Proof. Choose \(F\) as above and put \(n_0=2F\). Repeated finite Ramsey gives an \(n\) such that every coloring of the nonempty index subsets of \([n]\) has an \(n_0\)-element subset on which the color depends only on cardinality, when the number of colors is \(r\). More precisely, apply the finite Ramsey theorem successively to the subset sizes \(1,\ldots,n_0\), choosing the successive ambient bounds backward. Apply this to the coloring \(A\mapsto\chi(x_A)\), and call the resulting cardinality color \(\kappa(q)\), \(1\le q\le n_0\).

Color \(a\in[F]\) by \(\kappa(2a)\). Choose \(u_1,\ldots,u_m\) by the finite sums theorem and put \(q_d=2u_d\). Then \(q_d\ge2\), \(\sum_dq_d\le n_0\), and all nonempty subset sums of the \(q_d\) have the same \(\kappa\)-color. In the homogeneous ordered index set, choose successive groups of \(q_1-1,\ldots,q_m-1\) indices for \(T_1,\ldots,T_m\), followed by \(m\) indices for the pivots \(i_1<\cdots<i_m\). This uses \(\sum_dq_d\) indices and satisfies (3). A product of the corresponding block products is \(x_{\bigcup_{d\in J}B_d}\), whose index-set size is \(\sum_{d\in J}q_d\). All these sizes have the chosen color. ◻

Proof of Theorem 1, assuming the two principles. The case \(m=1\) follows by choosing any integer \(a_1>R\). For \(m\ge2\), fix \(s=s(m)\) from Principle 5 and \(n=n(m,r)\) from Lemma 7. Apply Principle 6 to fix \(\mathcal B,\mathcal A\) and \(\delta>0\). Let \(C\) be the number of triples \((B,a,c)\) indexing calibration events, and let \(K\) be the number of triples consisting of a scale vector, a chain of length \(m\), and a color. These are fixed finite numbers depending on data already chosen.

Choose \(0<\tau<1/4\) and then \(\eta>0\) so that \[ 3C\tau<\frac\delta4,\qquad C\eta<\frac\delta4,\qquad K\eta<\frac{\delta\tau^{2^m}}4. \tag{15}\] Now apply Principle 5. Define \(\mathsf G_w\) to be the event that none of its calibration failures occurs. The union bound and (8) give \[\lim_{\mathcal U}\mathbb P(\mathsf G_w^c) \le C(3\tau+\eta)<\frac\delta2.\] Since an ordinary lower limit bounds every ultrafilter limit, (14) yields \[ \lim_{\mathcal U}\mathbb P(\mathsf A_w\cap\mathsf G_w) \ge\frac\delta2. \tag{16}\]

On this intersection choose a witnessing \(b\in\mathcal B\) and apply Lemma 7 to \(x_i=h_ib_it_i\), which are positive integers for all sufficiently large \(w\). It supplies a chain and a color \(c\) for which \(U(z(t))=1\). In particular, the singleton product masks say \(\chi(h_{B_d}b_{B_d}t_{B_d})=c\) for every \(d\). Because \(\mathsf G_w\) holds, all centers \(S_d(t_{B_d})\) exceed \(2\tau\).

For a nonsingleton \(J\subseteq[m]\), let \(d=\max J\). Each \(B_k\), \(k\in J\setminus\{d\}\), belongs to \(\mathcal E_{B_d}\). The corresponding instance of (13) therefore gives \[S_d\bigl(L_J(z(t))\bigr)>\tau.\] There are \(q=2^m-m-1\) such factors. Thus at least one of the \(K\) nonnegative model integrands in (12) is greater than \(\tau^q\ge\tau^{2^m}\) on \(\mathsf A_w\cap\mathsf G_w\). Summing expectations and using (16) shows that their sum has ultrafilter limit at least \(\delta\tau^{2^m}/2\). It follows from (12) and (15) that \[ \lim_{\mathcal U} \sum_{b,c,\mathbf B}\mathcal I_{b,c,\mathbf B}>0. \tag{17}\] In particular, the finite sum is bounded below by a positive number on a set belonging to \(\mathcal U\). We did not need a fixed choice of witnessing chain across different samples or values of \(w\).

For some sufficiently large \(w\), one weighted count is positive. Since it is a finite nonnegative average, there is an actual tuple \(z\) in its support for which every factor is positive. Put \(a_d=c_dz_d\). The product mask gives every nonempty product of the \(a_d\) in color \(c\), including their singleton values. For every nonsingleton \(J\), positivity of its color factor gives \[\chi\left(c_{d(J)}L_J(z)\right) =\chi\left(\sum_{k\in J}a_k\right)=c.\] The extra divisor factors cannot create a false positive; they only restrict which tuples can contribute to the count.

It remains to obtain the prescribed separation. The finite rational scale lists give a constant \(C_0\) such that \(c_d\le C_0M\) for all these counts, and \(c_d\ge1\) once it is a positive integer. If \(d<e\), then \[a_d<C_0M X_{i_d}^2,\qquad a_e\ge X_{i_e}.\] For \(e\ge2\), put \(V_e=2+M+\prod_{j<i_e}X_j\). There is a constant \(C_1\), depending only on the fixed finite data, such that every support tuple satisfies \[\sum_{d<e}a_d+\prod_{d<e}a_d\le C_1 V_e^{3m}.\] Indeed each preceding \(a_d\) is at most \(C_0V_e^3\), and there are at most \(m-1\) such terms. Admissibility makes \(X_{i_e}\) dominate every fixed power of \(V_e\). Therefore, for the prescribed \(R,D\), all support tuples at sufficiently large \(w\) satisfy (1); the first element exceeds \(R\) since \(a_1\ge X_{i_1}\to\infty\). The positive-count set in (17) belongs to the nonprincipal ultrafilter and hence meets every sufficiently late range of \(w\). Choose \(w\) in that intersection. This proves both monochromaticity and separation, and completes Theorem 1. ◻

Proof of Corollary 2. Apply Theorem 1 with \(R=2\) and \(D=1\). Every \(a_j\) exceeds the sum and the product of all its predecessors, and every element is at least two. If two subset sums were equal, cancel their common terms. The greatest remaining index on either side would exceed the sum on the other side, a contradiction. For products, cancel common factors and compare the greatest remaining factor with the product of all preceding factors. An empty side after cancellation equals one, which is also smaller than every remaining factor. Thus both subset maps are injective.

To compare a subset sum with a subset product, first compare their greatest indices. If these differ, the expression with the larger index exceeds the other expression. If both have greatest index \(j\), the sum lies in \([a_j,2a_j)\), whereas a nonsingleton product is at least \(2a_j\). A singleton product equals \(a_j\), which equals a sum with greatest index \(j\) only for the singleton sum. Hence the intersection is exactly \(A\), giving the stated cardinalities. ◻

Proof. First color all of \(\mathbb N\), refining the given color by the residue modulo \(q\). Apply Theorem 1 to this finite coloring with \(\max(m,2)\) elements. Their common residue \(v\) also equals the residue of the sum of the first two elements, so \(v=2v\) modulo \(q\) and \(v=0\). Retain the first \(m\) elements. The separation and monochromaticity persist, and the preceding proof gives the distinctness assertions.

If no \(N\) worked, the finite colorings avoiding such a configuration would form a finitely branching tree, closed under restriction and with a node at every depth. An infinite branch would color \(\mathbb N\) without the configuration just proved to exist. Indeed each finite configuration and all its sums and products lie in some initial interval. This contradiction proves the finite interval assertion. ◻

Proof. The indicator in (18) is periodic modulo \(Wk\) and occupies exactly \(\phi(W)\) residue classes. On any interval, its counting function differs from \(\vartheta_W/k\) times length by at most \(\phi(W)\): apply the discrepancy bound of one to each occupied class. Partial summation against \(1/t\) gives (18). Apply it with \(k=1\) to obtain the bound on \(Z_X\). For a general class the error in its unnormalized mass is at most \(\phi(W)/X\); subtracting \(Z_X/k\) and using \(Z_X\ge\vartheta_W(L_X-W/X)\) proves (19). Sum the absolute class errors to obtain the total-mass assertion.

For translation put \(H=|h|\). Translation invariance of the norm reduces to \(h=H\ge0\). On the overlap of the supports, \(1/(n-H)\ge1/n\), and \(W\mid H\) preserves the unit condition. The negative part of \(\operatorname{Law}(Y+H)-\mu_X\) is therefore precisely the original mass on \([X,\min(X+H,X^2))\). Both measures have mass one, so the norm is twice that mass. If \(H\le X^2-X\), the interval \([X,X+H)\) has exactly \(\vartheta_WH\) units modulo \(W\), since its integer length is a multiple of \(W\); its harmonic mass is at most \(\vartheta_WH/X\). If \(H>X^2-X\), the norm is at most two, and the second bound in (20) is already at least two. This proves the translation estimate.

For dilation, at a multiple \(n\) of \(k\) the pushed-forward mass is \(k/(nZ_X)\), supported on \([kX,kX^2)\) and on \((n,W)=1\). Thus it agrees exactly with \(\eta_k\) on the overlap. Changing variables \(n=ku\) in the two boundary pieces, and using \(k\le X\), gives the exact formula \[\begin{align*} \|\operatorname{Law}(kY)-\eta_k\|_1 =\frac1{Z_X}\bigg( \sum_{\substack{X/k\le u<X\\(u,W)=1}}\frac1u +\sum_{\substack{X^2/k\le u<X^2\\(u,W)=1}}\frac1u \bigg). \end{align*}\] Each main term in (18) is \(\vartheta_W\log k\), and the two errors are at most \(\phi(W)k/X\) and \(\phi(W)k/X^2\), respectively. Dividing by the lower bound for \(Z_X\) proves (21). Its mass assertion follows from (19) with \(a=0\). Finally, integration against a function of supremum norm \(B\) costs at most \(B\) times the total-mass error. The stated growth conditions absorb every fixed choice of \(B=V^{O(1)}\). ◻

Proof. Condition on \(\sigma=t_T\). It is coprime to \(W\), independent of \(t_i\), and bounded by \(K_T\). Apply (21) with \(k=\sigma\), uniformly in this range, and average. The comparison measure is exactly \[\mu_i(y)\mathbb E_{\sigma}\sigma\mathbf 1_{\sigma\mid y} =\mu_i(y)\nu_B(y),\] proving (22). In particular its mass is \(1+o(V^{-C})\). Disjoint blocks use disjoint raw variables, so their original laws are independent. Telescoping a fixed tensor product bounds its total-mass error by the sum of the individual errors times the masses of the other factors, all of which are \(1+o(1)\). This proves the joint assertion. ◻

Proof. Set \(e_j=2^{N-j}\). The strict inequality \(e_j>\sum_{k>j}e_k\) shows that the sign of \(\sum_{j\in A}e_j-\sum_{j\in B}e_j\), for distinct subsets, is determined by the smallest index in their symmetric difference. For an adding pair that index belongs to \(A\); for two chained blocks it belongs to the earlier block. The positive difference is an integer, so the corresponding ratio of \(h\)-products is divisible by \(W^w\). Fixed rational multipliers have bounded valuations and only finitely many denominator primes. Consequently they are absorbed by these powers for all sufficiently large \(w\), leaving at least one factor of every prime dividing \(W\) in each required ratio of the \(c_d\)’s.

For completeness, a nonzero polynomial over \(\mathbb Q_p\) has a null zero set in a product of \(p\)-adic unit groups. In one variable it has only finitely many roots. In several variables, expand in a variable on which it depends and choose a nonzero coefficient polynomial. By induction, the set on which that coefficient vanishes has measure zero; off that set, each fiber has finitely many roots. Fubini’s Theorem proves the assertion. The decreasing events \(p^e\mid D\) therefore have probabilities tending to zero as \(e\to\infty\). At any fixed \(w\), the list of polynomials and primes is finite. Choose a common \(e_0\) making their union probability at most \(1/w\). Reduction of uniform units modulo \(W^{e_0}\) has exactly the required distribution modulo \(p^{e_0}\). We can now choose the power \(M\) large enough for all the finitely many stated smooth divisibilities and bounds.

Proceed in increasing order of \(l\). At this stage \(Q_l\) is a fixed integer. The prime number theorem in each reduced arithmetic progression modulo \(Q_l\) [32], followed by partial summation, says that the harmonic prime law on \([Y,2Y)\) tends to the uniform law on its unit classes as \(Y\to\infty\). Since there are finitely many classes, choose a dyadic lower endpoint so large that the total variation error is at most \(V_l^{-w}\) on every subsequent dyadic interval, and also \(P_l^-\ge wV_l^w\). Increasing the upper endpoint gives \(H_l^{\rm pr}\ge wV_l^w\), since the harmonic prime series diverges. A mixture of measures having the same error bound has that error bound. Thus this finite pool has the asserted residue distribution, however long the union of intervals is.

Only finitely many polynomial values occur in a finite pool. Take their nonzero absolute values, the earlier gap lengths, and \(M\), form the required least common multiple, and choose a multiple \(R_l\) at least \(w(P_l^++V_l)^w\). Finally choose a power of two \(X_l\) with \(\log X_l\ge wR_l^w\). There is no upper bound in any of these choices, and no later requirement changes an earlier choice.

