Proof. All comparisons are made directly with the original reference law. For a fixed base label, let \(p_I\) be the probability of a prefix \(I\in\mathcal D_{j_0}\), and let \(N_I(j)\) be its number of supported descendants at depth \(j\ge j_0\). The hypotheses give \[\max_Ip_I\le M^\epsilon2^{-sj_0}, \qquad \sum_I N_I(j)\le M^\epsilon2^{sj}.\] Thus, when the prefix is sampled according to its original law, \[\sum_I p_I N_I(j) \le M^{2\epsilon}2^{s(j-j_0)}.\] For a sufficiently large fixed constant \(C\), Markov’s inequality allows the removal of prefixes for which \[N_I(j)>M^{C\epsilon}2^{s(j-j_0)},\] at original mass at most \(M^{-(C-2)\epsilon}\). Summing over \(j\le n\) remains power-small, since \(n=M^{o_n(1)}\).

The prefixes for which \[p_I<2^{-sj_0}M^{-C\epsilon}\] have total original probability at most \(M^{-(C-1)\epsilon}\), by the prefix support bound. First remove these prefixes and the support-count exceptions at every prescribed depth. Then, from the deepest prescribed depth to the shallowest, discard each label-prefix fiber whose current retained fraction, relative to its original mass, is less than \(M^{-C\epsilon}\). Finally apply the same retained-fraction test to the base-label fibers. Each retained-fraction stage removes at most \(M^{-C\epsilon}\) times the original total mass. A later stage deletes whole fibers at every previously checked depth, so the retained fraction in each surviving fiber keeps its lower bound. The finite union of these losses and the original-law exceptions is negligible compared with \(M^{-o_n(1)}\).

Measured in the original conditional law given a surviving base label, the final retained mass of a surviving prefix is at least \[p_I M^{-C\epsilon}\ge 2^{-sj_0}M^{-2C\epsilon}.\] Dividing the original descendant weight bound by this retained denominator therefore gives \[\Pr(T\in J\mid\text{retained label and prefix}) \le M^{O(\epsilon)}2^{-s(j-j_0)}\] for a descendant \(J\) at depth \(j\). Its support count is bounded by the preceding Markov truncation. Parent cells determined by the label and prefix select entire such fibers. Finally, making every truncation against the reference law keeps the absolute constants from accumulating under successive restrictions; only a fixed finite union of power-small exceptional sets is removed. ◻

Proof. For each fixed \(i\), choose a finite power net in the permitted width range, including its endpoints. Its step tends to zero as \(i\to\infty\), and is small compared with the margin in the selected failure. Round both widths upward. This enlarges the test and increases \(W\) by only the allowed small power.

Seek the desired sublaw successively for each pair in this finite net, always testing against the original \(K\). If all steps succeeded, their successive subpower restrictions would give (3) for every width, contradicting the selected failure. Some step therefore fails on a retained law. Successful steps need only proceed along a subsequence, and there are finitely many steps for fixed \(i\). Lemma 6 preserves the time profile, and the fixed-power failure margin absorbs the finite net and normalization losses. ◻

Proof. Replace ellipses by comparable rectangles. Within the chosen bin, write \(Z=(Z^{(1)},Z^{(2)})\) for the two spatial variables; at lag \(y\), these are \((U,M^yX_v)\). Translations in the two components remain independent. Set \[\eta=\frac{w}{L\theta}.\] The bounded-factor cases with \(\eta\gtrsim1\) follow directly from \(\mu(S)\le\mu(C_0)\lesssim A\). We may therefore suppose \(\eta\lesssim1\).

For \(z,z'\in S\), let \(dL\) be the larger longitudinal separation of \(Z^{(j)}(z),Z^{(j)}(z')\), measured in the axis of \(C_0\). For \(d\gtrsim\eta\), the particles \(z'\) at separation at most \(dL\) lie in a product rectangle of radii \(O(dL),O(w)\). Consequently their mass is at most \[A d^{1+\kappa}.\] Suppose \(d\gg\eta\) and \(C_z\cap C_{z'}\ne\varnothing\). Choose a component realizing this longitudinal separation. The transverse separation of its anchors in the axis of \(C_0\) is \(O(w)\). Projection onto a normal to the axis of \(C_z\), using a common point of the two rectangles and their bounded lengths, gives \[dL\theta\lesssim w+L\gamma, \qquad \gamma=\angle(C_z,C_{z'}).\] Hence \(\gamma\gtrsim d\theta\).

In each component the intersection is contained in a rectangle with the axis of \(C_z\) and radii \[O\!\left(\frac{w}{d\theta}\right)=O(L\eta/d), \qquad O(w).\] The two component intersections use the same orientation, though their centers may differ. The assumed cap thus gives \[\mu(C_z\cap C_{z'}) \lesssim A(\eta/d)^{1+\kappa}.\] Every radius used lies between \(w\) and \(L\), up to bounded factors. This argument only uses that a box contains its anchor; the anchor need not be its center.

For fixed \(z\), the close pairs \(d=O(\eta)\) contribute at most \(A^2\eta^{1+\kappa}\) to the overlap integral. Each remaining dyadic shell contributes at most \[\bigl(A d^{1+\kappa}\bigr) \bigl(A(\eta/d)^{1+\kappa}\bigr) =A^2\eta^{1+\kappa}.\] There are \(O(1+\log(L/w))\) shells, so \[\int_S\int_S\mu(C_z\cap C_{z'})\,d\mu(z')\,d\mu(z) \lesssim mA^2\eta^{1+\kappa}\bigl(1+\log(L/w)\bigr).\] The union of the \(C_z\) lies in a product rectangle of radii \(O(L),O(\theta L)\), of mass at most \(O(A\eta^{-(1-\kappa)})\). For \[f(p)=\int_S\mathbf1_{C_z}(p)\,d\mu(z),\] heaviness and Fubini give \[\int f\,d\mu\ge mAM^{-e}, \qquad \int f^2\,d\mu =\int_S\int_S\mu(C_z\cap C_{z'})\,d\mu(z')\,d\mu(z).\] Cauchy–Schwarz on the enclosing box yields \[m^2A^2M^{-2e} \lesssim A\eta^{-(1-\kappa)} \,mA^2\eta^{1+\kappa}\bigl(1+\log(L/w)\bigr).\] After cancellation this is \[m\lesssim A M^{2e}\eta^{2\kappa}\bigl(1+\log(L/w)\bigr).\] The logarithm is subpower at fixed polynomial complexity, proving (5). Nothing else in the proof uses time. ◻

Proof. Write \(A=HW(E)\) and \(c_\kappa=4/\kappa\). Choose \(\tau_M\downarrow0\) sufficiently slowly that all bounded and logarithmic factors are \(M^{o(\tau_M)}\) and \(\tau_M\log M\to\infty\). Set \[\Theta_0=\rho M^{c_\kappa(e+\tau_M)}.\] From each \(C\in\mathcal H\), remove the particles belonging to another heavy box whose axis differs from that of \(C\) by more than \(\Theta_0\). Summing (5) over angular shells bounds the removed mass by \[A M^{2e-2\kappa c_\kappa(e+\tau_M)+o(\tau_M)} =A M^{-6e-8\tau_M+o(\tau_M)} =o(AM^{-e}).\] If the cutoff exceeds the angular range there is nothing to remove. The remaining set \(G_C\subset C\) has mass at least \(\tfrac12 AM^{-e}\).

A particle belonging to some \(G_C\) can belong to heavy boxes only at orientations within \(\Theta_0\) of \(C\). It therefore belongs to at most \[O(1+\Theta_0/\rho) \le M^{c_\kappa(e+\tau_M)+o(\tau_M)}\] grid boxes: there are this many possible orientations, bounded translated-grid overlap at each orientation, and only one time bin. Summing the masses of the \(G_C\) and using total mass at most one gives \[\tfrac12\,\#\mathcal H\,AM^{-e} \le M^{c_\kappa(e+\tau_M)+o(\tau_M)}.\] In particular, \[\#\mathcal H \le A^{-1}M^{(1+c_\kappa)e+c_\kappa\tau_M+o(\tau_M)},\] which proves (6). ◻

Proof. Keep \(A=HW(E)\), \(\rho=w/L\), \(c_\kappa=4/\kappa\), and take \[C_\kappa=\frac{c_\kappa+10}{2\kappa}.\] Write \(\nu(\text{whole space})=M^{-r_M}\), with \(r_M\to0\). Choose \(\tau_M\downarrow0\) slowly enough that \(r_M=o(\tau_M)\), all counting and logarithmic error exponents are \(o(\tau_M)\), and \(\tau_M\log M\to\infty\).

Fix a dilation \(Q=M^d\). Its baseline is \[A_Q=HW(QE)=Q^2A=M^{2d}A.\] Every enlarged heavy box has mass at least \[AM^{-e}=A_QM^{-(e+2d)}.\] For each \(C\in\mathcal H\), consider its particles that belong to some \(QD\), \(D\in\mathcal H\), at angle greater than \[\Theta_d=\rho M^{C_\kappa(e+d+\tau_M)}.\] Since \(C\subset QC\), apply Lemma 8 to the central box \(QC\), the tilted boxes \(QD\), baseline \(A_Q\), and tolerance \(e+2d\). Summing angular shells bounds this exceptional mass by \[A M^{2e+6d-2\kappa C_\kappa(e+d+\tau_M)+o(\tau_M)}.\] The application uses the caps between \(Qw\) and \(QL\), with bounded dilations.

Summing over the original heavy-box list, using the explicit count in the preceding proof, gives total exceptional mass at most \[\begin{split} &M^{(3+c_\kappa)e+6d+c_\kappa\tau_M -2\kappa C_\kappa(e+d+\tau_M)+o(\tau_M)}\\ &\hspace{2cm} =M^{-7e-(c_\kappa+4)d-10\tau_M+o(\tau_M)}. \end{split}\] The dyadic dilations in a fixed power range have \(M^{o(\tau_M)}\) members after the choice of \(\tau_M\). Their union has exceptional mass \[M^{-10\tau_M+o(\tau_M)} =o(M^{-r_M}).\] Removing this union proves the pointwise implication; its remaining tolerance \(C_\kappa\tau_M\) is \(o(1)\). ◻

Proof. Work at the fixed failing widths supplied by Lemma 7. Choose a fixed large integer \(J\), depending only on \(\kappa\), with \(2\kappa J>30\), and put \[f=\min(L,wM^J),\qquad F=(L,f).\] Here \(F\) denotes the ellipse of those radii in the orientation that will be specified for each group.

First suppose \(L>wM^J\). Call a source box of widths \(L,w\) high if its mass is at least \[KW(E)M^{-10}.\] Use boundedly enlarged grids so that every terminal test lies in a source test with these widths up to bounded factors: from \(U\in E,\ X_v\in M^{-1}E\) we obtain \(V=X_v-vU\in c+O(E)\).

Consider the particles whose \(P\) lies in two high source boxes at angular distance greater than \(M^Jw/L\). Inside any tested source box, at least one of those two boxes makes angle \(\gtrsim M^Jw/L\) with the test. Applying Lemma 8 at the source with \(e=10\), and summing angular shells, bounds this sublaw’s mass in the test by \[KW(E)M^{20-2\kappa J+o_n(1)}.\] Our choice of \(J\) makes this better than every failing exponent under consideration. The sublaw of particles in no high source box likewise has mass at most \(KW(E)M^{-10}\) in each source test. Each of these sublaws therefore satisfies the required terminal bounds. Neither can have large mass in the selected counterexample family. After their removal, a large sublaw remains, and all high source boxes through any one surviving \(P\) have axes within \(O(M^Jw/L)\).

Choose one such axis as a function of \(P\), round it at spacing comparable to \(f/L=wM^J/L\), and partition by the rounded direction and by translated \(U,V\) grid boxes of widths \(F\) in that direction. This partition depends on \(P\) alone. When \(L\le wM^J\), instead use isotropic \(F=(L,L)\), with no angular partition.

Any terminal test that violates the pre-partition threshold supplies a high source box through all of its retained particles. In the eccentric case, their rounded directions must therefore lie within \(O(f/L)\) of its direction. There are boundedly many such rounded directions. In either case, the terminal test meets boundedly many translated \(U,V\) group cells at each allowed direction, because it is contained in a source \(O(E),O(E)\) test. Thus each violating terminal test meets boundedly many groups. Tests already below the threshold remain below it under further restrictions.

We now normalize a group. Let \(A_F\) be the linear map taking the unit disk to \(F\), and subtract the two group centers before applying \(A_F^{-1}\). For a nonsingular linear map \(A\), extend the notation by \(W(A)=W(AB_2)\). The singular value formula is \[W(A)=|\det A|^{1-\kappa}\|A\|^{2\kappa},\] and consequently \[ W(AB)\le W(A)W(B). \tag{7}\] If the group has mass \(m_G\), its normalized input cap is thus at most \[\frac{KW(F)}{m_G}.\] This bound holds for every orientation of the normalized test.

For a previously violating terminal test at the selected widths that meets the group, its target ellipse \(E\) makes angle \(O(f/L)\) with the group axes, by the preceding partition argument. If \(A_E\) maps the unit disk to that ellipse, then \[\|A_F^{-1}A_E\|\asymp1, \qquad |\det(A_F^{-1}A_E)|=w/f.\] Its pullback therefore has radii comparable to \(1,w/f\). Moreover, \[W(F)W(A_F^{-1}E) \lesssim L^{1+\kappa}f^{1-\kappa}(w/f)^{1-\kappa} =W(E).\] The input floor needed for this target is a constant multiple of \(M^{-1}w/f\). Under \(A_F\), the smallest radius of any such normalized input ellipse is at least \[f(M^{-1}w/f)=M^{-1}w\ge M^{-D_i}.\] The original input cap is therefore available. The normalized complexity is \[1+\log_M(f/w)\le J+1.\] Bounded coordinate normalizations and endpoint roundings can be handled with arbitrarily small positive slack in complexity and the corresponding arbitrarily small loss in the input constant, as described above. For all sufficiently large indices, complexity at most \(J+3\) suffices.

If the limiting worst exponent for complexities bounded by \(J+3\) were strictly greater than \(q_*\), its universal bound would improve all these normalized groups beyond the selected failing threshold. Fixed-parameter uniformity supplies a large sublaw in every group with the same subpower losses. On multiplying back by group masses, the factors \(m_G^{-1}\) cancel, and the preceding inequality for the capacity weights returns the original \(KW(E)\). Recombining uses only boundedly many groups per previously violating test, while all other tests already obey the threshold. The recombined sublaw has large mass, a contradiction.

The limiting worst exponent at bounded complexities is therefore \(q_*\), and counterexample families may be chosen there. Increasing the fixed bound if necessary proves the lemma. ◻

Proof. Put \(\rho=M^{-D_i}\). Add independent uniform disk noises of radius \(\rho\) to \(U\) and \(V\), and append those noises to the base label. Their independence of the original law preserves the conditional time profile.

For an ellipse \(E\) of radii \(L,w\), let \(E^+\) have the same axes and radii \(\max(L,\rho),\max(w,\rho)\). For one noise variable, the probability of entering a translated \(E\), given the original point, is at most a constant times \(|E|/|E^+|\), and vanishes unless that point lies in a bounded enlargement of the corresponding translate of \(E^+\). Independence of the two noises yields \[\Pr(U_{\rm sm}\in E,\ V_{\rm sm}\in E) \lesssim KW(E^+)\left(\frac{|E|}{|E^+|}\right)^2 \lesssim KW(E).\] The final inequality follows because contracting a semiaxis gets exponent \(2\) from dilution, whereas its exponent in \(W\) is either \(1+\kappa\) or \(1-\kappa\), both less than \(2\). This extends the cap to smaller widths. Bounded support extends it to larger widths by clipping the long radius at a fixed support bound and covering by boundedly many tests. If both radii exceed that bound, the total-mass bound suffices: a fixed covering of the original bounded support by unit source boxes already gives \(K\gtrsim1\), with a constant depending only on that support bound.

