Proof. Normalize additive Haar measure on \(R_\ell\) to have mass one. The valuation shell \(\mathbf k_\ell\) has mass \(Z_\ell^{-1}/d_\ell(\mathbf k_\ell)\). Its conditional law is invariant under the transitive unit action, so \(P_\ell\) is exactly the fraction of this shell lying in \(L_\ell\). Since \(L_\ell\) has mass \(1/q_\ell\), summing over shells and then multiplying over \(S\) proves (8). Elements with a zero component have additive measure zero.

Choose an additive-Haar uniform element of \(L_\ell\), and denote its component valuations by \(v_{\mathfrak p}\). The condition \(R_\ell L_\ell=R_\ell\) implies that the projection to every component contains a unit. Modulo the \(k\)th power of its maximal ideal, the projection therefore contains all \(\mathbb Z_\ell\) multiples of that unit, at least \(\ell^{\lceil k/e_{\mathfrak p}\rceil}\) elements. Uniform measure on \(L_\ell\) induces uniform measure on this finite additive image. Consequently \[ \Pr(v_{\mathfrak p}\ge k) \le\ell^{-\lceil k/e_{\mathfrak p}\rceil}\qquad(k\ge1). \tag{11}\] This argument does not require the component projections to be independent, or even to equal their full integer rings.

Put \(j=\max_{\mathfrak p\mid\ell} \lceil v_{\mathfrak p}/e_{\mathfrak p}\rceil\). Then \(d_\ell(\mathbf v_\ell)\le\ell^{nj}\). If \(j\ge a\ge1\), some \(v_{\mathfrak p}\) is at least \(e_{\mathfrak p}(a-1)+1\). Equation (11) and a union bound give \(\Pr(j\ge a)\le n\ell^{-a}\). Hence \[ \mathbb E_{L_\ell}d_\ell(\mathbf v_\ell)^s \le1+n\sum_{a\ge1}\ell^{(ns-1)a} \le1+C_n\ell^{-1/2}. \tag{12}\] Using the shell masses once more yields the exact factorization \[\sum_{\mathbf k}P(\mathbf k)d(\mathbf k)^{s-1} =\frac1q\prod_{\ell\in S} \left(Z_\ell\,\mathbb E_{L_\ell}d_\ell(\mathbf v_\ell)^s\right).\] Each factor is at most \((1-\ell^{-1})^{-n}(1+C_n\ell^{-1/2})\). For fixed \(\varepsilon>0\) this is at most \(\ell^\varepsilon\) at all sufficiently large primes; the finitely many remaining primes contribute a constant depending only on \(n,\varepsilon\). As \(\prod_{\ell\in S}\ell\le q\), this proves (9). Finally, take \(\varepsilon=s/16\) and use \(q\le Q\) to obtain \[\sum_{d>Q^{1/8}}\frac{P(\mathbf k)}d \le Q^{-s/8}\sum_{\mathbf k}P(\mathbf k)d^{s-1} \ll_n q^{-1}Q^{-s/16}. \qedhere\] ◻

Proof. Put \(L=\log D\). Stark’s Lemma 3 gives at most one zero in \(\operatorname{Re}s\ge1-(4L)^{-1}\), \(|\operatorname{Im}s|\le(4L)^{-1}\); such a zero is real and simple. His Lemma 8, which also applies to nonnormal fields, says that a real zero \(\beta\ge1-(4n!L)^{-1}\) forces a quadratic subfield of \(K\) (Stark 1974, Lemmas 3 and 8). Since \(n\) is odd, there is no such subfield. Consequently, for a constant \(c_n>0\), every nontrivial zero \(\rho\) satisfies \(|1-\rho|\ge c_n/L\).

The real logarithmic derivative of the completed zeta function gives \[S_K(\sigma):=\sum_\rho\operatorname{Re}\frac1{\sigma-\rho} =\frac1\sigma+\frac1{\sigma-1}+\frac L2 +\frac{\zeta_K'}{\zeta_K}(\sigma)+G_n(\sigma),\qquad \sigma>1,\] where zeros are counted with multiplicity and \[G_n(\sigma)=\frac n2 \left(\frac{\Gamma'(\sigma/2)}{\Gamma(\sigma/2)}-\log\pi\right).\] This is the Hadamard identity in (Stark 1974, Equation (9)). The zero terms are nonnegative, \(G_n\) is bounded on \([1,2]\), and the Euler product gives \(\zeta_K'/\zeta_K\le0\) there. Thus, at \(\sigma_0=1+L^{-1}\), \(S_K(\sigma_0)\ll_n L\).

