Proof. Take a smooth bounded-overlap partition \(\{\chi_\omega\}\) of frequency space at resolution \(s^{-1}\). We can arrange that \(\chi_\omega(\omega+\eta/s)\) is supported in \([-1,1]^2\). Put \[g_\omega(\eta)=s^{-1}(\chi_\omega v)(\omega+\eta/s).\] Then \(\|g_\omega\|_2=\|\chi_\omega v\|_2\). On \([-\pi,\pi]^2\), expand \(g_\omega\) in the orthonormal Fourier basis: \[g_\omega(\eta)=\sum_{n\in\mathbb{Z}^2} a_{\omega,n}(2\pi)^{-1}e^{-in\cdot\eta}.\] Choose one smooth cutoff \(\psi\) supported in \((-\pi,\pi)^2\) and equal to one on \([-1,1]^2\). Multiplying the series by \(\psi\) leaves \(g_\omega\) unchanged, and gives the packet data \[v_{\omega,n}(\xi)=\frac{s}{2\pi} \psi\bigl(s(\xi-\omega)\bigr) e^{-isn\cdot(\xi-\omega)}.\] Parseval, followed by the bounded overlap of the frequency partition, gives (9). Multiplication by \(\psi\) is bounded on \(L^2\), and its translated frequency supports have bounded overlap. Applying these two facts to any selected Fourier coefficients proves (10). This also proves convergence of the decomposition in \(L^2\). Since all frequency supports remain in a fixed bounded set, the corresponding extensions converge uniformly in \((x,t)\).

Changing variables \(\xi=\omega+\eta/s\) gives \[ Ev_{\omega,n}(x,t)=\frac{s^{-1}}{2\pi} e^{i(x\cdot\omega-t|\omega|^2)} \int \psi(\eta) e^{i((x-sn-2t\omega)/s)\cdot\eta-i(t/s^2)|\eta|^2}\,d\eta. \tag{13}\] For \(|t|/s^2\leq C\), all derivatives of \(\psi(\eta)e^{-i(t/s^2)|\eta|^2}\) are bounded. Repeated integration by parts proves (11). In particular the profile in (13) is Schwartz in its first argument, uniformly on the prescribed time interval.

For clarity, the tail bound is uniform even for infinite packet expansions. At a fixed point, there are \(O(s^2)\) frequency centers. For each one the normalized spatial labels form a translate of \(\mathbb{Z}^2\), and, for \(N>1\), \[\sum_{n:\ |x-sn-2t\omega|>s^{1+\eta}} \left(1+\frac{|x-sn-2t\omega|}{s}\right)^{-2N} \leq C_Ns^{-\eta(2N-2)}.\] The factor \(s^{-2}\) from the squared packet bound cancels the \(O(s^2)\) frequency count. Cauchy–Schwarz and (9) therefore bound the discarded field by \(C_Ns^{-\eta(N-1)}\|v\|_2\). Choose \(N\) sufficiently large in terms of \(A\) and \(\eta\).

Finally, a bounded-velocity trajectory traverses distance \(O(s^2)\) in the time slab. Since \(s^{1+\eta}\leq s^2\), its padded trajectory meets only \(O(1)\) squares of side \(s^2\). Each coefficient is consequently assigned to only \(O(1)\) of the subfamilies defining \(v_Q\). Equation (10) proves their summed energy bound, and the preceding tail estimate proves the second part of (12). To center the time interval at \(t_0\), replace \(v(\xi)\) by \(e^{-it_0|\xi|^2}v(\xi)\) before decomposing it. ◻

Proof. For unit-ball frequency support, Theorem 1.6 of [13], specialized to spatial dimension \(2\) and counting exponent \(2\), gives \[\|Ev\|_{L^2(X)} \leq C_\epsilon S^\epsilon \gamma^{1/6}\lambda^{1/6}S^{1/6}\|v\|_2\] when every occupied square-root-scale lattice cube contains a comparable number \(\lambda\) of the unit cubes of \(X\). Its time sign is changed by reflection, and its physical-space input norm is converted to the frequency-space norm by Plancherel. A bounded frequency enlargement is reduced to unit frequency by a fixed parabolic dilation and bounded cube coverings. None of these operations changes a large-scale exponent.

For the asserted upper occupancy version, take an ambient radius comparable to \(S\) whose square root is an integer, large enough to contain the given ball. Its square-root grid then contains whole unit lattice cubes. The occupancy of each parent is at most \(C\lambda_0\), since such a parent is covered by a fixed number of \(\sqrt S\)-balls. Sort the parents by their dyadic occupancy \(\lambda\). Every resulting union retains the counting bound \(C\gamma r^2\). Apply the quoted theorem to each union, square it, and sum. The geometric sum of \(\lambda^{1/3}\) is at most \(C\lambda_0^{1/3}\); using a logarithmic bound would also suffice after decreasing the chosen small exponent. Renaming that exponent proves (15). Ball centers can be translated at the cost of replacing \(v(\xi)\) by \(e^{i(y\cdot\xi-u|\xi|^2)}v(\xi)\), which preserves its norm and support.

To obtain suprema, let \(F=Ev\). Its space-time Fourier transform is supported on a fixed compact portion of the paraboloid. Choose a Schwartz function \(\rho\) whose Fourier transform is one on that compact set, so that \(F=F*\rho\). For every unit cube \(Q\) and every \(z\in Q\), \[ \sup_{y\in Q}|F(y)| \leq C_N\int_{\mathbb R^3}(1+|w|)^{-N}|F(z+w)|\,dw. \tag{17}\] Indeed, taking the supremum over \(y-z\) in a fixed bounded set in the convolution formula leaves a rapidly decreasing majorant for the kernel. Cauchy–Schwarz with this integrable weight, followed by integration over \(z\in Q\) and summation in \(Q\), gives \[\sum_{Q\in\mathcal Q}\sup_Q|F|^2 \leq C_N\int_{\mathbb R^3}(1+|w|)^{-N} \int_{\bigcup\mathcal Q}|Ev(z+w)|^2\,dz\,dw.\] For each fixed \(w=(y,u)\) the inner field is the extension of the modulated datum above. Apply (15) on the unchanged domain and integrate the weight. This proves (16) without enlarging the ambient scale according to the translation. ◻

Proof. Fixed-radius enlargements of the second bound in (19) follow by a bounded covering with balls of radius \(\sqrt S\) centered on the support. This also extends it to arbitrary centers, up to a fixed constant. Every unit cube has mass at most \(Cm\). Split the positive-mass cubes into classes \[m\chi<\nu(Q)\leq2m\chi, \qquad \chi\in\{C_02^{-k}:k\geq0\},\] with harmless endpoint adjustments and fixed \(C_0\). Cubes in one class have counting parameter \(C\chi^{-1}\): the cubes meeting an \(r\)-ball have disjoint masses inside a fixed enlargement of that ball. Their square-root-scale occupancy is at most \(CSd\chi^{-1}\) by the same argument. Lemma 4 therefore gives \[\begin{align*} \sum_{Q\text{ in the }\chi\text{ class}}\nu(Q)\sup_Q|Ev|^2 &\leq C_\epsilon m\chi S^\epsilon \bigl(\chi^{-1}\cdot Sd\chi^{-1}\cdot S\bigr)^{1/3} \|v\|_2^2\\ &=C_\epsilon S^\epsilon mS^{2/3}d^{1/3} \chi^{1/3}\|v\|_2^2. \end{align*}\] The series \(\sum_\chi\chi^{1/3}\) converges. Summing proves both assertions, including measures with arbitrarily small positive cube masses; no lower mass cutoff is needed here. Finally the first ball bound, evaluated at \(r=\sqrt S\), supplies \(d=1\). ◻

Proof. Translate the space-time support ball to the origin, absorbing the translation in a unimodular factor on \(v\). Write \[g(\eta)=q\,v(v_0+q\eta),\qquad (y,\tau)=\bigl(q(x-2v_0t),q^2t\bigr),\qquad S'=Sq^2.\] Then \(\|g\|_2=\|v\|_2\), \(\mathop{\mathrm{supp}}g\subset B(0,1)\), and \[ Ev(x,t)=q\,e^{i(x\cdot v_0-t|v_0|^2)}Eg(y,\tau). \tag{22}\] Let \(\nu'\) be the pushforward of \(\nu\). The inverse map is \[t=q^{-2}\tau,\qquad x=q^{-1}y+2v_0q^{-2}\tau.\] Consequently, the inverse image of a ball of radius \(r\geq1\) is a bounded shear of a box with side lengths \(r/q,r/q,r/q^2\). It is covered by \(O(q^{-1})\) balls of radius \(Cr/q\). The original ball bound applies at these radii and gives \[ \nu'(B(z,r))\leq Cm q^{-3}r^2\qquad(r\geq1). \tag{23}\] There is no Jacobian multiplier on this pushforward measure.

The rescaled time interval has length \(O(S')\), but the spatial projection has diameter \(O(Sq)=O(S'/q)\). Partition the latter into squares \(Q\) of side \(S'\). On the common time slab, use Lemma 3 at packet width \(s=\sqrt{S'}\), with any fixed padding exponent in \((0,1)\). Let \(g_Q\) denote its localized data. Then \[\sum_Q\|g_Q\|_2^2\leq C\|g\|_2^2, \qquad |Eg-Eg_Q|\leq C_A(S')^{-A}\|g\|_2 \quad\text{on the corresponding prism},\] where \(A\) can be made arbitrarily large. Each restricted measure lies in a ball of radius \(CS'\), so Lemma 5 with \(d=1\) and (23) gives \[\sum_Q\int_{Q\times[-CS',CS']}|Eg_Q|^2\,d\nu' \leq C_\epsilon(S')^\epsilon m q^{-3}(S')^{2/3}\|g\|_2^2.\] The sum uses the localized energies, rather than the number of spatial squares. This distinction is essential for the cap power.

For completeness, \(\nu'(\mathbb R^3)=\nu(\mathbb R^3)\leq CmS^2\), whereas \(S'\geq S^{1-2\alpha_0}\). Thus the integrated squared error, after multiplication by the \(q^2\) in (22), is at most \[C_A m q^2 S^2(S')^{-2A}\|g\|_2^2.\] Choose \(A\) in terms of \(\alpha_0\) so that this is bounded by the desired right side; bounded \(S\) is absorbed in the constant. For the principal term, \[ q^2\,m q^{-3}(Sq^2)^{2/3} =mS^{2/3}q^{1/3}. \tag{24}\] Since \((S')^\epsilon\leq S^\epsilon\), this proves (21).

Finally, square (17) by weighted Cauchy–Schwarz, integrate in \(z\in Q\) against \(d\nu(z)\), and sum the original unit cubes. This bounds the sum of weighted suprema by \[C_N\int_{\mathbb R^3}(1+|w|)^{-N} \int_{\mathbb R^3}|Ev(z+w)|^2\,d\nu(z)\,dw.\] For every shift the field is an exact extension of modulated data with the same cap support and norm. Apply the already proved cap estimate and integrate the rapidly decreasing weight. This proves the supremum assertion and the asserted translated versions. ◻

Proof. Partition each direction support into neighborhoods of radius a sufficiently small fixed power of \(\nu\). There are at most \(\nu^{-O(1)}\) such neighborhoods. For each triple choose representative directions and send them to the coordinate axes by a linear map. Its norm, inverse norm, and volume distortion are bounded by fixed powers of \(\nu^{-1}\). The transformed direction neighborhoods lie in fixed small neighborhoods of the axes. The weighted multilinear Kakeya inequality of [1], applied there and summed over triples, proves (26). Weights follow first by repetitions and then by approximation. Widths larger than a fixed constant can instead be treated by the trivial bound, after increasing the inverse-power constant.

For completeness, use this Kakeya inequality to obtain restriction uniformity. Let \(A_B(S,\nu)\) be the least normalized product-integral constant for inputs supported in \([-B,B]^2\). Fix padding exponent \(\theta>0\) and Kakeya loss \(\sigma>0\). Cover a ball of radius \(S\) by cubes of side comparable to \(\sqrt S\). The exact packet decomposition restricts each input on a smaller cube to packets incident to its \(S^\theta\sqrt S\) enlargement. Its retained exact input has squared norm at most a fixed constant times the incident coefficient energy. Its support is within \(O_B(S^{-1/2})\) of the original support: only caps intersecting that support occur, and every enlarged cap lies within this distance of such an intersection. Thus determinant continuity preserves \(\nu/2\) transversality when \(S^{-1/2}\le c_B\nu\), without any openness or connectedness assumption on the supports.

Apply the smaller-scale product estimate in every cube. Its geometric coefficient is the product of the three square roots of incident energies. After division of coordinates by \(S\), the padded tubes have width \(\Delta=S^{-1/2+\theta}\). Enlarge them by a fixed factor, so an incident tube contains the whole smaller grid cube. Equation (26), divided by that cube’s volume \(S^{-3/2}\), gives the bound \[C_{B,\theta,\sigma}\nu^{-M_\sigma} S^{3/2}(S^{-1/2+\theta})^{3-\sigma} \prod_i\|v_i\|_2 \le C_{B,\theta,\sigma}\nu^{-M_\sigma} S^\beta\prod_i\|v_i\|_2,\qquad \beta=3\theta+\sigma/2,\] for the sum of these geometric coefficients. To bound discarded-field errors, normalize each parent input, use rapid tails for a discarded factor and the fixed-frequency \(L^\infty\) bound for the others, and expand the three-factor product. The seven error terms together contribute at most \(C_{B,\theta,N}S^{3-N}\prod_i\|v_i\|_2\), for every prescribed \(N\). This is relative to the current parent norms; no lower bound for a retained local norm is required. We obtain the recurrence \[ A_B(S,\nu)\le C_{B,\theta,\sigma}\nu^{-M_\sigma}S^\beta A_{B+1}(C_B\sqrt S,\nu/2) +C_{B,\theta,N}S^{3-N}, \qquad S^{-1/2}\le c_B\nu. \tag{27}\] Spatial translations only modulate the exact inputs. Taking \(N>3\) and setting \(\widetilde A_B=1+A_B\) absorbs the additive error in the same multiplicative recurrence for \(\widetilde A_B\).

Fix the target \(\epsilon>0\). Choose a finite \(J\) with \(3\cdot2^{-J}<\epsilon/2\), then choose \(\theta,\sigma\) with \(2\beta<\epsilon/2\). The finitely many radii defined by \(S_0=S\), \(S_{j+1}=C_{B+j}\sqrt{S_j}\) are comparable to \(S^{2^{-j}}\), with constants depending only on \(B,J\). If \(S\ge(C_{B,J}/\nu)^{2^J}\) for a sufficiently large constant, all \(J\) support-enlargement conditions hold. Iterating (27) and using the terminal trivial bound \(\widetilde A_{B+J}(R,\nu')\lesssim_{B,J}R^3\) gives \[\widetilde A_B(S,\nu) \le C_{B,\epsilon}\nu^{-M_{B,\epsilon}} S^{\beta\sum_{j=0}^{J-1}2^{-j}+3\cdot2^{-J}} \le C_{B,\epsilon}\nu^{-M_{B,\epsilon}}S^\epsilon.\] In the complementary range, the trivial bound \(A_B(S,\nu)\lesssim_B S^3\) is absorbed by increasing the fixed inverse-transversality exponent to at least \(3\cdot2^J\). This proves (25) for all \(S,\nu\). On a subpower-transverse sequence the threshold is eventually satisfied for each fixed \(J\), and every intermediate enlarged support remains subpower transverse. This is a finite version of the restriction bootstrapping in [1]; polynomial dependence is supplied by this argument, not by the statement of their restriction theorem alone.

Finally, fixed-frequency reproducing kernels give, for \(z\in Q\), \[\sup_Q|Ev_i|\lesssim_N \int(1+|w|)^{-N}|Ev_i(z+w)|\,dw.\] Integrate the product over \(Q\) and sum. For each triple of shifts the three shifted fields are exact extensions on the same domain, with independently modulated inputs of unchanged norms and supports. Apply (25) and integrate the Schwartz weights. ◻

Proof. Sort occupied \(L\)-boxes by their number of selected unit cubes. A dyadic occupancy class retains \(L^{4-o(1)}\) cubes. The squared supremum sum for each field on this class is at least \(L^{4/3-o(1)}\). By Lemma 4 at radius \(L^2\), a class with occupancy at most \(L^{2-\delta}\), for fixed \(\delta>0\), instead has squared sum at most \(L^{4/3-\delta/3+o(1)}\). Thus some retained class has \(L^{2-o(1)}\) cubes in each occupied box. The upper counting bound gives occupancy at most \(L^{2+o(1)}\). Divide coordinates by \(L^2\) and take the centers of these boxes. Counting cubes in an enlarged ball and dividing by the minimum occupancy proves (28). Translation of the original ambient ball merely modulates the exact data.

Apply Lemma 3 at packet width \(L\) on the time interval of length \(O(L^2)\). After division of coordinates by \(L^2\), the resulting tubes have width \(L^{-1+o(1)}\). Give each tube the squared coefficient of its packet as weight. On each retained box, packets reaching a padded enlargement give an exact local input and an error \(O(L^{-100})\). The padding exponent tends sufficiently slowly to zero; the rapid-tail estimates permit this by diagonalization. By (10), the squared norm of this exact local input is bounded by the incident coefficient energy \(e_i(p)=\sum_{T\ni p}w_i(T)\) up to a constant. On the at least \(L^{2-o(1)}\) selected unit cubes in this box the required squared supremum sum is \(L^{-2/3-o(1)}\). The weighted fractal estimate at radius \(L\) bounds it above by \(L^{2/3+o(1)}e_i(p)\), so \[e_i(p)\ge L^{-4/3-o(1)}.\] Lemma 9, in its Kakeya form and with \(\Delta=L^{-1+o(1)}\), yields \[\sum_{p\in P}\prod_{i=1}^3e_i(p)^{1/2}\le L^{o(1)}.\] For fixed \(\delta>0\), points at which some \(e_i(p)\ge L^{-4/3+\delta}\) therefore number at most \(L^{2-\delta/2+o(1)}\). Discarding these points by a diagonal choice leaves (28) and makes all three energies \(L^{-4/3+o(1)}\) uniformly.

