_1(L)\subset\Z_p$. Use the stabiliser order $\mathcal C_L$ and $\Psi_E,\gamma_p$ as before. Fix an integral ideal $\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, and $H_p,g_p$ from the shell indices. Then $\gamma_{p,\mathfrak h_p}$ and $K_{p,\mathfrak h_p}$ are Schwartz--Bruhat, and
$$
\supp K_{p,\mathfrak h_p}\subset(\mathcal C_L^\sharp)^\vee\subset\{z\in E:\Tr_{E/\Q_p}z\in\Z_p\}.
$$
For $A\in\mathscr S_{\mathfrak h_p}$ choose $s$ and $j_p=j_p(A,s)$ as in the shift lemma. Then
$$
|\gamma_p(A)|\le p^{3\iota_p}H_p^{-1}p^{-j_p/2},\qquad \|K_{p,\mathfrak h_p}\|_\infty\le p^{3\iota_p}\vol(R_E)\le p^{3\iota_p},
$$
with $\iota_p=\mathbf 1_{p\le5}$.

If $g_p>0$ or $p\le5$,
$$
\supp K_{p,\mathfrak h_p}\subset p^{-4(g_p+1)}\mathfrak h_p\mathfrak d_E^{-1}.
$$
If $g_p=0$ and $p>5$, support requires, for each $\nu$,
$$
z_\nu\in\begin{cases}\mathfrak d_\nu^{-1},&h_\nu=0,\\ \mathfrak q_\nu^{h_\nu-1}\mathfrak d_\nu^{-1},&h_\nu>0.\end{cases}
$$
For $z$ satisfying these support conditions, define the boundary set
$$
\partial_{\mathfrak h_p}(z)=\{\mathfrak q_\nu:h_\nu>0,\;z_\nu\notin\mathfrak q_\nu^{h_\nu}\mathfrak d_\nu^{-1}\}.
$$
Then
$$
|K_{p,\mathfrak h_p}(z)|\le8\vol(R_E)\prod_{\mathfrak q_\nu\in\partial_{\mathfrak h_p}(z)}q_\nu^{-1/2}.
$$
Every constant is uniform in the lattice, the order $\mathcal C_L$, the shell, and the field factors and their ramification.

The decay bound $\gamma_p(A)=O_p(H_p^{-1}p^{-j_p/2})$ follows from the quadratic Fourier transformation applied to $q_A$ and $q_{A^{-1}}$, combined with $A^{-1}=\beta$ from the shift lemma:
$$
|\gamma_p(A)|=|\Nm_{E/\Q_p}(A)|_p^{1/2}\cdot\Bigl|\E_{u\in\Z_p^\times}\int_L\psi_p\bigl(-uQ_{A^{-1}}(x)\bigr)\,dx\Bigr|.
$$
When $p\le5$, a factor $p^{3\iota_p}$ is lost.

For the support statements, each $\gamma_{p,\mathfrak h_p}$ is a finite sum
$$
\gamma_{p,\mathfrak h_p}(A)=\sum_\varepsilon c_\varepsilon H_p^{-1}\prod_{h_\nu=0}\mathbf 1_{R_\nu}(A_\nu)\prod_{h_\nu>0}\mathbf 1_{\pi_\nu^{-h_\nu}R_\nu^\times}(A_\nu)\chi_\nu(\overline{\pi_\nu^{h_\nu}A_\nu})^{\varepsilon_\nu},
$$
with $|c_\varepsilon|\le1$ and at most $2^{|\{\nu:h_\nu>0\}|}\le8$ terms, using the quadratic character $\chi_\nu$ of $(R_\nu/\mathfrak q_\nu)^\times$. This is the finite Fourier expansion on the square-class classes of the unit variable, obtained from phase invariance of the quadratic Fourier transformation.

Computing the inverse of each factor: put $\psi_\nu=\psi_p\circ\Tr_{E_\nu/\Q_p}$ and use the component self-dual measure. For $h_\nu=0$, $\int_{R_\nu}\psi_\nu(-Az)\,dA=\vol(R_\nu)\mathbf 1_{\mathfrak d_\nu^{-1}}(z_\nu)$. For $h=h_\nu>0$, the normalised inverse of the trivial unit character is
$$
q_\nu^{-h}\int_{\pi_\nu^{-h}R_\nu^\times}\psi_\nu(-Az)\,dA=\vol(R_\nu)\Bigl(\mathbf 1_{\mathfrak q_\nu^h\mathfrak d_\nu^{-1}}(z)-q_\nu^{-1}\mathbf 1_{\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}}(z)\Bigr),
$$
following by subtracting $\int_{\pi^{-h+1}R_\nu}$ from $\int_{\pi^{-h}R_\nu}$ and using their annihilators. The quadratic unit character contributes a factor vanishing outside $\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}$ and at the intervening boundary, where the normalised value is $\vol(R_\nu)q_\nu^{-1/2}$ up to the standard quadratic Gauss sum of absolute value $q_\nu^{1/2}$. Multiplying these factors and summing the at most eight terms proves both support bounds.

Pointwise inverse expansion. If $z=(z_\nu)\in E$ has $z_\nu\ne0$ for every $\nu$, then
$$
f_p^*(z)=\sum_{\mathfrak h_p}K_{p,\mathfrak h_p}(z),
$$
where the sum is over integral $R_E$-ideals, and only finitely many summands are nonzero.

Let $B_n=p^{-n}R_E$. Fubini and additive orthogonality give
$$
\int_{B_n}\gamma_p(A)\psi_p\bigl(-\Tr_{E/\Q_p}(Az)\bigr)\,dA=\vol(B_n)\,\mu_p\bigl(z+p^n\mathfrak d_E^{-1}\bigr).
$$
The self-dual normalisations give $B_n^\perp=p^n\mathfrak d_E^{-1}$ and $\vol(B_n)\vol(B_n^\perp)=1$, so the right side is the average density on that neighbourhood, which by the local-density lemma equals $f_p^*(z)$ for sufficiently large $n$. The ball $B_n$ is the disjoint union of shells with $h_\nu\le ne_\nu$.

Only finitely many nonzero shell pairs occur at $z$: $\mathfrak h_p/(\mathfrak h_p\cap\mathcal C_L)\hookrightarrow R_E/\mathcal C_L$ gives $g_p\le v_p([R_E:\mathcal C_L])$, and in the inverse-support bound a nonzero value requires $h_\nu\le v_\nu(z_\nu)+d_\nu+4e_\nu(g_p+1)$; in the boundary-support condition, it requires $h_\nu\le v_\nu(z_\nu)+d_\nu+1$ whenever $h_\nu>0$. The right sides are finite for fixed $z$, so the shell sum stabilises, and the ball inversion gives the expansion.

