Standard systems and the pointwise inputs

Put \(I=\{1,2,3,4\}\). It suffices to prove Theorem 1 on a standard probability space. Indeed, for a given tuple of bounded functions, the map recording all their integer translates takes values in a countable product of compact scalar discs. The pushforward measure and the shift give a standard factor, the functions descend to its coordinate functions, and mixing passes to this factor. Almost-sure convergence there pulls back to the original space. We henceforth write \((X,\mu,T)\) for this standard mixing system. All factors and conditional expectations are understood modulo null sets; probability spaces may be completed.

We use [22] in the following form. For every fixed finite family \(g_1,\ldots,g_s\in L^\infty(\mu)\), \[ \int_X\prod_{j=1}^s g_j(T^{n_j}x)\,\,d\mu(x) \longrightarrow \prod_{j=1}^s\int_Xg_j\,\,d\mu \quad\text{if}\quad \min_{j\ne l}|n_j-n_l|\longrightarrow\infty. \tag{2}\] The bounded-function formulation follows from the set formulation by simple-function approximation. All uses of this input involve fixed functions; no rate of mixing is assumed.

The other companion input will be used on systems without mixing. Its coefficient dependence is important, so we isolate its exact scope.

The proof, including the changes of coefficients in the analytic estimates of [21, 20], is given in Appendix 7. Neither ergodicity nor mixing is an assumption of Lemma 2.

Distal factors and relatively weakly mixing extensions

Let \((Y,m,R)\) be a standard probability-preserving system, and let \(B\) be a factor. We use the conditional-correlation characterization of relative weak mixing: for bounded \(f,g\), with \(\mathbb E[f\mid B]=0\), \[ \frac1H\sum_{h=1}^H \bigl\|\mathbb E[\overline f\,(g\circ R^h)\mid B]\bigr\|_2^2 \longrightarrow0. \tag{3}\] For \(F_1,\ldots,F_r\in L^2(m)\), the associated finitely generated \(L^\infty(B)\)-module is \[\left\{\sum_{j=1}^r a_j F_j:a_j\in L^\infty(B)\right\}.\] It is invariant if composition with \(R\) and with \(R^{-1}\) preserves this space. An extension is relatively compact if its \(L^2\) space is generated by these invariant modules. The individual modules need not be closed in \(L^2\); closure is taken after their union is spanned. This is the compact-module characterization in its corrected form [13][14]. A relatively distal extension is a tower of relatively compact extensions and inverse limits. Distality without a specified base means distality over the trivial factor. These definitions include nonergodic systems; in particular an identity action is compact.

The following norm estimate explains why proper subaverages have a particularly simple law over a distal factor. Its proof uses relative weak mixing alone.

In particular, weak mixing of \(T\) gives \(L^2\) convergence to the product of the means for all finite distinct nonzero linear powers on \(X\). In a flow, with \(B\) preserved by every time, the analogue of (5) for deterministic continuous-time averages follows by writing \(t=n+s\), \(0\le s<1\). Apply the lemma for each fixed \(s\) to \(g_i\circ S_Y^{a_i s}\), and integrate the resulting \(L^2\) estimate in \(s\). Conditional expectation commutes with these time shifts. Bounded convergence and the final interval of length less than one complete the passage to continuous time.

From failure to a sequence of selectors

Let \(C=X\times[0,1)\) have measure \(m_C=\mu\otimes\,du\) and the roof-one suspension flow \[ S_C^t(x,u)=\bigl(T^{\lfloor u+t\rfloor}x,\{u+t\}\bigr). \tag{6}\] Write \(\xi:C\to X\) for its spatial coordinate and \(\upsilon:C\to\mathbb R/\mathbb Z\) for its phase. The use of a flow will let us normalize the action at any one of the four outputs.

For the rest of the proof assume failure and fix the functions, selectors and phase test of Proposition 5. We use one fixed free ultrafilter \(\mathcal U\) on the indices \(a\).

A functorial law on four outputs

Consider a standard probability-preserving flow \((Y,m_Y,S_Y^t)\) with an equivariant factor \(\pi_Y:Y\to C\). The selector on \(Y\) is \(N_a\circ\pi_Y\). For bounded measurable functions \(G_i\) on \(Y\), define the selected law by \[ \int_{Y^I}\prod_{i\in I}G_i(y_i)\,\,d\lambda_Y =\lim_{a\to\mathcal U} \int_Y\frac1{N_a(\pi_Yy)} \int_0^{N_a(\pi_Yy)}\prod_{i\in I}G_i(S_Y^{it}y) \,\,dt\,\,dm_Y(y). \tag{9}\] The existence of a countably additive law with this formula is part of the next proposition.

Let \(\omega_Y:Y\to W_Y\) be the maximal distal factor of time one, write \(\nu_Y=(\omega_Y)_*m_Y\), and disintegrate \(m_Y=\int m_{Y,w}\,\,d\nu_Y(w)\). For a state \((y_i)_{i\in I}\) of the selected law, put \(P_i=\omega_Y(y_i)\) and \(P=(P_i)_{i\in I}\).

