Proof. The sum of the root lengths in generation \(n\) is at most \(\rho^n|J|\), by induction. Summing this geometric series proves both assertions. ◻

Proof. The private component of the sum has squared norm \(\sum_v\epsilon_v|c_v|^2\), giving the lower bound. The upper bound follows from Cauchy–Schwarz, since there are at most \(D\) columns and each has norm at most \(D\).

Let \(E_y:\ell^2(\mathcal A)\to\mathcal H\) be the evaluation map \(E_yc=\sum_vc_v\Phi_v(y)\), and let \[G=\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I E_y^*E_y\,dy\] be the normalized Gram matrix. The bounds just proved give \(G\ge D^{-2}\operatorname{Id}\) and \(\|E_y\|\le D^{3/2}\). Therefore the projection has the integral representation \[P_{I,\mathcal A}z(y) =\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I E_yG^{-1}E_t^*z(t)\,dt, \qquad \|E_yG^{-1}E_t^*\|\le D^5.\] This proves the supremum estimate. Finally, orthogonality to \(\Phi_v\) says that the physical pairing in the last assertion minus \(\epsilon_vc_v\) is zero. ◻

Proof. Put \(\Delta_0^i=P_0^i\) and \(\Delta_n^i=P_n^i-P_{n-1}^i\) for \(n\ge1\). Within each family these are pairwise orthogonal projections. A prefix of the index set \(\{0,\ldots,T\}\) can be partitioned into dyadic index blocks, with at most one block of each length. For a block \(A\) put \(\Delta_A^i=\sum_{n\in A}\Delta_n^i\). At any fixed block length, the operators \(\Delta_A^i\) are pairwise orthogonal projections.

Expand each factor of \(Z_{n(\nu)}\) using this binary decomposition. Fix a tuple of block lengths, one for each of the \(m\) factors. A term is zero if one of those lengths does not occur in the relevant prefix. Otherwise it has the form \[B_\nu\Delta_{A_m(\nu)}^m\cdots\Delta_{A_1(\nu)}^1g.\] Group rows with the same tuple of blocks. The row contraction bounds their sum of squared norms by \(\|\Delta_{A_m}^m\cdots\Delta_{A_1}^1g\|^2\). Summing over the block tuples that occur is bounded by the sum over all tuples of those fixed lengths. Successive Bessel inequalities give \[\sum_{A_1,\ldots,A_m} \|\Delta_{A_m}^m\cdots\Delta_{A_1}^1g\|^2\le\|g\|^2.\] This argument uses orthogonality within each layer, not commutation between layers. There are at most \(L_T^m\) tuples of block lengths. The triangle inequality in the Hilbert direct sum of the row spaces proves the first assertion.

For the second, telescope in the order \[Z_n-Z_{n-1} =\sum_{i=1}^m P_n^m\cdots P_n^{i+1}\Delta_n^i P_{n-1}^{i-1}\cdots P_{n-1}^1.\] Discard the left contractions and use Cauchy–Schwarz in the sum over \(i\). For a fixed \(i\), the rows \(B_n=\Delta_n^i\), \(1\le n\le T\), satisfy the row contraction hypothesis. Apply the first assertion to the \(i-1\) preceding families and the prefix index \(n-1\). The resulting bound is \[\sum_{n=1}^T\|(Z_n-Z_{n-1})g\|^2 \le m\sum_{i=1}^m L_T^{2(i-1)}\|g\|^2 \le m^2L_T^{2(m-1)}\|g\|^2,\] as claimed. ◻

Proof. For a nonempty product, we first divide the row cells into size segments and, on each segment, replace the input by a bounded function with the same first-layer moments. We then stop when the last-layer coefficient vector moves: nested-prefix estimates control the anchor products, and the pathwise birth-label budget controls the coefficient errors. Attempts are summed using both length and energy contraction. Finally, the size segments are summed using the bounds for their weighted averages and representative energies.

The assertion for an empty product is the row contraction hypothesis. Assume \(m\ge1\), and abbreviate the ordinary \(L^2(J;\mathcal H)\) norm to \(\|\cdot\|_2\). Add one final dyadic generation below all row cells, retaining columns and refitting there if necessary. This changes neither the hypothesis nor the quantity to be estimated.

Proof. If \(\mathcal E\) is empty there is nothing to prove. It is otherwise a finite set. We first bound coefficients in a minimal rational dependence. Suppose \(a_1,\ldots,a_k\) are minimally dependent vectors of the indicated height. Then \(k\le d+1\) and their rank is \(k-1\). For each \(i\), clear the denominators in its \(d\) entries by a common denominator \(q_i\le B^d\). The integer vector \(v_i=q_i a_i\) has each entry bounded by \(B^d\). Select \(k-1\) coordinate rows on which the vectors have rank \(k-1\). Their signed maximal minors give a null vector \((b_1,\ldots,b_k)\) with \[0<|b_i|\le d! B^{d^2}.\] None of these minors is zero: a zero component of a null vector would give a dependence on a proper subset, contrary to minimality. Returning to the original vectors gives \[\sum_{i=1}^k c_i a_i=0, \qquad c_i=b_iq_i\in\mathbb Z\setminus\{0\}, \qquad |c_i|\le H:=d! B^{d^2+d}.\]

Set \(R=4dH\). Suppose coefficient vectors have nonzero evaluation magnitudes \[0<s_1<s_2<\cdots<s_k, \qquad s_{i+1}>Rs_i.\] If these vectors were dependent, apply the preceding construction to a minimal dependent subset, ordered by evaluation magnitude, and let \(s_*\) be its largest magnitude. The coefficient of that term has absolute value at least one, whereas all the other terms together have absolute value at most \[dH\frac{s_*}{R}<s_*.\] They cannot sum to zero. Thus vectors whose evaluation magnitudes are successively separated by a factor greater than \(R\) are linearly independent, and there can be at most \(d\) of them.

Choose the smallest element \(s\) of \(\mathcal E\), cover \([s,Rs]\), and repeat with the smallest still uncovered element. The selected elements are successively separated by more than \(R\), so at most \(d\) intervals are needed. Their logarithmic lengths are \(\log_2R\), and \[\log_2R \le C(d+1)^2\log(2dB) \le C(d+1)^3\log(2B).\] Each logarithmic interval meets at most \(2+\log_2R\) unit bins. This proves the first assertion.

For the second, consider one logarithmic value interval \([s,s+\log_2R]\). The condition \(\alpha\le r_n^\ell v\le\beta\) confines \(n\) to a real interval of length at most \[\frac{\log_2R+\log_2(\beta/\alpha)}{\ell}.\] That interval has at most two plus its length integer points. Summing over the at most \(d\) value intervals and using the preceding bounds proves the claimed scale count. The factor \(r_0\) translates the interval and has no effect on its length.

For the third, partition \(\mathcal E\) by its occupied bins \([2^j,2^{j+1})\). For one such bin, a contribution at scale \(n\) is possible only if \(r_n^\ell2^j\le\varepsilon\), and its maximum is at most \(r_n^\ell2^{j+1}\). Let \(n_0\) be the first integer for which the former inequality holds. The sum of these upper bounds for \(n\ge n_0\) is at most \[2\varepsilon\sum_{a=0}^{\infty}2^{-\ell a} \le4\varepsilon.\] The maximum over all occupied bins is bounded by the sum of their separate maxima. Multiplying by the occupied-bin count proves the assertion.

Finally, the supremum norm of a nonzero evaluation vector is the magnitude of one of its nonzero coordinates, so the same set \(\mathcal E\) contains all such suprema. If there are at most \(h\) coordinates, its Euclidean norm lies between that supremum and \(\sqrt h\) times the supremum. Enlarging each of the \(d\) logarithmic intervals to the right by \(\tfrac12\log_2h\) covers all these Euclidean norms. The preceding bin, scale, and geometric-tail arguments then give exactly the additional costs stated in Lemma 10. ◻

Proof. For conjugation replace \(C,\lambda,\beta\) by their negatives and conjugate \(\chi\). For products use the direct sum of the horizontal coordinates and the block diagonal sum of the matrices. The product rule for derivatives gives the required Gevrey bounds.

It remains to check a translated horizontal argument. Let that argument be \(\theta y+a\). Choose a positive integer \(P\) such that every entry of \(PC\) is integral. The product of the denominators of the \(d^2\) entries is an admissible choice, so \(P\le H^{d^2}\). Write \(a=m_0+a_0\), with \(m_0\in P\mathbb Z^d\) and \(\|a_0\|_\infty\le P\), and replace the summation index by \(n+m_0\). With \(w=\theta y-n\), the new phase is \[(\theta y+m_0+a_0)^TC(w+a_0) =(\theta y)^TCw +(\theta y)^TCa_0+m_0^TC\theta y +a_0^TC(w+a_0)+m_0^TCa_0 \pmod{\mathbb Z}.\] The two middle terms linear in \(\theta y\) change the unrestricted linear phase. The last term is constant. The term \(a_0^TC(w+a_0)\) can be included in the smooth amplitude \(\chi(w+a_0)\). The support and derivatives of that amplitude have quasipolynomial bounds because \(a_0\), unlike \(m_0\), is bounded by \(P\). Translating an ordinary quadratic phase produces only a quadratic phase, a linear phase, and a constant. Rescaling simply rescales \(\theta,\lambda,\beta\). All these calculations hold with \(\theta y\) replaced by \(Vz\). ◻

Proof. We first show that every sufficiently small return has a bounded frequency increment. The lattice directions generated by these returns form a rational isotropic subspace with a bounded-height projection and bounded slow horizontal matrix. Returns in each coordinate direction then bound the remaining effective quadratic matrix. Let \(P\) clear the denominators of \(C\), as in Lemma 12. Put \(F=\vartheta Q\). The support of \(F\), its supremum norm, its \(L^2\) norm, and \(\int(1+\|z\|)|F(z)|\,dz\) are bounded by a quantity \(D\) quasipolynomial in \(H\). The value of \(D\) can be enlarged finitely many times below.

Suppose \[Vh=m+d_h,\qquad m\in P\mathbb Z^d.\] Translate \(z\) by \(h\) and replace the lattice index by \(n+m\). On the support of the translated amplitude, \(Vz-n\) has norm bounded in terms of the chart complexity. Expanding the phase therefore gives \[F(z+h)=e(c(h,m))e(\Omega(h,m)^Tz)F(z)+E_{h,m}(z), \qquad \Omega(h,m)=(2\Lambda+V^TCV)h-V^TAm,\] where \(c(h,m)\) is constant in \(z\), and \[\|E_{h,m}\|_{L^1}+\|E_{h,m}\|_\infty \le D(\|h\|+\|d_h\|)\] provided \(\|h\|+\|d_h\|\le1\). To verify that the error contains no unbounded coefficient, put \(u=Vz\), \(w=u-n\), and \(d=d_h\). The nonconstant increment of the chart phase is \[u^TCd+m^TCu+d^TCw.\] The first two terms, together with the ordinary quadratic increment, are exactly \(\Omega(h,m)^Tz\); the term \(m^TCn\) discarded in this calculation is integral. The remaining \(d^TCw\) is bounded by \(D\|d\|\). Translating the amplitude and \(\vartheta\) costs \(D(\|h\|+\|d\|)\). This proves the claimed estimate.

The Fourier transform of \(F\) is \(D\)-Lipschitz. Choose a point \(\xi_0\) where its absolute value is at least \(3\delta/4\), and put \(\rho=\delta/(100D)\). Choose an integer \[M\ge 100D^2\delta^{-2}\rho^{-k}.\] If necessary, increase \(D\) to cover the fixed volume constant in Plancherel’s theorem. Choose \(a>0\) so small that \[a\le\frac{\delta}{100DM},\qquad a\le\frac1{100(1+\|A\|)}.\] All of \(M,a^{-1},\rho^{-1}\) are quasipolynomial in \(H,\delta^{-1}\). For a return with \(\|h\|+\|d_h\|\le a\), the same translation calculation applies to \(\ell(h,m)\), \(1\le\ell\le M\). It produces Fourier values of absolute value at least \(\delta/2\) at \(\xi_0-\ell\Omega(h,m)\). If \(\|\Omega(h,m)\|\ge1\), the balls of radius \(\rho\) about those points are disjoint, and on each such ball the Fourier transform has magnitude at least \(\delta/3\). Their contribution to its squared \(L^2\) norm exceeds \(D^2\), by the choice of \(M\), contradicting Plancherel. Hence \[\|\Omega(h,m)\|<1 \quad\hbox{whenever}\quad \|h\|+\|d_h\|\le a.\] For two returns satisfying this condition, direct antisymmetrization gives \[h'^T\Omega(h,m)-h^T\Omega(h',m') =d_{h'}^TAd_h-m'^TAm.\] The integer \(m'^TAm\) has absolute value less than one after decreasing \(a\) by a fixed factor. It is therefore zero.

We next find all directions generated by short returns. For \(d\ge1\) let \[\mathcal K=P^{-1}V[-1,1]^k+[-1,1]^d.\] This is a symmetric convex body with nonempty interior. Write \(\lambda_1\le\cdots\le\lambda_d\) for its successive minima for \(\mathbb Z^d\), and \(\mu_1\le\cdots\le\mu_d\) for the successive minima of its polar body. We need the elementary transference bounds \[1\le\lambda_j\mu_{d+1-j}\le(d!)^2 .\] For the lower bound, independent lattice vectors in subspaces of dimensions \(j\) and \(d+1-j\) cannot have all cross pairings zero. One of their integer pairings has absolute value at least one. For the upper bound, Minkowski’s second theorem (Henk 2002, Theorem 1.3) gives \[\prod_{j=1}^d\lambda_j\mu_j \le\frac{4^d}{\operatorname{vol}(\mathcal K) \operatorname{vol}(\mathcal K^\circ)} \le(d!)^2.\] Here the last volume-product bound has a short direct proof. Choose maximal-determinant columns \(v_1,\ldots,v_d\) in \(\mathcal K\). Their cross-polytope is contained in \(\mathcal K\), and maximality places \(\mathcal K\) in their coordinate parallelotope. Taking polars gives an inscribed cross-polytope in \(\mathcal K^\circ\) with reciprocal determinant. The product of the two cross-polytope volumes is \(4^d/(d!)^2\). Since every opposite-minimum product is at least one, their product upper bound proves each individual upper bound.

Set \(L=16(d!)^2\), and choose \(b_{\max}\le a/(100P(d+k+1))\). The \(d+1\) disjoint intervals \[(b_{\max}L^{-j-1},\,b_{\max}L^{-j}], \qquad 0\le j\le d,\] cannot all contain a successive minimum. Select one containing none, write its left endpoint as \(b_0\), and put \[U=\mathop{\mathrm{span}}(\mathbb Z^d\cap b_0\mathcal K).\] The reciprocal of \(b_0\) is quasipolynomial in the required parameters. Every \(n\in\mathbb Z^d\cap b_0\mathcal K\) has a representation \(n=P^{-1}Vh+w\), with \(\|h\|_\infty,\|w\|_\infty\le b_0\). Thus \(m=Pn\), \(d_h=-Pw\) form a return satisfying the threshold \(a\). The integrality argument above, applied to any two such \(n\), shows that \(A\) vanishes on \(U\times U\).

Let \(r_0=\dim U\). If \(r_0<d\), then \(\lambda_{r_0+1}>Lb_0\); transference supplies \(d-r_0\) independent integer vectors \(q_\nu\) in \[\frac{(d!)^2}{Lb_0}\mathcal K^\circ .\] Each has zero pairing with \(\mathbb Z^d\cap b_0\mathcal K\), since that pairing is integral and has absolute value at most \((d!)^2/L<1\). They consequently span \(U^\perp\). The inclusion \([-1,1]^d\subset\mathcal K\) bounds their coordinates, while \(P^{-1}V[-1,1]^k\subset\mathcal K\) bounds each \(q_\nu^TV\), by quasipolynomial quantities. If \(S\) has the \(q_\nu^T\) as rows, \[P_s=S^T(SS^T)^{-1}S .\] Cramer’s rule, and the positive integral determinant of \(SS^T\), give quasipolynomial rational height and norm bounds for this matrix and for \(P_sV\). When \(r_0=d\), take \(P_s=0\); the same conclusions hold. Put \(P_f=1-P_s\).

Finally we bound the effective ordinary quadratic part. For each coordinate vector \(e_j\in\mathbb R^k\), sample \(P^{-1}V(te_j)\bmod\mathbb Z^d\) at equally spaced points \(t\in[0,b_0]\). Partition the horizontal torus into cubes of side at most \(b_0/4\). More than \((1+4/b_0)^d\) samples give two in the same cube. Their difference yields a return \[h=t_je_j,\qquad b_0/M_0\le t_j\le b_0,\qquad \|d_h\|_\infty\le Pb_0/4,\] where \(M_0\le2(1+4/b_0)^d\). Its lattice vector satisfies \(P^{-1}m\in b_0\mathcal K\), so \(m\in U\). In particular \(m-V_fh=-P_fd_h\in U\). Isotropy gives \[V^TA(m-V_fh)=V_s^TA(m-V_fh),\] which is quasipolynomially bounded. Direct matrix algebra gives \[2\Lambda+V^TCV-V^TAV_f =2\Lambda_*+V_s^TC_{\mathrm{sk}}V_s .\] The return bound for \(\Omega\) therefore bounds \(2\Lambda_*t_je_j\) quasipolynomially. Since \(t_j^{-1}\) is also quasipolynomial, this bounds every column of \(\Lambda_*\). This proves [eq:src-2].

For \(d=0\), the initial translation argument applies to every sufficiently small \(h\), with \(\Omega=2\Lambda h\); take \(h=ae_j/2\) and \(U=\{0\}\). This also proves the assertion in that case. ◻

Proof. Put \(u=Vz\), \(u_f=P_fu\), \(u_s=P_su\), and \(w=u-n\). For the symmetric part, \[u^TC_{\mathrm{sy}}(u-n) =\tfrac12u^TC_{\mathrm{sy}}u +\tfrac12w^TC_{\mathrm{sy}}w -\tfrac12n^TC_{\mathrm{sy}}n.\] For the skew part, isotropy on \(U\) gives \[u^TC_{\mathrm{sk}}w =u_f^TC_{\mathrm{sk}}u_s -u_f^TC_{\mathrm{sk}}P_sn +u_s^TC_{\mathrm{sk}}w.\] The first term is precisely the extra quadratic included in \(\Lambda_*\).

Choose an even integer \(P_0\) clearing the denominators of \(C_{\mathrm{sy}}\). Then \(-n^TC_{\mathrm{sy}}n/2\bmod\mathbb Z\) is determined by \(n\bmod P_0\mathbb Z^d\). Also choose a common positive denominator \(q_0\) for \(P_s\). On the box under consideration, \(u_s=V_sz\) is bounded, and \(w\) belongs to the support of \(\chi\). Thus \(P_sn=u_s-P_sw\) ranges over a bounded subset of \(q_0^{-1}\mathbb Z^d\). The size of this set, \(P_0^d\), and \(q_0\) are all quasipolynomial in \(H,B\).