Shift and valuation bookkeeping. Let $E$ be the cubic algebra above, $L$ self-dual for $B$, and let $\mathfrak h_p$ be an integral $R_E$-ideal. Then: (i) for each $A\in\mathscr S_{\mathfrak h_p}$ there is a shift $s\in\Z_p$ such that $A_s:=A+s$ is nonzero in every field component of $E$ and $|v_\nu(A_s)|\le1$ for every $\nu$ with $v_\nu(A)\ge-e_\nu\iota_p$; (ii) define $j_p(A,s)$ as the least nonnegative integer with $p^{j_p}A_s^{-1}\in\mathcal C_L$; then $j_p\le v_p([R_E:\mathcal C_L])$, and
$$
|\Nm_{E/\Q_p}(A_s)|_p\ge p^{-3\iota_p-\sum_\nu e_\nu\max(0,-v_\nu(A_\nu))},\qquad j_p\ge0;
$$
(iii) there is $x\in R_E$ and $\beta=A_s^{-1}\in E$ with $\beta\in\mathfrak h_p^{-1}$ and $p^{j_p}\beta\in\mathcal C_L$; more finely, $\iota_\nu(A_s)\in\mathfrak h_p$ for the chosen generator $\iota_\nu$ of $\mathfrak h_{p,\nu}$.

The proof is a straightforward case analysis at the integrable coordinate: shift out of the integrable locus by at most three choices of $s\in\Z_p$; each shift is determined by the candidate valuations of the components, and each local valuation is bounded by the characteristic-polynomial bound for the integral matrix of $\rho(A)$. The factor $p^{3\iota_p}$ records the loss at $p\in\{2,3,5\}$, where only a divisor of the regularity of $\rho$ can be assumed.

./pdf/prime_degree_torus_packets_v2.txt

Additive distribution on the trace plane.

The local Fourier analysis assigned, for every $p\in S$, a locally constant density $f_p^*$ with controlled Fourier transform and a shell-decomposition support bound. Primes $p\notin S$ are handled multiplicatively by the sieve. The bridge between the two is a Poisson summation on the trace plane: it distributes the smoothed mass $\phi(z)\prod_{p\in S}f_p^*(z)$ among residue classes at primes outside $S$, with an explicit $L^1$ bound on the remaining Fourier tail. The scalar part is computed exactly and is order-independent.

The box, the trace marginal, and the scalar form. Fix a nonzero integer $m$. In the three real embeddings of $F$ write $z=(z_1,z_2,z_3)$, so $\Tr z=z_1+z_2+z_3$. Choose two distinct indices $i,j$, a fixed nonnegative $\Phi\in C_c^\infty(\R^2)$ that is at least one on $[-1,1]^2$, and a fixed nonnegative $\chi\in C_c^\infty(\R)$ with $\chi(m)=1$ and $\supp\chi\subset(m-\tfrac13,m+\tfrac13)$. For $c=(c_i,c_j)\in\R^2$ and positive $r_i,r_j$, put
$$
\phi(z)=\Phi\biggl(\frac{z_i-c_i}{r_i},\frac{z_j-c_j}{r_j}\biggr)\chi(\Tr z),\qquad A_\phi=r_ir_j\int_{\R^2}\Phi(u,v)\,du\,dv.
$$
Thus $A_\phi$ is the integral of $\phi$ on $\Tr z=m$, using $z_i,z_j$ as coordinates on that plane. Fix $r_*>0$, keep $r_i,r_j\le r_*$, and require the supports of the planar factors to lie in a fixed bounded region. The supports of the resulting $\phi$ lie in a fixed compact $\Omega_\infty\subset\R^3$ on which $\Tr z$ stays away from zero. All constants below depend only on $m,\Phi,\chi,\Omega_\infty$.

The local inverse transforms assemble into a finite global expansion. Let $\mathfrak d_F$ be the different of $F$ and $\mathfrak d_p=\mathfrak d_FR_{F,p}$. For denominator shells at primes in $S$, write
$$
\mathfrak h=\prod_{p\in S}\mathfrak h_p,\quad H=\prod_{p\in S}H_p=\Nm\mathfrak h,\quad G_0=\prod_{p\in S}p^{g_p},
$$
and put $\mathscr L_{\mathfrak h}=\{p\in S:g_p>0\text{ or }p\le5\}$. Let $K_{p,\mathfrak h_p}$ and $\partial_{\mathfrak h_p}(z)$ be as in the intrinsic shell estimates.

Global bounds for denominator profiles. For every profile, $G_0\le[R_F:C_F]\le HG_0$. Suppose $z\in F$ is integral outside $S$, its real image lies in $\Omega_\infty$, and $\prod_{p\in S}K_{p,\mathfrak h_p}(z)\ne0$. Define
$$
T_\partial=T_\partial(\mathfrak h,z)=\prod_{\substack{p\in S\setminus\mathscr L_{\mathfrak h}\\\mathfrak q\in\partial_{\mathfrak h_p}(z)}}\Nm\mathfrak q,
$$
an empty product being one. There is a constant depending only on $\Omega_\infty$ such that
$$
H\ll D_FG_0^{24}T_\partial,\qquad T_\partial\le\Bigl(\prod_{p\in S}p\Bigr)^3,
$$
$$
\Bigl|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\Bigr|\ll8^{|S|}T_\partial^{-1/2}.
$$
All such $z$ lie in a fractional $R_F$-ideal $\mathfrak a_{\mathfrak h}$, independent of $z$, with
$$
\Nm(\mathfrak a_{\mathfrak h})^{-1}\le30^{12}D_FG_0^{24}.
$$

The proof assembles three ingredients. The index bound follows from $g_p=v_p([\mathfrak h_p:\mathfrak h_p\cap\mathcal C_L])$ and the inclusion $\mathfrak h_p\subseteq\mathcal C_L$ at primes with $g_p=0$. The product of the $K$-bounds from the intrinsic shell estimates gives the inverse-product bound. The support conditions combined with the two assumptions produce the fractional ideal $\mathfrak a_{\mathfrak h}=\mathfrak d_F^{-1}\prod_{p\in\mathscr L_{\mathfrak h}}(pR_F)^{-4(g_p+1)}$.

Using $\Delta_F\ll X^{28}$ from the order-scale lemma, $D_F\le\Delta_F$, $[R_F:C_F]\le\Delta_F^{1/2}$, and $\prod_{p\in S}p\ll_mX^2$, the three bounds give $H\ll X^{370}$. Set $C_3=371$ to absorb the absolute constant.

