Proof. The only roots of unity in a totally real field are \(1\) and \(-1\). The kernel of the absolute logarithmic map on \(U\) therefore has order two. Its image on \(U^+\) has index \(s/2\) in its image on \(U\). The unit theorem now gives covolume \((s/2)R_K\) for the positive-unit logarithms in the coordinates \(u_1,\ldots,u_5\).

We check that these are exactly the stabilizers and identifications in the packet. Suppose a real diagonal map sends \(\sigma(I)\) onto \(\sigma(J)\). Choose \(0\ne d\in I\), and let its image be \(\sigma(e)\) with \(e\in J\). All embeddings of \(d\) are nonzero, so the diagonal entries of the map are \(\sigma_j(e/d)\). It is multiplication by \(e/d\in K^\times\), and hence \(J=(e/d)I\). This observation also applies to normalized lattices after their positive covolume factors have been undone. It follows that different ordinary ideal classes give disjoint packet orbits. For a fixed ideal, the diagonal automorphisms are its units, because \((I:I)=\mathcal O_K\).

The stabilizer in \(A\) is consequently the image of \(U^+\). Two sign translates belong to the same \(A\)-orbit exactly when their signs differ by a unit signature. The signature image has cardinality \(s\), giving \(2^6/s\) orbits for each ordinary class and the stated orbit volume. The analytic class number formula, in this normalization, is \[Q\kappa=2^5h_KR_K.\] It agrees with the product of the orbit number and orbit volume.

For completeness, let \(\mathcal F\) be a fundamental domain for the positive-unit logarithms in the traceless diagonal algebra. Summing over ordinary ideal-class representatives and all sign matrices counts each orbit exactly \(s\) times. Thus for any nonnegative \(F\) on \(X\), \[ \int_XF\,\mathop{}\!d\mu_{K,\sigma} =\frac1{sQ\kappa}\sum_{[I]}\sum_w \int_{\mathcal F}F(\Lambda_{I,\sigma}w a(u))\,\mathop{}\!du. \tag{4}\] Here \(\mathop{}\!du=\mathop{}\!du_1\cdots\mathop{}\!du_5\) and the sixth coordinate is minus their sum.

Apply this identity to \(F=\widehat f\). A nonzero \(d\in I\) determines the integral ideal \[\mathfrak b=(d)I^{-1}, \qquad \prod_{j=1}^6 \left|\mathop{\mathrm{covol}}(\sigma(I))^{-1/6}\sigma_j(d)\right| =\frac{N\mathfrak b}{Q},\] because \(\mathop{\mathrm{covol}}(\sigma(I))=QNI\). The ideal \(\mathfrak b\) determines the class \([I]=[\mathfrak b]^{-1}\), and every integral \(\mathfrak b\) occurs. For that ideal, the possible \(d\) form a single \(U\)-orbit, or \(s\) orbits under \(U^+\). Summing a positive-unit orbit and integrating over \(\mathcal F\) unfolds to the entire traceless diagonal algebra. Multiplication by a representative of \(U/U^+\) translates the absolute logarithms and permutes the signs. After the sum over \(w\), each of these \(s\) orbits therefore contributes \(V_f(N\mathfrak b/Q)\). Their number cancels the factor \(s\) in (4), proving (3). All rearrangements are justified by nonnegativity. ◻

Proof. We first record the precise zero exclusion needed here. Stark’s Lemma 3 shows that the rectangle \[\Re s\ge1-\frac1{4\log D},\qquad |\Im s|\le\frac1{4\log D}\] contains at most one nontrivial zero of \(\zeta_K\), and that such a zero is real and simple. His Lemma 8 applies also to nonnormal fields: a real zero \(\beta\) with \[1-\frac1{4\cdot6!\log D}\le\beta<1\] would be a zero of the zeta function of a quadratic subfield of \(K\) (Stark 1974, Lemmas 3 and 8). Our field has no such subfield. Consequently, for an absolute \(c_0>0\), \[ |1-\rho|\ge \frac{c_0}{\log D} \quad\text{for every nontrivial zero $\rho$ of $\zeta_K$}. \tag{8}\] One can take \(c_0=1/2880\). This use of Stark’s result does not require \(K/\mathbb Q\) to be normal.

Set \(s_0=1+1/\log D\). The logarithmic derivative of the completed Dedekind zeta function gives, for real \(1<s\le s_0\), \[ S(s):=\sum_\rho\Re\frac1{s-\rho} =\frac1{s-1}+\frac12\log D +\frac{\zeta'_K(s)}{\zeta_K(s)}+O(1). \tag{9}\] The zeros are counted with multiplicity. The real-part sum converges, and the gamma-factor contribution is bounded uniformly because the degree and signature are fixed. Each summand on the left is nonnegative, and \(\zeta'_K(s_0)/\zeta_K(s_0)\le0\) by its Dirichlet series. Hence \(S(s_0)\ll\log D\).

The same estimate holds throughout \((1,s_0]\). Indeed, writing \(a=1-\Re\rho\ge0\), \(v=\Im\rho\), and \(u=s-1\), the relevant term is \[\frac{a+u}{(a+u)^2+v^2}.\] For \(0<u\le1/\log D\), its numerator does not exceed its numerator at \(s_0\), while (8) gives \(a^2+v^2\ge c_0^2/(\log D)^2\). The denominator at \(s_0\) is therefore at most a constant multiple of the denominator at \(s\). Termwise comparison proves the asserted uniform estimate for \(S(s)\). Integrating (9) after moving \(1/(s-1)\) to the other side now gives \[ \zeta_K(s_0)\asymp\kappa\log D. \tag{10}\] More explicitly, the logarithmic derivative of \((s-1)\zeta_K(s)\) is \(S(s)-\tfrac12\log D+O(1)\), whose integral over this interval is \(O(1)\), and its limit at \(s=1\) is \(\kappa\). Since \(\zeta_K(s_0)\ge1\), this proves (5).

The Euler product gives \[\sum_{p\le D}\frac{d_K(p)}{p^{s_0}} \le\log\zeta_K(s_0).\] Also \(d_K(p)\le6\) and \[\sum_{p\le D}d_K(p)\left(\frac1p-\frac1{p^{s_0}}\right) \le\frac6{\log D}\sum_{p\le D}\frac{\log p}{p}\ll1,\] where the final estimate follows from Chebyshev’s bound by partial summation. Equation (10) now proves (6).

To obtain the interval estimate, we use Shiu’s theorem (Shiu 1980), in the formulation of (Pollack 2020, Theorem 1.1) with empty excluded residue sets. For a nonnegative multiplicative function \(f\), the hypotheses are \[f(p^a)\le A_1^a, \qquad f(m)\le A_2(\epsilon)m^\epsilon \quad(\epsilon>0).\] For any fixed \(0<\beta<1/2\), the conclusion is \[ \sum_{x-y<m\le x}f(m) \ll\frac y{\log x} \exp\left(\sum_{p\le x}\frac{f(p)}p\right) \qquad(x^\beta<y\le x). \tag{11}\] Both the constant and the sufficiently-large threshold for \(x\) depend only on \(\beta,A_1\), and the function \(A_2\).

The coefficients \(d_K\) are multiplicative and are bounded by the ordinary six-fold divisor function \(d_6\). In particular, \[d_K(p^a)\le\binom{a+5}{5}\le6^a, \qquad d_K(m)\le d_6(m)\ll_\epsilon m^\epsilon.\] The hypotheses of (11) thus hold with constants independent of \(K\). We take, for example, \(\beta=1/3\); the assumed length \(y\ge x^{1/2}\) lies in the required range. Since \(x\le D\), (6) bounds the exponential factor by \(O(\kappa\log D)\). Finally \(x\ge D^{1/4}\) implies \(\log D/\log x\le4\), proving (7). ◻

Proof. Take \(f=\mathbf 1_{[-r,r]^6}\) and put \(X_r=Qr^6\). For \(0<p\le r^6\), the change of variables \(t_j=\log r-u_j\) in (2) identifies the positive-sign integration region with \[t_j\ge0\quad(1\le j\le5),\qquad \sum_{j=1}^5t_j\le\log(r^6/p).\] Thus \[V_f(p)=\frac{2^6}{5!}\bigl(\log(r^6/p)\bigr)^5 \quad(0<p\le r^6), \qquad V_f(p)=0\quad(p>r^6).\] Since the occurrence of a short vector is bounded by its count, (3) gives \[ \mu_{K,\sigma}\{m\le r\} \ll\frac1{Q\kappa} \sum_{l\le X_r}d_K(l) \bigl(1+\log(X_r/l)\bigr)^5. \tag{12}\] The sum is empty if \(X_r<1\).

Suppose first that \(r\ge D^{-1/48}\), so \(X_r\ge D^{3/8}\). For \(l\ge D^{1/4}\), divide the range of summation into intervals \((2^{-j-1}X_r,2^{-j}X_r]\). Whenever such an interval contains one of these \(l\), its upper endpoint is at least \(D^{1/4}\) and at most \(Q\). Lemma 3, with interval length half the upper endpoint, bounds its ideal count by \(O(\kappa 2^{-j}X_r)\). Its weight in (12) is \(O((j+1)^5)\). The total contribution of these terms before division by \(Q\kappa\) is therefore at most \[C\kappa X_r\sum_{j\ge0}2^{-j}(j+1)^5\ll\kappa X_r.\] For the remaining terms, the elementary divisor estimate \(\sum_{l\le Y}d_6(l)\ll Y(\log(2Y))^5\) gives an upper bound \(CD^{1/4}(\log D)^{10}\) before normalization. In view of (5), its normalized contribution is at most \[CD^{-1/4}(\log D)^{11}\ll r^6;\] the last inequality follows from \(r^6\ge D^{-1/8}\). This proves an \(O(r^6)\) bound in the present range.

Now let \(r<D^{-1/48}\) and \(X_r\ge1\). Dyadic summation of the same elementary divisor estimate yields \[\sum_{l\le X_r}d_6(l)\bigl(1+\log(X_r/l)\bigr)^5 \ll X_r(\log(2X_r))^5.\] Using \(X_r\le Q\) and (5), (12) is therefore at most \(Cr^6(\log D)^6\). But \(\log D<48\log(1/r)\) in this range, and \(r^6(\log(1/r))^6\ll r^3\). The claimed estimate follows, for example with \(c=3\), after increasing \(C\).

Mahler’s compactness criterion says that \(\{x:m(x)\ge r\}\) is compact for every \(r>0\). The bound just proved therefore gives tightness. Any weak limit has mass one, and \(A\)-invariance passes to the limit because right translation by each fixed element of \(A\) is continuous and preserves compact supports. ◻

Proof. At any lattice \(x_0\), choose a nonzero lattice vector none of whose coordinates is zero. Such a vector exists because a full-rank lattice cannot be contained in a finite union of proper linear subspaces. In an injective local chart at \(x_0\), continue this marked vector as \(v(x)\). By reducing the chart, its coordinates remain bounded and bounded away from zero. A finite collection of these charts covers \(\Omega\), so these bounds and all local derivative bounds can be taken uniform.