Here is the elementary zero-value estimate used for the final assertion. If independent variables each have maximum atom at most \(a\), a nonzero polynomial of total degree \(d\) vanishes with probability at most \(da\). Expand in its last relevant variable: outside the zero set of its leading coefficient there are at most as many roots as its degree in that variable, and induction bounds the leading-coefficient exception by its degree times \(a\). For our pool, \(a\le1/(P_l^-H_l^{\rm pr})\); the same estimate for \(x_i-x_j\) handles repeats. These bounds are smaller than every fixed inverse power of \(V_l\). Finally, the residue-law approximation modulo \(Q_l\), together with the Chinese remainder theorem (CRT), transfers (23) to the actual prime slots. ◻

Proof. The prime number theorem gives a maximum atom \(O(\log Y/Y)\) in the harmonic prime law on \([Y,2Y)\). The polynomial zero estimate in the proof of Lemma 11 proves (25), uniformly in the endpoints.

We shall also use the following consequence of the interval Brun–Titchmarsh inequality. If \(p\le\sqrt Y\) is prime, a harmonically sampled prime in \([Y,2Y)\) belongs to any prescribed nonzero class modulo \(p\) with probability \(O(1/p)\). Indeed Theorem 2 of [36], together with \(\log(Y/p)\ge\tfrac12\log Y\), bounds the number of primes in a reduced class in that interval by \(O(Y/(p\log Y))\). Divide by the prime count \(\gg Y/\log Y\). Harmonic and uniform sampling on that interval have densities within a factor of two. The zero class contains no prime in \([Y,2Y)\) when \(p\le\sqrt Y\).

Remove variables on which the polynomials do not depend, and argue by induction on the total number of remaining variables. A fixed nonzero constant has no prime factors larger than \(w\) once \(w\) is large, so it gives the base case. In each nonconstant polynomial choose as its main variable one whose interval has largest lower endpoint in that tuple. Denote those endpoints by \(Y_F,Y_G\), and the corresponding nonzero leading-coefficient polynomials by \(A,B\). The events \(A=0\) or \(B=0\) cost \(O_{F,G}(\log L/L)\). If a common prime divisor of \(F,G\) also divides \(A\), apply the induction hypothesis to \(A,G\); if it divides \(B\), apply it to \(F,B\). The two tuples in each application remain disjoint, and their total number of variables strictly decreases. It remains to consider common primes dividing neither leading coefficient.

Put \(Y_0=\min(Y_F,Y_G)\). For \(w<p\le\sqrt{Y_0}\), condition on all variables except the two main variables. When the respective leading coefficient is nonzero modulo \(p\), each polynomial has at most its fixed degree many roots modulo \(p\). The preceding Brun–Titchmarsh consequence bounds the two independent tests by \(O_{F,G}(p^{-2})\). Their sum is \(O_{F,G}(1/w)\).

For the remaining primes, expose the entire tuple whose main endpoint is \(Y_0\), say the \(F\)-tuple. Every variable in that tuple is at most \(2Y_0\), so, when \(F\ne0\), \(1\le |F|\le C_FY_0^{\deg F}\). It follows that \(F\) has only \(O_F(1)\) distinct prime divisors exceeding \(\sqrt{Y_0}\). Test each such prime \(p\) against the independent \(G\)-tuple, whose main endpoint \(Y=Y_G\) satisfies \(Y\ge Y_0\). If \(p\le\sqrt Y\), the cost is \(O_G(1/p)=O_G(Y_0^{-1/2})\), again excluding the already treated event \(p\mid B\). If \(p>\sqrt Y\), each of the boundedly many root classes contains at most \(Y/p+1\) integers of the interval. Dividing by \(\gg Y/\log Y\) gives \[O_G\left(\frac{\log Y}{\sqrt Y}\right) =O_G\left(\frac{\log Y_0}{\sqrt{Y_0}}\right).\] The last inequality holds for all sufficiently large \(Y_0\), since \(\log Y/\sqrt Y\) is then decreasing. Summing over the bounded number of large prime factors proves the induction and (26).

Conditioning pool samples on their individual dyadic intervals leaves independent harmonic prime laws. The established bounds are uniform over these choices, so averaging proves the pool assertion. Restricting to events of probabilities \(1-o(1)\) changes any probability by \(o(1)\). ◻

Proof. We first express the normalized divisibility count as a product of local kernel counts. Their excess over one is nonnegative; this lets us bound it without retaining the prime-only event. We then dominate the divisor draws by laws with independent prime valuations and sum the local excesses.

Proof. The dilation assertion is Lemma 9, applied at a cutoff whose logarithm dominates all fixed powers of \(P_l^++V\). The weighted law on the right has the same harmonic density as the law on the left; only the endpoints \([X_i,X_i^2)\) and \([kX_i,kX_i^2)\) differ. The logarithmic boundary mass is \(O(\log k/\log X_i)\), with the residue counting error from that Lemma. The scale separation makes their sum smaller than any prescribed negative power of \(P_l^++V\).

Averaging the multiplier in (32) gives \[A_l(Y)=\mathbb E_p p\mathbf 1_{p\mid Y} =\frac1{S_l}\sum_{p\in\mathcal P_l}\mathbf 1_{p\mid Y}.\] Periodic counting for the moduli \(p\) and \(pq\), including the \(W\)-unit restriction, gives their reciprocal probabilities with uniformly negligible relative errors. Therefore \[\begin{align*} \mathbb EA_l&=1+o(V^{-C}),\\ \mathbb EA_l^2 &=\frac1{S_l^2} \left(\sum_p\frac1p+\sum_{p\ne q}\frac1{pq}\right) +o(V^{-C})\\ &=1+\frac1{S_l}-\frac1{S_l^2}\sum_p\frac1{p^2}+o(V^{-C}) \end{align*}\] for every fixed \(C\). Thus \(\|A_l-1\|_{L^2(\mu_i)}\le S_l^{-1/2}+o(V^{-C})\). Cauchy–Schwarz bounds the error in replacing \(A_l\) by \(1\) by \(V^A(S_l^{-1/2}+o(V^{-C}))=o(1)\), since \(S_l\) dominates every fixed power of \(V\). The estimates are uniform in the parameters of \(F\), so one may integrate over those parameters afterwards. ◻

Proof. Initially the rows are the indicator vectors of the nonempty subsets \(J\) of \([m]\). They are pairwise nonparallel. We remove the masks in a fixed order, maintaining (33), the weight bounds, and a distinguished copy of \(g_*\).

Proof. Here is an explicit finite choice of a kernel vector. For a row \(A_R\), choose a column \(j\) with \(A_{R,j}\ne0\). The polynomial vectors \[b_k=A_{R,j}e_k-A_{R,k}e_j\qquad(k\ne j)\] span its kernel over the rational function field. Enumerate them as \(b_0,\ldots,b_{m-2}\) and set \(w_R(t)=\sum_{h=0}^{m-2}t^hb_h\). For \(I\ne R\), the polynomial \(A_Iw_R(t)\) is not identically zero: otherwise \(A_I\) would vanish on the kernel of \(A_R\) and hence be parallel to it. The product of these polynomials has degree at most \((|\mathcal R|-1)(m-2)\) in \(t\). At least one of the first \((|\mathcal R|-1)(m-2)+1\) positive integers is therefore not a root over this field. Use the first such integer. This is a deterministic choice, gives integer polynomial coordinates, and has every asserted nonzero response. Applying the same construction to \(A_*\) gives \(w_0\).

Include \(D\), all nonzero response polynomials, and the required row minors in the finite polynomial list used to choose \(M\) and the gap scales. There is no dependence on the functions in this list. Delete tuples with a zero polynomial value or a repeated prime entry, and tuples for which \(\pi^{e_0}\mid D\) for some \(\pi\le w\). The deleted probability is \(o(1)\); the first two parts even have superpolynomially small probability in \(V\). Deletion at this point remains legitimate for weighted expectations: for any product of distinct row weights its integral over a prime-only domain \(E\) equals \(2^t\mathbb P(E)+o(1)\), where \(t\) is the number of weights, by expansion and the restricted-domain assertion of Proposition 14. A domain of probability \(o(1)\) therefore has \(o(1)\) weighted mass. We henceforth normalize the prime law on the remaining good tuples, whose probability tends to one.

For integrality, write \(D=\epsilon S Q\), where \(\epsilon\in\{-1,1\}\), \(S=|D|_{\le w}\), and \(Q=|D|_{>w}\). On the good tuples \(S\mid W^{e_0-1}\). Since \(M/c_k\) is an integer divisible by \(W^{e_0+1}\), the expression in (39) can be written \[v_{R,k}= \epsilon\frac{M/c_k}{S}\,c_{a_*}w_{R,k}\frac{D}{d_R}.\] All its factors are integers, and \((M/c_k)/S\) is divisible by \(W\). The assertion for \(v_0\) follows directly from its formula. The degree and coefficient bounds for the polynomial vectors give \(|v_{R,k}|+|v_{0,k}|\le(P_l^++V)^{O_m(1)}\); all the scales \(c_k\) are at most \(M\le V\).

Direct substitution proves (40). More generally, \[\ell_I(v_R)=M(\boldsymbol p)\frac{c_{a_*}}{c_{a(I)}} \frac{A_Iw_R}{d_R},\qquad \ell_I(v_0)=M\frac{c_{a_*}}{c_{a(I)}}A_Iw_0.\] Outside primes dividing \(D(A_Iw_R)\), respectively \(A_Iw_0\), the first, respectively second, expression is a unit at \(w<\pi\le V\). The smooth factors are units there. This proves both the response claim and the finite-polynomial nature of all exceptional tests. ◻

Proof. Weighted Cauchy–Schwarz will leave a cube of \(g_*\) multiplied by retained row weights. We first form that cube, then use the direction \(v_0\) to remove the retained weights in a weighted second moment.

Every interval in (41) has length dominating all fixed powers of \(V\), uniformly over the prime tuples. Indeed \(M(\boldsymbol p)\) is bounded by a fixed power of \(P_l^++V\), whereas \(R\) dominates all such powers. Insert independent uniform \(u_R\) on these intervals and translate \[z\longmapsto z+\sum_{R\ne *}v_Ru_R.\] The translations are in \(W\mathbb Z^m\). Their sizes are at most \(R(P_l^++V)^{O_m(1)}\), and hence are negligible at every pivot cutoff, even after multiplying the total-variation errors by \(V^{O_m(1)}\). Lemma 9 therefore changes the average by \(o(1)\).

Eliminate the nondistinguished rows in a fixed order. At the step assigned to row \(R\), all its current copies are independent of \(u_R\), because \(\ell_R(v_R)=0\). Let \(H_{\mathrm{out}}\) be their product, and let \(\Omega_R\) be the product of their bounds \(W_R\). All other factors, including weights retained at earlier steps, belong to \(H_{\mathrm{in}}\). With all variables other than \(u_R\) outside, the exact inequality is \[\begin{align*} &\left|\mathbb E_{\mathrm{out}}H_{\mathrm{out}} \mathbb E_{u_R}H_{\mathrm{in}}\right|^2\\ &\hspace{6mm}\le (\mathbb E_{\mathrm{out}}\Omega_R) \mathbb E_{\mathrm{out},u_R^0,u_R^1} \Omega_R H_{\mathrm{in}}(u_R^0)H_{\mathrm{in}}(u_R^1). \tag{43}\end{align*}\] It follows by applying Cauchy–Schwarz with density \(\Omega_R\) and using \(|H_{\mathrm{out}}|\le\Omega_R\). In particular, the assigned functions disappear and their weights occur once, while every factor inside the average is duplicated. All prime parameters remain outside throughout this procedure.

We verify every use of Proposition 14 in (43). A current copy of a base row \(I\) has the form \[\ell_I(z)+\sum_Q\ell_I(v_Q)u_Q^{\eta_Q},\] where \(\eta_Q\) is one of the already duplicated choices or the single current variable. Distinct base rows are distinguished by their \(z\)-coefficient vectors. For two copies of the same row, some duplicated coordinate \(Q\ne I\) has different choices, and its response \(\ell_I(v_Q)\) is nonzero. Their difference therefore has a nonzero coefficient on an independent variable \(u_Q^0\) or \(u_Q^1\). Taking a minor with an anchor \(z\)-column proves pairwise independence at every relevant prime outside the response-polynomial tests of Lemma 17. A row’s anchor coefficient is still a monomial of pool primes, so each row is primitive at all \(w<\pi\le V\). Finally, conditional on the primes the base variables are independent harmonic variables or independent interval variables whose lengths dominate powers of \(V\). Their joint residues modulo the product of any fixed number of sampled divisors are uniform with error smaller than every fixed negative power of \(V\). These facts check primitivity, pair independence with polynomial exceptions, and the sampling hypotheses. Expanding the weights now shows that every prefactor in (43) is bounded by a constant depending only on the number of row copies.

After all \(d\) steps, the nondistinguished functions have disappeared. The remaining target factor is \[G(z,u)=\prod_{\omega\in\{0,1\}^d} g_*\left(\ell_*(z)+M(\boldsymbol p) \sum_Ru_R^{\omega_R}\right).\] For each eliminated row \(I\), there is one retained weight for each choice of the duplicated variables in the directions \(R\ne I\). Thus the retained product is \[ \Psi(z,u)= \prod_{I\ne *}\ \prod_{\eta\in\{0,1\}^{[d]\setminus\{I\}}} W_I\left(\ell_I(z)+\sum_{R\ne I}\ell_I(v_R)u_R^{\eta_R}\right), \tag{44}\] after identifying the nondistinguished rows with \([d]\). There are \(t=d2^{d-1}\) weight factors. Iteration has proved the left side of (42) is bounded by a constant times \(|\mathbb EG\Psi|+o(1)\).