For \(b\in I\) define the normalized flow \[ \mathsf M_b(Y)= \left(Y^I,\lambda_Y, (y_i)_i\longmapsto(S_Y^{it/b}y_i)_i\right). \tag{12}\] Its projection to slot \(b\) is an equivariant extension of \(Y\) and therefore of \(C\). This permits the selected-law construction to be iterated with the same selectors and the same ultrafilter.

Finally, the phase test in Proposition 5 can be approximated uniformly by finite sums of products of one-slot phase functions. Formula (9) therefore applies to its product with the four spatial tests. In particular, \[ \int_{C^I}\Psi\bigl((\upsilon(z_i))_i\bigr) \prod_i f_i(\xi(z_i))\,\,d\lambda_C\ge\epsilon_0. \tag{13}\] The phase array is part of the distal parameter array, since phase rotation is compact. Equation (13) is the nonzero centred correlation from which the argument of Section 6 will start.

Correlations and an upper bound on output roles

For \(A\subseteq I\), consider a correlation \[ \int \Phi(P)\prod_{i\in A}h_i(d_Y(y_i)) \prod_{i\notin A}h_i(x_Y(y_i))\,\,d\lambda_Y, \qquad P=(\omega_Y(y_i))_{i\in I}. \tag{30}\] Here \(\Phi\) is an arbitrary bounded measurable function of the complete distal parameter array. Each \(h_i\) is centred for its spatial law: \(\mu^{\otimes m}\) when \(i\in A\) and \(\mu\) otherwise. We call the indices in \(A\) the output roles. The functions in an output role may depend jointly on the whole output list. Lemma 11 implies that every factor in (30) is also centred conditionally on its own parameter \(P_i\).

The channel with no outputs is permitted by Definition 10. The phase test in Proposition 5 is measurable in the distal parameters of the suspension. Thus that proposition gives a nonzero correlation (30) with \(A=\varnothing\). Consequently, under the supposition of pointwise failure, there is a largest number \(r\in\{0,1,2,3,4\}\) of output roles in any nonzero correlation of this form, over all channels and all such tests.

Stabilizing conditional expectations under extension

The next extension construction prevents extra information in a later system from changing the conditional laws that we shall use. It is an energy-increment construction of the kind used for sated extensions [2]; we give the argument for the selected-law assignment needed here.

On \((Y^I,\lambda_Y)\) introduce the five sigma-algebras \[ \mathcal D_0(Y)=\sigma(P),\qquad \mathcal D_l(Y)=\sigma\bigl(P,(y_i)_{i\ne l}\bigr),\quad l\in I. \tag{31}\] If \(\rho:Y'\to Y\) is an extension over \(C\), functoriality gives the factor map \(\rho^I:(Y'^I,\lambda_{Y'})\to(Y^I,\lambda_Y)\). The pullback of each \(\mathcal D_l(Y)\) is contained in \(\mathcal D_l(Y')\), since the old distal factor is contained in the new one.

We call a system satisfying Proposition 15 saturated. A nonzero correlation persists under the passage to a saturated extension: its functions pull back, and its old parameter test remains measurable in the enlarged distal parameters.

A channel carrying the three other factors

Choose a nonzero correlation with the maximal number \(r\) of output roles, and replace its channel by a saturated extension \(Y\). Retain the notation \(A,\Phi,h_i\) for its tests. Proposition 14 gives \(r\le2\), so choose \(b\in I\setminus A\). Set \[Z=\mathsf M_b(Y),\qquad Z\longrightarrow Y,\quad (y_j)_{j\in I}\longmapsto y_b.\] Its state law is \(\lambda_Y\). We give \(Z\) a new channel structure adapted to this particular correlation. Keep the full old output list from slot \(b\), and for every \(j\ne b\) add the list \[ Q_j= \begin{cases} d_Y(y_j),&j\in A,\\ x_Y(y_j),&j\notin A, \end{cases} \tag{33}\] where in the second case the actual flow factor added is the copy of \(C\) in that slot. Index the new three groups by \(j\in I\setminus\{b\}\). Put \(Q_j\) in group \(j\) and assign the three old groups from slot \(b\) bijectively to these new groups. All speeds are positive rationals.

Write \[ q((y_j)_{j\in I})= \prod_{j\in A}h_j(d_Y(y_j)) \prod_{j\notin A\cup\{b\}}h_j(x_Y(y_j)). \tag{34}\] It is a function of the three added lists \(Q_j\) in a single \(Z\)-state. With \(y=y_b\), define \[ g(P,y)=\mathbb E_{\lambda_Y}[q\mid P,y_b],\qquad u(e,y)=\mathbb E_{m_Z}[q\mid\omega_Z,y_b],\quad e=\omega_Z. \tag{35}\] The original nonzero correlation equals \(\mathbb E[\Phi(P)h_b(x_Y(y_b))g(P,y_b)]\), so \(g\) is nonzero in \(L^2\). The two conditional expectations in (35) use the same state law, but different parameter information: \(e\) contains the old array \(P\) and may contain more.