For \(x\) in one of the charts and sufficiently small \(r\), every lattice in \(xB(r)\) has a nonzero vector in the ordinary Euclidean ball \(B_{\mathbb R^6}(v(x),C_1r)\). Let \(f_{x,r}\) be the indicator of that ball. Its support has absolute coordinate product in an interval \(J_{x,r}\) of length at most \(C_2r\); these intervals all lie in a fixed compact subinterval of \((0,\infty)\) when \(r<r_\Omega\). On this support the first five logarithmic coordinates each vary by \(O(r)\). Consequently \[V_{f_{x,r}}(p)\le C_3r^5\mathbf 1_{J_{x,r}}(p).\] Fix \(r>0\). For all sufficiently large \(i\), uniformly in the finitely many charts and in \(x\in\Omega\), the ideal norms in \(Q_iJ_{x,r}\) can be covered by an interval with upper endpoint comparable to \(Q_i\) and length at most \(C_4Q_ir\). We may choose that length comparable to \(Q_ir\), so it is at least the square root of its upper endpoint for large \(i\). The upper endpoint lies between \(D_i^{1/4}\) and \(D_i\). Lemma 3 and (3) therefore give \[\mu_i(xB(r)) \le\int_X\widehat f_{x,r}\,\mathop{}\!d\mu_i \le\frac{C_3r^5}{Q_i\kappa_i} \sum_{l\in Q_iJ_{x,r}}d_{K_i}(l) \le C_5r^6.\] The constants are independent of \(x\) and \(r\) in the stated range; the threshold for \(i\) may depend on \(r\). Since \(xB(r)\) is open, the open-set inequality for weak convergence gives the same bound for \(\mu(xB(r))\). This proves the assertion for each \(x\) and \(r\) with the uniform constants supplied by the finite cover. ◻

Proof. Since \(1\) is primitive in the free abelian group \(\mathcal O_K\), choose an integral basis \(\alpha_1=1,\alpha_2,\ldots,\alpha_6\). Let \(\beta_1,\ldots,\beta_6\) be its trace-dual basis, so \(\mathop{\mathrm{Tr}}_{K/\mathbb Q}(\alpha_j\beta_k)=\delta_{jk}\). By the definition of the codifferent, these \(\beta_k\) form a basis of \(\mathfrak D_K^{-1}\). In particular, \(\mathop{\mathrm{Tr}}_{K/\mathbb Q}\beta_k=\delta_{1k}\), which proves that the trace map onto \(\mathbb Z\) is surjective and that \(\sigma(\beta_2),\ldots,\sigma(\beta_6)\) form a basis of \(\mathcal L_K\). For \(2\le j,k\le6\), orthogonal projection gives \[\bigl\langle\operatorname{proj}_H\sigma(\alpha_j), \sigma(\beta_k)\bigr\rangle=\delta_{jk}.\] These projected vectors are therefore the dual basis inside \(H\). Since \(\operatorname{proj}_H\sigma(\alpha_1)=0\), we obtain \[ \mathcal L_K^*=\operatorname{proj}_H\sigma(\mathcal O_K). \tag{14}\]

Take a nonzero vector in the right side of (14), arising from \(\alpha\in\mathcal O_K\). The vector is nonzero precisely when \(\alpha\notin\mathbb Q\). Primitivity of the field implies \(\mathbb Q(\alpha)=K\), so the order \(\mathbb Z[\alpha]\) has discriminant at least \(D\). Differences of embedding coordinates are unchanged by orthogonal projection to \(H\). It follows that \[D\le |\mathop{\mathrm{Disc}}(\mathbb Z[\alpha])| =\prod_{j<k}|\sigma_j(\alpha)-\sigma_k(\alpha)|^2 \le C\|\operatorname{proj}_H\sigma(\alpha)\|^{30}.\] Thus the first successive minimum of \(\mathcal L_K^*\) is at least \(cD^{1/30}\). Transference of successive minima in dimension five gives \[\lambda_5(\mathcal L_K) \ll\lambda_1(\mathcal L_K^*)^{-1} \ll D^{-1/30}.\] A fixed-dimensional lattice basis bound then provides a basis for \(\mathcal L_K\) all of whose lengths are \(O(D^{-1/30})\); these standard geometry-of-numbers facts may be found in (Cassels 1959). The associated fundamental parallelotope has the asserted diameter.

The full codifferent lattice has covolume \(Q^{-1}\). Its trace planes have spacing \(1/\sqrt6\), since the trace takes every integer value and has normal vector \(\mathbf e\). Decomposing a basis into five vectors in \(H\) and a vector of trace one therefore gives \[Q^{-1}=\frac1{\sqrt6}\mathop{\mathrm{covol}}_H(\mathcal L_K).\]

The trace-\(m\) points, if any, form a translate of \(\mathcal L_K\). Tile that affine hyperplane by translates of the fundamental parallelotope, assigning boundaries measurably. A cell attached to a point counted in (13) lies in the \(O(D^{-1/30})\) enlargement of its bounded slice. The linear functional \(z\mapsto\sum_{j\in S}z_j\) is nonconstant on \(H\) because \(S\) is nonempty and proper. The five-dimensional volume of the enlarged slice is consequently \(O_M(\delta+D^{-1/30})\), uniformly in its location and in \(m\). Dividing by the cell covolume \(\sqrt6/Q\) proves the counting bound. ◻

Proof. For a lattice \(\Lambda\), its Euclidean dual is \(\Lambda^*=\{y:\langle v,y\rangle\in\mathbb Z\text{ for all }v\in\Lambda\}\). At each point of \(\Omega\), choose nonzero marked vectors in the lattice and its dual, both avoiding all coordinate hyperplanes. Continue the pair in a local lattice chart. Finitely many such charts cover \(\Omega\), and the marked vectors in these charts lie in fixed bounded boxes separated from the coordinate hyperplanes. Their inner product is an integer, constant in each chart. Indeed, under right multiplication by \(h\), the pair transforms as \(v\mapsto vh\), \(y\mapsto yh^{-\mathsf T}\).

Choose \(O\) sufficiently small in matrix coordinates. If \[h\in a(tb)Oa(-tb)\cap a(-tb)Oa(tb),\] then \(h\) is uniformly close to the identity, and each entry crossing the cut \(S\sqcup S^c\) is \(O(e^{-t})\). To verify the latter bound, the two membership conditions respectively bound its \((j,k)\) entry by a constant times \(e^{t(b_j-b_k)}\) and \(e^{-t(b_j-b_k)}\); their minimum is \(O(e^{-t|b_j-b_k|})\). Reducing \(O\) also makes \(h^{-1}\) uniformly bounded. Let \(P_S\) be the diagonal projection onto the coordinates in \(S\). Then \[hP_Sh^{-1}-P_S=[h,P_S]h^{-1}=O(e^{-t}).\] For the marked pair \(v_0,y_0\) at the center \(x\), put \(v=v_0h\) and \(y=y_0h^{-\mathsf T}\). These vectors remain in fixed bounded boxes avoiding the coordinate hyperplanes, and \[\begin{align*} \sum_{j=1}^6v_jy_j&=\langle v_0,y_0\rangle=:m\in\mathbb Z,\\ \sum_{j\in S}v_jy_j &=v_0hP_Sh^{-1}y_0^{\mathsf T} =v_0P_Sy_0^{\mathsf T}+O(e^{-t}). \end{align*}\] All constants are uniform over the finite cover of \(\Omega\). Thus membership in the tube forces the existence of an ordinary and dual vector pair satisfying these boundedness, trace, and partial-sum conditions. The short partial-sum interval may depend on \(x\).

We count such pairs in the packet. For a representative ideal \(I\) and a packet lattice \(c\sigma(I)wa\), where \(c=\mathop{\mathrm{covol}}(\sigma(I))^{-1/6}\), its dual is \[c^{-1}\sigma(\mathfrak D_K^{-1}I^{-1})wa^{-1}.\] Consequently a nonzero pair comes from \[d\in I\setminus\{0\},\qquad \ell\in\mathfrak D_K^{-1}I^{-1}\setminus\{0\}.\] The scalar, sign, and diagonal factors cancel in their coordinatewise product: \[ v_jy_j=\sigma_j(z),\qquad z=d\ell\in\mathfrak D_K^{-1}. \tag{17}\] The allowed \(z\) therefore belong to a fixed bounded embedding box, have trace \(m\), and have their \(S\)-sum in an interval of length \(O(e^{-t})\).

For a fixed nonzero allowed \(z\), the ordinary vector’s ideal \(\mathfrak b=(d)I^{-1}\) must divide \((z)\mathfrak D_K\). In fact \[\mathfrak b\bigl((\ell)I\mathfrak D_K\bigr) =(z)\mathfrak D_K,\] and both factors on the left are integral. Conversely, this divisibility ensures \(z/d\in\mathfrak D_K^{-1}I^{-1}\) whenever \((d)I^{-1}=\mathfrak b\). For a fixed \(d\) and \(z\), there is exactly one possible \(\ell\), namely \(z/d\).

Let \(f\) be the indicator of a fixed bounded box, or of a fixed finite union of boxes, containing all possible ordinary vectors \(v\) and separated from the coordinate hyperplanes. Its logarithmic fiber integrals are bounded uniformly in \(p>0\), because the first five absolute logarithms range over a fixed bounded set. Repeating the unfolding of Proposition 2, with the additional restriction \(\mathfrak b\mid(z)\mathfrak D_K\), gives \[ \int_X\#\{\text{admissible pairs with product }z\} \,\mathop{}\!d\mu_{K,\sigma} \le\frac1{Q\kappa} \sum_{\mathfrak b\mid(z)\mathfrak D_K} V_f(N\mathfrak b/Q) \ll\frac1{Q\kappa} \#\{\mathfrak b:\mathfrak b\mid(z)\mathfrak D_K\}. \tag{18}\] Here we have dropped the restrictions on the dual vector after fixing \(z\), which can only increase the count. The divisibility restriction is unchanged by multiplication of \(d\) by a unit, so the same unfolding and sign multiplicities apply.