This bound persists uniformly for \(1<\sigma\le1+4/L\). Indeed, for \(\rho=\beta+i\gamma\), set \(X=L(1-\beta)\), \(Y=L\gamma\), and \(U=L(\sigma-1)\). Then \(X\ge0\), \(0<U\le4\), and \(|X+iY|\ge c_n\). Hence \[\frac{\operatorname{Re}(\sigma-\rho)^{-1}} {\operatorname{Re}(\sigma_0-\rho)^{-1}} =\frac{X+U}{X+1} \frac{(X+1)^2+Y^2}{(X+U)^2+Y^2} \le4(1+3/c_n)^2.\] Summing gives \(0\le S_K(\sigma)\ll_n L\). For \(F_K(\sigma)=(\sigma-1)\zeta_K(\sigma)\), it follows that \[\frac{F_K'}{F_K}(\sigma) =S_K(\sigma)-\frac1\sigma-\frac L2-G_n(\sigma) =O_n(L).\] Since \(F_K\) extends analytically to \(1\) with \(F_K(1)=\kappa_K>0\), integration over an interval of length at most \(4/L\) proves the comparison in (13). At \(\sigma_0\), the inequality \(\zeta_K(\sigma_0)\ge1\) also gives \(\kappa_K\gg_n L^{-1}\).

For completeness, only finitely many fields of this fixed degree have discriminant below the cutoff. One can see this directly here: nonzero vectors of \(D^{-1/(2n)}\sigma(R)\) have absolute coordinate product at least \(D^{-1/2}\), so Mahler’s criterion puts these lattices in a fixed compact set. Local basis lifts and a finite cover give bases of \(R\) with uniformly bounded conjugates. A basis element outside \(\mathbb Q\) generates \(K\), since \(n\) is prime; its monic integral minimal polynomial therefore has bounded coefficients. There are only finitely many such polynomials. The simple pole of each of these finitely many zeta functions now gives the remaining claims uniformly for sufficiently small \(h>0\). ◻

Proof. Set \(b(m)=a_K(m)\mathbf 1_{(m,q)=1}\). This function is nonnegative, multiplicative, and bounded by \(d_n\). Indeed, at every prime \(\ell\), including the ramified primes, the local generating series for \(a_K\) is \(\prod_{\mathfrak p\mid\ell}(1-T^{f_{\mathfrak p}})^{-1}\); it is coefficientwise bounded by \((1-T)^{-n}\). Decomposing an ideal into its part over \(S\) and its coprime part, the sum in (15) becomes \[ \sum_{\mathbf k}P(\mathbf k) \sum_{(x-y)/d(\mathbf k)<m\le x/d(\mathbf k)}b(m). \tag{16}\]

First suppose \(d=d(\mathbf k)\le Q^{1/8}\), and put \(u=x/d\). Then \(u\ge Q^{5/8}\ge D^{1/4}\). For \(h=1/\log u\), Lemma 5 gives \(\zeta_K(1+h)\ll_n\kappa_K\log u\). There is no upper restriction on \(u\): increasing \(u\) only decreases \(h\), so the lemma applies even when \(D\) stays bounded and \(q\) grows. Positivity of the Euler logarithm yields \[\sum_{\ell\le u}a_K(\ell)\ell^{-1-h}\le\log\zeta_K(1+h).\] Since \(a_K(\ell)\le n\) and \(\sum_{\ell\le u}(\log\ell)/\ell\ll\log u\), replacing \(\ell^{-1-h}\) by \(\ell^{-1}\) costs only \(O_n(1)\). The latter prime-sum bound follows, for example, by comparing \(\sum_{\ell\le u}\lfloor u/\ell\rfloor\log\ell\) with \(\log(\lfloor u\rfloor!)\). Therefore \[\exp\left(\sum_{\ell\le u}\frac{a_K(\ell)}\ell\right) \ll_n\kappa_K\log u.\] Furthermore, \[\log Z_S=\sum_{\ell\in S}\frac{a_K(\ell)}\ell+O_n(1), \qquad \sum_{\substack{\ell\in S\\\ell>u}} \frac{a_K(\ell)}\ell \ll_n\frac{\log(2q)}{Q^{5/8}}\ll_n1.\] The first error is bounded by a constant times \(\sum_\ell\ell^{-2}\). Removing the primes of \(S\) thus gives \[ \exp\left(\sum_{\ell\le u}\frac{b(\ell)}\ell\right) \ll_n\frac{\kappa_K\log u}{Z_S}. \tag{17}\] For \(Q\) sufficiently large at fixed \(\eta\), the length \(v=y/d\ge\eta u\) is at least \(u^{1/2}\). Applying (14), then (17) and (8), bounds the contribution of these \(d\) by \[C_n\frac{\kappa_Ky}{Z_S} \sum_{d(\mathbf k)\le Q^{1/8}}\frac{P(\mathbf k)}{d(\mathbf k)} \le C_n\frac{\kappa_Ky}{q}.\]