It remains to justify replacing \(\Psi\) by \(2^t\) in this correlation; its mean alone would not justify that replacement. Translate \(z\) once more by \(v_0u_0\), with \(u_0\) uniform on \([0,R)\cap\mathbb Z\). The sampling error is again \(o(1)\), and the target \(G\) is unchanged because \(\ell_*(v_0)=0\). Put \[B(z,u)=\prod_{\omega\in\{0,1\}^d} \left(1+\nu_{a_*}\left(\ell_*(z)+M(\boldsymbol p) \sum_Ru_R^{\omega_R}\right)\right), \qquad H(z,u)=\mathbb E_{u_0}\Psi(z+v_0u_0,u).\] Then \(|G|\le B\). The same row check gives \[ \mathbb EB=2^{2^d}+o(1),\qquad \mathbb EBH=2^{2^d+t}+o(1),\qquad \mathbb EBH^2=2^{2^d+2t}+o(1). \tag{45}\] For completeness, in the last expression duplicate \(u_0\) independently as \(u_0^0,u_0^1\). Copies from distinct base rows are distinguished on \(z\); copies from one base row and different old shift choices are distinguished on an old \(u_R^\eta\); copies with identical old choices but different \(u_0\) branches are distinguished by the nonzero response \(\ell_I(v_0)\). Target rows have distinct cube shift choices and no \(u_0\) response. All forms in every expanded moment are thus distinct and satisfy exactly the primitivity and polynomial-minor conditions checked above. Expanding each \(1+\nu\) and replacing every product of \(\nu\)’s by its main term \(1\) gives all three powers of \(2\) in (45).

Consequently \[\mathbb EB(H-2^t)^2=o(1).\] Cauchy–Schwarz with density \(B\) now yields \[|\mathbb EG(H-2^t)| \le (\mathbb EB)^{1/2}\bigl(\mathbb EB(H-2^t)^2\bigr)^{1/2}=o(1).\] This is the required weighted replacement. Combining it with the \(d\) inequalities (43) proves (42). ◻

Proof. Only the pushforward of the cube root remains after Lemmas 16 and 18. For fixed primes and shifts, write \[ Y_* = \ell_*(z)+M(\boldsymbol p)\sum_Ru_R^0 = kz_{a_*}+h, \qquad k=A_{*,a_*}. \tag{47}\] The target support remains \(J_*\), so it contains some \(j<a_*\). The coefficient \(b=(c_j/c_{a_*})A_{*,j}\) is coprime to \(k\): the prime slots in different columns are disjoint, their numerical entries are distinct on the good tuples, and all scale ratios are \(w\)-smooth whereas all pool primes exceed \(V\ge w\). Also \(k\) is coprime to \(W\) and \(W\mid h\).

Condition first on every \(z\)-variable except \(z_{a_*}\). Put \(X=X_{i_{a_*}}\) and \(\delta_W=\phi(W)/W\). Uniformly in the variables being conditioned on, one may take \[0\le h\le H:=M(P_l^+)^{B_m}\sum_{j<a_*}X_{i_j}^2+dR/J_0\] for a fixed \(B_m\). The logarithm of \(X\) dominates every fixed power of \(H+k+W\), and in particular \(H<X/2\) eventually. The dilation and translation calculation of Lemma 9 gives \[ \mathcal L(kz_{a_*}+h) = k\mathbf 1_{y\equiv h\pmod{k}}\mu_{i_{a_*}}(dy)+\mathcal E_h, \qquad \|\mathcal E_h\|_{\mathrm{TV}}=o(V^{-C}) \tag{48}\] for every fixed \(C\). More explicitly, the total-mass error is at most \[ O\left(\frac{\log(2k)}{\log X}+\frac H X+ \frac{Wk^2}{\delta_W X\log X}\right). \tag{49}\] To see this directly, let \(Z_i\) be the normalizing constant of \(\mu_i\), so \(Z_i\asymp\delta_W\log X\). The exact image density is \(k/(Z_i(y-h))\) on \([kX+h,kX^2+h)\) in the class \(h\pmod k\), with \((y,W)=1\). The comparison density is \(k/(Z_i y)\) on \([X,X^2)\) in the same class. The \(W\)-unit restriction agrees because \(W\mid h\) and \((k,W)=1\). On the intersection of the intervals the relative discrepancy is at most \(H/X\). Harmonic progression counting on the two remaining boundary intervals gives a logarithmic main term \(O(\delta_W\log(2k))\) and error \(O(Wk^2/X)\) before normalization. This proves (49). All three terms are smaller than every fixed negative power of \(V\) by scale separation; the crude bound \(\delta_W^{-1}\le W\) already suffices here.

Now condition on all remaining variables except \(z_j\). Since \(i_j>l\), Lemma 9 makes \(z_j\) uniform modulo \(k\); its relative error is at most \[O\left(\frac{Wk^2}{\delta_W X_{i_j}\log X_{i_j}}\right)=o(V^{-C})\] for every fixed \(C\). The same holds for \(h=bz_j+h_0\) modulo \(k\), because \((b,k)=1\). Averaging the density in (48) therefore gives \[\mathbb E_{z_j} k\mathbf 1_{y\equiv h\pmod{k}}=1+o(V^{-C})\] uniformly in \(y\) and in the conditioned primes and shifts. Thus the conditional law of \(Y_*\), after integrating all \(z\)’s, differs from \(\mu_{i_{a_*}}\) by \(o(V^{-C})\) in total mass, uniformly in the primes and shifts. This argument also covers \(k=1\), when there is no residue condition to average.

Every remaining cube argument can be expressed using this root as \[Y_*+M(\boldsymbol p) \sum_{R\in\omega}(u_R^1-u_R^0).\] There is no other \(z\)-dependent factor left. The product of target bounds is at most \((1+V)^{2^d}\), so the total-mass error just proved is still \(o(1)\) in the cube expectation. Substitution in (42) and then (34) gives (46) with the stated exponent. All constants arise from fixed counts of row weights and squares, hence depend only on \(m\). At least one other row exists, so \(d\ge1\).

Finally, \(M(\boldsymbol p)\) divides \(M|D(\boldsymbol p)|\), whose divisibility into \(R_l\) was arranged in Lemma 11. The finite template assertion follows from the deterministic mask and kernel-vector choices. Every sampling estimate and every restricted linear-forms estimate used above was uniform over the functions; taking the maximum of the errors over finitely many \(J_0\)’s proves the last uniformity assertion. ◻

Proof. Expand the product using independent prime and shift replicas. In replica \(q\), of dimension \(d_q\) and modulus \(M_q\), the nonroot rows are \[Y_{q,\omega}=y+M_q\sum_{j\in\omega}(u_{q,j}^1-u_{q,j}^0), \qquad\varnothing\ne\omega\subset[d_q].\] The root row is \(y\). For a fixed nonempty \(\omega\), choose integers \(a_0,a_1,\ldots,a_{d_q}\) such that \[a_0\ne0,\qquad a_0+\sum_{j\in\omega}a_j=0,\qquad a_0+\sum_{j\in\omega'}a_j\ne0\quad(\omega'\ne\omega).\] Indeed, the hyperplane defined by the equality is not contained in any of the finitely many hyperplanes excluded by the inequalities: their normal vectors \((1,\mathbf 1_{\omega'})\) are pairwise nonparallel. A rational point avoiding their intersections exists, and clearing denominators gives the indicated integers. Insert a translation parameter for this row, acting by \[y\longmapsto y+a_0M_qv,\qquad u_{q,j}^1\longmapsto u_{q,j}^1+a_jv,\] and leaving all other shift variables unchanged. It fixes the chosen row. Every other row in its replica has a nonzero response, the root has response \(a_0M_q\), and a row in another replica has the same nonzero response \(a_0M_q\).

These directions are compatible even when the gaps differ. Put \(V_B=2+M+\prod_{b\in T}X_b^2\). The shortest shift interval in each replica dominates every fixed power of \(V_B\). The translation parameters for replica \(q\) may, for example, have lengths comparable to the square root of its shortest shift length. Their lengths still dominate powers of \(V_B\), whereas the resulting displacements of its shifts are negligible relative to their intervals, even after multiplication by any fixed power of \(V_B\). The sum of the root displacements is likewise negligible at \(X_i\). The parameters depend on the primes only through the lengths, which is permitted by the uniform box version of Proposition 14. Thus inserting all these averaged translations changes the original expression by \(o(1)\): use the crude bound \(1+\nu\leq1+\prod_{b\in T}X_b^2\) and the translation estimates of Lemma 9. Fixed direction constants do not affect these separations.

Eliminate the nonroot rows one at a time by the weighted Cauchy–Schwarz procedure of Lemma 18. At the step assigned to a row, all its current copies are independent of that row’s translation parameter. Bound those copies by their \(1+\nu\) weights and use their product as the outside density. Squaring duplicates that row’s assigned translation parameter; all other variables remain in the outside expectation, as in that procedure. Prime-only bounded factors can be discarded outside the inner average. Every prefactor is bounded, and one weight for each eliminated copy is retained in the final expression. The root, which is never eliminated, contributes copies of \((\nu-1)\) indexed by all choices of the duplicated translation parameters.

We verify the linear-forms hypotheses at every such application. Every form has coefficient \(1\) at \(y\). Different base rows have distinct coefficients on the original shifts. Copies of one base row are distinguished by a duplicated direction with nonzero response; for an eliminated row these are precisely directions other than its own. Root copies are distinguished because every inserted direction has nonzero root response. A distinguishing minor is a fixed nonzero integer times a single replica modulus \(M_q\); no difference of two moduli is required. At a rough prime, its possible vanishing is therefore included among the polynomial exceptional tests defining that modulus. Small-prime factors and fixed nonzero integer factors are covered by the smooth modulus and, eventually, by \(W\). All replicas use gaps after the common tail \(T\), so the multi-gap form of Proposition 14 applies with \(V_B\). The arbitrary prime-only restrictions in that Proposition include the separate good-tuple restrictions in each replica.

Expand the retained \(1+\nu\) weights and every root difference. Each resulting product has the main term obtained by replacing each \(\nu\) by \(1\), with a uniform \(o(1)\) error. In that substitution at least one root difference becomes zero. Thus the final Cauchy–Schwarz expression is \(o(1)\), proving (52). When \(b=0\) this is simply \(\mathbb E\nu=1+o(1)\).

For the moment bound, Jensen’s inequality and independent replicas give \[|\mathcal D(y)|^b\leq \mathbb E\prod_{q=1}^b\prod_{\varnothing\ne\omega\subset[d]} (1+\nu)(Y_{q,\omega}).\] Include the root weight \(1+\nu(y)\) and expand. These rows are already distinct without the additional translations. Proposition 14 gives \(2^{1+b(2^d-1)}+o(1)\), proving (53).

Let \(\mathcal D^{[K]}=\max(-K,\min(K,\mathcal D))\). For integers \(b>p\), \[\limsup_{w\to\infty} \mathbb E(1+\nu)|\mathcal D-\mathcal D^{[K]}|^p \leq 2K^{p-b}A_*^b.\] This estimate holds for every fixed \(b\); sending \(b\) to infinity after the limit proves that the left side is zero. Telescoping a fixed product and applying Hölder’s inequality with respect to \((1+\nu)\mu_i\) now transfers (52) to clipped tests, since \(|\nu-1|\leq1+\nu\) and all required fixed moments are bounded. No moment order growing with \(w\) has been used. ◻

Proof. First fix a finite collection of \(J_0\) values, a positive desired error, and a common clipping bound \(K\) from Lemma 21. The support of \(\mu_i\) is finite. Let \(\mathcal C\) be the closed convex hull of the signed clipped tests, viewed as vectors on that support. It is compact and lies in \([-K,K]\) coordinatewise. The candidate models form the compact convex cube \(\mathcal F=[0,1]^{\mathop{\mathrm{supp}}\mu_i}\). Finite-dimensional minimax for the bilinear pairing gives \[\inf_{F\in\mathcal F}\sup_{G\in\mathcal C}\langle\rho-F,G\rangle =\sup_{G\in\mathcal C} \bigl(\langle\rho,G\rangle-\langle1,G_+\rangle\bigr), \qquad G_+=\max(G,0),\] because \(\sup_{F\in\mathcal F}\langle F,G\rangle=\langle1,G_+\rangle\). Since \(0\leq\rho\leq\nu\), the right-hand side is at most \(\sup_G\langle\nu-1,G_+\rangle\).

To bound it uniformly, approximate \(t\mapsto\max(t,0)\) on \([-K,K]\) within \(\delta\) by a polynomial \(P(t)\). Every fixed power of a convex combination of signed tests is a convex combination of their signed products. Lemma 21, followed by continuity for the closed convex hull, therefore gives \(\langle\nu-1,P(G)\rangle=o(1)\) uniformly in \(G\in\mathcal C\). The polynomial approximation costs at most \(\delta\mathbb E(1+\nu)=2\delta+o(1)\). First take \(w\to\infty\) and then \(\delta\to0\). Minimax produces models with the desired error against the clipped tests; the clipping estimate gives the same conclusion for the original tests, since \(|\rho-F|\leq1+\nu\).

