Monochromatic finite sums and products in the positive integers

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 (2.2). 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 1.1, 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 2.3 and n=n(m,r) from Lemma 2.5. Apply Principle 2.4 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

Now apply Principle 2.3. Define \mathsf G_w to be the event that none of its calibration failures occurs. The union bound and (2.7) give

Since an ordinary lower limit bounds every ultrafilter limit, (2.13) yields

On this intersection choose a witnessing b\in\mathcal B and apply Lemma 2.5 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 (2.12) therefore gives

There are q=2^m-m-1 such factors. Thus at least one of the K nonnegative model integrands in (2.11) is greater than \tau^q\ge\tau^{2^m} on \mathsf A_w\cap\mathsf G_w. Summing expectations and using (2.15) shows that their sum has ultrafilter limit at least \delta\tau^{2^m}/2. It follows from (2.11) and (2.14) that

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

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

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

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 (1.1); the first element exceeds R since a_1\ge X_{i_1}\to\infty. The positive-count set in (2.16) 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 1.1.□

Proof of Corollary 1.2. Apply Theorem 1.1 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 \N, refining the given color by the residue modulo q. Apply Theorem 1.1 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 \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 (3.1) 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 (3.1). 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 (3.2). 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 (3.3) 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

Each main term in (3.1) 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 (3.4). Its mass assertion follows from (3.2) 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 (3.4) with k=\sigma, uniformly in this range, and average. The comparison measure is exactly

proving (3.5). 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 \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, Equation (1.1)], 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 (3.6) 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 3.3 proves (3.8), 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

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 (3.9).

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.

Local kernel counts. Expand each weight with its independent divisor draw. Replacing the base residue law by exact uniform measure modulo K costs at most 2\epsilon_{\rm base}\prod_u\sigma_u \le2V^q\epsilon_{\rm base}. The CRT theorem now factors the normalized divisibility count as

where a_u=v_p(\sigma_u) and A=\max_u a_u. Primes with all a_u=0 contribute one. Homogeneity makes the divisibility conditions the kernel of a homomorphism into \prod_u\Z/p^{a_u}\Z. Its image has size at most p^{\sum a_u}, so \alpha_p\ge1. With a single positive valuation, the corresponding primitive row is surjective modulo every power of p, giving \alpha_p=1. In general one primitive row with valuation A bounds the kernel probability by p^{-A}. If the two rows having the largest valuations A\ge B>0 are independent modulo p, one of their two-column minors is a unit. Fixing the other coordinates, the two selected coordinates are uniformly and bijectively mapped to the pair of row values modulo p^A. Their required divisibilities consequently have probability p^{-A-B}. Additional rows can only reduce the probability.

We bound the nonnegative excess uniformly, so that an arbitrary prime-only domain can subsequently be retained. Let \mathcal T_p be the test that at least one of the prescribed polynomial values vanishes modulo p. Set \beta_p=0 when fewer than two valuations are positive; otherwise set

On \mathcal G, the preceding kernel bounds give 0\le\alpha_p-1\le\beta_p. For every choice of the parameters and the true, bounded divisor draws,

In particular, we can first replace the prime residues by independent CRT-uniform units at cost at most 2V^q\epsilon_{\rm CRT}. This replacement is made while all divisor draws are still bounded.

Independent comparison laws. We next compare those draws with laws whose prime valuations are independent. For a raw factor at cutoff X_j, put \epsilon_j=1/\log X_j and define

Its denominator is (1+o(1))\vartheta_W\log X_j. Indeed \epsilon\zeta(1+\epsilon)\to1, and

Lemma 3.1 gives the same asymptotic normalization for the original harmonic law on [X_j,X_j^2), and n^{\epsilon_j}\le e^2 on that interval. Hence \mu_j(n)\le C_0\widetilde\mu_j(n) pointwise, with an absolute constant C_0 for sufficiently large w. For at most bq independent raw draws this incurs one overall constant C_0^{bq}, rather than a separate constant for each prime.

The Euler product defining \widetilde\mu_j shows that its valuations at different primes are independent, with

Thus a product of at most b raw factors has valuation mass at a at most (a+1)^b p^{-a}. This bound is uniform in the individual cutoffs. Divisors remain independent of each other and of all prime parameters.