The embedding bound on \(z\) implies \(N((z)\mathfrak D_K)=D|N_{K/\mathbb Q}z|\ll D\). For an integral ideal of norm \(N\), its number of ideal divisors is at most \(d(N)^6\), where \(d\) is the ordinary divisor function: over each rational prime there are at most six prime ideals, and every one of their exponents is at most the exponent of that prime in \(N\). The classical divisor bound consequently gives, uniformly for our allowed \(z\), \[ \#\{\mathfrak b:\mathfrak b\mid(z)\mathfrak D_K\} \le\exp\left(C\frac{\log D}{\log\log D}\right). \tag{19}\]

Lemma 6 bounds the number of allowed products by \(O(Q(e^{-t}+D^{-1/30}))\). Uniformly for \(t\le T_{+,i}\), \[D_i^{-1/30}=o(e^{-t}),\] because \(T_{+,i}/\log D_i\to0\). The product count is therefore \(O(Q_i e^{-t})\). The event defining the tube is bounded by the number of admissible pairs, so summing (18) over these products and using (19) gives \[\mu_i\bigl(x(a(tb)Oa(-tb)\cap a(-tb)Oa(tb))\bigr) \ll\kappa_i^{-1} \exp\left(-t+C\frac{\log D_i}{\log\log D_i}\right).\] Finally, (5) and \(t\ge T_{-,i}\) show that the logarithm of the factor multiplying \(e^{-t}\) is \(o(t)\), uniformly throughout the time range. Enlarging the discriminant threshold makes the last display at most \(e^{-t/3}\). The same choices work for finitely many cuts and diagonal elements by taking the smallest of their identity neighborhoods and the largest of their thresholds. ◻

Proof. Choose a sufficiently small fixed \(\epsilon>0\) and consider the two-sided tube \[\mathcal T_t=a(tb)B(\epsilon)a(-tb) \cap a(-tb)B(\epsilon)a(tb).\] If \(h\in\mathcal T_t\) and \(r=e^{-\lambda t}\), then \[|h_{jj}-1|<\epsilon,\qquad |h_{jk}|\le\epsilon r\quad(j\ne k).\] Every nondiagonal term in the determinant expansion contains at least two off-diagonal entries. Hence \(\prod_jh_{jj}=1+O(r^2)\). The matrix \[d(h)=\mathop{\mathrm{diag}}\left(h_{11},\ldots,h_{n-1,n-1}, \Bigl(\prod_{j<n}h_{jj}\Bigr)^{-1}\right)\] belongs to a fixed compact set \(\mathcal D\subset A\) and satisfies \(d(h)^{-1}h\in B(Cr)\). An \(r\)-mesh of the \((n-1)\)-dimensional set \(\mathcal D\) therefore covers \(\mathcal T_t\) by at most \(Cr^{-(n-1)}\) sets \(d_jB(C'r)\), with \(d_j\in\mathcal D\). For \(x\in\Omega\), all centers \(xd_j\) belong to the compact set \(\Omega\mathcal D\). Domination and (20) give \[ \xi(x\mathcal T_t)\le C_{\Omega,c}e^{-\lambda t}. \tag{21}\]

We apply the entropy criterion of Einsiedler–Lindenstrauss–Michel– Venkatesh (Einsiedler et al. 2009, Corollary 3.3). In the form needed here, it says that if probabilities \(\nu_i\) invariant under \(a(\mathbb Rb)\) converge to a probability \(\nu\), and \(t_i\to\infty\), then \[\nu_i\bigl(x[a(t_ib)Oa(-t_ib)\cap a(-t_ib)Oa(t_ib)]\bigr) \le C_\Omega e^{-2\eta t_i}\quad(x\in\Omega)\] for every compact \(\Omega\), with an identity neighborhood \(O\) allowed to depend on \(\Omega\), implies \(h_\nu(T)\ge\eta\). This criterion includes the noncompact space \(X\) and requires that the limit have total mass one. Take the constant sequence \(\nu_i=\xi\), \(t_i=i\), and \(\eta=\lambda/3\). Equation (21) verifies its hypothesis.

Write the \(A\)-ergodic decomposition as \(\mu=\int_Z\nu_z\,\mathop{}\!d\tau(z)\). The function \(z\mapsto h_{\nu_z}(T)\) is measurable: for increasing finite Borel partitions \(\mathcal P_k\) generating the Borel sets, it is given by \[h_{\nu_z}(T)=\sup_k\inf_{N\ge1}\frac1N H_{\nu_z}\left(\bigvee_{j=0}^{N-1}T^{-j}\mathcal P_k\right).\] The entropy integral formula (Rokhlin 1967, Theorem 9.8) applies to this decomposition into \(T\)-invariant probabilities. The component partition is fixed by \(T\), so the transformation on its quotient is the identity and has zero entropy. The formula remains valid after restricting to a measurable collection of components; the components need not themselves be \(T\)-ergodic.

If the zero-entropy components had weight \(w>0\), their normalized average \(\xi\) would satisfy \(\xi\le w^{-1}\mu\) and \(h_\xi(T)=0\). This contradicts the first part of the proposition. ◻

Proof. The Lie algebra \(\mathfrak j\) contains \(L\) and is invariant under \(\mathop{\mathrm{Ad}}(A)\). Since the off-diagonal weight spaces are one-dimensional, \(\mathfrak j\) is the direct sum of \(L\) and a collection of root lines \(\mathbb RE_{jk}\). Draw the edge \(j\to k\) when this line occurs. The bracket \([E_{ij},E_{jk}]=E_{ik}\) makes the graph transitively closed whenever \(i\ne k\).

A Lie group with a lattice is unimodular. Applying the vanishing of its modular character to \(A\) shows that \[\sum_{j\to k}(b_j-b_k)=0\qquad(b\in L).\] Thus each vertex has equal indegree and outdegree. Every strongly connected class is complete by transitivity. The graph of these classes is acyclic. If it had an edge, a source class of an edge-containing component would have strictly more outgoing than incoming edges in the original graph, contradicting the sum of the degree balances over that class. Hence there are no edges between classes. The resulting Lie algebra is precisely the block-diagonal one. A connected Lie subgroup is determined by its Lie algebra, giving the assertion. ◻

Proof. Write \(x=\Gamma g\) and \(\Delta=g^{-1}\Gamma g\cap J\). This is a lattice in \(J\). For the rational linear algebra in this proof, use the auxiliary column lattice \(\Lambda=g^{-1}\mathbb Z^n\), which \(\Delta\) preserves, and its rational span \(\Lambda_{\mathbb Q}\). This convention is separate from the row-lattice realization of the point \(x\). We first show that the real matrix commutant of \(\Delta\) equals the commutant of \(J\). For any matrix \(C\) commuting with \(\Delta\), the map \[\Delta\backslash J\longrightarrow\mathop{\mathrm{Mat}}_n(\mathbb R),\qquad \Delta j\longmapsto j^{-1}Cj\] is well-defined. It sends normalized Haar measure to a probability invariant under conjugation by \(A\). Choose a regular element of \(A\). Every off-diagonal matrix coordinate has a nonzero weight under its conjugation. Recurrence of an invariant probability forces all such coordinates to vanish almost surely: a nonzero coordinate expands geometrically in one of the two time directions and cannot recur to a bounded set. The map is continuous and Haar measure has full support, so \(j^{-1}Cj\) is diagonal for every \(j\in J\).

Taking \(j=1\) shows that \(C\) is diagonal. Conjugating by a root subgroup within one of the blocks of Lemma 10 forces the two corresponding diagonal entries of \(C\) to agree. Consequently the commutant of \(\Delta\) is exactly the algebra of block scalars, namely the commutant of \(J\).

Define \[F=\{C\in\mathop{\mathrm{End}}_{\mathbb Q}(\Lambda_{\mathbb Q}):C\delta=\delta C \text{ for all }\delta\in\Delta\}.\] In the lattice coordinates supplied by \(g\), these commutation relations are rational linear equations, since \(g\Delta g^{-1}\subset\Gamma\). Only finitely many of these equations are needed in the finite-dimensional matrix space. Thus \(F\) has real extension isomorphic to \(\mathbb R^d\), where \(d\) is the number of blocks. In particular \(F\) is a commutative, reduced, finite-dimensional \(\mathbb Q\)-algebra, a product of totally real fields.

Suppose \(F\) had a nontrivial idempotent. Its image in real coordinates would select a nonempty proper union of blocks and hence a subspace \(V\) defined over the rational lattice. The group \(\Delta\) preserves the full lattice in \(V\) and its inverse does too. Thus \[\chi(j)=|\det(j|_V)|\] is a character of \(J\) trivial on \(\Delta\). It is nontrivial on \(A\), because both the selected and the unselected union of blocks are nonempty. It is therefore onto \(\mathbb R_{>0}\). Pushing the finite invariant measure on \(\Delta\backslash J\) forward by \(\chi\) would give a finite nonzero multiplicatively invariant measure on \(\mathbb R_{>0}\), an impossibility. Hence \(F\) is a field.

The representation of \(F\) on the rational lattice space has some dimension \(m\) over \(F\). After extending scalars to \(\mathbb R\), each embedding of \(F\) has multiplicity \(m\), so all blocks have size \(m\) and \(dm=n\). The elements of \(F\) carrying \(\Lambda\) into itself form an order. Indeed, in lattice coordinates this multiplier set is represented by \((gFg^{-1})\cap\mathop{\mathrm{Mat}}_n(\mathbb Z)\), hence is discrete. Clearing the denominators of a rational basis gives a full-rank submodule. Thus the multiplier set is a finitely generated group of rank \(d\), and it is a subring containing \(1\). Dirichlet’s unit theorem for this order gives unit rank \(d-1\).

If \(J\ne G\), then \(d>1\). Choose a unit of infinite order and take an even power so that every real embedding is positive. Its action on the lattice is an automorphism with determinant \(N_{F/\mathbb Q}(u)^m=1\). In real coordinates it is a matrix \(a\in A\), constant on each block and therefore central in \(J\). It fixes \(x\), and centrality gives \((xj)a=(xa)j=xj\) for every \(j\in J\). Its entries are conjugates of an algebraic unit. The collection of possible such diagonal matrices is countable.

Finally, fixing \(a\) and lifting to \(G\), the fixed-point equation is \[ga g^{-1}=\gamma\qquad\text{for some }\gamma\in\Gamma.\] For each \(\gamma\), the nonempty solution set is a coset of the centralizer of \(a\). It is smooth, with tangent root directions exactly those satisfying \(a_{jj}=a_{kk}\). Restricting to countably many injective local quotient charts gives the asserted pieces in \(X\). If \(a_{jj}\ne a_{kk}\), two points of one local \(U_{jk}\) plaque cannot solve the same lifted equation: their relative element \(1+tE_{jk}\) would commute with \(a\), forcing \(t=0\). This proves the last assertion. ◻

Proof. Choose \(b\in L\) such that \(a(-tb)va(tb)\to1\) as \(t\to\infty\) for every \(v\in V\). At a point recurrent under \(a(b)\), a relation \(xv=x\) gives \[xa(kb)\,a(-kb)va(kb)=xa(kb).\] A subsequence of these points returns to a compact neighborhood of \(x\). Stabilizers of points in that neighborhood contain no nonidentity element in a fixed neighborhood of \(1\): this follows from the discreteness of the lattice and local quotient charts. The displayed relation is therefore impossible for \(v\ne1\). Poincaré recurrence applies at almost every point.

The orbit map \(v\mapsto xv\) is then an injective immersion. Its restriction to a compact set is an embedding. Take a slightly larger compact coordinate domain in \(V\) and use a complementary transverse slice. Compactness permits shrinking this slice until its product with the prescribed plaque domain is injective. Indeed, a failure along slices shrinking to \(x\) would give either two distinct points of the limiting embedded plaque with the same image, or contradict local injectivity near one of its points. Shrinking once more provides inner charts with margins. A countable subcover and continuity from below give the finite-cover assertion. ◻

Proof. Fix a chart, a finite bin partition, and a grid shift. If \(p_l\) is the conditional bin-probability vector, its coordinates form bounded martingales in the increasing transverse grid sigma-fields. Thus \[\int_{\mathcal B}\|p_v-p_u\|_2^2\,\mathop{}\!d\mu_i =\int_{\mathcal B}\|p_v\|_2^2\,\mathop{}\!d\mu_i -\int_{\mathcal B}\|p_u\|_2^2\,\mathop{}\!d\mu_i, \qquad u\le v.\] The sum of these differences over disjoint depth intervals is at most \(1\). This remains true after averaging in the shift. Enumerate all fixed chart/bin choices and assign them positive summable weights. Their weighted sum has the same bounded total variation.