Apply this argument successively to errors \(1/q\) and \(J_0\leq q\). For each fixed \(q\) it works for all sufficiently large \(w\), uniformly over the finite master data. Choosing the stage \(q=q(w)\) to increase sufficiently slowly gives (54) for every fixed \(J_0\). This diagonal choice is made together with the fixed-template diagonal in Lemma 11; it asserts no estimate for an arbitrary growing moment order. Extend the models outside the support of \(\mu_i\) with values in \([0,1]\). ◻

Proof. Lemma 31, proved independently in Section 7, supplies a compact set of nilmanifold cubes containing every linear orbit cube \[(x_\omega)_{\omega\subset[s+1]} =\left(g^{k+\sum_{j\in\omega}v_j}x\right)_{\omega\subset[s+1]}, \qquad k,v_1,\ldots,v_{s+1}\in\mathbb Z.\] On this set the nonroot vertices determine \(x_\varnothing\) continuously. Its Stone–Weierstrass consequence gives the following concrete input. For any accuracy \(\delta>0\), the observable at the root is uniformly approximated by a finite sum \[\sum_{t=1}^{q}\lambda_t \prod_{\varnothing\ne\omega\subset[s+1]}\phi_{t,\omega}(x_\omega)\] with \(|\phi_{t,\omega}|\leq1\). The same Lemma gives finitely many such recipes, with fixed total coefficient bounds, for our finite nilmanifold list and uniformly bounded Lipschitz observables. The recipes are valid for every \(g\) and \(x\), so they can be used on all pieces even though the translating elements and base points are unrestricted. It remains to realize these recipes as our dual tests and control the cubes that cross piece boundaries.

Use the extra dual test of dimension \(s+1\), modulus \(M\), and a fixed large \(J_0\). If \(y\) is farther than \((s+1)R_l/J_0\) from the endpoints of its \(R_l\) interval, all its cube vertices lie in that interval and have the same residue modulo \(M\). The recipe for that piece then reconstructs \(S'(y)\). Recipes can be selected through the input functions of the dual test: for a term belonging to recipe \(r\), put in every nonroot input the indicator that its piece selected \(r\), multiplied by its single-vertex factor evaluated on that piece’s orbit. Sum over the finite recipes and terms. On cubes staying in one piece this selects exactly its recipe. Each resulting input is bounded by \(1\), and hence is allowed in (51).

We record the weighted boundary estimate needed for the discarded roots. If \(E_l\) is the union of these boundary strips, then \[ \mathbb E_{\mu_i}(1+\nu_B)\mathbf 1_{E_l} \leq O_s(J_0^{-1})+o(1). \tag{56}\] For a fixed divisor draw \(\sigma=t_T\), divisibility by \(\sigma\) and coprimality to \(W\) are periodic with period \(\sigma W\). This period is negligible compared with \(R_l\), uniformly in \(\sigma\leq\prod_{b\in T}X_b^2\). Counting in each full \(R_l\) interval shows that a union of end strips of relative length \(O_s(J_0^{-1})\) has that proportion of its weighted mass, with relative error \(O(\sigma W/R_l)\). The harmonic factor varies by \(o(1)\) within a cell, since \(R_l/X_i=o(1)\); the two partial cutoff cells have \(o(1)\) mass by the same counting estimate. This proves the assertion for \(\sigma\mathbf 1_{\sigma\mid y}\mu_i\), uniformly in \(\sigma\). Average over \(\sigma\) and also take \(\sigma=1\) to obtain (56).

The recipe approximation costs \(O(\delta)\) against \(|\rho-F|\mu_i\leq(1+\nu_B)\mu_i\). For fixed recipes and \(J_0\), Proposition 22 makes every dual pairing \(o(1)\). The terms outside the good roots are bounded by a fixed constant (depending on the recipes) times the quantity in (56). First fix sufficiently accurate recipes, then take \(J_0\) sufficiently large for those recipes, and finally take \(w\to\infty\). Since \(\delta\) was arbitrary, this proves (55). ◻

Proof. The energies \(\|P_{i,l}F_{T\cup\{i\},a,c}\|_2^2\) lie in \([0,1]\). Color each ordered tuple consisting of the increasing members of \(T\), followed by \(l\) and \(i\), by their bins of width \(\varepsilon^2\), jointly for all \(a,c\). There are finitely many colors, with a bound independent of \(N\). Apply finite Ramsey successively for all tuple lengths \(3,\ldots,n+1\) to obtain a homogeneous set of size \(2n\). Label it alternately by padding and principal indices as displayed.

At each selected block approximate its coarse projection by a finite linear combination \(T'\) of representing families, to accuracy \(\varepsilon\). Its clipping \(S\) to \([0,1]\) is still a representing family. Pointwise clipping decreases the distance to \(F\in[0,1]\). Since both \(T'\) and \(S\) lie in the coarse subspace, orthogonality gives \[\|F-S\|_2^2=\|F-PF\|_2^2+\|PF-S\|_2^2,\] and the analogous identity for \(T'\). Thus clipping also decreases the distance to \(PF\), proving (58). There are only finitely many selected blocks and labels, so their representing families share a finite nilmanifold list and uniform observable bounds.

All tails of the chain precede its first pivot and hence precede \(l\); also \(l\leq k_u<j_u\). If \(l<k_u\), the tuples \((T,l,j_u)\) and \((T,k_u,j_u)\) lie in the homogeneous set and have the same color. Their energies differ by at most \(\varepsilon^2\). Equation (57) bounds the distance between the two projections by \(\varepsilon\). As \(S\) belongs to the coarse and hence the fine space, (59) follows by the triangle inequality. The case \(l=k_u\) follows directly from (58). ◻

Proof. We pass from the short increments to subgroup norms, combine the subgroups using rough coprimality, and apply the inverse theorem in each cyclic cell. We keep the cell probabilities throughout, so every threshold is independent of the number of cells and the master count.

Proof of Principle 5. Fix the data \(n,r,\chi,\mathcal B,\mathcal A,\tau,\eta\) of the Principle, with \(s=s(m)\) as in (50). There are \(q_m=2^m-m-1\) nonsingleton factors in each count. We spell out the order of choices before selecting master indices.

Choose a positive target cube error \(\zeta\) so small that \[C_m(2\zeta)^\theta<\eta/(2q_m)\] for every exponent and constant in Proposition 19. There are only finitely many orders and exponent choices, all controlled by \(m\); increasing the uniform constant if necessary makes this one finite set of requirements. Choose a fixed \(J_0\) so large that every periodization term \(O_d(J_0^{-1})\) in Lemma 25 is smaller than \(\zeta\). For the finitely many correlation-test types, apply that Lemma with \(\gamma=\zeta\), and choose \[0<\varepsilon<\eta, \qquad 2\varepsilon<\min_d\kappa(d,J_0,\zeta).\] These choices depend on the requested tolerances and on \(m\), and are independent of \(N\). Now choose \(N\) by Lemma 24, construct its arithmetic scales and dense models, and carry out that Lemma. Restrict \(h_i,X_i,t_i\) to the principal indices \(j_1,\ldots,j_n\), relabel them by \([n]\), and set \(H_u=R_{k_u}\). The intervening padding gaps and Lemma 11 give all conditions in Definition 3; the models \(S\) have exactly the pieces required by Definition 4.

For calibration at a selected block choose a Lipschitz function \(\psi:[0,1]\to[0,1]\) equal to \(1\) on \([0,2\tau]\) and to \(0\) on \([3\tau,1]\). The bounded distribution replacement in Corollary 10 gives \[\Pr\bigl(\chi(h_Bat_B)=c,\ S_{B,a,c}(t_B)\leq2\tau\bigr) \leq\mathbb E_{\mu_i}\rho\,\psi(S_{B,a,c})+o(1).\] The function \(\psi(S)\) is a bounded-complexity coarse piecewise nilsequence. Lemma 23 replaces \(\rho\) by \(F\). In the Hilbert limit it also lies in the coarse subspace, so, writing \(P=P_{i,k_u}\) and using (58), \[\langle F,\psi(S)\rangle =\langle PF,\psi(S)\rangle \leq\langle S,\psi(S)\rangle+\varepsilon \leq3\tau+\varepsilon.\] This is the first assertion of the Principle, since \(\varepsilon<\eta\).

For the count comparison, fix a chain on the principal indices, \(b\in\mathcal B\), and a color. Its block scale \(b_{B_d}\) belongs to \(\mathcal A\), so all its models have already been constructed. As shown after Proposition 22, telescope the nonsingleton color weights \(\rho_d(L_J)\) to \(F_d(L_J)\) at error \(o(1)\), retaining the center weights \(\nu_d(z_d)\) and every product mask, including the singleton masks.

Next telescope \(F_d(L_J)\) to \(S_d(L_J)\). Choose as fine gap the padding index immediately before the chain’s first pivot. It is valid for every block of the chain. Each individual difference has bounded target \(h=F_d-S_d\in[-1,1]\), and (59) gives \(\|P_{i_d,l}h\|_2\leq2\varepsilon\). All other factors, including the unbounded center weights, still satisfy the bounds of Proposition 19. The parameter choices and Lemma 25 bound its target cube mean by \(2\zeta\) in the ultrafilter limit. Equation 46 therefore bounds each telescoping error by less than \(\eta/(2q_m)\). Summing the finitely many differences gives a limit error at most \(\eta/2\) between the original count and \[\mathbb E_{z\sim\bigotimes_d\mu_{i_d}} U(z)\prod_d\nu_d(z_d) \prod_{|J|\geq2}S_{d(J)}(L_J(z)).\] All factors except the center weights are now bounded by \(1\). The blocks of the chain are disjoint. Corollary 10 thus replaces this last expectation, with error \(o(1)\), by \[\mathbb E_t U\bigl(z(t)\bigr) \prod_{|J|\geq2}S_{d(J)}\bigl(L_J(z(t))\bigr), \qquad z(t)_d=t_{B_d}.\] This proves the required count comparison with the stated tolerance \(\eta\). Every limit in this construction may be taken along the same fixed ultrafilter \(\mathcal U\), as allowed in Principle 5. In particular the auxiliary \(\varepsilon\) was chosen before the master Ramsey count \(N\), whereas the resulting model complexity was allowed to depend on all those fixed choices. The proof of the Principle is complete. ◻

Derivation of the box consequence. Normalize the observable by its Lipschitz bound. Choose a small positive \(\lambda\), depending only on \(d,c,C,B,\eta\), and tile the integer box by cubes of side \(\floor{\lambda Z}\), leaving strips along its boundary. The strips contain at most an \(O_{d,c}(\lambda+Z^{-1})\) fraction of its integer points. Choose \(\lambda\) so that their contribution to (65) is less than \(\eta/4\). Expressing the remaining average as a convex combination shows that one cube has Haar discrepancy at least \(\eta/2\). Apply the equal-side case of Theorem 26 to \(P(a+v)\) on this cube. Integer translations preserve filtered polynomiality because a group difference of a translated map is the corresponding translate of its group difference.

There is a fixed bound \(D_0\) on the scalar degree. Multiply the resulting character by \((D_0!)^d\). Indeed, on expanding the products \(\binom vI\), this multiplication clears every denominator, so each ordinary monomial coefficient is an integer plus an error \(O(Z^{-\abs I})\): a coefficient of degree \(I\) receives contributions only from binomial degrees \(J\geq I\), and \(Z^{-\abs J}\leq Z^{-\abs I}\). Finally expand \((x-a)^J\). The translation has integer entries and \(\abs a=O_{C,d}(Z)\); hence its integer coefficient contributions stay integral, while its errors at degree \(I\) are bounded by \[\sum_{J\geq I}O(Z^{\abs J-\abs I})O(Z^{-\abs J}) =O(Z^{-\abs I}).\] This proves (66). The bounded multiplication of the character still leaves a finite list. The bound on the location of \(\mathcal Q\) is used precisely in this last translation estimate. ◻

Proof. Write \(T(z)=(t_0+mz)^e\theta(z)\) and \(\epsilon=A(S/Z)^e\). Choose \(q+1\) integer nodes \(z_0,\ldots,z_q\) in \([0,H-1]\) with pairwise successive gaps at least \((H-1)/(2q)\), for example the nearest integers to \(j(H-1)/q\). Let \(n_j\) be a nearest integer to \(T(z_j)\), and let \(R\) be their degree-at-most-\(q\) Lagrange interpolant. On rescaling the index interval to \([0,1]\), the interpolation nodes remain separated by \(1/(2q)\). The Lagrange basis polynomials and all their coefficients in the rescaled variable are therefore bounded in terms of \(q\) alone. Since \(T\) already has degree at most \(q\), interpolation of the errors gives \[ \sup_{0\leq z\leq H-1}\abs{R(z)-T(z)}\leq C_q\epsilon. \tag{70}\] There is a positive integer \(D_H\), depending only on the chosen nodes, such that \(D_HR\in\mathbb Z[z]\). One may take the product of the nonzero node differences occurring in the Lagrange denominators. In particular, \(D_H\) is independent of \(w,t_0,m\) and the polynomial coefficients.