Summing the local excesses. For sufficiently large w, none of the fixed nonzero test polynomials is identically zero modulo any prime p>w. The elementary polynomial root bound on the product grid (\mathbb F_p^\times)^k gives \PP(\mathcal T_p)=O(1/p). Under the CRT-uniform law the tests at different primes are independent. Combining the valuation bounds with the definition of \beta_p, the contribution when \mathcal T_p fails is at most a constant times

To see this sum explicitly, choose the rows with the two largest valuations; all other valuations lie between zero and B, and summing their polynomial factors is bounded by a fixed power of B+1. The factors p^{-\sum a_u} cancel the normalizing power in \beta_p, leaving p^{-A-B}. On \mathcal T_p the same argument, with all lesser valuations bounded by A, gives

Its additional O(1/p) test probability again gives \E\beta_p\ll p^{-2}. The two displayed series bounds follow by factoring out respectively p^{-2} and p^{-1} and bounding the remaining geometrically convergent series using p\ge2.

We have now obtained independence across primes for every quantity in the nonnegative right side of (3.12). Consequently its expectation, after the single global divisor-domination constant, is at most

The original local product has lower bound one on \mathcal E. Integrating that lower bound and the estimated excess, and restoring the two residue-approximation errors, proves (3.11). Rational denominators which are units modulo the divisors induce the same homomorphisms after inversion, so the proof covers the stated rational-row extension.□

Proof. The dilation assertion is Lemma 3.1, 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 (4.3) gives

Periodic counting for the moduli p and pq, including the W-unit restriction, gives their reciprocal probabilities with uniformly negligible relative errors. Therefore

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 (4.4), the weight bounds, and a distinguished copy of g_*.

A substitution fixing the mask. Consider the mask with support U. If the distinguished support has an index u\notin U, use the change D(p)z given by z_u\mapsto pz_u. Otherwise choose distinct u,v in the distinguished support, necessarily both in U, and use

The first case has \eta_p=1. Lemma 4.1 justifies the first case directly. For the second, first insert p in coordinate v and then use (4.3) in coordinate u with the function read at z_u/p. Thus division by p is used only on the support of the displayed indicator, and its multiplier is retained. At this stage all unweighted functions and all products of row weights are bounded by a fixed power of V; even the intermediate fixed-prime multipliers are absorbed by the stronger endpoint error in (4.3). Both substitutions therefore have o(1) error. In either case z_U is unchanged.

Call a row invariant at this step when the coefficient vector of \ell_R(D(p)z) is a scalar multiple of that of \ell_R(z). The scalar is 1,p, or p^{-1}. All divisors in \nu_{a(R)} are smaller than the pool primes, so multiplication or admissible division by p preserves their divisibility conditions. Consequently

on the support in use. Put \Omega(z)=\prod_{R\text{ invariant}}W_R(\ell_R(z)). Divide each invariant function inside the fresh prime average by its corresponding weight, and call the remaining integrand H_p. Let \boldsymbol r denote all prime slots introduced at earlier steps. The current mask and row functions may depend on \boldsymbol r. All these previous slots are integrated together with z in the outer expectation. Thus the substituted correlation is

Here the new slot p is independent of (\boldsymbol r,z). Weighted Cauchy–Schwarz on the joint outer probability space gives

The old tuple \boldsymbol r is shared by the two branches, while p and q are fresh independent slots. This inequality applies to the full prime average.

If there are s_{\rm inv} invariant rows, the first factor equals 2^{s_{\rm inv}}+o(1), and hence is bounded in terms of the number of rows alone. Indeed, expand \Omega as a sum of products of \nu’s and apply Proposition 3.6 with \boldsymbol r among its prime parameters. Every anchor coefficient is a monomial of pool primes, hence a unit at every prime w<\pi\le V. A nonzero minor of two templates is multiplied in the actual rows by smooth row and column factors, also units at such \pi, so its possible vanishing is covered by a fixed polynomial test. This bound uses the average over \boldsymbol r.

The two prime branches. In the one-coordinate case the two branches have multipliers p and q on u. In the two-coordinate case, first discard p=q. Since all remaining factors are bounded by V^{O_m(1)}, its contribution is at most

For p\ne q, the multiplier is pq\1_{pq\mid z_u}. Absorb it by (4.3), replacing z_u by pqz_u. The two branch multipliers on (u,v) are then

Every surviving multiplicative mask is a bounded function of a scalar multiple of its original monomial; its two copies can be combined into one mask on that same support.