Split the sum over \(n\) according to \[P_sn=N,\qquad n\equiv a\pmod{P_0\mathbb Z^d}.\] For each pair \((N,a)\), remove the fixed residue phase and the rational linear phase \[-u_f^TC_{\mathrm{sk}}N.\] What remains is a smooth amplitude in the two vector variables \((u_s,w)\). To see explicitly that it admits a periodic Fourier expansion, let \(\eta(s)\) be a smooth cutoff equal to one on all values of \(V_sz\) in the box. Choose a smooth equality cutoff \(E\) with \(E(0)=1\), supported in the coordinate cube of radius \(1/(3q_0)\). Then the function \[\eta(s)\! \sum_{n\in a+P_0\mathbb Z^d} \chi(u-n)E\bigl(s-P_s(u-n)-N\bigr) e\!\left(\tfrac12(u-n)^TC_{\mathrm{sy}}(u-n) +s^TC_{\mathrm{sk}}(u-n)\right)\] has the required value when \(s=P_su\). Indeed its equality-cutoff argument then equals \(P_sn-N\); the lattice spacing makes the cutoff exactly the indicator of the specified equality. It is periodic in \(u\) with period \(P_0\) in each coordinate. By taking the support of \(\eta\) strictly inside a larger integer-period box, it extends smoothly and periodically in \(s\), with a period \(P_1\) quasipolynomial in \(H,B\).

The resulting Fourier modes have the form \[e\bigl(p^Ts/P_1+\ell^Tu/P_0\bigr),\qquad p,\ell\in\mathbb Z^d.\] Substitution of \(s=P_su\), and restoration of the rational linear phase for \(N\), gives \[\sigma=\sigma_N+P_s^Tp/P_1+\ell/P_0, \qquad \sigma_N^Tu=-(P_fu)^TC_{\mathrm{sk}}N.\] These frequencies have the height asserted in the lemma. The sum of the two ordinary quadratic terms is \(z^T\Lambda_*z\), proving the exact identity [eq:src-3].

Here are the quantitative convergence details. On each of the bounded supports above, all coefficients in the smooth phases, cutoff scales, periods, and lattice-point counts are bounded by a quasipolynomial quantity \(D_0\). The product rule and the chain rule imply, at order \(m\), derivative bounds of the form \[D_0^{C(m+d+1)}(m+1)^{C(m+1)} .\] This estimate also bounds derivatives after rescaling the periodic variables to unit periods. It follows either by differentiating the quadratic exponential, whose \(m\)-th derivative is a polynomial of degree at most \(m\) times that exponential, or directly by induction on \(m\). The cutoffs satisfy the same fixed-order Gevrey estimate.

For a nonzero integer Fourier index \(v=(p,\ell)\), select a coordinate with \(|v_j|\ge\|v\|/(2d)^{1/2}\) and integrate by parts \(m\) times in that coordinate. Thus \[|\widehat A(v)| \le D_0^{C(m+d+1)} (m+1)^{C(m+1)}(2d)^{m/2}(1+\|v\|)^{-m},\] after increasing the constant to include the zero index. A dyadic shell has at most \((CT)^{2d}\) indices. Taking \(m\ge J(d+1)^2+4d+4\) gives the displayed shell bound. Since \(d\le\log H\), the prefactor for this derivative order is quasipolynomial in \(H,B\). Summing the shells proves absolute convergence and the bound for the full coefficient sum.

There are at most \(B^{C(d+1)^2}\) rational projection matrices of height at most \(B\). For each such matrix there are quasipolynomially many choices \((N,a)\). All derivative estimates are uniform in the bounded slow parameters and in amplitudes with the stated complexity bounds. Consequently we may replace every coefficient by its uniform envelope before summing over this finite list. If different indices give the same frequency, use the sum of their envelopes at that frequency. Choosing the derivative order larger by \(C(d+1)^2\) absorbs, as well, the count \((D_0T)^{C(d+1)}\) of bounded-height rational frequencies at an index scale \(T\). This proves the pooling assertion in the form stated. Finally, the sum of the shells above a cutoff \(T_0\) is at most \(C D_JT_0^{-J(d+1)^2}\); choosing this to be at most \(\varepsilon\) gives the required quasipolynomial truncation. ◻

Proof. We first separate only \(\varphi_i\). Temporarily suppress its linear phase, put \(\alpha_i=1\), \(\alpha_k=0\), and choose \(\alpha_a,\alpha_b\) so that \[\sum_{j=0}^3\alpha_j=0,\qquad \sum_{j=0}^3j\alpha_j=0.\] The four indices lie in a fixed set, so these coefficients are fixed rationals of bounded height.

Choose a nonnegative, compactly supported Gevrey function \(\rho\) on \(\mathbb R^d\) whose integer translates sum to one. Insert its partitions of unity at \(\theta y_a,\theta y_b,\theta t\). Write the corresponding lifts as \[d_a=\theta y_a-m_a,\qquad d_b=\theta y_b-m_b,\qquad d=\theta t-m.\] For the original summation index \(m_i\), put \[d_i=\theta y_i-m_i,\qquad k_j=d_j-d_a-(j-a)d\quad(j=i,b),\qquad k_a=0.\] Each \(k_j\) is integral because \(y_j=y_a+(j-a)t\). The support restrictions bound the possible \(k_i,k_b\) quasipolynomially in \(H\). Conversely, fixing these integers determines \(m_i\) from \(m_a,m\), and the condition for \(k_b\) is enforced by a smooth equality cutoff in \(d_b-d_a-(b-a)d-k_b\). Such a cutoff can equal one at zero and vanish at every other integer vector, so this insertion is exact.

Put \(S=\sum_jj^2\alpha_j\). The two relations defining the \(\alpha_j\)’s imply the exact identity \[\sum_j\alpha_j \bigl(\lambda y_j^2+(\theta y_j)^TCd_j\bigr) =S\bigl(\lambda t^2+(\theta t)^TCd\bigr) +\sum_j\alpha_j(\theta y_j)^TCk_j .\] Indeed the coefficients of the common lift \(d_a\) and of \(x\) cancel, and the coefficient of \(t\) in \(\sum_j(j-a)\alpha_jy_j\) is \(S\). Solving this identity for the \(i\)-phase leaves the \(a\)- and \(b\)-phases with coefficients \(-\alpha_a,-\alpha_b\), and the \(t\)-phase with coefficient \(S\). The last displayed sum is linear in \(x,t\), and hence is a sum of a linear function of \(y_a\) and a linear function of \(y_b\). Restore the suppressed \(\beta y_i\) in the same way.

The remaining coupled amplitude is a smooth function of \((d_a,d_b,d)\) on a bounded box. It is a product of the partition bumps, the equality cutoff, and \(\chi(d_a+(i-a)d+k_i)\). Its derivative bounds are quasipolynomial versions of the fixed Gevrey bounds. Choose slightly larger product bumps equal to one on its support, extend the amplitude periodically inside that larger box, and take its Fourier series. Every Fourier term factors into a function of \(d_a\), a function of \(d_b\), and a function of \(d\). After the independent lattice sums, these are precisely three atoms of the form [eq:src-1], with rational chart matrices \(-\alpha_aC,-\alpha_bC,SC\). A Fourier index of size \(T\) contributes only derivatives of size \(T^m\) at order \(m\) to their smooth amplitudes, so its complexity is bounded by \((DT)^C\).

Apply the same construction to \(\varphi_k\), now setting the coefficient of the \(i\)-slot to zero, and combine the two triples by direct sums of horizontal coordinates. The number of integral choices and the dimensions are quasipolynomially bounded in the original complexity. The integration-by-parts calculation in the proof of Lemma 14, with derivative order larger than the Fourier dimension by at least \(110\), bounds shell coefficient sums by \(DT^{-100}\). The number of Fourier indices and integral choices in each bounded shell is also quasipolynomial. Normalize the three factors to have supremum norm one and put their bounded normalization factors into \(c\). The bounds remain of the stated form. Absolute convergence justifies all rearrangements and proves the identity for every \((x,t)\). ◻

Proof. Use \((y_a,y_b)\) as coordinates; their Jacobian is \(|b-a|\) and \(t=(y_b-y_a)/(b-a)\). After normalization by \(r\), the smooth kernel has a Fourier expansion on a fixed two-dimensional box whose coefficient sum is uniformly bounded. Each Fourier term merely modulates the two inputs. Insert \(\zeta(t/r)\), which is one on the relevant support. The remaining operator is convolution with \(Q(t)\zeta(t/r)\), after the fixed change of scale \(b-a\). Its Fourier multiplier is bounded by \(C r A_r\). Plancherel, the factor \(r^{-2}\) in the definition of \(H_I\), and \(\|f\|_{L^2(I)}=r^{1/2}\|f\|_{2,I}\) prove the first assertion. The factors \(\psi_a,\psi_b\) are bounded multipliers and do not increase the two \(L^2\) norms.

For the second assertion apply Lemma 13 to \(Q(r\,\cdot)\), with \(V=r\theta\) and \(\Lambda=r^2\lambda\). Lemma 14, truncated to a sufficiently small uniform error, then gives terms \[e(\lambda_*t^2+\gamma t),\qquad \gamma=\beta+\sigma^T\theta,\qquad |\lambda_*|r^2\le D .\] The formula for \(\Lambda_*\) in [eq:src-2] gives exactly the asserted rational dependence of \(\lambda_*\). Split the linear phase by \[e(\gamma t)= e\bigl(-\gamma y_a/(b-a)\bigr) e\bigl(\gamma y_b/(b-a)\bigr).\] The remaining smooth two-variable kernel, \(w_I(x,t)e(\lambda_*t^2)\), has derivatives in normalized coordinates bounded by \(D^{C(m+1)}(m+1)^{C(m+1)}\) at order \(m\). Its Fourier expansion, with fixed product cutoffs, has quasipolynomial coefficient sum and a quasipolynomial truncation giving the prescribed error. The resulting factors are a bounded one-variable smooth bump at scale \(r\), a linear phase, and \(\psi_e\).

For completeness, a smooth bump on \(I\), restricted to \(I\), is itself represented by a chart with zero matrix and one additional slow horizontal coordinate. Map \(I\) to an interval strictly inside one lattice cell, put the bump in the corresponding amplitude, and support that amplitude away from every other lattice translate. On \(I\) only the chosen translate contributes. Lemma 12 handles the affine offset and the product with \(\psi_e\). Outside \(I\) a bounded periodic extension is harmless. This realizes all separated factors as bounded atoms. Choose the successive truncation errors divided by the already bounded coefficient sums. Only a fixed number of quasipolynomial budgets is composed, giving all the stated estimates. ◻

Proof. The usual cube Cauchy–Schwarz inequality is valid on this circle. One direct derivation writes a vertex as \(x_1^{\omega_1}+x_2^{\omega_2}+x_3^{\omega_3}\), using three independent pairs of Haar variables. At the first Cauchy–Schwarz step the factors with the first coordinate equal to zero and one are put on opposite sides; repeating for the second and third coordinates gives \[|\Lambda_{\mathrm{cube}}(g_\omega)| \le\prod_\omega\|g_\omega\|_{\mathcal U}.\] The same iterations for a single function express its eighth power as an integral of squared absolute values, proving nonnegativity. Expanding the cube of \(f+g\), applying the displayed inequality to each mixed term, and summing the binomial coefficients gives the triangle inequality.

For the stated \(L^2\) estimate, split the vertices into \[\{000,100,010,001\} \quad\hbox{and}\quad \{111,011,101,110\}.\] Each four-tuple of corresponding linear forms is an integer linear automorphism of the four Haar variables, with determinant of absolute value one. Apply Cauchy–Schwarz to the product of these two four-factor groups, then change variables in each squared integral. Each squared integral factors into the four squared \(L^2\) norms. This proves the estimate. Expanding the difference of two mixed cube forms proves their continuity, first for bounded functions and then for all \(L^2\) functions by truncation. The fixed normalization between circle measure and interval measure proves the final assertion. ◻

Proof. Normalize \(I=[0,1)\). For \(L\ge1\), define the cell averages \[g_L(j)=L\int_{j/L}^{(j+1)/L}g(y)\,dy,\qquad 0\le j<L,\] and extend them by zero to \(\mathbb Z/(16L\mathbb Z)\). Their discrete raw cube averages converge to the continuous raw cube average above. Here is a justification that needs no regularity of \(g\). For a continuous bounded function the assertion is Riemann-sum convergence, because its cell averages differ from endpoint values by its modulus of continuity. For two functions bounded by one, telescoping the eight-factor product bounds the difference of their continuous cube averages by \(8\|g-b\|_{L^1(\mathbb T_{16})}\). The identical discrete bound holds for their cell averages, since \[\frac1L\sum_{j=0}^{L-1}|g_L(j)-b_L(j)| \le\|g-b\|_{L^1([0,1))}.\] Approximation in \(L^1\) by continuous functions bounded by one proves the assertion.

There is no wraparound contribution hidden in this comparison. For a nonzero supported cube choose \(x\in[0,1)\), and lift \(h_j\) as the difference between \(x+h_j\in[0,1)\) and \(x\). Then \(|h_j|\le1\), and every real vertex belongs to \([-3,4]\). Within that range, reduction modulo \(16\) lies in \([0,1)\) exactly when the real vertex does. Thus the circle and interval zero-extension cube averages differ only by their fixed normalizing factors. The denominator defining \(U^3[L]\), namely the cube norm of the interval indicator, likewise converges to a fixed positive constant. For all sufficiently large \(L\), therefore, \[\|g_L\|_{U^3[L]}\ge c\delta,\] after shifting the integer labels by one if needed. No bound on this choice of \(L\) is required.

Apply Theorem 17. Write its dimension as \(d_0\), its complexity as \(M_0\), its Lipschitz bound as \(K_0\), and its correlation as \(\varepsilon_0\). All these quantities have the bounds asserted there. We next convert its nilsequence into chart atoms, tracking only quantities independent of the real orbit coefficients.

In second-kind Malcev coordinates adapted to the central filtration subgroup \(G_2\), write a group element as \((z,s)\), with \(z\in\mathbb R^p\), \(s\in\mathbb R^{d_0-p}\). Since \([G,G]\subset G_2\) and \([G,G_2]=1\), commuting the one-parameter coordinate factors into their prescribed order gives the group law \[(z,s)(z',s')=(z+z',s+s'+B(z,z')).\] The coefficients of the bilinear map \(B\) are rational and have height bounded quasipolynomially in \(M_0\) and \(e^{d_0}\). Indeed, in step two every interchange contributes just its central commutator with coefficient \(z_i z'_j\); all further commutators vanish. The rational structure constants in the Malcev basis are part of the nilmanifold complexity. Integer coordinate tuples represent exactly the lattice \(\Gamma\).

Write the polynomial sequence in these coordinates as \((z(n),s(n))\). Its second group differences have no horizontal part, so \(z(n)=z_0+n\theta\). For the unit right difference \(Dg(n)=g(n+1)g(n)^{-1}\), the group law gives \[Dg(n)=\bigl(\theta,\Delta s(n)-B(\theta,z(n))\bigr),\quad D^2g(n)=\bigl(0,\Delta^2s(n)-B(\theta,\theta)\bigr),\quad D^3g(n)=\bigl(0,\Delta^3s(n)\bigr).\] The degree-two filtration makes \(D^3g=1\). Thus \(s(n)\) is an ordinary vector polynomial of degree at most two. No bound is imposed on the coefficients of \(z(n),s(n)\).

Choose \(\rho\) as the product of fixed one-dimensional nonnegative Gevrey partitions supported in \((-1,1)\), so that \(\sum_{m\in\mathbb Z^p}\rho(z-m)=1\). Right multiplication by the lattice element \((-m,0)\) gives \[F(g(n)\Gamma) =\sum_{m\in\mathbb Z^p}\rho(z(n)-m) \widetilde F\bigl(z(n)-m,s(n)-B(z(n),m)\bigr), \qquad \widetilde F(w,u)=F((w,u)\Gamma).\] The function \(\widetilde F\) is periodic in every central coordinate. On a bounded horizontal box its coordinate Lipschitz constant is quasipolynomially bounded in \(K_0,M_0,e^{d_0}\). To check this directly, the coordinate difference of two nearby group elements is \[(w,u)(w',u')^{-1} =\bigl(w-w',\,u-u'-B(w-w',w')\bigr);\] on a bounded box its coordinates are bounded by a controlled multiple of the coordinate displacement. The Malcev metric is bounded by the length of this coordinate displacement path. This proves the claimed coordinate Lipschitz estimate.

Multiply by a fixed horizontal cutoff equal to one on \(\mathop{\mathrm{supp}}\rho\) and supported strictly inside a larger period box. Periodize horizontally, and convolve in all coordinates with product Fejér kernels of order \(N_0\). For a Lipschitz function on a circle the error is bounded by its Lipschitz constant times \(C\log(2N_0)/N_0\), since this is an upper bound for the kernel’s first moment. Telescoping over the \(d_0\) coordinates gives the same estimate multiplied by \(d_0\). Choose \(N_0\) so that this error is at most \(\varepsilon_0/4\). It may be chosen quasipolynomially in \(K_0,M_0,e^{d_0},\varepsilon_0^{-1}\). The resulting trigonometric polynomial has the form \[\sum_{\ell,k}a_{\ell,k} e(\ell^Tw/P_2+k^Tu), \qquad \|\ell\|_\infty,\|k\|_\infty<N_0,\] where \(P_2\) is a fixed integer horizontal period. Each coefficient has absolute value at most a controlled multiple of \(K_0\), so its absolute coefficient sum is at most \(C K_0(2N_0+1)^{d_0}\). Because the partition \(\rho\) is nonnegative and sums to one, the approximation error remains at most \(\varepsilon_0/4\) after summing over \(m\).

For a fixed central frequency \(k\), define \(C_k\) by \(z^TC_kw=k^TB(z,w)\). The associated summand of this approximation is \[e\bigl(k^Ts(n)-z(n)^TC_kz(n)\bigr) \sum_{m\in\mathbb Z^p} \rho(z(n)-m)e\bigl(\ell^T(z(n)-m)/P_2\bigr) e\bigl(z(n)^TC_k(z(n)-m)\bigr).\] Its outside phase is an ordinary quadratic in \(n\). The matrix \(C_k\) is rational with controlled height, while the horizontal orbit is affine. Lemma 12 removes its constant offset, without putting an unbounded parameter in the amplitude. Thus each summand is an atom [eq:src-1]. The positivity of \(\rho\) shows that it has supremum at most one before the harmless normalization of the coefficients. Its complexity and the total absolute coefficient sum are quasipolynomial in \(\delta^{-1}\). It follows that one such discrete atom \(a(n)\) satisfies \[\left|\frac1L\sum_{j=0}^{L-1} g_L(j)\overline{a(j)}\right|\ge\kappa, \qquad \kappa^{-1}\le \exp\!\left(C(1+\log(1/\delta))^C\right).\] Here and below a shift of the polynomial argument incorporates the choice of integer labels.

We finish with an exact floor interpolation, which is necessary because the chosen atom may oscillate on the sampling scale. Write its parameters as \((\lambda,\beta,\theta,C,\chi)\). Let \(P\) clear the denominators of \(C\), and write \(\theta=\theta_0+Pq\), with \(q\in\mathbb Z^d\) and \(0\le(\theta_0)_j<P\). At integer arguments \(n\), reindexing the chart sum by \(Pqn\) replaces the ordinary quadratic coefficient by \(\lambda+(Pq)^TC\theta_0\); the other extra term is integral. Reduce this new coefficient and \(\beta\) modulo one. The same sampled atom consequently has a representation with \[0\le\lambda,\beta<1,\qquad 0\le\theta_j<P.\] Only this equality at integer arguments is needed.