It remains to remove concentration in direction without losing the local lower bound. Choose \(0<\kappa<1/3\) and \(0<u_0<1/4\). For each fixed rational \(0<u<u_0\), put \(q=L^{-u}\) and partition directions into bounded-overlap bins of diameter comparable to \(q\). Delete incident bins having weight greater than \(q^\kappa e_i(p)\). There are \(O(q^{-\kappa})\) such bins. On the original box of radius \(O(L)\), Lemma 6 applies with \(S=L\); the packet frequency enlargement \(O(L^{-1})\) is smaller than the bin width \(q\). Cauchy–Schwarz over the deleted bins then bounds the squared supremum sum of their combined field by \[L^{2/3+o(1)}q^{1/3-\kappa}e_i(p).\] Here the chosen unit cubes have local density \(L^{o(1)}\), and the energies of all deleted bins sum to \(O(e_i(p))\). The estimate also holds for arbitrary subfamilies of these bins. For every fixed \(u>0\) this is negligible relative to \(L^{-2/3-o(1)}\).

Delete successively for any fixed finite list of rational \(u\)’s, assigning a packet deleted more than once to just one bin. The triangle inequality in the square-sum norm shows that the surviving field retains its lower local norm. The unrestricted local estimate then gives surviving weight \(e_i^*(p)=L^{-4/3+o(1)}\). Every surviving bin has original weight at most \(q^\kappa e_i(p)\); normalizing by \(e_i^*(p)\) costs only \(L^{o(1)}\). Boundedly many neighboring bins cover a ball, and a diagonal choice proves (29) at all fixed rational scales. All estimates were uniform in \(p\), so this point-dependent pruning supplies legitimate conditional laws without requiring a single globally pruned family. ◻

Proof. A variable with at most \(L^{C+o(1)}\) labels satisfies, uniformly in conditioning, \[\mathbb P\{I_L(Y\mid V)>C+\eta\}\le L^{-\eta+o(1)}.\] The same counting argument with a varying threshold proves uniform integrability. For the conditioning assertion put \(p=\mathbb P(Y\mid V)\) and \(q=\mathbb P(Y\mid V,W)\) at the sampled labels. Conditional on \((V,W)\), the expectation of \(p/q\) is at most one. Thus \[\mathbb P\{I_L(Y\mid V)-I_L(Y\mid V,W)<-\eta\} \le L^{-\eta}.\] This proves the almost-sure sign in the limit; no finite-scale pointwise monotonicity is asserted. The chain rule is pointwise already at finite scale. The determination rule follows by summing over the stated small set of possible labels. This argument also works on restricted events, since it bounds the total mass of the bad low-probability labels in the event’s permitted set. If the event indicator is included in the joint limit, apply weak convergence to the open set where it is one and the information is strictly above \(\gamma+\eta\), and then let \(\eta\downarrow0\). For events defined through other limiting variables, the same argument uses strict inequalities and slack, as detailed in the zero-ratio application below.

Tightness and diagonal weak convergence on a countable product give the joint law. Include in this product all rational block and prefix parameters, all finitely many choices of reference and replacement tubes, their direction bins, coordinate grids, projections, the information differences used below, and the angular quantities to be introduced. Direction grids use deterministic representatives of occupied bins. Realized bases and inverse bases have norms \(L^{o(1)}\), so every coordinate partition has at most \(L^{C+o(1)}\) labels. Standard open-set inequalities for weak convergence justify the assertion about finite systems after enlarging their bounds by slack. Uniform integrability permits passage of expectations in the cost estimates below. ◻

Proof. The exact chain increment of \(G_i\) between \(s\) and \(s+t\) is \(I(X_{s+t}\mid X_s,T_i)\). Inside a parent, the tube meets at most \(L^{t+o(1)}\) child cubes, including its padding. The determination rule bounds this increment between zero and \(t\); the unrestricted spatial count gives the bounds zero and \(3t\) for \(F\). Rational comparisons simultaneously yield the Lipschitz extensions.

The ball count in (28) gives every \(X_s\)-cell mass at most \(L^{-2s+o(1)}\), proving \(F(s)\ge2s\). An \(X_1\)-cell contains boundedly many points and \(X\) is uniform, so \(F(1)=2\). Finally the retained weights give \[\mathbb P(X=p,T_i=T) =|P|^{-1}\frac{w_i(T)\mathbf1_{\{T\text{ retained at }p\}}} {e_i^*(p)} \le L^{-2/3+o(1)}w_i(T).\] The probability of sampling a tube whose marginal is smaller than \(L^{-\eta}w_i(T)\) is at most \(L^{-\eta}\sum_Tw_i(T)=O(L^{-\eta})\). Outside this event, every conditional point mass is at most \(L^{-2/3+\eta+o(1)}\). Letting \(\eta\downarrow0\) proves the conditional endpoint bounds, also for all primed tubes. This uses only summability of weights, so exact tube labels need not have bounded cardinality. ◻

Proof. The spatial child and the three coordinate labels determine one another up to \(L^{o(1)}\) choices. A reference tube determines its two off-axis coordinates to \(L^{o(1)}\) choices: both its width and the error in its approximate direction, relative to the parent, are at most \(L^{-t+o(1)}\). Conditional information calculus therefore gives \[J_i-\ell_i=I(Z_i\mid C,D,T_i) =I(Z_i\mid Z_{\widehat i},C,D,T_i) \le I(Z_i\mid Z_{\widehat i},C,D).\] For each ordering of the coordinates, its chain-rule term conditions on a subset of \(Z_{\widehat i}\), hence dominates the last expression. Sum the three terms to obtain (32).

At finite scale, the expected cost is the conditional mutual information, divided by \(\log L\), between the child and \(D\) given the parent (and, when present, the same fixed tube). Along the nested spatial filtration these mutual informations telescope. Inserting the omitted gaps between disjoint blocks only adds nonnegative expected terms. Their sum is at most \(H(D)/\log L=O(b_*)\), because there are \(L^{O(b_*)}\) direction-bin lists. Fixed tube conditioning is never changed within this telescoping argument. Passage to the limit follows from Lemma 11. ◻

Proof. Partition a fixed rational spatial interval into blocks of length at most \(b=b_*\). By (32), the positive part of the excess of the summed \(G_i\) increment over the \(F\) increment has expectation \(O(b)\). Let \(b\downarrow0\) to obtain the interval inequality almost surely. At \([0,1]\), (31) forces equality and \(G_i(1)=2/3\). The interval deficits are nonnegative and additive, so their zero full-interval total forces all rational subinterval deficits to vanish. Continuity gives \(F=\sum_iG_i\). Repeat for the finitely many replacement triples; subtracting identities gives \(G_i'=G_i\).

For any coordinate ordering, its \(i\)th chain term dominates \(I(Z_i\mid Z_{\widehat i},C,D)\) and then \(J_i-\ell_i\). The sum of the surpluses is exactly \(\sum_i\ell_i-\ell\le\sum_i\ell_i\), using \(J=\sum_iJ_i\). Take a chosen coordinate, or chosen pair, first in the ordering. This gives both bounds in (33), with each individual error controlled by the total cost. ◻

Proof. In the exact basis of \(T_1,T_2,T_3\), normalize the \(i\)th coordinate of the direction of \(T_i'\) to one, and call its other two ratios \(a_j\) (\(j\ne i\)). Replacement transversality and Cramer’s rule bound the normalizing denominator, its inverse, and all resulting ratios by subpowers. Include in the joint limit the values \(\min\{1,\max\{0,-\log_L|a_j|\}\}\), with value one at zero. The two limiting values cannot both be positive: that would put \(T_i'\) within a fixed positive-power angular ball about \(T_i\), with probability tending to zero by (29) and conditional independence given \(X\).

Let \(r_j\) be the limiting clipped value for \(a_j\). We make the restricted-event comparison on \(\{r_j=0\}\) precise. Fix a rational \(0<\eta<t/2\). At finite scale, \(r_{j,L}<\eta/2\) implies \(|a_j|>L^{-\eta/2}\). The approximate basis fixed by \(D\) differs from the exact basis by \(L^{-b_*+o(1)}\), including the subpower condition numbers. Its corresponding coefficient thus has magnitude at least \(L^{-\eta}\) for large \(L\). Given \((T_i',Z_j,C,D)\) alone, possible child positions on this event number at most \(L^{O(\eta)+o(1)}\): the approximate \(j\)th coordinate parametrizes the tube with inverse slope at most \(L^{\eta+o(1)}\). This permitted set depends only on those displayed labels. Exact reference directions need not be added to the conditioning.

Apply the restricted determination bound and the information conditioning rule. The probability that \(r_{j,L}<\eta/2\) and \[I_L(X_{s+t}\mid C,D,T_i')-I_L(Z_j\mid C,D)>C\eta+\eta\] tends to zero, with a fixed absolute \(C\). The joint limit therefore gives the corresponding comparison on the open event \(\{r_j<\eta/2\}\) with error at most \((C+1)\eta\); one can first use any strict upper error bound and then decrease it. This is an implication about the support of the joint limiting law and requires no coupling of finite-scale and limiting events. Let rational \(\eta\downarrow0\). On \(\{r_j=0\}\) we obtain \[G_i'(s+t)-G_i'(s)-\ell_i' \le I(Z_j\mid C,D),\] where \(D\) includes the alternative bin and \(\ell_i'\) is its analogous cost. Replacement invariance and (33) now compare the \(i\) and \(j\) profile increments up to the costs.

Take dyadic partitions of the exponent interval, with block size \(b=b_*\). On each block distribute its cost uniformly over its length. The integral of this density is the cost; hence its expected integral is \(O(b)\) by Lemma 13. Summability over dyadic \(b\) shows that all normalized costs tend to zero for almost every pair consisting of a path and an exponent. At simultaneous differentiability points it follows that \(\dot G_i\le\dot G_j\) whenever the corresponding limiting ratio information is zero. The restricted-event determination just used is applied before differentiation, so no uniform positive lower bound for the coefficient is required.

For almost every path each vertex \(i\) therefore has an edge \(i\to j\) for which \(\dot G_i\le\dot G_j\) almost everywhere. Since both integrals are \(2/3\), this forces \(G_i=G_j\). Every equivalence class of equal profiles has at least two members. With three families there is only one such class, proving (34). ◻

Proof. This is the \(n=2\), \(m=1\), \(\alpha=2d\) case of [17], a general form of the planar Bourgain discretized projection theorem. In dimension two its Grassmannian nonconcentration condition is the displayed angular-ball condition. That theorem gives the conclusion simultaneously for every large subset, for a direction set of probability at least one minus a positive power of \(\Delta\). Decreasing the tolerance relative to its exponent gives the stated formulation. ◻

Proof. The exact coordinate map and its inverse have subpower norm on the realized support. A joint ball in \((a_2,a_3)\) therefore pulls back to at most two direction balls of radius \(L^{-u+o(1)}\). The alternative direction law, conditional on \(X\) and the exact references, is still its original law given \(X\). Equation (29), enlarged by a subpower using covering, bounds this joint ball mass by \(L^{-\kappa u+o(1)}\).

Here ball and grid informations have the same limit. Conditional on the exact point and references, take grids at comparable sizes. If \(p_z\) is the mass of a sampled cell and \(q_z\) the total mass of its boundedly many neighbors, then \[\sum_{z:q_z>L^\eta p_z}p_z \le L^{-\eta}\sum_zq_z\lesssim L^{-\eta}.\] A boundedly finer grid and the determination rule handle balls not contained in one grid cell. The argument is identical for joint and marginal grids. The chain rule and (30) bound joint cell information by the sum of marginal informations. This proves (35); monotonicity follows directly from nesting of centered balls. ◻

Proof. Put \(K=L^b\) and use the same reference coordinates for \(D_0,D_1\). Let \(Y=(Z_1,Z_j)\) at the endpoint or a tested prefix. Equations (33)–(34) imply, with either conditioning \(C,D_0\) or \(C,D_1\), \[ I(Y_b\mid C,D_a)=2db+O(\zeta b),\qquad I(Y_u\mid C,D_a)\ge2d_0u-O(\zeta b),\quad a=0,1. \tag{36}\] Let \(\widetilde a_j\) be the approximate direction ratio fixed by \(D_1\), and let \(\Pi(Y_b)\) record the interval of length \(K^{-1}\) containing \(z_j-\widetilde a_jz_1\) at the pair-cell center. The exact and approximate ratios differ by at most \(L^{-b_*+o(1)}\). Given \(T_1',C,D_1\), this projection ranges over \(L^{o(1)}\) cells. Indeed, the approximate basis is already fixed by \(D_0\), and the exact central line of \(T_1'\) is part of its label. Its slope differs from the representative slope by \(L^{-b_*+o(1)}\), while its width in parent coordinates is at most \(L^{s-1+o(1)}\le L^{-b+o(1)}\). Along a segment of length \(L^{o(1)}\) both errors cause only \(L^{-b+o(1)}\) projection variation. Hidden exact reference directions do not enter this coordinate system. Also \(Y_b\) and \(X_{s+b}\) determine each other to \(L^{o(1)}\) choices with these labels: coordinate one parametrizes the alternative tube with subpower distortion. Consequently \[\begin{align*} I(Y_b\mid\Pi,C,D_1) &\ge I(Y_b\mid\Pi,C,D_1,T_1')\\ &=I(X_{s+b}\mid C,D_1,T_1') \ge G(s+b)-G(s)-\zeta b. \end{align*}\] Since \(\Pi\) is determined by \(Y_b,D_1\), its chain rule together with (36) gives \[ I(\Pi\mid C,D_1)\le db+O(\zeta b). \tag{37}\] All projection information variables are among those included in the joint limit.

We detail the passage from these sampled inequalities to one common set and one direction measure. Enlarge the inequalities by fixed slack and pass to finite \(L\). There is a success event of probability at least \(p_*>0\) along a subsequence. Fix parent and reference-bin labels \((c,d_0^{\rm bin})\) whose conditional success probability \(p\) is at least \(p_*\). No lower bound on the probability of this individual fiber is needed. Denote its conditional law by \(\mu\). Absorb all fixed slack into \(\alpha=C\zeta+o(1)\), enlarging the absolute constant \(C\) as necessary. On success, \[ \begin{split} K^{-2d-\alpha}\le\mu(Y_b=y)&\le K^{-2d+\alpha},\\ K^{-2d-\alpha}\le\mu(Y_b=y\mid A_{\mathrm{bin}}=a) &\le K^{-2d+\alpha},\\ \mu(Y_u=y_u)&\le K^{\alpha}L^{-2d_0u},\\ \mu(\Pi=\pi\mid A_{\mathrm{bin}}=a)&\ge K^{-d-\alpha}, \end{split} \tag{38}\] where \(A_{\mathrm{bin}}\) is the additional direction bin. These are statements about the indicated labels whenever they have any successful realization.

Let \(A\) consist of the centers of all endpoint pair cells with a successful realization in this fixed parent/reference fiber. The first line of (38) and total success mass at least \(p\) give \[ pK^{2d-\alpha}\le |A|\le K^{2d+\alpha}. \tag{39}\] A prefix cell containing a successful endpoint has mass at most the third line of (38). The nested grids put all mass of that endpoint cell in its prefix. Dividing by the minimum endpoint mass bounds the number of good endpoints in that prefix. A ball of radius \(r\) meets boundedly many cells at a comparable coarser tested prefix size; the finite mesh costs \(K^{O(\zeta)}\). Thus, for \(K^{-1}\le r\le1\), \[ \frac{\mathcal N_{K^{-1}}(A\cap B(z,r))} {\mathcal N_{K^{-1}}(A)} \lesssim p^{-1}K^{O(\zeta)+o(1)}r^{d_0}. \tag{40}\] In fact the preceding argument gives \(r^{2d_0}\) before weakening the exponent and absorbing rounding. For the untested shortest prefixes the allowed power factor suffices.

Retain bins \(a\) whose conditional success probability is at least \(p/2\). The successful mass in these bins is at least \(p/2\). For each retained bin, let \(A_a\subset A\) be its successful endpoint cells. The conditional upper pair-cell mass gives \[|A_a|\ge(p/2)K^{2d-\alpha},\qquad |A_a|/|A|\ge(p/2)K^{-2\alpha}.\] The lower projection-cell mass in (38) gives \[ \mathcal N_{K^{-1}}\bigl(\pi_{\widetilde a_j(a)}A_a\bigr) \le K^{d+O(\zeta)+o(1)}, \tag{41}\] where initially \(\pi_{\widetilde a_j}(z_1,z_j) =z_j-\widetilde a_jz_1\). Passing to a unit vector for this projection only shrinks interval lengths.

It remains to put a nonconcentrating probability on these directions. Restrict the actual samples to success and the retained bins, normalize, and push forward to the line spanned by \((-\widetilde a_j,1)\). This costs at most \(2/p\) in mass. To estimate a slope ball, first condition on the exact point \(X\) and exact reference tubes, before any restriction by the additional bin. These labels determine \((C,D_0)\), so fixing that fiber does not change the original alternative law once these exact labels are given. If that ball contains any successful alternative, the centered marginal bound at this alternative, supplied by (iii), bounds the original alternative probability of the entire ball after a fixed radius enlargement. Choosing a coarser tested radius costs \(K^{O(\zeta)+o(1)}\). Therefore its restricted mass is at most \[K^{O(\zeta)+o(1)}r^{\kappa_0}\qquad(K^{-1}\le r\le1).\] Integrate this bound over the exact points and references under the fixed fiber, and then normalize. This operation introduces no reciprocal probability of an individual additional bin; conditional independence supplies the original alternative law at the stage where the angular estimate is used.

All slopes have magnitude \(L^{o(1)}\). The sine-angle formula shows that the slopes in a projective ball of radius \(r\) occupy an interval of length \(rL^{o(1)}\); exact-to-approximate slope errors are \(K^{-1}L^{o(1)}\). Covering and radius rounding thus give, for the pushforward probability \(\sigma\), \[ \sigma(B(V,r))\le K^{O(\zeta)+o(1)}r^{\kappa_0}, \qquad K^{-1}\le r\le1. \tag{42}\] The coordinate range of \(A\) is at most \(L^{o(1)}\). Rescale it into the unit ball. At resolution \(K^{-1}\) this merges or covers at most \(L^{o(1)}\) grid cells per ball and changes every displayed power error by \(o(1)\) only. Use covering numbers after this rescaling.

Choose \(0<\rho_0<\min\{d_0,\kappa_0\}\), apply Lemma 16, and then take \(\zeta\) so small that every \(O(\zeta)\) error is below its tolerance and its projection gain. Equations (39), (40), and (42) meet its hypotheses. For every direction in the finite support of \(\sigma\), however, (41) gives a subset of the same \(A\) of relative covering number \(K^{-O(\zeta)-o(1)}\) with too small a projection. The robust large-subset conclusion gives the contradiction. ◻

Proof. First obtain a sufficiently reduced block-test tolerance \(\zeta_2\) for \(\kappa_0=\kappa/4\). Choose a positive rational \(c_1\ll\kappa\zeta_2\), and then a sufficiently reduced tolerance \(\zeta_1\) for \(\kappa_0=c_1/2\). Test \(h_2(u)\ge c_1u\) on a sufficiently fine finite rational grid from \(c\zeta_1b_0\) to \(b_0\), where \(c>0\) is fixed and small. If all tests pass, monotonicity fills the mesh gaps; below the first grid point nonnegativity gives the required inequality with the allowed additive error. Take \(b'=b_0\) and \(j=2\).