For an invariant row the two scalar-changed arguments are \alpha_p\ell_R(z) and \alpha_q\ell_R(z), after absorption if necessary. Its new function is

on the arguments actually used, and it is bounded by W_R(t). This formula records why only one weight survives. It is extended elsewhere with the same bound. In particular, in the two-coordinate case an invariant singleton on u acquires arguments qt and pt; the weight originally evaluated at pqt equals W_R(t). Every noninvariant row instead gives two functions, each with its original weight bound, on the two substituted rows.

Preserving the row structure. The new templates are pairwise nonparallel. For descendants of different old rows, a purported symbolic parallelity specializes at p=q=1 to parallelity of the old rows, a contradiction. For the two descendants of one row, use the exponent description of the substitution: a column is multiplied by p^{e_k}, with e_k\in\{0,1\} or \{-1,0,1\}. The p and q branches are parallel precisely when all e_k on the row support are equal. That is exactly the definition of an invariant row. The common column multiplication by pq in (4.8) is invertible over the rational function field and does not affect this argument. The distinguished row is noninvariant: it contains both an affected and a differently affected coordinate. Track a fixed one of its two copies. Its function is still g_* and its support is unchanged.

The new slots occur in at most one column per row, as is explicit in (4.8). Thus the template invariant persists. Repeated prime entries and exact polynomial coincidences have probability smaller than every fixed negative power of V. After absorption the integrands are bounded by V^{O_m(1)}, so these events can be deleted or reinstated at o(1) cost. The reinstated substitution-form terms on p=q have this same harmless bound; the larger, pre-absorption diagonal was estimated separately above. Consequently fresh slots can always be sampled independently. At the next step the entire existing tuple, including this step’s p and q, is the shared outer tuple \boldsymbol r. The deletion and reinstatement argument restores its independent product law before that next step. Any temporarily retained prime-only domain is covered by the restricted form of Proposition 3.6; the base variables and the independently expanded divisor draws retain their original laws.

Each step removes one mask, uses one square, and at most doubles the number of rows. There are q_{\mathrm{mask}} masks and initially q_{\mathrm{mask}} rows, giving at most K_m rows. Iterating (4.7), with its bounded prefactors and uniform errors, proves (4.5). Fixing the order of masks, the choices of coordinates, and the tracked branch makes every template depend only on m and J_*. The original and intermediate correlations are bounded by constants, by the same linear-forms estimates, so raising the inequalities to these fixed powers preserves o(1) errors.□

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

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\PP(E)+o(1), where t is the number of weights, by expansion and the restricted-domain assertion of Proposition 3.6. 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 (4.10) can be written

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 (4.11). More generally,

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 (4.12) 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

The translations are in W\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 3.1 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

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 3.6 in (4.14). A current copy of a base row I has the form

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 4.3. 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 (4.14) 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

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

after identifying the nondistinguished rows with [d]. There are t=d2^{d-1} weight factors. Iteration has proved the left side of (4.13) is bounded by a constant times |\E G\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\Z. The sampling error is again o(1), and the target G is unchanged because \ell_*(v_0)=0. Put

Then |G|\le B. The same row check gives

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 (4.16).

Consequently

Cauchy–Schwarz with density B now yields

This is the required weighted replacement. Combining it with the d inequalities (4.14) proves (4.13).□

Proof. Only the pushforward of the cube root remains after Lemmas 4.2 and 4.4. For fixed primes and shifts, write

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

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 3.1 gives

for every fixed C. More explicitly, the total-mass error is at most

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 (4.20). 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 3.1 makes z_j uniform modulo k; its relative error is at most

for every fixed C. The same holds for h=bz_j+h_0 modulo k, because (b,k)=1. Averaging the density in (4.19) therefore gives

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

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 (4.13) and then (4.5) gives (4.17) 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 3.3. 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

The root row is y. For a fixed nonempty \omega, choose integers a_0,a_1,\ldots,a_{d_q} such that

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,\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

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 3.6. 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 3.1. Fixed direction constants do not affect these separations.

Eliminate the nonroot rows one at a time by the weighted Cauchy–Schwarz procedure of Lemma 4.4. 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 3.6 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 (5.3). When b=0 this is simply \E\nu=1+o(1).