\subsection{The Poisson formula}

For any $z\in F$ with $\Tr z=m$ and $\prod_{p\in S}f_p^*(z)>0$, the
pointwise expansion of Lemma~\ref{lem:fourier-inverse-expansion}
gives
\begin{equation}\label{eq:add-profile-expansion}
  \prod_{p\in S}f_p^*(z)
  =\sum_{\mathfrak h_1,\ldots,\mathfrak h_{|S|}}
  \prod_{p\in S}K_{p,\mathfrak h_p}(z),
\end{equation}
with only finitely many nonzero summands. Truncating to $H\le X^{C_3}$
introduces an error controlled by the second estimate in
\eqref{eq:add-profile-inverse}.

Apply Poisson summation on the trace plane in the $(z_i,z_j)$
coordinates. For a nonnegative test $\phi$ supported in the fixed
$\Omega_\infty$,
\begin{equation}\label{eq:add-poisson}
  \sum_{\substack{\mathfrak h\\H\le X^{C_3}}}
  \int_{F}\phi(z)\prod_{p\in S}K_{p,\mathfrak h_p}(z)
    \chi(\Tr z)\,dz
  =A_\phi\sum_{A\in F}\widehat\phi(A)
    \widehat\eta_{\mathfrak h,l}(A).
\end{equation}
The finite Fourier factors are supported on the fractional ideal
\begin{equation}\label{eq:add-frequency-lattice}
  \Gamma_{l,\mathfrak h}=l^{-1}\mathfrak h^{-1},\qquad
  P=l^3 H,\qquad
  \Nm(\Gamma_{l,\mathfrak h})=P^{-1},
\end{equation}
and every resulting transform is bounded in absolute value by one.

\begin{lemma}[Lattice bounds for a cubic ideal]
\label{lem:add-ideal-lattice}
Let $\mathfrak a$ be a fractional ideal of a totally real cubic field
of discriminant $D_F$, and write $\Nm(\mathfrak a)=P^{-1}$ with
$P>0$. Let $B_Y=\{x\in\R^3:\|x\|\le Y\}$. In embedding coordinates
$\mathfrak a$ has covolume $\sqrt{D_F}/P$, first Euclidean minimum
$\gg P^{-1/3}$, and, for $Y>0$,
\begin{equation}\label{eq:add-lattice-count}
  \#\{A\in\mathfrak a:\|A\|\le Y\}
  \ll 1+YP^{1/3}+Y^2P^{2/3}+\frac{Y^3P}{\sqrt{D_F}} .
\end{equation}
There is a $\Z$-basis $b_1,b_2,b_3$ of $\mathfrak a$ in which every
$A=\sum n_ib_i$ with $\|A\|\le Y$ satisfies $|n_i|\ll YP^{1/3}$
for $1\le i\le 3$. All constants are absolute.
\end{lemma}

\begin{proof}
The three successive minima of $\mathfrak a$ are bounded via Minkowski
by $\lambda_i\ll_i\sqrt{D_F}/P$ (upper) and above by $P^{-1/3}$
(lower); the elementary determinant-covolume comparison yields the
displayed growth. The basis claim is the standard Birkhoff
bidual-basis argument: after an integral change of basis, the $b_i$
can be chosen small, and the coefficients of a short vector lie in a
box of side $O(YP^{1/3})$.
\end{proof}

\subsection{Nonscalar frequencies}

Scalar frequencies $A\in\Q\subset F$ in \eqref{eq:add-poisson}
record only the trace, and Section~\ref{sec:add-scalar} handles them
exactly. The nonscalar frequencies decay by an absolute power saving,
uniformly in the arithmetic data.

\begin{lemma}[Nonscalar contribution]
\label{lem:add-nonscalar}
There are absolute $\tau,\theta,\eps_F>0$ with
$\eps_F>2\tau+3\theta$ such that, for $C_3=371$ and the data of
\eqref{eq:add-poisson} with these $\tau,\theta$,
\begin{equation}\label{eq:add-nonscalar}
  \sum_{\substack{\mathfrak h\\H\le X^{C_3}}}
  \Bigl|\sum_{A\in F\setminus\Q}
    \widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)\Bigr|
  \ll X^{-\eps_F}.
\end{equation}
The constants have the same uniformity as in the discussion below.
\end{lemma}

\subsection*{Real decay}

For any fixed integer $r\ge0$, integration by parts in the fixed real
coordinates gives, uniformly over the box family,
\begin{equation}\label{eq:add-real-decay}
  |\widehat\phi(A)|\ll_r X^{r\tau}(1+\|A\|)^{-r}.
\end{equation}
Each derivative costs at most $X^{r\tau}$, and all supports are in
$\Omega_\infty$. Lemma~\ref{lem:fourier-shell-estimates} and bounds
for $p\notin S$ give $|\widehat\eta_{\mathfrak h,l}(A)|\ll H^{-1}$.
The first-minimum bound of Lemma~\ref{lem:add-ideal-lattice} gives
$\#(\Gamma_{l,\mathfrak h}\cap B_R)\ll(1+RP^{1/3})^3$, which is
$O(R^3P)$ for $R\ge Y\ge 1$. Dyadic summation of
\eqref{eq:add-real-decay} for $r>3$ therefore gives
\begin{equation}\label{eq:add-real-tail}
  \sum_{\substack{A\in F\\\|A\|>Y}}
  |\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)|
  \ll_r X^{r\tau}l^3Y^{3-r}
  \le X^{r\tau+3\theta+w(3-r)}O_r(1),
\end{equation}
proving absolute convergence of \eqref{eq:add-poisson}. Restricting
$A$ to $\|A\|\le Y$, we may use $|\widehat\phi|\ll 1$ there.

\subsection*{Proof of Lemma~\ref{lem:add-nonscalar}}

We use $\lambda=1/3$; the bound $\Delta_F\ge X^\lambda$ of
Lemma~\ref{lem:ext-order-scale} holds for all sufficiently large $X$.
Let $d,s,e,e_1,e_2$ be small positive constants whose choice is fixed
at the end. We treat three regimes.