Otherwise some grid point \(v\) satisfies \(h_2(v)<c_1v\). For rational \(u\le v\), monotonicity and (35) yield \[h_3(u)\ge\kappa u-h_2(u)>\kappa u-c_1v \ge\frac\kappa2u \quad\left(\frac{2c_1v}{\kappa}\le u\le v\right).\] Take \(b'=v\), \(j=3\). Because \(c_1\ll\kappa\zeta_2\), the missing smaller radii meet the weaker \(\kappa/4\) bound with additive error \(\zeta_2b\) for every \(b\in[v/4,v]\). Small enough mesh and cutoff constants leave spare slack in both cases. The possible \(v/b_0\) form a fixed finite list bounded away from zero. Reduce all constants once more to permit the stated small spatial variations. ◻

Proof. Suppose otherwise. Choose \(0<d<1\) and \(0<d_0<d\) such that a sufficiently small neighborhood of \(d\) contains \(\dot G(s)\) on a set of positive path-times-exponent measure. This is possible by choosing an interior point of the essential support. Apply Lemma 19 for these parameters. For dyadic \(b_0\downarrow0\), and each of its finite alternatives, form a rational mesh of test blocks of comparable length, so that each interior exponent is contained in an appropriate block, and include each block’s finite prefix grid. Use direction precision \(b_*=b_0\) for all alternatives. Within each size and prefix type, the candidate intervals have bounded overlap, with constants independent of \(b_0\). Explicitly, choose endpoints on one lattice of spacing \(q b_0\), where the positive rational \(q\) is fixed below all the relative mesh tolerances, and allow all candidate lengths between \(c_*b_0/4\) and \(b_0\). A point belongs to at most \(O(q^{-2})\) such candidates. Every tested prefix is contained in its candidate, and the number of prefix types is fixed. Color each of these interval families into finitely many disjoint lists.

Let \(\mathcal C_I\) be the sum of all reference, replacement, and extra-bin costs used by candidate block \(I\). By Lemma 13, \[ \sum_I\mathbb E\mathcal C_I\lesssim b_0. \tag{43}\] All candidate lengths lie between fixed positive multiples of \(b_0\). For the nonnegative function \[R_{b_0}(\omega,s)= \sum_{I\ni s}\mathcal C_I(\omega)/b_0\] we have \(\mathbb E\int_0^1R_{b_0}(\omega,s)\,ds\lesssim b_0\): integration cancels the normalizing block length. Dyadic summability therefore gives \(R_{b_0}\to0\) almost everywhere. In particular, all tested costs in a block containing such an \(s\) are \(o(b_0)\).

At a differentiability point \(s\), the approximation \(G(y)=G(s)+\dot G(s)(y-s)+o(b_0)\) is uniform for \(|y-s|\le Cb_0\). Hence all increments and finitely many prefix increments in these nearby blocks are approximated to \(o(b_0)\). If \(\dot G(s)\) lies in the chosen sufficiently small neighborhood of \(d\), the spatial hypotheses of the block test hold eventually, with its reduced tolerance. An angular alternative supplies its angular hypotheses, and the preceding cost estimate supplies the remaining ones. Only countably many fixed rational blocks and finite tests are involved. Positive path-times-exponent measure therefore implies that one fixed test holds on a positive-probability event, contradicting Lemma 18. A countable cover of interior derivative values proves the claim. ◻

Proof. Put \(H(s)=G(s)-s/2\). The density conditions give positive right slope at \(a\) and a positive drop towards \(b\) from the left. Thus \(\max_{[a,b]}H\) exceeds both \(H(a)\) and \(H(b)\): if the larger endpoint is \(a\) use its right increase, and if it is \(b\) use its left increase. A component of a suitable strict superlevel set therefore gives a closed hill \([a',b']\subset(a,b)\) with equal endpoint heights and strictly larger values inside.

Fix \(r<b'-a'\). Among levels having a closed subinterval of this hill of length at least \(r\), with both endpoints at the level and values no smaller inside, take the highest. Such a level exists: the admissible triples of level and endpoints form a nonempty compact set. At the maximizing level \(h\) take an admissible interval \([c,d]\). No component on which \(H>h\) inside it has length greater than \(r\), since such a component contains an interval of length at least \(r\) closed off at a slightly higher level. The closed set \(\{H=h\}\cap[c,d]\) thus contains both endpoints and has no gap longer than \(r\). Starting at \(c\), take its first point at or after \(c+r\). Its distance from \(c\) is between \(r\) and \(2r\). Between these two points \(H\ge h\), which gives the asserted total and prefix identities for \(G\). Apply this construction separately inside finitely many disjoint hills to obtain the final assertion. ◻

Proof. Suppose a path has arbitrarily many disjoint ordered density pairs of the type in Lemma 21. Use the angular alternatives for \(d=1/2\), \(d_0=1/4\) at dyadic \(b_0\downarrow0\). For its successful angular size \(b'\asymp b_0\), choose the balanced intervals with lengths between \(b'/3\) and \(2b'/3\). Round their endpoints to a sufficiently fine fixed relative rational mesh. Lipschitz continuity preserves the endpoint and finite prefix requirements of the block test with spare slack. For any prescribed number of disjoint density pairs this produces that many distinct spatially and angularly successful candidates at every sufficiently small \(b_0\); the rounding preserves distinctness inside their separate hills.

Consider all mesh candidates for the fixed finite list of angular alternatives, and all their prefix and direction costs, with \(b_*=b_0\). Equation (43) still holds. A failure of a candidate’s cost test requires one nonnegative cost at least a fixed positive multiple of \(b_0\). Hence the expected number \(N_{b_0}\) of such failures is bounded uniformly in \(b_0\). Every spatially and angularly successful candidate must fail a cost test almost surely, by Lemma 18 and the countability of the mesh collection. On the paths just considered, \(N_{b_0}\to\infty\). Fatou’s lemma contradicts their having positive probability, because \(\liminf\mathbb EN_{b_0}<\infty\).

Thus the number of alternations among density points of \(E\) and of its complement is bounded on almost every path. To see the finite-interval conclusion directly, assign to each ordered density point the maximum length of an alternating sequence ending there. These integers are bounded, nondecreasing along the ordered density points, and strictly increase when the type changes. Each integer level therefore has one type and occupies an ordered interval. The Lebesgue density theorem gives the asserted finite union modulo null sets.

The prefix inequality in (44) rules out an initial interval of zero measure, while \(|E|=2/3<1\) forces a subsequent nonempty interval of zero measure. There is therefore an interior full-to-empty jump. Around that jump choose a short block whose midpoint differs from the jump by at most \(\eta b\), with its left and right halves inside the adjacent full and empty intervals except for that discrepancy. Rational endpoints can be chosen with \(b<u_0\), giving (45). The countable family of rational blocks covers almost all paths; some fixed block has positive success probability. No direction cost is needed in this choice. ◻

Proof. Fix such a rational block and put \(K=L^b\). By (34) and the chain rule, its endpoint and midpoint information increments for \(X\) are both \(3b/2+O(\eta b)\), while the conditional endpoint increment given \(T=T_1\) is \(b/2+O(\eta b)\). Pass to finite \(L\) with fixed slack, and fix a parent \(C=X_s=c\) with conditional success probability bounded below by a constant \(p_*>0\) along the sequence. Write \[p_c(z)=\mathbb P(X_{s+b}=z\mid C=c),\qquad q_T(z)=\mathbb P(X_{s+b}=z\mid T,C=c).\] Call an incidence \((T,z)\) good if it has at least one realization in the success event. Let \(\mathcal Z\) be all endpoint labels with a good incidence, represented by their centers in rescaled parent coordinates. Every such endpoint satisfies \[ K^{-3/2-O(\eta)}\le p_c(z)\le K^{-3/2+O(\eta)}, \qquad q_T(z)\le K^{-1/2+O(\eta)} \quad\hbox{for a good }(T,z). \tag{46}\] Its midpoint cell has marginal mass at most \(K^{-3/2+O(\eta)}\). These estimates depend only on the displayed labels, so they remain true after defining goodness by existence of a successful realization. In particular \[ |\mathcal Z|\le K^{3/2+O(\eta)},\qquad \#(\mathcal Z\cap B(z,CK^{-1/2}))\le K^{O(\eta)} \tag{47}\] for every fixed \(C\). For the second bound divide the midpoint mass upper bound by the endpoint mass lower bound, then cover a ball by boundedly many midpoint cells. The central line of a tube with a good incidence passes within \(\delta=K^{-1}L^{o(1)}\) of the endpoint center. The parent has a fixed bounded diameter in these coordinates.

Choose a small fixed \(\varepsilon>0\), and set \(P=K^{1/2-\varepsilon}\). Polynomial partitioning in \(\mathbb R^3\) [15] gives a nonzero polynomial of degree \(D\lesssim P\) whose complementary open cells each contain at most \[ O(|\mathcal Z|/P^3)=K^{3\varepsilon+O(\eta)} \tag{48}\] of the centers. Take a wall of thickness a sufficiently large constant times \(\delta\) about its zero set. At midpoint resolution \(r=K^{-1/2}\), the number of boxes meeting the wall is \[ O(Dr^{-2}+D^2r^{-1}+D^3)=O(PK). \tag{49}\] Indeed \(\delta=o(r)\). Intersect the zero set with a fixed ball strictly larger than the bounded parent; every wall point relevant to that parent is within the required distance of this restricted zero set. Wongkew’s hypersurface neighborhood-volume bound [27], thickened to scale comparable to \(r\) in that fixed enlarged ball, is \(O(Dr+D^2r^2+D^3r^3)\). Divide by the disjoint box volume \(r^3\). The degree condition \(D\lesssim K^{1/2-\varepsilon}\) makes all three terms in (49) at most \(O(PK)\).

By (46)–(47), the marginal mass of good endpoints lost to the wall is at most \[ O(PK)\,K^{O(\eta)}K^{-3/2+O(\eta)} =K^{-\varepsilon+O(\eta)}. \tag{50}\] Take \(\eta\) small compared with \(\varepsilon\), so this tends to zero. The enlarged wall ensures that the nearby central-line point of each surviving incidence lies in the same open cell as its endpoint center. A line outside the polynomial zero set crosses it at most \(D\) times, and hence visits at most \(D+1=O(P)\) open cells. A line contained in the zero set has no off-wall incidence. Thus every tube’s good off-wall endpoints lie in at most \(O(P)\) open cells.

Draw \(T\) with its original marginal given \(C=c\), then draw \(z,z'\) independently with law \(q_T\). Let \(g(T)\) be the \(q_T\) mass of good off-wall endpoints. Its mean is at least \(p_*/2\) for large \(K\). Conditional on \(T\), Cauchy–Schwarz over its at most \(O(P)\) open cells gives probability at least \(g(T)^2/O(P)\) that both draws are good off-wall and belong to the same open cell. Jensen’s inequality therefore gives the unconditional lower bound \[ \mathbb P\{z,z'\text{ good off-wall in the same cell}\mid C=c\} \gtrsim P^{-1}=K^{-1/2+\varepsilon}. \tag{51}\] We bound this probability from above by separating near and far pairs.

For \(|z-z'|\lesssim K^{-1/2}\), the second bound in (47) gives at most \(K^{O(\eta)}\) possible good second endpoints. Each costs \(K^{-1/2+O(\eta)}\) under \(q_T\), so near pairs contribute at most \[ K^{-1/2+O(\eta)}. \tag{52}\] For a farther pair, a tube meeting both endpoint centers to error \(\delta\) must have direction within \(O(\delta K^{1/2})=K^{-1/2+o(1)}\) of their joining line, allowing two signs.

The conditioning at this point is essential. Under the double sampling, \[\mathbb P(T,z,z'\mid C=c)=\mathbb P(T\mid C=c)q_T(z)q_T(z').\] Thus its \((T,z)\) marginal is the original joint distribution given \(C=c\). Conditional on the first endpoint \(z\), the tube law is \[ \mathbb P(T\in\mathcal A\mid z,C=c) =\sum_{p:X_{s+b}(p)=z} \mathbb P(X=p\mid z,C=c)\mathbb P(T\in\mathcal A\mid X=p). \tag{53}\] The parent is determined by \(z\), and \(X\) determines all these spatial labels. Equation (29) therefore applies to every inner law. Use the slightly coarser rational angular scale \(L^{-b/3}=K^{-1/3}\), which covers the above cap for large \(L\) and has \(b/3<u_0\). For every fixed far pair the resulting angular probability is at most \[K^{-\kappa/3+o(1)}.\] For each first endpoint, (48) allows at most \(K^{3\varepsilon+O(\eta)}\) same-open-cell second endpoints. Each good second incidence costs an additional \(K^{-1/2+O(\eta)}\) under its original \(q_T\) law. For each first \(z\) this list is fixed before the tube is drawn. In the integral over the original law of \(T\) given \(z,C\), apply the bound on \(q_T(z')\) only where \((T,z')\) is good, and apply the angular cap bound to the resulting indicator. Sum over the fixed list of \(z'\) and then over the first-endpoint law in (53). The total far-pair contribution is at most \[ K^{-1/2-\kappa/3+3\varepsilon+O(\eta)+o(1)}. \tag{54}\] No angular regularity after conditioning on the success event has been used; success only restricts the pairs to which the original mass bounds apply.

Divide (52) and (54) by (51). The respective ratios are bounded by \[K^{-\varepsilon+O(\eta)},\qquad K^{-\kappa/3+2\varepsilon+O(\eta)+o(1)}.\] Choose \(\varepsilon>0\) sufficiently small relative to \(\kappa\), then \(\eta>0\) sufficiently small relative to \(\varepsilon\). Both ratios tend to zero, contradicting (51). ◻

Proof of Proposition 8. A countersequence gives the weighted tube configuration by Lemma 10. Its limiting information profiles satisfy (34). Lemma 20 makes the common derivative binary, and Lemma 22 supplies a positive-probability full-to-empty block for any prescribed fixed small tolerance. Lemma 23 rules out such a block. Hence no countersequence exists, which proves the existence of the fixed power \(c>0\) and the large-scale threshold. ◻

Proof. Use the Fourier transform in all native variables \((y,\sigma)\in\mathbb R^3\), and choose a smooth partition of unity \(\sum_{k\in\mathbb Z^2}\rho(\eta-k)=1\), with \(\rho\) supported in \([-2,2]^2\). With \[a_{\alpha,k,u}(\eta) =\rho(\eta-k)\widehat H_\alpha(\eta,u-|\eta|^2),\] Fourier inversion followed by \(u=\zeta+|\eta|^2\) gives the exact identity \[ H_\alpha(y,\sigma) =(2\pi)^{-3}\sum_k\int_\mathbb Re^{iu\sigma} \left[\int_{\mathbb R^2} e^{i(y\cdot\eta-\sigma|\eta|^2)} a_{\alpha,k,u}(\eta)\,d\eta\right]du. \tag{59}\] The carrier phase combines with the inner phase according to \[\begin{align*} x\cdot\omega_\alpha-t|\omega_\alpha|^2 +\frac{x-b_\alpha-2\omega_\alpha t}{L}\cdot\eta -\frac{t}{L^2}|\eta|^2 & =x\cdot(\omega_\alpha+\eta/L) -t|\omega_\alpha+\eta/L|^2 -b_\alpha\cdot\eta/L. \tag{60}\end{align*}\] Thus, with \(Ev(x,t)=\int e^{i(x\cdot\xi-t|\xi|^2)}v(\xi)\,d\xi\), the exact datum for one packet in a fixed layer is \[ v_{\alpha,k,u}(\xi) = L\,a_{\alpha,k,u}\bigl(L(\xi-\omega_\alpha)\bigr) e^{-ib_\alpha\cdot(\xi-\omega_\alpha)}. \tag{61}\] The factor \(L\) makes its \(L^2\) norm equal to \(\|a_{\alpha,k,u}\|_2\); the inverse Jacobian \(L^{-2}\) then produces the atom’s prefactor \(L^{-1}\) in the extension field.

Compact support and sufficiently many derivatives of \(H_\alpha\) give any prescribed finite collection of rapidly decaying bounds for its Fourier transform and its derivatives. Differentiating \(\widehat H_\alpha(\eta,u-|\eta|^2)\) introduces only polynomial factors in \(\eta\). On \(\eta\in k+[-2,2]^2\) these, including the case \(|u|\asymp |k|^2\), are absorbed by taking more original derivatives. Explicitly, on this support, \[(1+|k|)^P(1+|u|)^P \le C_P\bigl(1+|\eta|+|u-|\eta|^2|\bigr)^{3P}.\] After fixing the desired seminorms, the spatial decay orders, and \(C_{\rm nat}\), differentiation and integration by parts introduce only an additional factor \((1+|k|)^B\) for a fixed finite \(B\). Taking the original Fourier decay order larger than \(3P+B\) therefore permits, for any subsequently specified large finite \(P\), the common weight \[w(k,u)=C_P(1+|k|)^{-P}(1+|u|)^{-P},\] so that the desired derivatives of \(a_{\alpha,k,u}/w(k,u)\) are uniformly bounded. On the fixed bounded native time interval, integration by parts in \(\eta\) also gives the required finite-order spatial decay for the bracket in (59), after division by the same weight. Derivatives of \(e^{-i\sigma|\eta|^2}\) cost polynomial powers of \(k\), already included in \(B\). This also absorbs the displacement \(y\simeq2\sigma k\) of a layer’s own propagated center: spatial decay is asserted after normalization by the weight chosen for the prescribed decay order. In particular these exact packet fields decay about the original label trajectories to any prescribed finite order, uniformly in the layer after normalization.