For the moment bound, Jensen’s inequality and independent replicas give

Include the root weight 1+\nu(y) and expand. These rows are already distinct without the additional translations. Proposition 3.6 gives 2^{1+b(2^d-1)}+o(1), proving (5.4).

Let \mathcal D^{[K]}=\max(-K,\min(K,\mathcal D)). For integers b>p,

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 (5.3) 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 5.1. 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]^{\supp\mu_i}. Finite-dimensional minimax for the bilinear pairing gives

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 5.1, 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\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 (5.5) for every fixed J_0. This diagonal choice is made together with the fixed-template diagonal in Lemma 3.3; 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 7.2, proved independently in Section 7, supplies a compact set of nilmanifold cubes containing every linear orbit cube

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

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 (5.2).

We record the weighted boundary estimate needed for the discarded roots. If E_l is the union of these boundary strips, then

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\1_{\sigma\mid y}\mu_i, uniformly in \sigma. Average over \sigma and also take \sigma=1 to obtain (5.7).

The recipe approximation costs O(\delta) against |\rho-F|\mu_i\leq(1+\nu_B)\mu_i. For fixed recipes and J_0, Proposition 5.2 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 (5.7). 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 (5.6).□

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 \eps^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 \eps. 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

and the analogous identity for T'. Thus clipping also decreases the distance to PF, proving (5.9). 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 \eps^2. Equation (5.8) bounds the distance between the two projections by \eps. As S belongs to the coarse and hence the fine space, (5.10) follows by the triangle inequality. The case l=k_u follows directly from (5.9).□

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.

From short increments to subgroup norms. Discard the two partial R_l intervals at the ends of [X_i,X_i^2); their \mu_i mass is o(1). Within every remaining interval and every residue modulo M, harmonic measure differs from conditional uniform measure by relative o(1), uniformly in the cell, because R_l/X_i=o(1). Only residues coprime to W occur, and translation by M preserves them.

Write q_l=R_l/M. Each cell has exactly q_l points. Give it its original probability, normalized after discarding the partial cells, and identify its progression indices with G_l=\Z/q_l\Z. Their disjoint union, with these probabilities and conditional Haar measure in each cell, is a finite probability system \mathcal X_l. Translation by 1\in G_l means translation by M with wrapping inside that cell. Replacing the original cube by this periodized cube costs O_d(J_0^{-1})+o(1): every vertex moves by at most dR_l/J_0 from its root, so only roots within that distance of an interval endpoint can wrap. All functions in this argument are bounded by 1.

For a prime tuple put

The divisibility provisions in Lemma 3.3 give q_{\boldsymbol p}\mid q_l. Its subgroup has order q_l/q_{\boldsymbol p}, while L_{\boldsymbol p}=\floor{q_l/(J_0q_{\boldsymbol p})} tends to infinity uniformly over the pool. The pushforward of a uniform u\in[0,L_{\boldsymbol p}) to q_{\boldsymbol p}u\in Q_{\boldsymbol p} has density at most 2J_0 relative to subgroup Haar measure. The difference of two independent such variables has no larger density. Thus the d independent increments of the periodized cube have joint density at most (2J_0)^d on Q_{\boldsymbol p}^d.

For completeness, this density can be removed with a bounded number of Cauchy–Schwarz steps. Regard it as one factor, independent of the root, on Haar variables (x,v_1,\ldots,v_d), where x\in\mathcal X_l and v_j\in Q_{\boldsymbol p}. Insert one uniform subgroup direction translating the root alone; the density is invariant in that direction. For each nonempty \omega\subset[d], choose j\in\omega and insert a direction

which leaves that vertex fixed. These invariant changes of Haar variables commute. Successive Cauchy–Schwarz inequalities outside the assigned direction discard first the density and then each nonroot vertex function, including its copies from earlier steps. The root has response a in every direction. After k=1+(2^d-1)=2^d steps its retained copies are a Haar subgroup k-cube. Taking the 2^k-th root proves

The argument acts inside each cell and integrates its probability throughout; its constant is independent of the number of cells.