Failure persists under smoothing. A good sublaw of the smoothed joint law pushes back to a sublaw of the original law with the same mass. The perturbations of \(U\) and \(X_v\) are \(O(\rho)\), and the smallest selected terminal position radius is \(M^{-1}w\ge\rho\). Thus every original terminal test maps into boundedly enlarged smoothed tests, and their good estimates would give the original ones.

Finally, an isotropic power shrinking of both spatial variables puts the relevant widths a fixed power below the upper endpoint. If the shrinking factor is \(B^{-1}\), the transformed constant is \(KB^2\), while the ellipse weight becomes \(W(E/B)=B^{-2}W(E)\); their product, and therefore the failing comparison, is unchanged. After the bounded-complexity reduction, all required ranges and shrinking powers remain bounded. ◻

Proof. Call a prelaw box at epoch \(y'\), with both widths in (9), high when its mass is at least \[KW(F)M^{-q_*y'-e_{y'}}.\] Suppose a large sublaw consists of particles belonging to none of these high boxes. Every test in this band then has the improved cap with factor \(M^{-e_{y'}}\): a test above that threshold would itself be a high prelaw box containing those particles. The finite grids and bounded coverings make this statement simultaneous over all centers, orientations, and band widths.

Partition this contrary sublaw by the prefix \(v'=T_{y'}\) and by isotropic parent cells of widths \(L\) for \(U\) and \(h'L\) for \(X_{v'}\), where \[h'=2^{-\lfloor y'n\rfloor}.\] Let a parent have centers \(u_Q,x_Q\) and mass \(m_Q\). Normalize it by \[U'=\frac{U-u_Q}{L},\qquad V'=\frac{X_{v'}-x_Q}{h'L},\qquad S=\frac{T-v'}{h'}.\] The residual time \(S\) lies in \([0,1)\), and its terminal partition has base \[M_{\rm res} =2^{\lfloor yn\rfloor-\lfloor y'n\rfloor} =M^{y-y'+o_n(1)}.\] If \(s_0\) is the left endpoint of a tested residual interval of length \(M_{\rm res}^{-1}\), then on that event \(v=v'+h's_0=T_y\), and \[V'+s_0U' =\frac{X_v-x_Q-(v-v')u_Q}{h'L}.\] Both translations are fixed by the parent and this residual prefix. Moreover, \(h'/M_{\rm res}=2^{-\lfloor yn\rfloor}\asymp M^{-y}\), so the residual terminal position scale agrees with the original one, up to the declared dyadic rounding.

A normalized terminal test has ellipse widths \(1,w/L\), so the smallest input width required for its propagation is a constant multiple of \[M_{\rm res}^{-1}w/L.\] In original units this is exactly the lower endpoint \(wM^{-(y-y')}\) of (9), up to a bounded factor. The band cap hence supplies the full normalized input bound, with constant \[\frac{K L^2M^{-q_*y'-e_{y'}}}{m_Q}\] up to subpower factors, because \(W(LF)=L^2W(F)\). The parent time profile is supplied by Lemma 6. Its complexity is bounded for the fixed mesh and fixed \(i\).

Apply universal propagation on the residual interval. Multiplication by the original parent mass cancels \(m_Q^{-1}\); multiplication by \(W(E/L)\) cancels the factor \(L^2\). Thus the resulting sublaw in this parent satisfies the original terminal test with bound \[KW(E)M^{-q_*y-e_{y'}+o_i(1)+o_n(1)}.\] The relative retained masses and errors may be taken uniformly in the parents.

Recombination costs only a bounded factor. A terminal bin determines its prefix, and \[X_{v'}=X_v+(v'-v)U,\qquad |v'-v|\lesssim M^{-y'}.\] As \(U\) varies in a terminal ellipse of long radius \(L\), the rebased position varies by \(O(M^{-y'}L)\). The terminal cell consequently meets boundedly many of the isotropic parent cells in both variables.

Choosing \(e_{y'}\) larger than the relevant multiples of \(e_y\) and the propagation errors contradicts the hereditary obstruction at \(y\). The contrary particles cannot have large mass. We may assign high boxes to the remaining large sublaw and select a common dyadic pair of widths. The number of such pairs is subpower, so this selection retains large mass. For a fixed finite mesh the tolerances can be chosen in this backward order while all still tend to zero. Bounded rounding of the band endpoints contributes a vanishing error in logarithmic units. ◻

Proof. Let \(\mu\) be the prelaw and \(y'<y\). Put \[\begin{split} h&=y-y',\qquad a_0=|x(y)-x(y')|,\\ b_0&=|(x+d)(y)-(x+d)(y')|,\\ B_0&=(1+\kappa)a_0+(1-\kappa)b_0. \end{split}\] A later box has a single earlier prefix \(v'\), and \[M^{y'}X_{v'}=M^{-h}M^yX_v+M^{y'}(v'-v)U.\] The last coefficient is bounded, so its particles fit an earlier-bin box with the same axis and the later widths, up to bounded factors.

Let \(F\) have radii \(L_F=\max(L(y),L(y'))\) and \(w_F=\max(w(y),w(y'))\). Enlarge the transported box and the earlier assigned box along their own axes to these common widths. For \(j\in\{y,y'\}\), let \(E_j\) have radii \(L(j),w(j)\), and set \[A_j=KW(E_j)M^{-q_*j},\qquad A_F=KW(F)M^{-q_*y'}.\] The capacity ratios and the later box’s original heaviness give, up to the existing vanishing errors, \[\frac{A_F}{A_{y'}}\le M^{B_0},\qquad \mu(\text{transported later box}) \ge A_FM^{-e_y-q_*h-B_0},\qquad \frac{w_F}{L_F}\le M^{-d(y)+b_0}.\] The extended width range supplies the caps at these common widths and at the intermediate widths required by Lemma 8.

Apply that lemma inside each enlarged earlier box, assigning to each exceptional particle its transported later box. Use angular cutoff \(M^{-d(y)+C_\kappa\Delta+\sigma}\), where \(\sigma>0\) will tend to zero, and sum the angular shells. Sum next over the original earlier assigned list, whose count is at most \(A_{y'}^{-1}M^{(1+4/\kappa)e_{y'}+o(1)}\) by the proof of Corollary 9. The factor \(A_F\) in the hairbrush bound cancels this list baseline up to \(A_F/A_{y'}\le M^{B_0}\). Thus the total exceptional mass is at most \[M^{3B_0+2q_*h+2\kappa b_0+(1+4/\kappa)e_{y'}+2e_y -2\kappa C_\kappa\Delta-2\kappa\sigma+o(1)}.\] Since \(B_0,h,b_0=O(\Delta)\), a sufficiently large \(C_\kappa\) absorbs their contributions. Write the current retained mass as \(M^{-r_M}\), with \(r_M=o_n(1)\). At each fixed mesh, choose \(\sigma=\sigma_i\downarrow0\) slowly enough that \(e_y,e_{y'}\) and all \(o_i(1)\) errors are \(o(\sigma_i)\). With \(i\) fixed, take \(n\) sufficiently large that \(r_M\), the \(o_n(1)\) errors, and logarithmic counting losses are negligible compared with \(\sigma_i\), while \(\sigma_i\log M\to\infty\). The exceptional mass is then \(o(M^{-r_M})\), as required.

Perform these removals for each neighboring pair and apply Corollary 10 at each epoch, including its dyadic dilation grid. At each fixed mesh the number of requirements is subpower. Choose the vanishing margins also to dominate the logarithm of this number divided by \(\log M\); the union of the exceptional sets still has negligible relative mass. ◻

Proof. The two monotonicities give \(a\ge0\) and \(b\le1\). Also \(d'=\lambda\). Since \(d\ge0\), a differentiable interior point with \(d_0=0\) is a local minimum and has derivative zero. This proves the assertion when \(\lambda\ne0\).

Differentiability gives a common modulus \(\omega(h)\to0\) for the two depth functions, with affine approximation error at most \(|h|\omega(|h|)\). Choose \(H_\delta\to\infty\) so slowly that \(\delta H_\delta\to0\) and \[H_\delta\omega(\delta H_\delta)\to0.\] The errors are then \(o(\delta)\) uniformly for \(|h|\le\delta H_\delta\).

To transfer this conclusion to the finite-stage families, first fix \(\delta\) and a finite accuracy grid of continuity points. Convergence at these points, together with the monotonicity of \(x\) and \(x+d+y\), traps intermediate width values between nearby ones. Next take sufficiently advanced meshes, sufficiently large \(i\), and sufficiently large \(n\). A diagonal choice gives (11), makes every cap, list, and rounding error \(o(\delta)\), and makes the neighboring depth increments \(o(\delta)\). For the angular bounds, first fix \(\delta\) and prescribe a margin \(o(\delta)\). Choose the mesh and \(i\) so that \(\sigma_i\) lies below that margin and every \(e_y\) and outer error is \(o(\sigma_i)\); then choose \(n\) so that \(r_M\), the finite-base errors, and \(\log n/n\) are \(o(\sigma_i)\). Choose the single-epoch angular-exclusion margins with the same hierarchy. We may also impose \(\log n/n=o(\delta)\), \(\delta\log M\to\infty\), and a number of epochs subpower in \(M^\delta\).

If \(\lambda>0\), use the latest local epoch as reference; if \(\lambda<0\), use the earliest. Along the chain from a given epoch to that reference, every intermediate aspect depth is at least \(d_0+\delta\lambda v-o(\delta)\), by (11). Summing (10) along this chain gives (12): the worst angular error determines the logarithmic scale, and the number of links costs \(M^{o(\delta)}\). For \(\lambda=0\), all aspect depths are \(d_0+o(\delta)\), so the same argument works with any reference epoch.

Finally cover the space of unoriented directions by finitely many sectors. On one sector a large constant fraction of the law remains; after a fixed rotation, its slopes are uniformly bounded. This gives the stated representation of the reference axis. ◻

Proof. The two particles have exactly the same displayed point \((U,M^yX_{T_y})\) at the epoch in question. Each therefore lies in the other’s assigned high prelaw box. The pointwise exclusion of Corollary 10 bounds the distance between their assigned axes by the aspect ratio times \(M^{o(\delta)}\). Equation (12) transfers the same bound to their reference axes. Angle and slope are comparable in the fixed sector.

This is a pointwise membership argument against all high prelaw boxes. It does not require the two particles to carry the same auxiliary label, or their coincidence to have positive pair probability. Since the base label determines \(P\), the asserted conditional bin count follows. ◻

Proof. Write the spatial entries in the order of (13) as \((x,h,z,w)\). Changing the truncated parameters by \((\Delta t,\Delta r)\) gives the exact update \[ (x,h,z,w)\longmapsto \bigl(x,h-\Delta r\,x,z+\Delta t\,x, w+\Delta t\,h-\Delta r\,z-\Delta t\Delta r\,x\bigr). \tag{15}\] For component widths \(\beta_x,\beta_h,\beta_z,\beta_w\) and parameter widths \(\beta_t,\beta_r\), the errors in this update at the starting component resolutions are bounded precisely when \[\beta_x\beta_r\lesssim\beta_h,\qquad \beta_x\beta_t\lesssim\beta_z,\qquad \beta_t\beta_h\lesssim\beta_w,\qquad \beta_r\beta_z\lesssim\beta_w.\] These are (14), including the absolute background depths. They also bound the doubly sheared error, since \(\beta_t\beta_r\beta_x\lesssim\beta_t\beta_h\lesssim\beta_w\). The shifts of bin centers are known from the parameter labels and can be subtracted exactly. Only the within-bin errors require these inequalities; large absolute center coordinates cause no additional loss.

A finer state supplies finer parameter bins and finer spatial measurements. Applying (15) recovers a coarser state with a bounded number of possible bin labels. Conversely, inside a coarse state, each newly tested scalar coordinate and each parameter has at most a constant times its resolution ratio many refinements. This proves the first assertions.

For a coordinate rectangle, the side states supply the finer pair of parameter labels. Each finest spatial component is already measured by at least one side. Update that side to the finer parameter pair at its own component accuracy, using the inequalities valid at that side. This recovers the upper state. The same update proves that either side recovers the bottom state. Applying the usual entropy submodularity inequality to the two side labels proves the asserted inequality with an additive bounded error.

Finally let \(Q_0,Q_1\) be the coarse and fine parameter pairs in a parameter-only refinement, and let \(Z_0,Z_1\) be its states. The nesting just proved gives \[\begin{align*} \mathop{\mathrm{H}}(Z_1)-\mathop{\mathrm{H}}(Z_0) &\geq \mathop{\mathrm{H}}(Q_1\mid Z_0)-O(1)\\ &\geq \mathop{\mathrm{H}}(Q_1\mid Z_0,\mathcal B)-O(1)\\ &=\mathop{\mathrm{H}}(Q_1\mid Q_0,\mathcal B)-O(1). \end{align*}\] For the last equality, \(Z_0\) is determined by \(\mathcal B\) and \(Q_0\), whereas \(Z_0\) includes \(Q_0\). No independence between the two parameters, or between either parameter and \(\mathcal B\), is used. ◻

Proof. Fix a finite family first. The number of bins in each of its labels is at most \(\exp(O(\log M))\). Tails of probabilities smaller than an appropriately large negative power of \(M\) have negligible total mass, by this count. Partition the remaining probabilities into dyadic classes. There are \(O(\log M)\) relevant classes per label. Do the same for all the prescribed joint labels and, pointwise on the base label, for the prescribed conditional parameter probabilities. Pigeonholing selects common classes on an event of relative mass \[\exp\bigl(-O_J(\log\log M)\bigr) =\exp\bigl(-o(L_{\log})\bigr),\] where \(J\) is the fixed number of tests. We choose all diagonal parameters slowly enough that this remains true as the grids grow.

Suppose a selected unconditional class has original weight comparable to \(w\). It contains at most \(O(w^{-1})\) bins. After restriction and normalization by a subpower density, its maximal weight is at most \(w\exp(o(L_{\log}))\). Its new absolute entropy is therefore \(-\log w+o(L_{\log})\), by the upper support bound and the lower bound on entropy supplied by the maximal atom. This proves (16). For conditional classes, discard base-label fibers on which the relative retained mass is smaller than a sufficiently slowly vanishing power. Such fibers have negligible total retained mass. The identical support and maximal weight argument then applies on every remaining fiber. This procedure is also valid when \(\mathcal B\) takes values in a standard Borel space: the dyadic class of a conditional probability is a measurable finite label, and no count of base labels is involved.

Here is the conditional-parent deduction explicitly. Replace the coarse and fine labels by their joint label when necessary. By Lemma 17, this changes the fine support and entropy bounds by bounded factors and additive constants. Write the errors in (16) as \(\varepsilon L_{\log}\). If a parent has more than \(\exp(h_1-h_0+\eta L_{\log})\) descendants, then counting all joint cells and using the parent maximal weight bounds the total mass of such parents by \(O(\exp(-(\eta-2\varepsilon)L_{\log}))\). Parents of weight below \(\exp(-h_0-\eta L_{\log})\) have total mass at most \(\exp(-(\eta-\varepsilon)L_{\log})\). Outside these two exceptions, the conditional support and maximal weight bounds are the asserted ones, with errors \(O((\eta+\varepsilon)L_{\log})\). Choose \(\eta\downarrow0\) much more slowly than \(\varepsilon\). The discarded fractions can thereby be made negligible compared with any finite list of subpower densities needed later.

The support bound also shows that bins of absolute weight below \(\exp(-h-\eta L_{\log})\) have negligible total mass. For a finite list of joint labels, one may remove these bins successively, or iteratively remove bins whose remaining weight is below the chosen threshold. Charge each removed bin when it is first deleted. The sum of all charges is bounded by the sum of the corresponding support counts times their thresholds, and is still negligible if the slack dominates the existing errors. Dividing joint weights by parent weights gives the corresponding conditional lower-weight statements.