Let \(a=-t_0/m\). The polynomial \(T\) has a zero of multiplicity at least \(e\) at \(a\). The coefficient bounds in the rescaled Lagrange formula give, for \(0\leq j<e\), \[\abs{R^{(j)}(a)} =\abs{(R-T)^{(j)}(a)} \leq C_q\epsilon(H-1)^{-j} \left(1+\frac{\abs a}{H-1}\right)^q \leq C_q\epsilon(1+2S)^q.\] On the other hand \(R^{(j)}(a)\) is rational with denominator dividing \(D_Hm^q\). We have \(m\leq S\) because the progression lies in \([S,2S)\) and has at least two terms. Consequently a nonzero such rational has absolute value at least \(D_H^{-1}S^{-q}\). For fixed \(H\), the displayed upper bound is smaller than this number for large \(w\), by the assumed domination of every power of \(S\) by \(Z\). Thus \(R^{(j)}(a)=0\) for all \(j<e\), exactly. Polynomial division now gives \[ R(z)=(t_0+mz)^e M(z),\qquad M\in\mathbb Q[z],\qquad\deg M\leq q-e. \tag{71}\] In division by \((t_0+mz)^e\), the only denominators introduced, besides those already dividing \(D_H\), come from its leading coefficient \(m^e\). All their prime factors are therefore at most \(w\) once \(w\) exceeds the prime factors of the fixed integer \(D_H\).

For every integer \(z\in[0,H-1]\), (68) and (70) place \(R(z)\) within \((C_q+1)\epsilon\) of an integer. Since \(R(z)\in D_H^{-1}\mathbb Z\), it is itself an integer for large \(w\). Write \(M(z)=a_z/b_z\) in lowest terms. Every prime factor of \(b_z\) is at most \(w\), whereas \(\gcd(t_z,W)=1\). The integrality of \(t_z^eM(z)=R(z)\) then forces \(b_z=1\). Finally divide (70) by \(t_z^e\geq S^e\). This yields (69) with a constant depending on \(q\) alone. Only the rational exactness steps, not this constant, required a threshold depending on \(H\). ◻

Proof. The induction will turn a discrepancy on \(G/\Gamma\) into one on the kernel of a horizontal character. To preserve polynomial dependence during this reduction, we work on long progressions of exceptional values of \(t\) with smooth difference. Interpolation on such a progression will split the horizontal polynomial into an integer part and a slowly varying error. We first specify the stronger progression assertion to which this dimension induction applies.

A bad progression family consists of fixed bounds \(d,D,c,C,B,\eta>0\) and a fixed nilmanifold such that, for every prescribed integer \(H\), there are sequences with \(w,S\to\infty\) and \(Z/S^a\to\infty\) for every fixed \(a\), and progressions \[ t_z=t_0+mz\in[S,2S),\quad 0\leq z<H,\quad m\text{ is }w\text{-smooth},\quad \gcd(t_z,W)=1, \tag{74}\] on which every index has a discrepancy at least \(\eta\) as in (73). Here the logarithms of \(P_z(x)\) need only be jointly polynomial in \((z,x)\) with the fixed degree bound; the boxes, classes and tests may vary with \(z\). The sequences may depend on \(H\). It is equivalent to ask for arbitrarily large prescribed lengths: a longer progression can be restricted. We will prove that no such family exists, by induction on \(\dim G\).

First, failure of the Proposition would produce such a family. Indeed, suppose the proportion of bad values is at least \(\alpha>0\) along a subsequence. In the index coordinate \(t=r+Lu\), the set \(\mathcal T\) is an integer interval of length tending to infinity. For prescribed \(H\), finite Szemerédi’s Theorem [33] supplies a fixed integer \(K=K(H,\alpha)\) such that every subset of \([K]\) of density at least \(\alpha/2\) contains an \(H\)-term progression. Partition the index interval into blocks of length \(K\) and discard its final incomplete block. Some full block has bad density at least \(\alpha/2\) for large \(w\). Its bad progression has step \(jL\) with \(1\leq j\leq K\). This step is \(w\)-smooth once \(w\geq K\). Joint polynomiality is preserved by the affine substitution for \(t\). Thus (74) holds for each fixed \(H\).

We shall repeatedly thin a bad progression to one color in a finite coloring. Finite van der Waerden’s Theorem [35] makes this legitimate: to obtain a length \(H\) of one color, start with the fixed length prescribed by that Theorem for \(H\) and the number of colors. Under \(z=a+bu\), the new step is \(bm\); \(b\) is bounded in terms of the initial fixed length, so it remains \(w\)-smooth for large \(w\). The new step is still at most \(S\), and all scale and degree hypotheses remain valid. A finite choice of group data can also be fixed: first pass to a subsequence for each length, then choose one of the finitely many possibilities occurring for unbounded lengths. Bounds in all these colorings will be independent of \(H\).

For dimension zero the quotient is a point and every discrepancy is zero. Assume the assertion has been proved in smaller dimension, and suppose that a bad progression family on \(G/\Gamma\) exists. For each \(z\), at least one of its two samples has Haar discrepancy at least \(\eta/2\). Write its step as \(p_z\in\{1,t_z\}\) and its residue representative as \(b_z\in\{0,\ldots,p_z-1\}^d\). Substitute \(x=p_zv+b_z\). The resulting box lies in a fixed multiple of \([-Z/p_z,Z/p_z]^d\) and has comparable sides. Its common scale tends to infinity. Theorem 26, with the fixed stretched filtration (67), therefore gives a character from a fixed finite list. Thin to a common nonzero character \(\xi\), including the fixed denominator multiplication in that choice. Write \[ \xi(P_z(x))=\sum_I\theta_I(z)x^I. \tag{75}\] The coefficients \(\theta_I\) are polynomials of bounded degree in \(z\). The coefficient obstruction in the \(v\) variables is \[ \left\|p_z^{\abs I} \sum_{J\geq I}\binom JI b_z^{J-I}\theta_J(z) \right\|_{\mathbb R/\mathbb Z} \leq A_0(p_z/Z)^{\abs I} \qquad(\abs I>0), \tag{76}\] where \(A_0\) is independent of \(H,z\) and of every polynomial coefficient.

We claim, descending through the positive spatial degrees, that there are rational polynomials \(m_I(z)\) of bounded degree such that \[ m_I(z)\in\mathbb Z,\qquad \abs{\theta_I(z)-m_I(z)}\leq A_1 Z^{-\abs I} \quad(0\leq z<H,\ \abs I>0), \tag{77}\] with smooth coefficient denominators and a constant \(A_1\) independent of \(H\). Take \(H\) larger than the finitely many degree requirements of Lemma 27; this does not restrict the arbitrarily long progression assertion. Suppose higher degrees have already been treated and put \(e=\abs I\). Their integral parts in (76) contribute integers. Their error contribution has size at most \[C\sum_{J>I}p_z^e p_z^{\abs J-e}Z^{-\abs J} \leq C'(p_z/Z)^e,\] since \(p_z\leq2S=o(Z)\); here \(J>I\) means \(J\geq I\) and \(J\ne I\). It follows that \(\norm{p_z^e\theta_I(z)}_{\mathbb R/\mathbb Z}\leq C'(p_z/Z)^e\). If \(p_z=t_z\) this is already the needed bound. If \(p_z=1\), multiplication by the integer \(t_z^e\) gives the same conclusion: \[\norm{t_z^e\theta_I(z)}_{\mathbb R/\mathbb Z}\leq C''(S/Z)^e.\] Lemma 27 gives \(m_I\) and the claimed bound. There are only boundedly many spatial degrees and coefficients, so the descending induction terminates with \(A_1\) depending only on the fixed data. The length-dependent thresholds for rational exactness can be met simultaneously.

The positive-degree coefficients now have the required integer parts and small errors. To obtain a bounded horizontal error on the whole box, we must also split the spatially constant coefficient. This requires a finite coloring argument. Let \(q_0\) bound \(\deg\theta_0\). Color \(z\) by the interval containing \(\{\theta_0(z)\}\) in a partition of \([0,1)\) into intervals of length less than \(2^{-q_0-2}\). Thin to a long monochromatic progression and reparametrize it. On each consecutive \((q_0+2)\)-term segment of the new progression, \[\Delta^{q_0+1}\floor{\theta_0(z)} =-\Delta^{q_0+1}\{\theta_0(z)\}\] is an integer of absolute value less than one: the constant part of the common fractional-part interval cancels, and the sum of the absolute difference coefficients is \(2^{q_0+1}\). It is therefore zero. Newton interpolation, or induction using this difference identity, shows that a rational polynomial \(m_0(z)\) of degree at most \(q_0\) agrees with \(\floor{\theta_0(z)}\) on the whole new progression. Its Newton coefficients are integer differences; hence its ordinary coefficient denominators divide \(q_0!\). All previously constructed \(m_I\) retain (77) after the same affine reparametrization.

Set \[ m_z(x)=\sum_I m_I(z)x^I, \qquad a_z(x)=\xi(P_z(x))-m_z(x). \tag{78}\] For integer \(z\) in the progression, \(m_z\) has integer coefficients as a polynomial in \(x\). By (77), its error satisfies, throughout \([-CZ,CZ]^d\), \[ \abs{a_z(x)}\leq C_1, \abs{\partial_{x_j}a_z(x)}\leq C_1/Z \quad(1\leq j\leq d), \tag{79}\] where \(C_1\) is independent of \(H\). To see this explicitly, the constant term lies in \([0,1)\), and an error coefficient at degree \(e\) contributes at most \(A_1Z^{-e}(CZ)^e\) to the first bound and at most \(eA_1Z^{-e}(CZ)^{e-1}\) to a derivative. Only boundedly many monomials occur.

Choose a rational vector \(V\in\mathfrak g\) with \(\xi(\exp V)=1\). Such a vector exists because a nonzero horizontal character has nonzero rational differential. There is a fixed positive integer \(K\) with \(\exp(KV)\in\Gamma\). For completeness, the Mal’cev coordinates of \(\exp(tV)\) are rational polynomials in \(t\) with zero constant term, by the finite BCH formula. Taking \(K\) divisible by their finitely many denominators makes these coordinates integral, which places \(\exp(KV)\) in the lattice. All choices depend only on the fixed character and rational group data.

Put \(H_\xi=\ker\xi\) and \[ Q_z(x)=\exp(-a_z(x)V)P_z(x)\exp(-m_z(x)V). \tag{80}\] Then \(\xi(Q_z(x))=0\) identically, for real \((z,x)\), and \[ P_z(x)=\exp(a_z(x)V)Q_z(x)\exp(m_z(x)V). \tag{81}\] The group \(H_\xi\) is connected and simply connected: the exponential map identifies it with the kernel of the linear differential of \(\xi\). That kernel is rational, so \(H_\xi\) is a rational subgroup of dimension \(\dim G-1\). Formula (80) and the finite BCH formula show that \(\log Q_z(x)\) is jointly polynomial in \((z,x)\) with a degree bound depending only on the old degree and nilpotence step. No bound on its coefficients is needed.

The polynomial now takes values in a smaller group. To apply the induction hypothesis, we must also express both sampling laws and their test functions on a fixed compact quotient of \(H_\xi\). Partition each test box into a bounded number of rectangular subboxes, each of side between \(\lambda cZ/2\) and \(2\lambda CZ\), where the sufficiently small constant \(\lambda>0\) is chosen below. For example divide each side into the same sufficiently large fixed number of intervals. On a subbox choose a point \(x_*\) and replace the left factor \(\exp(a_z(x)V)\) in (81) by \(\ell_{z,*}=\exp(a_z(x_*)V)\). The error in the test is at most \(C_2\lambda\): the derivative estimate (79) bounds the change in \(a_z\), and the action map on the product of a fixed compact subset of \(G\) and \(G/\Gamma\) has bounded derivative. The constant \(C_2\) depends on the original Lipschitz bound, but not on \(H,z\) or the choice of subbox. Choose \(\lambda\) so that twice this error is less than \(\eta/8\).

Split each subbox further according to \(x\bmod K\). At a fixed \(z\) and residue vector \(u\), the integer polynomial \(m_z(x)\) has a fixed value \(j\bmod K\), and hence \[\exp(m_z(x)V)\Gamma=\sigma_j\Gamma, \qquad\sigma_j=\exp(jV),\quad0\leq j<K.\] The resulting test on the smaller group is \[ h\bigl(H_\xi\cap\sigma_j\Gamma\sigma_j^{-1}\bigr) \longmapsto F_z(\ell_{z,*}h\sigma_j\Gamma). \tag{82}\] The intersection is a lattice in \(H_\xi\): conjugation by rational \(\sigma_j\) preserves the rational structure, and a connected rational subgroup meets a rational lattice in a lattice. This can also be seen by taking a rational Mal’cev basis of the subgroup and clearing the finite coordinate denominators. The displayed map from its compact quotient to \(G/\Gamma\) is smooth. Since there are only finitely many \(j\) and \(\ell_{z,*}\) lies in a fixed compact set, all the tests (82) have one uniform Lipschitz bound.

We justify the sampling weights in this splitting. For an interval of length comparable to \(Z\) and any residue modulo \(q\), its count is its length divided by \(q\), with error at most two. Multiplying this formula over the fixed number of coordinates shows that the normalized weights of the boundedly many subboxes under the unrestricted and step-\(t_z\) samples differ in total by \(O(t_z/Z)\). For \(w\geq K\), \(\gcd(t_z,K)=1\). Thus both samples assign weight \(K^{-d}+O(K^dt_z/Z)\) to each \(K\)-residue vector inside a subbox: in the step-\(t_z\) coordinate, reduction modulo \(K\) is a bijection. The total difference of the weights of all the pieces is consequently \(o(1)\), uniformly in \(z\) and in the box endpoints.