Combining the subgroups. For independent tuples, Lemma 3.5, also after the good-tuple restrictions, gives \gcd(q_{\boldsymbol p},q_{\boldsymbol p'})=1 with probability 1-o(1). On this event Bézout’s identity implies Q_{\boldsymbol p}+Q_{\boldsymbol p'}=G_l. Consequently

Together with (5.11) and (5.13), this says: for every \gamma>0 a bound on the global U^t norm, t=2k-1, sufficiently small in terms of d,J_0,\gamma makes the periodized cube mean at most \gamma. This assertion is uniform in all finite system data.

From the global norm to a nilsequence projection. We next show that a fixed positive global norm forces a fixed positive fine projection. We first justify use of the interval inverse theorem on one cyclic cell, without requiring the resulting nilsequence to be periodic. Let v be 1-bounded on \Z/q\Z and extend it periodically to [0,Kq). For L=Kq define the integer cube domain

The 2^t-th power of the interval norm is the mean of the periodic cube product over A_L. Averaging a parameter translate r\in\{0,\ldots,q-1\}^{t+1} gives exactly the cyclic cube mean at every parameter point. The discrepancy is therefore bounded by \E_r|(A_L+r)\mathbin\triangle A_L|/|A_L|. For a vertex \omega, its displacement is r_0+\sum_{j\in\omega}r_j\leq(t+1)(q-1). The boundary discrepancy at that vertex has size at most this quantity times 2L^t: once that vertex and its t neighboring vertices are specified the integer cube is determined, and each neighbor has at most L choices on the relevant side of the symmetric difference. Summing over vertices bounds the numerator by O_t(qL^t). Nonnegative increments and root with total less than L already give |A_L|\geq\binom{L+t}{t+1}\geq L^{t+1}/(t+1)!. Thus, uniformly in q,K,v,

This is an estimate for the powers of the norms.

If the cyclic norm is at least \delta, choose a sufficiently large fixed K=K(t,\delta) in (5.14). The interval norm is then at least \delta/2. The inverse theorem gives a uniformly bounded complexity correlator on those K periods. Splitting the correlation into its K period averages shows that one period has correlation of at least the same modulus. Translating that period back to [0,q) only replaces the nilsequence base point by a translate g^{jq}x. It leaves its manifold, step and observable bounds unchanged. We have proved a cyclic-to-interval correlation statement on fixed representatives of \Z/q\Z; the correlator itself need not have period q.

Now write the cell weights of \mathcal X_l as \alpha_C. The exact identity

shows that, if the left-hand norm is at least \delta, cells with \|h_C\|_{U^t(G_l)}\geq\delta/2 have total probability at least \delta^{2^t}/2. On each such cell choose its inverse-theorem correlator, rotate its complex phase and take the real part so that its pairing with the real function h_C is positive. Put zero on the remaining cells. Every chosen sequence comes from the same finite nilmanifold list and has the same uniform observable bounds. They form one permitted real piecewise nilsequence family V at gap l, with |V|\leq1. Translating from conditional uniform measure back to \mu_i costs o(1). Thus along any set of w on which the global norm is at least \delta,

The harmless smaller inverse threshold allows slack in all these inequalities. The step is at most t-1=2(2^d)-2\leq s, so V represents an element of \mathcal N_{i,l}. If the stated norm bound holds on a set belonging to \mathcal U, select the correlators there and set the family to zero on its complement. Orthogonality and \|V\|_2\leq1 give the positive lower bound

Choose \delta small enough that Bessel and (5.13) give the prescribed \gamma, and choose \kappa smaller than the displayed positive projection bound. If the projection is less than \kappa, the global norm cannot exceed that threshold along \mathcal U. Restoring the periodization error proves (5.12).□

Proof of Principle 2.3. Fix the data n,r,\chi,\mathcal B,\mathcal A,\tau,\eta of the Principle, with s=s(m) as in (5.1). 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

for every exponent and constant in Proposition 4.5. 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 5.5 is smaller than \zeta. For the finitely many correlation-test types, apply that Lemma with \gamma=\zeta, and choose

These choices depend on the requested tolerances and on m, and are independent of N. Now choose N by Lemma 5.4, 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 3.3 give all conditions in Definition 2.1; the models S have exactly the pieces required by Definition 2.2.