The square and its conditional laws

The selected law \(\lambda_Z\) is a law on a square of \(Y\)-states. Write \(\mathbf z_i=(y_{ij})_{j\in I}\) for row \(i\), a \(Z\)-state, and set \[P_{ij}=\omega_Y(y_{ij}),\qquad E_i=\omega_Z(\mathbf z_i),\qquad E=(E_i)_{i\in I}.\] Each \(E_i\) contains the complete old parameter row \((P_{ij})_j\), by Lemma 7. In column \(b\) put \[y_i=y_{ib},\qquad p=(P_{ib})_{i\in I},\qquad w=P_{bb}.\] Functoriality for \(Z\to Y\) says that this column has law \(\lambda_Y\). Row \(b\) has the same law because a single \(Z\)-state has law \(\lambda_Y\). Our aim is to show that \[\mathbb E\bigl[q((y_i)_i)q(\mathbf z_b)\mid E\bigr]\not\equiv0.\] The row factor \(q(\mathbf z_b)\) is read from the added outputs, so it is the candidate for the additional output role at \(b\). To establish nonvanishing, saturation will first identify the column law given \(E\) and the conditional expectation of the row factor given that column. Figure 1 distinguishes these two uses of the old selected law.

We can now compute the correlation between the two highlighted lines. First condition on the column in Lemma 17, then condition within its old law on its corner state. The result is \[ \mathbb E\bigl[q((y_i)_i)q(\mathbf z_b)\mid E\bigr] =\int g(p,y)u(E_b,y)\,m_Y(\,dy\mid w). \tag{40}\] The right side is not manifestly positive. The next lemma supplies the parameter independence that will show it cannot vanish identically.

Nonvanishing and the additional role

We record the elementary Hilbert-space fact used to finish the argument. Its separability hypothesis is automatic for the probability fibres occurring here.

The local estimates used from H and P

For this appendix, a role set is a set \(S\) of four distinct integer coefficients. Initially \(S=\{0\}\cup J\), where \(J\subset\{1,2,3,4\}\) has three elements. If \(K\) is a dyadic interval of length \(r\), put \[ \mathcal H_{K,S}(z) =r^{-2}\int_{\mathbb R^2}w_K(x,t)\prod_{j\in S}z_j(x+jt)\,\,dx\,\,dt. \tag{43}\] Fix \(c_0>0\) and \(C_0\ge1\). The admissible kernels are the smooth, compactly supported functions \(w_K\) for which every \(x+jt\) on the support lies in \(K\) at distance at least \(c_0r\) from its boundary, \[ \int_{\mathbb R}w_K(v-jt,t)\,\,dt=0 \quad(v\in\mathbb R, j\in S), \qquad \|\partial_x^a\partial_t^b w_K\|_\infty \le r^{-a-b}\bigl(C_0(a+b+2)\bigr)^{C_0(a+b+2)}. \tag{44}\] Kernels may be chosen independently at the different intervals.

H’s Uniform local estimate [20] asserts, for \(S=\{0,1,2,3\}\) and a suitable \(q\in(2,3)\), that \[ \sum_{K\in\mathcal K}|K|\,|\mathcal H_{K,S}(z)| \le C\int_{\mathbb R}\prod_{j\in S}M_qz_j(y)\,\,dy, \qquad M_qz=(M(|z|^q))^{1/q}, \tag{45}\] for every finite family of dyadic intervals and compactly supported scalar \(L^q\) inputs. Here \(M\) is the Hardy–Littlewood maximal operator. Its proof, in H, Sections 3–10, supplies the quantitative chart, projection, separation, counting, and side estimates used in P.

The extension needed for oscillation is P’s Local estimate with variation in depth, Theorem 3.2. To state it, let \(K_*\) be a root interval and let \(s(K)=\log_2(|K_*|/|K|)\) be the depth of a descendant. For a finite integer block \(B=\{a,\ldots,b\}\) and a scalar sequence \((z_s)\), write \[\mathcal V_B(z)(y) =\max_{s\in B}|z_s(y)|+ \sum_{s=a}^{b-1}|z_{s+1}(y)-z_s(y)|.\] For the original role set \(\{0,1,2,3\}\), P proves the assertion below. We establish it for the four required role sets.

The proof of Proposition 21 is the proof-by-modification below. The analytic inputs are H’s local estimate and its proofs of the intermediate estimates, and P’s depth-local construction in Sections 4–7. Proposition 21 is the sole analytic input to the local-to-orbit argument in Subsection 7.6; Subsection 7.7 then proves Lemma 2. The intervening subsections establish the proposition by checking, in order, the coordinate and quadratic identities, the integer-lift and counting arithmetic, and their use in P’s depth scheduling. We retain their definitions of quadratic chart atoms, inherited projection families, curvature tags, and finite progression tests. The following computations specify their dependence on the four role coefficients. In particular, the proposition is not an assertion of uniform continuity of a Hilbert-transform bound in its slopes.