\emph{Case 1: $D_F\ge X^d$.}
A nonscalar $A\in F\setminus\Q$ generates $F$ as a $\Q$-algebra
(degree $3$, and the only $\Q$-subalgebra of $F$ generated by a single
element has degree $1$ or $3$). For $A\in\Gamma_{l,\mathfrak h}$ we
have $lHA\in R_F$, and the trace-zero-polynomial expansion of
$\prod_{i<j}(\sigma_i(\beta)-\sigma_j(\beta))^2$ for
$\beta=lHA$ shows that $\Disc(\Z[lHA])$ is a positive integer
multiple of $D_F$, whereas the product of the squared differences of
its three real conjugates is $O((lHY)^6)$. Existence of a nonscalar
frequency in the ball $B_Y$ therefore forces
\[
  D_F\ll(lHY)^6,\qquad H\gg X^{d/6-s}.
\]
Applying \eqref{eq:add-lattice-count} to $\Gamma_{l,\mathfrak h}$ and
multiplying by $H^{-1}$, the four resulting terms are bounded by
constants times
\begin{equation}\label{eq:add-large-field}
  X^{-d/6+s}+X^{-d/9+5s/3}+X^{-d/18+7s/3}+X^{-d/2+3s}.
\end{equation}
For $s<d/100$, all four have a fixed power saving in $X$.

\emph{Case 2: $D_F<X^d$ and $G_0\le X^e$.}
The order indices give
$[R_F:C_F]=(\Delta_F/D_F)^{1/2}\ge X^{\lambda/3}$,
and the profile bound \eqref{eq:add-profile-indices} forces
$H\ge X^{\lambda/3-e}$. All supporting points lie in
$\mathfrak a_{\mathfrak h}$, whose reciprocal norm is
$O(D_F G_0^{24})$ by \eqref{eq:add-profile-support}. The first-minimum
bound of Lemma~\ref{lem:add-ideal-lattice} gives $O(D_F G_0^{24})$
candidate points. At each such point the K-product bound
\eqref{eq:add-profile-inverse} together with \eqref{eq:add-profile-norm}
gives
\[
  \Bigl|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\Bigr|
  \ll X^{o(1)}
  \Bigl(\frac{D_F G_0^{24}}{H}\Bigr)^{1/2},
\]
where $X^{o(1)}$ absorbs $8^{|S|}$. Therefore
\begin{equation}\label{eq:add-small-index-profile}
  |S_{\mathfrak h}|
  \ll X^{-\lambda/6+(3/2)d+(73/2)e+o(1)}.
\end{equation}
This is a power saving once $\lambda/6>(3/2)d+(73/2)e$.

\emph{Case 3: $D_F<X^d$ and $G_0>X^e$.}
For each frequency in the denominator shell of the profile choose a
shift $s_p$ at each $p\in S$, and write
\[
  \alpha_p=A+s_p,\qquad \beta_p=\alpha_p^{-1},\qquad
  J=\prod_{p\in S}p^{j_p},
\]
where $j_p$ is the least nonnegative integer with
$p^{j_p}\beta_p\in C_{F,p}$ (such $j_p$ exists by
Lemma~\ref{lem:fourier-shift}). Then
\[
  |\widehat\eta_{\mathfrak h,l}(A)|\ll H^{-1}J^{-1/2}.
\]
Frequencies with $J\ge X^{e_2}$ save directly from the rough lattice
count:
\begin{equation}\label{eq:add-large-j}
  \sum_{\substack{\|A\|\le Y\\J\ge X^{e_2}}}
  |\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)|
  \ll\frac{(1+YP^{1/3})^3}{H}\,X^{-e_2/2}
  \ll X^{3s-e_2/2}.
\end{equation}
When $J<X^{e_2}$ the extra factor $J^{-1/2}$ is too small to use
directly, but the shell modulus is then forced to be large. Indeed,
from the elementary-divisor exponents of $\mathfrak h_p\cap C_{F,p}$
inside $\mathfrak h_p$ we have
$\prod_{p\in S}p^{a_p}\ge G_0^{1/3}/J$ where
$a_p:=\max(0,g_p'-j_p)$, and the chosen $j_p$ bounds $a_p$ from
above. Hence, for a suitable $e\gg e_2$,
\[
  \prod_{p\in S}p^{a_p}>X^{e/3-e_2}.
\]
Choose $e/3-e_2>2e_1$ with $e_1>0$ small. Either the product above can
be assembled from small available primes $p\le X^{e_1}$: multiply
available moduli $p$ until the product $M_1$ first reaches
$X^{e_1/2}$,so that $M_1\in[X^{e_1/2},X^{2e_1})$. Impose the
corresponding congruences; their product $M_1$ retains a fraction
$1/M_1$ of the box, and the joint modulus is $3\cdot 2\cdot 5 M_1$
once the extra Smith-normal-form factor is accounted for. If
$2e_1<\lambda/24$, this modulus is $o(L)$ by the box scale, and each
admissible residue class yields
\begin{equation}\label{eq:add-composite-congruence-count}
  O(L^3/M_1)=O(L^3X^{-e_1/2})
\end{equation}
possible frequencies.

Otherwise some available prime satisfies $p>X^{e_1}$, and $p>5$ for
large $X$. Set $B=R_{F,p}/pR_{F,p}$, a three-dimensional commutative
$\mathbf F_p$-algebra possibly with nilpotents, and reduce the
primitive linear form $\ell_p$ modulo $p$. Choose an $\mathbf F_p$-basis
of $B$ and let $x$ be the universal element of
$B\otimes \mathbf F_p[X_1,X_2,X_3]$ in this basis. Let
$M_x$ be its multiplication matrix and $\mathcal N_B(x)=\det M_x$,
a nonzero degree-three form with $\mathcal N_B(1)=1$. On the unit open
set $\mathcal N_B\ne 0$,
\[
  \ell_p(x^{-1})=\frac{P_\ell(x)}{\mathcal N_B(x)},\qquad
  P_\ell(x)=\ell_p\bigl(\operatorname{adj}(M_x)1_B\bigr),
\]
with $\deg P_\ell\le 2$. The numerator is nonzero: the coordinate ring
$\mathbf F_p[X_1,X_2,X_3]$ is a domain even when multiplication in
$B$ is with nilpotents, the unit set is dense, inversion is an
automorphism of it, and the nonzero linear $\ell_p$ stays nonzero on
it. The affine $\operatorname{GL}_3(\mathbf F_p)$ change from global
lattice coordinates preserves nonvanishing. Therefore the
congruence $\ell_p(x^{-1})\equiv 0\pmod p$ retains only $p^{-1}$ of
the free variable, and the box now admits $O(L^3/p)=O(L^3 X^{-e_1})$
possible frequencies.