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 3.2 gives

The function \psi(S) is a bounded-complexity coarse piecewise nilsequence. Lemma 5.3 replaces \rho by F. In the Hilbert limit it also lies in the coarse subspace, so, writing P=P_{i,k_u} and using (5.9),

This is the first assertion of the Principle, since \eps<\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 5.2, 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 (5.10) gives \|P_{i_d,l}h\|_2\leq2\eps. All other factors, including the unbounded center weights, still satisfy the bounds of Proposition 4.5. The parameter choices and Lemma 5.5 bound its target cube mean by 2\zeta in the ultrafilter limit. Equation 4.17 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

All factors except the center weights are now bounded by 1. The blocks of the chain are disjoint. Corollary 3.2 thus replaces this last expectation, with error o(1), by

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 2.3. In particular the auxiliary \eps 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 (6.2) 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 6.1 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

This proves (6.3). 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

There is a positive integer D_H, depending only on the chosen nodes, such that D_HR\in\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,

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

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], (6.5) and (6.7) place R(z) within (C_q+1)\epsilon of an integer. Since R(z)\in D_H^{-1}\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 (6.7) by t_z^e\geq S^e. This yields (6.6) 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

on which every index has a discrepancy at least \eta as in (6.10). 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 (6.11) 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 6.1, with the fixed stretched filtration (6.4), 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

The coefficients \theta_I are polynomials of bounded degree in z. The coefficient obstruction in the v variables is

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

with smooth coefficient denominators and a constant A_1 independent of H. Take H larger than the finitely many degree requirements of Lemma 6.2; 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 (6.13) contribute integers. Their error contribution has size at most

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)}_{\R/\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:

Lemma 6.2 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,

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 (6.14) after the same affine reparametrization.

Set

For integer z in the progression, m_z has integer coefficients as a polynomial in x. By (6.14), its error satisfies, throughout [-CZ,CZ]^d,

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

Then \xi(Q_z(x))=0 identically, for real (z,x), and

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 (6.17) 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 (6.18) by \ell_{z,*}=\exp(a_z(x_*)V). The error in the test is at most C_2\lambda: the derivative estimate (6.16) 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

The resulting test on the smaller group is

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 (6.19) 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

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\Z^d with b\equiv a\pmod q. Under x=a+qy, its condition is exactly

This identity does not use invertibility of either factor. Apply Proposition 6.3 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

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 (7.2). 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 (7.2). 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 jth 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 (7.5) fails, fix one of its characters \xi and pass to a subsequence on which

Choose a fixed rational V\in\mathfrak h_j with \xi(\exp V)=1. Replace the polynomial R_L by

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

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 (7.5) 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

takes values in C. To see this and record the needed coefficient information, expand

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 \ellth 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 (7.3): 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 6.1, applied to the fixed rational nilmanifold and the filtration (7.3), 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\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 (7.5).

Applying \chi to (7.10) gives the scalar polynomial

All terms with \ell>j vanish by maximality. In its degree-j part, the coefficient of

Indeed the top part of the binomial polynomial is the jth 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 (7.11).

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 6.1, clearing the fixed binomial-to-monomial denominators if necessary, would therefore give

for a fixed nonzero rational number a. This contradicts (7.5) 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 (7.9), 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 7.3 therefore gives

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 (7.9), 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 (7.8). This integral is well defined because the Haar images depend continuously on \beta. No further subsequence depending on q was used: (7.5) 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 (7.1), 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

The ambient automorphism

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

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:

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

Lemma 7.1 puts this element in J_{i+1}. Differentiating at t=0 proves (7.16). In particular \Phi preserves every filtration level.

Choose filtration-adapted coordinates on \mathfrak j. The polynomial

has weighted degree strictly less than the output weight in each coordinate. For its linear part this follows directly from (7.16): 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

This is a shear of the form (7.12). 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 (7.14). Finally a map of (7.12) is inverted successively in increasing weight, so it is a polynomial bijection of the cover.□

Proof. Composition preserves (7.12), 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 (7.20), 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

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 (7.20) 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 (7.20) 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 (8.2) gives the exact identity

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,

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 (8.6) give

These are exactly the requirements after the update. Thus (8.7)–(8.8) 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.