Put \(L_i=\log(T_{+,i}/T_{-,i})\). Divide the interval of logarithmic depths into \(m_i\) consecutive intervals, where \(m_i\to\infty\) and \(m_i=o(L_i)\). Discard its first and last intervals. Some remaining interval has weighted martingale cost \(O(m_i^{-1})\). Its endpoints, rounded to integers, give \(l_-\) and \(l_+\). The discarded end intervals and the length \(L_i/m_i\to\infty\) give all three ratios in (23). Each fixed chart/bin choice has cost tending to zero; Cauchy–Schwarz gives the asserted \(\ell^1\) conclusion. Orthogonality of martingale increments gives it for intervening depths as well.

For the boundary assertion, fix a point \(q\) in a Euclidean transverse coordinate domain. Under a uniform grid shift, the probability that \(q\) is within \(\rho 2^{-l}\) of a depth-\(l\) grid wall is \(O(\rho)\), uniformly in \(q,l\), and the sampling measure. Integrating first in the shift gives the same estimate for any measure on that domain. Choose \(\rho_i\downarrow0\) slowly. Markov’s inequality, followed by a diagonal choice over the chart/bin and depth lists, selects shifts with both the conditional-probability estimates and boundary masses tending to zero. Common labels can use common shift variables: each marginal shift remains uniform, which is all the argument uses.

Here is the comparison with an auxiliary grid at an interior depth \(d\). Outside negligible boundary mass, one of its cells lies inside one depth-\(l_-\) cell and is a union of depth-\(l_+\) cells, apart from the latter cells meeting its walls. The relevant relative collar widths are \(O(2^{l_--d})\) and \(O(2^{d-l_+})\). Both tend to zero much faster than the chosen \(\rho_i\). Summing the martingale bin errors over the contained fine cells compares the auxiliary conditional law to the coarse one. The discarded mass changes the weighted conditional \(\ell^1\) error by at most twice that mass. This proves the last assertion. The same argument compares any two of the selected grids by sandwiching them between these endpoint grids. ◻

Proof. Let \(q_i^0\) be a representative of the cell containing \(q\). At the sequence level, \(P^{\mathcal B}_{i}\) is a function of that cell. Consequently, for bounded continuous tests \(f\) and \(F\), \[\int_{\mathcal B} f(h)F(q_i^0,P^{\mathcal B}_{i})\,\mathop{}\!d\mu_i =\int_{\mathcal B}P^{\mathcal B}_{i}(f) F(q_i^0,P^{\mathcal B}_{i})\,\mathop{}\!d\mu_i.\] On compact inner boxes, replacing \(q_i^0\) by \(q\) incurs an error tending to zero. Let \(i\to\infty\) and then exhaust the chart by inner boxes. Its boundary has zero limiting mass. This proves the conditional-law identity. The standard support assertion for a sample from a probability measure follows, for example, by using a countable basis of the plaque domain. ◻

Proof. For a trimmed patch, sum the errors in (24) over \(e\in I_p\) and \(j\in J(p)\). The resulting comparison is between its unnormalized bin-mass vector and \[\left(\sum_{e\in I_p}M_e\right) \mathbf 1_{B_p}P_{E(p)}.\] Every pair \((e,j)\) occurs in at most one trimmed patch, since those sets are disjoint. Hence the total unnormalized error is at most \(\delta_i\). Restoring discarded mass adds at most \(\epsilon\). For two nonnegative finite vectors \(v,w\), comparison after normalization, weighted by \(\|v\|_1\), costs at most \(2\|v-w\|_1\); this follows by adding and subtracting \(w/\|v\|_1\). The same inequality covers zero mass by an arbitrary choice of normalized vector. It proves the first assertion and, on multiplying by \(P_{E(p)}(B_p)\le1\), proves (25).

Apply the same argument separately to patches and subpatches and use their common restricted vector \(P_E|_{B_p}\) to obtain their comparison. Approximating a continuous test by its values on bins gives the remaining test assertion. The bound is in \(\ell^1\), so it is uniform over all choices of bounded bin values, including values depending on the sample. Finally, the plateau controls the expected fixed-bin distance between the coarse conditional vector and the recorded vector. Restricting to disjoint patches does not increase that expectation. ◻

Proof. We prove one local comparison that covers all three assertions. The motion is either the identity, a fixed \(a(b)\), or \(a(b_i)\) with \(b_i\in\ker\beta\) and \(V=U_\beta\). Denote its induced conjugation on \(V\) by \(c\). In the first two cases \(c\) is fixed; in the third case \(c\) is the identity.

We display the geometry in group-element plaque coordinates. For a reparametrized chart, write its original coordinate as \(t\in B_0\) and its group element as \(h=r_q(t)\), where the maps \(r_q\) and their inverses are smooth on the fixed extended chart. Its group-coordinate domain is \(B(q)=r_q(B_0)\); for a multiplication chart \(B(q)\) is constant. The conditional probabilities and bins remain in the original fixed coordinates. When converting a probability to group coordinates at a sample, use \(r_{q_s}\). Across a depth-\(F\) cell this conversion differs from the actual \(r_q\) by \(O(2^{-F})\), which has vanishing effect on every continuous test considered below.

In the long-motion case choose an input transverse depth \(F\) with \[ l_-\prec R\prec F\prec l_+, \tag{26}\] For a fixed motion, it suffices to take \(l_-\prec F\prec l_+\); all factors \(\exp(O(R))\) below are then bounded constants. The depth-\(F\) input grid may be an auxiliary grid selected for this comparison. Thus \(R\) and \(F\) need not belong to the countable list used to define the states; Lemma 13 compares the auxiliary laws back to the recorded ones. Its boundary selection uses \(\mu_i\) and the scale bound, and does not depend on the particular \(b_i\) within that bound. The input patches are the depth-\(F\) transverse cells times the full plaque-coordinate domain. Their conditional laws are the input probabilities at depth \(F\).

Fitting the images into output charts. Choose input inner charts and fixed displacement test ranges first. An input chart may have any fixed plaque extension needed for these ranges. By Lemma 12, output inner charts can be chosen whose outer charts accommodate the entire image plaque range, with a margin. A finite such cover omits arbitrarily small limiting mass. The output distribution is again \(\mu_i\), so the same is true for the sequence. If an image patch meets one of these inner charts, its whole image fits in the corresponding outer chart for all sufficiently large \(i\). This last assertion follows from the transverse diameter estimate below. Assign such patches to output charts, or work with each applicable pair of charts; the finite number of pairs only multiplies error bounds by a fixed constant.

More precisely, let \(B(q)\) and \(B'(q')\) be the full input and output domains in group coordinates, and let \((q_s,h_s),(q'_s,h'_s)\) be the two samples. The output extension is required to contain \[ h'_s\,c\bigl(h_s^{-1}\overline{B(q_s)}\bigr)\subset B'(q'_s) \quad\text{with a positive boundary margin}. \tag{27}\] This condition concerns the entire input plaque domain. It is an open condition on the paired sample coordinates: the domains are smooth images of fixed compact closures. Their possible centered ranges lie in a fixed compact subset of \(V\), so the extension lemma supplies output covers with this property.

In the two charts the map has the form \[ (q,h)\longmapsto\bigl(\theta_i(q),\,d_i(q)c(h)\bigr). \tag{28}\] The first coordinate is independent of \(h\), because the motion normalizes \(V\). The factor \(d_i(q)\) is a left translation in \(V\). It remains in a fixed compact set on the comparisons in question: both plaque-coordinate ranges and \(c\) are fixed. For the varying motion, right multiplication and its inverse have differential norm at most \(\exp(O(R))\) in a left invariant metric on \(G\) and the induced quotient metric. Fixed chart changes have bounded differentials. Consequently both \(\theta_i(q)\) and \(d_i(q)\) vary over an input cell by \[ O\bigl(2^{-F}\exp(O(R))\bigr)=o(2^{-l_-}). \tag{29}\] For reparametrized charts, the input domains \(B(q)\) also vary by \(O(2^{-F})\) over the cell, and their admitted output ranges vary by \(O(2^{-F}\exp(O(R)))\). The fixed boundary margins absorb these variations. Thus the same estimate applies to those charts. The same bounds hold for pullbacks of paths in the extended charts. They justify following one plaque sheet throughout its prescribed range, rather than including a different sheet at a return of the orbit. They also justify the fitting assertion: vary \(q\) along the short path in its input cell and then follow the fixed plaque range inside the output extension.

On one such patch, \(\theta_i\) is injective. To see this, the admitted output plaque ranges have a common open subset, by the margin and the small variation of \(d_i(q)\). If two transverse inputs had the same image, one could choose the same output plaque point in this common subset, contradicting injectivity of the motion and of the input chart. This observation permits product trims in output coordinates, even if many distinct input patches return to the same output box.

Trimming to fine columns and fixed plaque bins. Discard image patches meeting a coarse transverse wall. By (29), every point of such a patch is in a collar whose relative width tends to zero. The grid convention, applied to the output distribution \(\mu_i\), makes the discarded mass \(o(1)\). Each remaining patch lies transversely in one coarse cell \(E\).

Take fine transverse cells whose closures lie in its projected transverse range, and plaque bins admitted over the whole range. Their products form an inner set as in Lemma 15. Here are the two loss estimates. First, a path varying the transverse output coordinate through one fine cell pulls back a distance \[ O\bigl(2^{-l_+}\exp(O(R))\bigr)=o(2^{-F}). \tag{30}\] Away from the input cell walls and a fixed small collar of the input chart boundary, it therefore stays in the input transverse range and in the chart. Input grid-wall loss is \(o(1)\) by the selected shifts; chart-boundary loss can be made arbitrarily small by its null limiting mass. This verifies that almost all sampled fine transverse cells are admitted.

Second, on each plaque the transformation in (28) has bounded distortion, uniformly in \(i\), and its dependence on \(q\) tends to zero over the patch. A sample away from a small collar of the input plaque boundary thus lies in an output plaque bin admitted over the whole transverse range, provided the fixed bins are chosen sufficiently fine. The collar has arbitrarily small sampling mass, again by the fixed null-boundary choice. For a reparametrized chart these collars are pulled back to its original fixed coordinate boundaries, which have zero limiting mass. Continuous tests expressed in the fixed coordinates pull back with uniform continuity moduli on the compact chart ranges. This gives arbitrarily small total trim loss. No bound on the number of patches is involved.

Comparing the conditional laws. The image of \(\mu_i\) restricted to an input patch is exactly \(\mu_i\) restricted to its image, by \(A\) invariance (or trivially for the identity motion). Apply Lemma 15 to its trim. The input patch law, converted by (28), agrees with \(P_E\) restricted to the admitted bins, up to arbitrarily small weighted error.

To express this without dividing by a possibly small admitted mass, take two continuous displacement tests \(f,\phi\) supported with margin inside the input plaque range. Their output bin approximations are wholly inside the admitted bins. The unnormalized comparison says that each input test integral, multiplied by \(P_E(B_p)\), agrees with the corresponding output \(P_E\) integral. Cross-multiplication therefore gives \[ P_{\mathrm{in}}(f)P_{\mathrm{out}}(\phi\circ c^{-1}) -P_{\mathrm{in}}(\phi)P_{\mathrm{out}}(f\circ c^{-1}) \longrightarrow0 \tag{31}\] in probability on the compared samples. All integrals here are in displacement coordinates centered at the respective samples. The small \(q\) variation changes the tests by a uniformly vanishing amount; sample-dependent centering is permitted by the uniform test assertion in Lemma 15. The input depth-\(F\) law can be replaced by its recorded probability by Lemma 13; the output coarse law can be replaced by the output recorded probability by the same lemma. This proves (31) for the data that actually define the states.