For completeness, the energy bound uses the full phase-space separation, not bounded multiplicity of frequencies alone. In the Gram matrix of (61), the entry indexed by \(\alpha,\widetilde\alpha\) vanishes unless \(L|\omega_\alpha-\omega_{\widetilde\alpha}|\le C\), since the same \(k\) occurs in both frequency cutoffs. On an overlap, integration by parts in the rescaled frequency variable gives, for any fixed large \(N\), \[ |\langle v_{\alpha,k,u},v_{\widetilde\alpha,k,u}\rangle| \le C_N w(k,u)^2 \mathbf1_{\{L|\omega_\alpha-\omega_{\widetilde\alpha}|\le C\}} \left(1+\frac{|b_\alpha-b_{\widetilde\alpha}|}{L}\right)^{-N}. \tag{62}\] The bounded number of labels in every unit phase-space ball makes each row and column summable. Schur’s test proves (58), for an arbitrary subarray as well as for the entire array. ◻

Proof. Suppose no positive \(c_b\) works. After normalizing \(m_0=\mathcal E=1\) and dividing all fields by a fixed bound for their input norms, this gives a sequence \(s\to\infty\), with \(K=s^{o(1)}\), for which the broad sum is at least \(s^{4/3-o(1)}\). Fixed constants in the radius, support box and norm bounds are harmless. There are polynomially many cubes, and the fields have bounded supremum by Cauchy–Schwarz in frequency. We may discard polynomially tiny mass and field levels, and then pigeonhole one active triple and dyadic values \[\nu(Q)\asymp\chi,\qquad \|V\|_{L^6(Q)}\asymp b\] on \(N\) cubes. The number of triples and all factors lost here are \(K^{O(1)}\) or logarithmic, hence \(s^{o(1)}\). We obtain \[ \chi^{2/3}b^2N\ge s^{4/3-o(1)},\qquad \chi\le s^{o(1)},\qquad \chi N\lesssim s^4. \tag{66}\] The last two statements follow respectively from the ball bound on a \(K^2\)-cube and on the ambient ball.

We explain carefully the fourth inequality, furnished by multilinear restriction. Each chosen factor has \(L^6(Q)\) norm at least \(K^{-A}b\), so some unit cube in \(Q\) has supremum at least \(K^{-A-1}b\): here \(|Q|^{1/6}=K\). For each of the three factors, choose one such unit cube. Pigeonholing their three relative integer positions within \(Q\) leaves a common triple of offsets, with cost \(K^{O(1)}\). Translate the three fields separately by these fixed offsets. They now all have supremum at least \(b\,s^{-o(1)}\) on the same unit cube attached to each surviving \(Q\). These attached cubes are disjoint. Translation only modulates the frequency inputs and preserves both their norms and their supports.

The separate-supremum version of Lemma 9 therefore yields \[ b^3N\le s^{o(1)}. \tag{67}\] Equivalently, this use of the lemma follows by applying a rapidly decreasing reproducing kernel to each factor separately, integrating the product on each attached cube, and averaging the trilinear integral bound over the three independent shifts. For every fixed triple of shifts the integration domain is the same and the inputs are exact with unchanged supports and norms. The determinant lower bound is \(K^{-2}=s^{-o(1)}\), precisely the regime of that lemma. Thus an unspecified dependence of a multilinear constant on transversality is not being used here.

The four inequalities force saturation of all three sizes. For clarity, write \((x,y,n)\) for limiting base-\(s\) logarithms of \((\chi,b,N)\), passing to a subsequence if needed. The retained levels and the first inequality give polynomial bounds, so such subsequences exist. Equations (66) and (67) imply \[x\le0,\qquad x+n\le4,\qquad 3y+n\le0, \qquad 2x+6y+3n\ge4.\] Consequently \[4\le2x+6y+3n\le2x+n\le4+x\le4.\] Equality holds throughout, giving \(x=0\), \(n=4\), and \(y=-4/3\). In the original sequence this says \[ \chi=s^{o(1)},\qquad N=s^{4+o(1)},\qquad b=s^{-4/3+o(1)}. \tag{68}\] In particular the needed lower bounds are \(\chi\ge s^{-o(1)}\), \(N\ge s^{4-o(1)}\) and \(b\ge s^{-4/3-o(1)}\).

The attached unit cubes also have the required ball counting bound. If their locations lie in a radius-\(r\) ball, their associated full \(K^2\)-cubes lie in a ball of radius \(r+CK^2\). Each associated cube has mass at least \(s^{-o(1)}\), and they are disjoint. For \(r\ge1\) their number is therefore at most \[s^{o(1)}(r+CK^2)^2\le s^{o(1)}r^2.\] The three translated exact fields, the cubes in (68), and the determinant bound thus contradict Proposition 8. A fixed enlargement of the ambient ball can be removed by replacing \(s\) by a fixed multiple and absorbing the constants into the subpower factors.

This contradiction supplies an absolute positive \(c_b\), reducing it if necessary so that the same exponent appears in both its hypothesis and its conclusion. The argument used only the maximum of the total input norm and the individual subfield input norms. Normalizing by that maximum preserves activity ratios. Thus it is uniform for exact fields and its exponent is independent of the high decay orders used to localize packet arrays later. ◻

Proof of Proposition 25. We apply the preceding estimates on the scales just specified. At any node the preceding transport calculation gives a measure \(\nu\) with time range \(O(s^2)\), ball parameter \(m_j\), and packet moment bound \(m_js^3/D\), where \[ m_j\le C^j h^3m L^{Cj\eta}. \tag{79}\] All support boxes and cell counts are polynomial in \(L\). We may apply the estimates at radius \(O(s^2)\) spatial box by spatial box. Here exact packets replace the compact packets used at the root: retain only packets whose trajectories meet a fixed enlargement of the box, and discard the others using their prescribed rapid decay. On the time range \(O(s^2)\), each retained packet is used in boundedly many boxes. Its coefficient energy therefore has bounded overlap, also for every child subarray. A discarded packet has native transverse distance \(\gtrsim s\) on the box. Cauchy–Schwarz, the phase-space counting condition and a sufficiently high decay order make the total error an arbitrarily small negative power of \(L\), since \(s\ge L^{\tau_0/2}\). This is the localization mechanism of Lemma 3, applied to the exact arrays constructed above.

Whenever this localization is performed, use the same retained labels in the total field and in all its subfields. Negligible field-level cubes are absorbed in the additive error; on the remaining cubes the activity tests tolerate that error, or can be made afresh for the localized exact array. Lemma 27 then uses the maximum of its total and subfield input norms, bounded by a fixed multiple of the retained coefficient norm by (58). In particular increasing the later decay orders cannot reduce the absolute broad saving.

At every nonterminal node the broad contribution is bounded by (65), and the narrow contribution is passed to children by (75). Child arrays partition the parent labels. Hence their squared coefficient norms sum to the parent energy; there is no loss from the number of children.

It remains to estimate a terminal node. By rapid decay keep only \(K^2\)-cubes lying within transverse distance \(L^\eta s\) of at least one label trajectory. Taking the decay order sufficiently high again makes the omitted contribution negligible. Restrict \(\nu\) to these retained cubes. If a ball \(B\) of radius \(s\) meets the retained support, choose one retained cube it meets and a packet near that cube. Since \(K^2\ll s\) and velocities are bounded, every point of \(B\) is at normalized transverse distance \(O(L^\eta)\) from that one packet. Its moment bound therefore gives \[ \nu(B)\le C L^{CM\eta}\frac{m_js^3}{D}. \tag{80}\] The same estimate holds for fixed enlargements of \(B\). No sum over packets occurs: one packet suffices for each ball.

Normalize the node coefficient energy to one and split retained cubes into dyadic classes \(\nu(Q)\asymp m_j\chi\). The upper range is \(\chi\lesssim K^4\) by the ball bound. Polynomially tiny levels are negligible: there are polynomially many tested cells, and a uniform polynomial supremum bound follows from the exact input energy. Thus only \(O(\log L)\) levels need be kept. In each \(Q\) choose a unit cube \(q_Q\) with the largest field supremum. Then \[ \sum_{Q\text{ in the class}}\nu(Q)^{2/3}\|V\|_{L^6(Q)}^2 \lesssim K^2(m_j\chi)^{2/3} \sum_Q\sup_{q_Q}|V|^2. \tag{81}\] The selected unit cubes need not themselves carry measure. Their counts are instead controlled by their disjoint full \(Q\)-cubes, each of mass comparable to \(m_j\chi\). For \(r\ge1\) this gives \[\#\{q_Q:q_Q\cap B(z,r)\ne\varnothing\} \lesssim \chi^{-1}(r+CK^2)^2 \lesssim K^4\chi^{-1}r^2.\] At radius \(s\), using (80) and \(K^2\ll s\) instead gives occupancy at most \(C L^{CM\eta}\chi^{-1}s^3/D\). Thus the parameters in Lemma 4, with ambient radius \(S\asymp s^2\), can be taken as \[\gamma_0\lesssim K^4\chi^{-1},\qquad \lambda_0\lesssim L^{CM\eta}\chi^{-1}s^3/D.\] Its unit-cube supremum version and (81) yield \[\begin{align*} \mathcal T_\nu(V) &\lesssim_\epsilon L^{CM\eta}K^{10/3}L^\epsilon m_j^{2/3}s^{5/3}D^{-1/3}\|c_{\rm node}\|_{\ell^2}^2 \\ &=L^{CM\eta}K^{10/3}O_\epsilon(L^\epsilon) m_j^{2/3}s^{4/3}(s/D)^{1/3} \|c_{\rm node}\|_{\ell^2}^2. \tag{82}\end{align*}\] Here the logarithmic number of weight classes is absorbed in \(L^\epsilon\). The cancellation of the weight parameter is complete: \[\chi^{2/3}(\chi^{-1}\chi^{-1})^{1/3}=1.\] The displayed \(K^{10/3}\) consists of \(K^2\) in (81) and \(K^{4/3}\) from the counting parameter. Allowing additional fixed cube or grid enlargements merely replaces it by \(K^{C_1}\) for an absolute \(C_1\). Spatial localization is as above, so this argument applies even when the node’s spatial support extends beyond a single \(s^2\)-box.

We finish with an explicit order of choices. Bounded \(L\) causes no difficulty, and \(\tau_0>1\) permits no unbounded admissible scales, so assume \(0<\tau_0\le1\). Fix first the absolute numbers \[ a_* =\min\{c_b/2,1/6\},\qquad c_s=a_*/8. \tag{83}\] At a broad leaf \(s\ge D^{1/2}\) gives \(s^{-c_b}\le D^{-c_b/2}\), while (63) gives the terminal gain \[(s/D)^{1/3}\le K^{1/3}D^{-1/6}.\] Thus both gains are at least \(D^{-a_*}\) before the losses just recorded.

Now choose \(\beta>0\) sufficiently small in terms of \(\tau_0,c_b,a_*\). It can satisfy, with fixed strict margins, \[\beta<\frac{\tau_0c_b}{4},\qquad 2\beta<\frac{\tau_0}{4},\qquad (C_1+1/3)\beta<\frac{\tau_0a_*}{16}.\] For all large \(L\), these imply \(K\le s^{c_b}\) at every broad node and allow the terminal \(K\) factors to consume at most \(D^{a_*/8}\), after fixed constants are absorbed. The depth bound \(J=O(\beta^{-1})\) is now fixed. Choose the small losses in decoupling and the refined fractal bound so that their total product throughout a path is at most \(D^{a_*/8}\). Next choose \(\eta>0\), depending also on \(J\) and \(M\), so that all padding losses together cost at most \(D^{a_*/8}\) and \(2\beta+\eta<\tau_0/2\). This last inequality implies (77) at every existing child scale, since each such scale is at least \(D^{1/2}\ge L^{\tau_0/2}\). Finally choose \(\gamma<\tau_0/4\), so that \(|k|/s=o(1)\) for every retained layer, and then choose the frequency-tail exponent, the physical decay orders, and the original profile differentiability. The latter choices make every discarded tail as small a negative power as required; they do not change \(c_b\) or \(c_s\).

To see that these choices close the entire tree, a depth-\(j\) path contributes, before its broad or terminal gain, at most \[C^j L^{Cj\eta+O(\epsilon)}h^{-2/3}, \qquad h=K^j,\] times its node estimate. On the other hand (79) gives \[m_j^{2/3}s^{4/3} \le C^jL^{Cj\eta}h^{2/3}m^{2/3}L^{4/3}.\] The \(h\) powers cancel, as anticipated in (78). At a fixed depth the total branch energy is bounded by a fixed constant to that depth times the original energy, including the spatial localizations. Summing over the bounded depth therefore costs only a constant depending on \(\tau_0\). The allotted losses leave more than the saving \(D^{-c_s}\) in (83). Finally, all tail errors can be made negligible relative to \(m^{2/3}L^{4/3}D^{-c_s}\|c\|_{\ell^2}^2\), even after summing the tree: after normalization there are only polynomially many supported cells, boxes and branches, while the tail exponent is chosen last.

This proves the desired bound for \(\mathcal T\). Minkowski in Lemma 26, the Sobolev reduction (57), and the initial spatial energy summation give (55), with the stated derivative and uniformity properties. ◻

Proof. Smoothly partition the spatial Fourier transform of each native profile into layers \(k+[-2,2]\), \(k\in\mathbb Z\), writing \(\langle k\rangle=1+|k|\). A fixed, sufficiently large number of derivatives gives any prescribed fixed polynomial layer decay together with the finitely many spatial decay and derivative estimates that we will use. Assign the new carrier center \(\xi_\ell+k/h\). In its native coordinates the profile undergoes a translation by \(2k\) times native time and acquires a temporal oscillation of frequency \(k^2\). Every required derivative and envelope comparison therefore costs only a fixed polynomial in \(\langle k\rangle\). Choose the original decay orders larger than these costs and put the remaining decay into the layer amplitudes. A common frequency shift preserves the phase-space counts.

At a witness the sum of all layers with \(|k|\geq c\delta h\) is bounded by \[C_N(\delta h)^{-N}\sqrt h\,B.\] Indeed, in each original frequency bin of width \(h^{-1}\), the spatial tails sum uniformly by bounded offset counts; weighted Cauchy–Schwarz over the \(O(h)\) frequency bins gives \(\sqrt h\). Since \(\delta h\geq\sqrt h\), the displayed error is negligible for large \(h\). Bounded scales are vacuous once the required ratio constant is large, by the same envelope Cauchy–Schwarz estimate. Choose the fixed \(c>0\) small enough that the retained frequency layers still lie in separated intervals of length \(O(\delta)\).

If the retained sum is large, some layer has modulus at least \(c'\langle k\rangle^{-2}A\). For any prescribed fixed large \(N\) its envelope bound and energy can be taken to be \(C_N\langle k\rangle^{-N}B\) and \(C_N\langle k\rangle^{-2N}H^*\), respectively. Thus the ratio condition persists, and applying the desired estimate at these thresholds and summing the two layer indices converges. For example, the ratio factors gain \(\langle k\rangle^{-2(N-2)}\) and the corresponding energy-to-level ratio gains the same power. Constants and an initial finite set of layers are absorbed into the fixed ratio threshold.

We may consequently work with one exactly cap-supported layer, divided by \(B\). If \(|s_\ell|>\lambda\), then \(|s_\ell|\leq\lambda^{-1}|s_\ell|^2\). The pointwise profile decay, chosen stronger than the envelope decay, bounds the deleted sum by \(C\lambda^{-1}S^2\leq C\lambda^{-1}\) at witnesses.

For a fixed fine frequency bin, transported spatial intercepts still have bounded counts per interval of length \(h\) on \(|t|\lesssim h^2\): velocity variation inside that bin changes positions by only \(O(h)\). Summing profile tails over these offsets gives (86), since all retained amplitudes are at most \(\lambda\). Each derivative \(D_b\) costs \(\Delta_0^2\), and each \(\partial_x-ib\) costs \(\Delta_0\), including the harmless change from an individual center to the bin center. Weighted Cauchy–Schwarz within a bin gives (87); a single packet has squared spacetime measure scale \(h^3\). Spatial frequencies of distinct bins have bounded overlap. Enlarging \(D\) by a fixed factor proves all the stated assertions. ◻

Proof. Construction of the masks. Summing the preceding caps that meet \(P_a\) gives \[ \|c_a\|_\infty\leq CK\lambda,\qquad \|\mathcal Lc_a\|_\infty \leq CC_1\lambda\Delta^2/K. \tag{97}\] There are \(O(K)\) preceding normalized centers, and their curvature scale is \((\Delta/K)^2\). The initial bounds give the same estimates at the first merge.

Take a nonnegative smooth partition of unity \(\sum_e\chi_e(\Delta^2t)=1\) by translates with bounded support and overlap, each bounded below on a fixed inner interval. In the co-moving coordinate \(r_x=\Delta(x-2at)\), mark every unit cell on which \(|c_a|>\lambda\) somewhere during the open positive support of the \(e\)th time cutoff, and let \(A_e\) be their union. With a fixed nonnegative \(\psi\in C_c^\infty((-1,1))\) of integral one, take \[m_e=\psi*\mathbf1_{\{r:\mathop{\mathrm{dist}}(r,A_e)<K+2\}}.\] For \(K\geq4\), this function is one within distance \(K\) of \(A_e\), zero outside its \(2K\) enlargement, and lies in \([0,1]\); its fixed-order derivative bounds are independent of \(K\). Define \[1-\rho_a=\sum_e\chi_e(\Delta^2t)m_e(r_x).\] Choose the time cutoffs flat to all required orders at the endpoints of their positive supports. Marking can be tested on dense countable sets in the open slabs, using continuity, so the construction is measurable also in \(a\). Half-open cell conventions, or fixed slight enlargements, leave all bounds unchanged. At a point with \(|c_a|>\lambda\) every active index is marked, giving \(\rho_a=0\) and therefore \(|b_a|\leq\lambda\).

The curvature bound. Where a derivative of \(\rho_a\) is nonzero, there is no same-time peak within co-moving distance \(K/2\). Otherwise every active index would be identically fully marked locally, and all derivatives of the mask would vanish. Local Bernstein, using \(|c_a|\leq\lambda\) in this neighborhood and the global bound \(CK\lambda\) against rapidly decreasing reproducing-kernel tails outside it, gives \(|(\partial_x-ia)c_a|\lesssim\Delta\lambda\) there. Now \[\mathcal L(\rho_ac_a)=\rho_a\mathcal Lc_a+ \bigl(i(\partial_t+2a\partial_x)\rho_a+ \partial_x^2\rho_a\bigr)c_a +2(\partial_x\rho_a)(\partial_x-ia)c_a.\] Scaled derivatives of the masks cost \(\Delta\) spatially and \(\Delta^2\) in co-moving time. Thus \(\|\mathcal Lb_a\|_\infty\lesssim \lambda\Delta^2(1+C_1/K)\). The outer projection has bounded \(L^1\) kernel. Choose \(C_1\) first sufficiently large independently of \(K\), and then \(K\) sufficiently large, to close (93).