After freezing, the original discrepancy is still at least \(7\eta/8\). Write the two averages as convex combinations over these pieces and replace one set of combination weights by the other. The error is \(o(1)\) times the uniform supremum bound on the tests. It follows from the triangle inequality that at least one piece has a conditional discrepancy at least \(\eta/2\), for large \(w\). This lower bound does not depend on the number of pieces: it follows from a convex combination, not from an unnormalized sum.

For each \(z\) choose such a piece. Its subbox and frozen observable may vary arbitrarily with \(z\), as allowed in the induction assertion. Thin once more to make its residue vector \(u\bmod K\) and its coset index \(j\) common. In this subbox substitute \(x=u+Ky\), taking \(u\in\{0,\ldots,K-1\}^d\). The unrestricted sample becomes all integer \(y\) in a box of side comparable to \(Z/K\), within a fixed multiple of that scale. The step-\(t_z\) sample becomes one residue vector modulo \(t_z\), because \(K\) is invertible modulo \(t_z\). The new map \(Q_z(u+Ky)\) is still jointly logarithmically polynomial, and \(Z/K\) still dominates every fixed power of \(S\). We have therefore obtained a bad progression family on \[H_\xi/\bigl(H_\xi\cap\sigma_j\Gamma\sigma_j^{-1}\bigr)\] of strictly smaller dimension.

All the new degree, box, observable and discrepancy bounds are independent of the progression length. Characters, rational directions, residue vectors and cosets range over fixed finite lists; as explained at the start, one such quotient works for unbounded lengths. The sole length-dependent quantities were interpolation denominators and asymptotic thresholds. Those denominators enter only the unrestricted coefficients of the kernel polynomial. At every retained integer index its spatial integer part has integer coefficients, so the fixed residue splitting and the finite list of quotient lattices are unchanged. This contradicts the induction hypothesis and proves that bad progression families do not exist. Failure of the Proposition produced precisely such a family, so the Proposition follows, with the uniformity asserted in its statement. ◻

Proof. Write the finer class as \(b+qt\mathbb Z^d\) with \(b\equiv a\pmod q\). Under \(x=a+qy\), its condition is exactly \[y\equiv(b-a)/q\pmod t.\] This identity does not use invertibility of either factor. Apply Proposition 28 to the normalized polynomial and the scale \(Z_0/q\). Its uniformity permits arbitrary compatible \(b\) and arbitrary endpoints of the normalized boxes. ◻

Proof. For \(X\in\mathfrak h_{|D|}\) write \(X_D\) for the corresponding face array. The Boolean monomials \(\prod_{j\in D}e_j\) are linearly independent, so \[\mathfrak c^q =\bigoplus_{D\subseteq[q]} \mathfrak h_{|D|}\prod_{j\in D}e_j\] is a direct sum of vector spaces. The entrywise Lie bracket satisfies \([X_D,Y_E]=[X,Y]_{D\cup E}\). The filtration places this bracket in the summand indexed by \(D\cup E\). This remains true when a set is empty, because \(H_0=H_1\) and the filtration is decreasing. Thus \(\mathfrak c^q\) is a rational Lie subalgebra.

Every suffix in the chosen order of summands is an ideal: a bracket with a summand indexed by \(E\) stays at \(E\) or moves to a strictly larger set. Successively passing to the quotients by these suffix ideals gives (84). Equivalently, the coefficient at \(E\) is recovered from the entry at vertex \(E\) after the coefficients at proper subsets of \(E\) have been removed. This also proves uniqueness and smoothness of the inverse. The connected subgroup with Lie algebra \(\mathfrak c^q\) is closed and simply connected, as follows from the exponential diffeomorphism for a simply connected nilpotent group. It is exactly the group generated by the faces.

The same recovery at successive vertices proves the assertion about an array supported at the top vertex. Reflecting a coordinate replaces \(e_j\) by \(1-e_j\). The resulting face indicator is a linear combination of upper-face indicators of no greater cardinality; the decreasing filtration therefore preserves \(\mathfrak c^q\), and hence its exponential group. Permutations are immediate. Finally rationality gives a cocompact lattice \(C^q(H_\bullet)\cap\Gamma^{\{0,1\}^q}\), so the indicated image is compact. ◻

Proof. Reflect the missing vertex to the all-ones vertex. If two group cubes \(a,b\in C^{s+1}(G_\bullet)\) agree modulo \(\Gamma\) at all other vertices, then \(z=a^{-1}b\) is a group cube whose entries at all proper subsets of \([s+1]\) belong to \(\Gamma\). Recover its face coefficients by (84). Inductively the coefficient at a proper subset \(E\) lies in \(\Gamma\), because its value is the entry at \(E\) multiplied by inverses of previously recovered lattice elements. The top coefficient belongs to \(G_{s+1}=\{1\}\). The top entry of \(z\) is therefore also a product of lattice elements. The original two top vertices agree modulo \(\Gamma\).

The deletion map from the compact cube set to the space of partial cubes is now a continuous injection into a Hausdorff space. It is a homeomorphism onto its image, proving continuity of reconstruction. For a representative \(x_0\in G\) of \(x\), the displayed linear cube is represented by the product of the constant arrays \(g^{b_0}\) and \(x_0\), with the codimension-one arrays \(g^{b_j}\) between them. It therefore lies in the cube group.

On the compact image of the deletion map, products of continuous functions of individual coordinates form an algebra containing constants and separating points. Stone–Weierstrass approximates the reconstructed observable by finite sums from this algebra. Lipschitz functions are dense in the continuous functions on each compact nilmanifold, so the factors can be made Lipschitz and then normalized to have sup norm at most one. Finally the family of observables with the prescribed sup and Lipschitz bounds is compact in the uniform norm. A finite uniform net, followed by the construction for each member of that net, proves the last assertion. ◻

Proof. We first justify Taylor coordinates. For any adapted \(R\) write \(a_0=R(0)\) and remove this constant on the left. Suppose that, after removing the factors of orders below \(j\), the remaining adapted polynomial \(R_j\) is the identity at \(0,\ldots,j-1\). The \(j\)th forward difference of \(\log R_j\) at zero equals \(\log R_j(j)\): all its other evaluated terms vanish. Terms of degree less than \(j\) in the logarithm make no contribution, and all terms of degree at least \(j\) have coefficients in \(\mathfrak h_j\). Thus \(a_j=R_j(j)\in H_j\). Multiplication on the left by \(a_j^{-\binom bj}\) leaves an adapted polynomial vanishing at the first \(j+1\) integers. After order \(s\), its logarithm has degree at most \(s\) and \(s+1\) zeros, so it is identically zero. This proves the ordered Taylor representation.

Choose \(\lambda_L\in\Lambda\) so that \(c_L=P_L(0)\lambda_L\) lies in a fixed compact set of representatives. Normalize by \(R_L(b)=c_L^{-1}P_L(b)\lambda_L\). It is adapted, has constant term the identity, and \(P_L(b)\Lambda=c_LR_L(b)\Lambda\). Adaptation is preserved here either by the preceding BCH observation, or by noting that conjugation by \(\lambda_L\) preserves all terms of \(K_\bullet\).

We describe one descent step for a fixed current rational filtration \(H_\bullet\) and a normalized adapted polynomial with Taylor coefficients \(a_{j,L}\). If (87) fails, fix one of its characters \(\xi\) and pass to a subsequence on which \[\xi(a_{j,L})=m_L+\lambda'_L,\qquad m_L\in\mathbb Z,\qquad \abs{\lambda'_L}\leq C L^{-j}.\] Choose a fixed rational \(V\in\mathfrak h_j\) with \(\xi(\exp V)=1\). Replace the polynomial \(R_L\) by \[ R'_L(b) =\exp\bigl(-\lambda'_L\tbinom bj V\bigr) R_L(b) \exp\bigl(-m_L\tbinom bj V\bigr). \tag{88}\] It remains adapted to the old filtration. Its Taylor coefficients of orders less than \(j\) are unchanged, since both multipliers are the identity at \(0,\ldots,j-1\). Modulo \(H_{j+1}\), the order-\(j\) coefficient is the old coefficient with the two displayed corrections. To verify this even in the presence of lower Taylor factors, commute those factors past the left correction at \(b=j\); the commutators lie in \([H,H_j]\subseteq H_{j+1}\). Its \(\xi\)-value is consequently \(-\lambda'_L+\xi(a_{j,L})-m_L=0\).

Replace \(H_j\) by \(H_j\cap\ker\xi\) and leave all other positive levels unchanged. When \(j=1\) replace \(H_0\) by the same kernel. The kernel is a connected rational subgroup, since in exponential coordinates it is the kernel of a rational linear form. Nesting is preserved because \(\xi\) kills \(H_{j+1}\). A bracket inclusion whose target is the changed level is preserved because \(\xi\) kills each \([H_i,H_{j-i}]\). Inclusions whose target has larger index already land in \(H_{j+1}\), and the other inclusions are unchanged or have a smaller source. Thus this is again a filtration. The Taylor calculation shows that \(R'_L\) is adapted to it. In the case \(j=1\), all higher coefficients already belong to \(H_2\), so its entire image lies in the new \(H\).

The removed factors give \[R_L(b)= \exp\bigl(\lambda'_L\tbinom bj V\bigr) R'_L(b) \exp\bigl(m_L\tbinom bj V\bigr).\] The left factor is bounded and has derivative \(O(L^{-1})\) on every interval \(\abs b\leq A L\), with constants allowed to depend on \(A,C,V\). The right factor is an integer power of the fixed rational element \(\exp V\) at integer inputs. Repeating the procedure strictly reduces \(\sum_{i=1}^s\dim H_i\) at every step. It therefore terminates after finitely many steps. If any fixed rational level character still failed (87) on the final subsequence, the same operation would give another strict reduction. Hence all the asserted character limits hold.

There is no conjugation of accumulated smooth factors by rational factors in this procedure: from \(P=\epsilon_1R_1\gamma_1\) and \(R_1=\epsilon_2R_2\gamma_2\) one gets \(P=(\epsilon_1\epsilon_2)R_2(\gamma_2\gamma_1)\). Thus \(h_L\) is the product of \(c_L\) and finitely many of the bounded smooth factors just described. Pass to a subsequence on which \(c_L\) converges and every bounded scalar \(L^j\lambda'_L\) converges. Since \(L^{-j}\binom{L\beta}{j}\) converges with all derivatives on compact sets to \(\beta^j/j!\), this gives the first conclusion.

For completeness, the right factors have a common bounded period modulo \(\Lambda\), even though the integers \(m_L\) need not be bounded. Finitely many fixed rational elements, together with a finite generating set of \(\Lambda\), generate a discrete group \(\Lambda'\) containing \(\Lambda\). Here is the denominator argument. Collect a word into integer powers of the finitely many generator and iterated-commutator types of length at most the nilpotence class. Collection terminates because a commutation error has strictly greater commutator length. The logarithms of these finitely many types have rational coordinates. Applying the finite BCH formula to their ordered powers bounds every coordinate denominator by one fixed integer, independent of the word and its integer exponents. This proves discreteness. A compact fundamental set for \(\Lambda\) then shows that \([\Lambda':\Lambda]<\infty\).

The normal core \(\Lambda''=\bigcap_{g\in\Lambda'}g\Lambda g^{-1}\) has finite index in \(\Lambda'\), so \(\Lambda'/\Lambda''\) is a finite group. Let \(a\) be a common multiple of its element orders. Each \(\binom bj\) modulo \(a\) has a fixed period, for example \(a s!\) for \(0\leq j\leq s\): Vandermonde’s identity shows that \(\binom{b+a s!}{j}-\binom bj\) is divisible by \(a\). Consequently every accumulated right factor is periodic in this finite quotient with one common period \(d_0\). There are only finitely many possible maps from its residue classes to \(\Lambda'/\Lambda''\). Pass to a subsequence on which this map is fixed and choose rational representatives \(\sigma_r\). This gives the second conclusion and completes the proof. ◻

Proof. We first prove equidistribution before inserting the smooth factor. Fix \(q,r\) and abbreviate \(C=C^q(H_\bullet)\). The cube polynomial \[\mathbf Q_L(\mathbf b) =\bigl(Q_L(b_0+\textstyle\sum_{j=1}^q e_jb_j)\bigr)_e\] takes values in \(C\). To see this and record the needed coefficient information, expand \[ \binom{b_0+\sum_{j=1}^q e_jb_j}{\ell} =\sum_{D\subseteq[q]} \Bigl(\prod_{j\in D}e_j\Bigr)F_{\ell,D}(\mathbf b). \tag{92}\] Each \(F_{\ell,D}\) has degree at most \(\ell\), and every one of its monomials uses exactly the increment variables indexed by \(D\), each with positive exponent. In particular it vanishes for \(|D|>\ell\). Thus the \(\ell\)th Taylor factor on the cube is the product of the face arrays \((a_{\ell,L})_D^{F_{\ell,D}(\mathbf b)}\) with \(|D|\leq\ell\). These are in \(C\), because \(H_\ell\subseteq H_{|D|}\). This expansion also shows adaptation to (85): a coefficient of total polynomial degree \(k\) in a factor of order \(\ell\geq k\) lies in \(\mathfrak h_\ell\subseteq\mathfrak h_{\max(k,|D|)}\). BCH preserves this assertion on multiplying the Taylor factors.