For completeness, the errors just used are localized as follows. Fix a pair of charts and open conditions placing the samples in inner boxes, requiring the full-plaque margin (27), and placing the displacement test supports inside the input plaque range with positive margins. The patch fits and trims hold for all sequence samples satisfying these conditions, uniformly for the chosen motion bound. Trim accuracy may be made arbitrarily good independently of the fixed cover or these margin conditions. Thus the cross-product discrepancy tends to zero in probability on each such event. Exhaust by stricter margins and use continuous tests on compact inner boxes. Every joint distributional limit satisfies the corresponding exact identity there, since its tests and coordinate changes are continuous. Countably many charts, margin conditions, and tests suffice.

Gluing along one plaque. Apply this result first to the identity motion. Compare any chart at the sample with a chart extending far enough to contain its bounded plaque domain. By Lemma 14, each probability has positive mass on every neighborhood of the sampled point. Choose a nonnegative continuous test supported in a common such neighborhood and positive at that point. Its integral is positive for both probabilities. The cross-product identities then give projective agreement for every continuous test with compact support in the first plaque domain.

To compare two bounded domains, use a single extension containing both, on the same plaque sheet. Exhaust \(V\) by compact connected coordinate ranges containing \(1\) in their interiors. Successive extensions agree after normalization near \(1\), and so define a locally finite measure on the union. Local finiteness follows because each compact set lies in one extension carrying a finite chart probability with a finite positive normalization factor. The identity lies in its support by the sampling lemma. This construction does not require chaining through intermediate support points: each comparison uses a chart containing both the identity neighborhood and the entire displacement range in question. Using countably many tests and choosing normalizing bumps from a countable collection makes the resulting projective measure measurable.

Equivariance, including the long motion. For a fixed diagonal motion, the same cross-product identities give pushforward by \(c(v)=a(-b)va(b)\), first on bounded test ranges and then globally by the extensions just constructed. For the varying motion in \(\ker\beta\), \(c\) is the identity, and all estimates above are uniform for \(\|b_i\|\le CR_i\): the only growing chart cost is \(\exp(O(R_i))\), dominated by (26). In a paired-state limit the second base point need not be a fixed function of the first. This causes no difficulty. The local identities hold on every paired open margin event, and these events exhaust probability one because both marginals admit the required extended charts. Their exact identities therefore hold in every such limit. Since the argument applies to an arbitrary sequence \(b_i\) under the displayed bound, it proves the last assertion with the stated quantifiers. ◻

Proof. Choose interior scales \(l_-\prec T\prec F\prec l_+\), and apply the diagonal motion \(a_0^{\lfloor T\rfloor}\). In one fixed input chart, a patch specifies a \(q\)-cell at depth \(F\), with the whole \(w\) and \(u\) ranges. A subpatch also specifies a \(w\)-cell at depth \(F\). Denote these two labels by \(Q_F\) and \(W_F\).

Fix an arbitrarily small permitted coverage loss. Take finitely many output \(U\)-charts, with inner boxes and sufficiently large plaque extensions, covering all but that loss of output probability. The law of the output point is still \(\mu_i\). An input patch whose image meets one of these inner boxes lies in the corresponding extended chart, for all sufficiently large \(i\): its \(U\) range has bounded fixed size, its \(W\) displacements shrink by \(O(e^{-cT})\), and its \(q\) variation contributes at most \[O(2^{-F}e^{CT})\] for fixed constants \(c,C>0\). Assign such patches to output charts and omit the others. The omitted mass is bounded by the coverage loss.

In an assigned \(U\)-chart, the transverse image of a patch has diameter \[O(e^{-cT}+2^{-F}e^{CT}) =o(2^{-l_-}).\] The transverse projection is independent of \(u\). Its restriction to the input \((q,w)\) range is injective: the chart has a common open \(U\) range for all these inputs, and two equal transverse coordinates would otherwise give overlapping images of two different points of the injective input chart. In the output plaque coordinate, the input \(u\) interval is translated by a parameter varying by \(o(1)\) across the patch. Thus patches and their subpatches admit a common collection of inner \(U\) bins.

Here are the trimming details needed for Lemma 15. A patch meeting a coarse output wall has its sampled point in a collar of relative width \(o(1)\); the grid convention makes the total mass of these patches \(o(1)\). Fine output transverse cells can be pulled back at cost at most \(e^{CT}\). Since \(l_+\succ F\), the pullback variation is negligible compared with the input \(q\)- and \(w\)-meshes. Away from their grid walls, the whole fine cell therefore stays inside the prescribed input labels. Fixed small collars of the input chart boundary have arbitrarily small mass. After removing these collars, sufficiently fine fixed output \(U\) bins lie wholly in the admitted plaque interval, except for arbitrarily small sampling mass. One may first trim each subpatch into products of fine transverse cells and these common plaque bins, and take their union as a trim of the patch. Both levels of the comparison use the same coarse output cell and the same plaque bins.

The patch/subpatch conclusion of Lemma 15 now gives, for every continuous function \(f\) on the closure of the input \(u\) interval, \[ \int \left| \mathbb E_{\mu_i}\bigl(f(u)\mid Q_F,W_F\bigr) -\mathbb E_{\mu_i}\bigl(f(u)\mid Q_F\bigr) \right|\,\mathop{}\!d\mu_i \longrightarrow0. \tag{36}\] To obtain the limit zero, first make the bin and chart-collar losses arbitrarily small for the fixed finite cover, and then make the coverage loss arbitrarily small. The common translation of the \(u\) coordinate causes no extra error, because its variation tends to zero and \(f\) is uniformly continuous.

The plateau comparison replaces the displayed depths by the depths defining the state data. It follows that the \(u\) marginal of the \(V\) data is the \(U\) probability at the sample. For independence, let \(g\) be a bounded continuous function of \(w\), with modulus of continuity \(\omega_g\). Approximate it by a constant function \(g_F\) on each \(W_F\)-cell. Conditional expectation, followed by the triangle inequality, gives \[\begin{split} &\int\left| \mathbb E_{\mu_i}(f(u)g(w)\mid Q_F) -\mathbb E_{\mu_i}(f(u)\mid Q_F)\mathbb E_{\mu_i}(g(w)\mid Q_F) \right|\,\mathop{}\!d\mu_i\\ &\quad\le \|g\|_\infty \int\left| \mathbb E_{\mu_i}(f(u)\mid Q_F,W_F) -\mathbb E_{\mu_i}(f(u)\mid Q_F) \right|\,\mathop{}\!d\mu_i +2\|f\|_\infty\omega_g(C2^{-F}). \end{split}\] The right side tends to zero by (36). Passing to the state limit therefore gives \[P^V(f(u)g(w))=P^V(f(u))P^V(g(w))\] where \(P^V\) denotes the limiting \(V\)-chart probability. A countable dense family of continuous tests proves that the limiting probability is a product.

For a sampled plaque point with coordinates \(w_s,u_s\), the group displacement from that point to \(wu\) has \(W,U\) coordinates \[ \left(\mathop{\mathrm{Ad}}_{u_s^{-1}}(w_s^{-1}w),\ u_s^{-1}u\right). \tag{37}\] This changes the two coordinates separately, so the product assertion holds in displacement coordinates as well. Proposition 16 identifies overlapping chart probabilities projectively. Apply this comparison on product ranges containing the identity and exhausting \(W\times U\); charts with the required plaque extensions are available. The resulting \(W\) marginals agree projectively and define \(\nu^W_\omega\). This proves (35). ◻

Proof. For this proof write \(d_\alpha\) for the positive integral weight of a pattern entry \(\alpha\), and put \(B_\alpha=2^{d_\alpha}\). The digit set consists of integral pattern matrices \(d\) whose \(\alpha\) entry belongs to \(\{0,\ldots,B_\alpha-1\}\). Define \[ \mathcal F= \left\{ \prod_{\nu=1}^{\infty}\Delta^\nu(d_\nu): d_\nu\in\mathcal D \right\}, \tag{38}\] with factors ordered by increasing \(\nu\). Contraction and the finite nilpotent multiplication formulas imply uniform convergence of these products. More explicitly, in any entry the product is a finite sum of convergent series of products of digit entries, with positive geometrically decaying weights. Thus \(\mathcal F\) is compact.

Order the pattern entries by increasing root height after putting the pattern in upper triangular form. In an entry \(\alpha\), the coordinate in (38) is \[\sum_{\nu\ge1}B_\alpha^{-\nu}(d_\nu)_\alpha +C_\alpha,\] where \(C_\alpha\) is determined entirely by digits in entries of smaller height. At height one, this is the usual base-\(B_\alpha\) expansion of a number in \([0,1]\). Inductively, once the lower-height digits have been specified outside their null sets of ambiguous expansions, the possible \(\alpha\) coordinates form the interval \([C_\alpha,C_\alpha+1]\), with a unique digit expansion except at its countably many base endpoints.

Left multiplication by an integral pattern matrix has the same triangular form: the \(\alpha\) coordinate changes by its integral \(\alpha\) entry plus a function of already chosen lower-height entries. Reduce entries in this order. At each step exactly one integer places the remaining coordinate in an interval of length one, except at an endpoint, and its base expansion is then unique almost everywhere. Fubini’s theorem at each of the finitely many entries proves that almost every point of \(W\) belongs to exactly one translate \(\gamma\mathcal F\). It also proves that \(\mathcal F\) has Haar measure one.

The translates are locally finite, because \(\mathcal F\) is compact and \(\Gamma_W\) is discrete. Their union is consequently closed, and the almost-everywhere covering just proved implies that they cover all of \(W\). Splitting the first digit from (38) gives \[ \mathcal F =\bigcup_{d\in\mathcal D} \Delta(d)\Delta(\mathcal F). \tag{39}\] Triangular reduction also shows that \(\mathcal D\) represents the right cosets of \(\Delta^{-1}\Gamma_W\) in \(\Gamma_W\): \[\Gamma_W= \bigsqcup_{d\in\mathcal D} (\Delta^{-1}\Gamma_W)d.\] Applying \(\Delta\) shows that the origins in the subdivision of all level-zero tiles are exactly \[\Delta\Gamma_W =\bigsqcup_{d\in\mathcal D}\Gamma_W\Delta(d).\] Iteration proves the asserted level-\(k\) subdivision and nesting.

For completeness, these compact tiles have the regularity needed for boundary trimming. The closed translates cover \(W\), so the Baire category theorem gives nonempty interior to \(\mathcal F\). Every point of \(\mathcal F\) lies in a descendant tile in the iterated subdivision (39). Such descendants have nonempty interior and diameters tending to zero. Therefore \(\mathcal F\) is the closure of its interior. An interior intersection of two translates would then give a positive-volume overlap, contradicting almost-everywhere uniqueness. A boundary point of one translate belongs to another: otherwise local finiteness would give a neighborhood covered only by the first translate, making that point interior. Thus the boundary is contained in the union of the pairwise overlaps, which is null. Boundary assignments can be chosen measurably and consistently through the nested subdivisions.