Uniform higher derivatives. Order zero follows from bounded masks and frame contraction; spatial derivatives of projected pieces follow from Bernstein. For higher \(D_a\) orders use (85) to express the incoming piece in preceding centers \(\widetilde a\). After normalization by \(\Delta^{2r}\), the coefficient of the preceding top-order supremum bound is at most \(CK^{1-2r}\). In integrated center \(L^2\) norm it is at most \(CK^{1/2-2r}\): Cauchy–Schwarz in the preceding-center window costs \(O(\sqrt K)\), while the current-center window for each fixed preceding center has \(da/\Delta\) measure \(O(1)\). Every other term in the center change uses a lower affine time order; its spatial derivatives cost the preceding Bernstein scale. In \(D_a^r(\rho_ac_a)\) the top term is \(\rho_aD_a^rc_a\), with \(|\rho_a|\leq1\), and all remaining terms use lower orders and controlled co-moving derivatives of \(\rho_a\). Fix the maximum required derivative order in advance. The top coefficients are contractive for every \(r\geq1\) in this finite list after increasing \(K\) once. This enlargement leaves \(C_1\) unchanged and only improves both the curvature closure and the Duhamel source smallness. Induction on \(r\), and then on stages with that contraction, proves (94). The first step uses the fine lattice-bin estimates. The same reasoning gives mixed spatial and affine supremum bounds for \(c_a,d_a\), with constants depending on \(K\).

Persistence of a marked peak. A peak in a marked cell forces spatial squared mass at least \(c\lambda^2/\Delta\) within co-moving distance \(K/2\) of that cell at every comparison time in its time slab. To see this, propagate from the comparison time \(t\) to the peak time \(t_0\) by Duhamel, inserting a smooth cap cutoff equal to one on the spatial Fourier supports of both \(c_a\) and \(\mathcal Lc_a\). Since \(|t-t_0|=O(\Delta^{-2})\), the co-moving propagator kernels are uniformly rapidly decreasing at scale \(\Delta^{-1}\), with \(L^1\) norm \(O(1)\) and \(L^2\) norm \(O(\sqrt\Delta)\). The source contribution is at most \(CC_1\lambda/K\) by (97). The main term outside distance \(K/2\) is negligible by its kernel tails and the global \(CK\lambda\) bound. With \(K\) fixed sufficiently large, the remaining main term has modulus at least \(c\lambda\). Cauchy–Schwarz with its \(O(\sqrt\Delta)\) kernel norm proves the asserted mass lower bound. This argument explicitly permits nonzero \(\mathcal Lc_a\).

Charging the activated sets. We have \(L_a\geq\sum_e\chi_em_e|c_a|^2\). For each marked cell and slab, its enlarged activated region has spacetime volume \(O_K(\Delta^{-3})\). Integrating the preceding persistence bound over the inner times where \(\chi_e\) is bounded below gives a charge \(c\lambda^2\Delta^{-3}\) in that cell’s fully marked neighborhood. These neighborhoods have overlap \(O_K(1)\), including across slabs. Consequently the total activated volume, including the supports of mask derivatives counted slab by slab, is at most \(C_K\lambda^{-2}\int L_a\). The mixed derivative supremum bounds for \(d_a\) are \(C_{K,r,s}\lambda\Delta^{s+2r}\) on this activated set and vanish off it. Multiplying by its volume proves both inequalities in (95).

Finally fix a time with \(\chi_e>0\). The spatial projection of \(\chi_em_ec_a\) is bounded by summing its rapidly decreasing kernel over activated cells, using \(|c_a|\leq CK\lambda\). Persistence gives at that same time a lower squared-mass bound in each corresponding fully marked neighborhood. Since the positive kernel of \(W_j\) decays more slowly than the projection kernel, and the enlarged neighborhoods have bounded overlap, this gives \[|P_a(\chi_em_ec_a)| \leq C_K\lambda^{-1}W_j(\chi_em_e|c_a|^2).\] The common factor \(\chi_e(t)\) causes no difficulty even near slab endpoints. More explicitly, with \(u=\Delta(x-2at)\) and marked cell indices \(q\), the kernel upper bound is \(C_K\lambda\sum_q(1+|u-q|)^{-P_0}\). Persistence, kernel comparability on the fully marked neighborhoods, and their \(O_K(1)\) overlap give \[\lambda^2\sum_q(1+|u-q|)^{-P_0} \lesssim_K W_j(m_e|c_a|^2)(x,t).\] Multiplying by \(\chi_e(t)\), summing \(e\), and integrating \(a\) proves (96). ◻

Proof. Write \(g=g_{j-1}\), \(R=R_j\), and \(\Delta=\Delta_j\). Expanding \(B_j^0\) and using \(g_j=g-R\) gives \[\begin{align*} \Phi_j={}&-2\operatorname{Re}\left(g\overline R- \int_a(P_ag)\overline{d_a}\right)\\ &+2\operatorname{Re}\left(|R|^2- \int_ad_a\overline{P_aR}\right) +\int_a|P_aR|^2-|R|^2. \end{align*}\] For either of the first two differences the product multiplier contains, up to sign, \(p((\xi_1-a)/\Delta)-p((\xi_2-a)/\Delta)\). On the output band \(|\xi_1-\xi_2|\asymp r\) it costs \(O(r/\Delta)\). Both inputs may be restricted to \(O(\Delta)\) about \(a\): the cutoff on one factor and the output support force the same localization on the other, including the otherwise unprojected \(d_a\). The last difference has symbol \[\int_\mathbb Rp((\xi_1-a)/\Delta)p((\xi_2-a)/\Delta) \frac{da}{\Delta}-1,\] which vanishes at \(\xi_1=\xi_2\) and has the same bound. After bin and band localization, all fixed rescaled derivatives in \((\xi_1-a)/\Delta\) and \((\xi_1-\xi_2)/r\) satisfy that bound. Fourier expansion therefore represents each multiplier by spatially translated products with total coefficient or kernel mass \(O_K(r/\Delta)\).

Here is the loss-sensitive norm bound for those products. Let \(Q_b\) be smooth lattice projections of width \(\Delta\), allowing fixed enlargements. From (94) and the center-change identity, \[\|D_b^sQ_bg\|_\infty+ \|D_b^sQ_bR\|_\infty\leq C_{s,K}\lambda\Delta^{2s}.\] Moreover, \[ \sum_b\|D_b^sQ_bR\|_2^2 \leq C_{s,K}\Delta^{4s}l_j. \tag{99}\] Indeed \(Q_bR=\int_aQ_bP_ad_a\) uses only \(|a-b|\lesssim\Delta\); apply Cauchy–Schwarz in this normalized window and (95), observing boundedly many \(b\) per \(a\). The terms with an explicit \(d_a\) are handled in the same way after grouping the continuous centers into lattice bins and inserting the permitted spatial projections. All these estimates persist under the spatial translations in the multiplier expansion. Indeed, for \(T_yf(x,t)=f(x-y,t)\), one has \(D_bT_y=T_yD_b\); translations preserve both spatial cap support and the global \(L^\infty,L^2\) norms used here. Thus even translations of size \(r^{-1}\) in the output-band variable have no cost depending on \(\Delta/r\).

Thus the output from a bin centered at \(b\) has \(L^2\) norm at most \(C_K\lambda(r/\Delta)l_b^{1/2}\), with \(\sum_bl_b\lesssim_K l_j\). It has the same bound after any of the fixed normalized product derivatives \(((\partial_t+2b\partial_x)/\Delta^2)^s\) that we need. On a product with a conjugated factor this derivative splits as \(D_b\) on the factors, with the carrier constants canceling. On spacetime Fourier variables \((\xi,\tau)\), the weighted overlap of the bin strips obeys \[\sum_b\left(1+\frac{|\tau+2b\xi|}{\Delta^2}\right)^{-4} \lesssim1+\frac\Delta r.\] The centers \(b\) have spacing \(\Delta\), so their strip centers in \(\tau\) have spacing comparable to \(\Delta r\) and width \(\Delta^2\). Weighted Cauchy–Schwarz, followed by Plancherel and the differentiated product bounds, gives \[\|\operatorname{Proj}_rW_j\Phi_j\|_2^2 \lesssim_K\lambda^2(r/\Delta)^2(1+\Delta/r)\sum_bl_b,\] which is (98). ◻

Proof. For a spatial Gaussian multiplier \(q_{j,a}\) of width \(\Delta_j\) centered at \(a\), define the normalized potential \[\mathcal P_j(f)=\int_\mathbb R\int_{\mathbb R^2}|q_{j,a}f(z)|^4\,dz \frac{da}{\Delta_j}.\] On quartets with alternating sums of both spatial and temporal frequencies equal to zero, put \(u=\xi_1-\xi_2\) and \(v=\xi_1-\xi_4\). The common midpoint is \((\xi_1+\xi_3)/2=(\xi_2+\xi_4)/2\); the sum of the four squared deviations from it is \(u^2+v^2\). Integrating the Gaussian center therefore gives exactly the multiplier \(G_j(u)G_j(v)\), after fixed normalization. Expanding this product yields \[ \mathcal P_j(g)-\mathcal P_{j-1}(g) =2\int(G_j-G_{j-1})(u) |\widehat{|g|^2}(u,\tau)|^2\,du\,d\tau +\operatorname{Err}_j, \tag{101}\] where \(g=g_{j-1}\) and the error multiplier is \[E_j(u,v)=(1-G_j(u))(1-G_j(v)) -(1-G_{j-1}(u))(1-G_{j-1}(v)).\] The two quadratic terms in the expansion are equal by interchanging \(u\) and \(v\).

We establish the quantitative error estimate \[ |\operatorname{Err}_j|\lesssim_K\lambda^2 \left[\left(\frac{\Delta_0}{\Delta_{j-1}}\right)^2D+ \sum_{m<j}\left(\frac{\Delta_m}{\Delta_{j-1}}\right)^2l_m\right]. \tag{102}\] Write \(\Delta'=\Delta_{j-1}\) and partition every input into smooth spatial caps \(Q_b\) of width \(\Delta'\). The quartet identity \[ 2uv=\xi_1^2-\xi_2^2+\xi_3^2-\xi_4^2 =\sum_{\nu=1}^4(-1)^{\nu+1}(\tau_\nu+\xi_\nu^2) \tag{103}\] allows division of \(E_j\) by \(2uv/(\Delta')^2\), placing a normalized curvature derivative \(\mathcal L/(\Delta')^2\) on one input. The quotient is smooth at \(u=0\) or \(v=0\), since \(E_j\) vanishes quadratically in each variable. The sign of the multiplier \(\mathcal L\), namely \(-(\tau+\xi^2)\), is irrelevant to the bounds.

Norm bookkeeping for a distinguished derivative. On the input that receives this first derivative write \(g=g_0-\sum_{m<j}R_m\). For every required fixed \(q\geq1\), \[\begin{align*} \left(\sum_b\|Q_b\mathcal L^qg_0/(\Delta')^{2q}\|_2^2\right)^{1/2} &\lesssim_K(\Delta_0/\Delta')^{2q}\sqrt D, \tag{104}\\ \sum_b\|Q_b\mathcal L^qR_m/(\Delta')^{2q}\|_1 &\lesssim_K(\Delta_m/\Delta')^{2q}\lambda^{-1}l_m. \tag{105}\end{align*}\] The first assertion is the initial mixed-derivative estimate and bounded spatial Fourier overlap. For the second use \(R_m=\int_aP_ad_a\) at scale \(\Delta_m\) and \(\mathcal L=iD_a+(\partial_x-ia)^2\). Each derivative costs \(\Delta_m^2\) in the \(L^1\) bound (95). The projection kernels have bounded \(L^1\) norm, and each earlier synthesis center meets only boundedly many current bins, proving (105). On every other input keep \(g=g_{j-1}\) itself. Its normalized curvature derivatives in \(\Delta'\) caps have supremum \(C_K\lambda\) and square-summed \(L^2\) norm \(C_K\sqrt D\), by (94), spatial Bernstein, and the bounded preceding-center window. These bounds hold for the fixed extra derivative orders introduced below.

For one fixed cap-offset pattern, Fourier expansion of a smooth rescaled multiplier produces integrals of spatially shifted products. Hölder with a distinguished \(L^2\) input, another \(L^2\) input and two \(L^\infty\) inputs gives the base term in (102). For an earlier loss, use its distinguished \(L^1\) input and three \(L^\infty\) inputs. In summing the common cap vertex, Cauchy–Schwarz handles the two \(L^2\) sequences, or direct summation handles the \(L^1\) sequence. There is no additional factor counting free caps. Any further derivative falling on the distinguished input preserves its gain, since \(\Delta_m\leq\Delta'\).

The off-diagonal multiplier summation. On \(|u|,|v|\lesssim_K\Delta'\) the divided symbol, with smooth cutoffs, has bounded Fourier-kernel \(L^1\) norm in its two rescaled variables, and only boundedly many cap offsets occur. For explicit rescaled bounds, write \(s=u/\Delta'\), \(t=v/\Delta'\), \(F_A(s)=1-e^{-cs^2/A^2}\), and \(E(s,t)=F_K(s)F_K(t)-F_1(s)F_1(t)\). For the remaining regions, by symmetry let \(|v|\asymp Z\Delta'\) be the larger variable, with \(Z\) dyadic and large. If \(|u|\lesssim\Delta'/Z\), the once-divided symbol has kernel cost \(O_K(Z^{-2})\) at these derivative scales. There are \(O(Z)\) possible offsets for each cap vertex, giving total cost \(O_K(Z^{-1})\). In fact \(E(s,t)=s^2e(s,t)\), with every required derivative of \(e\) in coordinates \((Zs,t/Z)\) bounded uniformly in \(Z\). Consequently \(E(s,t)/(2st)=(s/(2t))e(s,t)\) has fixed rescaled derivative bounds \(O_K(Z^{-2})\) after the cutoffs. Anisotropic rescaling preserves the \(L^1\) norm of its Fourier kernel.

In the remaining dyadic regions \(|u|\asymp U\Delta'\), \(1/Z\lesssim U\lesssim Z\), use (103) three additional times. After four divisions the symbol has kernel cost \[O_K\bigl((U/Z)(UZ)^{-3}\bigr) =O_K(U^{-2}Z^{-4}).\] Indeed the undivided error and its fixed derivatives in coordinates \((s/U,t/Z)\) cost \(O_K(\min(U^2,1))\leq O_K(U^2)\), while each division costs \((UZ)^{-1}\). Smooth Gaussian derivatives give these estimates uniformly on overlapping rectangles; the very small \(u\) region uses a single cutoff at scale \(\Delta'/Z\). The spatial sum constraint on the quartet leaves at most \(O(Z\max(1,U))\) offset patterns per fixed vertex. Fixing the first cap leaves \(O(\max(1,U))\) choices for the second and \(O(Z)\) for the fourth, and then \(O(1)\) for the third by the sum constraint. The multiplier-induced spatial translations only modulate Fourier factors and do not enlarge their supports, so they do not change these counts. Their combined cost is therefore \(O_K(Z^{-3}U^{-2}\max(1,U))\). For fixed \(Z\) the dyadic sum over \(1/Z\lesssim U\leq1\) is \(O_K(Z^{-1})\), and that over \(1\leq U\lesssim Z\) is \(O_K(Z^{-3})\). Including the very small \(u\) part and then summing dyadic \(Z\) converges. At least the first curvature derivative remains distinguished in every term of the expanded quartet identities. The norm bounds above thus prove (102), with no temporal cap-support assumption and no dependence on the number of scales.

Telescoping the potential. At fixed scale \(j\), \[ |\mathcal P_j(g_j)-\mathcal P_j(g_{j-1})| \lesssim_K\lambda^2l_j. \tag{106}\] Both Gaussian-filtered states have supremum \(C_K\lambda\) by their nearby compact cap pieces and rapid Gaussian overlap. Also \[\int_\mathbb R\|q_{j,a}R_j\|_1\frac{da}{\Delta_j} \lesssim\int_a\|d_a\|_1\lesssim_K\lambda^{-1}l_j.\] The first inequality follows by composing the Gaussian with the compact synthesis projections: their \(L^1\) kernels have rapidly decaying overlap in the normalized difference of centers. The inequality \(\bigl||u|^4-|v|^4\bigr|\lesssim(|u|+|v|)^3|u-v|\) then proves (106). Similarly \(\mathcal P_n(g_n)\lesssim_K\lambda^2D\). In summing (101), insert and subtract \(\mathcal P_j(g_j)\) at each stage. The remaining boundary terms are \(\mathcal P_n(g_n)-\mathcal P_0(g_0)\), with the initial term favorable. The errors in (102) sum because \[\sum_{j>m}(\Delta_m/\Delta_{j-1})^2\lesssim_K1, \qquad \sum_j(\Delta_0/\Delta_{j-1})^2\lesssim_K1, \qquad \sum_ml_m\leq D.\] This proves (100).

All identities are global spacetime integrals. There are finitely many scales, and cap supremum bounds together with \(L^2\) bounds give finite quartic integrals. Smooth localized approximations, if needed, justify Fourier expansions; the established fixed derivative and decay bounds pass to the limit. No derivative in the center variable \(a\) is used. ◻

Proof. Recovering partial sums without a logarithm. Decompose the spatial outputs of \(H_j\) and \(W_j\Phi_j\) into smooth Littlewood–Paley bands \(r\asymp2^{-\ell}\Delta_j\), with \(\ell\) bounded below by a fixed integer. Lemma 31 and \(\sum_jl_j\leq D\) bound the square sum of flux band norms by \(C_K\lambda^2D\,2^{-\ell}\). For the growth term, evenness of \(w\) at zero and its compact support give, on the corresponding band, \[|w(u/\Delta_j)-w(u/\Delta_{j-1})|^2 \leq C_K2^{-2\ell}(G_j-G_{j-1})(u).\] Near zero the squared difference vanishes to fourth order, whereas the Gaussian gap vanishes to second order; on the rest of the fixed compact support their ratio is bounded. Hence (100) bounds the square sum of growth band norms by \(C_K\lambda^2D\,2^{-2\ell}\).

For a fixed \(\ell\), successive stage bands have scale ratio \(K\). Choose \(K\) larger than their fixed support ratio, so the spectra are disjoint and ordered. If \(F_\ell\) is the sum over all stages in that relative band, a smooth spatial low pass between successive bands recovers each initial segment exactly and kills the later terms. Its pointwise maximal value is bounded by a constant times the spatial Hardy–Littlewood maximal function of \(F_\ell\). Plancherel for the disjoint spectra and the maximal \(L^2\) bound, integrated also in \(t\), therefore bound maximal initial segments by the square sum of their norms. Summing \(\ell\) by the triangle inequality converges, with leakages \(2^{-\ell/2}\) and \(2^{-\ell}\). Zero spatial frequency contributes no separate \(L^2\) piece. This proves (92); no maximal inequality for arbitrary orthogonal series is being used.