After any further restriction of density \(\rho=\exp(-o(L_{\log}))\), the support bounds remain valid and maximal weights grow by at most \(\rho^{-1}\). Thus the absolute entropies change by \(o(L_{\log})\). The same parent and base-fiber trims recover the conditional statements. This proves heredity. Finally apply the construction on larger finite grids, with their size increasing sufficiently slowly. Off-grid tests are sandwiched by admissible finer and coarser grid states. The domain has interior shift vectors with every coordinate positive, and others with every coordinate negative, so such sandwiches exist with arbitrarily small offset errors. The Lipschitz estimates of Lemma 17 complete the diagonal extension. ◻

Proof. The finite-state comparisons give equicontinuity, after reference subtraction, with errors tending to zero in the present log unit. One first takes a subsequence on a countable dense grid, and then uses the admissible coordinatewise joins and bounded comparable shifts to obtain local uniform convergence on the full domain. Monotonicity and Lipschitz bounds pass to the limit. The entropy rectangle inequalities give submodularity. Equivalently, \(-\partial_{ij}F\) is a nonnegative distribution, and hence a nonnegative Radon measure, for \(i\ne j\) in the interior. Pinning coordinates or summing disjoint nonnegative coordinate increments preserves those inequalities. Comparisons on faces follow by translating the valid rectangle into the interior and passing to the limit; convexity of the domain makes these translations available.

To verify the normalization in (17), put \[B_0=-\log K+ \bigl(2x_0+(1-\kappa)d_0+q_*y_0\bigr)\log M.\] At full alignment, a state atom is contained in a tested ellipse cell, with its displayed time bin and axis. The simultaneous cap from Section 2 therefore gives \[\mathop{\mathrm{H}}(Z(z))\geq B_0+L_{\log}(k_C C+k_B B+q_*Y)-o(L_{\log}).\] For a nonmesh time use a nearby earlier mesh time, with the rebasing and width errors controlled by (10). On the calibrated path, Corollary 9 supplies the opposite support estimate. Given an assigned high box and its time bin, the reference-axis approximation at path aspect accuracy leaves only \(\exp(o(L_{\log}))\) possibilities for the labels (13). The same is true of the slight width mismatches and time rebasing. Thus the support logarithm is at most the displayed lower bound plus \(o(L_{\log})\). In particular, \(\mathop{\mathrm{H}}(Z(0))=B_0+o(L_{\log})\). Subtract this equality and divide by \(L_{\log}\). Equality first holds on the fine mesh and then everywhere on the line by continuity. All the cap and list estimates survive the subpower restriction used to regularize. ◻

Proof. Given \(P\) and a time prefix at a local epoch, all particles in that fiber have the same measured line state at that epoch. Their assigned high boxes therefore pass through that state. The pointwise exclusion in Corollary 10 applies to all high prelaw boxes, irrespective of subsequent restrictions or additional labels. It implies that the assigned axes in the fiber have only subpower many possibilities at path aspect precision. Corollary 16 transfers this conclusion to the common reference axis. The conditional entropy excess of the axis over the time prefix is consequently \(o(L_{\log})\) at that precision.

If \(\lambda>0\), the epoch of offset \(Y\) predicts the reference axis at offset depth \(\lambda Y\), hence also at every \(R\leq\lambda Y\). If \(\lambda<0\), for any fixed \((Y,R)\) choose an earlier fixed epoch \(v\leq Y\) with \(\lambda v>R\). Its time prefix is determined by the one at \(Y\), and its aspect precision is finer than the requested axis cutoff. The common reference epoch is still earlier, on the deeper-aspect end of the stretched interval. Chaining (10) from that reference to \(v\) loses only the worst aspect precision, namely the precision at \(v\). Thus this epoch also predicts the common reference axis to the requested accuracy. This proves the asserted zero excess entropy, including its absolute \(o(L_{\log})\) bound before reference subtraction. Lemma 17 proves the increment comparison with \(F\), and hence \(\partial_YF\geq s\) in the decreasing-aspect case.

We give the spatial prediction argument for (18) with all its buffers. Fix a state in that open region and choose a sufficiently small \(\eta>0\). Test the path epoch \[v_*=-\frac{R+\eta}{c}.\] Two particles with the same displayed state have a common truncated slope \(r\) and time \(t_Y=T_{y_0+\delta Y}\). Their differences satisfy \[\begin{align*} |\Delta U_1|&\lesssim M^{-x_0-\delta C},& |\Delta U_2-r\Delta U_1| &\lesssim M^{-x_0-d_0-\delta B},\\ |\Delta(X_{t_Y})_1|&\lesssim M^{-x_0-y_0-\delta D},& |\Delta(X_{t_Y})_2-r\Delta(X_{t_Y})_1| &\lesssim M^{-x_0-y_0-d_0-\delta A}. \end{align*}\] Since \(Y>v_*\), they also have the same epoch-\(v_*\) prefix. Choose the assigned axis \(r_*\) of either particle at that epoch. Its distance from \(r\) is at most \(M^{-d_0-\delta R+o(\delta)}\): compare first with that particle’s reference axis at depth \(R+\eta\), and then with its common prefix \(r\) at depth \(R\).

In the \(r_*\) frame, the velocity errors have the two offset depths \[C,\qquad \min(B,C+R),\] with absolute background depths \(x_0\) and \(x_0+d_0\). If \(t_*=T_{y_0+\delta v_*}\), then \(|t_*-t_Y|\lesssim M^{-y_0-\delta v_*}\). Using \(X_{t_*}=X_{t_Y}+(t_*-t_Y)U\), the position errors in that frame have offset depths \[\min(D,C+v_*),\qquad \min(A,D+R,B+v_*,C+R+v_*),\] with background depths \(x_0+y_0\) and \(x_0+y_0+d_0\). The path targets at this epoch are \[-av_*,\quad -bv_*,\quad (1-a)v_*,\quad (1-b)v_*.\] Every displayed error depth is strictly finer than its target for small \(\eta\). Indeed this follows from the five inequalities in (18); the additional comparisons \(C+R>bR/c\) and \(D+R>(b-1)R/c\) follow from \(c=b-a\).

Thus peers in this state lie in bounded enlargements of one another’s assigned epoch boxes. The all-high-box angular exclusion forces their assigned axes to agree up to a subpower factor at offset depth \(R+\eta\). Comparison with each particle’s reference axis gives the same conclusion for the finer reference-axis bins. The state therefore determines this extra axis precision with entropy cost \(o(L_{\log})\). After refining the parameter, its spatial labels at the original cutoffs can be updated with bounded ambiguity by (14). Hence increasing \(R\) a little at fixed other offsets costs zero in the limiting profile. The strict margins hold throughout a neighborhood, proving local independence. ◻

Proof. First fix a radius \(h\) and a finite test grid for the desired tangent. Approximate the original profile on the corresponding grid of absolute cutoffs to an error much smaller than \(h\). Choose the original stage sufficiently far out that \(hL_{\log}\to\infty\), and that every previous logarithmic error and every probability-class loss is \(o(hL_{\log})\). Lemma 18 can be imposed on this finite list and on its joint and conditional labels. Now pass to a diagonal sequence as \(h\downarrow0\) and the grid grows. This realizes the tangent in its new log unit. Repeating the construction handles iterated tangents. A finite collection of sites is treated by putting all its cutoffs in the same list. Coordinates prescribed to be physically common are represented by the same absolute cutoff, rather than by separately rounded approximations.

Inside a common parent bin, subtract the known bin centers and use its spatial and parameter widths as units. Formula (15) still applies. For instance a time increment changes position by its normalized scalar increment times normalized velocity, with known center terms subtracted exactly. At an aligned triple the position unit equals the velocity unit times the time unit. The defining affine equalities of these depths are preserved under a centered rescaling through alignment. Dyadic rounding changes the units only by bounded factors.

For comparable parent and refinement states, the conditional support and maximal-weight estimates now use their entropy difference, by Lemma 18. This applies as well after a fine parameter bin is included in the parent tuple: the old spatial labels can be rebased to that parameter bin with bounded ambiguity. Lower typical weights, when needed, are obtained by the same trims.

To see the resulting product density, fix a typical parent and let \(\mathsf A,\mathsf B\) be the supported bin sets of two prescribed refinement labels. Let \(\mathsf E\subset \mathsf A\times\mathsf B\) consist of pairs with positive conditional joint probability. Use joint labels with the parent where nesting has bounded ambiguity; this does not change the exponents. Write \(L\) for the final logarithmic unit. If the two marginal entropy exponents are \(\alpha,\beta\) and the joint exponent is \(\alpha+\beta\), regularization gives \[\begin{gather*} |\mathsf A|\leq e^{(\alpha+o(1))L},\qquad |\mathsf B|\leq e^{(\beta+o(1))L},\\ \max_{(a,b)\in\mathsf E}\Pr(a,b\mid\text{parent}) \leq e^{-(\alpha+\beta-o(1))L}. \end{gather*}\] The conditional law has mass one, so \(|\mathsf E|\geq e^{(\alpha+\beta-o(1))L}\) and hence \[\frac{|\mathsf E|}{|\mathsf A|\,|\mathsf B|} \geq e^{-o(L)}.\] This is subpower density of supported bin pairs, with no independence assertion. All these statements hold at every fixed finite power grid, and hence on successively finer grids by the same diagonal choice. ◻

Proof. We prove the assertion with an interior mollification first. Write \(\mu_{ij}=-\partial_{ij}F\geq0\) for \(i\ne j\). Choose a compact sheet patch, a smooth cutoff on a slightly larger patch, and an offset compact set lying strictly inside the tangent domain. Translate the sheet patch by \(hu\), with \(u\) in that offset set and \(h>0\) small. If \(i\) belongs to the group being separated and \(w\) is the chosen sheet-tangent direction, then \(w_i=0\) and \[-\partial_{iw}F=\sum_{j\ne i}w_j\mu_{ij}.\] The integral of the left side on the translated patch against the cutoff is bounded uniformly in \(h\) and \(u\): integrate by parts in the sheet direction \(w\) and use the bounded first derivative \(\partial_iF\). Terms from previously separated groups have bounded absolute integrals by the estimates at their earlier separation steps, using symmetry of mixed derivatives. All the remaining terms outside the current group are nonnegative, with strictly positive coefficients \(w_j\). Consequently each such \(\mu_{ij}\) has a uniformly bounded integral on the translated sheet patch. Use nested, slightly larger patches at successive steps. This establishes all the required cross-pair bounds before any tangent is taken.

Average these bounds also over compact interior offsets \(u\). For the rescaled function \(F_{x,h}\) the corresponding mixed derivative has density \(h\mu_{ij}(x+hu)\) in the offset variable. Thus its averaged mass on an offset compact set, integrated over sheet centers \(x\), is \(O(h)\). These bounds are summable for dyadic \(h\). By positivity, summability, and exhaustion by countably many compact subsets, at almost every sheet center all the cross-pair measures tend to zero on compact subsets in every limiting dyadic tangent.

For an unmollified profile, the off-diagonal negative mixed derivatives are Radon measures. Mollify at a radius much smaller than \(h\) times the strict interior margin of the offset compact set. Enlarge that compact set slightly when testing the measure. The same averaged bounds pass to the measures and give the conclusion just obtained. This argument never takes a Hessian trace on a binding face.

Vanishing cross derivatives give additive separation on the convex open tangent domain. One way to see this is to mollify locally again: the gradient in a given group is then constant along each fiber of all the other coordinates. Connected convex fibers show that it depends only on that group’s coordinates. The resulting conservative field has a primitive on their convex projection. Patching these primitives gives a sum of group functions, up to a constant; Lipschitz continuity extends the identity to the closed domain.

Apply Rademacher’s theorem in intrinsic Euclidean coordinates on the relative interior of \(\Sigma\) (Heinonen 2005, Theorem 3.1). At almost every sheet center the restriction of the original profile to \(\Sigma\) is differentiable. Its locally uniform linear blowup shows that the restriction of any tangent there is affine on the whole tangent sheet. This uses intrinsic sheet measure, not ambient almost-everywhere differentiability. After the preceding splitting it is a sum of Lipschitz functions of the distinct group forms. Such a sum can be affine only if each summand is affine along its form. For completeness, fix one form and apply successive finite differences in directions annihilating each of the other forms but not the fixed one. Additional differences annihilate the affine right side. The result says that a sufficiently high distributional derivative of the chosen one-variable function vanishes. It is a polynomial, and its global Lipschitz bound makes its degree at most one. A zero group form is constant on the sheet and requires no assertion beyond that. For a singleton with nonzero form, its projected coordinate runs through the real line on the sheet, so its full summand is affine and its full partial is constant. ◻

Proof. Fixed exterior slices and their rates. For each fixed later parameter \(\beta\), the function \(y\mapsto F_0(N(y),Q(\beta))\) is Lipschitz on \(y<\beta\). Choose a countable dense family of parameters containing all dyadic partition endpoints used below. At common differentiability points of the corresponding restrictions, define \[r_\beta(y)=-\frac{d}{dy}F_0(N(y),Q(\beta)),\qquad m(y)=\lim_{\beta\downarrow y}r_\beta(y),\] where the limit is taken through that countable family. Submodularity and monotonicity of \(Q\) show that \(r_\beta(y)\) increases as \(\beta\) decreases to \(y\). All these rates are bounded, so \(m\) is bounded and measurable.

Proof. Include time among the state coordinates, so the terminal face is part of the compact reachable set. Call a state good if it admits a positive-length straight step before the terminal time with gain rate greater than \(g_0+\varepsilon/2\), and call all other states bad. Goodness is relatively open. Indeed a strict comparison persists under small perturbations. If a step ends on the terminal face or on the boundary of the half-space, shorten it slightly; the comparison remains strict. A step starting on the half-space boundary either enters the interior or follows the boundary, and remains admissible when its start is moved slightly into the half-space. For an interior start, shortening or a small convex perturbation toward a boundary-following velocity keeps the perturbed step admissible. Thus the bad set is closed; it includes the terminal face.

By hypothesis, the bad set occupies zero time on every admissible path. Its shrinking neighborhoods consequently have uniformly vanishing occupation time over all such paths. To prove this, otherwise choose shrinking radii and paths with a fixed positive occupation time. The paths have a common Lipschitz bound. A uniformly convergent subsequence has an admissible limit, because the velocity box is compact and convex and the half-space is closed. Uniform convergence shows that the limiting path spends at least that positive time in every fixed neighborhood of the closed bad set. Continuity from above of Lebesgue measure then gives positive time in the bad set itself, a contradiction.

Fix a small neighborhood of the bad set. Outside it, compactness and openness of goodness give finitely many comparison steps, each valid on an open set of starting states and with its length bounded below by a positive number. At a state outside the neighborhood, take one of these good steps. Inside the neighborhood, take an arbitrary viable straight step of a fixed sufficiently small length, clipped at terminal time. Choose that length so that such a step stays in a slightly larger neighborhood, using the common speed bound. This constructs a path reaching the terminal time in finitely many steps. The total time of its nongood steps is at most the occupation time of the larger neighborhood, hence is uniformly small. Its good steps gain at least \(g_0+\varepsilon/2\) per unit time, whereas the Lipschitz bound gives a uniform lower rate on the remaining steps. Taking the neighborhood small enough makes the total gain at least \(g_0\) times the full duration.