Put \(\alpha=2\lambda+\theta^TC\theta\), and define the enlarged horizontal vector and matrix \[\Theta=\begin{pmatrix}\theta\\1\\\alpha\end{pmatrix}, \qquad \widetilde C= \begin{pmatrix} C&0&0\\ 0&0&0\\ 0&-1&0 \end{pmatrix}.\] Choose \(0\le\eta\le1\), Gevrey of the fixed order at cutoff scale \(\epsilon\), supported in \((0,1)\), and equal to one outside endpoint strips of total length at most \(2\epsilon\). Choose a one-dimensional Gevrey partition \(\sum_{\ell\in\mathbb Z}\rho_0(r-\ell)=1\). For lift variables \((w,h,r)\), define \[\widetilde\chi(w,h,r)= \chi(w-\theta h)\eta(h)\rho_0(r) e\!\left(-h\theta^TCw +(\lambda+\theta^TC\theta)h^2-\beta h\right).\] The atom \[A(v)=e(\lambda v^2+\beta v) \sum_{n,m,\ell} \widetilde\chi(\theta v-n,v-m,\alpha v-\ell) e\!\left((\Theta v)^T\widetilde C (\Theta v-(n,m,\ell))\right)\] satisfies the exact identity \[A(v)=\eta(\{v\})a(\lfloor v\rfloor) \qquad(v\in\mathbb R),\] with value zero at integer \(v\). Indeed the support of \(\eta\) forces \(m=\lfloor v\rfloor\); the \(\ell\)-sum is one. With \(h=v-m\) and \(w=\theta v-n\), the phase in the remaining summand is \[\lambda v^2+\beta v+(\theta v)^TCw -(2\lambda+\theta^TC\theta)vh-h\theta^TCw +(\lambda+\theta^TC\theta)h^2-\beta h,\] which equals \(\lambda m^2+\beta m+(\theta m)^TC(\theta m-n)\). This proves the identity.

All parameters occurring in the new amplitude are now bounded: the reductions above bound \(\lambda,\beta,\theta\) and hence \(\alpha\), while its support bounds \(h,r,w-\theta h\). The cutoff \(\eta\) has Gevrey derivative bounds with scale \(\epsilon^{-1}\). Thus the complexity of \(A\) is quasipolynomial in the old complexity and \(\epsilon^{-1}\); its two extra horizontal coordinates and its new matrix entry \(-1\) introduce no uncontrolled height. Moreover \(|A|\le1\), because its values are a cutoff times sampled values of the bounded atom.

Take \(v=Ly\). This changes only unrestricted real oscillation parameters. Exact cell averaging gives \[\int_0^1 g(y)\overline{a(\lfloor Ly\rfloor)}\,dy =\frac1L\sum_{j=0}^{L-1}g_L(j)\overline{a(j)}.\] Replacing the sampled atom by \(A(Ly)\) changes this integral by at most \(2\epsilon\), since the total measure of the removed cell endpoint strips is at most \(2\epsilon\). Choose \(\epsilon\le\kappa/8\). The resulting correlation is at least \(3\kappa/4\), and its complexity is quasipolynomial in \(\delta^{-1}\), independently of \(L\). This proves the normalized-interval assertion. Lemma 12 transfers it to the original interval. ◻

Proof. All functions and inner products in this proof use normalized measure on \(I\), and all functions are extended by zero for their \(\mathcal U(I)\) norms. Put \(\sigma=\tau/2\). Let \(\mathcal A\) be the bounded atoms of the complexity supplied by Lemma 19 at threshold \(\sigma^2\), restricted to \(I\), and let \(\kappa>0\) be the corresponding correlation bound. Choose \(M\ge\max(1,2\sigma/\kappa)\), quasipolynomial in \(\sigma^{-1}\). Consider the convex balanced set \[\mathcal B= M\operatorname{absconv}(\mathcal A) +\{u\in L^2(I):\|u\|_{\mathcal U(I)}\le\sigma\} +\{v\in L^2(I):\|v\|_{1,I}\le\sigma\}.\] Here the absolute convex hull consists of finite combinations. This set contains an \(L^2\) ball about zero.

Suppose \(f\) were outside the \(L^2\) closure of \(\mathcal B\). The real Hahn–Banach separation theorem, and the complex Riesz representation of a real functional as a real part, give \(h\in L^2(I)\), normalized so that \[\operatorname{Re}\langle f,h\rangle>1,\qquad \sup_{z\in\mathcal B}\operatorname{Re}\langle z,h\rangle\le1.\] Balancedness of each summand then implies \[\|h\|_{\infty,I}\le\sigma^{-1},\qquad |\langle u,h\rangle| \le\sigma^{-1}\|u\|_{\mathcal U(I)} \quad(u\in L^2(I)),\qquad \sup_{\varphi\in\mathcal A}|\langle h,\varphi\rangle|\le M^{-1}.\] The first inequality follows by testing bounded simple functions in the \(L^1\) ball; no assumption that the \(L^1\) dual is represented by an arbitrary measure is needed. Since \(\|f\|_{2,I}\le1\), the first strict pairing inequality gives \(\|h\|_{2,I}>1\). Testing the second bound with \(u=h\) yields \[\|h\|_{\mathcal U(I)} \ge\sigma\|h\|_{2,I}^2>\sigma.\] Thus \(\sigma h\) is bounded by one and has \(\mathcal U(I)\) norm greater than \(\sigma^2\). Lemma 19 supplies \(\varphi\in\mathcal A\) with \(|\langle h,\varphi\rangle|\ge\kappa/\sigma>M^{-1}\), a contradiction.

Therefore \(f\) belongs to the closure of \(\mathcal B\). Choose an element of \(\mathcal B\) within \(\tau/2\) in \(L^2(I)\). Its finite structured combination has coefficient sum at most \(M\). Put the approximation error into its \(L^1\) error term, using \(\|\cdot\|_{1,I}\le\|\cdot\|_{2,I}\). The \(L^1\) error is then at most \(\tau\), and the uniformity error is at most \(\tau/2\). All components lie in \(L^2(I)\), as required. ◻

Proof. Normalize \(I=[0,1)\). Extend the kernel smoothly to a sufficiently large fixed periodic \((x,t)\) box, supported where its real arguments lie in \(I\). The interior support margin permits this extension without changing the kernel on its support. Its Fourier coefficient sum is bounded by a constant determined by the fixed kernel bounds. Thus it suffices to prove the first estimate for each character-weighted progression average on the fixed circle.

Let \(R\) occupy slot \(j\), \(g\) slot \(i\), and the two bounded inputs slots \(k,\ell\). Disintegrate Haar measure along \(y_j=x+jt\) and apply Cauchy–Schwarz in that variable. The first cost is \(\|R\|_2^2\), and translation along the free direction \((-j,1)\) introduces an independent Haar difference parameter \(h_1\) in every remaining factor. Next disintegrate along \(y_k\), keeping \(h_1\) on the base. The repeated \(b\) factors are unchanged under translations in \((-k,1)\); Cauchy–Schwarz therefore removes them at cost \(\|b\|_\infty^4\) and introduces \(h_2\). The corresponding step at slot \(\ell\) removes the four repeated \(c\) factors at cost \(\|c\|_\infty^8\) and introduces \(h_3\). After these three steps the eighth power of the original average is bounded by \[\|R\|_2^8\|b\|_\infty^8\|c\|_\infty^8 \int\prod_{\omega\in\{0,1\}^3} \mathcal C^{|\omega|} g\bigl(y_i+\omega_1(i-j)h_1 +\omega_2(i-k)h_2+\omega_3(i-\ell)h_3\bigr).\] Every coefficient \(i-j,i-k,i-\ell\) is a nonzero integer. Multiplication by it preserves Haar probability measure, and the three parameters remain independent. The last integral is exactly \(\|g\|_{\mathcal U}^8\).

A Fourier character of \((x,t)\) may be distributed among slots \(0,1\), since \(x=y_0\), \(t=y_1-y_0\). Multiplication by a circle character preserves the \(L^2\) and supremum norms and the \(U^3\) norm, so the same estimate applies to every Fourier term. The argument starts with bounded inputs; the mixed-cube estimate in Lemma 18 and \(L^2\) approximation extend it to the stated class. Summing the kernel’s absolutely convergent Fourier series proves the first bound.

For the second bound, use the two distinct coordinates occupied by \(R\) and \(g\). Their Jacobian is a fixed nonzero integer. Bound the other two inputs and the kernel in supremum norm. The double integral then factors into the two \(L^1\) norms, giving the assertion after normalization. ◻

Proof. We keep track of the finite sequence of tolerances in the argument. Write \(q'=q/(q-1)<2\), and truncate each opposing input at absolute value \(B\ge1\), by putting its tail equal to zero. For the \(h\) tail, \[\|h\mathbf 1_{\{|h|>B\}}\|_{2,I}\le B^{-(q-2)/2}.\] For the \(u,v\) tails, their normalized \(L^2\) bounds give \[\|u\mathbf 1_{\{|u|>B\}}\|_{q',I}, \ \|v\mathbf 1_{\{|v|>B\}}\|_{q',I} \le B^{-(q-2)/q}.\] Use the local absolute form estimate with exponents \((2,2,2,2)\) for the first tail and \((2,q,q',2)\) for either of the other tails. In each case the reciprocal exponent sum is at most two. It follows, by replacing the inputs successively, that the total clipping error is at most \(C C_1B^{-(q-2)/q}\). Choose \[B=\max\!\left(2,(100C C_1/\delta)^{q/(q-2)}\right).\] The clipped form has magnitude at least \(3\delta/4\). All clipped inputs have supremum norm at most \(B\), and retain their original \(L^q\) or \(L^2\) bounds.

Apply Lemma 20 to the clipped \(h/B\) with \[\tau_1=\frac{\delta}{100C C_1B^3}.\] On multiplying back by \(B\), write its structured part as \(s_1=\sum_\nu a_\nu\varphi_\nu\), with \[\sum_\nu|a_\nu|\le A_1,\qquad A_1=B D_{\tau_1}.\] Lemma 21 bounds the two error contributions by \(C C_1B^3\tau_1\) each: the other two opposing inputs are bounded by \(B\), and \(\|R\|_{1,I}\le C_1\). The resulting form with \(s_1\) in this slot still has magnitude greater than \(\delta/2\).

Next apply Lemma 20 to the clipped \(u/B\) with \[\tau_2=\frac{\delta}{100C C_1A_1B^2}.\] Its structured part \(s_2=\sum_\mu b_\mu\psi_\mu\) has coefficient sum \[\sum_\mu|b_\mu|\le A_2,\qquad A_2=B D_{\tau_2}.\] Sum the error estimates separately over the atoms of \(s_1\). The two new error contributions are each at most \(C C_1A_1B^2\tau_2\). Therefore, denoting the remaining clipped input by \(v_B\), \[|H_I(R,s_1,s_2,v_B)|>\delta/4.\] By the absolute coefficient bounds, some bounded atom pair \(\varphi_i,\varphi_k\) satisfies \[|H_I(R,\varphi_i,\varphi_k,v_B)| >\delta_0,\qquad \delta_0=\frac{\delta}{4A_1A_2}.\] The complexities of this pair and \(\delta_0^{-1}\) are quasipolynomial in \(C_1,\delta^{-1}\), with a \(q\)-dependent constant but an absolute log exponent. Notice also that \(\|v_B\|_{2,I}\le1\).

Apply Lemma 15 to this pair. Let \(D_0\) bound its absolute coefficient sum and its shell constants. For each term without its coefficient the absolute local form is at most \(C C_1\), since its three atom factors are bounded and the two free inputs have the indicated \(L^2\) sizes. Discard shells above \(T_0\), with \(T_0\) chosen so that \(C C_1D_0T_0^{-100}<\delta_0/4\). The retained terms have quasipolynomial complexity. At least one such term has form magnitude at least \(\delta_1=\delta_0/(2D_0)\). The first part of Lemma 16 forces its Fourier supremum \(A_r\) to be at least \(\delta_1/(C C_1)\).

Use the second part of that lemma with a uniform kernel error at most \(\delta_1/(10C C_1)\). Pair independence bounds the resulting form error by \(\delta_1/10\). The surviving form is a finite sum \[\sum_\nu c_\nu \left(\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I R(y)\phi_{\nu,a}(y)\,dy\right) \left(\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I v_B(y)\phi_{\nu,b}(y)\,dy\right),\] with a quasipolynomial bound \(D_1\) for \(\sum_\nu|c_\nu|\) and for all atom complexities. Fixed coordinate Jacobians have been included in the coefficients. Every second moment has magnitude at most one, by the \(L^2(I)\) bound on \(v_B\) and the boundedness of its atom. Consequently one of the first moments has magnitude at least \(\delta_1/(2D_1)\). Conjugating its atom if needed gives the stated normalized moment.

There are exactly two bounded-input decompositions, followed by the progression separation and the single-scale separation. Thus all budgets above involve a fixed number of quasipolynomial compositions. The only \(q\)-dependent exponent used before these compositions is the power defining \(B\), whose logarithm is a \(q\)-dependent constant times \(1+\log C_1+\log(1/\delta)\). This proves the final quantitative assertion. ◻

Proof. Set \(\mathcal K=1+m+\log(2H)\). Throughout this proof, \(D_1,D_2,\ldots\) denote quantities with logarithms bounded by a polynomial in \(\mathcal K\). The single-scale part of Lemma 16 gives the bilinear bound \(CA_r\) before any expansion. When \(A_r>0\), Lemmas 13 and 14 give an expansion into terms \[e(\lambda_*t^2+\gamma t),\qquad |\lambda_*|r^2\le D_1.\] The possible unscaled \(\lambda_*\) and \(\gamma\) are fixed rational-height evaluations of the original parameters. In particular, they do not change their underlying parameter vector when \(r\) changes.

We first justify a finite truncation in a norm appropriate for a scale sum. Use the pooled tail conclusion of Lemma 14 to make the sum of the worst-case absolute tail coefficients, over all the effective quadratic and linear choices, smaller than \(e^{-2m}(1+D_1)^{-2}\), with any additional fixed polynomial reserve needed below. For each fixed tail pair \((\lambda_*,\gamma)\), write \(e(\lambda_*t^2)=1+(e(\lambda_*t^2)-1)\). The linear part is estimated by the mean-zero square functions proved below, with a fixed modulation \(\gamma\). The second part has smooth-kernel norm at most \(C|\lambda_*|r^2\operatorname{poly}(D_1)\); increasing the tail reserve handles this polynomial. At a fixed \(\lambda_*\), \[\sum_{r:\,|\lambda_*|r^2\le D_1}|\lambda_*|r^2\le C D_1.\] At each depth, contractions and Cauchy–Schwarz give the requisite product of ordinary \(L^2\) norms; for an energy sequence the same argument costs at most \(1+L\). Thus the tail is acceptable in the absolute scale sum, not merely at an individual interval.

There is now a finite main catalog, with absolute coefficient norm and rational heights bounded by \(D_2\). Choose a sufficiently small \(\epsilon>0\) after that catalog, with \(\log(1/\epsilon)\) polynomial in \(\mathcal K\). Mark as exceptional every scale where a possible nonzero \(|\lambda_*|r^2\), or a possible nonzero \(r|\gamma-\gamma'|\), belongs to a prescribed intermediate annulus between \(\epsilon\) and \(\epsilon^{-1}\). Enlarge the annulus by fixed powers of the catalog bounds when necessary. The height-bin observation from Section 3 bounds the number of exceptional scales by a polynomial in \(\mathcal K\). At these scales use the original single-scale bound \(CA_r\le Ce^{-m}\), before summing coefficients. This gives the desired estimate for the exceptional scales.

At each remaining scale the effective quadratic is negligible, and the linear centers partition into clumps of diameter at most \(\epsilon/r\), separated by more than \(\epsilon^{-1}/r\). One way to arrange the stated diameter is to use a smaller tiny threshold than \(\epsilon\) by the main catalog count, and then take connected components of the tiny-distance relation. The enlarged intermediate annulus excludes all intermediate distances. Replace every effective quadratic by zero and every center in a clump by one representative. The height-bin observation gives \[\sum_r\max\{r^2|\lambda_*|:\ r^2|\lambda_*|\text{ is tiny at }r\} +\sum_r\max\{r|\gamma-\gamma'|:\ r|\gamma-\gamma'|\text{ is tiny at }r\} \le \epsilon\operatorname{poly}(\mathcal K),\] after readjusting \(\epsilon\) by catalog factors. Multiplication by the absolute coefficient norm therefore makes the total replacement error smaller than the right side of [eq:src-5].

Every combined clump coefficient is \(O(e^{-m})\). Indeed, test the original expansion of \(Q(r\,\cdot)\) against \(\zeta(u)e(-r\gamma_{\mathcal C}u)\), where \(\gamma_{\mathcal C}\) is the chosen representative. The coefficient of its own clump is multiplied by the nonzero constant \(\int\zeta\). All other clumps give rapidly decreasing Fourier tails, and the quadratic, truncation, and replacement errors have already been made smaller than \(e^{-m}\) by the choice of \(\epsilon\) and the tail reserve. The defining Fourier supremum is at most \(e^{-m}\).

For completeness, we prove the square-function estimate used for these linear clumps. Make the fixed change of coordinates \((x,t)\mapsto(y_a,y_b)\). In these coordinates, \(w_I\) has zero integral in either variable while the other is fixed. Its double primitive has compact support in an interior rectangle of \(I\times I\). Expand that primitive in a smooth Fourier series on a fixed larger rectangle, multiply by interior cutoffs equal to one on its support, and differentiate once in each variable. This expresses the kernel as a sum of tensor products of smooth, interior, mean-zero bumps. Their coefficients decrease faster than every fixed polynomial in the tensor indices, while their fixed-order derivative bounds grow at most polynomially. It is consequently enough to consider rows \[\mathcal W_{I,\nu}F =|I|^{-1/2}\int_I e(\nu y)\omega_I(y)F(y)\,dy, \qquad \nu=\pm\gamma/(b-a),\] where \(\omega_I\) is such a bump. The fixed bounded multiplier \(\psi_a\) or \(\psi_b\) can be incorporated into each row.

These rows are Bessel with squared bound polynomial in \(\mathcal K\), jointly over the retained scales, intervals, and clump representatives. Here are explicit frequency estimates. Decompose the input for a row into the demodulated shells \(|\xi+\nu|\asymp2^\ell/|I|\). At fixed scale and center, mean zero and the \(L^2\) Poincaré inequality give row-operator norm \(O(2^\ell)\) when \(\ell<0\). When \(\ell\ge0\), two frequency antiderivatives and integration by parts give \(O(2^{-2\ell})\). These estimates are summed over the disjoint intervals at that scale. For a fixed Fourier variable \(\xi\) and shell index \(\ell\), the height-bin observation, applied to the original horizontal parameters, the fixed modulation, and \(\xi\), bounds the number of possible scales by \(\operatorname{poly}(\mathcal K)\). At one scale, separation of the centers bounds their shell multiplicity by \(O(1+2^\ell)\). Thus the total frequency multiplicity is \(\operatorname{poly}(\mathcal K)(1+2^\ell)\). Taking its square root and summing the preceding row norms over \(\ell\) proves the Bessel estimate. Exact center frequencies form a null set and cause no additional term. A fixed bounded multiplier preserves the estimate.

For a projection sequence, apply Lemma 9 to these rows. Its cost is polynomial in \(\log(2D_{\mathrm h})\). For an energy sequence, use instead the per-interval Bessel estimate over separated centers and then sum its given energy bound; no composition lemma is needed. Cauchy–Schwarz over the two sets of rows, together with the \(O(e^{-m})\) bound for each combined coefficient, proves [eq:src-5]. If \(A_r=0\), the cutoff Fourier transform is zero, so \(Q\) vanishes on the part of the \(t\)-axis relevant to the kernel. ◻

Proof. Expand the low pair by Lemma 15. Its dyadic shell coefficients have the stated rapidly decreasing absolute sums. Apply Lemma 23 to each shell and sum its Fourier-supremum bands. First discard sufficiently high separation shells; then, for each retained term, discard sufficiently small \(A_r\) bands. The shell and band thresholds have logarithms polynomial in \(1+B+\log(2H_{\rm low})\). The remaining scales admit the finite chart expansion from Lemma 14. Truncate its pooled tails in the absolute scale-sum norm exactly as in the proof of Lemma 23. Distribute the fixed modulation \(e(\gamma t)\) between positions \(a,b\). The surviving rational-height choices of quadratic parts and linear shifts form a fixed finite list. All of its bounds have the asserted logarithmic size. The discarded errors have the required sum by [eq:src-5]. ◻

Proof. The predictor identity expresses the multiplier-weighted mean of the physical residual as \(\epsilon\) times the predictor’s column coefficient. Smooth-bump approximation below the delayed partition supplies the \(2^{-K_0}\) term, while subtracting the coarse mean and summing private-coordinate increments gives the \((1+K_0)\sqrt\epsilon\) term.