Coordinate changes and the quadratic cancellation vector

Every set \(S\) above contains an adjacent pair. Translate all role labels by the same integer so that one such pair is \(0,1\). This is the change of variables \(x'=x+mt\) in (43), with Jacobian one. The four spatial coordinates themselves are unchanged. Consequently the support and cancellation conditions remain the same, and the derivative bounds retain their form after increasing the fixed constant \(C_0\). During the local argument we use these translated labels. Their pairwise differences have absolute value at most four, and their diameter is either three or four. Thus \[ |t|\le |K|/3 \quad\text{whenever every }x+jt\text{ lies in }K. \tag{47}\]

The vector \((-1,3,-3,1)\) in H is replaced by a primitive integer vector \((c_j)_{j\in S}\) proportional to \[\left(\prod_{\ell\in S\setminus\{j\}}(j-\ell)\right)^{-1}.\] The overall sign is immaterial. Before translation, convenient choices are \[ \begin{array}{c|rrrr} S&\multicolumn{4}{c}{(c_j)_{j\in S}\text{ in increasing role order}}\\ \hline \{0,1,2,3\}&-1&3&-3&1\\ \{0,1,2,4\}&-3&8&-6&1\\ \{0,1,3,4\}&-1&2&-2&1\\ \{0,2,3,4\}&-1&6&-8&3 \end{array} \tag{48}\] Lagrange interpolation of the polynomials \(1,u,u^2\) gives \[ \sum_{j\in S}c_j=\sum_{j\in S}jc_j= \sum_{j\in S}j^2c_j=0. \tag{49}\] These identities are preserved by translation of the role labels. Every \(c_j\) is nonzero. All coefficients and denominators arising from this finite list of coordinate systems are bounded absolute constants.

We distinguish numerical role labels from enumerations of four objects. A coefficient such as \(j\), \(j-i\), or \(c_j\) is changed according to \(S\). An ordering of four input sizes, four leaves, or four positions in a multilinear expansion still enumerates four objects. This convention leaves those parts of H’s proofs unchanged.

Chart separation, detection, and the scale estimates

The local absolute estimate, H, Lemma 2.1, uses the map \((x,t)\mapsto(x+it,x+jt)\), whose determinant is \(j-i\). Its endpoint bounds and interpolation therefore hold for the new roles. In particular the initial bound for the finite-depth constant in (45), and H’s stopping reduction to normalized local sizes, remain valid. The coefficient and projection bounds, compression and height-bin estimates, H, Lemmas 3.2–3.5, are statements about finite families of vectors or real frequencies and do not specify a progression coefficient.

We next check the part of H, Section 4, that converts two chart atoms in distinct roles into separated factors. A chart phase has the form \[\lambda y^2+(\theta y)^TCd+\beta y,\] where \(\theta\) is a vector of real frequencies, \(C\) is the rational chart matrix, and \(d\) is a bounded lift of \(\theta y\) modulo the integer lattice. The source definition also includes a smooth amplitude on the lift. Its one-variable Weyl estimates, Fourier expansion, and uniform tail estimates are unchanged.

For H, Lemma 4.5 (Progression separation), fix source roles \(i,k\) and spectator roles \(a,b\). Set \(\alpha_i=1\), \(\alpha_k=0\), and solve \[\sum_{j\in S}\alpha_j=\sum_{j\in S}j\alpha_j=0\] for \(\alpha_a,\alpha_b\). This is possible because \(a\ne b\). Writing \(y_j=x+jt\), insert the same lift partitions as in H at \(\theta y_a\), \(\theta y_b\), and \(\theta t\). If their lifts are \(d_a,d_b,d\), then \[k_j=d_j-d_a-(j-a)d\] is an integer vector. For instance, if \(d_j=\theta y_j-m_j\) and \(d=\theta t-m\), the identity \(y_j=y_a+(j-a)t\) gives \(k_j=-m_j+m_a+(j-a)m\). Thus the equality cutoffs in H still test integer carries exactly. Their support bounds change only by a fixed factor. With \(A_2=\sum_jj^2\alpha_j\), direct expansion gives \[ \begin{split} \sum_{j\in S}\alpha_j [\lambda y_j^2+(\theta y_j)^TCd_j] &=A_2[\lambda t^2+(\theta t)^TCd]\\ &\quad+\sum_{j\in S}\alpha_j(\theta y_j)^TCk_j. \end{split} \tag{50}\] The last sum is linear in \(x,t\) and hence separates in \(y_a,y_b\). The smooth amplitude on the bounded lift box has the same Gevrey bounds up to fixed constants. Fourier expansion therefore gives H’s absolutely convergent separation into two spectator atoms and one atom in \(t\), with the same coefficient-tail exponents and quasipolynomial complexity bounds. H, Lemma 4.6 (Single-scale separation), then uses \(t=(y_b-y_a)/(b-a)\); every added rational height is bounded by an absolute constant.