Finally, a strict endpoint comparison from the tangent origin passes back along its defining blowup sequence: the endpoint error is little-oh of the step length, while the gain margin is a fixed positive multiple of that length. Admissibility under the linear domain constraint is retained in this passage. Any additional perturbation must remain admissible and have velocity in the given box; sufficiently small such perturbations preserve the strict margin. This gives the last assertion. ◻

Proof. First discard direction fibers and pin fibers whose relative masses are too small, using a threshold with exponent tending to zero sufficiently slowly. Normalize the restrictions of the reference laws \(\mu\) and \(\zeta\) to the retained pin and direction sets, and keep \(E\) as a subset of their product. We continue to denote these reference laws by \(\mu,\zeta\); their Frostman bounds lose only subpower factors, and \(E\) still has dense product measure. We now reduce these weighted laws to the finite sets in Theorem 28.

Sample the reference direction law \(\zeta\) globally, before drawing the pencils, so every pencil uses the same sampled population. For small fixed \(0<\rho<\sigma\), take \(n\asymp m^{\sigma-\rho}\) independent samples. Choose a permitted empirical loss \(m^a\) with \(a>\rho\), with both exponents below the input allowance in Theorem 28. The required count in an interval of radius \(h\) is at most \(m^a n h^\sigma\). At \(h=m^{-1}\) this threshold is of order \(m^{a-\rho}\), whereas the expected count is at most \(n h^\sigma m^{o(1)}\). Binomial tails therefore give the count bound simultaneously over the polynomial family of intervals with mesh endpoints. Enlargement then gives it for all angular intervals. The expected proportion of samples sharing a mesh bin with another sample is at most \[O\bigl(m^{\sigma-\rho}\sup_\omega \zeta(B(\omega,O(m^{-1})))\bigr) \le m^{-\rho+o(1)}.\] Thus repetitions can be removed at a cost negligible compared with the dense incidence. The tail failures can also be made negligible relative to that density. Averaging gives a sample on which dense incidence survives, with dense direction fibers at a dense set of pins. Apply the same construction to the reference pin law, using \(m^{k-\rho}\) samples after also requiring \(\rho<k\); mesh balls have a polynomial-size covering family. Restriction to a dense fiber divides a normalized ball bound by at most a subpower factor. Finally pigeonhole the pencil cardinalities. Their exponent is \(\sigma-\rho-o(1)\), and the pin exponent is \(k-\rho-o(1)\); the normalized nonconcentration losses remain arbitrarily small.

For each retained point–direction pair, draw the line of constant projection through that point. The sampled directions number at most \(m^{\sigma+o(1)}\) in total. Hence the projection support hypothesis gives the global line-count upper bound \(m^{\sigma+u+o(1)}\). On the other hand, Theorem 28, with output error \(\varepsilon\), gives lower exponent \[\sigma-\rho+\min\{k,(k+\sigma)/2,1\}-\varepsilon-o(1).\] To make this comparison quantitative, first prescribe an output error, then choose all sampling, restriction, and Frostman losses below the input allowance in that theorem, and finally let \(m\) tend to infinity. Sending the output error and \(\rho\) to zero proves (22).

In a bounded pin chart, the map from a direction to the line through a given pin is uniformly bi-Lipschitz in its slope coordinate. The passage between angular bins, lines, and dyadic tubes therefore changes only bounded covering constants. If angular bounds are initially known only for balls centered at surviving directions, an arbitrary ball meeting the support is contained in the ball of twice its radius centered at one of those directions. This also proves the stated variants. ◻

Proof. At a fixed power scale \(\beta\), all original subpower losses are also \(\beta^{o(1)}\). It is enough to rule out a sequence on which the bad pair mass is \(\beta^{o(1)}\): failure of any power bound would supply such a subsequence. We use angular grids with bounded overlap, choosing strict exponent margins so that fixed constants in the definition of \(A_\beta\) are harmless.

We first prove the assertion for \(2\sigma<s_0\). At any pin there are at most \(O(\beta^{-\sigma})\) angular bins of mass at least \(\beta^\sigma\). Suppose that pairs in these bins have dense incidence. Cauchy–Schwarz, applied to target fibers, gives two pins with a dense common target set. They can be required to be at distance at least \(\beta^\alpha\), for any fixed sufficiently small \(\alpha>0\), because the Frostman estimate makes closer pin pairs power-small. Likewise, discard the strip of width \(\beta^\alpha\) about their joining line. The strip hypothesis makes the total discarded triple mass power-small, whereas the common-target triple mass is the square of the dense incidence and is still subpower.

For a target outside this strip, the two joining directions make an angle whose sine is at least a constant times \(\beta^{2\alpha}\). Indeed the cross product is the pin separation times the distance to their joining line, and all distances in the denominator are bounded. An intersection of two pencil tubes is therefore contained in a ball of radius \(O(\beta^{1-2\alpha})\). The common targets lie in at most \(O(\beta^{-2\sigma})\) such balls. Their total mass is at most \[\beta^{-2\sigma+(1-2\alpha)s_0-o(1)},\] which is power-small when \(\alpha\) is sufficiently small. This contradicts dense common incidence and proves the initial range.

For the improvement step, suppose the assertion is already known at an exponent \(\sigma<s_0\), at every fixed power scale. Choose \(\alpha,\eta>0\) so that \[ 3\eta+\alpha<\frac{s_0-\sigma}{2}, \qquad \eta<\frac{\alpha(s_0-\sigma)}2. \tag{23}\] We prove it at \(\sigma+\eta\). A sufficiently fine finite power grid, the previous assertion in both orders of a pair, and removal of power-small exceptions give the following inherited estimates: at every retained pair the angular masses, at all radii from \(\beta\) to \(1\), are bounded by \(\beta^{-\tau}\) times the radius to the power \(\sigma\). Here \(\tau>0\) is arbitrarily small and is chosen after the strict margins in (23). Bounds centered at surviving directions extend to all angular balls by enlargement.

Suppose that angular bins of unrestricted target mass at least \(\beta^{\sigma+\eta}\) carry dense incidence even after this removal. Call such a bin dispersed if its containing tube has \(\nu\times\nu\) mass at least \(\beta^{2\sigma+3\eta}\) in pairs separated by at least \(\beta^\alpha\). If the dispersed bins carry dense incidence, discard pins whose surviving dispersed target fiber has less than a subpower fraction of the target mass. The normalized remaining pin law satisfies the planar \(s_0\) bounds. At each such pin \(x\), normalize its surviving target fiber and push it forward by \(y\mapsto[y-x]\). The inherited angular bounds make this pin’s direction law Frostman \(\sigma\), with arbitrarily small restriction losses. Sample the pins and then sample each of these direction laws, using the sampling and collision estimates above; choose comparable pencil cardinalities. Every selected tube is dispersed, up to a bounded enlargement, and the direction sets may vary with the pin. These are the pin set and incident pencils in Theorem 28. As its input losses tend to zero, it produces at least \[\beta^{-(s_0+3\sigma)/2+o(1)}\] distinct dispersed tubes. Here \(\min\{s_0+\sigma,(s_0+3\sigma)/2,1+\sigma\} =(s_0+3\sigma)/2\), since \(\sigma<s_0\le1\).

A separated subcollection of tubes gives the opposite bound. A pair of points at separation at least \(\beta^\alpha\) belongs to at most \(O(\beta^{-\alpha})\) tubes of this subcollection: the admissible slopes occupy an interval of length \(O(\beta/|z-z'|)\), and for each slope only boundedly many intercept bins are possible. Integrating the separated-pair masses of the tubes therefore gives at most \[O(\beta^{-2\sigma-3\eta-\alpha})\] tubes. The first strict inequality in (23), followed by sufficiently small input losses in Theorem 28, contradicts the lower bound.

It remains to consider nondispersed bins. If such a bin has mass \(b\ge\beta^{\sigma+\eta}\), averaging the separated-pair mass inside the bin gives a ball of radius \(O(\beta^\alpha)\) containing all but \[O(\beta^{2\sigma+3\eta}/b) \le O(\beta^{\sigma+2\eta})\] of its unrestricted target mass. There are at most \(O(\beta^{-\sigma-\eta})\) heavy bins per pin, so removing these outside portions costs \(O(\beta^\eta)\) in pair mass. Dense incidence survives.

At a target \(y\) with a dense reverse fiber, each surviving pin \(x\) now determines a tube carrying unrestricted target mass \(\gtrsim\beta^{\sigma+\eta}\) inside \(B(y,C\beta^\alpha)\): the surviving \(y\) lies in the ball chosen for its bin, and doubling that ball about \(y\) retains the same original target mass. If \(z\) belongs to the same angular bin from \(x\), boundedness of the ambient region gives \[\left|\det\left(z-y,\frac{y-x}{|y-x|}\right)\right| \lesssim\beta.\] For fixed \(z\), the admissible reverse directions \([x-y]\) therefore lie within \(O(\beta/|z-y|)\) of \([z-y]\), with the trivial bound when \(|z-y|\lesssim\beta\). Apply the inherited reverse angular estimate to this interval and integrate in \(z\). Normalizing the dense reverse fiber costs only a subpower factor, so the average is at most \[\begin{align*} &\beta^{-2\tau} \int_{|z-y|\lesssim\beta^\alpha} \min\{1,(\beta/|z-y|)^\sigma\}\,d\nu(z)\\ &\hspace{25mm}\lesssim \beta^{\sigma-3\tau+\alpha(s_0-\sigma)}. \end{align*}\] To obtain the last line, split the integral into the ball of radius \(\beta\) and dyadic annuli between \(\beta\) and \(C\beta^\alpha\), then use the planar Frostman bound; its exponent exceeds \(\sigma\). Taking \(\tau\) sufficiently small, the second inequality in (23) makes this upper bound power-smaller than \(\beta^{\sigma+\eta}\). This is again a contradiction.

The averaging in the nondispersed case uses the original targets inside each angular bin; thus its discarded mass estimate does not presuppose that a restricted bin retains its original weight. On a dense reverse fiber the prior centered angular estimates remain valid after normalization, with only a subpower loss. These facts justify both removals used above.

The improvement \(\eta\) can be chosen bounded below while \(\sigma\) stays a fixed distance below \(s_0\). Starting from any exponent below \(s_0/2\) and iterating finitely many times proves the assertion at every fixed exponent below \(s_0\). At each step the desired output gap is fixed first, then the old angular losses and the Furstenberg-theorem input losses are chosen. A finite union of the exceptional sets proves the simultaneous-grid assertion. ◻

Proof. On the full alignment the six coordinate forms are \[C,\quad C+R,\quad C+Y,\quad C+Y+R,\quad Y,\quad R.\] They are pairwise nonproportional and positive on a common vector. The peeling rule of Lemma 23, applied successively to extreme rays of the cone generated by these forms, makes each coordinate a singleton summand in a tangent. Hence the tangent is affine with the stated full partials.

We give the normal-velocity/time test in detail. Choose a block length \(\ell>0\) and a parent state \(z_0\) on \(A=B+Y\), with each of the other three slacks in (14) larger than \(\ell\). Keep \(C,D,R\) fixed. Writing \(\mathbf e_i\) for the coordinate vectors, compare these four refinements of \(z_0\): \[\begin{array}{c|c} \text{state}&\text{entropy increment from }z_0\\ \hline z_{\rm sp}=z_0+\ell(\mathbf e_B+\mathbf e_A)&\ell(n_1+g)\\ z_{\rm par}=z_0+\ell\mathbf e_Y&\ell t\\ z_{\rm joint}=z_0+\ell(\mathbf e_B+\mathbf e_A+\mathbf e_Y) &\ell(n_1+g+t)\\ z_{\rm out}=z_0+\ell(\mathbf e_A+\mathbf e_Y)&\ell(g+t). \end{array}\] The spatial and output refinements preserve \(A=B+Y\); the parameter and joint states lie on its admissible side. The chosen slacks make all four states admissible. Their increments are those of the affine tangent.

Inside a common parent cell, subtract the known bin centers and normalize by the parent units. Let \(v,w\) be the two skew-normal spatial entries in the starting parameter frame, and let \(a\) be the normalized time increment. The matched equality makes the position unit the product of the velocity and time units, so the output measurement is the scalar shear \(w+av\). The spatial state measures \((v,w)\) and the parameter state measures \(a\). Lemma 17 identifies their joint labels with \(z_{\rm joint}\) up to bounded ambiguity: the fine time label permits rebasing with the finely measured velocity. Center-dependent terms are known translations, and (14) bounds the remaining errors.

Lemmas 18 and 22 turn these increments into hereditary support and weight bounds. At every fixed block subresolution, the spatial law is planar Frostman \(n_1+g\), the coefficient law is Frostman \(t\), and their joint exponent is \(n_1+g+t\).

We make the comparison with the actual marginal laws explicit. Let \(\mu_1,\mu_2\) be the marginals of the same joint law, with support exponents \(\alpha,\beta\) and maximal joint-bin weight \(m^{-\alpha-\beta+o(1)}\). In any incidence of joint mass \(\rho=m^{-o(1)}\), discard marginal bins with reference weights below \(m^{-\alpha-o(1)}\) and \(m^{-\beta-o(1)}\), choosing the threshold slack to tend to zero sufficiently slowly. The support bounds make their marginal masses, and hence the joint mass incident to them, \(o(\rho)\). Since the paired labels encode the joint cells up to bounded multiplicity, at least \(\rho m^{\alpha+\beta-o(1)}\) pairs remain. Each has \(\mu_1\times\mu_2\) weight at least \(m^{-\alpha-\beta-o(1)}\). Their total product mass is therefore at least \(\rho m^{-o(1)}=m^{-o(1)}\). For the present labels take \(\alpha=n_1+g\) and \(\beta=t\). This proves dense incidence for the actual Frostman marginals, also after the required subpower joint restrictions.

Conditioning \(z_{\rm out}\) on \(z_{\rm par}\) leaves exponent \(g\), so the projections occupy at most \(m^{g+o(1)}\) bins for typical fine time labels, where \(m\) absorbs the block length. The parent and time-bin trims can be made negligible relative to this dense incidence. If \(n_1,t>0\), Lemma 29 gives \[g\ge\min\{n_1+g,(n_1+g+t)/2,1\}.\] If \(g<1\), the first and third entries of this minimum exceed \(g\), so \(g\ge n_1+t\). If \(g=1\), the desired inequality is automatic. This proves \(g\ge f_t(n_1)\); the case \(n_1=0\) is also automatic by the definition of \(f_t(0)\).

The other three tests use the following triples and shear maps: \[\begin{array}{c|c|c} \text{cutoffs}&\text{matched face}&\text{projection}\\ \hline (C,D,Y)&D=C+Y&z+ax\\ (C,B,R)&B=C+R&y-ax\\ (D,A,R)&A=D+R&w-az. \end{array}\] Here the unused cutoffs are fixed with strict slack, and \(a\) denotes the parameter increment in the corresponding normalized units. The same joint-count and conditional-support calculation proves the remaining assertions. The shear update (15) controls the labels at the unused cutoffs; it does not require the normalized spatial law to be independent of the coefficient law. Only dense product incidence is used.

The tangent is affine throughout its admissible domain. Placing a test on its matched face with slack at the other constraints leaves all six rates unchanged. The zero-input and saturated-output cases are automatic; in every other case the projection minimum gives the linear inequality that the output rate is at least the sum of the input and parameter rates. Thus each displaced test proves the assertion for the original constant rates. ◻

Proof. The predictability statement of Section 3 gives \(K(Y,R)=sY\) for \(R\le\lambda Y\). Apply the increasing-coordinate trace rule proved with Lemma 25 to the two conditional parameter coordinates. A generic tangent along their increasing tie is affine on each of the two ordering half-planes. Continuity along \(R=\lambda Y\) forces its upper-side expression to be \(sY+\theta(R-\lambda Y)\). Monotonicity in \(R\) and \(Y\) gives \(\theta\ge0\) and \(s-\lambda\theta\ge0\), respectively. This proves (25). Conditional parameter increments lower-bound the corresponding full-state increments, giving all the asserted rate bounds without an independence assumption.