Let \(E_h\) denote conditional expectation on the depth-\(h\) partition of \(J\), and write \(P_h\) for the direct sum of its local projections. On descendants where the predictor is present, put \[D_h=(f_a-(P_hF_a)_{\mathrm{phys}})\psi_a, \qquad \mu_h=E_hD_h.\] If \(a_{h,p}\) is the predictor coefficient on a depth-\(h\) cell, orthogonality to \(\Phi_p\) gives \[\mu_h=\epsilon a_{h,p}.\] Consequently \(\mu_{h+1}-\mu_h\) is \(\sqrt\epsilon\) times the change of the \(p\)-th private coordinate of \(P_hF_a\). This identity continues to hold after arbitrary further column insertions.

For a smooth normalized bump \(\omega_I\) on an interval of depth \(s\), set \(H=s+K_0\). The residual moment has the exact telescoping representation \[\begin{split} \int_I D_H\omega_I ={}&\int_I\mu_H\omega_I\\ &+\sum_{h\ge H}\int_I(D_h-D_{h+1})(\omega_I-E_h\omega_I)\\ &+\sum_{h\ge H}\int_I(\mu_{h+1}-\mu_h) (E_{h+1}\omega_I-E_h\omega_I). \end{split}\] To verify it, telescope \(\int_I D_h(\omega_I-E_h\omega_I)\) and use \(E_{h+1}D_{h+1}=\mu_{h+1}\); the term involving \(\mu_h\) and \(E_{h+1}\omega_I-E_h\omega_I\) vanishes by conditional expectation. The final remainder tends to zero because \(D_h\) has uniformly bounded \(L^2(I)\) norm and the smooth bump’s conditional-expectation error tends to zero. Thus no convergence of \(P_hF_a\) to \(F_a\) is being assumed.

The smoothness of the bump gives a factor \(O(2^{s-h})\) in each of the two infinite sums. Its other factor is a localized orthogonal projection increment, with an additional \(\sqrt\epsilon\) for the private-coordinate term. At a fixed depth offset \(h-s\), sum squared row norms over intervals of depth \(s\), then sum over \(s\) using orthogonality of the increments \(P_{h+1}-P_h\). Minkowski’s inequality over the offsets gives an aggregate row norm \[C2^{-K_0}\|\mathbf 1_JF_a\|_2.\] This estimate uses only the bounded multiplier \(\psi_a\), and is unaffected by the size of the enlarged column catalog.

If \(\omega_I\) has mean zero, the first term can be written with \(\mu_H-\mu_s\). Decomposing it into the \(K_0\) successive private increments gives aggregate row norm at most \[C K_0\sqrt\epsilon\,\|\mathbf 1_JF_a\|_2.\] Apply the mean-zero tensor decomposition from the preceding proof to \(w_I\), including the fixed distributed modulation in the fixed multipliers. Pair the resulting error rows in position \(a\) with the Bessel rows in position \(b\). Compression handles a current projection in that position; an energy sequence is handled by its per-interval Bessel estimate. This proves [eq:src-7] for the part of the kernel with \(e(\lambda_*t^2)\) replaced by one.

For the difference kernel \(w_I(e(\lambda_*t^2)-1)\), an ordinary smooth tensor expansion suffices. Its derivative costs contain the factor \(|\lambda_*|r^2\operatorname{poly}(D_{\rm pred})\). The same telescoping argument estimates its expectation-free part. For the initial \(\mu_H\) term, which need not be tested against a mean-zero bump, use the per-depth bound \(C\sqrt\epsilon\|\mathbf 1_JF_a\|_2\) and sum \(|\lambda_*|r^2\) geometrically over the scales satisfying \(|\lambda_*|r^2\le D_{\rm pred}\). This accounts for the additional \(\sqrt\epsilon\) term in [eq:src-7].

For sequential replacement, write a previously delayed projection as its current projection plus a rolling projection difference. The latter has a square-energy bound with \(L\le C(1+K_0)\), by telescoping its orthogonal increments. It is therefore an admissible opponent in the second application of the argument. ◻

Proof. Start with no columns at a segment root \(J\) and proceed in depth order. At a cell whose residual fails the desired inequality, Lemma 22 supplies a bounded atom \(\varphi\) with physical residual correlation at least \(\rho\), where \(\log(1/\rho)\le k^{O(1)}\) and the atom complexity obeys the same logarithmic bound. Adjoin \((\varphi,\mathbf e_v)\) with a fresh private coordinate and refit. The old residual has zero component in this fresh direction. The squared projection energy therefore increases by at least \(c\rho^2|I|\). Repeated fits at one cell and refining fits at successive depths form one nested sequence of subspaces in \(L^2(J)\), so their total energy gain is at most \(\|\mathbf 1_JF_j\|_2^2\le|J|\).

Stop at the first cells where the cumulative number of insertions along a path would exceed \(C\rho^{-2}\). Integrating the insertion count over the disjoint first-excess cells bounds their total length by \(C^{-1}\) times a fixed multiple of \(|J|\). Choose \(C\) so this is at most \(|J|/4\), and restart the same procedure on those roots. Iteration gives a geometric bound for the sum of segment-root lengths. On every retained cell the greedy fit has terminated with the required residual estimate.

With private weight one, the coefficient \(\ell^2\) norm is bounded by the normalized projection norm. A catalog of at most \(\exp(k^{O(1)})\) bounded scalar columns consequently gives the claimed physical supremum bound by Cauchy–Schwarz. Complexity and count bounds follow from detection and the insertion cap. The uniform polynomial-degree assertion is inherited from Lemma 22. ◻

Proof. For a finite terminal level, decompose each actual input into its increments up to that level and its terminal residual. In every term having at least two increments, record the earliest two levels \(u<t\). Summing all later choices in either other position gives exactly the input minus its last update before \(t\). This yields [eq:src-8] for all such terms. The terms with at most one increment consist of either four terminal residuals, or one whole terminal projection and three terminal residuals.

We record the comparison that controls these remainders and the claimed pair estimates. If \(h>\ell\) are updates in distinct positions, replacing their pair of projections by the corresponding actual inputs has form-norm error at most \[C_q\bigl(e^{-k_h}\exp(C_q k_\ell^{d})+e^{-k_\ell}\bigr)\] for an absolute \(d\). Replace the more accurate projection first, using the physical supremum bound of the other as its opposing \(L^q\) norm, and then replace the less accurate projection against the actual \(L^q\) input. Choose \(A>d\), with fixed reserve. The same reasoning applies when a projection is already replaced by an actual input. In particular, a pair of terminal residuals has form norm tending to zero. Their two remaining opponents have bounded \(L^2\) size, proving the assertion about the terminal remainder on a finite interval collection.

For the tuple estimate, first suppose \(u<t-4\). Whenever the early argument in position \(i\) is paired with a later difference or residual, compare that later argument directly with its actual input against each of the two earlier projections. The actual terms cancel in the later difference or residual. Its smallest accuracy index is at least \(t-4\), and exceeds the relevant earlier indices; the stagger therefore defeats their supremum bounds. Pairs not involving this early position use only indices at least \(t-4\), apart from the preceding projection in position \(j\), whose index is exactly \(t-4\).

If instead \(u\ge t-4\), distinct positions force \(u\ge t-3\); all nonzero projections in the tuple then have indices at least \(t-7\). Expand a selected pair into projection/actual pairs and use the preceding comparison. At least one selected full argument has coefficient sum zero after replacing projections by their actual inputs, so the two-actual terms cancel. The same argument covers the pairs in the first case that do not contain the early position. Each error is bounded by a fixed multiple of \(\exp(-c_q k_{t-7})\) for large starting log. Since \(k_{t-7}=k_t^{A^{-7}}\), decreasing a fixed exponent below \(A^{-7}\) gives the stated \(e^{-2k^{\theta_0}}\) bound. The \(L^2\) size assertion follows from contraction and the triangle inequality. ◻

Proof. We give the choices in their dependency order. On a common older segment, the low columns, their coefficients, and their path counts have a bound \(D_{\rm old}=\exp(k^{O(1)})\). Apply Corollary 24 to every available pair of low labels, using a precision log \(B\) that is a sufficiently large fixed polynomial in \(k\). Insert the conjugate predictor columns in positions \(a,b\) at the first joint availability of that pair, and retain them thereafter. Pairs inherited at the segment root have their predictors present from that root. Use private weight \(\sqrt\epsilon\) for these predictors.

Choose \(K_0\) and \(\log(1/\epsilon)\) to be sufficiently large fixed polynomials in \(k\), after \(D_{\rm pred}\) and the original low coefficient/count bounds have been fixed. The choices make both the error of Corollary 24 and the error of Lemma 25 smaller than \(e^{-2k}\) after all original-label sums. Only a polynomial in the logarithm of a subsequently enlarged high catalog occurs in these errors. Thus the choices remain valid under any later fixed-polynomial growth of that log catalog bound, by increasing the starting \(k\).

Here is the summation over low labels in detail. Expand the two low projections into their original labels only in this application. A fixed pair is available on the subtree rooted at its deeper birth cell, or on the segment root if it was inherited. Its coefficients on a use cell are constant and bounded by a fixed power of \(D_{\rm old}\). The number of such pair-birth roots on a path is also bounded by a fixed power of \(D_{\rm old}\). Consequently, on a normalized segment root \(R\), \[\sum_{J\text{ pair-birth root}} \|\mathbf 1_JF_a\|_2\|\mathbf 1_JF_b\|_2 \le D_{\rm old}^{C}|R|.\] Indeed Cauchy–Schwarz bounds this sum by the product of the square roots of the corresponding two sums of squared masses, and the path multiplicity bounds each by \(D_{\rm old}^C\|\mathbf 1_RF_e\|_2^2\). Apply Corollary 24 on both sides of each replacement, and then Lemma 25 to each retained term. A previously delayed opposing projection is its current projection plus a rolling square-energy sequence with \(L=O(1+K_0)\). This proves the absolute-sum error assertion, even after the extra fits now described.

The predictors alone need not give the pair estimates in the statement. We therefore add tuple-specific accurate fits. Proceed chronologically in projection depth \(h\). At depth \(h\), first admit any predetermined predictor columns becoming available there. A coarse decision cell at that step has depth \(s=h-K_0\). Test the physical residual \[f_e-(P^{\rm new}_{e,h}F_e)_{\mathrm{phys}}\] on that coarse cell in the norm \(\mathfrak d_{e,I}\). The normalized \(L^2(I)\) size of this residual is bounded, because \(P^{\rm new}_{e,h}\) is a contraction on the partition of \(I\) into its depth-\(h\) descendants. Require a precision whose log is a sufficiently large fixed power of \(k\). On failure, detection gives a bounded scalar column with a coarse residual moment at least \(\rho'\), where \(\log(1/\rho')\le k^{O(1)}\). Insert one new common label with private weight one on all the depth-\(h\) subcells of \(I\), and retain that label on their descendants.

The joint gain from this insertion is at least \(c(\rho')^2|I|\). To see this, let \(m_Q\) be the normalized residual moment on a depth-\(h\) subcell \(Q\subset I\). The fresh private direction is absent from every old tested residual, so the gain on \(Q\) is at least \(c|Q||m_Q|^2\). Cauchy–Schwarz and the coarse moment bound give \[\sum_{Q\subset I,\,\operatorname{depth}(Q)=h} c|Q||m_Q|^2 \ge c|I|\left|\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I\text{residual}\,\overline{\varphi}\right|^2 \ge c(\rho')^2|I|.\] All ancestor extra requests affecting a coarse cell are common to its fine partition. Cap their cumulative number along a coarse path at a sufficiently large multiple of \((\rho')^{-2}\). First excess coarse cells have small total relative length, by the joint gains and the orthogonal energy bound for the entire chronological iteration. Send their coarse summands and descendants to fresh attempts. Predictor admissions have their separate path cap from the older lists and the finite predictor catalogs.

There is no conflict between a coarse cut and a delayed projection already used above it. If a coarse cell is cut when projection depth is \(h\), all its strict ancestors have already had their delayed evaluations computed, since their projection depths are smaller than \(h\). Continue the stopped estimator on the cut subtree by inheritance and refitting, with its predictor moments retained; its later residual tests are no longer needed for the summands sent to the new attempt. This extension supplies all finer projections needed in the telescoping proof of Lemma 25. A fresh attempt for the descendant coarse summands may use a new extra-label catalog. Intersect the resulting coarse segment families between the two high positions and the older lists. Their roots still have bounded total length, and each resulting piece uses restrictions of genuine nested projection sequences, including their extensions beyond its boundaries.

The accuracy order is as follows. Choose the first new high precision to dominate all previous projection and predictor costs. Detection fixes the first new catalog bound. Choose the second new high precision to dominate that first catalog’s physical supremum bound as well. For a fine subcell \(Q\subset I\) at depth \(s+K_0\), \[\|F_e\|_{2,Q}\le 2^{K_0/2}\|F_e\|_{2,I}.\] The private-coordinate coefficient estimate thus bounds the first new projection’s physical supremum on \(I\) by \(2^{K_0}\) times a fixed power of its own count and conditioning bound. It does not involve the next high precision. The two choices are therefore successive fixed-polynomial log choices.

Apply the comparison argument of Lemma 27 to these two new approximants and the older projections. The new fits are more accurate than every older projection, and the second is more accurate than the first with sufficient reserve for its supremum bound. In each pair of full arguments in [eq:src-9], the actual-input contributions cancel exactly as before. If the earliest low position has a much earlier index, the other selected full argument is a difference of later approximants and is tested directly against that earlier position. Otherwise all older indices involved are at least \(t-7\). Decreasing the absolute exponent from Lemma 27 yields the claimed \(\eta\) bound. Contraction on the fine partition and on the older coarse cells gives the local \(L^2\) bounds. Finally, the new high fits are specific to this tuple; their larger precisions do not enter the global update recurrence \(k_{t+1}=k_t^A\). ◻

Proof. Fix a path point \(y\), and let the fourth position be \(l\). Writing the transpose at \(y+h\), its squared \(L^2\) norm is an integral in the scaled variables \((h,t,t')/r\), \(r=|I|\), containing the fixed triple and its conjugate copy and the two smooth kernels at the shared anchor. The six atoms combine into the multivariable chart setting of Lemma 13. Recenter at the fixed \(y\). All unscaled horizontal, linear, and quadratic parameters are now fixed along the path, before multiplication by \(r\) or \(r^2\).

If \(\mathcal A_I>\delta\), apply Lemma 13 to this integral at Fourier frequency zero, and then Lemma 14 with an error small compared with \(\delta^2\). Every effective quadratic entry and every linear \(t\)-frequency in the truncated expansion is of the form \(r^2 a\cdot\chi_*\) or \(r a\cdot\chi_*\), respectively, for a fixed finite parameter vector \(\chi_*\). The coefficient vectors have rational height quasipolynomial in \(D_{\rm base}\) and \(\delta^{-1}\); the dimension is polynomial in the original chart dimension. This description also covers the choices of rational splitting.

Choose tiny/large gap thresholds after the finite truncation, with reserve for its absolute coefficient norm. Outside polynomially many intermediate scales, the height-bin observation makes every effective quadratic entry that is bounded in the Weyl expansion tiny, and every retained \(t\)-frequency either tiny or large. Remove the effective quadratic by uniform approximation. Large \(t\)-frequencies integrate to a negligible amount by smoothness of the kernel. For a tiny \(t\)-frequency, replace its oscillation by a constant; the remaining \(t\)-integral is zero by the exact marginal \[\int w_I(y+h-lt,t)\,dt=0\] at fixed \(h,t'\). The other kernel and the remaining linear oscillations are independent of this \(t\) variable. All errors, after coefficient summation, are smaller than \(\delta^2\). This contradicts \(\mathcal A_I>\delta\) and proves the count.

Apply that count to the successive bands \(2^{-n-1}\delta<\mathcal A_I\le2^{-n}\delta\), using threshold \(2^{-n-1}\delta\). Their total contribution is bounded by \[\sum_{n\ge0}2^{-n}\delta\, \operatorname{poly}\bigl(\log D_{\rm base} +\log(1/\delta)+n\bigr),\] which is the asserted small-norm sum. The kernel may vary with \(I\) within its fixed class: only its uniform smoothness and exact marginal were used. ◻

Proof. Declare \(I\) active if some triple of available pure basic scalar columns in three positions has transpose norm greater than \(D_{\rm base}^{-C'}\), with \(C'\) a sufficiently large fixed constant. Along a path there are at most \(D_{\rm base}^3\) triples, and Lemma 29 gives only a logarithmic-polynomial number of active scales for each one. Enlarging the absolute power proves [eq:src-10].

On an inactive interval, expand three of the \(B\) arguments into their basic columns. Their coefficient sums and the local \(L^2\) size of the remaining argument cost only a fixed power of \(D_{\rm base}\). For a fixed triple, the small-norm assertion of Lemma 29 bounds its path sum by \(D_{\rm base}^{-C'} \operatorname{poly}(\log D_{\rm base},C')\). Sum over the pathwise triple count and integrate over path points. Choosing \(C'\) larger than all these fixed power costs, and using \(D_{\rm base}\ge e^k\), gives the stated \(e^{-k}|R|\) error.

For each fixed leading level, all budgets and the multiplicity in [eq:src-10] are fixed constants. Local absolute \(L^2\) estimates and bounded sizes thus give a bound independent of the number of depths. For later levels, the pair estimate in Proposition 28 and the bounded sizes of the other two arguments give \(|H_I(Y)|\le C\eta\). Decrease \(\theta\) once more, and increase the starting log, to absorb this fixed constant into the definition of \(\eta\). Multilinearity expands \(H_I(Y)\) into the retained patterns and the already estimated term with both \(U\) arguments. For nonnegative quantities, \(\min(\eta,\sum_j a_j)\le \sum_j\min(\eta,a_j)\). The absolute errors and the two-\(U\) term may therefore be taken outside the minimum, leaving [eq:src-11] separately for the remaining patterns.

This reduction retains the whole coarse arguments \(B_l\). In particular, later filter removal in two nonside positions returns exactly these arguments; substituting \(Y_l=B_l+U_l\) changes the term only by terms with an additional basic side. The full complementary pair then has the pair-smallness already proved. All relevant current families remain inherited under restriction to a normalized subroot, and their rolling square estimates follow anew from orthogonal energy there. Hence a later stopping attempt can use the same base data. ◻

Proof. For the inner product, smooth the interval indicator with \(L^1\) error much smaller than \(\tau_2\) and derivative bounds polynomial in \(\tau_2^{-1}\). The physical columns have bounded supremum norm. A larger inner product would therefore give a Fourier coefficient of their conjugate product above a fixed multiple of \(\tau_2\) after recentering and rescaling. Lemma 13 gives [eq:src-2] for the difference data. The fixed factor \(c_i\) in a same-slot product only changes the allowable height by a fixed power. Choosing \(H_{\rm ed}\) above this bound proves the first assertion.