It remains to check the law of a shifted tile coordinate; this check matters because the sampling measure on \(W\) need not have a density. Since the weights are positive integers, \(\Gamma_W\subset\Delta^k\Gamma_W\). The projection \[W/\Gamma_W\longrightarrow W/\Delta^k\Gamma_W\] sends Haar probability to Haar probability. Inversion and right multiplication by \(w\) send the latter distribution to invariant volume on the left quotient \(\Delta^k\Gamma_W\backslash W\). The representative of \(g^{-1}w\) in \(\Delta^k\mathcal F\) therefore has normalized Haar distribution on that tile, and rescaling gives Haar measure on \(\mathcal F\). This is precisely the coordinate of \(w\) in its tile \(g\gamma\Delta^k\mathcal F\). The null boundary implies the collar assertion by continuity from above. Fubini’s theorem permits the same conclusion after sampling \(w\) from any probability measure. ◻

Proof. The number of \(u\)-cells meeting its fixed bounded interval is \(O(2^S)\), so \(H(U_S)\le C(1+S)\). Because the tile labels are nested, the chain rule gives \[\sum_{k=0}^{L-1} I(K_{k+1};U_S\mid Q_F,K_k) = I(K_L;U_S\mid Q_F,K_0) \le H(U_S).\] For a jump of length \(j\), expand its mutual information into these one-stage increments. Each increment occurs at most \(j\) times in the sum over starting stages, proving (40). Pinsker’s inequality and Cauchy–Schwarz bound the average conditional \(\ell^1\) error. For \(T\ge4j\), write \(\mathcal K_T=\{k\in\mathbb Z:T/2\le k\le3T/4\}\). The bound is \[\begin{split} &\frac1{|\mathcal K_T|}\sum_{k\in\mathcal K_T} \mathbb E\bigl\| \mathcal L(K_{k+j}\mid Q_F,K_k,U_S) -\mathcal L(K_{k+j}\mid Q_F,K_k) \bigr\|_1\\ &\hspace{25mm}\le C\sqrt{\frac{j(1+S)}{T}}. \end{split}\] This tends to zero for fixed \(j\). The entropy bound is uniform in the shift, so it can also be averaged over shifts. ◻

Proof. Let \(m(x)\) denote the shortest-vector length. Divide \(X\) into layers on which \(m(x)\) is comparable to \(e^{-k}\), for integers \(k\geq0\), using a first layer that also contains the thick part. Layer boundaries may be chosen \(\mu\)-null. Successive minima and lattice basis reduction give a representative matrix \(g\) for each point of the \(k\)th layer such that \[\|g\|+\|g^{-1}\|\leq e^{C_1(1+k)}.\] Consequently the injectivity radius for relative group coordinates on that layer is at least \(e^{-C_2(1+k)}\). Indeed, if a nonidentity integral matrix \(\gamma\) gave a smaller return, conjugating the return by \(g\) would give \(\|\gamma-1\|<1\), which is impossible.

We impose a quantitative margin condition on these charts. Write \(d_G\) for the fixed left-invariant Riemannian distance on \(G\). In a fixed identity neighborhood of \(G\), adapted product coordinates give a smooth retraction \(\pi\) onto \(VA\), with \(\pi(h)=h\) for \(h\in VA\) and \[d_G(h,\pi(h))\leq C_{\pi}\mathop{\mathrm{dist}}_G(h,VA), \qquad d_G(1,\pi(h))\leq C_{\pi}d_G(1,h).\] After shrinking the neighborhood, \(1\) and \(\pi(h)\) are joined inside \(VA\) by a path of length at most \(C_{\pi}d_G(1,h)\); one may use the local exponential coordinates of \(VA\).

Choose outer \(VA\) product charts in the \(k\)th layer of radius comparable to \(r_k=e^{-C_3(1+k)}\), with \(C_3\) sufficiently large for injectivity, and inner boxes whose relative-coordinate distance from the outer boundary is at least a fixed multiple of \(r_k\). Use translates of a fixed smooth group-coordinate chart, so that its local product and metric comparison constants are uniform. Cover the layer by still smaller symbol boxes inside these inner boxes. Require their relative-group diameter to be at most \(c_*r_k\), where \(c_*>0\) is small enough compared with both the inner-to-outer margin and the radius of the domain of \(\pi\). In particular, whenever \(x,y=xh\) are in one symbol box, the whole path from \(x\) to \(x\pi(h)\) just described stays inside its outer chart. It follows that \(x\pi(h)\) is on the same chart plaque as \(x\), rather than on another local sheet of that orbit.

The number of symbol boxes needed is at most \(e^{C_4(1+k)}\). Indeed, subdivide the bounded-matrix representatives at an exponentially small mesh, and refine by the fixed factor \(c_*\) when necessary. These operations preserve an exponential bound in \(k\). Passing from the cover to a partition by successive differences does not increase the number of pieces or their diameters. The charts and their cuts can be chosen from the stock with \(\mu\)-null boundaries: perturb centers within a small fixed fraction of the covering radius and choose each coordinate width in a fixed comparable interval, avoiding the at most countably many widths that give a boundary of positive measure. These choices preserve the exponential bounds on sizes and on the number of boxes, as well as the inner-chart margins. Thus each partition atom remains inside a chart with a margin, although it need not itself be a coordinate rectangle.

The shortest-vector length changes by at most a fixed multiplicative factor under \(T\). We can therefore choose the boxes so small that both a relative lift in one symbol and its conjugate by \(a(b)\) lie in an injective neighborhood at the next symbol. Uniqueness in that neighborhood identifies the latter lift with the lift chosen there. Induction gives the consistency in assertion 2. Decreasing all box sizes by a common fixed factor ensures that every lift lies in the prescribed \(O\).

By Proposition 4, a probability \(\nu\leq C\mu_i\) assigns the \(k\)th layer mass at most \(C_5e^{-c k}\), uniformly for sufficiently large \(i\). If that mass is \(p_k\) and there are at most \(M_k\) symbols in the layer, their entropy is bounded by \[p_k\log M_k-p_k\log p_k.\] Since \(\log M_k=O(1+k)\) and \(p_k\leq C_5e^{-ck}\), the sum of these bounds converges uniformly, and its tail tends to zero uniformly. Keeping all symbols from the first finitely many layers proves both entropy assertions. ◻

Proof. First sample \(x\) under \(\nu_i\), and draw the second plaque coordinate from the \(\mu_i\) probability conditioned only on its transverse cell in the whole chart. The depth-comparison rules allow this conditional probability to be replaced, in the limit, by the chart probability in the state data. Any joint limit of the state with \(x\) sampled under \(\nu_i\) is dominated by \(C\) times the usual state law. Its crossing-root indicators vanish almost surely, since their nonnegative sum has expectation tending to zero. Lemma 22 therefore puts the second point on the same \(H_s\) plaque as the sample. Its \(z\) coordinate equals the sampled \(z\). The transverse mesh tends to zero; continuous coordinate tests on the fixed compact chart boxes now show that the prelimit \(z\) difference tends to zero in probability. This argument uses the probability attached to one chart and its statewise support. It does not condition the sampled point on the other plaque probabilities in the state.

To change the second draw to the stated \(\nu_i\) conditional law, fix \(\varepsilon>0\) and discard events \(E=S\cap\{q\text{ in a given cell}\}\) such that \[\nu_i(E)<\varepsilon\mu_i(\{q\text{ in that cell}\}\cap\text{chart}).\] Their total \(\nu_i\) mass is at most \(\varepsilon\) times the number of symbols under consideration. On every remaining event the conditional \(\nu_i\) law has density at most \(C/\varepsilon\) with respect to the \(\mu_i\) chart-cell law. The preceding convergence thus remains valid for the second draw. Let \(i\to\infty\) and then \(\varepsilon\to0\).

For the averaged statement, first observe a uniform sequential consequence of the first assertion: no sequence of dominated probabilities can have indicator expectation tending to zero while the asserted pair error stays bounded away from zero. Let \(a_i\to0\) be the weighted mean of the indicator expectations in a list. Discard members whose expectation is greater than \(\sqrt{a_i+i^{-1}}\). Their total weight is at most \(a_i/\sqrt{a_i+i^{-1}}\to0\) by Markov’s inequality. On the remaining members, the pair errors tend uniformly to zero by the sequential consequence: otherwise a sequence of members violating it could be selected. This proves the averaged assertion. ◻

Proof of Proposition 25. Suppose, along a subsequence, that the expression in (52) tends to zero. Choose even integers \(N_i\sim\tau R_i\) with \(N_i+2<\tau R_i\). Averaging a starting shift \(s\in[0,1]\) shows that we may replace \(\xi_i\) by its translate through some \(s_i\in[0,1]\) so that \[ \frac1{N_i+1}\sum_{t=0}^{N_i} \int A_{J,i}(T^t x)\,\mathop{}\!d\xi_i(x)\longrightarrow0. \tag{54}\] All these translated probabilities still satisfy \(\xi_i\leq C\mu_i\). More generally, every time-\(t\) distribution \(\nu_{i,t}=(T^t)_*\xi_i\) has the same domination bound.

Choose a compact set \(K\) with \(\nu_{i,N_i/2}(K)\geq1-\varepsilon_0\) for all sufficiently large \(i\), where \(\varepsilon_0>0\) will be small. This follows from the uniform cusp estimate and domination. Take the neighborhood \(O\) in Lemma 26 small enough to use Proposition 7 uniformly on \(K\). If a name atom \(B\in\mathcal P_{[0,N_i]}\) meets \(T^{-N_i/2}K\), choose a reference point in that intersection. Consistency of the relative lifts puts every midpoint of \(B\) in the tube about its reference midpoint with time \(N_i/2\). Since \(T_{-,i}\prec R_i\prec T_{+,i}\), this time is in the arithmetic range for all sufficiently large \(i\). Hence \[\xi_i(B)\leq C e^{-\eta N_i/2}.\] Such atoms have total \(\xi_i\) mass at least \(1-\varepsilon_0\). Integrating the negative logarithm of the atom mass gives, after decreasing \(\varepsilon_0\) and taking \(i\) large, \[ \frac1{N_i}H_{\xi_i}(\mathcal P_{[0,N_i]})\geq\frac\eta3. \tag{55}\]

We next derive the incompatible upper bound. Fix a small target error \(\varepsilon>0\). First choose a finite coarsening \(\mathcal P^0\) so that the conditional entropy in (53) is as small as required below. Let \(q_0\) be its alphabet size. Choose also a finite collection \(\mathcal S\) of original symbols whose total complement has arbitrarily small probability under every \(\nu\leq C\mu_i\). The symbols in \(\mathcal S\) need not be the symbols retained in \(\mathcal P^0\).

The boundary of \(\mathcal P^0\) is a finite union of compact \(\mu\)-null sets. We can choose a fixed distance accuracy \(\rho>0\) so that its \(2\rho\) neighborhood has arbitrarily small probability under every \(\nu\leq C\mu_i\) for all sufficiently large \(i\). To justify this uniformly, take decreasing closed neighborhoods of these boundaries; their \(\mu\) masses tend to zero, and the weak convergence upper bound applies to each fixed neighborhood. All subsequent orbit predictions will be within distance \(\rho\) of the actual point. Incorrect symbols can then occur only near these boundaries, or on the exceptions specified below.