For \(d>Q^{1/8}\), terms with \(d>x\) vanish because the inner sum in (16) then contains no positive integer. For the remaining terms, \(x/d\ge1\), and the elementary bound \(\sum_{m\le z}d_n(m)\le z(1+\log z)^{n-1}\) for \(z\ge1\) follows by summing reciprocals in the first \(n-1\) factors. Using (10), with \(s=1/(2n)\), the remaining contribution to (16) is at most \[C_n\frac{x}{q}Q^{-s/16}(1+\log x)^{n-1} =o\left(\frac{\kappa_Kx}{q}\right).\] This is uniform in the stated range: Lemma 5 gives \(\kappa_K^{-1}\ll_n\log(3D)\), and \(D\le Q^2\). At fixed \(A_0,\eta\), increase the threshold until this term is at most \(\kappa_K\eta x/q\le\kappa_Ky/q\). The implied constant has not changed with \(\eta\), proving the proposition. ◻

Proof. For \(f=\mathbf 1_{[-\theta,\theta]^n}\), the first \(n-1\) logarithms in (6) are at most \(\log\theta\). The last coordinate bounds the sum of their gaps from \(\log\theta\) by \(\log(\theta^n/z)\). Thus \[\mathcal V_f(z)=\frac{2^n}{(n-1)!} \bigl(\max\{0,\log(\theta^n/z)\}\bigr)^{n-1}.\] Put \(X=\theta^nQ\) and \(U=Q^{3/4}\). Up to an \(n\)-dependent constant, Proposition 3 reduces the first moment to \[\frac1{\sqrt D\,\kappa_K} \sum_{\mathop{\mathrm{N}}\mathfrak a\le X} P(\mathfrak a)\left(\log\frac X{\mathop{\mathrm{N}}\mathfrak a}\right)^{n-1}.\] If \(X<1\), the sum is empty. Otherwise its logarithmic factor is at most \((\log Q)^{n-1}\), since \(\theta<1\). The terms with \(\mathop{\mathrm{N}}\mathfrak a\le\min(U,X)\) contribute at most \(C_nQ^{-1/4}(\log Q)^{n-1}\), by applying Proposition 6 at \(x=y=U\). When \(X>U\), define the truncated cumulative sum \[A_U(t)=\sum_{U<\mathop{\mathrm{N}}\mathfrak a\le t}P(\mathfrak a), \qquad U\le t\le X.\] Writing each logarithmic power as an integral and applying Tonelli gives \[\begin{align*} \sum_{U<\mathop{\mathrm{N}}\mathfrak a\le X} P(\mathfrak a)\left(\log\frac X{\mathop{\mathrm{N}}\mathfrak a}\right)^{n-1} &= (n-1)\int_U^X A_U(t) \left(\log\frac Xt\right)^{n-2}\frac{dt}{t} \\ &\ll_n \frac{\kappa_K}{q} \int_U^X\left(\log\frac Xt\right)^{n-2}\,dt \ll_n \frac{\kappa_K}{q}X. \end{align*}\] Here the same proposition at \(x=y=t\) bounds \(A_U(t)\), and the last inequality follows from \(\int_0^1(\log(1/v))^{n-2}\,dv<\infty\). There is no lower boundary term, since \(A_U(U)=0\), and the logarithmic weight vanishes at \(X\). If \(X\le U\), this part is absent. Dividing by \(\sqrt D\,\kappa_K\) gives \(O_n(\theta^n)\). Since \(E_f\ge1\) whenever the displayed cusp event occurs, this proves (18).