For the generalized von Neumann bound, H, Lemma 4.11, let \(i\) be the role whose input is controlled by its \(U^3\) norm. Apply Cauchy–Schwarz in the directions \((-j,1)\) for the other three roles, beginning with the role containing the \(L^2\) input. Each direction holds \(x+jt\) fixed and changes \(x+it\) by \((i-j)h\). Multiplication by the nonzero integer \(i-j\) preserves Haar measure on a circle. The three independent circle parameters therefore produce the same cube integral as in the source proof. Character weights split using the marked pair \(x=y_0\), \(t=y_1-y_0\). Enlarging the fixed periodic boxes accommodates all translated role sets. This verifies the coefficient-dependent part of the generalized von Neumann bound. In H, Lemma 4.12 (Local detection), the potentially unbounded \(L^2\) residual occupies a different role. After clipping the opposing inputs, the source proof decomposes two of them using its continuous inverse theorem and bounds the uniform errors by Lemma 4.11. The separated chart moments then detect the residual. Those separations were verified above; the clipping, inverse theorem, and mixed-norm exponents are unchanged.

H, Lemma 5.1 (A scale sum for one separated term), uses the spectator coordinates \((y_a,y_b)\). Holding one fixed, the integral of the kernel over the other is zero by (44), with a constant Jacobian factor \(|b-a|\). The double primitive of that kernel consequently has the same compact support and smooth bounds. Its tensor expansion into mean-zero bumps, the Bessel estimates for those bumps, and the frequency-height bounds apply with the new pair. The only additional demodulations have frequencies \(\pm\gamma/(b-a)\); their rational multipliers have bounded height, although \(\gamma\) can be arbitrary real. This also verifies the coefficient changes in the finite predictor list and the delayed predictor replacement. The scale bands, accuracy gaps, and projection-energy bounds in the remaining base construction depend on these estimates and the four-slot multilinearity, rather than the values of the labels. There is also a multivariable use of the chart estimates in H, Lemma 5.7 (Few active scales for a fixed triple). Squaring the transpose at the remaining role \(\ell\) gives six atoms with arguments \(h+(j-\ell)t\) and \(h+(j-\ell)t'\). These are still fixed integer linear forms in three variables, so the three-variable Weyl alternative and chart expansion of H, Lemmas 4.3–4.4, apply with the same complexity bounds. Its final kernel cancellation is the marginal at anchor \(\ell\) in (44).

Curvature tags, short blocks, and integer heights

H’s localization in Section 6 groups chart atoms by their normalized quadratic phase and horizontal chart data, called their curvature tags. Normalize the parameters \((\lambda,C)\) in role \(j\) by dividing both by \(c_j\), and retain the horizontal vector \(\theta\). The coefficient calculation in H, Lemma 6.1 (Separation of different tags), can then be checked directly. For source roles \(i,j\) and spectator roles \(a,b\), put \[\xi_\ell=\frac{b-\ell}{b-a},\qquad \zeta_\ell=\frac{\ell-a}{b-a}; \qquad y_\ell=\xi_\ell y_a+\zeta_\ell y_b.\] The barycentric formula gives \[ c_i\xi_i\zeta_i=-c_j\xi_j\zeta_j\ne0. \tag{51}\] Indeed the factors \((i-a)(i-b)\) cancel from the product defining \(c_i\), leaving a nonzero multiple of \((i-j)^{-1}\); interchanging \(i,j\) changes its sign. The return calculation in that lemma therefore has the same nonzero prefactor and the same separation conclusion. Its period is enlarged to clear these fixed denominators.

The residue-template calculation in H, Lemma 6.6 (A uniform same-tag test), uses the marked adjacent pair. Its lifts satisfy \[ d_\ell=(1-\ell)d_0+\ell d_1+k_\ell, \qquad k_\ell\in\mathbb Z^d. \tag{52}\] Substitution of (52) and (49) cancels the common quadratic terms and leaves only linear phases from the carries. Their Fourier expansion has the same residue-template bound after a fixed enlargement of the lift box. It introduces no new curvature tag. Thus the source localization proof receives the same tag-separation and transfer estimates. Its graph distances, color histories, masks, and projection bounds are then used with those estimates and their original parameter order.

H, Lemma 7.2 (Short-block comparison), uses two sampling windows to compare a finite lattice form with a form on a connected torus. Place those windows on roles \(0,1\). The lattice and lifted torus coordinates in the other roles are respectively \[n_j=(1-j)n_0+jn_1, \qquad w_j=(1-j)u+jv.\] Both formulas have integer coefficients. A fixed torus of length \(64\) and lift boxes in \([-12,12]\) accommodate the present role sets; the source support parameter is decreased by a fixed factor if necessary. Maps from a pair of variables to two distinct role coordinates are injective on their lattice images and surjective on connected tori. These are the pair-coordinate properties needed in the comparison.