For the second assertion, call the source slots \(i,j\) and the spectator slots \(a,b\). In centered, rescaled spectator coordinates \((z,w)\), the linear parts of the source coordinates are \(r(\xi_i z+\zeta_i w)\) and \(r(\xi_j z+\zeta_j w)\), where \[\xi_l=\frac{b-l}{b-a},\qquad \zeta_l=\frac{l-a}{b-a}.\] All four coefficients for \(l=i,j\) are nonzero fixed rationals. The identities \(\sum_l c_l l^u=0\) for \(u=0,1,2\) give \[c_i\xi_i\zeta_i=-c_j\xi_j\zeta_j=:\kappa\ne0.\] First clip each spectator at a sufficiently large inverse polynomial in \(\tau_2\). The discarded part has \(L^1\) norm at most the inverse clipping height, since its \(L^2\) norm is bounded. The absolute form estimate with exponents \(1,2,\infty,\infty\) bounds this error. We can therefore suppose that the spectators have both bounded \(L^2\) norm and supremum norm at most an inverse polynomial in \(\tau_2\).

Use the return calculation from the proof of Lemma 13 for the displayed difference data. Choose an integer period clearing the chart and interpolation denominators. For a small return \(Vh=m+d_h\), with \(m\) divisible by this period, translate \(z\) by \(h\) and translate the source lift indices by \(\xi_i m_v,\xi_j m_w\). The amplitude and kernel errors are bounded by \(\lvert h\rvert+\lvert d_h\rvert\) times a quasipolynomial in the available complexity. The resulting modulation in \(w\) is \[\kappa\Omega(h,m),\qquad \Omega(h,m)=(2\Lambda+V^T C V)h-V^T(C-C^T)m.\] Indeed the mixed quadratic contribution in each source contains \(c_l\xi_l\zeta_l\); factoring out \(\kappa\) leaves exactly the difference of the two curvature data. The modulation in \(z\) can be included in the first spectator and has no effect on its \(L^2\) norm.

Suppose the original correlation is larger than \(\tau_2\). Applying the calculation to \(\ell(h,m)\), for a sufficiently large quasipolynomial number of integers \(\ell\), produces correlations of size at least a fixed multiple of \(\tau_2\) between one fixed bounded kernel \(K_1(z,w)\) and tests \(f_\ell(z)g_\ell(w)\). Here the \(f_\ell\) have bounded \(L^2\) norms, and \[g_\ell(w)=g(w)e\bigl(\ell\kappa\Omega(h,m)w\bigr)\] up to a common bounded multiplier. The error remains small by choosing the return threshold sufficiently small before choosing these multiples. If \(\lvert \Omega(h,m)\rvert\) exceeds a fixed constant, the separated exponentials, multiplied by the clipped spectator, are Bessel on the fixed box, with bound controlled by that spectator’s squared supremum norm. Cauchy–Schwarz in \(z\) gives \[\sum_\ell\left|\int K_1(z,w)f_\ell(z)g_\ell(w)\,dz\,dw\right|^2 \lesssim \int\sum_\ell\left|\int K_1(z,w)g_\ell(w)\,dw\right|^2dz.\] The right side is bounded by the same Bessel constant times \(\lVert K_1\rVert_2^2\), while the left side grows with the number of returns. This is a contradiction. We have proved a uniform bound for \(\Omega(h,m)\) on every return below a quasipolynomially small threshold.

The final, algebraic part of the proof of Lemma 13 now applies to these difference data: the alternating return identity gives an isotropic lattice space, successive minima give a rational fast projection and a bounded slow component, and one return in each coordinate direction bounds the effective quadratic form. This part uses only the just-established return bound. It yields [eq:src-2] at quasipolynomial height, contradicting \(d_s(v,w)>1\) when \(H_{\rm ed}\) is chosen sufficiently large. ◻

Proof. List the path tags as \(v_0,\ldots,v_L\), and put all their horizontal coordinates in one direct sum. For each edge choose its fast space \(U_e\) from the definition of an edge. Let \(U\) be the subspace of the joint coordinate space whose restriction to each consecutive pair belongs to \(U_e\). These are rational linear equations. Clearing denominators and applying determinant bounds to one maximal independent system shows that the orthogonal projection onto \(U\) has logarithmic height polynomial in \(L\), the dimensions, and \(\log H_{\rm ed}\). The same determinant estimate bounds the norm of a right inverse on the range of this system.

Let \(V\) be the concatenation of the normalized horizontal vectors. Each defining equation has bounded error on \(V\), because the edge slow component is bounded. The right-inverse bound therefore shows that \(V^F\), the orthogonal projection of \(V\) onto \(U\), satisfies \(\lVert V-V^F\rVert\le H_1\), where \(\log H_1\le\operatorname{poly}(P,k)\). On an edge, both its chosen fast component and the restriction of \(V^F\) lie in \(U_e\), and their difference \(\Delta\) is bounded by this new budget. Since \(U_e\) is isotropic for the edge skew form, replacing its fast component by this restriction changes the effective correction by \[\operatorname{sym}(\Delta^T C_{{\rm sk},e}V_e) =\operatorname{sym}(\Delta^T C_{{\rm sk},e}V_{s,e}),\] which is bounded at the new budget. In particular, no unbounded fast coordinate enters this error.

Sum the edge effective quadratic forms using these common joint fast and slow components. The diagonal symmetric terms, the skew mixed terms, and the original \(\lambda\) terms telescope, leaving only the two endpoints. Let \(U'\) be the image of \(U\) under endpoint restriction. It is rational with the same type of height bound. For two vectors of \(U\), summing their edge skew pairings cancels all intermediate coordinates; each edge pairing is zero. Hence \(U'\) is isotropic for the endpoint difference skew form. The restricted joint fast vector lies in \(U'\) and is within the new bounded error of the endpoint \(V\). Replacing it by the orthogonal projection of endpoint \(V\) onto \(U'\) changes it by a bounded vector of \(U'\). The preceding isotropy calculation again bounds the change in effective curvature. These are all the assertions of [eq:src-2] in endpoint coordinates. ◻

Proof. At a present vertex a round’s top score lies in \(\{0,\ldots,d_0\}\): the vertex itself is a competitor of distance zero. Individual primary competitor scores only increase with depth. At a fixed top score a marked winner can occur in at most one episode, with a fixed identity. Once a rival has destroyed its gap, that rival’s score cannot decrease; a different marked winner at the same top value would have the old winner as a rival of that value. Thus the first marked primary round, its score, and its winner have at most \(5d_0^2\) episodes.

Within a primary episode there are at most \(d_0\) cohorts. Within one cohort all profile metrics are fixed. The vector of floor log-sizes of all the balls has at most \(2d_0^2\) epochs, including its initial value, since its coordinates increase and each has at most \(O(\log D)\) possible log-sizes. Within each such epoch the selected \(l,n\) are fixed. In that epoch the high metric is fixed, so high scores increase only by admission of competitors. The same marked-episode argument gives at most \(5d_0^2\) episodes of selected high data. Altogether there are at most \(50d_0^7\le d_0^8\) changes when \(k\) is large. Seven metadata coordinates, each with at most \(d_0+2\) states, and the change counter require fewer than \(256d_0^{15}<d_0^{30}\) ordered states. Decrementing the counter at each change makes the assigned color nonincreasing even if the other metadata increase.

For completeness, a fixed untruncated score situation is marked with probability at least \(2^{-5}\). Choose a canonical winner in every delay outcome, resolving ties deterministically, and increase only its delay by \(5\). The resulting outcome has a unique winner with gap at least \(5\). This map is injective, because its image identifies the increased coordinate; geometric probabilities decrease by exactly \(2^{-5}\). Truncation changes this probability by at most \(D2^{-d_0}\). In particular it remains at least \(1/64\) for large \(k\).

There are only polynomially many situations to check on a fixed spatial path. For the primary stage cap \(\lfloor d_s(z,v)/b_1\rfloor\) at \(d_0+5\); larger values cannot affect any top or gap decision. The capped pair distances change only \(O(D^2(d_0+1))\) times, by monotonicity. Counting the vertices to test gives \(O(D^3(d_0+1))\) fixed primary situations. Conditional on all primary arrays, the primary, cohort, and ball epochs number at most \(O(Dd_0^5)\) across vertices. Each has at most \(D\) present prefixes, so at most \(O(D^2d_0^5)\) high situations require checking. These situations are fixed before revealing the independent high arrays. The probability that all \(d_0\) rounds fail at any such situation is at most \((1-1/64)^{d_0}\). The union bound tends to zero with \(D\), since \(d_0\ge1000\log(2D)\).

Integrate this pathwise failure probability over \(R\). Some realization of the arrays has a small measure of failed paths. The first failure cells are disjoint, and their total length is at most that measure. This proves the claimed exceptional-root bound without a bound on the number of vertices across different spatial branches. ◻

Proof. At a marked primary center \(v\) at depth \(s\), each rival’s true score is at most \((g-5)b_1\), since its quantized score is at most \(g-5\). Consequently \[f_v(s')\le(g-5)b_1<T_l\quad(s'\le s).\] The winning true score is greater than \((g-1)b_1\). If \(d_s(b,v)\le h\), the winner score at \(b\) loses at most \(h\), and \(f_b(s)\le(g-5)b_1+h<T_l\). Both profiles are below the clipping range at every earlier depth. At all later depths the graph distance remains at most \(h\), so their true profiles, and then their clipped profiles, differ by at most \(h\). Thus both profile distances from \(b\) to \(v\) are at most \(h\), which proves the first assertion with witness \(v_0=v\).

Next compare a historical central witness \(v_0\) with the current center \(v\) of the same group. Their common high winner \(a\) has delay at most \(d_0\), and the marked top score \(g_2\) is nonnegative. Hence \[d^{(l+1)}(a,v_0),\ d^{(l+1)}(a,v)<(d_0+1)b_2.\] The earlier low ball at \(v_0\), on the vertices present when it was chosen, is contained in the current high ball at \(v\): use \(d^{(l+1)}\le d^{(l)}\) and \(R_l+2(d_0+1)b_2<R_{l+1}\). This earlier low ball and the current low ball each have at least \((3/2)^n\) vertices, whereas the current high ball has fewer than \((3/2)^{n+1}\) vertices. The two low balls intersect. The triangle inequality in the low metric gives \[d^{(l)}(v_0,v)\le2R_l.\] For a pad-\(h\) vertex \(b\) it follows that \(d^{(l)}(b,v)\le A_h:=10R_lh+2R_l\). Our constants give \[A_h<T_{l+1}-T_l-10\qquad(h\le H_{\max}).\] For example the gap on the right before subtracting \(10\) is \(\tfrac12d_0^9b_2(H_{\max}+1)\), whereas \(A_{H_{\max}}\le10d_0(d_0+1)b_2(10H_{\max}+2)\). Since the clipped profile of \(v\) at the use is \(T_l\), this proves the true rival bound \[f_b(s)\le T_l+A_h<T_{l+1}-10.\]

Let \(b'\) be a present graph neighbor of \(b\) at this use. Its winner score loses at most one. Both \(f_b\) and \(f_{b'}\) are below \(T_{l+1}\) at all depths up to \(s\), and their difference is at most one at all depths after \(s\). Thus \(d^{(l+1)}(b,b')\le1\). For the low metric, after \(s\) the distance from \(b'\) to \(v_0\) costs at most \(10R_lh+1\). Before \(s\), its clipped profile is between \(T_l\) and \(T_l+A_h+1\), while the clipped profile of \(v_0\) is between \(T_l\) and \(T_l+2R_l\). Their difference is at most \(A_h+1\le10R_l(h+1)\). All three predicates of [eq:src-12] therefore hold at pad \(h+1\). Finally, at admission the first predicate gives \[d_s(z,b)\le(d_0+1)b_1+H_{\max},\] which is polynomial in \(P,k\) and proves the path bound.

It remains to prove the usable-endpoint assertion. Let \(I,J\) be overlapping endpoints for distinct same-color groups \(G,G'\), with \(J\) the finer endpoint, and choose an actual-use certificate \(u\) for \(G'\) at or below \(J\). Let \(b,b'\) be the respective mask vertices. Suppose, for a contradiction, that \(d_{\operatorname{depth}(J)}(b,b')\le1\). This distance bound persists to \(u\). The vertex \(b'\) remains in its mask at \(u\). No future use of \(G\) on this same branch is required.

First suppose their primary winners \(z,z'\) differ. Their colors give the same primary round, score \(g\), and equality case or separating length level. At a fixed length level, upper-half index cohorts on an increasing path either use the same dyadic block or increasing disjoint blocks. In either case the earlier winner belongs to the later frozen prefix \(W'\). This is immediate also for equality cohorts. The earlier winner score at \(b\) persists, so adjacency at \(u\) gives \[f'_{b'}(u)\ge(g-1)b_1-h-1.\] This contradicts the true rival bound for \(G'\) at \(u\), which is less than \(T_{l+1}-10\le(g-5/2)b_1-10\), since \(b_1\) is much larger than \(H_{\max}\).

Now suppose \(z=z'\). Its fixed admission index and the common separating level determine the same cohort block; its prefix \(W\) is inherited on the overlapping path. The metrics coincide, and distinct groups must have distinct high winners \(a,a'\). Historical witnesses \(v_0,v'_0\) for the two masks lie on the common ancestor path of \(J\) and can be ordered along it. At the later historical witness the earlier high winner remains available in the same high round and metric. The earlier winner’s true score is greater than \((g_2-1)b_2\) at its own witness, whereas at the later witness it is at most \((g_2-5)b_2\). Therefore \[d^{(l+1)}(v_0,v'_0)>4b_2>2H_{\max}+1.\] On the other hand, at the certificate \(u\) the true rival bound puts \(f_{b'}(u)\) below \(T_{l+1}-10\), and adjacency puts \(f_b(u)\) below \(T_{l+1}-9\). Both profiles were below the high clipping range at all earlier depths, and they differ by at most one at all later depths. It follows that \(d^{(l+1)}(b,b')\le1\). The two mask predicates then imply \[d^{(l+1)}(v_0,v'_0)\le h+1+h'\le2H_{\max}+1,\] a contradiction. This proves separation, including the case where the two endpoints have later certificates on different branches. ◻

Proof. For a fixed label \(v\) in \(G\), combine all its occurrences into the scalar coefficient function \[a_{v,G}(y)=\sum_I a_{v,G,I}\mathbf 1_I(y).\] The private coordinate of label \(v\) in \(Z_G(y)\) is precisely \(\epsilon_v^{1/2}a_{v,G}(y)\). Thus, if \(W_0\) bounds inverse square-root private weights, \[\sum_v|a_{v,G}(y)|^2\le W_0^2\lVert Z_G(y)\rVert^2.\] This uses the combined coefficients, so any cancellation between repeated uses of a label is retained.

Fix labels \(v,w\) in distinct groups. Their tags have a deterministic last separating depth: by monotonicity, the depths where \(d_s(v,w)>1\) form an initial interval. Cap this interval at the last depth in the finite calculation if needed, and call its endpoint \(s_*\). If two supporting endpoint cells \(I,J\) overlap, their intersection is the finer cell. Lemma 34 says that its depth is at most \(s_*\). Therefore every nonzero term in \[a_{v,G}\overline{a_{w,G'}} =\sum_{I,J}a_{v,G,I}\overline{a_{w,G',J}}\mathbf 1_{I\cap J}\] is measurable on the grid at depth \(s_*\). The whole product is measurable on that grid. Endpoint terms finer than \(s_*\) have no overlap with any term of the other label and contribute zero. If no separating depth occurs within the root, there are no overlaps at all.

Apply Lemma 31 on every grid cell. The two private labels cannot coincide on an overlap, since equal tags have distance zero. Hence \[|\langle Z_G,Z_{G'}\rangle| \le\tau_2\sum_{v,w}\int|a_{v,G}(y)a_{w,G'}(y)|\,dy.\] Pointwise Cauchy–Schwarz in groups and labels bounds the sum of all these cross terms by \(\tau_2\) times a fixed power of the pathwise count and private-weight bounds, multiplied by \(\sum_G\lVert Z_G\rVert^2\). Taking \(\tau_2\) smaller than the reciprocal of this factor proves the assertion. There is no factor counting repetitions or endpoint times. ◻

Proof. Fix a nonnegative compact smooth partition function \(\rho\) with \(\sum_{n\in\mathbb Z^d}\rho(u-n)=1\). Its support and Gevrey bounds can be chosen within the basic budget. Choose a sufficiently large integer period \(P_0\), fixed for this vertex, accommodating the supports of both its original amplitude and these partition functions. The templates have amplitude \[\rho(d)e(\ell^T d/P_0),\qquad d=\theta_v y-n,\] where \(\ell\) ranges over residue vectors modulo \(P_0\), in addition to an arbitrary external linear modulation. Their physical supremum norms are bounded by one because the partition weights are nonnegative.

Insert this partition in the other three slots and use lifts \(d_l=\theta_v y_l-n_l\) in all four slots. Extract the matching phase \(c_l(\lambda_vy_l^2+(\theta_vy_l)^TC_vd_l)\) from each input, putting its conjugate into the corresponding input multiplier. On the progression, \[d_l=(1-l)d_0+ld_1+k_l,\] with bounded integer carries \(k_l\). The common quadratic terms cancel because \(\sum_l c_l l^u=0\) for \(u=0,1,2\). The phase left by the carries is \[\sum_l c_l(\theta_v y_l)^T C_v k_l,\] which is a sum of linear modulations. There are quasipolynomially many possible carries within the fixed support bounds.

To separate the lifts, impose each specified carry equality by a smooth function of \(d_l-(1-l)d_0-ld_1\) that is one at that integer and zero at all other possible integers. Such a function is exact here, since its argument on the progression is integer valued. Expand these coupled smooth functions in Fourier series on a box of period \(P_0\) containing the bounded lifts. After multiplication by the partition functions, every nonsource factor has the required template amplitude, possibly with an unrestricted character index. Write an index as \(\ell=P_0q+r\); then \[e(\ell^Td/P_0)=e(q^T\theta_v y)e(r^Td/P_0),\] since \(q^Tn\) is integral. Thus only the residue \(r\) enters the template, and the other part is an allowed linear modulation. The source factor, after cancellation of its phase, periodizes a smooth horizontal function; expand that function in ordinary torus characters. Its remaining dependence on \(y_i\) is only a linear modulation. Derivatives of sufficiently high order, polynomial in the dimension, bound all these absolute Fourier coefficient sums by a quasipolynomial in the fixed basic budget. For a template source, reducing its own character to a residue gives the same bound. There is no iterative increase of nonlinear complexity.

It remains to prove the assertion for one separated term. Express \(y_i,y_j\) in the two spectator coordinates \(y_a,y_b\). Separate the smooth rescaled kernel by an absolutely summable Fourier expansion in these coordinates; source linear modulations can be absorbed into the two spectator tests. The target coordinate has the form \[y_j=\alpha y_a+\beta y_b,\] with both \(\alpha,\beta\) nonzero fixed rationals. Zero-extend the demodulated target, consisting of the physical residual multiplied by the conjugate of its bounded template, from \(I\) to a fixed larger interval and Fourier-expand it there. Its coefficients are normalized moments of the residual with same-tag templates and arbitrary linear modulations. If their supremum is \(M\), the resulting integral is bounded by \[M\sum_m|A_mB_m| \le M\Bigl(\sum_m|A_m|^2\Bigr)^{1/2} \Bigl(\sum_m|B_m|^2\Bigr)^{1/2}.\] Here \(A_m,B_m\) are the spectator coefficients at frequencies affine in \(m\), with nonzero spacings proportional to \(\alpha,\beta\). These families are Bessel on the fixed integration intervals. Multiplication of a spectator by its bounded demodulating template preserves its \(L^2\) bound. The last display is therefore at most a fixed constant times \(M\). Fourier expansions can first be finite; \(L^2\) convergence and the same Bessel estimate pass to the limit. Summing the absolutely convergent separated terms gives the form bound \(H_0M\). A form norm larger than \(t_0\) consequently forces a target moment at least \(t_0/H_0\), after enlarging \(H_0\) by a fixed factor if necessary to avoid an unattained supremum. ◻

Proof. Hold the original graph vertices, masks, and named histories fixed. The numbers of masks and histories at one cell are \(\exp(k^{O(1)})\): there are polynomially many original-vertex contexts, polynomially many ranks and colors, \(P\) pad choices per factor, and \(O(\log\mathcal L)\) factors. This count uses names of contexts and masks, not their numeric distance values or the number of inserted columns. The helper bases have the stipulated same count.