Local constancy. In each \(H_j\) and \(W_j\Phi_j\), spatial output support and the flux expansion permit grouping into \(O(1/\Delta_j)\) matched product bins at bounded centers \(b\). The unprojected \(d_a\) factors can again be restricted to matched bins by the other input and output supports. Every such product has \[\| (\partial_t+2b\partial_x)^s(\text{product})\|_\infty \leq C_{s,K}\lambda^2\Delta_j^{2s}.\] This follows from (94), the mask derivative bounds, and bounded spatial \(L^1\) kernels; the preceding caps per current bin cost only a fixed factor depending on \(K\). Cut a product to affine temporal frequency \(|\tau+2b\xi|\lesssim\Delta_j\). The smooth high pass divided by the \(s\)th power of the rescaled affine frequency has integrable kernel, for a fixed sufficiently large \(s\). The resulting error is \(C_{s,K}\lambda^2\Delta_j^s\) per bin, hence at most \(C_{s,K}\lambda^2\Delta_j^{s-1}\) for the stage. These errors sum geometrically for \(s>1\).

The sum \(\widetilde Y_k\) of the truncated products differs from \(Y_k\) by \(O_K(\lambda^2)\) uniformly and has full spacetime Fourier support in a ball of radius \(O(\Delta_k)\): its spatial frequencies are \(O(\Delta_j)\) and its velocities are bounded. A smooth spacetime low pass at scale \(\Delta_k\) recovers \(\widetilde Y_k\), so it also recovers \(Y_k\) up to \(O_K(\lambda^2)\). Translating its Schwartz kernel from \(z'\) relative to a reference point \(z\), and comparing dyadic kernel annuli with the maximal function, proves (107). ◻

Proof. Put \(\Delta=\Delta_k\). Partition each state by smooth spatial lattice caps \(Q_b\) of width \(\Delta\). Cut each piece to affine modulation \[|\tau+2b\xi-b^2|\lesssim\Delta.\] Its \(D_b^r\) estimates show that this cutoff makes a uniform error \(C_{r,K}\lambda_i\Delta^r\) per cap, by the same inverse-derivative high-pass argument as in Lemma 33. There are \(O(1/\Delta)\) caps. For a fixed sufficiently large \(r\), the total supremum error is \(O_K(\lambda_i)\) and the sum of squared errors is \(O_K(\lambda_i^2)\). Thus the truncated fields, denoted \(\widetilde g^{(i)}=\sum_bf_{i,b}\), still have modulus comparable to \(\alpha_i\) at the designated points.

Choose \(\eta_Q\) as stated, with arbitrarily high fixed Schwartz decay at the scale of \(Q\). Such a weight can be made by scaling a function with compact smooth Fourier support and positive lower bound on the unit square. The full spacetime Fourier support of \(\eta_Q\widetilde g^{(1)}\widetilde g^{(2)}\) is bounded independently of scales, since the frequency centers and velocities are bounded. A Schwartz reproducing kernel at unit scale therefore bounds its squared value at any point of a unit square by a rapidly weighted local squared integral. Summing over the distinct unit squares gives \[\alpha_1^2\alpha_2^2\#\mathcal Z_Q \lesssim\|\eta_Q\widetilde g^{(1)}\widetilde g^{(2)}\|_2^2.\] This permits arbitrary designated points, including points on cell closures, with only a fixed multiplicity adjustment.

The support of \(f_{i,b}\) lies within \(O(\Delta)\) of \((b,-b^2)\) in spacetime frequency. Hence the pair products after multiplication by \(\eta_Q\) have Fourier overlap bounded by a fixed power of \(\delta^{-1}\). In detail, the map \[(b,b')\longmapsto(b+b',b^2+(b')^2)\] has determinant \(2(b'-b)\) on the two ordered separated intervals. If two such mapped pairs are within \(O(\Delta)\), their sums and squared differences imply that their original ordered centers are within \(O(\delta^{-1}\Delta)\). Since the cap lattice spacing is \(\Delta\), the overlap is at most \(C\delta^{-C}\), uniformly in the number of caps. Plancherel then gives \[\|\eta_Q\widetilde g^{(1)}\widetilde g^{(2)}\|_2^2 \lesssim\delta^{-C}\int|\eta_Q|^2 \left(\sum_b|f_{1,b}|^2\right) \left(\sum_{b'}|f_{2,b'}|^2\right).\]

For completeness, the local spatial Bessel bound is \[\sum_b|Q_bg(x,t)|^2 \lesssim\int_\mathbb R(1+|y|)^{-P_0}|g(x-y/\Delta,t)|^2\,dy \lesssim_K W_{k-1}|g|^2(x,t).\] To obtain the first inequality, write the projection kernels as modulations of a common Schwartz kernel. The values \(Q_bg(x,t)\) are Fourier coefficients on a fixed period of the periodization of \(y\mapsto\check\phi(y)g(x-y/\Delta,t)\). Parseval and weighted Cauchy–Schwarz for the periodization tails give the displayed positive convolution. The second inequality compares its scale \(\Delta^{-1}\) with \(\Delta_{k-1}^{-1}=K\Delta^{-1}\). The bounded squared sum of the temporal-cutoff errors consequently gives \(\sum_b|f_{i,b}|^2\lesssim_KW_{k-1}|g_{k-1}^{(i)}|^2+1\), proving (108). ◻

Proof of Theorem 28. Apply Lemma 29 and first work with one retained layer of each field, normalized by its fine envelope bound. Use the same scales and masks above, with \(\lambda_i=C_*/\alpha_i\) and \(D_i\lesssim H_i^*/B_i^2\). The constants are chosen in this order: the fixed profile and decay bounds, then \(C_1\) and \(K\) as in Lemma 30, then a trigger level \(C_0\), then \(C_*\), and finally the required lower bound for \(\alpha_i\). Choose \(C_0\) sufficiently large relative to the uniform \(O_K(\lambda_i^2)\) errors in (107), using \(\lambda_i\leq1\).

At a witness, until the first index \(k\) at which either \(|Y_k^{(i)}|>C_0\), positivity in (91) and the witness baseline (90) give \[\sum_{j<k}W_jL_j^{(i)}\lesssim1+C_0.\] By (96), the total change in each field through those stages, including the initial deletion, is at most \(C_K(1+C_0)\lambda_i^{-1}\). Choose \(C_*\) large enough that this is a small fixed fraction of \(\alpha_i\), since \(\lambda_i^{-1}=\alpha_i/C_*\). Both preceding states therefore retain modulus at least \(c\alpha_i\) at the witness. A trigger must occur by stage \(n\): otherwise both final states retain those values, whereas the terminal synthesis has only \(O_K(1)\) normalized centers and hence supremum \(O_K(\lambda_i)\). The last assertion uses the interval size \(O(\delta)\) and \(\Delta_n\asymp_K c_0\delta\).

Use nested dyadic square grids of side \(\Delta_k^{-1}\); this is possible because \(K\) and \(\Delta_0\) are dyadic. Each witness selects the square of its first-trigger scale. Take the maximal squares \(Q\) among these selected squares. They are disjoint and cover all witnesses; assign a witness to the unique maximal square containing it. If such a square has scale \(k\), an assigned witness cannot have triggered at a smaller index \(j<k\): its larger dyadic square would strictly contain \(Q\) and contradict maximality. Thus both \(g_{k-1}^{(i)}\) are large at every witness assigned to \(Q\). At least one witness in \(Q\) triggers at index \(k\). Applying (107) to that point yields, almost everywhere on \(Q\), \[ \mathcal M_1(z)+\mathcal M_2(z)\gtrsim_K1, \qquad \mathcal M_i=\mathcal M\bigl(\sup_j|Y_j^{(i)}|\bigr). \tag{109}\]

Apply Lemma 34 to the assigned witnesses. We explain how to bound its weighted integral using an arbitrary reference point \(z\in Q\). Fix a witness \(z_0\in Q\), common to both fields. The recurrence at stage \(k-1\) gives \[W_{k-1}|g_{k-1}^{(i)}|^2(z') \leq S_0^{0,(i)}(z')+|Y_{k-1}^{(i)}(z')|.\] The baseline (90), and the fact that \(h\gtrsim\Delta_k^{-1}\) up to a fixed constant, bound its first term by a fixed polynomial in \(1+\Delta_k\mathop{\mathrm{dist}}(z',Q)\). For the second term use (107) at the reference point \(z\), noting \(\Delta_{k-1}\leq\Delta_k\); if \(k=1\), it is simply zero. We obtain \[W_{k-1}|g_{k-1}^{(i)}|^2(z')+1 \lesssim_K(1+\Delta_k\mathop{\mathrm{dist}}(z',Q))^{C} (1+\mathcal M_i(z)).\] Choose the fixed Schwartz decay of \(\eta_Q\) large enough to absorb this polynomial. Its integral scale is \(|Q|\), and \((1+\mathcal M_1)(1+\mathcal M_2)\lesssim 1+\mathcal M_1^2+\mathcal M_2^2\). Thus the local sampling estimate is at most \[C_K\delta^{-C}|Q| (1+\mathcal M_1(z)^2+\mathcal M_2(z)^2)\] for almost every reference point \(z\in Q\). Average over \(Q\) and absorb the constant using (109). This proves \[\alpha_1^2\alpha_2^2 \#\{\text{witnesses assigned to }Q\} \lesssim_K\delta^{-C}\int_Q(\mathcal M_1^2+\mathcal M_2^2).\] Summing the disjoint squares, then applying the maximal \(L^2\) inequality and (92), gives \[\#\lesssim_K\delta^{-C}\alpha_1^{-2}\alpha_2^{-2} (\lambda_1^2D_1+\lambda_2^2D_2).\] Finally \(\lambda_i^2D_i\lesssim H_i^*/A_i^2\) in the original units. The summable threshold splitting and excess amplitude decay in Lemma 29 allow summation over all initial layer pairs. This proves (84).

The construction used only a fixed finite list of profile derivatives and decay orders. The derivative inductions are contractive, the potential errors and relative bands sum geometrically, and all coverings have fixed overlap. Consequently no constant grows with the number of stages or the array cardinality. The only variable separation loss is the fixed polynomial overlap bound in Lemma 34. ◻

Proof. The zero-coefficient case is immediate. Homogeneity reduces the proof to \(\sum_\ell|c_\ell|^2=1\), but we retain the homogeneous form (112) in recursive applications to subarrays. We will obtain a broad bound with constant depending on \(K\) and a narrow contraction with a constant independent of \(K\). We then choose \(K\), the base scale, and the induction constant in that order.

First, the hypotheses give two elementary bounds: \[ \nu(\mathbb R^2)\leq C m_1L^3,\qquad \|V\|_\infty\leq C \quad\text{on } |t|\leq L^2. \tag{113}\] For the mass bound, cover the output box by a fixed number of tests (111) with \(r=\rho=L^2\) and any fixed \(v\in I\). For the field bound, a packet contributing at \((x,t)\) must satisfy \(|b_\ell/L+2(t/L^2)L\xi_\ell-x/L|\leq C\). Because \(\xi_\ell\in I\) and \(|t|/L^2\leq1\), this part of phase space is covered by \(O(L)\) unit balls. At most \(O(L)\) labels contribute, and Cauchy–Schwarz cancels the normalization \(L^{-1/2}\). It follows that a nonempty level set has \[ g\leq C L^{5/6}. \tag{114}\] The mass bound settles \(g<1\), and also settles all bounded scales once the induction constant is chosen sufficiently large. The same tube test, with \(r=\rho=1\) and a fixed bounded covering when \(v\neq0\), shows that every unit lattice square has \(\nu\)-mass at most \(C m_1\).

Proof. Rotate the spatial and frequency coordinates so that the frequency line has constant normal coordinate. Subtract this central normal frequency by the corresponding Galilean shear and modulation. These transformations preserve the field modulus, up to composing with the change of variables, and change the ball bound by a fixed factor: the spatial rotation is orthogonal and the space-time shear and its inverse have bounded norms. They also preserve uniform profile and phase-space bounds. In the new coordinates write the spatial variables as \((x,y)\) and the frequency labels as \((\xi_\alpha,\eta_\alpha)\), where \[ |L\eta_\alpha|\leq C. \tag{129}\] The packet fields, including their coefficients, are now \[L^{-1}c_\alpha e^{i(x\xi_\alpha+y\eta_\alpha -t(\xi_\alpha^2+\eta_\alpha^2))} H_\alpha\left( \frac{x-b_{\alpha,x}-2\xi_\alpha t}{L}, \frac{y-b_{\alpha,y}-2\eta_\alpha t}{L}, \frac{t}{L^2}\right).\] We keep \(m\) for the ball parameter after its fixed enlargement.

Partition the normal coordinate into half-open slabs of width \(L\), and let \(y_s\) denote the center of slab \(s\). By (129) and compact profile support, only labels in a set \(\mathcal A_s\) with \[ |b_{\alpha,y}-y_s|\leq C L \tag{130}\] can contribute in this slab during \(|t|\leq L^2\). Every label belongs to only boundedly many such sets. Moreover, the projected pairs \((b_{\alpha,x}/L,L\xi_\alpha)\) have bounded counts in unit balls for \(\alpha\in\mathcal A_s\). Indeed, over any such ball, the remaining coordinates \((b_{\alpha,y}/L,L\eta_\alpha)\) lie in a bounded box by (129) and (130). Its center may depend on \(s\), but it is covered by a fixed number of unit balls. The original four-dimensional phase-space bound therefore bounds the number of projected labels uniformly. Projection does not cost a power of \(L\).

Let \(\nu_s\) be the pushforward to \((x,t)\) of the restriction of the transformed measure to slab \(s\). For any bounded tangential velocity \(v\) and \(1\leq\rho\leq r\leq L^2\), the inverse image of a test in (111) has dimensions comparable to \(r\), \(\rho\), and \(L\), after a shear of bounded norm. It can be covered by \[C\frac r\rho\left(1+\frac L\rho\right)\] balls of radius \(C\rho\). These radii are at least one. Applying the ball bound gives \[ \nu_s\{|t-t_0|\leq r,\ |x-b-2vt|\leq\rho\} \leq Cmr(L+\rho) =CmL\,r\left(1+\frac\rho L\right). \tag{131}\] Thus Proposition 36 applies to \(\nu_s\) with \(m_1=CmL\) and any fixed tangential frequency interval containing the labels.

It remains to handle the dependence of the field on the normal coordinate. A pointwise choice of a normal slice would not suffice for an arbitrary measure, so we use a single Fourier expansion valid throughout the slab. Put \[z=\frac{x-b_{\alpha,x}-2\xi_\alpha t}{L},\qquad \tau=\frac{t}{L^2},\qquad w=\frac{y-y_s}{L},\qquad \zeta_{\alpha,s}=\frac{y_s-b_{\alpha,y}}L, \quad a_\alpha=L\eta_\alpha.\] Choose a fixed smooth cutoff \(\chi(w)\) equal to one for \(|w|\leq1/2\) and supported strictly inside one period of fixed length \(T\). Expand \[\begin{align*} F_{\alpha,s}(z,\tau,w) &:=\chi(w)e^{i a_\alpha w-i a_\alpha^2\tau} H_\alpha(z,w+\zeta_{\alpha,s}-2a_\alpha\tau,\tau) \\ &=\sum_{n\in\mathbb Z}e^{2\pi i n w/T} F_{\alpha,s,n}(z,\tau). \tag{132}\end{align*}\] The constants \(a_\alpha\) and \(\zeta_{\alpha,s}\) are bounded on \(\mathcal A_s\). Consequently, if \(D\) is the finite derivative order needed in Proposition 36, integration by parts in \(w\) gives, for any sufficiently large fixed \(N\) for which the original profiles have \(D+N\) derivatives, \[ \max_{j+k\leq D} \|\partial_z^j\partial_\tau^kF_{\alpha,s,n}\|_\infty \leq C_N(1+|n|)^{-N}. \tag{133}\] All these profiles have support in a common bounded \((z,\tau)\) box. The residual normal phase has bounded derivatives in these scaled variables; this is the reason for first removing the central normal frequency. The remaining constant phase \(e^{iy_s\eta_\alpha}\) can be included in the coefficient.

Absorb \(C_N(1+|n|)^{-N}\) from (133) into the coefficients and normalize each remaining profile. This produces one-dimensional fields \(V_{s,n}\) with uniform admissible profiles and the already verified projected phase-space counts, such that in slab \(s\) \[ U(x,y,t)=L^{-1/2}\sum_{n\in\mathbb Z} e^{2\pi i n (y-y_s)/(TL)}V_{s,n}(x,t), \qquad \|c_{s,n}\|_2\leq C_N(1+|n|)^{-N}e_s, \tag{134}\] where \(e_s^2=\sum_{\alpha\in\mathcal A_s}|c_\alpha|^2\). The series converges absolutely, and all its external phase factors have modulus one at every witness. The factor \(L^{-1/2}\) outside the series is the difference between the two-dimensional packet normalization \(L^{-1}\) and the one-dimensional normalization \(L^{-1/2}\).