On \(\Gamma\) the spatial forms are \[C,\qquad C+\lambda Y,\qquad C+Y, \qquad C+(1+\lambda)Y.\] For \(\lambda>0\), \(\lambda\ne1\), they are distinct, nonproportional forms, and the only remaining proportional group consists of the two parameters. Peeling separates the four spatial singletons, whose rates are constant throughout this tangent. At a generic full-alignment point on the lower side \(R<\lambda Y\), take a further realized tangent. It has the same four spatial rates, and the conditional-parameter increment bound gives its time rate at least \(s\). Proposition 31 therefore gives the time inequalities with subscript \(s\). At a generic full-alignment point on the upper side \(R>\lambda Y\), the same construction gives tilt rate at least \(\theta\) and the tilt inequalities with that subscript. The two further tangents may have different parameter partials, but both conclusions concern the original four constant spatial rates. They therefore hold simultaneously at the tie. ◻

Proof. The dense-strip alternative. Suppose first that a strip of width \(m^{-\chi}\) has dense mass. Its slope \(a=dr/dt\) satisfies \[m^{-o(1)}\le |a|\le m^{o(1)}.\] Indeed, if \(|a|\le m^{-\varepsilon}\) along a subsequence, its intersection with the bounded parameter region lies in a tilt interval of length \(O(m^{-\varepsilon}+m^{-\chi})\). The positive tilt Frostman exponent makes its mass power-small. If \(|a|\ge m^\varepsilon\), including the vertical case, use the time marginal instead. Since this applies to every fixed \(\varepsilon>0\), it proves the claim. Conditioning on the strip costs only a subpower factor in either marginal bound.

On the strip write \[r=at+b+e, \qquad |e|\le m^{-\chi}\sqrt{1+a^2}=m^{-\chi+o(1)}.\] Uniformly in its pins and targets, \[Q_f(d)=a(t_d-t_f)^2+O(m^{-\chi+o(1)}).\] The time Frostman bound permits removal of pairs with \(|t_d-t_f|<m^{-o(1)}\), with the cutoff decreasing sufficiently slowly that the lost mass is negligible compared with every dense incidence being retained. On each sign branch of \(t_d-t_f\) the quadratic has derivative of magnitude between \(m^{-o(1)}\) and \(m^{o(1)}\). At radii \(\rho\ge m^{-h}\), with fixed \(h<\chi\), the inverse image of a coefficient interval is therefore contained in at most two time intervals of length \(m^{o(1)}\rho\). The time marginal bound proves (i), also at all the intermediate radii needed for a projection test.

Proof. On \(\Gamma\) the coordinate forms are \(C\), \(C+Y\) twice, \(C+2Y\), and \(Y\) twice. Peeling separates the singleton groups \(C,A\) and the paired groups \((B,D)\) and \((Y,R)\); each group has an affine restriction to its diagonal. This gives the claimed form. Write \(J(B,D)\) for the separated middle-pair summand of this actual tangent. On lower-side full alignment, \(R<Y\) is equivalent to \(B<D\). Further full-alignment tangents there preserve \(p\) and have time rate at least \(s\), so the scalar time test gives \(\partial_DJ\ge f_s(p)\) almost everywhere in \(B<D\). For \(h>0\), monotonicity in \(B\) and integration of this \(D\)-bound give \[Hh=J(b+h,b+h)-J(b,b) \ge J(b,b+h)-J(b,b)\ge h f_s(p).\] The integration first holds on almost every vertical segment; continuity includes every such segment and its boundary endpoint. This proves the inequality for \(H\) without assigning separate \(B,D\) partials at the tie. If \(p=0\), both assertions in (31) are immediate. Assume from now on that \(p>0\).

Take a block in a typical fully aligned parent on the tie. In normalized units write the spatial entries in its starting frame as \(x,y,z,w\), and fine parameter pairs as \(d=(r_d,t_d)\) with law \(\nu\). The parameter diagonal rate is at least \(s\). Heredity makes its law Frostman of that diagonal dimension, also inside intermediate parameter cells at every later fixed subresolution. The time marginal is Frostman \(s\), and the tilt marginal has exponent at least \(\theta>0\), by (25) and the conditional-parameter lower bound. Thus the parameter laws satisfy the hypotheses of Lemma 33.

Denote the parent by \(z_0\) and the block length by \(\ell>0\). With \(\mathbf e_i\) denoting the coordinate vectors, set \[\begin{aligned} z_{\rm sp}&=z_0+\ell(\mathbf e_C+\mathbf e_B+\mathbf e_D+\mathbf e_A),\\ z_{\rm par}&=z_0+\ell(\mathbf e_Y+\mathbf e_R),\\ z_{\rm joint}&=z_{\rm par} +\ell(\mathbf e_C+\mathbf e_B+\mathbf e_D+\mathbf e_A). \end{aligned}\] The spatial increment of \(z_{\rm sp}\) has exponent \(p+H+g\). Group additivity gives the same exponent for \(z_{\rm joint}\) conditional on \(z_{\rm par}\). The marginal-weight comparison in the proof of Proposition 31, with spatial exponent \(p+H+g\) and the parameter diagonal exponent, therefore gives dense incidence under the product of the actual spatial and parameter marginals. Given the fine parameter label \(d\), write the spatial entries in its frame as \(x,y_d,z_d,w_d\). The two further states \[z_{\rm par}+\ell(\mathbf e_C+\mathbf e_A),\qquad z_{\rm par}+\ell(\mathbf e_B+\mathbf e_D+\mathbf e_A)\] have increments \(\ell(p+g)\) and \(\ell(H+g)\) from \(z_{\rm par}\). The remaining refinement from the first of these states to \(z_{\rm joint}\) costs \(\ell H\). All these states satisfy (14); in particular the fine parameters provide the slack for these partial spatial refinements.

Lemma 17 rebases the full spatial labels using the known fine parameters, with bounded ambiguity. Lemmas 18 and 22, applied to these state pairs at every required block subresolution, make the conditional \((x,w_d)\) law planar Frostman \(p+g\). Its additional \((y_d,z_d)\) fibers have support upper exponent \(H\), and the output triples \((y_d,z_d,w_d)\) have support upper exponent \(H+g\). We use these conditional laws before any overlap restriction.

We spell out how the dense spatial overlaps carry the weights needed for Lemma 29. This also ensures that subsequent choices of parameters do not replace a weighted projection problem by a mere cardinality assertion. Let \(m'\) be the base of the positive-length block presently used; it may be a fixed fractional block. In a typical parent, let \(S_f\) be the coarse-frame spatial bins incident to a fine parameter bin \(f\). Coarse-frame and fine-frame full spatial bins determine one another up to bounded coverings when \(f\) is given. The exact truncated parameters are used in subtracting center-dependent translations, so large absolute centers create no new error.

Put \(L=p+H+g\). The common spatial universe is the support of the spatial label at \(z_{\rm sp}\), of size at most \((m')^{L+o(1)}\). For almost every tested \(f\), \(S_f\) has size \((m')^{L+o(1)}\), and every retained joint cell has pre-trim conditional weight at least \((m')^{-L-o(1)}\) given \(f\). These conclusions follow from the support and maximal-weight bounds for \(z_{\rm joint}\) conditional on \(z_{\rm par}\), followed by discarding cells below the reciprocal-support threshold. The lower weights refer to this conditional law before that final discard. The exceptional fraction can be made negligible compared with any prescribed dense parameter restriction, by taking the threshold slack to zero sufficiently slowly. In particular the entire sets \(S_f\) remain available after such a restriction.

For any dense restriction of the parameter law, Cauchy–Schwarz in the common spatial universe gives \[\begin{align*} \iint |S_f\cap S_d|\,d\nu(f)d\nu(d) &=\sum_x\left(\int\mathbf1_{S_f}(x)\,d\nu(f)\right)^2\\ &\ge \frac{\left(\int |S_f|\,d\nu(f)\right)^2} {|\text{spatial universe}|} \ge (m')^{L-o(1)}. \end{align*}\] Here \(x\) in the sum denotes a full spatial cell, not only its first coordinate. Since the overlap upper bound is \((m')^{L+o(1)}\), a dense set of parameter pairs has overlaps of full exponent \(L\). Power-small exceptional parameter pairs can be removed before selecting these overlaps.

For one such pair \(f,d\), put \(\Delta r=r_d-r_f\) and \(\Delta t=t_d-t_f\). The shear update gives, to mesh accuracy, \[\begin{align*} y_d&=y_f-\Delta r\,x, &z_d&=z_f+\Delta t\,x, \\ w_d-\Delta t\,y_d+\Delta r\,z_d &=w_f+\Delta t\Delta r\,x. \tag{32}\end{align*}\] The last identity follows by substituting the first two into the full shear update, including its mixed term. Bounded bin errors produce only bounded mesh errors in normalized units.

There are at most \((m')^{p+g+o(1)}\) \((x,w_f)\) bins in the overlap. Discard those carrying fewer than \((m')^{H-o(1)}\) additional full cells, with the slack chosen sufficiently slowly. They account for a negligible fraction of an overlap of exponent \(L\). Call the remaining input bins rich. Inside one rich input bin, the different full cells map to different triples \((y_d,z_d,w_d)\) up to bounded multiplicity: once \(x\) is fixed to mesh accuracy, the frame change can be inverted from that triple. Thus a scalar value on the right of (32), attained in one rich input bin, uses at least \((m')^{H-o(1)}\) output triples. Scalar values separated by a sufficiently large mesh constant use disjoint triples, because the left side of that identity is determined to mesh accuracy by a triple. The total output support has size at most \((m')^{H+g+o(1)}\). Consequently the scalar projection \[(x,w_f)\longmapsto w_f+\Delta r\Delta t\,x\] of the rich subset has at most \((m')^{g+o(1)}\) bins.

These rich subsets are dense for the original conditional \((x,w_f)\) law. Indeed they retain \((m')^{L-o(1)}\) joint cells, each with pre-trim conditional weight at least \((m')^{-L-o(1)}\). Their total weight is therefore \((m')^{-o(1)}\), and their image consists of the corresponding coarse-accuracy input bins. The planar Frostman bound is the one before this final restriction, obtained from the matched \(C,A\) advances. The rich restriction costs only a subpower factor. This is the required passage from overlap counts to weighted point fibers.

We now obtain enough coefficient directions. If \(\nu\) has dense mass in a strip of width \(m^{-\chi}\) for some fixed \(\chi>0\), first restrict the parameters to this strip and repeat the overlap argument with the full spatial sets of the typical parameter bins. Use a fixed positive fractional block ending at depth \(h<\chi\) sufficiently small. Lemma 33(i) gives a Frostman \(s\) law of \(\Delta r\Delta t\) on dense pairs. The removed time-diagonal pairs have negligible mass compared with the dense overlaps. All spatial rate, fiber, and weight statements above apply at this fractional depth with \(m'=m^h\).

If there is no such strip restriction along the realizing sequence, pass to a subsequence on which every strip of each fixed power width has power-small mass. This dichotomy is obtained by inspecting the maximum strip mass on countably many fixed power grids; bounded mesh enlargements suffice. Lemma 33(ii) gives coefficient exponents arbitrarily close to \(s\) off power-small exceptional pairs. Remove these pairs on the finite fine grid of coefficient resolutions before choosing a pin with dense overlap.

In either case, averaging selects \(f\) with densely many targets \(d\) that have the coefficient Frostman bounds and the rich projected-input subsets just constructed. Parameter binning changes the coefficient only by mesh order. If several target bins produce the same coefficient bin, choose one corresponding rich input subset for that bin; each has the required dense mass for the same pre-trim conditional \((x,w_f)\) law. Thus the coefficient marginal together with these rich subsets supplies dense product incidence for Lemma 29. The map from a bounded scalar coefficient to the associated projection direction is bi-Lipschitz in a bounded chart.

Applying that lemma and letting all coefficient and restriction losses tend to zero yields \[g\ge\min\{p+g,(p+g+s)/2,1\}.\] Since \(p>0\), either \(g=1\), or the first and third terms exceed \(g\) and the middle term gives \(g\ge p+s\). In both cases \(g\ge f_s(p)\), completing the proof. ◻

Proof. Here \(0\le a=b\le1\), and we use the isotropic profile \(F(u,z,Y)\) of (20), on \(z\le u+Y\). The reference-axis label is retained with the exact entropy cancellation (19); the later choice of a strip axis will also retain its earlier parent. Suppose first that \(a>0\). Take a generic tangent on the calibrated line and apply Lemma 25, with \(u\) the only negative coordinate. Its pure profile is \(Pu\), where \(P\) is constant and \(\partial_uF\ge P\) throughout the domain. For \(T>0\), run backwards from \((0,0,0)\) to \((T,0,-T)\), keeping \(z=0\), with the frozen velocity cutoff in the upper envelope sufficiently fine for this segment. The remaining time rate is at least \(s\), so the support inequality and upper envelope give \[(2-q_*)T\le F(T,0,-T)\le(P-s)T.\] Consequently \[q_*\ge s+2-P, \qquad\text{and hence}\qquad P>1-s.\]

On the early side of the calibrated line, namely \(u+aY<0\), \(z=u+Y\), the support inequality and equality on the line imply an average bound \(\partial_uF+\partial_zF\le2\) when integrating toward the line at fixed \(Y\). Since \(\partial_uF\ge P>1-s\), there is a generic aligned point on this side with position rate \(E:=\partial_zF<1+s\). Lemma 23 gives an affine tangent in the three isotropic coordinates there; denote its time rate by \(\tau\ge s\). Simultaneously retain a tangent at the calibrated point with the same velocity depth \(u\), whose time depth is \(-u/a\). This is later than the early point and has finer position depth. The two sites have the same physical velocity units, and the later profile still satisfies \(\partial_uF\ge P\).

If \(a=0\), first peel off the constant coordinate on the line \(u=0,z=Y\). In a tangent, \[F(u,z,Y)=J(u)+H_0(z,Y),\] where the second factor is defined on the full projected \((z,Y)\)-plane. At a further generic point on the line, the two-coordinate order rule makes \(H_0\) affine on each order half-plane. On \(z\ge Y\), write \[H_0(z,Y)=Ez+\tau Y.\] Then \(\tau\ge s\) and \(E+\tau=q_*\), so \(E<1+s\). At \(Y=0,u=z>0\), the support inequality gives \[ J(u)\ge(2-E)u. \tag{34}\] The following projection block is based at the origin in this case. Choose any fixed point on \(u=0,z=Y\) with \(Y>0\) as the later site; the tests near it will use velocity depths near zero.

We give the common block argument. Write \(m\) for its base and let \(b_0\in\mathbb R^2\) be normalized velocity. Let \((d,c_0)\in\mathbb R\times\mathbb R^2\) be normalized time and position in the fine-time frame, after subtracting known translations. Position in the parent frame is therefore \[A_{b_0}(d,c_0):=c_0-d b_0\] to mesh accuracy. The profile and its regularization give the following properties inside typical parents.

1. Velocity has a Frostman ball bound of some exponent strictly greater than \(1-s\). For \(a>0\) this follows from \(\partial_uF\ge P\); for \(a=0\) it follows from (34) at every tested prefix depth.

2. Velocity cells and time-position cells have dense product incidence. Indeed the refinements \((u,z,Y)=(1,0,0)\), \((0,1,1)\), and \((1,1,1)\) have additive joint exponents. The time-position exponent is \(E+\tau\).

3. Given a fine velocity bin \(b_0\), the image under \(A_{b_0}\) has a Frostman \(E\) bound and at most \(m^{E+o(1)}\) bins. The fiber in one image bin has at most \(m^{\tau+o(1)}\) time-position cells. These follow by testing \(u=1\), \(0\le z\le1\), \(Y=0\), and the joint refinement, rebasing with the known velocity.