Mahler’s compactness criterion (Mahler 1946, sec. 3, Theorem 2) says that the complement of this cusp event is compact. Given a desired mass error, choose \(\theta\) first and then \(Q\) sufficiently large. The finitely many earlier packet supports are compact by Lemma 2, so the whole sequence is tight. The lattice space is locally compact and second countable; weak compactness for tight probabilities gives probability subsequential limits. Invariance passes to them by testing against continuous compactly supported functions. ◻

Proof. First fix a compact \(V\subset(\mathbb R^\times)^n\) and \(b_0\ge1\). For \(w\in V\) and small \(r\), take \(f=\mathbf 1_{w+[-b_0r,b_0r]^n}\). Each coordinate has a fixed sign, is uniformly separated from zero, and has logarithmic width \(O_{V,b_0}(r)\). Its absolute product lies within \(Cr\) of \(z_w=\prod_j|w_j|\), so \[\mathcal V_f(z)\le Cr^{n-1} \mathbf 1_{[z_w-Cr,z_w+Cr]}(z)\] for some \(C\ge1\) depending only on \(V,b_0,n\). Choose \(r_0\) small enough that \(z_w>4Cr_0\) on \(V\). Use Proposition 6 with \[x=(z_w+2Cr)Q,\qquad y=4CrQ.\] This slightly enlarged interval contains every contributing ideal norm. Uniformly in \(w\), for sufficiently large \(Q\) it has \(Q^{3/4}\le x\le A_0Q\) with a fixed \(A_0\ge1\), and \(\eta x\le y\le x\) with \(\eta=4Cr/A_0\le1\), after decreasing \(r_0\) if necessary. For every fixed \(r\in(0,r_0)\), Proposition 3 therefore gives, eventually, \[ \int E_{\mathbf 1_{w+[-b_0r,b_0r]^n}}\,d\mu_* \ll_{V,b_0,n} \frac{r^{n-1}}{\sqrt D\,\kappa_K} \frac{\kappa_KrQ}{q}=r^n . \tag{20}\] Here \(Q=q\sqrt D\). The implicit constant is independent of \(r\). The threshold for \(Q\) may depend on \(r\), which is all that will be needed.

We next choose a suitable lattice vector at each \(x\in\Omega\). Every full lattice has a vector with no zero coordinate: for any basis, its combination with coefficients \((1,N,\ldots,N^{n-1})\) has coordinates that are nonzero polynomials in \(N\), so an integer \(N\) can avoid their finitely many roots. Near each center choose a continuous local lift to basis matrices, and continue the chosen vector using its fixed integer basis coefficients. On a sufficiently small neighborhood with compact closure in that chart, its coordinates remain bounded and bounded away from zero. A finite cover of \(\Omega\) by such neighborhoods therefore gives choices \(w_x\) lying in one compact \(V\subset(\mathbb R^\times)^n\). This pointwise selection need not be continuous or measurable.

For \(h\) in \(B(r)\), each coordinate of \(w_xh-w_x\) has absolute value at most \(r\sum_i|(w_x)_i|\). Thus one fixed \(b_0\) makes every lattice in \(xB(r)\) contain a nonzero vector in \(w_x+[-b_0r,b_0r]^n\). Equation (20) bounds the packet mass of \(xB(r)\) by \(C_\Omega r^n\). This set is open, so Portmanteau gives \[\mu(xB(r))\le\liminf_i\mu_i(xB(r))\le C_\Omega r^n.\] The argument holds for every fixed \(x\) and \(r\) with the same constant on \(\Omega\). It proves (19) for all \(0<r<r_\Omega\), without any exchange of shrinking-radius and packet limits. ◻

Proof. Fix \(0<b<1/4\) and consider the two-sided neighborhoods \[T_t=a(t)B(b)a(-t)\cap a(-t)B(b)a(t),\qquad t\ge0.\] If \(h\in T_t\), then \[ |h_{ii}-1|<b,\qquad |h_{ij}|<b e^{-|\lambda_i-\lambda_j|t}\quad(i\ne j). \tag{21}\] Set \(r=e^{-\lambda t}\). Every term in the determinant expansion other than the product of the diagonal entries contains at least two off-diagonal entries. Since \(\det h=1\), it follows that \(\prod_i h_{ii}=1+O_{n,b}(r^2)\). Thus \[d(h)=\operatorname{diag}\left(h_{11},\ldots,h_{n-1,n-1}, \Bigl(\prod_{i<n}h_{ii}\Bigr)^{-1}\right)\] belongs to a fixed compact subset \(\mathcal D\subset A_n\), and \(d(h)^{-1}h\in B(Cr)\) for a constant depending only on \(n,b\).