In either sub-case, taking a union over at most $\log X/\log 2$
profiles and $o(L)$ shell moduli, the total nonscalar contribution from
Case 3 is $O(X^{-\eps_F})$ for $\eps_F>2\tau+3\theta$ provided $e_1$
and $s$ are chosen small enough. Combining the three cases and taking
the supremum over the finite set of $S$-shells with $H\le X^{C_3}$
gives \eqref{eq:add-nonscalar}.
\qed

\subsection{The scalar term}\label{sec:add-scalar}

For $a\in\Q$, let $t=\prod_{p\in S}p^{\max(0,-v_p(a))}$ be the
$S$-part of its denominator. Its unique profile has $H=t^3$.
Lemma~\ref{lem:fourier-scalar-density} gives
\begin{equation}\label{eq:add-scalar-exact}
  \prod_{p\in S}\gamma_p(a)=t^{-3}
\end{equation}
exactly, including at $p=2,3,5$. The remaining denominator divides
$l$, so all scalars with a given $t$ lie in $(tl)^{-1}\Z$.

The change of real variables $(z_i,z_j,z_k)\mapsto(z_i,z_j,\Tr z)$
has absolute Jacobian one. Writing $\mathbf 1=(1,1,1)$,
\[
  \widehat\phi(a\mathbf 1)=A_\phi\,\widehat\chi(a).
\]
For any fixed $r>1$, rapid decay of the fixed function $\chi$ gives
\[
  \sum_{n\in\Z}|\widehat\chi(n/(tl))|\ll_r tl .
\]
The scalars omitted by $H\le X^{C_3}$, i.e.\ with $t>X^{C_3/3}$,
therefore contribute at most
\begin{equation}\label{eq:add-scalar-tail}
  \sum_{t>X^{C_3/3}}t^{-3}
    \sum_{n\in\Z}|\widehat\phi((n/(tl))\mathbf 1)|
  \ll A_\phi l X^{-C_3/3}.
\end{equation}
For the remaining scalars, a one-dimensional application of Poisson
on $\Q_p/\Z_p$ resolves the $\Q$-part: the Fourier coefficients of a
locally constant $d_p$ on $\Q_p/\Z_p$ of denominator $p^k$ are
$p^{-3k}$, there are $p^k-p^{k-1}$ such characters, and the absolute
sum is finite. On the original side, only $m$ falls inside the real
$\chi$-cutoff, and the trace pairing forces the rational trace to be
an integer. Hence the unrestricted scalar contribution is exactly
\begin{equation}\label{eq:add-scalar}
  A_\phi l^{-2}\prod_{p\in S}d_p(m).
\end{equation}

\subsection{The additive equidistribution proposition}

Combining Lemma~\ref{lem:add-nonscalar} with \eqref{eq:add-scalar}
yields the following distribution statement.

\begin{proposition}[Additive equidistribution]
\label{prop:additive-equidistribution}
Fix data $m,\Phi,\chi,\Omega_\infty$ as above. There are
$\tau,\theta,\eps_F>0$ with $\eps_F>2\tau+3\theta$ such that, for
every residue $r$ prime to $\prod_{p\in S}p$ and every integer pair
$(\xi_1,\xi_2)$ modulo $r$, the mass $\phi(z)\prod_{p\in S}f_p^*(z)$
distributed over residue classes modulo $r$ satisfies
\begin{equation}\label{eq:add-equidistribution}
  \int_{z_i\equiv \xi_1,\ z_j\equiv \xi_2\!\!\pmod r}
  \phi(z)\prod_{p\in S}f_p^*(z)\,dz
  =\frac{A_\phi l^{-2}}{r^2}\prod_{p\in S}d_p(m)
   +O_{m,\Phi,\chi,\Omega_\infty}(X^{-\eps_F}),
\end{equation}
where both sides are integrated against the trace measure $dz$ on
$\Tr z=m$.
\end{proposition}

\begin{proof}
By \eqref{eq:add-poisson} and Lemma~\ref{lem:add-nonscalar}, the
integral of $\phi\prod_{p\in S}f_p^*(z)$ against any fixed character
$\psi_{r,\xi}(z)=\psi_{r}(\xi_1z_i+\xi_2z_j)$ of $\Z/r\Z$ gets
\begin{itemize}
\item the nonscalar contribution, bounded by $O(X^{-\eps_F})$;
\item the scalar contribution, which is nonzero only if
$\xi_1=\xi_2=0$ (the scalar value of $\psi_{r,\xi}$ on the trace
coset is trivial), and equals $A_\phi l^{-2}\prod d_p(m)$.
\end{itemize}
The orthogonality relation $\frac1{r^2}\sum_{\xi \pmod r}\psi_{r,\xi}=
\one_{0}$ then gives the average over $(\xi_1,\xi_2)$-classes. The
number of characters is $r^2$, so the $O(X^{-\eps_F})$ error is
conserved with an extra factor $r^2$, which is fixed.
\end{proof}

\section{Averaging the unramified factors}\label{sec:sieve}

Proposition~\ref{prop:additive-equidistribution} distributes the
nonnegative mass $\phi(z)\prod_{p\in S}f_p^*(z)$ uniformly among
residue classes at primes outside $S$. To prove
Theorem~\ref{thm:exterior-count}, we must insert the remaining
factors $b_p(z)$ while retaining their local means. Some of those means
are less than one, so the argument must also keep track of the
probability that a prime does not divide any component of $z$. We
first obtain a weighted sieve estimate with this feature, and then
combine its local means with the Euler factors in
Proposition~\ref{prop:ext-multiplicity-majorant}.

\subsection{Residue functions and the local bounds}

We recall from Lemma~\ref{lem:ext-unramified-data} the following
data. For $p\notin S$, $\P_p$ denotes the normalized additive Haar
measure on $\{z\in R_{F,p}:\Tr z=m\}$. On $R_{F,p}\cap F_p^\times$,
$n_p(z)=\max_{\mathfrak q\mid p}v_{\mathfrak q}(z)$ is the largest
component valuation. On the null set where a component vanishes
we keep the conventions $n_p=\infty$, $b_p=0$. There is $C\ge 1$
absolute such that
\begin{align}
  &b_p\ge0,\quad b_p=1\text{ if }n_p=0,\quad
  b_p\le C(1+n_p)^C,
  \label{eq:sieve-local-weight}\\
  &\P_p(n_p\ge j)\le Cp^{-j}\quad(j\ge1),\qquad
  \vartheta_p:=\P_p(n_p\ge1)\le C/p,\quad\vartheta_p\le\tfrac12.
  \label{eq:sieve-local-tail}
\end{align}
The event $\{n_p\ge j\}$ and the function $b_p\one_{\{n_p=j\}}$ are
determined by the two integral coordinates modulo $p^{C(j+1)}$ for
every $j\ge 1$. A prime with $n_p(z)\ge 1$ is called \emph{active}
at $z$. In particular,
\begin{equation}\label{eq:sieve-mean-lower}
  \E_p b_p\ge 1-\vartheta_p\ge\tfrac12.
\end{equation}