We claim that \(\mathbf Q_L\) is equidistributed on \(C/\Gamma^{[q]}_{\sigma_r}\) on every box whose sides are fixed positive fractions of \(L\), whose origin is \(O(L)\), and with any fixed progression restrictions. The error tends to zero uniformly for fixed bounds on these data and on the Lipschitz tests. Otherwise Theorem 26, applied to the fixed rational nilmanifold and the filtration (85), supplies along a subsequence a nonzero horizontal character from a fixed finite list. Pass to a further subsequence on which this character, denoted \(\chi:C\to\mathbb R\), is fixed. It annihilates commutators and takes integer values on \(\Gamma^{[q]}_{\sigma_r}\).

For a face \(D\), write \(\chi_D(y)=\chi(y_D)\) wherever this is defined. Choose the largest \(j\geq1\) such that for some \(|D|\leq j\) the restriction \(\xi=\chi_D|_{H_j}\) is nonzero. Such a pair exists because the face subgroups generate \(C\), including the empty face and \(H_0=H_1\). The restriction is a rational homomorphism: it is integer-valued on the lattice \(H_j\cap\sigma_r\Lambda\sigma_r^{-1}\), and conjugation by \(\sigma_r\) preserves rationality. Maximality makes it vanish on \(H_{j+1}\). For \(1\leq i<j\), choose \(A,B\subseteq D\) with \(A\cup B=D\), \(|A|\leq i\), and \(|B|\leq j-i\); one or both sets may be empty. If \(u\in H_i\) and \(v\in H_{j-i}\), their face arrays on \(A,B\) lie in \(C\) and \([u_A,v_B]=[u,v]_D\). Since \(\chi\) kills this commutator, \(\xi\) kills \([H_i,H_{j-i}]\). Thus \(\xi\) is exactly a character covered by (87).

Applying \(\chi\) to (92) gives the scalar polynomial \[\chi(\mathbf Q_L(\mathbf b)) =\sum_{\ell,D}\chi_D(a_{\ell,L})F_{\ell,D}(\mathbf b).\] All terms with \(\ell>j\) vanish by maximality. In its degree-\(j\) part, the coefficient of \[ b_0^{j-|D|}\prod_{k\in D}b_k \quad\hbox{is}\quad \frac{\xi(a_{j,L})}{(j-|D|)!}. \tag{93}\] Indeed the top part of the binomial polynomial is the \(j\)th power divided by \(j!\). No other face can contribute to this monomial: its increment support must be exactly \(D\). No lower Taylor order has degree \(j\). This proves both the value and the absence of cancellation in (93).

On a fixed residue progression, substitute \(b_\nu=r_\nu+d t_\nu\) with fixed \(d\). Its top-degree coefficient is multiplied by \(d^j\); translations of the box origin do not alter it. The bounded scalar coefficient obstruction in Theorem 26, clearing the fixed binomial-to-monomial denominators if necessary, would therefore give \[L^j\norm{a\,\xi(a_{j,L})}_{\mathbb R/\mathbb Z}=O(1)\] for a fixed nonzero rational number \(a\). This contradicts (87) for the rational character \(a\xi\). The claim follows. The argument also covers \(q=0\), when only the empty face occurs. If \(C\) is trivial, the claim is immediate.

In the averages in (91), all vertex inputs have the same residue \(r\bmod d_0\). Consequently their right factors can be replaced exactly by \(\sigma_r\Lambda\). For fixed \(\alpha\), the preceding claim gives the Haar law for the \(Q_L\) cube. At every vertex, \(b/L=\beta+O_q(\alpha)\); Lemma 32 therefore gives \[h_L(b)=h(\beta)+O(\alpha)+o_{L\to\infty}(1)\] in any fixed local metric, uniformly on the sampled box. Multiplication by these bounded elements and passage to the compact nilmanifold changes a bounded Lipschitz test by \(O_F(\alpha)+o(1)\), uniformly over the remaining entries. The latter uniformity follows from smoothness of the action on the compact nilmanifold. First letting \(L\to\infty\) and then \(\alpha\downarrow0\) proves (91), including its stated uniform versions.

Finally partition \([0,1]\) into finitely many intervals of length at most \(\alpha\). On each interval and each residue class, apply the same argument with \(q=0\) and freeze \(h\) at an endpoint. The proportions of the residue classes tend to \(1/d_0\), and the interval proportions tend to their lengths. Letting \(\alpha\downarrow0\) gives the Riemann integral in (90). This integral is well defined because the Haar images depend continuously on \(\beta\). No further subsequence depending on \(q\) was used: (87) proves the equidistribution assertion for each fixed dimension. ◻

Proof. Put \(J_i=\pi(H_i)\) and \(J=J_1\). These are connected rational groups; their Lie algebras are rational images of those of \(H_i\). Their filtration satisfies (83), and projection of the face generators shows \(\pi(C^q(H_\bullet))=C^q(J_\bullet)\). In the remainder of the proof use \(h,\sigma,\Lambda\) for the projected elements and lattice. The projected point coset \(hJ\sigma\Lambda/\Lambda\) is preserved by \(S\): each of its points can occur at a vertex on the specified face of a constant cube, and face preservation, also for the inverse map, gives equality of the point sets. Choose \(c\in J\) such that \[ S(h\sigma)\Lambda=hc\sigma\Lambda. \tag{97}\] The ambient automorphism \[\Psi(y)=h^{-1}S(h)S(y)S(h)^{-1}h\] satisfies \(S(hy\sigma)\Lambda=h\Psi(y)c\sigma\Lambda\) for \(y\in J\). We next verify on the group cover that \(\Psi(J)=J\).

The map \(y\mapsto y\sigma\Lambda\) on \(J\) has stabilizer \(\Gamma_J=J\cap\sigma\Lambda\sigma^{-1}\). Rationality makes \(J/\Gamma_J\) a compact embedded orbit. Given a path \(y(t)\) in \(J\) from the identity to \(y\), the path \(\Psi(y(t))c\sigma\Lambda\) stays in this orbit. Lift it there starting at \(c\in J\). It is also a lift under the ambient covering \(K'\to K'/\Lambda\) after multiplying on the right by \(\sigma\). Uniqueness of lifts under this covering forces the ambient lift \(\Psi(y(t))c\) to lie in \(J\). Thus \(\Psi(J)\subseteq J\); equality follows because its differential is injective and both connected groups have the same dimension. Consequently \[\Phi(y)=c^{-1}\Psi(y)c\] is an automorphism of \(J\), and the action on the point coset is represented by the affine map \(y\mapsto c\Phi(y)\).

We claim that its linear part is the identity on each associated graded space: \[ (\Phi_*-\mathrm{id})\mathfrak j_i\subseteq\mathfrak j_{i+1}, \qquad 1\leq i\leq s. \tag{98}\] Fix \(U\in\mathfrak j_i\) and work in dimension \(i+1\). Put \(\exp(tU)\) on the face with its first \(i\) coordinates equal to one, and the identity elsewhere. This is a path \(z(t)\) in \(C^{i+1}(J_\bullet)\). Apply the affine map \(c\Phi\) on the upper face of the last coordinate. The resulting array \(z'(t)\) has its image in the same cube coset by hypothesis. At \(t=0\) it is the constant-\(c\) upper-face array, which itself belongs to \(C^{i+1}(J_\bullet)\).

This quotient assertion lifts to \(z'(t)\in C^{i+1}(J_\bullet)\) for every \(t\). Indeed this cube group is rational with lattice \(C^{i+1}(J_\bullet)\cap\Gamma_J^{\{0,1\}^{i+1}}\), so its quotient is an embedded compact subspace of the product point orbit. Lift the path through that quotient beginning at \(z'(0)\) and use uniqueness in the ambient product covering, exactly as in the preceding paragraph.

Multiply \(z'(t)\) on the left by the inverse constant-\(c\) last-face array, and then on the left by \(z(t)^{-1}\). The resulting group cube is the identity except at the top vertex, whose value is \[\exp(-tU)\Phi(\exp(tU)).\] Lemma 30 puts this element in \(J_{i+1}\). Differentiating at \(t=0\) proves (98). In particular \(\Phi\) preserves every filtration level.

Choose filtration-adapted coordinates on \(\mathfrak j\). The polynomial \[ D(Y)=\log\bigl(c\Phi(\exp Y)\bigr)-Y \tag{99}\] has weighted degree strictly less than the output weight in each coordinate. For its linear part this follows directly from (98): an input from level \(i\) moves only to levels greater than \(i\). For the remaining terms use BCH with \(\log c\) and \(\Phi_*Y\). Every nonconstant bracket correction contains at least one copy of the fixed vector \(\log c\), which spends at least one unit of filtration weight but contributes no polynomial degree in \(Y\). A bracket contributing to output weight \(j\) therefore has input weighted degree at most \(j-1\). The constant term also has the required degree.

The linear projection \(p=\pi_*:\mathfrak h\to\mathfrak j\) has a filtration-preserving linear right inverse \(R\). To construct one, choose an adapted basis downstairs and lift each basis vector of weight \(i\) to \(\mathfrak h_i\), which is possible because \(p(\mathfrak h_i)=\mathfrak j_i\). Set \[ T(u)=u+R D(pu). \tag{100}\] This is a shear of the form (94). Explicitly, a filtration-preserving linear map can send an input coordinate of weight \(i\) only to an output of weight at least \(i\). Substitution in \(D\) and then application of \(R\) thus preserve the strict inequality between input degree and output weight. Moreover \(p(Tu)=pu+D(pu)=\log(c\Phi(\exp(pu)))\), proving (96). Finally a map of (94) is inverted successively in increasing weight, so it is a polynomial bijection of the cover. ◻

Proof. Composition preserves (94), and inversion does so by solving one weight at a time. Let \(V_d\) be the space of polynomials of weighted degree at most \(d\), including the constants, and put \(V_{-1}=0\). Substitution by a shear preserves \(V_s\), and substitution minus the identity maps \(V_d\) into \(V_{d-1}\). Indeed every term in the difference of a substituted monomial uses at least one strict lower-degree displacement in place of its coordinate. Use inverse substitution if necessary to obtain a group representation rather than an opposite representation. It is faithful because all coordinate functions belong to \(V_s\).

Let \(\mathcal I\) denote the algebra of linear operators on \(V_s\) which send each \(V_d\) into \(V_{d-1}\). Then \(\mathcal I^{s+1}=0\), and the representation lies in \(1+\mathcal I\). The elementary identities for inverses in a nilpotent algebra give \([1+\mathcal I^a,1+\mathcal I^b]\subseteq1+\mathcal I^{a+b}\). Thus every \((s+1)\)-fold group commutator is the identity. Extra translation coordinates can all be given weight one, which proves the direct-product assertion as well.

For a fixed shear \(T\), its displacement in a weight-\(j\) coordinate is \(O_T(E^{j-1})\) on \(B_E\). Its derivative matrix is block triangular by weight with identity diagonal, so its Jacobian determinant is one. The image \(T(B_E)\) is contained in the box with side bounds \(E^j+C_T E^{j-1}\). The latter box exceeds \(B_E\) in relative volume \(O_T(E^{-1})\). Since \(T\) preserves Lebesgue measure, the symmetric difference of \(T(B_E)\) and \(B_E\) has the same bound. This proves (102), uniformly over a fixed finite list by taking the largest of its constants. The assertion is deliberately for fixed shears; no uniform coefficient bound over all shears is needed.

For \(a\in H\) fixed, left multiplication in logarithmic coordinates is \[u\longmapsto\log(a\exp u).\] BCH makes this another shear: every displacement bracket uses the fixed vector \(\log a\) and hence has input weighted degree strictly below its output weight. Equation (102) therefore implies that every weak subsequential limit of the projected boxes on \(H/(H\cap\sigma\Lambda\sigma^{-1})\) is invariant under all left translations by \(H\). This compact homogeneous space has a unique such probability measure, its Haar probability. Compactness supplies subsequential limits, and uniqueness shows that the entire sequence converges. Translating on the left by \(h\) and on the right by \(\sigma\Lambda\) gives the stated point-coset Haar law. Applying (102) once more proves the identical limit after each fixed shear or word. ◻

Proof. We give the backward recursion, including the requirements protecting an earlier target. At a fixed pivot let its ordered targets be \(B_1,\ldots,B_q\). Suppose the required lists after update \(j\) have already been specified for \(B_1,\ldots,B_j\); denote them by \(\mathcal L_{j,B}\) and \(\mathcal W_{j,B}\). At \(j=q\) these are the terminal lists in the statement. Taking a full product of the leading and word lists only strengthens the requirements.

Target \(B_j\) at update \(j\). Its leading product does not change. For a word \(w\), let \(I_k(w)\) denote the old-scale template obtained by replacing each letter \((e,\gamma)\) by \((e,\gamma k^{\lambda_e})\). Equation (104) gives the exact identity \[ w[b\kappa(k)]=I_k(w)[b]. \tag{108}\] For \(B_j\) and a fixed current center \(x\), color the acting group by \(g\mapsto(C_{B_j,a b_{B_j}}(g x))_{a\in\mathcal L_{j,B_j}}\). Its size is at most \(2^{r|\mathcal L_{j,B_j}|}\). Regard each rational old-scale letter needed in \(\mathcal W_{j,B_j}\) as a separate abstract generator. The words \(I_k(w)\) are ordered products of these generators to powers \(\pm k^{\lambda_e}\), with all exponents vanishing at zero. The finite version of recurrence therefore supplies a finite set \(\Omega_j\) of alternatives \((k,J)\) satisfying the required equalities for \(B_j\) at \(J[b]x\) and scale \(b\kappa(k)\). This set is independent of the incoming numerical scale.