An exposure retaining polynomial dependence. We prove the assertion in probability by changing the order of exposure. We must freeze the target’s model formula while leaving its tail variables available for progression comparison. A fixed pivot microcell alone does not do this as the tail varies. Instead we condition on a short bin for tp; the eventual microcell origin will enter the base coordinate x_0, not a coefficient of the polynomial family. Expose the earlier variables outside T, the residues modulo L=MW^w of those inside T, dyadic bins [S_\ell,2S_\ell) for \ell\in T, the pivot residue r_0\bmod M, and a bin for tp of width t_{\rm lo}R, where t_{\rm lo}=\prod_{\ell\in T}S_\ell. Do not expose the exact pivot microcell at this stage. Then t_{\rm lo}\le t\le2^{|T|}t_{\rm lo}. Every r_e is fixed: earlier residues determine q_e\bmod L and t_C\bmod L. For own pairs e=(B,A) the full integer q_e is fixed, because A\cap T=\varnothing. Hence all slot shifts s(v) are fixed. So are

We may discard exposed bins whose entire preimage in p, for every t in the indicated product of dyadics, is not inside [X_i+2R,X_i^2-2R]. Such a discard forces the actual p into a fixed multiplicative neighborhood of a cutoff endpoint, up to an O(R) enlargement. Its harmonic probability is O_n(1/\log X_i)+o(1). Also discard tp bins within a sufficiently large fixed multiple of t_{\rm lo}R of an H_i boundary. The boundary count preceding the Lemma gives probability o(1) for this discard. The constant is chosen to cover the given bounded-multiple boxes, all slots, and all required bounded array indices. On the remaining bins the interval and residue selecting every formula for this target are fixed throughout the exposure. Thus all its parameters F,g,x are fixed as the variables in T vary.

For later use, the conditional law of those variables is dominated by a constant, depending only on n, times the product of uniform laws on their dyadic bins and prescribed L residues. Before conditioning on the tp bin, their harmonic densities in each dyadic differ from uniform by at most a factor 2 per variable. If the exposed bin is [Q,Q+t_{\rm lo}R), its preimage at a given t has length t_{\rm lo}R/t, between 2^{-|T|}R and R. It lies fully inside the pivot cutoff. On the residue r_0\bmod M the unnormalized pivot mass there is

uniformly in t. The discretization error is relatively O(M/R), and the common normalization cancels. In particular conditioning on this bin changes the preceding product comparison by a bounded factor. This is the required domination; no exact microcell has been fixed in deriving it.

Putting the tail product in the sampling step. Let p_* be the first point of r_0\bmod M in the actual microcell eventually selected. Put Y_*=\lfloor Q/M\rfloor and change the base and increment coordinates by

The unknown cell origin has entered x_0 rather than a polynomial coefficient. Since p_* is within R of the actual pivot, these coordinates lie in boxes in a fixed multiple of

with all sides comparable to Z. The original residue restrictions on \ell_0,\ldots,\ell_a become one residue vector modulo dt in x. Changing an endpoint by less than dt has vanishing relative counting cost, so these are precisely the permitted normalized box averages.

At vertex \omega\in\{0,1\}^a, set X=x_0+\sum_{j=1}^a\omega_jx_j. Its array exponent is

Every parameter here except X and t was fixed by the exposure. The displayed exponent is a polynomial of degree at most two, with its only possible quadratic terms of the form tu_e. BCH then shows that the logarithms of the arrays, viewed in the fixed array group, are ordinary polynomials jointly in x and the variables t_\ell, \ell\in T, of bounded degree. There is no bound required on their coefficients. This joint polynomiality is the hypothesis that permits Proposition 6.3.

Removing each factor of the step. We compare the sampling in a class modulo dt with the sampling in the same class modulo d, removing the factors of t in a fixed order. Consider removal of t_\ell, and condition on all other factor values. Let w' be the product of the factors still in the step after this removal. The current step is dw't_\ell. There is a representative a_0 for its base coordinate modulo dw' which can be chosen independently of t_\ell, after allowing all the finitely many residues modulo d. Indeed the base congruence modulo w' is

and the increment congruences are x_j\equiv0\pmod {w'}. Since w' is a product of W-units, (M,w')=1, and also (d,w')=1 for large w. The Chinese remainder theorem therefore gives representatives a_j\in[0,dw') determined by these congruences and the chosen d residues, independently of t_\ell.

Substitute x_j=a_j+dw'y_j. The current restriction becomes a single class vector modulo t_\ell in y, while removing it gives unrestricted integer y. No coprimality of w' and t_\ell is used: both original classes are nested congruences, and division of x_j-a_j by the integer dw' is exact. The new common box scale is Z/(dw'). It dominates every fixed power of S_\ell, uniformly over the other factor values, by the choice of R. The substitution has coefficients independent of t_\ell, so joint log-polynomiality in (y,t_\ell) is retained. The variable t_\ell lies in a fixed residue modulo L=MW^w in [S_\ell,2S_\ell), with

All hypotheses of Proposition 6.3 hold.