Choose transverse grids in the charts for \(\mathcal S\) at an interior depth \(l_i\prec R_i\). Let \(M\) be a large integer, fixed while \(i\to\infty\). Its size will be chosen after the charts and \(\rho\); the concentration tolerances used below may then depend on \(M\). Split the name into blocks of \(M\) future symbols given the past through the block’s initial time. We may omit an initial \(t_i=o(N_i)\) symbols and a final remainder of length less than \(M\): their entropy cost divided by \(N_i\) tends to zero by the uniform single-symbol entropy bound.

Here is why the extra transverse-cell label costs only a bounded amount at every remaining block. In a past-name atom at time \(t\), the negative root entries of its relative lift are \(O(e^{-c_b t})\), where \(c_b=\min_{\beta(b)>0}\beta(b)>0\). Triangular coordinates in a small group neighborhood therefore put the lift at distance \(O(e^{-c_b t})\) from \(VA\). Let \(x,y=xh\) be two current points in one past-name atom, and use the retraction \(\pi\) from the partition construction. The quantitative diameter condition ensures that \(x\pi(h)\) lies on the same plaque of the current outer chart as \(x\). Thus \(q(x\pi(h))=q(x)\). Lipschitz continuity of the transverse coordinate on that chart gives \[\begin{aligned} |q(y)-q(x)| &=|q(xh)-q(x\pi(h))|\\ &\leq C\,d_G(h,\pi(h)) \leq C'\mathop{\mathrm{dist}}_G(h,VA) \leq C''e^{-c_b t}. \end{aligned}\] The constants are uniform over the fixed finite collection of current charts. Choose \(t_i\) to be a sufficiently large fixed multiple of \(l_i\); then \(e^{-c_b t}<2^{-l_i}\) for \(t\geq t_i\), and \(t_i=o(N_i)\). Consequently the possible transverse coordinates meet only a bounded number \(B_0\) of neighboring cells. Given the past, their label has entropy at most \(\log B_0\). For current symbols outside \(\mathcal S\) use a single dummy label.

Let \(\mathcal Q_i\) record this cell label in the current symbol, with the dummy label outside \(\mathcal S\), and write \(\mathcal F_t=\mathcal P_{[0,t]}\). After telling the cell label, we may discard the rest of the past when upper-bounding the entropy of the next coarsened block. Precisely, the chain rule and monotonicity under conditioning give \[ \begin{aligned} &H_{\xi_i}\bigl((\mathcal P^0)_{[t+1,t+M]}\mid\mathcal F_t\bigr)\\ &\quad\leq H_{\xi_i}\bigl(T^{-t}\mathcal Q_i\mid\mathcal F_t\bigr) +H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]} \mid\mathcal P\vee\mathcal Q_i\bigr)\\ &\quad\leq\log B_0+ H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]} \mid\mathcal P\vee\mathcal Q_i\bigr). \end{aligned} \tag{56}\] The conditioning in the last entropy is just the current symbol and its cell. By (54), the mean crossing-root activity at the block starts tends to zero. Indeed their nonnegative sum is bounded by the full sum in that equation, and normalizing by the number of blocks loses at most the fixed factor \(M\). The averaged form of Lemma 27 applies to their distributions. For each symbol-cell event we may therefore choose a \(z\) center so that, apart from a probability tending to zero on average, \(z\) lies within any prescribed fixed tolerance of that center. Such a center exists by averaging the two-draw error and choosing one of the points as center.

The \(d\) coordinates lie in a fixed compact part of \(H_s\) for each of the finitely many charts. Every root occurring in \(H_s\) has expansion rate at most \(\delta\), and \(A\) commutes with the motion. A mesh in these coordinates, of accuracy a fixed multiple of \(\rho e^{-\delta M}\), therefore predicts the trajectory through the next \(M\) steps to the required distance accuracy, provided the \(z\) and transverse coordinates are sufficiently close to their centers. The number of mesh points is at most \[ C_0\exp(C_6\delta M). \tag{57}\] Here \(C_6\) depends only on the dimension, while \(C_0\) depends on the fixed charts and \(\rho\), not on \(M\). This follows by using finitely many smooth coordinate meshes on the compact \(d\) ranges: conjugation expands every entry of a group-relative error by at most \(e^{\delta M}\). Left invariance of the metric transfers the resulting relative-group error to an error on \(X\).

The cross-root center tolerance may need to be exponentially small in \(M\), because cross-root directions expand faster. This causes no additional mesh labels: \(M\) is fixed before applying the concentration lemma, so any such positive tolerance is allowed. The transverse width also tends to zero for each fixed \(M\). Thus \(C_0\) in (57) remains independent of \(M\).

Encode the next \(M\) symbols of \(\mathcal P^0\) by a mesh label, with a single additional label for an omitted chart or an outlier, and then by corrections to the predicted symbols. Let \(e_{i,t}\) be the average probability of an incorrect prediction over these \(M\) positions, with the starting point sampled under \(\nu_{i,t}\). Put \(h_2(e)=-e\log e-(1-e)\log(1-e)\), with the usual zero conventions. The mesh and correction costs give \[ \begin{aligned} &H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]} \mid\mathcal P\vee\mathcal Q_i\bigr)\\ &\quad\leq\log\bigl(C_0e^{C_6\delta M}+1\bigr) +M\bigl(h_2(e_{i,t})+e_{i,t}\log q_0\bigr). \end{aligned} \tag{58}\] For completeness, reveal the error indicator at each position and, on an error, reveal the actual symbol; subadditivity and concavity of \(h_2\) give this bound also when the error probabilities differ across positions and blocks. The chart omissions and boundary neighborhoods were chosen with arbitrarily small probabilities. The additional concentration errors tend to zero on average. Consequently the mean of \(e_{i,t}\) over the block starts can be made as small as required; concavity of \(h_2\) gives the same small correction bound after averaging blocks.

Finally, the original block is recovered from its coarsened block at the following cost: \[ \begin{aligned} &H_{\xi_i}\bigl(\mathcal P_{[t+1,t+M]}\mid\mathcal F_t\bigr)\\ &\quad\leq H_{\xi_i}\bigl((\mathcal P^0)_{[t+1,t+M]}\mid\mathcal F_t\bigr) +\sum_{r=1}^M H_{\nu_{i,t+r}}(\mathcal P\mid\mathcal P^0). \end{aligned} \tag{59}\] Indeed, conditional on the entire coarsened block, revealing each original symbol costs at most its one-step conditional entropy given its own coarsened symbol. Every summand on the right is controlled by (53), because \(\nu_{i,t+r}\leq C\mu_i\).

Sum (59) over the blocks and use (56) and (58). The chain rule, together with the negligible initial and final pieces, now gives \[ \limsup_{i\to\infty}\frac1{N_i} H_{\xi_i}(\mathcal P_{[0,N_i]}) \leq C_6\delta+\varepsilon+ \frac{\log B_0+\log(C_0+1)}{M}. \tag{60}\] All constants in the last numerator were fixed before \(M\) was chosen. Choose \(\delta\) first so that \(C_6\delta<\eta/12\), then choose the truncation, charts and distance accuracy to make \(\varepsilon<\eta/12\), and finally choose \(M\) so large that the last term is less than \(\eta/12\). Letting \(i\to\infty\) contradicts (55). This proves the proposition, including its subsequence formulation. ◻

Proof. For \(v\) in a fixed bounded subset of \(L\), the vector \(v-\beta(v)b_\beta\) belongs to \(\ker\beta\) and remains bounded. Wall invariance (50), followed by invariance of \(\mu_i\) under the common translation, gives \[ \sup_{v\text{ in a fixed bounded set}} \int_X\left| G_{\beta,i}^{x}\bigl(\beta(v)\bigr) -g_{\beta,i}\bigl(xa(R_i v)\bigr) \right|\,\mathop{}\!d\mu_i(x)\longrightarrow0. \tag{65}\]

Fix the parameters in assertion 1. The left side of (63) is a continuous function of the weak-* array, because \(\beta(b_J)\ne0\) for every crossing root. By (65), its prelimit differs in \(L^1(\mu_i)\) by \(o(1)\) from \[Y_i(x)=\sum_{\beta\in\mathcal E_J}\int_0^\tau g_{\beta,i}\bigl(xa(R_i(v_0+t b_J))\bigr)\,\mathop{}\!dt.\] Suppose that the limiting observable were zero on a set of probability \(p>0\). Convergence in law and nonnegativity allow us to choose thresholds \(\varepsilon_i\downarrow0\) and events \(E_i\) with \(\mu_i(E_i)\geq p/2\) such that the original prelimit observable is at most \(\varepsilon_i\) on \(E_i\). To see this directly, first choose a fixed sufficiently small continuity threshold, use convergence in law, and then decrease the threshold along a diagonal subsequence. The normalized restrictions \(\mu_i|_{E_i}/\mu_i(E_i)\) consequently have \(Y_i\) expectation tending to zero. Translate these probabilities by \(a(R_i v_0)\) and call the results \(\xi_i\). They satisfy \(\xi_i\leq(2/p)\mu_i\), whereas the change of variable \(u=R_i t\) gives \[\frac1{R_i}\int_0^{\tau R_i}\int_X A_{J,i}\bigl(xa(u b_J)\bigr)\,\mathop{}\!d\xi_i(x)\,\mathop{}\!du \longrightarrow0.\] This contradicts Proposition 25. The parameters in assertion 1 range over a countable set, so the assertion holds simultaneously almost surely.

For assertion 2, the roots \(\beta_{jk}\) and \(\beta_{k\ell}\) are linearly independent. Choose \(v(s,t)\) linearly in \((s,t)\) with \[\beta_{jk}(v(s,t))=s,\qquad \beta_{k\ell}(v(s,t))=t.\] It follows that \(\beta_{j\ell}(v(s,t))=s+t\). On a bounded rectangle, (65) permits replacing all three functions in the defect by their indicators at the common point \(xa(R_i v(s,t))\), with total expected error \(o(1)\). The expected defect there is \(o(1)\) by the approximate transitivity in Lemma 23; its distribution is unchanged by the translation. Integrating over the rectangle proves the claim. ◻

Proof. Pass to a subsequence on which \(\int_X(1-g_{\beta_0,i})\,\mathop{}\!d\mu_i\geq\varepsilon\) for a fixed \(\varepsilon>0\). Invariance of \(\mu_i\) and Fubini’s theorem imply, for any fixed interval \(I\), \[\mathbb E\int_I(1-G_{\beta_0,i}^{x}(s))\,\mathop{}\!ds =|I|\int_X(1-g_{\beta_0,i})\,\mathop{}\!d\mu_i \geq\varepsilon|I|.\] The integral is a bounded weak-* continuous observable, so the same inequality holds for the limiting law. This law therefore gives positive probability to a positive deficiency. Choose a point \(F\) in its support that has a positive deficiency and satisfies assertion 1 of Lemma 29.

For each integer \(q\), choose a neighborhood \(U_q\) of \(F\) of diameter less than \(1/q\). It has positive limiting probability, and thus the prelimit probabilities of \(U_q\) are bounded below by a positive number for all sufficiently large indices. There are finitely many root compositions. By (64), the sum of their defects on \([-q,q]^2\) has expectation tending to zero. Choose an index so large that, by Markov’s inequality, the probability that this sum exceeds \(1/q\) is less than half the probability of \(U_q\). There is therefore a sampled array in \(U_q\) with total defect at most \(1/q\). Make these choices with increasing indices. They converge to \(F\) and have vanishing defects on every bounded rectangle, by nonnegativity and containment in some \([-q,q]^2\). ◻

Proof. Fix a root \(\beta_{jk}\) and a desired interval \(I\); replacing it by a bounded nonempty open subinterval only strengthens what we must prove. Choose a rational \(v_0\in L\) so that \(\beta_{jk}(v_0)\) lies in \(I\). Choose \(r>0\) small enough that \[[\beta_{jk}(v_0)-6r,\ \beta_{jk}(v_0)+6r]\subset I.\] For every root \(\alpha\), put \(I_\alpha=(\alpha(v_0)-r,\alpha(v_0)+r)\). Draw the directed edge \(p\to q\) if \(\int_{I_{\beta_{pq}}}F_{pq}>0\).