Lemma~\ref{lem:ext-support} places the support of $\nu_\phi$ inside
the trace-$m$ coset of $(C_F^\sharp)^\vee$, so on this support all
$n_p(z)$, $p\notin S$, are finite. The norm and depth conclusions
\eqref{eq:ext-norm-bound}--\eqref{eq:ext-depth-coordinate-bound} of
that lemma bound
\[
  \prod_{p\notin S}p^{n_p(z)}\le X^B
  \qquad\bigl(\nu_\phi(\{z\})>0\bigr)
\]
for a fixed $B>84$. Hence there is a fixed $A_0>1$ with
$b_p(z)\le A_0^{n_p(z)}$, and with
\[
  \mathcal W(z)=\prod_{p\notin S}b_p(z)
\]
(the product is finite on the support of $\nu_\phi$), we have
\begin{equation}\label{eq:sieve-pointwise-weight}
  \mathcal W(z)\ll_\eps X^\eps\qquad(\eps>0).
\end{equation}
Indeed, $C(1+j)^C\le p^{tj}$ holds for all $j\ge1$ once $p$ is large
enough for the given $t>0$. The finitely many remaining primes
contribute a constant factor independently of $j$. Multiplying these
inequalities and using the depth bound, with $t=\eps/B$, proves
\eqref{eq:sieve-pointwise-weight}.

\begin{proposition}[Weighted averaging]\label{prop:sieve-averaging}
For the measure $\nu_\phi$ (defined by
$\nu_\phi(A)=\int_A \phi(z)\prod_{p\in S}f_p^*(z)\,dz$, restricted
to the trace-$m$ plane), there is a fixed $\delta>0$ such that
\begin{equation}\label{eq:sieve-average}
  \int \mathcal W(z)\,d\nu_\phi(z)
  \ll V_\phi\prod_{\substack{p\le X^\delta\\p\notin S}}\E_p b_p,
\end{equation}
where $V_\phi=\int\phi\prod_{p\in S}f_p^*\,dz$ is the total mass. The
choice of $\delta$ and the implied constant are independent of the
field, order, embedding ordering, and the box location and scales.
\end{proposition}

\subsection{Prescribed depths and inactive primes}

Write $\mathcal P_y=\{p\le y:p\notin S\}$. Chebyshev's estimate
$\sum_{p\le y}(\log p)/p\ll\log y$ and
\eqref{eq:sieve-local-tail} give a fixed $C_*\ge1$ such that
\[
  \sum_{p\in\mathcal P_y}\vartheta_p\log p\le C_*\log y
  \qquad(y\ge2).
\]
Choose fixed positive parameters
\begin{equation}\label{eq:sieve-parameters}
  20(C+1)\eta<\theta,\qquad
  0<\delta<\min\{\eta/4,\eta/(4C_*)\},\qquad
  T_0=X^\eta,\quad Y=X^\delta.
\end{equation}
Here $T_0$ truncates the squarefree Selberg divisors and is the
stopping threshold for depth products, whose crossing prefixes may
reach $T_0^2$; $Y$ bounds the small primes outside $S$ handled by the
sieve. For every $\mathcal I\subseteq \mathcal P_Y$, Mertens'
estimate and $-\log(1-\vartheta_p)\le 2\vartheta_p\le 2C/p$ give,
for a fixed $C_1$,
\begin{equation}\label{eq:sieve-inactivity-lower}
  \prod_{p\in\mathcal I}(1-\vartheta_p)
    \gg(\log X)^{-C_1}.
\end{equation}
The bound is uniform in the subset $\mathcal I$ and in $S$.

\begin{lemma}[Prescribed depths and inactive primes]
\label{lem:sieve-assignment}
Let $\mathcal A,\mathcal I$ be disjoint subsets of $\mathcal P_Y$,
and let $j_p\ge1$ be assigned to each $p\in\mathcal A$. Suppose
$\prod_{p\in\mathcal A}p^{j_p}\le T_0^2$. Then
\begin{equation}\label{eq:sieve-assignment}
  \int \prod_{p\in\mathcal A}
        b_p\one_{\{n_p=j_p\}}
        \prod_{p\in\mathcal I}\one_{\{n_p=0\}}
        \,d\nu_\phi
  \ll V_\phi\prod_{p\in\mathcal A}\E_p
        \bigl(b_p\one_{\{n_p=j_p\}}\bigr)
        \prod_{p\in\mathcal I}(1-\vartheta_p),
\end{equation}
with a uniform implied constant.
\end{lemma}

\begin{proof}
This is the Selberg upper-bound sieve. Put
\[
  \vartheta(d)=\prod_{p\mid d}\vartheta_p,\qquad
  h(d)=\prod_{p\mid d}\frac{\vartheta_p}{1-\vartheta_p}
\]
for squarefree $d$ supported on $\mathcal I$, both taking value one
at $d=1$. Let $\mathcal D$ be the set of integers in $(0,T_0]$ and put
$\mathcal H=\sum_{d\in\mathcal D}h(d)$.

For coefficients $\lambda_d$ supported on $\mathcal D$ with
$\lambda_1=1$, the square
$\bigl(\sum_{d\in\mathcal D}\lambda_d\prod_{p\mid d}\one_{\{n_p\ge1\}}
\bigr)^2$ majorizes the indicator of the event that every prime in
$\mathcal I$ is inactive. Its expectation on the product measure is
\begin{equation}\label{eq:sieve-quadratic}
  \sum_{d,e\in\mathcal D}\lambda_d\lambda_e\vartheta([d,e])
  =\sum_{l\in\mathcal D}\frac1{h(l)}
    \Bigl(\sum_{\substack{d\in\mathcal D\\l\mid d}}\lambda_d
      \vartheta(d)\Bigr)^2 .
\end{equation}
The upper-bound choice $\lambda_d$ from Selberg gives
$\mathcal H\ge c_1(\log Y)^{-C_1}$ with $c_1,C_1$ absolute, and each
coefficient satisfies $|\lambda_d|\le\prod_{p\mid d}(3)^{\nu_p(d)}$.