That Proposition bounds the exceptional proportion uniformly in the polynomial coefficients, all permitted boxes, class vectors, and tests. Its uniformity in endpoints is essential: the exact microcell and hence the endpoints may depend on the pivot sampled after the exposure. For each choice of other factors the exceptional proportion in t_\ell is o(1) with a uniform bound. Integrate first against the independent uniform dyadic/residue laws and then use the conditional domination already proved. Each removal therefore fails with probability o(1). A union bound over the finitely many factors and the triangle inequality give agreement, to error tending to zero in probability, between the original marginal cube law and the law with only its d-residue restrictions. The same reasoning applies to a test composed with the fixed face automorphism; its Lipschitz bound is still fixed.

In this latter law, applying \mathcal S_e on the upper face \omega_j=1 replaces x_j by x_j+\Delta_e in (8.16), at all slots and for all observables of this component simultaneously. By (8.10), d\mid\Delta_e eventually. Moreover |\Delta_e|/Z\to0 uniformly, since its size is bounded by a fixed power of M+D_i^- whereas R dominates all those powers. Translation of the corresponding box changes normalized counting averages by O(|\Delta_e|/Z+d/Z)=o(1). Comparing on both sides establishes the asserted face invariance. Taking the finite union over targets and own pairs is harmless. This argument proves marginal invariance; it makes no assertion about the other components.

The countable diagonal and the period. Enumerate the conditions consisting of a positive integer modulus, a cube dimension at most s+1, bounds C\in\N, lower side bounds c=1/j, Lipschitz bounds in \N, and tolerances 1/j. The assertion just proved holds uniformly over the uncountably many endpoints and tests within each such condition. Each of the first finitely many conditions fails with probability tending to zero. Given events of probability at least \epsilon>0 along a subsequence, choose an increasing number of initial conditions whose union of failures has probability less than \epsilon/2, and select one conditioning from the event outside that union. By making the initial list and the accuracy increase successively, we obtain a sequence satisfying every fixed condition with error tending to zero.

Now fix the finite choices of ambient groups along a subsequence and apply Proposition 7.4. Its rational filtration, period d_0, and rational representatives may depend on this sequence. The face invariances have already been secured for every fixed positive integer modulus, so they hold for d_0. For any fixed small relative box size \alpha>0, they hold for base boxes near a given macroscopic point \beta and increment boxes near zero, with the residue restrictions specified in that Proposition. First take the sequence limit and then let \alpha\downarrow0. The projected local cube Haar measure is therefore invariant under the face automorphism. Since it has full support on its compact cube coset and the automorphism is a homeomorphism, that coset is preserved. This proves the last assertion with the required order of limits.□

Proof. Suppose the uniform assertion fails on sets of conditionings of probability bounded below along a subsequence. By Lemma 8.2, choose a sequence in those sets with all its empirical face invariances. Pass to a subsequence fixing the ambient array nilmanifolds and using Proposition 7.4. 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

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,

with its Haar image measure \mu_{\beta,r}. The preceding Lemma and Proposition 7.5 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 7.7 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

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

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

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 (8.21) gives exactly (8.18) 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

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 (8.22), including at equality with the threshold.

Sample z from the expanding weighted boxes of Lemma 7.7. 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

Using the closed-set direction of Portmanteau separately for the finitely many summands yields

The last implication also follows by summing the indicator inequality \sum_o\1_{F_o}\le Q_i\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 (8.23) integrates over the macroscopic-position/residue mixture without a measurable choice of lifts.

Replace each comparison (8.22) by the open comparison

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 7.4 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 (8.24); 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 (8.21) 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 (8.12) 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 2.4. For the given n,r,s take the lists and all budgets from Lemma 8.1. Let I be the set of pivots having targets and define

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 2.2, 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 8.3, integrated over the pivot microcell and residue, implies

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 (8.25) therefore changes each summand by o(1) when p+s(v) is replaced by p. There are at most V_i slots, and consequently

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 (8.28) 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 (8.18) 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 (8.7)–(8.8).

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,

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 2.4.□