Suppose that every strip of power-small width has power-small velocity mass. By Lemma 30, applied with a slightly smaller ball exponent at most one, directions \(b-b_0\) have an angular Frostman exponent \(\sigma>1-s\) outside removable exceptional pairs. Product density and Cauchy–Schwarz give a pin \(b_0\) and a dense set of such \(b\)’s that share a dense set of time-position cells with \(b_0\). We may also arrange \(|b-b_0|\ge m^{-o(1)}\).

Fix one of these overlaps. Keep its rich \(A_b\)-bins, each containing at least \(m^{\tau-o(1)}\) cells. The support upper bound for \(A_b\) shows that these retain full overlap exponent \(E+\tau\). Inside a fixed \(A_b\)-bin the different cells give, up to bounded mesh multiplicity, different \(d\)-bins: once \(d\) and \(A_b\) are given, \(c_0=A_b+db\) is known to the tested accuracy. Moreover \[A_{b_0}=A_b+(b-b_0)d.\] Since \(|b-b_0|\ge m^{-o(1)}\), the rich bin therefore yields at least \(m^{\tau-o(1)}\) distinct \(A_{b_0}\)-bins in one thick row parallel to \(b-b_0\). Select perpendicular row labels separated by a sufficiently large mesh constant. Their collections of point bins are disjoint up to bounded multiplicity. There are at most \(m^{E+o(1)}\) point bins in all, so the perpendicular projection of the retained \(A_{b_0}\)’s has at most \(m^{E-\tau+o(1)}\) bins.

This restriction is dense for the Frostman image law, not merely large in cardinality. Indeed full overlap exponent \(E+\tau\), together with the fiber upper bound \(\tau\), gives \(E-o(1)\) image exponent. The typical image bins have pre-trim weights at least \(m^{-E-o(1)}\), by Lemma 18. Thus the image restriction has subpower mass. We can consequently apply Lemma 29 to these direction-dependent restrictions. It is contradicted by \[E-\tau<E,\qquad E-\tau<1,\qquad E-\tau<(E+\sigma)/2.\] The first inequality uses \(\tau\ge s>0\), the second uses \(E<1+s\), and the third follows from \(E-2\tau<1-s<\sigma\). If \(E-\tau<0\), the support upper bound itself is already impossible.

It follows that, in a dense family of parents, a dense amount of velocity mass lies in strips of width \(m^{-\chi}\) for some fixed \(\chi>0\), after subselection. Here the choice need not use one exceptionally rare parent. To see this, average the maximal conditional strip mass over the parents, using finite grids and bounded strip enlargements. If no fixed width power had subpower average along a subsequence, each such average would be power-small. Markov’s inequality and diagonal selection over countably many width powers would produce typical parents with all strips power-light, contrary to the block argument. We retain the selected strip mass across the parents, choosing its axis separately in each parent.

Choose a fixed neighborhood of the later site in which the time and position cutoffs remain finer than those of the earlier parent. Use the resulting parallel and normal coordinates there, and include the earlier parent label. At later isotropic states with positive velocity refinement, this label is recoverable up to bounded ambiguity. Both time and position are finer than at the earlier parent; rebasing position backwards multiplies velocity uncertainty by at most the earlier time resolution. The label includes the earlier time bin, spatial bins, and any fixed original slope label. Once it is known, the strip axis is fixed, and switching isotropic grids costs only bounded multiplicity. Thus the later isotropic profile and calibration are preserved under the dense restriction.

The conditional time rate is also preserved. Given the base label, the fixed original slope label, and a later time prefix, the earlier parent time and its spatial bins, hence its selected axis, are already determined. Conditioning on these extras therefore consumes no further time increment. The linear frame change in velocity is used for position in the matched units as well: position units are velocity units times the corresponding time unit. In the two-site construction only the time and position units change between the sites. This retains the scalar shear relation. Lemma 22 allows these tests to be realized simultaneously, with errors negligible in the final block units.

Now take mixed velocity depths \(l_1,l_2\in(0,\chi)\), with position depths \(l_i+Y\). Given the parent, the normal velocity rate is zero throughout this depth window. The parallel velocity rate is at least \(P\) if \(a>0\), and at least \(J'(l_1)\) almost everywhere if \(a=0\). For completeness, this lower bound is valid also at the aligned faces used in projection tests. Compare a short parallel increment with the same increment after making time and both position cutoffs finer, with the two position cutoffs equal and with strict slack. At the finer state change the normal velocity cutoff to the parallel cutoff at both endpoints; this costs zero while both depths remain in \((0,\chi)\). The resulting isotropic increment has the asserted lower rate. Dropping the extra conditioning can only increase that increment by submodularity. Valid interior comparisons pass to the aligned tangents by continuity.

Whenever this parallel rate exceeds \(1-s\), its aligned position rate equals one by the scalar projection test (24), now in the fixed axes. The distinct velocity, position, and time forms can be peeled at generic mixed aligned states, so these are full partial rates there.

For \(a>0\), start on the later calibrated path with equal velocity depths \(l_1=l_2=l\in(0,\chi)\). Choose the backward duration small enough to stay in the chosen neighborhood and to keep both depths in \((0,\chi)\). Hold the parallel depth \(l_1\) fixed and increase the normal depth \(l_2\) at speed one. Then the normal position depth \(l_2+Y\) is fixed, while the parallel position depth decreases at speed one. The normal velocity has zero rate, so time contributes at least \(s\) and parallel position contributes one. The advancing normal depth is the larger width depth and therefore has coefficient \(1-\kappa\) in the ellipse support term. Thus the profile minus the ellipse width term decreases backwards at rate at least \[ 2+s-\kappa. \tag{35}\] To integrate generic derivatives, perturb the isotropic starting depth and starting time slightly. The resulting segments sample all three mixed alignment parameters. Integrate on generic segments and pass back by Lipschitz continuity. Parent recoverability holds throughout because the starting velocity depths remain inside the strip window. The initial equality and support now contradict (33).

For \(a=0\), (34) supplies arbitrarily small positive \(l\) with \(J'(l)>1-s\), on a set of positive measure. Fix a positive comparison duration within the chosen later neighborhood and smaller than \(\chi/2\). For each sufficiently small such generic \(l\), set both initial depths to \(l\) and use generic later starting times. The error of the initial isotropic state against the calibrated value is \(O(l)\), and including the parent adds no error in exponent. Sending \(l\) to zero with the duration fixed again contradicts (33), since \(2+s-\kappa>1+2s-10\kappa\) for \(s\le1\). ◻

Proof. Apply Lemma 25 separately along the path under consideration. The negative coordinate is \(X\); \(y,Z\) strictly increase. Slack is obtained by shifting \(y\) more than \(Z\). At almost every path point the resulting tangent has pure slope \(p_0\), and \(d_X\ge p_0\). If \(p_0>u_*/2\), velocity zero gives rate at least \(t_0+h\) by (37).

If \(p_0\le u_*/2\), use velocity one. In centered offsets its ray is \(X=-y,Z=0,y<0\). If \(m\in[\rho,1-\rho]\) is the original path speed, choose \(\beta\in(y/m,y)\). Then \[-y>-m\beta,\qquad y>\beta,\qquad0>(1-m)\beta.\] These inequalities give the overlap condition of Lemma 25, locally up to alignment. Thus \(d_X=p_0\) on the comparison ray, and integration gives rate at least \[t_0+k-p_0\ge t_0+1-\kappa-u_*/2=t_0+h-\kappa.\] Perturb velocities zero and one into the permitted interval. The loss is arbitrarily small by the Lipschitz bounds. The strict comparison in the tangent gives arbitrarily short comparisons in the original profile. Any additional linear contribution to the gain is included in these same endpoint calculations. ◻

Proof. For the low comparison use the scalar groups \[X=B,\qquad y=Y,\qquad Z=A=R+d_0'.\] For the upper envelope (21), the remaining function is \[F^+(Q)=F(n(-H_0),Q)-G(n(-H_0)).\] Its rates are those of the actual frozen-cutoff slice \(Q\mapsto F(n(-H_0),Q)\), since the subtracted term is constant. Choose \(H_0\) sufficiently large for all the following compact tests; in particular \(C\) is fixed sufficiently fine in this slice. Any additional negative \(D\) is held fixed in the pure part; in the remaining part it is either \(d_0'\) or fixed finer, with \(D<C+Y\). In the exact-slope version \(F-PC\) is locally independent of \(C\). Freeze \(C\) at a slightly finer value to obtain strict slack for \(B<C+R\) and \(D<C+Y\); on alignment this auxiliary slice agrees with the remainder up to a constant.

To verify (37), retain \(R,A\) separately with \(A\le D+R\), \(A\le B+Y\), while the other faces have slack. At a generic scalar aligned point in any path tangent, peel off the singleton rates for \(B,Y\); the \((R,A)\)-summand has affine diagonal rate. Increase \(R\) separately to open \(A<D+R\), retaining \(A=B+Y\). The time-projection block for \((B,Y,A)\) gives \(g\ge f_s(n_1)\). Its fixed \(D\)-measurement rebases under time refinement with bounded ambiguity because \(C\) is fine. The two marginal laws and their dense product relation are the comparable-cutoff laws of Lemma 18; thus this is the actual sheared block test. If \(d_X=n_1>u_*/2\), it follows that \(g\ge h\). The direction increasing \(B,R,A\) together preserves \(A=B+Y\) and has rate \(n_1+r+g\ge n_1+h\). Integrate this inequality on the displaced matching slices and pass back by continuity to the \((R,A)\)-grouped slice. There the same direction has rate \(n_1+d_Z\). The peeled singleton \(n_1\) is constant on both slices, so subtraction gives \(d_Z\ge h\). The conditional parameter increment with \(C,D\) fixed gives \(d_y\ge s\).

These constructions also apply after the signed-path tangent: take simultaneous tangents of the indicated auxiliary slices, which have actual realizing states. When freezing \(C\) is only local, cover the open path portion by countably many neighborhoods with a single valid frozen value on each. Lemma 36 with \(t_0=s\) now gives total rate at least \[s+h-\kappa+k_C-P\ge1+2s-\kappa.\]

For the high comparison the scalar groups are \[X=B=R+c_0',\qquad y=Y=D-c_0',\qquad Z=A.\] At full-aligned generic states, \(e\ge f_s(p)\ge h\). Hence the \(y\)-group has rate at least \(s+h\). If \(g<h\), the projection inequalities imply \(n_1\le u_*/2\). They also force \(r=0\): otherwise \(g\ge f_r(e)\ge h\), a contradiction.

We justify transfer to the grouped slice, including its path tangents. At a generic scalar aligned point, first peel off the fixed coordinate \(C\) by a direction increasing every other coordinate; then separate the \(X,y,Z\) groups. Their distinct forms are positive on a common vector. The \(A\)-rate is therefore a constant singleton in this first tangent. Test full-aligned generic points on nearby translated sheets, allowing \(C\) to vary. The aligned \(y\)-direction also advances \(A\); subtracting its constant rate transfers \(d_y=t+e\ge s+h\). If this singleton rate is less than \(h\), the aligned \(X\)-direction, with the same subtraction, transfers \(d_X=n_1+r\le u_*/2\). Thus (37) holds with \(t_0=s+h\), and Lemma 36 gives \(s+2h-\kappa=1+2s-\kappa\). ◻

Proof. Take the conditional parameter tangent (25) and a further simultaneous signed-envelope tangent on the calibrated line. The negative coordinates are \(C\), together with \(B\) if \(b>0\) and \(D\) if \(a>1\). Their pure domain is full: the parameters \(R,Y\) in mixed constraints increase, and negative cutoffs at an earlier path time can be paired with the other cutoffs at a later time. Large positive shifts in \(R,Y\) open the required slack. On compact aligned tests all negatives can be replaced by \(n(-H_0)\) in (21) for sufficiently large \(H_0\). When relevant, the differences \(C-B\) and \(C-D\) of the shifted negatives increase strictly, since \(a>b\) and \(a>a-1\).

Let \(P\in[0,1]\) be the pure marginal rate of \(C\). If \(P=0\), test the upper envelope backwards for unit time with \(v_1=M_0,v_2=0\), where \(M_0\) is large and fixed. The \(C\)-cost is zero, \(B\) is constant, and \(Y,A,R\) decrease backwards. The only possible positive backward cost is that of \(D\), at most \(M_0-1\), whether it belongs to the pure or remaining profile. The gain is at least \[k_CM_0-(M_0-1)=1+\kappa M_0,\] contradicting support for large \(M_0\). Comparisons along binding faces follow by translating the segments into the relative interior first. Henceforth \(P>0\).

We divide the remaining proof into three cases. When \(\theta=0\), we use longitudinal comparisons, upper-envelope tests, and full-alignment projection bounds. When \(\theta>0\) and \(\lambda\ne1\), we use the projection inequalities on the tied plane and then separate the possible velocities. When \(\theta>0\) and \(\lambda=1\), the coincident spatial forms require the grouped projection bounds.

Proof. Use the paths (45) from a small generic \(C=B=L,Y=0\). In the strict causal interior the constant full \(C\)-slope and \(t\ge s\) supply the low longitudinal comparison. At generic boundary points, the tie inequalities give \(g\ge s+P\), and the spatial singleton rates in the first tangent are constant. The backward direction \(v_1=1,v_2=0\) enters the strict causal side and has gain at least \(s+k_C-P+g\ge1+2s+\kappa\). Perturbing to \(v_2=\rho<1-\lambda\) retains this strict gain. The generic-\(L\) argument in Proposition 38 gives these steps almost everywhere along every admissible path, including its boundary epochs. Lemma 27 therefore gives a fixed-duration endpoint comparison, with arbitrarily small additional loss.

Translate both endpoints by the same aligned depth shift to move the start from \(L\) to zero. This changes the gain by \(O(L)\); it does not alter \(R,Y\), so it preserves the half-space. Choose \(L\) and the further clipping and control losses so that, together with the \(\kappa\)-loss in the low longitudinal comparison, the total loss per unit time is less than \(4\kappa\). The straight endpoint displacement lies in the convex velocity box and respects the half-space. It consequently certifies a short step upon passing back from a tangent. ◻

Proof. Here \(b=-\lambda\). Take (25) and peel off the constant coordinate \(C\) in a generic calibrated tangent; all other coordinates strictly increase. Its additive contribution is a function of \(C\) alone. Use aligned backward paths through zero with \[\rho\le v_1\le M_0,\qquad -(\lambda+1)\le v_2\le1,\] where \(M_0\) is fixed sufficiently large relative to \(\lambda\) and \(1/\kappa\). If \(\theta>0\), impose \(R\le\lambda Y\). Along every such path, the \(C\)-summand is differentiable almost everywhere because \(C\) moves monotonically with speed at least \(\rho\). In a tangent there its full slope \(p\) is constant. We verify the control property in this tangent.

If \(p=0\), take \(v_1=M_0,v_2=0\). After removing \(C\), the only backward cost is at most \(M_0-1\), from \(D\), and the gain is at least \(1+\kappa M_0\). Taking \(M_0\ge\lambda\) also preserves the causal half-space at its boundary. Suppose \(p>0\). If \(\theta=0\), or at a strictly causal state, \(t\ge s\) throughout a neighborhood and its tangents. Use the low longitudinal comparison for \(p\le u_*/2\) and the high comparison for \(p>u_*/2\), perturbing \(v_1=0\) to \(\rho\) in the latter.

It remains to consider boundary states with \(\theta>0\). The path point need not be generic on \(\Gamma\). Its tangent nevertheless has constant full \(C\)-slope \(p\). Each of the first three cases below chooses a fixed tie direction using only \(p,\lambda\). Its gain bound holds at generic points of the tangent’s tie; integrate on nearby generic parallel tie segments and pass by continuity to the segment from the tangent origin. This gives the endpoint comparison required by Lemma 27. The fourth case supplies such a comparison directly.

1. For \(\lambda=1\), use (31). If \(0<p\le u_*\), the tie test \(v_1=1,v_2=0\) has gain at least \(s+k_C-p+(s+p)=1+2s+\kappa\). If \(p>u_*\), the test \(v_1=0,v_2=-1\) has gain at least \(s+1+2-k_B\). Perturb along the tie; in the second test use \(v_1=\rho,v_2=\rho-1\).