For each previously processed \(B=B_l\), \(l<j\), define predecessor lists by including, for every \((k,J)\in\Omega_j\), \[\begin{align*} \mathcal L_{j-1,B}&\supset \{a\kappa_B(k):a\in\mathcal L_{j,B}\}, \tag{109}\\ \mathcal W_{j-1,B}&\supset \{J\}\cup\{I_k(w)J:w\in\mathcal W_{j,B}\}. \tag{110}\end{align*}\] The unions over all alternatives are finite. If these predecessor requirements hold at \((x,b)\), then for each new leading index \(a b_B\kappa_B(k)\) they compare the same old center \(x\) both with \(J[b]x\) and with \(I_k(w)[b]J[b]x\). Equality through the old center and (108) give \[C_{B,a b'_B}(J[b]x) =C_{B,a b'_B}(w[b']J[b]x).\] These are exactly the requirements after the update. Thus (109)–(110) prove preservation, with no assumption that the jump fixes earlier targets. At \(j=1\) there is no earlier target to protect. Backward induction defines the whole plan, and forward induction executes it.

Next plan the pivots in decreasing order. Once the plans at pivots larger than \(i\) are fixed, take all coordinatewise products of their possible multiplier tuples; call this finite list \(\mathcal D_i\). Use \(\mathcal D_i\) as the list of \(d\)’s in the terminal requirements at pivot \(i\). The preceding construction returns multiplier options depending only on that list and the fixed data, so there is no dependence on an unknown incoming scale. The last pivot starts with the singleton future list \(\{(1,\ldots,1)\}\).

Starting all scales at \(1\), enumerate all intermediate and terminal paths through these finite alternatives. This gives finite scale lists and finitely many rational palette indices \(a b_B\) at every checkpoint. Define \(\mathcal B\) to contain the final scales and \(\mathcal A\) to contain all these indices and all needed subset products of final scales. Now choose \(q_0\) divisible by the denominators of every rational number \(\gamma b_A/b_U\) used in any evaluated word on any enumerated path. This does not affect the abstract recurrence choices: \(q_0\) merely changes the homomorphism interpreting each old-scale generator.

Finally enumerate all prefixes of all words, their cumulative jumps, and the terminal tests, with the now integral exponents. Their sums in the auxiliary translation coordinates give a finite slot list, which we enlarge to contain \(0\). Intermediate word evaluations use only this list. One may assign arbitrary palettes at other slots when defining the coloring on the whole acting group. Every step of the construction used only \(n,r,s\). ◻

Proof. Fix a target \(B=T\cup\{i\}\) and write \(t=t_T\). Formulae outside the conditioned cell mean the extensions of the already selected nilsequence formulae, so all bounded-multiple boxes in the statement are defined. We will compare them with averages whose step has no factor \(t\).

Proof. Suppose the uniform assertion fails on sets of conditionings of probability bounded below along a subsequence. By Lemma 38, choose a sequence in those sets with all its empirical face invariances. Pass to a subsequence fixing the ambient array nilmanifolds and using Proposition 33. There are only finitely many observable formulae in a conditioning. Their common supremum and Lipschitz bounds give, by compactness, a further subsequence on which all of them converge uniformly. Denote the limiting observables by \(F_{B,a,c,v}\).

For each stored factor and integer input \(u\), define the reading of a joint state \(y\in K/\Lambda\) by \[ \mathcal R_{B,a,c;v,u}(y) =F_{B,a,c,v}\bigl(\operatorname{ev}_u\pi_{B,a,c,v}y\bigr). \tag{121}\] Here \(\pi_{B,a,c,v}\) selects the indicated array factor and \(\operatorname{ev}_u\) evaluates it at the integer input \(u\). The coordinates of \(u\) are indexed by the own pairs \((B,A)\); when adding \(u\) to a slot vector, insert zeros at all other pairs. These readings are continuous functions of \(y\) and are defined before choosing any lifts on a point-law coset.

Consider a point-law coset of that Proposition, \[h(\beta)H\sigma_r\Lambda/\Lambda,\] with its Haar image measure \(\mu_{\beta,r}\). The preceding Lemma and Proposition 34 lift each own-pair array shift to a polynomial shear of the joint cover coordinates \(z\in\mathfrak h\). Such a lift may move other target components. Adjoin its translation \(v\mapsto v+\mathbf e_e\) and call the result \(L_e\). Lemma 36 puts these transformations in a nilpotent group of step at most \(s\), including the ordinary slot translations. Write \(\Theta(z)=h(\beta)\exp(z)\sigma_r\Lambda\). At slot \(v\), color a state by the palette \[ C_{B,a}(z,v)= \bigl(\mathbf 1_{\{\mathcal R_{B,a,c;v,0}(\Theta(z))>3\tau/2\}}\bigr)_{c\in[r]}. \tag{122}\] These palettes have at most \(2^r\) values, as required by the finite plan.

Apply that plan pointwise starting at any \((z,0)\). It returns one of the \(Q_i\) options, a jumped center, and terminal palette equalities there. Only own letters occur in a terminal word for \(B\). Write \(m=(m_e)_e\) for their integer powers. Their projection on this entire component is exactly the commuting array shift \(u\mapsto u+(m_e)_e\), while their slot effect is \(v\mapsto v+m\), with the zero extension just specified. Consequently at a center state \(y\) and slot \(v\) the two readings are \(\mathcal R_{B,a,c;v,0}(y)\) and \(\mathcal R_{B,a,c;v+m,m}(y)\). This statement holds even if earlier jumps involved letters belonging to other targets.

To check the numerical meaning, use the corresponding prelimit observables and evaluate the same readings on an actual array state \(Y(\ell)\), with \(p=p_*+M\ell\). The center value is the model at \(t(p+s(v))\). The translated value is the model at \[\begin{align*} t\left(p+s(v)+\sum_e m_er_e\right) +M\sum_e m_e\Delta_e &=t(p+s(v))+\sum_e m_eq_e. \tag{123}\end{align*}\] Each occurrence uses the correct residue-specific observable at its new slot. For the terminal word at outcome \(b'\) and future multiplier \(d\), the powers are \[m_e=q_0\frac{b'_A}{b'_B}\frac{d_A}{d_B}\quad(e=(B,A),\ A\in D), \qquad m_e=0\quad(A\notin D).\] They are integers by the plan, and \(m_eq_e=(h_A b'_A d_A/(h_B b'_B d_B))t_A\). The leading index is \(b'_Bd_B\). Therefore (123) gives exactly (120) with \(x=p+s(v)\). This numerical interpretation is used for terminal own-letter words. An arbitrary cumulative jump need not be a translation of the actual pivot orbit.

For an option \(o\) with scale outcome \(b'\) and slot \(v\), let \(F_o\subset K/\Lambda\) consist of the states \(y\) satisfying every required comparison. Explicitly, for each \(B,d,c,D\), put \(a=b'_Bd_B\) and take \(m\) as above, and impose \[ \mathcal R_{B,a,c;v,0}(y)\le3\tau/2\quad\text{or}\quad \mathcal R_{B,a,c;v+m,m}(y)\ge3\tau/2. \tag{124}\] These finitely many conditions are closed by continuity of the readings. They depend only on the state, the option’s finite data, and the limiting formulae. In particular they do not depend on the choice of lifts. For every starting cover coordinate, the pointwise palette construction puts the \(K/\Lambda\)-valued coordinate of the jumped state into \(F_o\) for at least one option \(o\). Equality of palettes implies (124), including at equality with the threshold.

Sample \(z\) from the expanding weighted boxes of Lemma 36. For each fixed option \(o\), let \(Y_o\) be the \(K/\Lambda\)-valued coordinate of the state after its cumulative jump. The jump acts on the cover coordinate \(z\) by a fixed shear, so the law of \(Y_o\) converges to \(\mu_{\beta,r}\). Pointwise coverage gives \[1\le\sum_o\Pr\{Y_o\in F_o\}.\] Using the closed-set direction of Portmanteau separately for the finitely many summands yields \[ 1\le\sum_o\mu_{\beta,r}(F_o),\qquad \mu_{\beta,r}\left(\bigcup_o F_o\right)\ge\frac1{Q_i}. \tag{125}\] The last implication also follows by summing the indicator inequality \(\sum_o\mathbf 1_{F_o}\le Q_i\mathbf 1_{\cup_oF_o}\). This is the only quantitative loss in passing from pointwise recurrence to the point-law measure. Although the shears may depend on the coset, the sets \(F_o\) do not. Therefore (125) integrates over the macroscopic-position/residue mixture without a measurable choice of lifts.

Replace each comparison (124) by the open comparison \[ \text{center}<7\tau/4\quad\text{or}\quad \text{translated value}>5\tau/4, \tag{126}\] and take the same finite intersections and union. This defines an open set containing \(\bigcup_oF_o\), so it has mixture measure at least \(1/Q_i\). Proposition 33 gives weak convergence of the actual uniform conditional state laws to that mixture. Open-set Portmanteau therefore gives lower limit at least \(1/Q_i\) for this open event. For sufficiently late terms, uniform convergence of all observables has error less than \(\tau/4\). If the actual center is greater than \(2\tau\), its limiting reading is greater than \(7\tau/4\), forcing the second alternative in (126); the actual translated reading is then greater than \(\tau\). The good boundary condition ensures that these actual readings use exactly the selected model formulae. By (123) the open event implies success at some \(p+s(v)\).

This contradicts the selected conditional probabilities being less than \(1/(2Q_i)\). Hence the exceptional sets have probability tending to zero. The uniform-to-harmonic total variation estimate following (114) proves the last statement. The compactness subsequence and how late the estimates hold may depend on \(\tau\) and the fixed model complexity. The constants \(Q_i\) and the displayed lower bounds do not. ◻

Proof of Principle 6. For the given \(n,r,s\) take the lists and all budgets from Lemma 37. Let \(I\) be the set of pivots having targets and define \[ \delta=\prod_{i\in I}\frac1{3Q_iV_i}>0, \tag{128}\] with the empty product equal to \(1\). This definition precedes the choice of admissible parameters, models, and \(\tau\). Now fix any such parameters, any family of models of step at most \(s\) and bounded complexity in the sense of Definition 4, and any fixed \(\tau>0\).

Suppose \(E\) is an event of the earlier raw variables on a given finite path, with incoming scale \(b\) and all the earlier pivot requirements secured. Lemma 39, integrated over the pivot microcell and residue, implies \[ \frac1{3Q_i}\mathbb P(E)-o(1) \le\sum_v\mathbb P\bigl(E,\mathcal P_i(b;p+s(v))\bigr). \tag{129}\] The exceptional set of conditionings has probability \(o(1)\) unconditionally, so its intersection with \(E\) also costs \(o(1)\). Conditional on the earlier variables each \(s(v)\) is fixed. Equation (127) therefore changes each summand by \(o(1)\) when \(p+s(v)\) is replaced by \(p\). There are at most \(V_i\) slots, and consequently \[ \mathbb P\bigl(E,\mathcal P_i(b;p)\bigr) \ge\frac1{3Q_iV_i}\mathbb P(E)-o(1). \tag{130}\] This estimate is uniform over the finite incoming-scale list and the finite path events. No conditional density assumption on \(E\) has been used.

On the successful extension choose the first successful scale outcome in a fixed enumeration. This is a measurable choice from a finite list and partitions that event into finitely many new path events. They depend only on raw variables through the current pivot. Apply (130) to each path at the next pivot. Pivots without targets impose no condition and can be skipped. Summing the finitely many path estimates and iterating yields total success probability at least \(\delta-o(1)\).

For completeness, consider explicitly a requirement secured at pivot \(i\) when its resulting scale was \(b^{(i)}\). Every later update multiplies the scale by one of the finite tuples used to construct \(\mathcal D_i\). On any ensuing path, their complete product is some \(d\in\mathcal D_i\). Requirement (120) at pivot \(i\) was required for this very \(d\), at the original earlier raw variables and the original raw pivot. Later steps change none of those raw variables. Therefore at the final scale \(b^{\rm fin}=b^{(i)}d\) its leading index is exactly \(b^{\rm fin}_B\), and every offset ratio is exactly \(h_A b^{\rm fin}_A/(h_B b^{\rm fin}_B)\). This verifies preservation between pivots in addition to the within-pivot preservation proved in (109)–(110).

Every successful final path thus gives \(b^{\rm fin}\in\mathcal B\) such that, simultaneously for every block \(B\), color \(c\), and \(D\subset\mathcal E_B\), \[S_{B,b^{\rm fin}_B,c}(t_B)>2\tau \quad\Longrightarrow\quad S_{B,b^{\rm fin}_B,c}\left(t_B+ \sum_{A\in D}\frac{h_A b^{\rm fin}_A} {h_B b^{\rm fin}_B}t_A\right)>\tau.\] For blocks with no adding pairs the sole comparison is the identity, which holds automatically. Taking the ordinary lower limit as \(w\to\infty\) gives probability at least \(\delta\). The lists and \(\delta\) depend only on \(n,r,s\), whereas the rate of convergence may depend on the fixed threshold and model family. These are exactly the quantifiers of Principle 6. ◻