Upon a failed test, Lemma 36 supplies a bounded physical column with moment at least the threshold in [eq:src-13] divided by \(H_0\). Insert a fresh copy at the target rank. Its new private coordinate is zero in the old tested state and in the old projection. Thus that physical moment equals the vector inner product with the old orthogonal residual. If \(P_{\rm old},P_{\rm new}\) are the target projections before and after insertion and \(X\) is the old tested state, then \[\lVert (P_{\rm new}-P_{\rm old})X\rVert_{2,I} \gtrsim \frac{\tau_3}{D\sqrt{n_{r-1}}H_0} \lVert \text{base}\rVert_{2,I}.\] Indeed the fresh vector column has uniformly bounded norm, and projection onto its component orthogonal to the old range gives this inequality. If the base has zero norm no test fails.

Although \(X\) can change when other projections change, its squared increments can still be charged. Fix a named history, a target mask, and a trial sequence of at most \(T\) insertions. Every individual positive projection in that history is nested in the insertion index. Expand complements and use the binary-prefix argument of Lemma 9. The target increments are orthogonal rows; the right-hand history is evaluated at the old insertion index. The sum of their squared actions on this changing history is at most \[(C\log(2T))^{O(\log\mathcal L)}\lVert \text{base}\rVert_{2,I}^2.\] This is precisely the row version of that binary argument, and it does not require treating a changing projected state as a fixed vector.

It follows that rank-\(r\) insertions at a cell are bounded by \[n_{r-1}\exp(k^{C_1}) (C\log(2T))^{O(\log\mathcal L)}.\] The exponential factor includes the configuration count, the squared inverse tolerance, and \(H_0^2\). It does not include the number of possible triggering source columns. For a fixed target history, any triggering source produces a fresh moment in that history’s old residual; the resulting target increment is charged to the same orthogonal row sequence regardless of that source’s identity.

There are at most \(M\) active cells on a path. Absorb this factor, and fixed factors for ranks and slots, in \(\exp(k^{C_1})\). Starting with a basic bound \(n_0\), choose \[n_r/n_{r-1}=\exp(k^{C_2})\] with \(C_2\) sufficiently large, and set the global trial cap to the resulting \(n_{\max}\), with spare fixed factors. Since there are only \(\mathcal L+5=k^{O(1)}\) ranks, \(\log n_{\max}=k^{O(1)}\). With \(T=n_{\max}\) the binary-prefix factor is much smaller than the chosen ratio for large \(k\). Thus no trial cap is attained. This argument also accommodates simultaneous changes of histories caused by insertions at other ranks: all the underlying positive projections remain nested along the common insertion sequence. It proves termination of the main closure, and the identical argument proves termination of the later helper closure on its fixed bases.

All constants used in this count were determined by the basic template bounds and the number of named graph contexts. They do not depend on \(H_{\rm ed}\). We may therefore first fix these bounds, then choose \(\tau_2\) sufficiently small for Lemma 35 and all pair-mismatch errors, and finally choose \(H_{\rm ed}\) by Lemma 31. This order is not circular. ◻

Proof. Maintain a range of colors \([\ell,h]\), an upper primal guard slot, and when \(\ell>1\) a lower residual guard slot. Initially the range is \([1,\mathcal L]\); use any of the at least three basic coarse arguments as the upper guard and take no lower guard. At a midpoint \(m=\lfloor(\ell+h)/2\rfloor\), choose a third slot distinct from the guards and split it by the smeared cumulative filter at \(m\) and its complement. In the positive term that slot becomes the upper guard, the former upper guard is released, and the range becomes \([\ell,m]\). In the residual term it becomes the lower guard, the former lower guard is released, and the range becomes \([m+1,h]\). A guard is not postcomposed before it is released. Use a deterministic slot-choice rule for each branch of this binary search, even if some colors are empty at a particular cell. There are \(O(\log\mathcal L)\) splits per path and at most \(\mathcal L\) leaves.

Consider a leaf with color \(k'\). Its upper guard is either the initial basic coarse argument or has outer cumulative rank \(\mathcal L-k'+1\) and the corresponding pad. Expand this one outer primal support, keeping its average when present. For the initial basic guard, set the upper pad to zero. Call a source tag old if it lies within this upper pad of a vertex of color at most \(k'-1\). If \(k'>1\), the lower residual guard has rank \(\mathcal L-k'+2\) and a base pad larger by at least \(10P\). Every old source therefore lies in all its residual masks and has lower rank. Equation [eq:src-13] kills it, with arbitrary normalized \(L^2\) spectators. For \(k'=1\) there are no old tags.

Every remaining upper source is near a current vertex of color \(k'\). In the initial basic case this is the source’s own original vertex; in the other case it follows from membership in the cumulative mask and exclusion of old tags. Let \(G\) be the current group of such a central vertex. The upper pad is smaller than the first ball pad, so Lemma 34 puts the source in every needed mask of that ball filter. The source cannot match two distinct groups: it would then violate same-color mask separation at the current cell. Moreover it is separated from the primal sources of every other current group’s ball filter.

Choose a third slot, distinct from the upper and lower guards, and insert its first-rank ball projection, summed over current groups. For the matching group, the difference from the identity is bounded by [eq:src-13], using the upper source. For any other group, expand the new primal ball source and apply Lemma 31. Old-source terms on either side are still bounded using the untouched lower residual guard. This yields an identity, with small error, in which the third slot has pure support in one group’s first ball mask.

Keep that third slot as the source, and postcompose the other three slots with their own larger-rank, larger-pad ball filters in order. The first ball source has lower rank than each new target, and its mask is contained in all the new target masks. Thus [eq:src-13] bounds each residual insertion. The third slot is left unchanged. The result is [eq:src-14]; each slot’s inner cumulative history depends only on the search path, and only its outer ball filter depends on the group.

We specify why the numerical errors are summable without spending a full coefficient bound in a main term. Each test using a rank-\(r\) residual expands only an outer lower-rank source. Its coefficient bound is at most \(D\sqrt{n_{r-1}}\) times the local input norm, canceled by the denominator in [eq:src-13]. Basic local sizes and all same-time contractions are bounded. The remaining counts of active cells on a path, leaves, current groups, and filter-insertion steps are fixed powers of \(D\); choosing \(C_\tau\) sufficiently large makes their total length-weighted contribution at most \(e^{-k}|R|\). Pair-mismatch errors can additionally contain fixed powers of the full main count, but \(\tau_2\) was chosen after that count and makes the same bound hold. No expansion of a full column coefficient sum is retained in [eq:src-14]. Finally, \(\min(\eta,\sum_j a_j)\le\sum_j\min(\eta,a_j)\) for \(a_j\ge0\), so the minimum can be distributed over the fixed leaves as claimed. ◻

Proof. Expand each complementary factor \(1-\overline Q\) as a difference of identity and a positive average. The number of resulting products is at most \(2^{O(\log\mathcal L)}=\operatorname{poly}(k)\). In any such product, remove the larger rank among the current outermost filters of \(c,d\), regarding an unfiltered basic input as rank zero. Positive ranks are distinct: each cumulative rank was inserted in exactly one slot at one search split, and the four final ball ranks are distinct. Thus the other complementary source always has strictly lower rank than the filter being removed.

Suppose the next filter is on \(c\), and use the projections and arguments defined in the statement. The \(A_l^p\) are orthogonal band projections at the current cell and sum to the identity.

On band \(p\), the positive average \(P^{-1}\sum_{u=1}^P Q_{l,u}\) has weight \((P-p+1)/P\), including weight zero at \(p=P+1\). Therefore the difference in the statement is exactly Equation [eq:src-15], with the arguments written in role order. This proves the algebraic identity, including the terminal bands.

If \(p>q+2\), the lower \(c\) band is a difference of primals, so expand its pure sources. Their rank is a main rank below the helper rank, and their tags belong to both helper masks needed to write the higher \(a\) band as a difference of residuals. The helper version of [eq:src-13], applied to the history on the fixed base \(S_a\), bounds the resulting terms. The two other arguments are arbitrary \(L^2\) spectators.

If \(q>p+2\), instead expand the outer primal source of \(x_d\), using the basic expansion if it is already unfiltered. This is a main source of rank strictly below the outgoing rank on \(c\). The lower \(a\) band is primal. Let \(\mathcal M_p\) be the outgoing vertex mask at its pad \(h+p\). Classify a pure source of \(x_d\) as near if its tag is within one edge of some vertex in \(\mathcal M_p\), and as far otherwise. For a near source, neighbor closure gives membership in \(\mathcal M_{p+1}\), hence in both masks needed for the higher \(c\) residual band. Apply the main test with the entire helper band \(A_a^p x_a\) as one \(L^2\) spectator. For a far source, expand the helper primal: every pair of source tags is separated, so Lemma 31 applies. Neighbor closure is the triangle inequality for a cumulative mask and Lemma 34 at the current actual use for a ball mask. The gap \(q-p>2\) leaves at least this one-pad margin. The main version of [eq:src-13], already imposed on the inner history \(x_c\), then bounds it. Thus helper insertions need not alter any main catalog.

In the first direction the coefficient bound of the main source is canceled by the helper test’s denominator. In the second direction the lower-rank main source coefficient is canceled by the main test’s denominator; the helper primal serves only as an \(L^2\) spectator. For pair mismatch, expanding the two primal supports can cost fixed powers of the helper and main counts. Those counts were anticipated before \(\tau_2\) was selected. There are only \(O(P^2)\) pairs of bands and polynomially many transfer positions. Since \(\log P=O(k)\), these factors are absorbed by the prior accuracy choices. All filters are contractions, so the remaining norm factors are \(\lVert S_a\rVert_{2,I}\lVert S_b\rVert_{2,I}\) and bounded complementary basic sizes. This proves the stated far-error estimate. Averages in \(\mathcal D\) are bounded directly by their tested averaged histories and do not cost the number of pad choices.

Retain the near bands and forward only the transferred main term to the next step. Iterating deletes all complementary filters and gives the asserted final main term. The number of helper factors remains \(O(\log\mathcal L)\). Restoring \(B_c,B_d\) to \(Y_c,Y_d\) in that main term uses the full pair estimate; every resulting difference contains an additional basic side \(U\). These are the three-side terms estimated in Section 7. ◻

Proof. Here \(v\in A_a\) means that \(v>0\) almost everywhere and \[[v]_{A_a} =\sup_J\left(\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_Jv\right) \left(\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_Jv^{-1/(a-1)}\right)^{a-1}<\infty.\] We use the scalar weighted maximal theorem and the following family-of-pairs form of Rubio de Francia extrapolation: if a family \(\mathcal F\) of nonnegative pairs \((U,V)\) satisfies, for some \(a_0\in(1,\infty)\) and every \(v\in A_{a_0}\), \[\int U^{a_0}v \le C_{a_0}([v]_{A_{a_0}})\int V^{a_0}v \qquad ((U,V)\in\mathcal F),\] uniformly in the pair, then the corresponding estimate holds at every \(a\in(1,\infty)\) and every \(v\in A_a\). No operator relation between \(U\) and \(V\) is assumed; see (Cruz-Uribe et al. 2011, Theorem 3.9).

Initially let \(g\geq0\) be bounded and supported in a bounded interval in \(y\). All the weighted left-hand sides used in the following applications are finite: the inner norms are bounded by a constant times the scalar maximal function of the indicator of that interval. Thus the version of extrapolation with finite left-hand sides suffices. The scalar weighted \(L^t\) maximal inequality, integrated in \(w\), gives the weighted \(L^t\) hypothesis for the family of pairs \[U_\xi(y)=\|M_y g(y,\xi,\cdot)\|_{L_w^t}, \qquad V_\xi(y)=\|g(y,\xi,\cdot)\|_{L_w^t}.\] Extrapolation gives their weighted \(L^a\) inequality for every \(a>1\). Choose \(a=s\) and integrate this inequality in \(\xi\). The resulting pairs \[U(y)=\|U_\xi(y)\|_{L_\xi^s},\qquad V(y)=\|V_\xi(y)\|_{L_\xi^s}\] satisfy the weighted \(L^s\) hypothesis for every \(A_s\) weight. A second application of extrapolation, now with target exponent \(r\), proves the first assertion. This argument never interchanges the three norms and requires no ordering among \(r,s,t\).

For general nonnegative \(g\), apply the result to \(\min(g,n)\mathbf 1_{[-n,n]}(y)\) and use monotone convergence, including \(M_y g_n\uparrow M_y g\). Absolute values treat general \(g\). Finally apply the first assertion to \(g=|f|^q\), with \[r=p/q,\qquad s=b/q,\qquad t=h/q,\] and take the \(q\)th root. Each exponent is strictly greater than one, and the power identity for the iterated norms is exact. ◻

Proof. By homogeneity it suffices to take \(s_j=1\). A zero size gives a zero input almost everywhere and hence a zero form. To estimate the expected absolute value, insert a common \(\sigma\)-dependent scalar of modulus one into the kernel signs. It is therefore enough to estimate the absolute value of the expected signed sum.

Proof. For this estimate expand complements by linearity and couple each pad choice across all times in an average. Jensen permits us to treat a fixed choice. We have \[Z_{G,s}=L_{G,s}B'_sF,\] where \(L_{G,s}\) is an individual nested ball projection and \(B'_s\) is a shared product of \(O(m)\) nested main projection families. Basic \(B\) inputs are fixed signed sums of at most two such current histories, which changes only a fixed factor. Split \[Z_{G,t}-Z_{G,r} =(L_{G,t}-L_{G,r})B'_rF +L_{G,t}(B'_t-B'_r)F.\]

For a fixed group, the rows \(\mathbf 1_{E_{G,r,t}}(L_{G,t}-L_{G,r})\) have a square Bessel bound, since increments of a nested orthogonal projection family on disjoint depth intervals are orthogonal. Their adjoints have ranges in sums of whole ball ranges at the two endpoints. A right-endpoint certificate also certifies the required earlier input cell when its mask is nonempty. Splitting at the \(r\)-cells if necessary therefore places these adjoint ranges in the usable endpoint spaces of Lemma 35. That lemma makes the rows jointly Bessel across groups. They are local to their \(r\)-cells, so Lemma 9 inserts the shared product \(B'_r\) at a cost \(G_0\).

For the second term, the usable-range Bessel estimate at each common right endpoint first sums the outer ball projections over groups. Telescope \(B'_t-B'_r\). In each telescoping term, discard the left contractions. The changed middle factor is an orthogonal increment on \([r,t]\), and the factors to its right are evaluated at \(r\). These middle increments form Bessel rows, and Lemma 9 inserts their right factors. Summing the \(O(m)\) telescoping terms is again absorbed in \(G_0\).

An interval coloring treats bounded overlap. Expansions of the \(O(m)\) complements cost \(2^{O(m)}\), and averages are restored by Jensen, without a factor counting pad choices. The proof without an outer ball is the same middle-increment argument. Intervals crossing the start of the attempt occur with bounded overlap; the zero convention and the one-time projection bound treat them. ◻

Proof. For a sequence \(W_s\), \(b\le s<b+K\), write \[W_s=W_b+\sum_{[r,t]\in\mathcal P(b,s)}(W_t-W_r),\] where \(\mathcal P(b,s)\) is the usual disjoint binary decomposition of the prefix \([b,s]\). There is at most one interval of each length in this decomposition. For a fixed length, its intervals are disjoint as depth intervals, and the selected endpoint is at most \(s\). Restrict each resulting fixed function to the union of the use cells selecting it. This preserves its value at every required use and only decreases its norm. The base position and the choices of lengths account for logarithmic factors.

For a basic-\(U\) input, use each depth \(s\) directly as a position. The current side square bound controls the sum of its energies; subsequent helper products are local contractions. On the banded side, paying at most \(P+1\) indices is permitted in the first inequality of Equation [eq:src-18].

For a fixed linear side \(V_{G,b}\), let \(T_s\) be the product of its subsequent averaged projections. At a fixed band index, the outer band difference is a difference of two such products. The binary-prefix product estimate in the proof of Lemma 9, now on a sequence of at most \(K+1\) times, gives \[\|T_bV_{G,b}\|_2^2+ \sum_{[r,t]\ \mathrm{at\ a\ fixed\ length}} \|(T_t-T_r)V_{G,b}\|_2^2 \le (C\log(K+2))^{O(m)}\|V_{G,b}\|_2^2.\] Individual projections are nested; averages are coupled across times and restored by Jensen. Sum the length levels and use Equation [eq:src-17]. Pay at most \(P+1\) on the banded side.

For a rolling input write \(W_s=T_s(Z_{G,s}-Z_{G,u_b})\). The base term is controlled by Lemma 42 on the intervals \([u_b,b]\), whose overlap across blocks is bounded. Its increments split exactly as \[W_t-W_r =T_t(Z_{G,t}-Z_{G,r}) +(T_t-T_r)(Z_{G,r}-Z_{G,u_b}).\] The first term is bounded directly by Lemma 42. When estimating norms, keep the entire endpoint cells containing the selected use cells: these cells still have downstream certificates, and locality of \(T_t\) makes this enlargement legitimate.

For the second term, also expand \([u_b,r]\) by binary prefix intervals in a common window of \(O(K)\) depths. Fix an outer length, an inner length, and an inner interval \([a,v]\). For all outer positions \([r,t]\) using this same inner interval, use the fixed input \[X=\mathbf 1_{\Omega}(Z_{G,v}-Z_{G,a}),\] where \(\Omega\) is the union of the needed \(r\)-cells. This input is independent of the outer position. Since \(v\le r\), locality on \(r\)-cells shows that it gives the required outputs of \(T_t-T_r\). Its norm is bounded by keeping the union of the \(v\)-cells containing those \(r\)-cells; all of these have downstream use certificates. For this fixed input, the product-jump estimate bounds the sum over outer positions by \((C\log(K+2))^{O(m)}\|X\|_2^2\). Now sum the squared inputs over groups and inner intervals at the fixed inner length, using Lemma 42. The windows for different blocks overlap a bounded number of times. Finally sum the two length choices. This proves the rolling estimate with only small overhead, and with the allowed factor \(P\) for a banded side.

The number of search leaves, transfer steps, choices of side component, and length configurations is small overhead. Individual pad choices are not additional configurations: they remain inside the actual averaged operators throughout. Thus summing all required configurations preserves the asserted bounds after increasing \(G_0\). ◻

Proof. The three-side power bound was proved above using the path square caps and \(3p>2\). Lemma 39 gives the decomposition into transferred main terms, numerical errors, and coupled near bands. Restoring the full complementary pair in a main term introduces a third basic side and leaves the pair-small estimate \(\eta\) on its two arbitrary side inputs. The frozen-complement argument accounts for every freeze error and every discarded large complement as a third strong factor. Lemma 43, the common maximal cuts, and the sparse comparison then give Equation [eq:src-19] for the remaining terms.