2. For \(\lambda\ne1\), \(p>u_*/2\), the spatial inequalities (40) give \(e,g\ge h\) and \(K_N\ge K_L\ge1\). Formula (41) with \(m=0\) gives at least \(s+2h=1+2s\). A small tie perturbation makes \(v_1\ge\rho\).

3. For \(\lambda>1\), \(0<p\le u_*/2\), take \(m=(\lambda+1)/2\) on the tie. Formula (41) gives \[\mathcal D_\Gamma(m)\ge s+1-p+g+\frac{\lambda-1}{2}(K_N-K_L)+\kappa\lambda.\] Here \(g\ge p+s\), since \(p+s\le h\le1\), and the difference term is nonnegative. The rate is at least \(1+2s\), and the velocities fit the chosen box.

4. For \(0<\lambda<1\), \(0<p\le u_*/2\), use Lemma 39, with its constant full \(C\)-slope \(P=p\). This enters or follows the causal region and gives a strict endpoint comparison in the tangent.

All these comparisons, including any secondary control construction inside a tangent, retain a fixed positive margin over \(q_*\) after the chosen losses. Their endpoint displacements lie in the convex velocity box and respect the constant half-space constraint. Lemma 27 therefore contradicts (17) for the original paths. ◻

Proof. Write \(c=b-a=-\lambda>0\). Then \(0\le a<b\le1\). We have \(t\ge s\) everywhere. The negative coordinates on the calibrated line are \(B,R\), and \(C\) when \(a>0\). Take the signed-envelope tangent with the slack shifts described in Lemma 25. In this case it is essential to retain the additional prediction (18).

First suppose \(a>0\). The pure domain is \(B\le C+R\). Let \(P\) be the constant pure \(C\)-partial on \(C/a>\max\{B/b,R/c\}\), and let \(S\) be the constant pure \(R\)-partial on \(R/c>\max\{C/a,B/b\}\). Lemma 25 supplies these constants and makes each a global upper bound for its corresponding pure partial. On the aligned sector in which \(C\) is ahead, \(C/a>R/c\) and \(B=C+R\), the pure profile has the form \[ G(C,B,R)=PC+Q_2B \tag{46}\] after referencing at zero, with constant \(Q_2\ge0\). To prove this identity, note first that \(C/a>B/b\) as well, since \(B/b=(a/b)(C/a)+(c/b)(R/c)\). The pure \(C\)-rate is consequently \(P\). The pure \(R\)-rate is zero there. Indeed in the external slices used to construct the pure profile, (18) gives flatness in \(R\) when \(C/a,B/b>R/c\); the other coordinates are fixed strictly finer. This flatness holds on an open order region of the slice tangents and passes to \(G\).

Thus varying \(C\) from the order tie at fixed \(B\), with \(R=B-C\), has slope \(P\). On the tie, \(C=(a/b)B\), the pure total is affine. Integrating from it proves (46). Nonnegativity of the \(B\)-increment gives \(Q_2\ge0\). The same \(B\)-rate holds locally in the interior: the \(C,R\) partials there are constant, and one can approach the aligned boundary keeping \(B\) fixed. Thus the partials of \(G\) near this aligned sector are \((P,Q_2,0)\) in the \((C,B,R)\) coordinates. The signed-envelope lower comparison gives \(p\ge P\) and \(n_1\ge Q_2\) for the actual profile.

The shift to \(n(-H_0)\) in (21) is valid on compact aligned tests because \(a,b,c>0\). If \(P=0\), use \(v_1=M_0,v_2=0\) backwards from zero. The target pure tuple is \(C=M_0,B=0,R=-M_0\), in the \(C\)-ahead sector, so (46) gives zero pure cost. The remaining cost is at most \(M_0-1\), from \(D\); the other varying remaining coordinates decrease backwards. Thus the gain is at least \(1+\kappa M_0\), contradicting support for large fixed \(M_0\).

If \(P+Q_2\le u_*\), use \(v_1=v_2=1\). Its pure tuple \(C=B=1,R=0\) is again in the \(C\)-ahead sector and costs \(P+Q_2\). The coordinates \(D,A\) are fixed, while the remaining time rate is at least \(s\). The gain is therefore at least \(2+s-(P+Q_2)\ge1+2s\).

If \(P,Q_2>0\) and \(P+Q_2>u_*\), use the sheet (38) with \(y<\alpha\), and the comparison \(v_1=v_2=0\). At generic sheet points the same peeling separates \(e,g,t\): the free \(y\)-direction first splits off \(C,B,R\), after which the three remaining forms are distinct. The first perturbation in (39) enters the \(C\)-ahead sector while preserving full alignment. Formula (46), the pure lower bounds, and the full-aligned projections give simultaneously \[e\ge f_s(P),\qquad g\ge f_s(Q_2),\qquad t\ge s.\] As above, the two positive input rates with sum greater than \(1-s\) force \(e+g\ge1+s\). Integrate on generic parallel segments and pass to \(\alpha=0\), obtaining gain at least \(1+2s\).

The only remaining possibility with \(a>0\) is \(P>u_*\), \(Q_2=0\). On the same sheet the \(C\)-ahead perturbation gives \(e\ge1\). If \(S>0\), the second perturbation in (39) enters the \(R\)-ahead sector \(R/c>C/a\), still with \(B=C+R\). There the actual tilt rate is at least \(S\). Since \(e\) is a constant singleton in the first tangent, the tilt projection inequality \(g\ge f_r(e)\) gives \(g\ge1\). Transfer this to the sheet and use \(v_1=v_2=0\), whose gain is at least \(s+2\ge1+2s\).

If \(S=0\), the global upper bound for the pure \(R\)-partial makes \(G\) independent of \(R\). It therefore projects to an ordinary submodular profile on full \((C,B)\)-space: any \((C,B)\) is feasible after taking \(R\) sufficiently fine, and the value does not depend on that choice. If necessary take a further simultaneous generic tangent on the calibrated line so that this two-variable pure profile has ordinary order slopes, by the full-domain order rule of Lemma 25. Its lower comparisons with the actual profile and (46) are preserved. If the resulting \(B\) marginal is positive, the second perturbation in (39) enters its \(B\)-ahead order, which is the \(R\)-ahead order on alignment. The normal time projection gives \(g\ge s\), while the singleton \(e\ge1\) persists. The same zero-velocity test then has gain at least \(1+2s\). If the \(B\)-marginal vanishes, \(G\) depends on \(C\) alone and has global slope \(P\), by (46). Thus the actual profile satisfies \(p\ge P>u_*/2\), and the high longitudinal comparison contradicts calibration.

It remains to treat \(a=0\), so \(c=b>0\). Put the static coordinate \(C\) in the nonnegative group. The external slices now have strict mixed slack also for \(B\le C+R\), and the pure domain in \(B,R\) is full. Prediction (18) gives zero pure \(R\)-rate on \(B>R\). The two-variable pure order rule therefore gives \[ G(B,R)=Q_2 B\qquad(B\ge R), \tag{47}\] with \(Q_2\ge0\); the equality at \(B=R\) follows by continuity. In particular this applies on alignment when \(C\ge0\). Both negative coordinates can be shifted to equal finer tied cutoffs in the upper envelope; the other mixed constraints have large slack after that shift.

Suppose \(Q_2\le u_*/2\). Test the upper envelope backwards from zero with \(v_2=1\) and \(v_1\in[\rho,1-\rho]\). At interior negative times \(C>0\), while \(A\) is fixed. The pure backward cost per unit time is \(Q_2\), by (47). With \(B,R\) frozen at their finer tied shift, the remaining scalar coordinates are \[X=C,\qquad y=Y,\qquad Z=D,\] with \(A\) fixed. Because \(C>0\), the first shear constraint has slack; the only binding local constraint is \(D\le C+Y\). The direct longitudinal velocity-time projection gives \(e\ge f_s(p)\), and the time rate is at least \(s\). The fixed normal measurements rebase with bounded ambiguity by (14), with \(B,R\) fine. Thus (37) holds with \(t_0=s\), also in the path tangents. The scalar comparison gives \(s+h-\kappa\); the additional normal contribution is \[k_B-Q_2\ge1-\kappa-u_*/2=h-\kappa.\] The total is at least \(1+2s-2\kappa\), before an arbitrarily small clipping and control loss, contradicting (33).

Finally suppose \(Q_2>u_*/2\). Use actual paths backwards from zero with \(v_2=0\) and \(v_1=m\in[\rho,1-\rho]\). At negative times \(C>0\), so the pure bound gives \(n_1\ge Q_2\) locally and therefore \(g\ge h\) at generic aligned states and in their inherited tangents. Along each path, \(C\) alone decreases; \(B\) is static, while \(R,Y,D,A\) strictly increase. Apply the signed rule at almost every point of that path, with pure \(C\)-slope \(p_0\). The required overlap and slack hold: increase \(R,Y\) substantially more than the shifts of the other nonnegative coordinates.

If \(p_0>u_*/2\), use \(v_1=v_2=0\) in this tangent. The pure and projection bounds give \(e,g\ge h\), and \(t\ge s\), yielding \(1+2s\). If \(p_0\le u_*/2\), use \(v_1=1,v_2=0\). We verify exactness of \(p=p_0\) where the gain is transferred. The centered comparison ray has \[C=-y,\qquad B=0,\qquad R=y,\qquad y<0.\] Perturb to \(B>0\), \(R=B-C\), with \(C\) near \(-y\). Away from \(y=0\), choose \[\beta\in(-C/m,(B-C)/m),\qquad \beta<y.\] This is possible for sufficiently small positive \(B\), since at \(B=0,C=-y\) both endpoints tend to \(y/m<y\). Then \(C>-m\beta\), and the remaining cutoffs are strictly finer than their path values \[0,\quad m\beta,\quad\beta,\quad(1-m)\beta,\quad\beta\] for \(B,R,Y,D,A\), respectively. In particular \(R=B-C>m\beta\); \(D=C+y\) is near zero and exceeds \((1-m)\beta<0\); and \(A=B+y>\beta\). Overlap therefore gives exact \(p=p_0\) at generic nearby aligned states. Integrate on these perturbed segments and pass to the ray. Discarding the nonnegative \(r\)-term gives gain at least \[s+k_C-p_0+h\ge1+2s+\kappa.\] Clip endpoint velocities into the prescribed box and use Lemma 27. This proves the contradiction in the last decreasing-aspect case. ◻

Completion of the proof of Theorem 5. If the claimed exponent bound failed, the calibrated construction would give the profile used at the start of this section. Proposition 35 excludes \(\lambda=0\). Propositions 38 and 40 exclude \(\lambda>0\), and Proposition 41 excludes \(\lambda<0\). These exhaust \(a\ge0,b\le1,\lambda=a-b\).

Each comparison is in a fixed tangent problem. Fix the finite parameters and any large auxiliary velocity bound first. For an upper-envelope comparison, then choose \(H_0\) sufficiently large for the resulting compact test region. The admissible depth perturbations and velocity clipping parameters may subsequently depend on these choices. The scalar losses are at most \(2\kappa\) in the uses above, and the local constrained comparison reserves less than \(4\kappa\). Choose the additional endpoint, clipping, and control losses so that, together with the scalar or local constrained loss already incurred, the total in the selected comparison is strictly below \(10\kappa\). These are alternative comparisons, so their losses are not summed over the mutually exclusive cases. Static coordinates at \(a=1\) or \(b=1\) belong to the nonnegative group, as permitted in Lemma 25; no division by a zero negative speed is used. When \(s=1\), the branches requiring \(0<P\le u_*/2\) are empty, and the zero-rate and high-rate comparisons cover the remaining possibilities. The finite realizing grids and large-scale limit are chosen after these finite tests, as in Lemma 22. The strict inequality (33) is therefore impossible, proving (4). ◻

Proof. The identity \(N^2w\theta=1\) shows that, as \(z\) varies over its grid, \(N^2z\) varies over \((L_{\mathrm f}\theta)^{-1}\mathbb Z^2\). Thus the analysis entries for one frequency window are the unnormalized Fourier coefficients on a square of side \(L_{\mathrm f}\theta\). Parseval gives \[\sum_z|(\mathsf P_{m,I}f)_{\nu,z}|^2 =(L_{\mathrm f}\theta)^2\|f\chi_\theta^\nu\|_2^2.\] Sum over \(\nu\). Fourier inversion, with its factor \((L_{\mathrm f}\theta)^{-2}\), gives \(\mathsf S_{m,I}\mathsf P_{m,I}f =f\sum_\nu(\chi_\theta^\nu)^2=f\). Finally, \(\mathsf S_{m,I}\) is \((L_{\mathrm f}\theta)^{-2}\) times the adjoint of \(\mathsf P_{m,I}\). The operator norm statements follow. ◻

Proof. A translated disk of radius \(\theta\) can include any chosen entry in both tests in (51). Since its biased area is \(\theta^2\), we have \[K(c)\ge\|c\|_\infty^2, \qquad G(c)^3\ge w^2\|c\|_2^2\|c\|_\infty \ge w^2\sum_v|c_v|^3.\] The triangle inequality in \(\ell^3\) transfers this lower bound to every atomic decomposition and hence to \(A\). A coordinate atom has cost \(w^{2/3}|c_v|\), so \(A\) is finite and positive away from zero. Homogeneity, the triangle inequality, and absolute monotonicity follow directly from its definition. The weighted \(\ell^3\) sum in (52) proves the corresponding statements for \(\mathcal A\) and gives (53).

Atoms in a decomposition can first be restricted to \(\mathop{\mathrm{supp}}d\), since \(G\) is monotone. If \(|d|\le\sum_j|a_j|\), choose, at each coordinate, complex numbers \(d_j\) with the phase of \(d\), with \(\sum_jd_j=d\) and \(|d_j|\le|a_j|\). For a linear map \(T\) and an output norm \(\mathcal B\), a bound \(\mathcal B(Tb)\le C G(b)\) therefore gives \[\mathcal B(Td)\le\sum_j\mathcal B(Td_j) \le C\sum_jG(a_j).\] Taking the infimum proves the last assertion. In particular, support restrictions imposed on the input are inherited by all atoms. ◻

Proof. Consider a parent bin \(I'\) at factor \(m'\), of length \(r'=(m')^{-2}\) and left endpoint \(t_0\). Make the change \[t=t_0+r'\widetilde t,\qquad z=r'\widetilde z,\qquad N'=N/m'.\] The factor \(e^{2\pi iN^2t_0\phi}\) is removed by rebasing the data. The phase is again \(N'^2(\widetilde x\cdot\xi+ \widetilde t\phi(\xi))\). At a descendant factor \(m''\), the normalized factor is \(\widetilde m=m''/m'\). Its frequency width is unchanged, whereas its spatial width and time length become \[\widetilde w=w/r',\qquad \widetilde r=r/r'.\] The spatial grids consequently become exactly the grids prescribed above. The capacity is unchanged: the condition \(z_v\in z+rE\) becomes \(\widetilde z_v\in\widetilde z+\widetilde rE\). Thus \[ \widetilde K=K,\qquad \widetilde G^3=(r')^{-2}G^3,\qquad \widetilde A^3=(r')^{-2}A^3. \tag{57}\] The last identity follows by applying the same common scalar factor to every atomic cost.