This directed graph is strongly connected. Indeed, for any nonempty proper subset \(J\), choose a positive rational \(\tau\) sufficiently small that \(\alpha(v_0+t b_J)\in I_\alpha\) for every \(\alpha\in\mathcal E_J\) and \(0\leq t\leq\tau\). Equation (63) then supplies an edge from \(J\) to its complement. A finite directed graph with an outgoing edge across every nontrivial cut is strongly connected: otherwise the set of vertices reachable from a suitable vertex is a nonempty proper set with no outgoing edge.

Take a simple directed path \(j=p_0,p_1,\ldots,p_m=k\), where \(m\leq5\). Write \(\alpha_h=\beta_{p_{h-1}p_h}\) and \(I_h=I_{\alpha_h}\). Weak-* convergence gives positive lower bounds, for sufficiently large \(i\), on \[\int_{I_h}G_{\alpha_h,i}(s)\,\mathop{}\!ds, \qquad 1\leq h\leq m.\] The product of these integrals is consequently bounded below by a positive constant. Interpret it as the measure of tuples \((s_1,\ldots,s_m)\in\prod_h I_h\) on which all the path labels equal one.

Except for a set of tuple measure tending to zero, successive composition of these labels gives \[G_{p_0p_h,i}(s_1+\cdots+s_h)=1, \qquad 1\leq h\leq m.\] Here is the integration detail. At step \(h\), the failure of this implication, if all earlier implications held, is bounded by the transitivity defect for \(p_0,p_{h-1},p_h\). The path is simple, so these are distinct vertices. Integration in \(s_1,\ldots,s_{h-1}\) pushes Lebesgue measure on bounded intervals to a measure of bounded density in their sum. Multiplying by previous indicator restrictions can only decrease this measure. The defect integral on a fixed bounded rectangle therefore bounds the failure measure by a fixed multiple of an \(o(1)\) quantity. Summing over the finitely many steps proves the assertion. If \(m=1\), no composition is necessary.

The final sum \(s_1+\cdots+s_m\) lies within \(mr\) of \(\alpha_1(v_0)+\cdots+\alpha_m(v_0)=\beta_{jk}(v_0)\), hence in \(I\). Its pushforward density is bounded above by a fixed constant. Since a positive measure of tuples has final label one, we obtain \(\liminf_i\int_I G_{jk,i}>0\). Weak-* convergence proves the lemma. ◻

Proof. If the family is empty there is nothing to prove. Otherwise write \(\sigma_p^2=\int x^2p(x)\,\mathop{}\!dx\), so \(v\leq\sigma_p^2\leq B^2\), and let \[\phi_p(t)=\int e^{itx}p(x)\,\mathop{}\!dx.\] Bounded support and the zero mean give the uniform expansion \[ \phi_p(t)=1-\frac12\sigma_p^2t^2+O(B^3|t|^3). \tag{67}\] Applying the same expansion to \(|\phi_p(t)|^2\), the characteristic function of the difference of two independent variables with density \(p\), shows that for a sufficiently small \(\delta>0\), depending only on \(B,v\), \[ |\phi_p(t)|\leq e^{-vt^2/4}\qquad(|t|\leq\delta). \tag{68}\]

There is also a uniform gap below one outside this neighborhood. For \(|t|\geq\delta\), an angle \(\theta\), and \(0<a<\pi\), the set \[\{x\in[-B,B]:\operatorname{dist}(tx-\theta,2\pi\mathbb Z)<a\}\] has Lebesgue measure at most \(a(2B/\pi+4/\delta)\). Indeed, each complete period contributes the fraction \(a/\pi\) of its length, and at most two incomplete periods contribute at the ends. Choose \(a\) so small that \(M\) times this bound is at most \(1/2\). With \(\theta=\arg\phi_p(t)\), at least half of the probability lies outside this arc. Taking the real part after rotation by \(e^{-i\theta}\) gives \[ \sup_p\sup_{|t|\geq\delta}|\phi_p(t)| \leq\frac{1+\cos a}{2}=:\rho<1. \tag{69}\] If \(\phi_p(t)=0\), the inequality is immediate and the angle is irrelevant. Notice that the estimate holds at arbitrarily large frequencies; no uniform version of the Riemann–Lebesgue lemma is needed.

Plancherel’s identity yields \[\int_\mathbb R|\phi_p(t)|^2\,\mathop{}\!dt =2\pi\int_\mathbb Rp(x)^2\,\mathop{}\!dx\leq2\pi M.\] For \(m\geq2\), \(\phi_p^m\) is integrable. Fourier inversion therefore defines a continuous representative of \(p^{*m}\). We claim the following uniform local central limit estimate: \[ \sup_p\sup_{x\in\mathbb R} \left| \sqrt m\,p^{*m}(x) -\frac1{\sqrt{2\pi}\,\sigma_p} \exp\left(-\frac{x^2}{2m\sigma_p^2}\right) \right|\longrightarrow0. \tag{70}\] To verify it, change variables \(u=t\sqrt m\) in the inversion formula. The difference in (70) is at most \(1/(2\pi)\) times \[\int_\mathbb R\left| \phi_p(u/\sqrt m)^m-e^{-\sigma_p^2u^2/2} \right|\,\mathop{}\!du.\] On each fixed bounded \(u\) interval, the expansion (67) proves uniform convergence of the integrand to zero; the logarithm may be taken near one, and its error is \(O(B^3|u|^3/\sqrt m)\). On \(|u|\leq\delta\sqrt m\), (68) bounds the first term by \(e^{-vu^2/4}\), while the Gaussian term is bounded by \(e^{-vu^2/2}\). Thus the contribution outside a large fixed bounded interval but inside this region is uniformly small. On \(|u|>\delta\sqrt m\), (69) and Plancherel give \[\int_{|u|>\delta\sqrt m}|\phi_p(u/\sqrt m)|^m\,\mathop{}\!du \leq2\pi M\sqrt m\,\rho^{m-2}\longrightarrow0.\] The Gaussian tail there also tends to zero uniformly. These three estimates prove (70).

For \(|x|\leq L\) and all sufficiently large \(m\), the Gaussian term in (70) is at least \(1/(2\sqrt{2\pi}B)\), uniformly in \(p\). Increase \(m\) so that the error is at most \(1/(4\sqrt{2\pi}B)\). Then \[p^{*m}(x)\geq\frac1{4\sqrt{2\pi}B\sqrt m} \quad(|x|\leq L),\] which is the required conclusion. ◻

Proof of Theorem 28. Suppose the theorem fails, and take the deterministic arrays and limiting functions from Lemma 30. Fix an edge \(j\to k\), and choose \(\ell\) distinct from \(j,k\). Lemma 31 supplies, for each of the roots \(\ell j\) and \(j\ell\), positive limiting integral on each of the intervals \((-2,-1)\) and \((1,2)\). Accordingly, for all sufficiently large \(i\), each of the four sets where the relevant \(G\) equals one has length bounded below by a positive constant.

For each auxiliary root, normalize Lebesgue measure on its two sets to probability densities \(p_i^-\) and \(p_i^+\). Their means belong to \([-2,-1]\) and \([1,2]\), respectively. There is a convex combination \[p_i=\lambda_i p_i^-+(1-\lambda_i)p_i^+\] with mean zero. The coefficients can be chosen in \([1/3,2/3]\) by solving the mean-zero equation. Thus these densities have support in \([-2,2]\), a common finite upper bound, and variance at least one. They are supported entirely where the corresponding auxiliary indicator is one. Denote the density for \(\ell j\) by \(p_i\) and that for \(j\ell\) by \(q_i\).

Starting from \(G_{jk,i}(t)=1\), composition with a parameter \(s\) in the support of \(p_i\) gives the edge \(\ell k\), and composition with a parameter \(s'\) in the support of \(q_i\) returns to \(jk\): \[\ell j+jk=\ell k,\qquad j\ell+\ell k=jk.\] Both are root compositions with three distinct vertices. Their defect estimates imply, for every fixed bounded interval \(I\), \[ \int_I\int_\mathbb R\int_\mathbb R G_{jk,i}(t)\bigl(1-G_{jk,i}(t+s+s')\bigr) p_i(s)q_i(s')\,\mathop{}\!ds\,\mathop{}\!ds'\,\mathop{}\!dt\longrightarrow0. \tag{71}\] Indeed, except on null sets of parameters, both auxiliary labels are one. The indicator in (71) is then bounded by the sum of the two composition defects. For the first defect, integrate out \(s'\) and use the uniform density bound on \(p_i\). For the second, the variable \(t+s\) has a bounded density on a fixed bounded interval when \(t\in I\) is integrated with Lebesgue measure and \(s\) with \(p_i\). The bound on \(q_i\) and the defect estimate for \((j\ell,\ell k)\) finish the verification.

Put \(r_i=p_i*q_i\). These cycle-increment densities have support in \([-4,4]\), a common density bound, mean zero, and variance at least two. We can iterate (71) any fixed number \(m\) of times to get \[ \int_I\int_\mathbb R G_{jk,i}(t)\bigl(1-G_{jk,i}(t+u)\bigr) r_i^{*m}(u)\,\mathop{}\!du\,\mathop{}\!dt\longrightarrow0. \tag{72}\] For clarity, along a sequence of \(m\) increments, a path starting with label one and ending with label zero has a first transition from one to zero. Bound its indicator by the sum of the indicators of these transitions. The intermediate positions lie in \(I+[-4m,4m]\). After integrating all preceding increments, their distributions have densities bounded by one relative to Lebesgue measure on that larger interval, because the initial \(t\) is integrated over \(I\) and all increments have probability laws. Each term is therefore bounded by the one-cycle error on the larger interval. There are only \(m\) terms, proving (72).

Fix a bounded interval \(I\) of positive length and apply Lemma 32 to the \(r_i\), with \(L\) large enough that \(I-I\subset[-L,L]\). Choose its fixed power \(m\) and lower bound \(c>0\). In (72), retain only the pairs \(t,t+u\in I\). We obtain \[c\left(\int_I G_{jk,i}(t)\,\mathop{}\!dt\right) \left(\int_I(1-G_{jk,i}(t))\,\mathop{}\!dt\right) \longrightarrow0.\] The first factor has positive lower limit by Lemma 31. Hence \(\int_I(1-G_{jk,i})\to0\) for every such interval. Weak-* convergence shows that \(F_{jk}=1\) almost everywhere. The argument applies to every root, contradicting the deficiency (66). Thus (61) holds.

By Lemma 23, the sequence indicators converge jointly in law to the activity indicators of the limiting state. Since each has expectation tending to one, every limiting root indicator equals one almost surely. There are only thirty roots, so this holds for all of them simultaneously. ◻