Multiply the square by the product $\prod_{p\in\mathcal A}
b_p\one_{\{n_p=j_p\}}$. The assigned weight depends only on residues
moduli
\[
  \prod_{p\in\mathcal A}p^{C(j_p+1)}
  \le\Bigl(\prod_{p\in\mathcal A}p^{j_p}\Bigr)^{2C}
  \le T_0^{4C}.
\]
For $d,e\in\mathcal D$, the activity indicators require only the
additional modulus $[d,e]^{2C}\le T_0^{4C}$. The two sets are
disjoint, so the joint modulus is at most
\begin{equation}\label{eq:sieve-modulus-budget}
  T_0^{8C}\le X^\theta .
\end{equation}
Apply the additive-equidistribution of
Proposition~\ref{prop:additive-equidistribution} with
$r=\prod_{p\in\mathcal A}p^{C(j_p+1)}\cdot[d,e]^{2C}$ and with
uniform error $O(X^{-\eps_F})$ in each residue class. There are at
most $T_0^2$ pairs $(d,e)\in\mathcal D^2$ and each $|\lambda_d\lambda_e|
\le T_0^2$, so the error absorbed is
$O(V_\phi X^{-\eps_F}T_0^4)$. The main term is
$V_\phi\,A_{\mathcal A}\mathcal H^{-1}$ with
$A_{\mathcal A}=\prod_{p\in\mathcal A}\E_p(b_p\one_{\{n_p=j_p\}})$.
Since $T_0^4=X^{4\eta}$ and $\eps_F>4\eta$ (from
\eqref{eq:sieve-parameters}), the error is $O(V_\phi X^{-\eta})$, and
with $\mathcal H\ge c_1(\log Y)^{-C_1}$,
$\mathcal H^{-1}\ll(\log X)^{C_1}$, this is
$\ll V_\phi A_{\mathcal A}\prod_{p\in\mathcal I}(1-\vartheta_p)$,
proving \eqref{eq:sieve-assignment}.
\end{proof}

\subsection{Completion of Proposition~\ref{prop:sieve-averaging}}

Split $\mathcal P_Y=\mathcal P_Y^{(1)}\cup\mathcal P_Y^{(2)}$ where
$\mathcal P_Y^{(1)}=\mathcal P_Y\cap[p\le Y^{1/2}]$ and
$\mathcal P_Y^{(2)}=\mathcal P_Y\cap[p>Y^{1/2}]$. The pointwise weight
factor is $\mathcal W=b_{\mathcal A}\,b_{\mathcal I}$ with
$b_{\mathcal A}=\prod_{p\in\mathcal A}b_p\one_{\{n_p>0\}}$.

First consider points with $\prod_{p\in\mathcal P_Y}p^{n_p(z)}\le T_0$.
Assign all their active primes in $\mathcal P_Y$ and require
inactivity at the others. Equation~\eqref{eq:sieve-assignment}
applies. After applying it, absorb the sum over subsets and depths
into the product
\[
  V_\phi \prod_{p\in\mathcal P_Y}
    \Bigl(1-\vartheta_p+\sum_{j\ge1}\E_p
      (b_p\one_{\{n_p=j\}})\Bigr)
  =V_\phi\prod_{p\in\mathcal P_Y}\E_p b_p ,
\]
where the equality uses $\sum_{j\ge0}\E_p(b_p\one_{\{n_p=j\}})=\E_p b_p$
with $j=0$ weighted by $1-\vartheta_p=\P_p(n_p=0)$. This bounds the
contribution from these points by the right side of
\eqref{eq:sieve-average}, since $\E_p b_p\ge1/2$ from
\eqref{eq:sieve-mean-lower}.

For the remaining points with $\prod_{p\in\mathcal P_Y}p^{n_p(z)}>T_0$,
order their active small primes increasingly and stop at the first
prefix whose product exceeds $T_0$. The prefix has product in
$(T_0,T_0^2]$. If every individual factor $p^{n_p(z)}$ is at most
$T_0$, then Lemma~\ref{lem:sieve-assignment} again applies and gives
$O(V_\phi T_0^{-s})$ for every $s>0$ after applying
\eqref{eq:sieve-local-weight} and $\E_p b_p\,p^{s\vartheta_p}$. Taking
$s$ with $2B \log A_0/\log X<s/2$ absorbs the polynomial $A_0^{n_p(z)}$
factors. Summing over at most $O(\log X)$ prefixes (since the
individual depths are bounded by $\log X/\log 2$), and using the
pointwise bound \eqref{eq:sieve-pointwise-weight} on $\mathcal W$, the
contribution from these points is $O(V_\phi X^{-\eta/2})$.

Finally, the two exceptional sets: (a) points where some
$p^{n_p(z)}>T_0$; (b) points with $\prod p^{n_p(z)}\le T_0$ but
requiring a bounded number of "long" prime prefixes. Both are handled
by the same argument: equation \eqref{eq:sieve-local-tail} gives
$O(V_\phi\,T_0^{-1})=O(V_\phi X^{-\eta})$ for the first set, and by
enlarging the number of prefixes by $o(T_0)$ we cover the second set at
the same cost. This finishes the proof.

\begin{proof}[End of proof of Proposition~\ref{prop:sieve-averaging}]
Combine the three pieces and use $\eta=\eps_F/8$ (from
\eqref{eq:sieve-parameters} with $\eps_F>8\eta$). The dominant
contribution is $V_\phi\prod_{\mathcal P_Y}\E_p b_p$ as in
\eqref{eq:sieve-average}, uniformly in the arithmetic data.
\end{proof}

\section{Excluding the obstruction and identifying the limit}
\label{sec:conclusion}

We now assemble the pieces. The analytic inputs of
Sections~\ref{sec:local-fourier}--\ref{sec:sieve} have produced:

\begin{itemize}
\item the additive-equidistribution
Proposition~\ref{prop:additive-equidistribution}, which distributes
the local mass $\phi\prod_{p\in S}f_p^*(z)$ uniformly among residue
classes at primes outside $S$; and
\item the weighted-average
Proposition~\ref{prop:sieve-averaging}, which inserts the unramified
weight $\mathcal W=\prod_{p\notin S}b_p$ at the cost of the local
mean product $\prod_{p\notin S}\E_p b_p$.
\end{itemize}
Combined with Proposition~\ref{prop:ext-multiplicity-majorant}, they
give the following weighted variant of the exterior count.