Choose positive numbers \(a_n=c_*(1+|n|)^{-2}\) with \(\sum_n a_n=1/2\). If \(|U(x,y,t)|\geq gL^{-4/3}\) in slab \(s\), then (134) forces, for at least one \(n\), \[|V_{s,n}(x,t)|\geq a_ngL^{-5/6}.\] Taking the projected measure of this union and applying Proposition 36 gives \[\begin{align*} \mu\bigl(\{|U|\geq gL^{-4/3}\}\cap\text{slab }s\bigr) &\leq\sum_{n\in\mathbb Z} \nu_s\{|V_{s,n}|\geq a_ngL^{-5/6}\} \\ &\leq C mL^4g^{-p_0}e_s^{p_0} \sum_{n\in\mathbb Z}a_n^{-p_0}(1+|n|)^{-Np_0} \\ &\leq C mL^4g^{-p_0}e_s^{p_0}, \tag{135}\end{align*}\] provided \((N-2)p_0>1\). Only finitely many derivative bounds are needed to choose such an \(N\). Finally, \[\sum_s e_s^2\leq C\|c\|_2^2, \qquad \sum_s e_s^{p_0} \leq\left(\sum_s e_s^2\right)^{p_0/2} \leq C\|c\|_2^{p_0},\] by bounded slab overlap and \(p_0>2\). Summing (135) proves (128). ◻

Proof. For a square centered at \(v\), use \((x',t')=(r(x-2vt),r^2t)\), up to fixed normalization conventions. The child packet size is \(L'=rL\), the field factor is \(r\), and the pushed measure has ball parameter \(m'\leq Cm r^{-3}\). The new threshold parameter is comparable to \(g r^{1/3+\eta}\), since \[r^{-1}r^\eta gL^{-4/3} =g r^{1/3+\eta}(rL)^{-4/3}.\] After a fixed inflation of \(m'\), covering tubes and their transverse annuli supplies (6) with \(D=1\); equivalently \(m'(L')^3\asymp mL^3\). The time interval is still \(|t'|\leq(L')^2\). Partition the transformed spatial range into boxes of side comparable to \((L')^2\). A compact child packet meets only boundedly many such boxes during this time interval. If their relevant coefficient energies are \(e_j\), then \(\sum_j e_j\lesssim\sum_{\mathcal C}|c_\alpha|^2\) and \(\sum_j e_j^{p/2}\leq(\sum_j e_j)^{p/2}\). Consequently outer induction costs no number of boxes. Its dimensional factor is \[(mr^{-3})(rL)^4(g r^{1/3+\eta})^{-p} =mL^4g^{-p}r^{1-p/3-p\eta},\] as asserted. These parabolic transformations translate the pairs \((b_\alpha/L,L\omega_\alpha)\) and leave the normalized profiles unchanged. Enclosing a specified cap in a comparable square does not add any other labels. ◻

Proof. Zero coefficients may be omitted, and \(E=0\) is trivial. Place successive intervals of lengths \(|c_\alpha|^2/E\) in \([0,1]\). For a dyadic interval \(I\subset[0,1]\), apportion \(c_\alpha\) by the fraction of its interval lying in \(I\). If that fraction is \(f_{\alpha,I}\), its apportioned energy obeys \[\sum_\alpha |f_{\alpha,I}c_\alpha|^2 \leq\sum_\alpha f_{\alpha,I}|c_\alpha|^2=E|I|.\] These apportioned blocks satisfy the assumed distribution estimate, up to a constant depending on \(p\). Each is either a single apportioned coefficient or the sum of a whole interior block and at most two fractional endpoint coefficients. Homogeneity gives the estimate for the fractional singletons. Split the threshold equally among these at most three pieces and use \(\sum_i\|c_i\|_2^p\leq(\sum_i\|c_i\|_2^2)^{p/2}\); the pieces have disjoint coefficient supports. The resulting bound uses exactly the apportioned energy above.

Every prefix interval \([0,s)\) is a disjoint union of at most one dyadic interval at each depth \(k\geq0\). Every coefficient contributing to a used interval belongs to the prefix, because \(s\) is between whole coefficient intervals. The sum of these block fields equals the prefix field, also when the binary expansion is infinite; the original array is finite and the coefficient fractions sum exactly.

Choose \(0<\epsilon<(p/2-1)/p\) and positive thresholds \(t_k=c_\epsilon t2^{-\epsilon k}\) with \(\sum_k t_k=t\). If a prefix field exceeds \(t\), some used block exceeds \(t_k\). There are at most \(2^k\) blocks at depth \(k\), each with energy at most \(E2^{-k}\). Restricting the measure separately to points using a block, and summing the distribution bounds, gives \[3^pKt^{-p}E^{p/2}c_\epsilon^{-p} \sum_{k\geq0}2^{k(1-p/2+p\epsilon)} \leq C_pKt^{-p}E^{p/2}.\] This argument also applies inside each fixed frequency cap, with its own total energy \(E\). ◻

Proof. At a proposed exit, subtract the selected prefix from \(U\). The remaining field has modulus at least a fixed fraction of \(\lambda\), and every fine cap sum has modulus at most \(2r^\eta\lambda\). Only physically neighboring remaining labels can contribute at the point.

Mark the terminal cells of the finite direction tree that contain such a neighboring label. Give unmarked terminal cells capacity zero, fine cells of side \(r\leq\delta_0\) capacity \(r^\eta\), and coarser cells capacity \(1\). A cut covering the marked leaves either contains a coarse cell, and hence has capacity at least \(1\), or consists of fine cells. In the latter case its cell fields sum to the remaining field, so the triangle inequality gives total capacity at least an absolute positive constant. Nonneighboring labels in these cells contribute zero at the point.

For completeness, the maximal mass that can pass through a node is the minimum of its capacity and the sum of the maximal masses of its children. Induction up the finite tree shows that this is the minimum cut capacity. Thus a mass bounded below can be sent from the root to marked leaves. Choose a neighboring label in each such leaf and normalize the resulting flow to a probability. A fine cell of side \(r\) then has probability \(O(r^\eta)\). A ball of radius \(r\geq L^{-1}\) is covered by boundedly many cells of comparable size. At coarser scales \(r\geq\delta_0\), the desired bound follows from \(\delta_0^{-\eta}r^\eta\geq1\). Taking \(C_{\rm ang}\) large enough gives (163), a contradiction to the exit test. The assertion about the final field follows by subtracting its prefix. ◻

Proof. We first justify spatial almost orthogonality for arbitrary smooth packet profiles. At native time \(t'=O(a^2)\), a child atom and all its fixed derivatives have spatial support within \(O(a)\) of its transported intercept \(b'_\alpha+2\omega'_\alpha t'\). Spatial integration by parts in the difference of the carrier frequencies gives Gram entries bounded by \[ C_N(1+a|\omega'_\alpha-\omega'_\beta|)^{-N}, \tag{177}\] and the entries vanish unless the transported intercepts differ by \(O(a)\). The joint centers \[(b'_\alpha/a,a\omega'_\alpha) \longmapsto \big(b'_\alpha/a+2(t'/a^2)a\omega'_\alpha, a\omega'_\alpha\big)\] undergo a uniformly bounded shear. Thus their phase-space counts remain bounded. Summing (177) over these centers gives uniformly summable Gram rows when \(N\) is sufficiently large. It follows that, for every required derivative, \[\|\partial^\beta U'_\theta(\cdot,t')\|_{L^2(\mathbb R^2)}^2 \lesssim \sum_{\alpha\in\theta}|c_\alpha|^2.\] Time derivatives are included: bounded child frequencies and factors \(a^{-1}\) or \(a^{-2}\) from differentiating profiles preserve the same packet bounds. Unit-cell Sobolev estimates, bounded overlap of the enlarged cells, and integration over the \(O(1)\) native time layers in \(J\) now give the first assertion of (176). Summing the coefficient energies across fine arrays and clusters costs at most \(1\).

For the second assertion, let \(\mu_\theta=F_{\theta\#}\mu\) in the coordinates of the relevant cluster. Covering the inverse image of a ball shows that its ball parameter is at most \(Cmh^3\) for radii at least \(1\). For every child label, the transverse ratio in (6) is unchanged by (172), and \[ (mh^3)a^3/D=mL^3/D. \tag{178}\] Thus \(\mu_\theta\) satisfies the same sparsity condition with \(D\) after a fixed inflation of its ball parameter.

Convolve \(\mu_\theta\) with \(K_P(z')=(1+|z'|)^{-P}\) and restrict the resulting measure to \(|t'|\leq C'a^2\), with a fixed enlargement sufficient for the cells and derivatives in use. Call the resulting measure \(\widetilde\mu_\theta\). Its ball parameter remains \(Cmh^3\). Its smoothed tube moments are still bounded as in (178): translating a child tube kernel by \(w\) changes it by at most a factor \(C(1+|w|)^M\), and \[\int_{\mathbb R^3}(1+|w|)^{M-P}\,dw<\infty.\] Every relevant unit cell \(q\) in a fixed enlargement of \(T\) has \[ \widetilde\mu_\theta(q)\gtrsim mh^3d(T), \tag{179}\] after using the time enlargement; this follows by comparing the kernel at all points of \(q\) with its distance to \(T\).

Apply Proposition 25, including its derivative version, at packet size \(a\) to \(U'_\theta\) and \(\widetilde\mu_\theta\). Unit cells in \(T^*\), of which there are boundedly many, control the suprema defining \(s_{\theta,T}\). Using (179) and dividing out \((mh^3)^{2/3}\) yields \[\sum_T d(T)^{2/3}s_{\theta,T}^2 \lesssim h^{-2}a^{4/3}D^{-c_s} \sum_{\alpha\in\theta}|c_\alpha|^2.\] The transformed spatial domain may exceed size \(a^2\). To use the local form of the sparse estimate, divide it into boxes of side comparable to \(a^2\) and retain only intersecting compact child packets. Each such packet meets boundedly many boxes on \(|t'|\leq C'a^2\); summing the squared estimates therefore uses only bounded total coefficient energy.

This application requires a fixed positive-power lower bound on \(D\) in terms of \(a\). By \(a\asymp(BD)^{C_a}\) and \(a\geq D\), we have \(a^{1/(2C_a+2)}\leq D\leq a\) once \(D\) exceeds a constant depending on \(B\). Use this fixed exponent in Proposition 25. For the remaining bounded values of \(D\), also \(a\) is bounded in terms of \(B\). The first assertion summed over \(O(a^2)\) time layers, and \(d(T)\lesssim1\), gives the second assertion with a different allowed constant. Finally sum over the partitioned coefficient arrays. ◻

Proof. Partition the normal coordinate into slabs of width \(h\). In a slab centered at \(y_0\), only boundedly many normal cell indices \(T_y\) can contribute, and on \(J\) only boundedly many time indices \(T_t\) can contribute. Write \(t=t_0+s\) with \(t_0\) an endpoint of \(J\). For a composite cell in (175), the tangential carrier has frequency \(v_\theta\), and its intercept in the time coordinate \(s\) is \[ b_{\theta,T}=hT_x+2v_\theta t_0. \tag{188}\] The fine centers are spaced at scale \(h^{-1}\). At each such center, \(T_x\) runs over an integer lattice translated by \(2v_\theta t_0/h\); the bounded number of normal and time indices thus gives the required one-dimensional phase-space counts uniformly in \(t_0\). The packet profiles are compact and smooth on the \(h,h^2\) scales, by the definition of \(s_{\theta,T}\).

Expand these profiles, with a fixed smooth cutoff on an enlargement of the slab, in the variable \((y-y_0)/h\). The mode packets can be written with scalar amplitudes at most \[C s_{\theta,T}\langle n\rangle^{-N_1}, \qquad \langle n\rangle=(1+n^2)^{1/2},\] and uniformly bounded required tangential profile derivatives, for a fixed large \(N_1\). This follows by integrating the Fourier coefficients by parts, using enough derivatives in the definition of \(s\). The external Fourier phases have modulus one at every witness.

For the cells belonging to this slab and time bin, the omitted normal and native time displacements in \(F_\theta(z)-T\) are bounded at a witness. Hence the one-dimensional envelope weight in Theorem 28, after (188), is at most a fixed multiple of the full weight defining \(S_i\) there. On the set retained in (184), the \(n\)th mode envelope is consequently bounded by \(CB_i\langle n\rangle^{-N_1}\). Its energy parameter \(H_i^*\) in (84) is at most \(Ch^3E_{i,\mathrm{slab}}\langle n\rangle^{-2N_1}\), where \(E_{i,\mathrm{slab}}\) is the part of (182) near this slab.

Whenever a band field has modulus at least \(A_{\ell_i}\), some Fourier mode has modulus at least \(cA_{\ell_i}\langle n\rangle^{-2}\). For each pair of selected modes apply Theorem 28 at separation \(\delta_0\). The threshold-to-envelope ratios only improve with \(|n|\) when \(N_1>2\), so (185) verifies its ratio hypothesis uniformly in the modes. The factors in the frame estimate are summable over the two mode indices: for example its first energy term has powers \(\langle n_1\rangle^{-4N_1+8} \langle n_2\rangle^{-2N_1+4}\). Take \(N_1>3\), with sufficient additional regularity for all derivatives.

The projection to \((x,t)\) of the measure restricted to a slab has mass at most \(Cmh\) on any unit square: its inverse image has dimensions \(O(1),O(h),O(1)\) and is covered by \(O(h)\) unit balls. This is an upper bound on the full projected measure, regardless of which normal coordinate supplied a witness. Multiplying the unit-square count from (84) by \(Cmh\), and summing the modes, bounds the retained witness mass in this slab by \[C\delta_0^{-C}mh \frac{B_1^2B_2^2}{A_{\ell_1}^2A_{\ell_2}^2} \left(\frac{h^3E_{1,\mathrm{slab}}}{A_{\ell_1}^2} +\frac{h^3E_{2,\mathrm{slab}}}{A_{\ell_2}^2}\right).\] Each composite cell belongs to boundedly many enlarged slabs, and therefore \(\sum_{\mathrm{slab}}E_{i,\mathrm{slab}}\lesssim E_i\). Summing costs no number of slabs and proves (186), since at least half of \(M_0\) was retained by Markov.

Finally substitute (184) and (181). The product of the two squared envelope bounds supplies \(m^2h^6d_1d_2E_1E_2/M_0^2\); the projected square mass and frame energy supply \(mh\cdot h^3\). Thus the total factor is \(m^3h^{10}/M_0^2\). The six threshold powers contribute \(\lambda^{-6}\) times a fixed power of \(\delta_0^{-1}\) and of \(1+\ell_1+\ell_2\). Rearranging proves (187). ◻

Proof. We first apply Theorem 41 at \(D=1\) to an arbitrary measurable spatial graph in unit-frequency variables. Let \(\mathcal B\) be a spatial ball of radius \(O(L^2)\), let \(T:\mathcal B\to[0,L^2]\) be measurable, and define \[ \mu(F)=\int_{\mathcal B}{\bf1}_F(X,T(X))\,dX. \tag{193}\] This is the pushforward of spatial area, regardless of the regularity of \(T\). A space-time ball of radius \(r\) projects into a spatial disk of the same radius, so \(\mu(B_r)\le\pi r^2\). A tube of width \(L\) and time length \(L^2\), with one of the bounded velocities in the packet class, is covered by \(O(L)\) space-time balls of radius \(CL\). Its \(\mu\)-mass is therefore \(O(L^3)\). A transverse enlargement by \(2^\ell\) can be covered by \(O(L2^{2\ell})\) such balls. Summing these bounds against the decay \(2^{-M\ell}\), where \(M\) is the fixed large exponent in (6), proves that condition with \(D=1\) after a fixed enlargement of \(m\). All constants here are uniform over \(T\), the packet intercepts, and the center of \(\mathcal B\).

We explain why the compact packet estimate applies to an exact unit-frequency extension \[Ev(X,T)=\int e^{i(X\cdot\eta-T|\eta|^2)}v(\eta)\,d\eta, \qquad \mathop{\mathrm{supp}}v\subset\{c_0\le |\eta|\le C_0\}.\] Apply Lemma 3 at packet width \(L\). It gives \(v=\sum_{\omega,n}c_{\omega,n}v_{\omega,n}\), with \(\omega\in L^{-1}\mathbb Z^2\), \(n\in\mathbb Z^2\), and \[\sum_{\omega,n}|c_{\omega,n}|^2\le C\|v\|_2^2.\] Its exact profile formula (13) reads \[\begin{gathered} Ev_{\omega,n}(X,T) =L^{-1}e^{i(X\cdot\omega-T|\omega|^2)} H\left(\frac{X-Ln-2\omega T}{L},\frac{T}{L^2}\right), \\[2pt] H(z,s)=\frac1{2\pi}\int\psi(\zeta) e^{i(z\cdot\zeta-s|\zeta|^2)}\,d\zeta, \end{gathered}\] where \(\psi\) is the fixed smooth frequency cutoff in that lemma. The phase-space counts are bounded, and, on every fixed bounded \(s\)-interval, all required derivatives of \(H\) decrease faster than any fixed power of \(|z|\). The remaining issue is its noncompact spatial support; we resolve it by summing compact layers.

To obtain the compact profiles of (4), choose a smooth time cutoff equal to one on \([0,1]\), and a spatial partition of unity \(\sum_{q\in\mathbb Z^2}\rho(z-q)=1\) with \(\rho\) compactly supported. Fix the finite profile derivative order required by Theorem 41. In the \(q\)th term replace the intercept \(Ln\) by \(Ln+Lq\). The new profile is the old profile evaluated at \(z+q\), multiplied by \(\rho(z)\) and the time cutoff. For any prescribed large \(A\), its required derivative bounds are \(O((1+|q|)^{-A})\). Put this factor into the coefficients. For each fixed \(q\), all phase-space intercepts undergo the same translation, so the counting bounds are preserved, and the new coefficient norm is at most \[C_A(1+|q|)^{-A}\|v\|_2.\] Thus every layer is an admissible array for Theorem 41.

The exponent \(A\) is chosen once, after the required derivative order; the fixed profile bound may depend on both, but never on \(L\) or \(q\). For completeness, let \(b_q>0\) satisfy \(\sum_q b_q=1\) and \(b_q\asymp(1+|q|)^{-3}\). Apply (157) to the \(q\)th layer at threshold \(b_q\lambda\), using homogeneity in its coefficients. Choosing \(A\) sufficiently large makes \(\sum_q b_q^{-p}(1+|q|)^{-Ap}\) finite and yields \[ \mu\{|Ev|>\lambda\} \le C L^4\bigl(\lambda L^{4/3}\bigr)^{-p}\|v\|_2^p. \tag{194}\] These operations can first be made on finite packet sums. At a point, each frequency label has only \(O(1)\) intercepts contributing to a fixed compact layer, and there are \(O(L^2)\) frequency labels. Cauchy–Schwarz therefore bounds that layer by \(C(1+|q|)^{-A}\|v\|_2\), uniformly in \(L\); the factor \(L\) from the square root of the label count cancels the packet amplitude \(L^{-1}\). The same bound holds with the coefficient norm of any tail in place of \(\|v\|_2\). Taking \(A>2\) makes the layer sum absolutely and uniformly convergent, and square-summability makes finite coefficient truncations converge uniformly as well. Fourier-series partial sums also converge in \(L^2\) on a fixed bounded frequency set, hence their exact extensions converge uniformly by Cauchy–Schwarz in frequency. These observations identify the compact layer sum with the exact extension. Passing to the limit in the distribution bound, first with a slightly smaller threshold, proves (194) for arbitrary \(v\in L^2\) with the indicated support. This passage uses exact extensions and therefore introduces no time-dependent choices of representatives.