The quadratic randomization in role \(j\) becomes \(e(\gamma m^2/c_j)\), where \(e(v)=\exp(2\pi i v)\) and \(\gamma\) is uniform modulo a common multiple of all \(|c_j|\); the value \(24\) works for (48). On the Fourier support of the four-linear form, \[\sum_jm_j=\sum_jjm_j=0.\] The two vectors \(c=(c_j)\) and \((jc_j)\) are independent and span this two-dimensional nullspace. Hence \(m_j=c_j(v+jv')\) for some real \(v,v'\), and \[\sum_j\frac{m_j^2}{c_j} =\sum_jc_j(v+jv')^2=0\] by (49). The randomization preserves the form. For each single role, averaging its fourth moment in the spatial variable and in \(\gamma\) imposes equality of a pair’s sum and sum of squares; the same two pairings as in H remain. Thus its fourth-moment and clipping estimates, including their powers of \(q-2\), are unchanged.

The return from lattice points to small intervals also holds with these coordinates. If four real role coordinates are within \(e\) of integers, the marked adjacent pair reconstructs an integer progression. The discrepancy at any other role is an integer of magnitude less than \(8e\). Choosing \(e<1/100\) forces it to vanish. The admissible real offsets have positive area, computed explicitly below in (58). All support and overlap losses in the short-block comparison are therefore fixed constants.

The remaining arithmetic change is the height base. In H, Sections 8–9, replace \(L_h=6^h\) by \(L_h=12^h\), and replace the factor \(36\) in the finite progression tests by \(144\). Anchor changes still have integer coefficients. The common progression multiplier in H, Section 8.3, uses exactly (52) and (49), so its smoothness and Fourier truncation are unchanged in type. For the divisibility step, a coordinate in the free lattice quotient in H, Lemma 8.3 (Height stabilization), is an integral rational multiple of \(12^{R'}\). Here the quotient is the integer coordinate lattice modulo its intersection with the source’s small rational subspace, \(R'\) is the height gap, and \(C_*\) bounds the denominators before multiplication by \(12^{R'}\). The intersection is saturated, so this quotient is free abelian. Writing \(\Delta\) for the required drop in height, its \(2\)- and \(3\)-adic valuations are at least \[2R'-\log_2 C_*,\qquad R'-\log_3 C_*.\] Thus \(R'>\log_2C_*+2\Delta\) ensures divisibility by \(12^\Delta\). No assumption that the initial denominator has only primes \(2,3\) is needed: the resulting coordinate is already integral. The reserves and the later growth constant \(A_0\), used in the source bounds \(E_h=A_0^{h+1}\), are chosen using the new base.

Progression counting and the depth-varying extension

We finish the coefficient check at the finite counting step because it requires more than the identities for quadratic phases. In H, Section 9, eliminating a role \(\ell'\) gives the three coefficients \[\alpha_i=i-\ell',\qquad i\in S\setminus\{\ell'\}.\] They are nonzero and have absolute value at most four. The horizontal frequency tests use sums of role frequencies, while their anchored tests use the weights \(i-\ell'\). Their kernel cancellations are precisely (44). Accordingly the source’s Fourier expansions, coefficient tails, and annular tests keep their forms after these replacements.

In H, Lemma 9.1 (Sparse mass of separated progression tests), the shift data consist of a vector \(\zeta\in\mathbb R^d\) and integer vectors \(n\) with \(\|n\|_\infty\le B\). Replace the shifts \(36n\cdot\zeta\) in its annular relations by \(144n\cdot\zeta\), and the neighborhood shifts \(6n\cdot\zeta\) by \(12n\cdot\zeta\). The parameter \(N_0\) is the coefficient reserve for the latter neighborhoods. The proof uses a fine common grid; choose its step on a multiple clearing the integers \(1,2,3,4\).

For a four-step path, alternately add its four annular relations. The intermediate label in role \(i\) and the two labels in a second role cancel, leaving \(\alpha_i(a-a')\) and an expression in the four color labels. Dividing by \(\alpha_i\) constructs the source’s difference cover. The required lattice compatibility is \[ \frac{144}{\alpha_i}\in12\mathbb Z, \qquad \frac{12\alpha_j}{\alpha_i}\in\mathbb Z. \tag{53}\] Both assertions hold because \(|\alpha_i|\) divides \(12\). The integer vector obtained by adding four original shifts has norm at most \(4B\). In units of the new neighborhood shift \(12\), division by \(\alpha_i\) therefore gives a vector of norm at most \(4B\cdot12/|\alpha_i|\le48B\). The reserve \(N_0\ge64B\) accommodates it. The scalar error is at most four times one relation’s error after division by a nonzero integer.