All cuts were imposed on nonnegative partial ancestor quantities and removed the original summands at and below their first violation. The normalized base hypotheses, the basic square estimate, and the root-uniform hypothesis [eq:src-17] are available anew when the construction is restarted on such subroots. The fixed functions in the parent attempt remain unchanged. The thresholds may be chosen to make the union of the bad roots an arbitrarily small fixed fraction of \(|R|\). This proves the asserted uniformity and support convention.

Finally, the estimates have the required relation to the original cell minimum. The subadditivity of \(x\mapsto\min(\eta,x)\) removes absolute numerical errors and the critical unpowered terms. For the remaining cell sum use \[\min(\eta,x)\le\eta^{1-p}x^p.\] The group sum and the prescribed coupled bands are formed before this inequality is used. Thus every power cost gains \(\eta^{1-p}\), whereas Equation [eq:src-19] is used exactly as an absolute estimate. This completes the conditional side argument. ◻

Proof. The branch cutoff makes the conjugating factor a smooth quadratic chart supported in a bounded branch cube. At \(I_0\), the joint difference data of \(v\) and \(z\) obey the endpoint version of Equation [eq:src-2], with the harmless fixed factor \(c_l\). Recenter at \(I_0\), choose one rational splitting there, and apply Lemma 14. Its tail estimate gives the stated uniform error and finite coefficient and count bounds. A single splitting gives a single effective quadratic coefficient for this primary. Recentring adds a common unrestricted linear shift; the differences of the remaining linear shifts are rational combinations of the two endpoint horizontal vectors. Clearing \(Q_0\) gives Equation [eq:src-21]. All these choices are made once.

Here is an explicit lift, which will also explain its operator properties. Write the main atom as \(\Phi=(\varphi_\Phi,\epsilon_\Phi^{1/2}\mathbf e_\Phi)\), and set \[q_\Phi(y)= \sum_m a_{\Phi,m}(\psi_m)_{\mathrm{phys}}(y) -(\zeta^\rho(y))^2\varphi_\Phi(y).\] It satisfies \(|q_\Phi|\le\delta_c\) on the matched subtree. For a main augmented input \(F\), define the physical part of \(\mathscr R_{l,G}F\) by \[(\zeta^\rho)^2 F_{\mathrm{phys}} +\sum_{\Phi\ {\rm matched\ at}\ y} \frac{F_\Phi(y)}{\epsilon_\Phi^{1/2}}q_\Phi(y).\] Its mode coordinate \(m\), belonging to \(\Phi\), is \[\epsilon_{\mathrm m}^{1/2}a_{\Phi,m} \frac{F_\Phi(y)}{\epsilon_\Phi^{1/2}},\] and is zero before the appropriate match. The main count and inverse weight bounds, followed by Cauchy–Schwarz, bound the physical correction’s pointwise operator norm by \(\delta_c D_{\mathrm m}^{C}\). Choose \(\epsilon_{\mathrm m}>0\) so small that the squared norm of the mode-coordinate map is bounded by one. The preceding count and coefficient estimates allow \(\log(1/\epsilon_{\mathrm m})\le\operatorname{poly}(P,k)\). Thus the whole lift has a uniform pointwise norm.

An outer ball output on a cell has a representation \(\sum_\Phi\beta_\Phi\Phi\) with constant coefficients, including when the projection is averaged. Its main private coordinate is exactly \(\epsilon_\Phi^{1/2}\beta_\Phi\). Substitution in the displayed definition gives \(\sum_{\Phi,m}\beta_\Phi a_{\Phi,m}\psi_m\), as required. The lift is pointwise and hence cell-local; it has not been commuted through any projection. ◻

Proof. Fix the primary, put \(r_s=r_0 2^{-s}\), and write \(\tau=\log_2\mathcal T\). For a nonzero evaluation set \(x(a)=\log_2(|a\cdot\Theta|r_0)\). Exception at block start \(b\) means \[b-2K-\tau<x(a)<b+K+\tau.\] Lemma 10 places all these \(x(a)\) in at most \(d_{\max}\) packets of width \(W\). A packet \([\alpha,\alpha+W]\) can cause an exception only when \[\alpha-K-\tau<b<\alpha+W+2K+\tau.\] This interval has length at most \(4K\) by our choice of \(K\), so it contains at most five starts of the \(K\)-spaced block schedule. Zero evaluations are always tiny.

For the second assertion, choose a basis of tiny vectors from the defining set. Cramer’s rule expresses any height-\(\mathcal H\) vector in its span with coefficients of total absolute value at most \((d_{\max}\mathcal H)^{C d_{\max}^3}\). Its evaluation is consequently bounded by this factor times \(\mathcal T^{-1}/r_{v_b}\). The choice of \(\mathcal T\) makes this strictly smaller than \(\mathcal T/r_{t_b}\). The gap in Equation [eq:src-23] then forces the tiny alternative. ◻

Proof. Abbreviate \(R'=R_*^{\mathrm{ht}}\), and discard \(h<R'\). If two modes of this primary are connected in CURRENT, sum the edge witnesses along a simple connecting path. There are at most \(D_*\) edges. Their quadratic parts cancel between the endpoints, because the primary has one effective quadratic coefficient. With \(a\) from Equation [eq:src-21], this gives an integral \(m_h\) with \(|m_h|_\infty\le D_*E_h\) and \[|a\cdot\Theta+m_h\cdot\vartheta/L_h| \le D_*\Xi/r_{t_b}.\] The vector \(a+(0,m_h/L_h)\) has height at most \(\mathcal H\). Since \(D_*\Xi<\mathcal T\), the gap implies \[a+(0,m_h/L_h)\in K_{\mathrm{rat}}.\]

For each height \(i\), let \(V_i\) be the rational span of the height-\(D_*\) vectors \(q'\) for which \[q'+(0,m/L_i)\in K_{\mathrm{rat}} \quad\hbox{for some }m\in\mathbb Z^{\dim\theta_z}, \qquad |m|_\infty\le D_*E_i.\] Let \(U_i\) be the rational span of the integer vectors in \(K_B\) with size at most \(C_*^3D_*E_i\). Both families increase. Discard every \(h\) for which the rank of either family changes between \(h-R'\) and \(h\). There are at most \(2d_{\max}R'\) such heights, since each rank can increase only \(d_{\max}\) times.

At a remaining height, \(a\in V_h=V_{h-R'}\). Express \(a=\sum_jb_ja_j\) using a basis from the defining vectors at \(h-R'\), with witnesses \(m_j\). The \(b_j\) and their common denominator are bounded by \(C_*\). Subtracting the corresponding relations in \(K_{\mathrm{rat}}\) shows that \[m_h-6^{R'}\sum_jb_jm_j\in K_B.\] After clearing the common denominator this is an integer vector whose size is at most \(C_*^3D_*E_h\); this follows directly from the witness bounds and the choice of \(A_0\). It belongs to \(U_h=U_{h-R'}\), which we denote by \(U\).

The lattice \(U\cap\mathbb Z^{\dim\theta_z}\) is saturated: if a nonzero integer multiple of an integer vector belongs to \(U\), the vector itself belongs to the rational space \(U\). Thus the quotient \(\mathbb Z^{\dim\theta_z}/(U\cap\mathbb Z^{\dim\theta_z})\) is free. In an integer basis of this quotient, \[[m_h]=6^{R'}\sum_jb_j[m_j].\] The right side has integral coordinates. The common denominator bound and \(R'>\log_2C_*+2\Delta\) show, coordinate by coordinate, that each is divisible by \(6^\Delta\): the valuations at both 2 and 3 are at least \(\Delta\). Other denominator primes do not affect these divisibilities. Choose an integral lift \(m'\) of \([m_h]/6^\Delta\).

Then \(m'-6^{R'-\Delta}\sum_jb_jm_j\in U\). Choose linearly independent integer vectors of size at most \(C_*^3D_*E_{h-R'}\) spanning the rational space \(U\), from the definition of \(U_{h-R'}\). Subtract integer multiples of these vectors from \(m'\), rounding the coefficients of the preceding difference. The result remains an integer lift and satisfies \[|m'|_\infty \le 6^{R'-\Delta}\sum_j|b_j|\,|m_j|_\infty +d_{\max}C_*^3D_*E_{h-R'} \le E_{h-\Delta}.\] The last inequality is one of the permitted fixed-power choices of \(A_0\). This rounding argument uses a small integer spanning set of the rational space, not a lattice basis, and therefore incurs no covolume factor.

Finally, \(m_h-6^\Delta m'\in U\subseteq K_B\), so \[a+(0,m'/L_{h-\Delta})\in K_{\mathrm{rat}}.\] Its height remains at most \(\mathcal H\). Lemma 46 makes its evaluation at most \(\mathcal T^{-1}/r_{v_b}\), which is below the PRE threshold. Equation [eq:src-21] gives the required direct edge. ◻

Proof. Write \(F=\pi(\mathcal F_h)\). Each predecessor of a good mode has more than \(|F|/1.1\) primaries. Two such predecessor primary sets intersect in more than \((2/1.1-1)|F|\), and thus their intersection contains a nonexceptional height-good primary, since fewer than one tenth of the primaries are excluded. The witnessing modes of that primary belong to \(\mathcal F_h\). Lemma 47 joins them directly in PRE at \(h-\Delta\), so the two predecessor components agree.

Modes in the same minimal PRE component have the same components at every larger height and in CURRENT, so all their predicates agree. This proves the second assertion. For the last assertion, two core modes have at most \(2\delta_*(J_*+1)\) bad heights between them. Our lower bound on \(J_*\) ensures that the interval of integer heights below \(4\delta_*J_*\) contains more heights than this. They therefore have a common good height \(h_0\) in that interval. Their CURRENT components at \(h_0\) agree, since their minimal CURRENT components agree. The first assertion identifies their PRE components at \(h_0-\Delta\). Nesting then identifies all later PRE and CURRENT components, which also identifies the corresponding predicates. Finally, \(|\mathcal H_{\mathrm{up}}|\ge(1-4\delta_*)J_*\), so the full-range bad count \(\delta_*(J_*+1)\) is at most \(2\delta_*|\mathcal H_{\mathrm{up}}|\) for our small \(\delta_*\) and sufficiently large \(J_*\). ◻

Proof. Enumerate elements by first entry into the increasing sets, breaking ties arbitrarily. A containing set for the element of entry rank \(r\) has at least \(r\) elements. Its total contribution is at most \(A/r\). Sum this bound over the elements. ◻

Proof. Write \(p_{i,h}\) and \(c_{i,h}\) for the two floor logs at height \(h\). Their values are integers between \(0\) and \(L\). For \(h\ge\Delta\), the first failure indicator is bounded by \[c_{i,h}-p_{i,h-\Delta} =(p_{i,h}-p_{i,h-\Delta})+(c_{i,h}-p_{i,h}).\] At fixed \(i\), the sum of the first term over \(h\) is at most \(\Delta L\), because each increment of the increasing sequence \(p_{i,h}\) is counted at most \(\Delta\) times. At fixed \(h\), the temporal nesting gives \[\sum_i(c_{i,h}-p_{i,h}) \le c_{N,h}-p_{1,h}\le L.\] Including the initially bad heights, this contributes at most \(N\Delta(L+1)+(J+1)L\) bad incidences.

The second failure indicator is bounded by \(\beta^{-1}\) times the sum of the exceptional and height-bad fractions. At fixed \(i\), Lemma 49, applied as \(h\) increases, bounds the height-bad fractions by \(AH_M\). At fixed \(h\), apply that lemma as \(i\) increases to bound the exceptional fractions by \(BH_M\). The total bad count is therefore at most \[NQ+(J+1)(L+\beta^{-1}BH_M).\] On the other hand it is greater than \(\delta N(J+1)\). The assumed lower bound for \(J+1\) absorbs \(NQ\) into half of this last quantity, proving the result. ◻

Proof. First verify the asserted oscillatory bound. The common factor \(e(c_lp^\rho)\) cancels between two physical atoms. Fourier-expand \((\zeta^\rho)^2\) on the \(\vartheta\) torus, retaining frequencies of size at most \(B_*\), with absolute coefficient sum at most \(D_*\) and tail at most \(D_*^{-300}\). At height zero these shifts are allowed because \(E_0=A_0\) exceeds \(B_*\). At a larger height the corresponding shifts are multiplied by \(L_h\), and \(B_*L_h\le E_h\), by the choice of \(A_0\).

For each retained shift, nonadjacency gives a point of the relevant pre-cell where the absolute phase derivative exceeds its graph threshold. The affine-slope oscillation bound makes the absolute derivative at every point of that cell at least half the threshold. The derivative is affine and of constant sign, so the elementary first-derivative oscillatory estimate on any interval applies. For PRE on a pre-cell it bounds the normalized integral by \(C/\Xi\). For CURRENT on an actual cell \(I\), it gives the same bound, since \(|I|\ge r_{t_b}\). The cutoff coefficient sum and tail give \[C D_*/\Xi+D_*^{-300}\le D_*^{-200}\] after increasing the starting budget. Distinct labels have orthogonal private coordinates, so they contribute no other inner product. Earlier modes satisfy the same slope-variation estimate on the later pre-cell by inheritance.

For the Bessel conclusion it suffices, by duality, to bound the synthesis of arbitrary \(f_e\) in these projection ranges. Write \(f_e=\mathbf 1_{J_e}\sum_{m\in\mathcal M_e}\beta_{e,m}\psi_m\), and put \(b_e=(\sum_m|\beta_{e,m}|^2)^{1/2}\). Private coordinates give \[|J_e|b_e^2\le D_*^2\|f_e\|^2.\] For an overlapping pair, constancy of these coefficients on their cells and the cross-pair estimate bound its inner product by \[D_*^{-199}|J_e\cap J_f|\,b_eb_f.\] There are at most \(D_*\) participating rows at a path point. Consequently \[\begin{split} \sum_{e\ne f}|\langle f_e,f_f\rangle| &\le D_*^{-199} \int_R\left(\sum_{e:x\in J_e}b_e\right)^2\,dx\\ &\le D_*^{-198}\sum_e|J_e|b_e^2 \le D_*^{-196}\sum_e\|f_e\|^2 . \end{split}\] Adding the diagonal proves the synthesis bound and hence the row Bessel bound. This calculation uses a pathwise label count; it does not count the number of unused depths between rows. ◻

Proof. Initially omit the shared histories. For one group, \(P_e\mathscr R_{l,G}\) is a Bessel family with the same type of bound, since \(\mathscr R_{l,G}\) is one fixed bounded operator. It depends only on \(J_e\), and therefore also only on the containing decision cell \(u_e\). Lemma 9 applies to the outer individual main ball projection at that cell. It permits arbitrarily many rows with the same decision cell. Its loss is a fixed power of \(\log(2D_{\mathrm m})\). Averaging the pad choices uses the triangle inequality in the Hilbert sum of row outputs and costs no number-of-pads factor.

Let \(A_G\) be the resulting row operator for this group. For an individual outer projection its row adjoint has the order \[L_{l,G,u_e}\,\mathscr R_{l,G}^*P_e.\] Its range is thus in the main ball range on the entire cell \(u_e\). With an averaged projection, it lies in the sum of these usable ranges. Lemma 35 gives \[\left\|\sum_G A_G^*w_G\right\|^2 \le C\sum_G\|A_G^*w_G\|^2 \le G_0\sum_G\|w_G\|^2.\] Duality proves the joint Bessel bound before the shared histories. This is the step for which full endpoint cells and their downstream certificates are needed.

Finally apply Lemma 9 again to these joint rows, with decision cells \(u_e\), to insert the shared histories. Expand complements and the fixed signed sum in \(B_l\), and keep the positive averages as averages. Their product lengths are \(O(\log\mathcal L)\), their column log-budgets are polynomial in \(k\), and the number of signed terms is a small overhead. The compression loss is therefore \(G_0\). The original input has squared norm at most \(|R|\). All operators have remained in their indicated order. ◻

Proof. Fix \(l,G\). An event is a triple \((b,J,\mathcal E)\), where \(J\) is a retained \(v_b\)-cell and \(\mathcal E\) is a minimal PRE component of side modes. Such a component consists entirely of side modes by Lemma 48. Split block indices into their ten congruence classes. Within one class, link an event to a later event on a descendant path when some of their modes agree or are adjacent in the later minimal PRE graph.

For two linked events, every earlier CURRENT edge persists in the later PRE graph. Indeed the later pre-cell is contained in the earlier pre-cell, the mode lists persist, and the separation of block starts gives \(v_{\rm later}\ge t_{\rm earlier}\). The minimal PRE link also persists at every height. Consequently, choosing any mode in each event of a linked chain gives \[\pi(\mathcal E^{(i)}_h) \subseteq\pi(\mathcal F^{(i)}_h) \subseteq\pi(\mathcal E^{(i+1)}_h)\] at every height.

Apply Lemma 50 with \[M=D_{\mathrm m},\quad \beta=\tfrac1{10},\quad A=C d_{\max}R_*^{\mathrm{ht}},\quad B=C d_{\max}, \quad\delta=\delta_*.\] The height multiplicity comes from Lemma 47; the exceptional-block multiplicity comes from Lemma 46. Every event mode has more than \(\delta_*(J_*+1)\) bad heights. Our choice of \(J_*\) meets the lemma’s lower bound. Thus every chain has length at most \[C_{\delta_*}d_{\max}\log(2D_{\mathrm m}).\] For clarity, the inequality being absorbed is \[\delta_*N(J_*+1) \lesssim N d_{\max}R_*^{\mathrm{ht}}\log(2D_{\mathrm m}) +(J_*+1)d_{\max}\log(2D_{\mathrm m}).\] Although \(R_*^{\mathrm{ht}}\) depends on the mode budget, it disappears from the final chain-length bound.

Give each event its longest-chain level. Linked events have different levels. Distinct overlapping events in one level therefore have no shared or adjacent modes in the later minimal PRE graph. Events on one cell are distinct components and have the same separation. Their label sets are pairwise disjoint on a path, with total count at most \(D_*\). Lemma 51 applies to the orthogonal projection \(P_{\mathcal E,J}\) onto the augmented columns of \(\mathcal E\), restricted to \(J\). Summing the ten congruence classes and the bounded number of levels gives squared Bessel bound \(G_0\) for the event rows.

The decision cell for a row is its containing \(u_b\)-cell. Apply Lemma 52 to obtain \[\begin{equation*} \sum_{G,b,J,\mathcal E} \|P_{\mathcal E,J}\widetilde Z_{l,G,b}\|^2 \le G_0|R|. \tag{24} \end{equation*}\] Zero outputs from pre-depths before the attempt were omitted.

It remains to pass from projections to exact coefficient subtotals. Write \(T_{\mathcal E,J}\) for that subtotal. Every mode outside \(\mathcal E\) is PRE-nonadjacent to every mode inside it. The oscillatory estimate in Lemma 51, the private-coordinate lower bound, and \(\sum_m|\alpha_m|\le D_*\) give \[\|P_{\mathcal E,J}\widetilde Z_{l,G,b} -T_{\mathcal E,J}\|_{2,J} \le D_*^{-190}.\] For example, pairing the complementary subtotal against a unit vector in this range costs its coefficient \(\ell^1\) norm, at most \(D_*^{3/2}\), times the outside coefficient sum and the pair bound \(D_*^{-200}\). Within one event level the number of occurrences on a path is at most \(D_*\). The number of levels is \(G_0\), and the budgets also include all groups present on these full-cell paths. Thus the squared additive errors summed over all events are negligible compared with \(|R|\).