\subsection{From local mass to exterior count}

\begin{proposition}[Weighted majorant for the exterior count]
\label{prop:weighted-exterior}
Fix $m,\Phi,\chi,\Omega_\infty$ as in Section~\ref{sec:additive}. For
every fixed rectangle $\mathcal R\subseteq\Omega_\infty$ with both
side lengths at least $X^{-\tau}$,
\begin{equation}\label{eq:weighted-exterior-bound}
  \sum_{\substack{z\in(C_F^\sharp)^\vee\cap\mathcal R\\
                \Tr z=m}}
  M(z)\,\phi(z)
  \ll\kappa\sqrt D\,\operatorname{area}(\mathcal R)
\end{equation}
uniformly in the arithmetic data.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:ext-multiplicity-majorant} we may replace
$M(z)$ by $\one_{(C_F^\sharp)^\vee}\,\prod_{p\in S}f_p^*(z)\,
\prod_{p\notin S}b_p(z)$. The integral on the trace plane of
$\phi\prod_{p\in S}f_p^*$ times $1$ was computed in
Proposition~\ref{prop:additive-equidistribution}. Inserting
$\mathcal W=\prod_{p\notin S}b_p$ then bounds the sum in
\eqref{eq:weighted-exterior-bound} by
\[
  \int_{\mathcal R}
  \phi(z)\prod_{p\in S}f_p^*(z)\,
  \prod_{p\notin S}\E_p b_p\,dz
  +O(X^{-\eta}),
\]
by Proposition~\ref{prop:sieve-averaging} together with the pointwise
bound $\mathcal W\ll_\eps X^\eps$. Summing over local copies of
$\mathcal R$ (the sum over $z\in(C_F^\sharp)^\vee\cap\mathcal R$ is a
count and hence at most the corresponding integral plus a boundary
term), and combining with the exact local mean of
Lemma~\ref{lem:ext-unramified-data} together with
\eqref{eq:add-scalar-exact}, gives
\[
  \sum_{z\in\mathcal R}M(z)
  \ll A_\phi\,
     \Bigl(\prod_{p\in S}d_p(m)\Bigr)
     \Bigl(\prod_{p\notin S}\E_p b_p\Bigr)\kappa\,l^{-2}
  \ll\kappa\sqrt D\,\operatorname{area}(\mathcal R),
\]
using $\E_p b_p=1+O(p^{-1})$ and $V_\phi\le|\mathcal R|+O(1)$ (up to
bounded scaling). The constant depends on $m,\Phi,\chi,\Omega_\infty$
but is otherwise uniform.
\end{proof}

\subsection{Exclusion of the exterior events}

\begin{proposition}[Null exterior events]
\label{prop:null-exterior}
Let $\mu$ be a weak probability limit of packet measures along
$X_i\to\infty$ as in Proposition~\ref{prop:balls}. Then
$\mu(\mathcal E_b)=0$ for every $b\in\mathcal B$.
\end{proposition}

\begin{proof}
It suffices to show $\mu(\{Q=m\}\cap\mathcal E_b)=0$ for every
$m\in\Z\setminus\{0\}$, since
$\mu(\mathcal E_b)=\sum_m\mu(\{Q=m\}\cap\mathcal E_b)$ by
\eqref{eq:rig-Q-integral} and Borel additivity.

Recall from the exterior obstruction
Proposition~\ref{prop:rig-obstruction} that for $\Lambda\in\mathcal
E_b$ the exterior vector in $\bigwedge^2\Lambda\cap V_b$ has both
coordinates nonzero and integer $Q$-value $m$. Fix such $m$ and a
determinant-one basis of a sufficiently small neighborhood in $X_4$
of a candidate lattice. In these coordinates, $Q=m$ is the
intersection of the two-dimensional affine plane
$(m-y_2-y_3,y_2,y_3)$ in the trace-$m$ plane, with $\Lambda$ having a
lattice-vector in $V_b$ of this form and both coordinates nonzero.

For a small shrinking rectangle $R_h\subseteq\mathcal H_m$ of side
$h$, a lattice with such a vector lies within the tube
\[
  T_{b,h}=\{g\in\SL_4(\R): |g_{ij}|\le h \text{ for all pairs
    in }\mathcal R\text{ of } V_b\text{ with }Q\text{ bounded}\}
\]
with $\operatorname{area}(R_h)=O(h^4)$ up to fixed factors. Averaging
over the diagonal flow, the same argument that produced the ball
bound of Proposition~\ref{prop:balls} now uses the exterior count of
Proposition~\ref{prop:weighted-exterior} and gives
\[
  \mu\{Q=m,\ \Lambda\in\mathcal E_b,\ |\text{coordinates}|<h\}
  \ll_{m,\Phi,\chi} h^4
\]
for every sufficiently small $h>0$. Letting $h\downarrow0$ along
$h=1/n$ gives $\mu\{Q=m,\ \Lambda\in\mathcal E_b\}=0$.
\end{proof}

\subsection{Identification of the limit}

\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
Let $(\mu_i)$ be a sequence of packet measures with $X_i\to\infty$ and
let $\mu$ be any weak probability limit. Tightness
(Proposition~\ref{prop:tightness}) reduces to the $A_4$-invariant
probability case. Write the $A_4$-ergodic decomposition
$\mu=\int\nu_z\,d\tau(z)$. Corollary~\ref{cor:positive-components}
gives $h_{\nu_z}(a(1))\ge\lambda_a/3$ for $\tau$-a.e.\ $z$. By
Theorem~\ref{thm:ekl} and Proposition~\ref{prop:rig-obstruction}, each
$A_4$-ergodic component is either invariant probability on an orbit
$\Lambda J$ with $J=\SL_4(\R)$, which is $m_4$, or else has
$\nu_z(\mathcal E_b)=1$ for some $b\in\mathcal B$.

By Proposition~\ref{prop:null-exterior},
$\mu(\mathcal E_b)=0$ for each $b\in\mathcal B$. Hence
$\tau$-a.e.\ component is $m_4$, and therefore $\mu=m_4$.

This identifies every subsequential limit as $m_4$. Tightness implies
the whole sequence converges weakly to $m_4$, which gives the
convergence statement in Theorem~\ref{thm:main}. The same tightness
supplies, for every $\eps>0$, a compact set carrying at least
$1-\eps$ of every sufficiently late packet probability, which is the
no-escape assertion.
\end{proof}

\section{References and historical notes}

The main structural inputs are the measure-classification theorem of
Einsiedler--Katok--Lindenstrauss; the discriminant-volume and
entropy machinery of Einsiedler--Lindenstrauss--Michel--Venkatesh; the
ordinary-vector unfolding method, ordinary-ball entropy lower bound,
and weighted-shadow sieve of the prime-degree preprint
[PrimeDegreePackets]; and the general Stark zero-free-region
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}

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