For the slab cover, write the first two coordinates as their box centers plus shifts \(12n_i\cdot\zeta\) and scalar interval errors. Substitute them in the three-role relation and solve for the third coordinate. The resulting shift coefficients are \(12\alpha_1/\alpha_3\), \(12\alpha_2/\alpha_3\), and \(144/\alpha_3\). By (53) they are integers, and the resulting shift vector has norm at most \(C(d+1)^2B\) as in H. Choose the separation reserve and the excluded pure-shift interval constants larger by the fixed factors arising from these substitutions. The covering iterations use the same orders of sumsets, the graph count uses the same four roles, and the block and scale counts use the same covers. Their exponents are consequently unchanged. In particular the source factor \(L^{29}\delta^{1/2080}\) remains valid, where \(L\) bounds the number of scale occurrences of a fixed pair of labels and \(\delta\) is the sparse tuple-density threshold. In the application, let \(\xi\) be the vector of frequencies obtained from the height construction and take \(\zeta=\xi/144\), so the available heights start at \(12^{\Delta-2}\) instead of \(6^{\Delta-2}\). The preceding height reserves supply the required shift sizes. Pair determination in the application divides only by role differences, and the short-block windows were checked above. This verifies the coefficient-dependent hypotheses of the counting estimate and its application, not merely its formal conclusion.

We can now apply the remaining operator and summation arguments of H. The main projection families, Fourier modes, and counting tests have the same bounds in powers of logarithms of their budgets. The coordinate changes have inserted only fixed rational heights, fixed derivative losses, and the replacement of one fixed exponential height base by another. None changes a budget exponent as a function of the final \(q\). H’s Section 10 chooses those exponents before taking \(q\in(2,3)\) sufficiently close to two; its absorption of the finite-depth constant therefore applies to the modified estimates. This proves (45) for the four role sets together with the intermediate estimates used in P.

It remains to carry these inputs through P’s depth-local proof. The Maximal estimate for local catalog rows, The depth-varying row estimate, and Smooth mean-zero rows on depth inputs in P, Section 4, concern Hilbert-space operators supported on nested dyadic cells. Here a catalog consists of finitely many vector-valued column functions, each with a dyadic birth cell; descendants use restrictions of the same functions. A local projection is the orthogonal projection in the cell’s \(L^2\) space onto constant-coefficient combinations of a specified set of available columns. Their hypotheses specify cellwise constant coefficients in a persistent catalog, an ordinary fixed-input Bessel bound, and nondecreasing input-read indices. They contain no role coefficients. These are P, Lemmas 4.3, 4.4, and 4.6. The replacement just proved supplies their same scalar and Bessel inputs.

More concretely, P, Lemma 5.3, Corollary 5.4, and Lemma 5.5, use \((y_a,y_b)\) and the demodulations \(\pm\gamma/(b-a)\) already checked. Their cutoff equal to one on \([-1/3,1/3]\) still covers the support by (47). Separating the kernel gives exactly mean-zero bump rows, with fixed complete carriers. Clump replacement and predictor rows have the same Bessel bounds because the cancellation and height estimates have just been verified. The support and observation partitions of the rows, and their nondecreasing reads of the depth inputs, are unchanged. In P, Section 6, use the normalized tags from (49); the mask, template-transfer, short-block, and counting estimates are then the versions established above. Scheduling the column updates and summing their row bounds introduces no new arithmetic operation on a role label.

For completeness, the final parameter dependence can be seen in the retained estimates of H, Section 10.3, and P, Section 7. Write \(k\) for the accuracy logarithm, \(\mathfrak p\) for their growth parameter, \(K_0\) for the resulting depth-block length, and \(G_0\) for the strong energy and projection overhead. The term requiring \(q\) close to two has the form \[\mathfrak p^{-1/2}G_0(K_0+2)^{C_{19}(q-2)},\qquad K_0\le C_q(1+k+\mathfrak p)^{C_5},\qquad G_0=\exp\!\bigl(C_q(1+\log(2k))^C\bigr).\] Here we have renamed the source parameter \(P\) to avoid confusion with the manuscript P. The degrees \(C,C_5,C_{19}\) have absolute bounds fixed before \(q\). The short-block estimate feeding this term has the explicit exponent \(11(q-2)/(q-1)\) in H, Lemma 7.2; the coefficient changes above affect its fixed constants, not this exponent. Taking \(q\) so close to two that \[\max\{C_{19}^{\rm H}C_5^{\rm H}, C_{19}^{\rm P}C_5^{\rm P}\}(q-2)<1/8\] is therefore permitted in both proofs. Here the accuracy scheme fixes \(0<\theta<1\) before \(q\), and the positive constant \(\alpha\) is chosen sufficiently small after \(q\), as in the source proofs. Their later choice \(\mathfrak p=\lceil\exp(\alpha k^\theta)\rceil\) gives \(G_0=\mathfrak p^{o(1)}\) and makes the displayed term at most \(C_q\mathfrak p^{-1/4}\). The remaining accuracy exponents and summations retain their source order of choice. P’s depth-local proof consequently gives (46).

Translating the role labels back restores \(S=\{0\}\cup J\). P’s depth-local theorem permits variation in any role, and the local constructions allow any role as testing slot. Thus this translation does not require the testing role to remain the marked role \(0\) during the local proof. This completes the proof of Proposition 21.