Now take \(L=\sqrt N\) and \(v(\eta)=N\widehat f_N(N\eta)\). Then \(\|v\|_2=\|\widehat f_N\|_2\), and \[ S(t)f_N(x)=(2\pi)^{-2}N\,Ev(Nx,N^2t). \tag{195}\] A unit spatial ball becomes a ball of radius \(N=L^2\), and a time interval of length \(1/N\) becomes one of length \(N=L^2\). For a measurable physical graph \(t=t(x)\), its lifted spatial area in the new coordinates is \(N^2\) times the original area. This cancels \(L^4\) in (194). The threshold scale in the physical field is \[N L^{-4/3}=N^{1/3}.\] Consequently, for every such graph and every \(\lambda>0\), \[\bigl|\{x\in B:|S(t(x))f_N(x)|>\lambda\}\bigr| \le C\left(\frac{N^{1/3}\|f_N\|_2}{\lambda}\right)^p.\] For finitely many times choose, measurably, the first time attaining the largest modulus. Apply this graph bound, then increase the finite list to a countable dense set. Continuity of the bounded-frequency evolution identifies the resulting supremum with the supremum over all times and allows the endpoints. Finally, replacing \(f_N\) by \(S(t_0)f_N\) translates the interval without changing either its Fourier support or its \(L^2\) norm. ◻

Proof. For two intervals whose indices differ by at most a sufficiently large fixed integer \(C_1\), Hilbert-space Cauchy–Schwarz bounds their pairing by the geometric mean of their squared norms. Each interval has only \(O(C_1)\) such neighbors, so their total contribution is at most \(C\sum_I\|G_N(I)\|_2^2\).

For all other pairs, \(|t(x)-t(y)|\ge(C_1-1)/N\), while \(|x-y|\le2\). Write \(z=x-y\) and \(\tau=t(x)-t(y)\). On the support of \(\chi\), the gradient \[\nabla_\eta(Nz\cdot\eta-N^2\tau|\eta|^2) =Nz-2N^2\tau\eta\] has magnitude at least \(cN^2|\tau|\) if \(C_1\) is large enough, since the annulus has a positive inner radius. Dividing the phase by \(N^2|\tau|\) gives a smooth phase with uniformly bounded derivatives and a uniformly nonvanishing gradient. Repeated integration by parts gives, for every fixed integer \(M\), \[|K_N(z,\tau)|\le C_M N^2(N^2|\tau|)^{-M} \le C_M N^{2-M}.\] Integrate this bound once over the full set of far pairs, whose spatial measure is at most \(|\Omega|^2\). There is no additional factor for the number of interval pairs. After normalization by \(N^{-2/3}\), taking \(M=4\) gives an error \(O(N^{-8/3})\), which is bounded by \(O(N^{-2})\). ◻

Proof. For the first estimate, dualize the frequency \(L^2\) norm in (196). Against a frequency test function of norm one, the spatial integrand is an annular Schrödinger extension with input norm bounded by a fixed constant. Its values at \(t(x)\) are bounded by its short-time maximal function on the interval \(J\), whose length is at most \(1/N\). Proposition 47 and (197) therefore give \[\|G_N(J)\|_2\le C N^{1/3}m_J^{1-1/p}.\] Squaring and normalizing proves (200).

For the second estimate, partition space into half-open squares \(Q\) of side \(r\), and write \(G_N(J,Q)\) for the same dual integral over \(E_J\cap Q\). Set \(m_{J,Q}=|E_J\cap Q|\). Declare two squares neighbors if their grid indices differ by at most a sufficiently large fixed constant \(C_2\). Neighbor interactions are bounded by \(C\sum_Q\|G_N(J,Q)\|_2^2\), using Cauchy–Schwarz and the bounded number of neighbors of each square.

For non-neighboring squares, \(|x-y|\ge cC_2r\), whereas \(|t(x)-t(y)|\le |J|=r/N\). Choose \(C_2\) using the outer radius of the annular support. The spatial part now dominates the gradient of the phase in (198), and integration by parts gives \[|K_N(x-y,t(x)-t(y))| \le C N^2(1+N|x-y|)^{-4}.\] The right side has a uniformly bounded spatial integral. Since the graph measure is projected spatial area and \(|w|\le1\), the total normalized contribution of these far pairs is at most \[ C N^{-2/3}\int_{E_J}\int_{\mathbb R^2} N^2(1+N|x-y|)^{-4}\,dy\,dx \le C N^{-2/3}m_J\le C r^{2/3}m_J. \tag{203}\] The last inequality is exactly where \(k\le j\), equivalently \(Nr\ge1\), is used.

It remains to estimate a single square contribution. Let \(x_Q\) be the center of \(Q\) and \(t_J\) the left endpoint of \(J\). Make the change \[x=x_Q+rX,\qquad t=t_J+r^2s,\qquad \xi=\eta/r.\] The new graph lies in a unit spatial ball, has spatial area \(r^{-2}m_{J,Q}\), and has times in an interval of length \[\frac{|J|}{r^2}=\frac1{Nr}.\] The new annular frequency is \(Nr\ge1\). The dual function acquires the factor \(r^2\) from \(dx\) and a unimodular factor from \(x_Q,t_J\); changing frequency measure gives \[\|G_N(J,Q)\|_2^2=r^2\|\widetilde G_{Nr}\|_2^2.\] Apply Proposition 47 and (197) directly at the normalized scale, including its lowest possible value \(Nr=1\). This gives \[\begin{align*} N^{-2/3}\|G_N(J,Q)\|_2^2 &\le C N^{-2/3}r^2(Nr)^{2/3} (r^{-2}m_{J,Q})^{2-2/p}\\ &=C r^{-4/3+4/p}m_{J,Q}^{2-2/p} \le C r^{2/3}m_{J,Q}, \end{align*}\] where the last step uses \(m_{J,Q}\le r^2\) and \(2-2/p>1\). Sum this estimate over \(Q\), include the bounded neighbor interactions and (203), and obtain (201). Finally, the geometric mean of (200) and (201) is (202). ◻

Proof. Fix \(j\ge1\) and \(I\in\mathcal D_j\). Starting from \(H_{I,0}=I\), select at each step a child with larger mass, breaking ties consistently by choosing the left child. Call the selected child \(H_{I,k}\) and the other child \(P_{I,k}\), for \(1\le k\le j\). Thus \[m_{P_{I,k}}\le m_{H_{I,k}},\qquad I=\left(\mathop{\dot\bigcup}_{k=1}^jP_{I,k}\right) \mathbin{\dot\cup}H_{I,j}.\] The second identity is an exact partition of time, including any atoms of the time measure. Figure 3 records this partition and the two different depths used in the summation.

By additivity of \(G_N\) and the triangle inequality in frequency \(L^2\), followed by (202), \[e_j(I)^{1/2} \le C\sum_{k=1}^j2^{-k/6}m_{P_{I,k}}^{\alpha/2} +C2^{-j/6}m_{H_{I,j}}^{\alpha/2}.\] Use Cauchy–Schwarz with weights \(2^{-k/6}\), including one additional terminal weight \(2^{-j/6}\). Their sum is bounded independently of \(j\). Since \(m_{H_{I,j}}\le m_I\), we obtain \[ e_j(I)\le C\sum_{k=1}^j2^{-k/6}m_{P_{I,k}}^\alpha +C2^{-j/6}m_I^\alpha. \tag{205}\]

We give the convexity calculation behind the sum of the first terms. If a parent has child masses \(a\ge b\ge0\), then \[ (a+b)^\alpha-a^\alpha-b^\alpha \ge(2^\alpha-2)b^\alpha. \tag{206}\] For \(b>0\), divide by \(b^\alpha\) and put \(u=a/b\ge1\). The function \((u+1)^\alpha-u^\alpha-1\) is increasing for \(u\ge1\), so its minimum is \(2^\alpha-2\); the case \(b=0\) is immediate. Summing parent-minus-children deficits through depth \(d-1\) telescopes to \[m_{[0,1)}^\alpha-\sum_{Q\in\mathcal D_d}m_Q^\alpha \le m_{[0,1)}^\alpha.\] The deficits are nonnegative. Letting \(d\to\infty\) shows that the sum of the lighter-child \(\alpha\)-powers over any collection of distinct dyadic parents is at most \(m_{[0,1)}^\alpha/(2^\alpha-2)\).

Now fix \(k\ge1\) and range over all \(j\ge k\) and all \(I\in\mathcal D_j\). The parent of \(P_{I,k}\) is \(H_{I,k-1}\), at absolute depth \(j+k-1\). Different \(j\) give different absolute depths; for the same \(j\), different \(I\) have disjoint descendants. These parents are therefore distinct. Consequently, \[\sum_{j\ge k}\sum_{I\in\mathcal D_j}m_{P_{I,k}}^\alpha \le \frac{m_{[0,1)}^\alpha}{2^\alpha-2}.\] For the terminal terms, superadditivity of the \(\alpha\)th power gives \(\sum_{I\in\mathcal D_j}m_I^\alpha\le m_{[0,1)}^\alpha\). Thus summing (205), first at fixed \(k\) and then in \(k\), gives \[\sum_{j\ge1}\sum_{I\in\mathcal D_j}e_j(I) \le C m_{[0,1)}^\alpha \left(\sum_{k\ge1}2^{-k/6}+\sum_{j\ge1}2^{-j/6}\right) \le C.\] All sums are of nonnegative terms, so these rearrangements also follow from finite truncations. Finally apply Lemma 48 and sum its \(O(2^{-2j})\) errors. ◻

Proof. First suppose \(\widehat f\) has compact support. Take a smooth frequency partition \[\psi_0+\sum_{j\ge1}\psi_j=1,\] where \(\psi_0\) is supported in a fixed ball and \(\psi_j\) is supported on an annulus \(|\xi|\asymp2^j\), with uniformly bounded overlap. Put \(\widehat f_j=\psi_j\widehat f\). Choose enlarged smooth annular cutoffs \(\chi_{2^j}\) equal to one on \(\mathop{\mathrm{supp}}\psi_j\). These are the cutoffs used in Lemma 50.

For any common graph \(t(x)\) and phase \(w(x)\), Fourier inversion and the definition of \(G_N\) give \[\int_\Omega w(x)\sum_{j\ge1}S(t(x))f_j(x)\,dx =(2\pi)^{-2}\sum_{j\ge1} \int\widehat f_j(\xi)G_{2^j}([0,1))(\xi)\,d\xi.\] Only finitely many summands are nonzero. Cauchy–Schwarz in frequency and in \(j\) bounds the modulus by \[\begin{align*} C\left(\sum_{j\ge1}2^{2j/3}\|\widehat f_j\|_2^2\right)^{1/2} \left(\sum_{j\ge1}2^{-2j/3} \|G_{2^j}([0,1))\|_2^2\right)^{1/2} &\le C\|f\|_{H^{1/3}}. \tag{208}\end{align*}\] The first factor is bounded by the Sobolev square norm because on the support of \(\psi_j\) one has \(2^{2j/3}\asymp(1+|\xi|^2)^{1/3}\), and the supports overlap boundedly. The second factor is bounded by (204). The low-frequency term satisfies \(\sup_{x,t}|S(t)f_0(x)|\le C\|f_0\|_2\) by Cauchy–Schwarz on its fixed frequency ball, so its integral over \(\Omega\) has the same bound.

To recover the maximal function, take any finite list of times in \([0,1)\) and let \(t(x)\) be the first time attaining the maximal modulus of \(S(t)f(x)\) on that list. Set \(w(x)=\overline{S(t(x))f(x)}/|S(t(x))f(x)|\) where the denominator is nonzero, and set \(w(x)=0\) otherwise. These are measurable choices, and the linearized integral is exactly the integral of the finite maximum. Increase the list to a countable dense subset of \([0,1)\) containing zero. Continuity in time and monotone convergence prove (207). The estimates above used only fixed ball radii and spatial differences, so the constant is uniform in \(x_0\).

For general \(f\), choose \(f_n\) with \(\widehat f_n\in C_c^\infty(\mathbb R^2)\) and \(\|f_n-f\|_{H^{1/3}}\le2^{-n}\). In particular, \[\sum_{n\ge1}\|f_{n+1}-f_n\|_{H^{1/3}}<\infty.\] Define the measurable functions \[d_n(x)=\sup_{0\le t<1} |S(t)f_{n+1}(x)-S(t)f_n(x)|.\] The supremum can be taken over rational times, together with zero. By (207), \[\sup_{x_0\in\mathbb R^2}\int_{B(x_0,1)}\sum_{n\ge1}d_n(x)\,dx \le C\sum_{n\ge1}\|f_{n+1}-f_n\|_{H^{1/3}}<\infty.\] A countable covering of \(\mathbb R^2\) by unit balls therefore gives a single full-measure set \(E_0\) on which \(\sum_n d_n(x)<\infty\). For each \(x\in E_0\) the sequence \(S(t)f_n(x)\) converges uniformly over \(0\le t<1\). Denote its limit by \(v_f(x,t)\), and define \(v_f(x,t)=0\) for \(x\notin E_0\). It is jointly measurable, and its time dependence is continuous at every point of \(E_0\), as a uniform limit of continuous functions.

For each fixed \(t\), unitarity gives \(S(t)f_n\to S(t)f\) in \(L^2\). The pointwise limit on \(E_0\) must therefore agree almost everywhere with that \(L^2\) limit. This identifies the \(L^2\) class of \(v_f(\cdot,t)\) for every fixed \(t\); it does not assert a pointwise intersection over uncountably many times. Finally, Fatou’s lemma in the maximal estimate gives \[\int_{B(x_0,1)}\sup_{0\le t<1}|v_f(x,t)|\,dx \le C\lim_{n\to\infty}\|f_n\|_{H^{1/3}} =C\|f\|_{H^{1/3}}.\] ◻

Proof. For a fixed \(\rho>0\), cover each annulus \(2^\ell\rho\le |z|<2^{\ell+1}\rho\) by at most \(C(1+2^\ell\rho)^2\) unit balls. Positivity gives \[\begin{align*} \int_{|z|\ge\rho}K_a(z)h(x-z)\,dz &\le \frac{CA}{a}\sum_{\ell\ge0}(1+2^\ell\rho)^2 e^{-4^\ell\rho^2/(4a)}\\ &\le C_\rho A a^{-1}e^{-\rho^2/(8a)}, \qquad 0<a\le\min\{1,\rho^2\}. \tag{212}\end{align*}\] The series in the first line is finite for every fixed \(a>0\). Together with local integrability this proves finiteness of the convolution, and the second line shows that its distant contribution tends to zero.

At a Lebesgue point \(x\), put \(H_x(r)=\int_{|z|<r}|h(x-z)-h(x)|\,dz=o(r^2)\). Given \(\varepsilon>0\), choose \(\rho>0\) so that \(H_x(r)\le\varepsilon r^2\) for \(0<r\le\rho\). With \(k_a(r)=(4\pi a)^{-1}e^{-r^2/(4a)}\), radial integration by parts gives \[\begin{align*} \int_{|z|<\rho}K_a(z)|h(x-z)-h(x)|\,dz &=k_a(\rho)H_x(\rho) +\int_0^\rho(-k_a'(r))H_x(r)\,dr\\ &\le\varepsilon\left(k_a(\rho)\rho^2 +\int_0^\rho(-k_a'(r))r^2\,dr\right) \le C\varepsilon. \end{align*}\] Outside this neighborhood, (212) controls the \(h\) term, and the constant \(h(x)\) is controlled by the vanishing Gaussian mass outside \(B(0,\rho)\). Let \(a\downarrow0\) and then \(\varepsilon\downarrow0\). ◻

Proof of Theorem 1. At time zero, the fixed-time \(L^2\) identification in Theorem 52 gives \(v(\cdot,0)=f\) in \(L^2\). Since \(f^*\) represents the same \(L^2\) function on the set of Lebesgue points of \(f\), intersect \(E_0\) with the full-measure set where \(v(x,0)=f^*(x)\) and with that Lebesgue-point set. Next intersect with the Lebesgue-point sets of the countably many functions \(M_n\). Call the resulting full-measure set \(E\). For every \(x\in E\) we now have, simultaneously, \[ \begin{gathered} v(x,\cdot)\text{ is continuous on }[0,1),\qquad v(x,0)=f^*(x),\\ M_n(x)\longrightarrow0,\qquad (K_a*M_n)(x)\longrightarrow M_n(x) \quad\text{as }a\downarrow0\text{ for every }n. \end{gathered} \tag{213}\] The last assertion follows from (210) and Lemma 53. Only countably many exceptional sets were removed.

For each fixed \(t\in[0,1)\), the function \(v(\cdot,t)\) represents \(S(t)f\) in \(L^2\). Since \(K_a(x-\cdot)\in L^2\), convolution at a given \(x\) is an \(L^2\) pairing and is independent of the representative of this \(L^2\) class. Plancherel’s identity therefore gives \[ (K_a*v(\cdot,t))(x) =\frac1{(2\pi)^2}\int e^{ix\cdot\xi-it|\xi|^2-a|\xi|^2} \widehat f(\xi)\,d\xi =U_af(x,t) \tag{214}\] for every \(x\in\mathbb R^2\), every \(a>0\), and every \(0<t<1\). The integral is absolutely convergent by Cauchy–Schwarz. The quantifiers in this step are significant: for each arbitrary \(t\), an identity of \(L^2\) classes gives the convolution identity at every evaluation point. There is no intersection of time-dependent pointwise exceptional sets in (214).

For the approximants, absolute Fourier integrability yields \[\begin{align*} R_n(a)&:=\sup_{x\in\mathbb R^2,\,0<t<1} |U_af_n(x,t)-S(t)f_n(x)|\\ &\le\frac1{(2\pi)^2}\int |1-e^{-a|\xi|^2}|\,|\widehat f_n(\xi)|\,d\xi \longrightarrow0\qquad(a\downarrow0). \tag{215}\end{align*}\] Positivity of \(K_a\), the error envelope, and (214) give, for each \(x\in E\), \[ \sup_{0<t<1}|U_af(x,t)-v(x,t)| \le (K_a*M_n)(x)+M_n(x)+R_n(a). \tag{216}\] Indeed the difference between the two convolutions is bounded by \(K_a*M_n\), because \(|v(y,t)-S(t)f_n(y)|\le M_n(y)\) for almost every \(y\), simultaneously for all \(t\); the remaining two differences are bounded by \(R_n(a)\) and \(M_n(x)\). All these convolutions are finite by the preceding lemma or by the \(L^2\) pairing.

Fix \(n\) and take \(\limsup_{a\downarrow0}\) in (216). By (213) and (215), \[\limsup_{a\downarrow0}\sup_{0<t<1}|U_af(x,t)-v(x,t)| \le2M_n(x).\] Let \(n\to\infty\). Thus for every \(x\in E\), \[ \lim_{a\downarrow0}\sup_{0<t<1}|U_af(x,t)-v(x,t)|=0. \tag{217}\] In particular, at each such \(x\) the finite limit defining \(u_f(x,t)\) exists for every real \(0<t<1\), with \(t\) fixed before \(a\) tends to zero, and it equals \(v(x,t)\). Continuity at zero now gives \[\lim_{\substack{t\downarrow0\\t\in\mathbb R}}u_f(x,t) =\lim_{t\downarrow0}v(x,t)=v(x,0)=f^*(x).\] This proves Theorem 1. ◻