Finally, on each \(J\) the different minimal PRE subtotals are almost orthogonal by the same coefficient calculation. Their sum is \(V_{l,G,b}\). Summing their squared norms in Equation [eq:src-24] therefore proves the assertion. ◻

Proof. Say an earlier successful component links to a later one if a mode of the earlier component is equal or CURRENT-adjacent, at the later height-\(h\) graph, to a mode of the later component. Persistence of the earlier CURRENT edges then gives \(\mathcal F_{\rm early}\subseteq\mathcal F_{\rm late}\). Write \(F=\pi(\mathcal F_{\rm late})\). Later goodness and the common first type coordinate imply \[|\pi(\mathcal E_{\rm late})|>|F|/1.1,\qquad |\pi(\mathcal E_{\rm early})| >|\pi(\mathcal E_{\rm late})|/1.1 >|F|/1.1^2 .\] Both predecessor primary sets lie in \(F\). Their intersection therefore has more than \((1/1.1+1/1.1^2-1)|F|\) elements, a proportion greater than one tenth. It contains a later nonexceptional height-good primary. Two witnessing modes of that primary lie in \(\mathcal F_{\rm late}\), one from each predecessor. Lemma 47 joins them in the later PRE graph at \(h-\Delta\). Earlier PRE edges persist, so \[\mathcal E_{\rm early}\subseteq\mathcal E_{\rm late}.\] The same argument at one block reduces to uniqueness of the good predecessor.

Now process the considered pre-cells in block order. If a current predecessor contains an earlier successful predecessor of this type on its path, inherit its representative. Otherwise choose any mode of the current predecessor by a fixed ordering of labels. To prove consistency, take any two earlier predecessors contained in the current one. Equality of the second type coordinate gives each more than \(1/1.1\), and in particular more than one half, of the current mode count. They intersect. They lie on one path, so their shared mode is a link from the earlier to the later of them. The inclusion just proved makes them nested. By induction all their inherited representatives agree. Comparable pre-cell data occur on ancestors, and the list and graph are constant throughout each pre-cell, so this defines representatives consistently for all its actual descendants.

If two components on comparable uses with this type had equal or adjacent modes in the later CURRENT graph, the same inclusion argument would force inheritance of the earlier representative. Thus distinct representatives imply the asserted nonadjacency of every cross pair of modes. Suppose \(I'\) is the later, smaller actual cell, in a block with pre-cell \(J'\) and comparison depth \(t_{b'}\). For each allowed \(n\), nonadjacency gives \[\sup_{x\in J'} |\omega_a(x)-\omega_{a'}(x)+n\cdot\vartheta/L_h| >\Xi/r_{t_{b'}}.\] The slope oscillation on \(J'\) is negligible compared with this threshold, so its value at \(x_*\) is greater than \(\Xi/(2r_{t_{b'}})\). Since \(|I'|\ge r_{t_{b'}}\), this implies Equation [eq:src-25]. ◻

Proof. Modes in different minimal CURRENT components are nonadjacent in that graph. The oscillatory proof of Lemma 51 applies on \(I\), because \(|I|\ge r_{t_b}\). Its synthesis calculation on this single cell shows that for arbitrary subtotals \(f_{\mathcal D}\) in distinct components, \[\left\|\sum_{\mathcal D}f_{\mathcal D}\right\|_{2,I}^2 \asymp\sum_{\mathcal D}\|f_{\mathcal D}\|_{2,I}^2 .\] The constants are uniform, since the off-diagonal norm is at most a fixed small inverse power of \(D_*\) after the private-coordinate bound. In particular this is a two-sided estimate and applies to any subset of labels inside each component.

Let \(T_{\mathcal F}\) be the full coefficient subtotal on \(\mathcal F\). Its core subtotal is \(A_{\mathcal F}=T_{\mathcal F}-V_{\mathcal F}\). The preceding estimate bounds the sum of the squared core component norms in Equation [eq:src-26] by \(C\|A_{\mathcal F}\|_{2,I}^2\). The contribution of the coefficient floors is at most \[C D_*^{-60}\left(\sum_m|\alpha_m|\right)^2 \le C D_*^{-58}.\] Every mode outside \(\mathcal F\) is CURRENT-nonadjacent to the modes inside. Pairing against a unit vector of its augmented span, as in the proof of Proposition 53, gives \[\|P_{\mathcal F,I}\widetilde Z_{l,G,b} -T_{\mathcal F}\|_{2,I}\le D_*^{-190}.\] Combine these observations with \(\|T_{\mathcal F}-V_{\mathcal F}\|^2 \le2\|T_{\mathcal F}\|^2+2\|V_{\mathcal F}\|^2\) to obtain the first bound, with its stated generous error.

Finally decompose \(V_{l,G,b}\) by all CURRENT components at height \(h\), whether successful or not. They too are separated, so the two-sided almost orthogonality gives the second bound after discarding the nonsuccessful terms. ◻

Proof. First fix one height, type, and dyadic threshold \(D_*^{-1}\le\sigma\le D_*^2\). For each group and representative, take the first actual-use cells on each spatial branch for which \(\sigma_{\mathcal F}\ge\sigma\). Call them its first tops. First tops of the same representative do not overlap. For two overlapping first tops of different representatives, Lemma 54 gives CURRENT nonadjacency of all their component modes at the later block. The label sets are therefore disjoint on a path, with total mode count at most \(D_*\). Lemma 51 proves a uniform Bessel bound for the rows \(P_{\mathcal F,I}\) at these tops.

The row is supported on its actual cell \(I\), which is contained in the \(u_b\)-cell of its frozen list. This remains a valid decision cell for Lemma 52, even if many rows share it. All these endpoint cells have the top itself as a downstream actual-use certificate. That lemma gives \[\sum_{\rm tops} \|P_{\mathcal F,I}\widetilde Z_{l,G,b}\|^2 \le G_0|R|.\] No restriction of the input ball projection to the smaller top cell is made in this step.

For the side contributions in Equation [eq:src-27], sum first at one depth of one block. The actual cells there are disjoint; at each cell the component sum is bounded by \(\|V_{l,G,b}\|_{2,I}^2\). Summing the at most \(K\) depths and then applying Proposition 53 gives \[\sum_{\rm tops}|I|\,\|V_{\mathcal F}\|_{2,I}^2 \le C K\sum_{G,b}\|V_{l,G,b}\|^2 \le K G_0|R|.\] Since \(\sigma_{\mathcal F}\ge\sigma\), the size comparison now yields \[\sigma^2\sum_{\rm tops}|I| \le C(1+K)G_0|R| +C D_*^{-50}\sum_{\rm tops}|I|.\] The final term can be absorbed, because \(\sigma^2\ge D_*^{-2}\). Thus \[\sigma^2\sum_{\rm tops}|I| \le C(1+K)G_0|R|.\]

There are only polynomially many choices in \(P,k\) of height, type, and threshold: the counts are controlled by \(J_*\) and powers of \(\log(2D_*)\). For each choice let \(N_\sigma(x)=\sum_{\rm tops}\mathbf 1_I(x)\), summing also over groups. The displayed integral bound and Markov’s inequality give a pathwise bound \(N_\sigma(x)\le P^C\sigma^{-2}\) outside a set of as small a fixed measure as desired, after enlarging \(C\) and the constant in the stopping threshold. Summing these exceptional measures over all the choices still costs a small fraction of \(|R|\). Implement the cut by stopping at the first actual cells where the corresponding partial top count exceeds its threshold. All the estimates are valid again on normalized subroots, so these summands are assigned to fresh attempts.

If a representative is used in the bin \([\sigma_j,2\sigma_j)\) on a retained path, its first \(\sigma_j\)-top is an ancestor of that use. It too is retained; otherwise the descendant summand would already have been sent to a fresh attempt. Different representatives have different such tops on the path, so their count is at most \(N_{\sigma_j}(x)\). The full mode-label budget gives the additional bound \(D_*\). For \(\sigma_j<D_*^{-1}\), that full bound alone implies the minimum in Equation [eq:src-28]. Above the tested threshold range the bins are empty by the coefficient and atom bounds, after the allowed budget enlargement.

The side square bound is the pruning already made following Proposition 53. For amplitude control, \(Z_{u_b}=Z_s-(Z_s-Z_{u_b})\), the bounded pointwise lift, and the side bounds give \(\|\widetilde Z_{l,G,b}\|_{2,I} +\|V_{l,G,b}\|_{2,I}\le P^C\). Projection is a contraction on \(I\), so Lemma 55 bounds \(\sigma_{\mathcal F}\) by \(P^C\) as well. ◻

Proof. We give the cardinality argument on a common fine grid. This entails no restriction on the real data. There are finitely many labels, scales, and shifts of size at most \(Q\). Approximate their real coordinates on one grid, taking label and frequency approximants on a fixed multiple of its step so that the divisions by \(1,2,3\) below remain on the grid. Separation remains valid with, for example, \(\Sigma/2\); the annular bounds remain valid with \(\mu/2\) and \(2\nu\); and the wide pure-shift gap retains the narrower interval \([\mu R_s/100,100\nu R_s]\) needed after relaxing the parameters. These assertions follow by choosing the step after all the finite data, and sufficiently small relative to the least \(R_s\). Round interval widths upward by at most one grid step. All such changes cost fixed constants, absorbed in \(C_{\mathrm{res}}\) and \(C\). No final estimate depends on the step. It therefore suffices to prove the grid version with these harmless fixed margins. We keep the original letters for the relaxed parameters.

Proof. Identify a label \(a\) with its fixed value \(\omega_a(x_*)\). Different labels in one fixed role and type have distinct values by [eq:src-25] with zero shift. We first establish pair determination. In the four-core case, subtract the two frequency tests for two incident tuples that have a given pair of labels in common. At a cost \(O(\nu\max(R_s,R_t))\), the two remaining frequencies solve the linear equations for \(Q\) and \(T\), modulo the bounded shifts. The inverse determinant divides by an integer of magnitude at most \(3\). Separation [eq:src-25] forces the two alternative labels in each remaining role to agree. In the three-core case, the single equation for \(T_l\) similarly determines the third core label from the other two, with a divisor of magnitude at most \(3\). These conclusions hold across different scales, because the separation threshold uses the larger reciprocal scale. All shift differences and divisions fit the height reserve.

In addition, for any triple of core roles let \(l'\) be the excluded fourth role and put \(\alpha_i'=i-l'\) on that triple. The tests imply \[\begin{equation*} c\mu R_s\leq \left|\sum_{i\ \mathrm{in\ the\ triple}} \alpha_i'a_i+n\cdot\xi\right| \leq C\nu R_s, \qquad n\in\mathbb Z^{\dim\theta_z},\quad |n|_\infty\leq CB'_h. \tag{35} \end{equation*}\] For four cores this is \(t'-l'q'\), using both bounds in [eq:src-31]; for three cores it is precisely the nonzero \(T_l\) test. Fixing a pair fixes the whole core tuple, and hence the label values in this annulus. The height-bin observation, applied to those fixed values together with the coordinates of \(\xi\), bounds the number of scale occurrences by \(P^C(1+\log(\nu/\mu))^C\). We use \(L_0\geq1\) for a bound of this form. This proves the first count for any pair of core roles.

For a pair consisting of a core role and the side, fix the core label and a scale. The windows in [eq:src-32] associated with different incident triples are disjoint. If two windows overlapped, their \(Q\) values would be close modulo bounded shifts, with error \(O(\Lambda R_s)\). Combining this with the two \(T_l\) relations determines the two remaining core labels, again by a two-by-two system with divisor at most \(3\). Since \(\nu,\Lambda\leq CU_0\ll\Xi\), separation forces those labels to be identical. Couple the fresh offset in all these windows. Parseval for each offset, followed by expectation, shows that the number of incident triples in the chosen side bin is at most \[w_s=\sigma_l^{-2}\|S_{l,G,s}\|_{2,I_s}^2.\] This remains valid on any subset of scales after summing. Summing over the fixed core labels proves the first inequality of [eq:src-34] for a core/side pair as well.

We now apply Lemma 57 with \(M=M'\), \(L=L_0\), \(d=\dim\theta_z\), and \[\zeta=\xi/36.\] The fixed constants in [eq:src-35] may be absorbed by decreasing \(\mu\) and increasing \(\nu\) by absolute factors. The pair and side-budget hypotheses were just verified. For separation, note that \[\frac{\xi}{36}=6^{\Delta-2}\frac{\vartheta}{L_h}.\] Thus the shifts in the finite lemma are among those of [eq:src-25], with \(Q=\lfloor E_h/6^{\Delta-2}\rfloor\) and \(\Sigma=\Xi/2\). The choice of \(A_0\) and \(B'_h=D_*^5E_{h-\Delta}\) ensures \(Q\geq C_{\mathrm{res}}(d+1)^2 CB'_h\). Likewise \(\Xi=D_*^{1000}\) and \(\nu\leq CD_*^{250}\) ensure the separation reserve, with room for the fixed annular constants.

Finally, any successful component contains a nonexceptional primary, even when its representative is itself exceptional. For that primary the pure-horizontal cases of [eq:src-23] apply to every shift in the finite lemma: their rational heights are smaller than \(\mathcal H\). They are either at most \(\mathcal T^{-1}/r_{v_b}\) or at least \(\mathcal T/r_{t_b}\). Since \(U_0^{-1}\leq\mu\leq1\leq\nu\leq CU_0\) and the test scale lies between these depth scales, the prescribed choice of \(\mathcal T\) excludes the entire intermediate interval required by the pure-shift hypothesis. This checks every hypothesis of the finite lemma, uniformly over the current group.

It follows that, for an absolute \(c'>0\), \[\begin{equation*} \sum_{s:\ N_s^G\leq\delta(M')^2}N_s^G \leq P^C(1+\nu+\mu^{-1}+\Lambda)^C \delta^{c'}(M')^2 \qquad(0<\delta\leq1). \tag{36} \end{equation*}\] For example, \(c'=1/2080\) is admissible after changing the absolute constants. Indeed Lemma 57 gives the stronger coefficient \(C\ell L_0^{29}\); the dimension and the logarithms in this expression fit the displayed budget.

Pair determination also gives \(N_s^G\leq(M')^2\). Choose \(p_0\in(0,1)\) with \(1-p_0<c'\), for instance \(p_0=1-c'/2\). On the set of scales with \[2^{-j-1}(M')^2<N_s^G\leq2^{-j}(M')^2\] we may bound \((N_s^G)^{p_0}\) by \([2^{j+1}/(M')^2]^{1-p_0}N_s^G\). Sum this inequality and use [eq:src-36] with \(\delta=2^{-j}\). The remaining geometric series has ratio \(2^{1-p_0-c'}<1\), and proves the second inequality of [eq:src-34]. ◻

Proof. Set \[p=1-\varepsilon=(1-\theta')+\theta'p_0,\] where \(\theta'>0\) is a sufficiently small absolute number, to be fixed below. Hölder interpolation between the two inequalities in [eq:src-34] gives, for any distinct roles \(i,j\), \[\sum_s(N_s^G)^p \leq (L_0M_i^GM_j^G)^{1-\theta'} \big[P^C(1+\nu+\mu^{-1}+\Lambda)^C (M')^{2p_0}\big]^{\theta'}.\] Order the amplitude bins from largest to smallest, temporarily calling their roles \(1,2,3,4\), so that \(\sigma_1\geq\sigma_2\geq\sigma_3\geq\sigma_4\). Use the pair \(1,2\) in the first factor. At least one of this pair is a core role, because there is at most one side.

The path bounds [eq:src-28] and the side square budget give \[\sum_G M_i^G\leq CP^C\sigma_i^{-2} \quad\text{in every role},\qquad \sum_G M_i^G\leq D_* \quad\text{in a core role}.\] Expand the maximum \(M'\) into at most four choices of the role attaining it. The total exponent of the resulting product of \(M_i^G\) is \(2p>1\). For nonnegative sequences \(m_i^G\) and exponents \(\beta_i\geq0\) with \(\sum_i\beta_i\geq1\), one has \[\sum_G\prod_i(m_i^G)^{\beta_i} \leq\prod_i\left(\sum_Gm_i^G\right)^{\beta_i}.\] Indeed normalize each nonzero sum to one, apply Hölder with exponents \((\sum_i\beta_i)/\beta_i\), and use that the normalized sequences have \(\ell^{\sum\beta_i}\) norm at most one. This proves the needed group summation.

The largest size-to-count factor is \(\sigma_4^{-2}\). In the core member of the first pair we may also use the count cap \(D_*\). If this member is role \(2\), its factor \(\min(1,D_*\sigma_2^2)\) is bounded by the corresponding factor with \(\sigma_1\). Multiplying by \((\prod_i\sigma_i)^p\) from [eq:src-33], the amplitude dependence is consequently at most a \(P^C\) factor times \[\min(1,D_*\sigma_1^2)^{1-\theta'} \sigma_1^{p-2(1-\theta')} \sigma_2^{p-2(1-\theta')} \sigma_3^p\sigma_4^{p-4\theta'p_0}.\] Let the four displayed power exponents be \(e_1,e_2,e_3,e_4\). Their sum is zero, and their three preceding partial sums are \[-1+\theta'(1+p_0),\qquad -2+2\theta'(1+p_0),\qquad -1+\theta'(1+3p_0).\] All are strictly negative when \(\theta'\) is sufficiently small. Writing \(g_j=\log_2(\sigma_j/\sigma_{j+1})\geq0\), the remaining power factor is \(2^{\sum_{j=1}^3g_j(e_1+\cdots+e_j)}\). Thus all three amplitude-gap sums are convergent geometric series.

Only a common shift of all the bins remains. The retained upper amplitude bound is \(P^C\). Below \(\sigma_1=D_*^{-1/2}\) the prefactor \(\min(1,D_*\sigma_1^2)^{1-\theta'}\) decays geometrically; above it there are only \(O(1+\log D_*+\log P)\) possible dyadic bins up to that upper bound. The common-shift cost is therefore polynomial in the allowed logarithmic budgets. There are only \(24\) amplitude orderings, so this completes the amplitude summation.

It remains to sum the frequency bands. The first count loses only powers of \(1+\log(\nu/\mu)\). The second count, after interpolation, loses at most \(\mu^{-C\theta'}\nu^{C\theta'}\Lambda^{C\theta'}\). The powered kernel factor in [eq:src-33] is \(\mu^p\nu^{-Lp}\Lambda^{-Lp}\). First take \(\theta'\) small enough that \(p-C\theta'>0\), and then take the fixed derivative order \(L\) sufficiently large. The sums over dyadic \(0<\mu\leq1\) and \(1\leq\nu,\Lambda<\infty\) converge, including the logarithmic losses. Extending the actual truncated ranges to these ranges only increases the bound. The number of heights, types, role patterns, and retained leaves has an allowed polynomial cost. We have proved a bound \(C_qP^{C_4}\) for the path sum of the \(p\)th powers of all the principal core contributions, keeping each \(N_s^G\) inside its power.

To finish, let \(\mathcal C_I\) be the sum of their absolute principal contributions at a cell. Subadditivity for \(p<1\) and \(\min(\eta,t)\leq\eta^{1-p}t^p\) give \[\min(\eta,\mathcal C_I) \leq\eta^\varepsilon\mathcal C_I^p \leq\eta^\varepsilon \sum_{\mathrm{configurations}} (\text{principal contribution at }I)^p.\] Integrating the path estimate uses the exact identity \(\sum_I|I|F_I=\int_R\sum_{I\ni x_*}F_I\,dx_*\) for nonnegative cell data. It gives \(C_q\eta^\varepsilon P^{C_4}|R|\). The previously separated absolute errors contribute \(O(e^{-k}|R|)\), which proves [eq:src-37]. All polynomial degrees in the finite counting lemma, and hence \(p_0\), \(\theta'\), \(\varepsilon\), and \(C_4\), are absolute. Taking \(\theta'\) smaller if necessary gives \(p>2/3\) as required by the side estimates. ◻