From local forms to orbit oscillation

An averaging profile is a real compactly supported smooth function \(\varphi\) satisfying \[ \int\varphi=1,\qquad \int t^a\varphi(t)\,\,dt=0\quad(1\le a\le3), \qquad \|\varphi^{(n)}\|_\infty\le (G(n+2))^{G(n+2)}\quad(n\ge0) \tag{54}\] for some \(G\ge1\). Profiles may have either sign. Write \(\varphi_A(t)=A^{-1}\varphi(t/A)\). These are the profiles of P, Section 2.

Convergence on arbitrary systems

We now prove Lemma 2 from Proposition 22. The use of quantitative oscillation bounds to prove pointwise convergence belongs to the classical approach developed in [15]. The criterion needed here has the following short proof. A uniformly bounded sequence of measurable functions satisfying (55) is Cauchy almost everywhere. Indeed, if its tail oscillation exceeds \(4\epsilon\) on a set \(E\) of positive measure, then for every fixed initial endpoint \(k\) the supremum of \(|V_\ell-V_k|\) over \(\ell\ge k\) exceeds \(2\epsilon\) on \(E\). By monotone convergence one can choose a deterministic later endpoint so that the corresponding integral maximum is at least \(\epsilon\nu(E)\). Recursively choosing endpoints gives a lower bound \(W\epsilon\nu(E)\), contradicting the upper bound \(C_\varphi\sqrt W\) for large \(W\). The finite sums defining \(V_k^\varphi\) are uniformly bounded because the profile is bounded and compactly supported. Thus every fixed profile gives almost everywhere convergence.

P, Lemma 2.8 (Approximation with three vanishing moments), states that for every \(c\in[1,2]\) and every \(\epsilon>0\) there is an averaging profile with \[ \|\varphi-h_c\|_1<\epsilon, \qquad h_c(t)=c^{-1}\mathbf1_{(0,c]}(t). \tag{59}\] The use of signed profiles is essential to this assertion. One first smooths \(h_c\), preserving integral one. The other three moments can be canceled, without changing the integral, by four broad translated bumps at distance and width comparable to a large parameter \(A\). Their moment matrix for orders \(0,1,2,3\), after division of row \(j\) by \(A^j\), is a fixed invertible Vandermonde-type matrix. The prescribed zeroth correction is zero and the remaining normalized corrections are \(O(A^{-1})\). The correcting coefficients are therefore \(O(A^{-1})\), and their total \(L^1\) cost tends to zero. The bumps retain the derivative bounds in (54) with a profile-dependent constant.

Fix a triple \(J\) and write \(a_n(y)=\prod_{j\in J}g_j(R^{jn}y)\), so \(|a_n(y)|\le1\). For \(N_k=\lfloor c2^k\rfloor\), the Riemann-sum comparison gives, uniformly in \(y\), \[ \limsup_{k\to\infty} \left|\frac1{N_k}\sum_{n=1}^{N_k}a_n(y)-V_k^\varphi(y)\right| \le\|h_c-\varphi\|_1. \tag{60}\] To see this, first replace \(N_k^{-1}\) by \((c2^k)^{-1}\), at cost \(O(2^{-k})\), then bound the difference by \(2^{-k}\sum_{n\in\mathbb Z}|h_c(n2^{-k})-\varphi(n2^{-k})|\). This converges to the displayed integral because the integrand is compactly supported and piecewise continuous.

Choose countably many profiles with errors tending to zero for each rational \(c\in[1,2]\). On their common conull set, the existence of all the smooth-profile limits and (60) show that the ordinary averages along \(N_k\) are Cauchy. The Host–Kra norm convergence theorem [11], stated for arbitrary invertible probability preserving systems, gives an \(L^2\) limit for the averages at all integer lengths: apply it to the four consecutive powers, inserting the constant function one at the omitted role. The bounded pointwise limit along each \(N_k\) must equal this \(L^2\) limit almost everywhere, by bounded convergence and uniqueness of norm limits. Thus all rational-\(c\) subsequences have the same limit on one conull set. No identification of that limit and no mixing assumption is needed.

Finally, for \(M\le N\) and \(|a_n|\le1\), \[\left|\frac1N\sum_{n=1}^Na_n-\frac1M\sum_{n=1}^Ma_n\right| \le 2\frac{N-M}{N}.\] For any rational mesh of \([1,2]\), every sufficiently large integer \(N\) lies within its relative mesh size, up to \(O(N^{-1})\), of some \(\lfloor c2^k\rfloor\) from that finite mesh. Its finitely many subsequences have the common pointwise limit. The preceding inequality, followed by meshes of size tending to zero, proves convergence along all positive integers. Homogeneity removes the normalization \(|g_j|\le1\). A proper subset with fewer than three roles is contained in one of the triples already treated; assign the constant function one to its additional roles. This proves Lemma 2 in its full stated scope.