In logarithmic coordinates, \(\mathcal D\) is a bounded subset of \(\mathbb R^{n-1}\). Covering it by a mesh of size comparable to \(r\), and enlarging \(C\) to allow for matrix multiplication, gives \[ T_t\subset\bigcup_{j=1}^{N_t}d_jB(Cr), \qquad d_j\in\mathcal D,\qquad N_t\le Cr^{-(n-1)}. \tag{22}\] For \(x\) in a compact set \(\Omega\), all the centers \(xd_j\) lie in the fixed compact set \(\Omega\mathcal D\). The assumed ball estimate and domination therefore imply, for all sufficiently large \(t\), \[ \xi(xT_t)\le C_{\Omega,b,c}\,r^{-(n-1)}r^n =C_{\Omega,b,c}e^{-\lambda t} \qquad(x\in\Omega). \tag{23}\] Since \(\xi\) is a probability, the constant can be enlarged to cover the finitely many integer times before this estimate holds.

We use the entropy criterion of Einsiedler–Lindenstrauss–Michel–Venkatesh (Einsiedler et al. 2009, Corollary 3.3). In the notation \[B^{(s,t)}=a(-s)Ba(s)\cap a(-t)Ba(t),\] the criterion states the following: if flow-invariant probabilities \(\xi_i\) converge weakly to a probability \(\xi\), and \(t_i\to\infty\), then \(h_\xi(a(1))\ge\eta\) provided that, for every compact set \(\Omega\), there are an identity neighborhood \(B\) and a constant \(C_\Omega\) such that \[\xi_i\bigl(xB^{(-t_i,t_i)}\bigr) \le C_\Omega e^{-2\eta t_i}\qquad(x\in\Omega).\] This criterion applies on the noncompact lattice space; its requirement that the limit be a probability is essential. Here take the constant sequence \(\xi_i=\xi\), the neighborhood \(B(b)\), \(t_i=i\), and \(\eta=\lambda/3\). Since \(B(b)^{(-t,t)}=T_t\), Equation (23) proves the required estimate. ◻

Proof. The group \(A_n\simeq\mathbb R^{n-1}\) is locally compact and second countable, and its action on the standard Borel space \(X_n\) is continuous. The ergodic decomposition theorem (Greschonig and Schmidt 2000, Theorem 5.2), specialized to invariant probabilities, therefore supplies a measurable decomposition \[\mu=\int_Z\nu_z\,\,d\tau(z)\] into \(A_n\)-invariant, \(A_n\)-ergodic probabilities over a factor fixed by \(A_n\). Set \(T(x)=xa(1)\). The function \(z\mapsto h_{\nu_z}(T)\) is measurable. Indeed, choose increasing finite Borel partitions \(\mathcal P_k\) generating the Borel \(\sigma\)-algebra of \(X_n\). For every \(T\)-invariant probability \(\nu\), \[h_\nu(T)=\sup_k\inf_{N\ge1}\frac1N H_\nu\left(\bigvee_{j=0}^{N-1}T^{-j}\mathcal P_k\right),\] where \(H_\nu\) denotes the Shannon entropy of a finite partition. The right side is a countable combination of Borel functions of \(\nu\).

The component partition is fixed by \(T\), since \(a(1)\in A_n\). The entropy integral formula for a fixed invariant partition (Rokhlin 1967, Theorem 9.8) consequently gives \[ h_\mu(T)=\int_Z h_{\nu_z}(T)\,\,d\tau(z). \tag{24}\] The same formula holds after restricting to any measurable collection of components. It does not require the \(\nu_z\) to be \(T\)-ergodic: the induced transformation on the component space is the identity and contributes zero entropy.

If \(Z_0=\{z:h_{\nu_z}(T)=0\}\) had weight \(w=\tau(Z_0)>0\), then \[\xi=\frac1w\int_{Z_0}\nu_z\,\,d\tau(z)\] would be an \(A_n\)-invariant probability with \(\xi\le w^{-1}\mu\). Equation (24), applied to this restricted decomposition, would give \(h_\xi(T)=0\), contradicting Proposition 9. ◻

Proof. Write \(M=M(g_f)\) and \(\widehat M=\widehat M(g_f)\). First identify the ambient quotient with \(X_n\). Given a finite matrix lift \(g_f\), choose a basis matrix \(c\in\operatorname{GL}_n(\mathbb Q)\) for \(\mathbb Q^n\cap\widehat\mathbb Z^{\,n}g_f^{-1}\). Then \(cg_f\in\operatorname{GL}_n(\widehat\mathbb Z)\), and the corresponding real lattice is \(\mathbb Z^ncg_\infty\), normalized to covolume one. Changing this basis or either matrix lift has no effect on that lattice. This constructs the usual quotient identification directly. For odd \(n\), the signed real \(n\)th root gives \([h]\mapsto(\det h)^{-1/n}h\), identifying \(G(\mathbb R)\) with \(\mathop{\mathrm{SL}}_n(\mathbb R)\) and \(G(\mathbb Z)\) with \(\mathop{\mathrm{SL}}_n(\mathbb Z)\).

Multiplication matrices modulo scalars give \[ \mathbf T_K(\mathbb Q)\backslash\mathbf T_K(\mathbb A_\mathbb Q) \simeq K^\times\backslash\mathbb A_K^\times/\mathbb A_\mathbb Q^\times. \tag{26}\] Here the pointwise lifts can also be seen by elementary linear algebra: over any ground field \(F_0\) containing \(\mathbb Q\), a matrix whose projective class lies in \(\mathbf T_K\) becomes a scalar multiple of a multiplication operator over an algebraic closure. It therefore commutes exactly with the regular representation already over \(F_0\). The centralizer of that representation is the algebra of multiplication operators, so the matrix belongs to \((K\otimes_\mathbb QF_0)^\times\), with scalar ambiguity \(F_0^\times\). Finite lifts may be chosen integral at almost all primes.

A finite idele \(t_f\) acts on the completed lattice by \(\widehat M\mapsto t_f^{-1}\widehat M\). Hence its translates are exactly the prescribed local homothety type. The stabilizer of \(\widehat M\) among finite ideles is \[\{z:z\widehat M=\widehat M\} =(\mathcal O(M)\otimes\widehat\mathbb Z)^\times:\] equality holds precisely when both \(z\) and \(z^{-1}\) preserve \(\widehat M\). In the projective torus the stabilizer is the image of this group, since a representative that preserves the lattice up to a scalar idele can be rescaled to preserve it exactly. The unit subgroup is compact open in the finite ideles and has finite index in \(\widehat R^\times\); at each prime not dividing \(q\) it equals \(R_\ell^\times\). Its projective image is compact open as well. No invertibility of \(M\) over \(\mathcal O(M)\) is used.

To describe the quotient and its measure, first divide (26) by the image of \(\widehat R^\times\). Fractional-ideal factorization and \(\mathbb A_\mathbb Q^\times =\mathbb Q^\times\mathbb R^\times\widehat\mathbb Z^\times\) give one real fiber \(E\backslash C\) for each ordinary ideal class. These fibers are compact by the unit theorem, and the omitted finite-unit group is compact. Thus the torus quotient is compact. At the stabilizer of \(\widehat M\), the finite fibers instead retain the translates \(\mathcal N(I)\). The same decomposition becomes \[\coprod_{[I]}(\mathcal N(I)\times C)/E,\] with exactly the action in Lemma 2. Haar measure pushes to uniform counting measure on each finite unit orbit and to the stated logarithmic Haar measure on \(C\). After normalization it is therefore precisely the probability in that lemma. For \(g_\infty=[B_\sigma]\), (25) identifies the resulting real lattices with its displayed map. This proves both the packet identification and the equality of measures, including the orbit volume weights.

Finally, every full \(M\) is obtained by choosing a basis matrix for its completed lattice and taking its inverse as \(g_f\). Any other allowed real diagonalizer has the form \([B_\sigma dP]\), where \(d\) is diagonal and \(P\) is a permutation matrix, because it must permute the joint eigenspaces. The diagonal factor translates the real torus and permutes its sign components, preserving its Haar probability; the permutation reorders the embeddings. If a rational change of basis has matrix \(c\), then \(B'_\sigma=cB_\sigma\) and \(\rho'=c\rho c^{-1}\), while the same packet data have representatives \(g'_\infty=cg_\infty\) and \(g'_f=cg_f\). Thus the torus is conjugated and the translated packet is left-multiplied by \(c\), which disappears in the ambient rational quotient. ◻