An estimate with normalized cubed norm constant \(C_{m''/m'}\) becomes \[\frac{r''}{r'}\sum_{I''\subset I'}A(c_{I''})^3 \le C_{m''/m'}A(d_{I'})^3.\] Multiply by \(r'\) and sum over parent bins. This gives the desired global cubed bound at that transition. Iterate it, using mask contraction at every stage. Taking logarithms in base \(m\) gives the stated weighted sum of exponents. In particular, a fixed decrease in \(\beta_j\) decreases the full exponent by that amount times the positive logarithmic length \(a_{j+1}-a_j\). At each such fixed split, all normalized position and frequency ranges still have fixed polynomial complexity in that step’s scale ratio. The uniformity in Definition 44 therefore applies simultaneously to all bins. Finally, the last assertion follows from the support-preserving atomic decomposition in Lemma 43. ◻

Proof. The matrix entry is \[(L_{\mathrm f}\theta_-)^{-2} \int\chi_{\theta_-}^{\nu_v}(\xi) \chi_{\theta_+}^{\nu_b}(\xi) e^{2\pi iN^2((z_b-z_v)\cdot\xi+ (t_{I_b}-t_{I_v})\phi(\xi))}\,d\xi.\] Disjoint frequency supports give the angular assertion. On overlap, rescale \(\xi=\nu_v+\theta_-\zeta\). The amplitude and all its derivatives are uniformly bounded, since \(\theta_-\le\theta_+\). After the constant and linear phase terms are removed, the phase derivatives are bounded because \[N^2\theta_-^2(t_{I_b}-t_{I_v}) =\frac{t_{I_b}-t_{I_v}}{r_-}\in[0,1).\] The remaining linear frequency is \((z_b-z_v-(t_{I_b}-t_{I_v})U_{\nu_v})/w_-\). Repeated integration by parts proves (58).

For finite retained arrays, sum the decay bound over the polynomial number of entries counted after Definition 44. An inserted frame identity instead uses the full position lattice; its infinite tail is controlled by summing the same decay over lattice annuli. Fix the number of transitions, the admissible complexity, and the positive beam-radius power first. Then enlarge the intermediate position ranges by fixed polynomial powers, and choose the integration-by-parts order \(L\) for the prescribed final error. Frequency labels spread by only a bounded amount at each transition, so a fixed number of insertions and truncations has negligible total error. The polynomial norm comparisons transfer this statement to all the norms in use. This estimates the error of a finite approximation to an exact matrix product; it does not replace a transition by an arbitrarily truncated set of edges. ◻

Proof. The mass estimate is Lemma 42, since \(w=\delta/m\). For the capacity estimate, fix an output test ellipse \(E\), with both radii at least \(\theta=m\delta\), and choose an arbitrarily small fixed \(\eta>0\). By Lemma 47, the input entries contributing to this test, apart from negligible error, have velocities in a translate of \(C m^\eta E\). Their initial positions lie in another translate of the same enlarged ellipse. Indeed, a contributing position equals its child position minus elapsed time times its velocity, up to \(m^\eta\delta\); the elapsed time is at most one and both radii of \(E\) are at least \(\delta\). Both translations are allowed in the definition of \(K(d)\).

Restrict the input to this one product of two translated tests. Its squared sum is at most \[\delta^{-2}K(d)W(Cm^\eta E) \le C m^{2\eta}\delta^{-2}K(d)W(E).\] Apply the global squared operator norm bound \(m^2\) to this restricted input, and then multiply by the output factor \(\theta^2/W(E)\). Since \(\theta^2=m^2\delta^2\), the result is \(C m^{4+2\eta}K(d)\), with negligible error. The estimate is uniform over translations, orientations, and eccentricities. Tail errors are uniform too: \(\theta^2/W(E)\le1\), and \(K(d)\ge\|d\|_\infty^2\). Let \(\eta\) be arbitrarily small to obtain the stated exponent bound.

Finally, \(A(c_I)\le G(c_I)\). Multiply the mass bound by the uniform upper bound for \(K(c_I)^{1/2}\) and sum in time with weight \(r\). There are \(r^{-1}\) bins, giving \(\mathcal A(c)^3\le m^{2+o(1)}G(d)^3\). ◻

Proof. The grids give at most \(C R^2\) packets. Write \(\mathcal C_R=R^2\sum_T|d_T|^4\). For \(|t|\le3R\), integration by parts in the normalized cap gives, for every fixed \(L\), \[|F_T(X,t)| \le C_L|d_T| \left(1+\frac{|X-z_T+t\nabla\Phi(\xi_T)|}{\sqrt R}\right)^{-L}.\] Outside the counted transverse radius this gains \(R^{-L\delta_0}\). The polynomial packet count can therefore be included before choosing \(L\) to obtain any prescribed power of decay. Choose a fixed band-limited Schwartz function \(\psi\), bounded away from zero on \([-4,4]\), and put \(P_T=\psi(t/R)F_T\). The spatial \(L^\infty\) bound is \(C|d_T|\), and Plancherel gives \(\|F_T(\cdot,t)\|_2^2\le C|d_T|^2R\). Hence \[\|P_T\|_4^4\asymp |d_T|^4R^2.\] For the lower bound, use a short interval \(|t|\le cR\) and a ball of radius \(c\sqrt R\) about the core. The normalized integral is bounded below there because \(a\) is fixed, nonnegative and has positive integral. Thus no tail estimate is normalized by the possibly small norm of an arbitrary oscillatory profile.

Let \(D(R)\) be the best constant in \[\Bigl\|\sum_TF_T\Bigr\|_{L^4(Y)}^4 \le D(R)k\mathcal C_R\] over the fixed classes just specified. For the induction, strengthen the allowed time slab slightly to \(|t|\le(2+R^{-1/16})R\); keep the counted tubes in \(|t|\le3R\). The same core bounds apply. This harmless buffer accommodates cells meeting the endpoints after rescaling: the added length is \(O(R^{3/4}+R^{1/2+\rho})\), and, at \(r=\sqrt R\), \[R^{-1/16}+O(R^{-1/4}+R^{-1/2+\rho})<r^{-1/16}\] for sufficiently large \(R\), once \(\rho<1/16\). For bounded \(R\), the packet count, the preceding cost bound and Hölder’s inequality give a finite bound for \(D(R)\). The zero array is immediate, so assume \(\mathcal C_R>0\).

Partition the frequency centers into caps \(\tau\) of diameter \(R^{-1/4}\). For each \(\tau\), partition the time-zero position centers into squares of side \(R^{3/4}\). Assign each packet once to its frequency cap and position square; denote the resulting disjoint packet groups by \(\mathcal W_l\), and set \(P_l=\sum_{T\in\mathcal W_l}P_T\). The centerlines in a group remain in a fixed enlargement of the corresponding coarse cylinder, of radius \(R^{3/4}\) and length \(O(R)\). At any \(\sqrt R\)-cube, only boundedly many such cylinders per coarse cap are nearby. Far cylinders have separation comparable to \(R^{3/4}\) from their \(\sqrt R\) cores and contribute arbitrarily small relative errors. No packet is multiplied by a coarse spatial cutoff. The fine cap supports lie in fixed enlargements of their assigned coarse caps. These enlargements have bounded overlap; a fixed coloring separates them before ordinary decoupling, at a constant cost.

Here is the exact normalization of a group. If \(c_\tau\) is its frequency center and \(z_l\) its position-square center, put \[\xi=c_\tau+R^{-1/4}\zeta,\qquad X'=R^{-1/4}\bigl(X+t\nabla\Phi(c_\tau)-z_l\bigr),\qquad t'=R^{-1/2}t,\qquad r=\sqrt R.\] After removing a common unimodular factor and absorbing each packet’s constant phase into its coefficient, the packets have the same form at scale \(r\), with the same profile \(a\) and \(|d'_T|=|d_T|\). Indeed \(R\,d\xi=r\,d\zeta\), their frequency and position meshes become \(r^{-1/2}\) and \(\sqrt r\), and the new position centers have size \(O(r)\). The new phase has derivatives \(R^{(2-j)/4}D^j\Phi\) for \(j\ge2\), so it stays in the fixed elliptic class. The cutoff becomes \(\psi(t'/r)\). The physical Jacobian is \(R\); consequently the inner cost \(|d_T|^4r^2\), after returning to the original coordinates, is exactly \(|d_T|^4R^2\).

Apply the inductive estimate at scale \(r\). Its counted radius \(r^{1/2+\delta_0}\) pulls back to the narrower radius \(R^{1/2+\delta_0/2}\). A pulled-back inner cube has transverse size \(R^{1/2}\) and longitudinal size \(R^{3/4}\). Directions in one coarse cap differ by \(O(R^{-1/4})\), so their relative displacement across that cube is \(O(\sqrt R)\). Choose \[0<\rho<\min(\delta_0/8,\,1/16).\] If a pulled-back inner cube \(P\) meets a translate of an outer cube \(Q\) by distance at most \(C R^{1/2+\rho}\), then every inner counted tube meeting \(P\) belongs to an original tube meeting \(Q\). The transverse distance is bounded by \[C\bigl(R^{1/2+\delta_0/2}+R^{1/2+\rho}+\sqrt R\bigr) <R^{1/2+\delta_0}.\] This is where the distinction between the core and the counted tube is used. Each induction step rescales the core and chooses its own counted radius; it does not rescale the parent’s already enlarged tube.

For clarity, we give the local decoupling bookkeeping, including the weights. For every outer cube \(Q\), choose a translate and dilate \(\eta_Q\) of one band-limited Schwartz function which is bounded below on \(Q\). Its spatial and temporal scale is \(\sqrt R\). Applying ordinary global decoupling to \(\eta_Q\sum_lP_l\) is legitimate: multiplication broadens Fourier support by \(O(R^{-1/2})\), exactly the thickness at the coarse decoupling scale \(S=\sqrt R\). Keep the boundedly many nearby coarse cylinders per cap. Their fourth-power estimate is \[\Bigl\|\sum_TF_T\Bigr\|_{L^4(Q)}^4 \le C_\beta R^\beta \left(\sum_l\|\eta_QP_l\|_4^2\right)^2 +\text{relative tail errors},\] where \(\beta>0\) is arbitrarily small.

Truncate each local integral to cubes \(Q+j\sqrt R\) with \(j\in\mathcal J\subset\mathbb Z^3\), \(|j|\le C R^\rho\). The set of shifts is the same for every \(Q\), and \(\#\mathcal J\le C R^{3\rho}\). On the retained region \(|\eta_Q|\le C\). For each group \(l\), keep only inner cubes meeting one of these retained translates, with \(Q\) ranging over the cubes of \(Y\). The time-buffer inequality above places every such whole inner cube in the strengthened child slab. Partition these cubes dyadically by the number of inner counted tubes they meet. Write \(Y_{l,h}\) for the pulled-back union with between \(h\) and \(2h\) tubes, where \(h\) ranges over \(O(\log R)\) positive dyadic values. The zero-count cells give only relative tail errors by the core bound. Set \[A_{l,j,h}(Q) =\int_{Q+j\sqrt R} \mathbf1_{Y_{l,h}}\,|P_l|^4.\] If this is nonzero, the preceding incidence comparison supplies at least \(h\) packets from \(\mathcal W_l\) whose outer tubes meet \(Q\). The groups are disjoint, so there are at most \(Ck/h\) such groups \(l\). Therefore \[\left(\sum_l A_{l,j,h}(Q)^{1/2}\right)^2 \le \frac{Ck}{h}\sum_l A_{l,j,h}(Q).\] Two finite Cauchy–Schwarz sums over \(j,h\) cost at most \(C R^{6\rho}(\log R)^2\). For each fixed shift \(j\), the cubes \(Q+j\sqrt R\) are disjoint. Sum their integrals before estimating them by an entire inner-cell union: \[\sum_Q A_{l,j,h}(Q)\le\int_{Y_{l,h}}|P_l|^4 \le C D(\sqrt R)\,h R^2 \sum_{T\in\mathcal W_l}|d_T|^4.\] The last inequality is the child estimate and the exact Jacobian calculation above; the bounded common cutoff costs only a constant. The factor \(h\) cancels the preceding \(h^{-1}\), and summation over the disjoint groups preserves \(\mathcal C_R\). This order avoids counting one long inner cell once for each outer cube it meets.

All discarded terms are homogeneous in the coefficients. For example, \(\#\{T\}\le CR^2\) and \(|\sum_{T\in\mathcal W_l}P_T|^4 \le(\#\mathcal W_l)^3\sum_{T\in\mathcal W_l}|P_T|^4\). The Schwartz bound on \(\eta_Q\), summed over the cube grid outside \(R^\rho Q\), then bounds the off-neighborhood error by \(C_LR^{8-(4L-3)\rho}\mathcal C_R\), before harmless fixed powers. Choose its decay order after \(\rho,\delta_0\) and the desired error. The core bound treats far coarse cylinders and zero-count child cells in the same way. Summing over space uses integrable decay; no bound on the number of distant selected cubes is needed. Thus, for every prescribed \(A>0\), these terms are at most \(C_A R^{-A}\mathcal C_R\) at the current induction scale.

We have proved the recurrence \[D(R)\le C R^{\beta+6\rho}(\log R)^2D(\sqrt R)+C_A R^{-A}.\] Choose \(\rho\) smaller if necessary, after \(\delta_0\) and \(\varepsilon\), and then \(\beta\), so that \(\beta+6\rho<\varepsilon/4\). At a sufficiently large fixed base, the logarithm and the fixed constant fit within another \(R^{\varepsilon/8}\). The resulting exponent, including the child bound \(R^{\varepsilon/2}\), is less than \(7\varepsilon/8\). Induction from that base gives \(D(R)\le C_{\varepsilon,\delta_0}R^\varepsilon\). The additive errors are absorbed at their own scale; no error at the last small scale is asserted to be a rapid-decay error in the original scale. The assumption \(k\ge1\) allows their absorption into \(k\mathcal C_R\). This proves (60). The weighted version is the stated packet-cost comparison. ◻

Proof. Fix a parent time bin and one child frequency window. Only parent windows overlapping this child window contribute. Select these parents, but keep their synthesis packets intact. In particular, we do not multiply individual parent windows by the child cutoff before applying refined decoupling.

Let \(\nu_+\) be the child frequency center and \(t_-=t_{I_-}\). Set \[\xi=\nu_++\theta_+\zeta,\qquad T=\frac{t-t_-}{r_+},\qquad X=\frac{x-(t-t_-)U_{\nu_+}}{w_+}.\] The identities \(N^2w_+\theta_+=N^2r_+\theta_+^2=1\) give the normalized phase, after removing the constant and linear terms, \[\Phi_+(\zeta)=\theta_+^{-2} \bigl(\phi(\nu_++\theta_+\zeta)-\phi(\nu_+) -\theta_+\nabla\phi(\nu_+)\cdot\zeta\bigr).\] It has uniformly definite Hessian and bounded higher derivatives on a fixed compact set. In these units the parent slab has length \[R=\frac{r_-}{r_+}=\ell^2,\] the parent frequency caps have diameter \(O(\ell^{-1})\), and the parent packets have transverse width \(\ell\). More precisely, the normalized synthesis packet has the canonical form in Theorem 49 with the fixed profile \(a=L_{\mathrm f}^{-2}\chi\) and centers \[\zeta_v=\frac{\nu_v-\nu_+}{\theta_+},\qquad Z_v=\frac{z_v}{w_+}.\] Here the constant phase \(e^{-2\pi iN^2z_v\cdot\nu_+}\) is absorbed into \(d_v\), preserving its absolute value. The position mesh is \(L_{\mathrm f}^{-1}\sqrt R\). The normalized velocities are bounded, because all selected parent windows overlap the child window. The synthesis factor in (50) cancels the area of a parent frequency window, so each parent packet has amplitude \(O(|d_v|)\), with no extra scale factor.

For completeness, localize time by a fixed Schwartz function \(\psi(T/R)\) whose Fourier transform is compactly supported and which is bounded away from zero on an interval containing \([0,1]\) in its argument. Denote the resulting parent packets by \(P_v\). Their Fourier support lies in an \(O(R^{-1})\) neighborhood of the normalized graph, over caps of diameter \(O(R^{-1/2})\). On \(|T|=O(R)\), integration by parts on the parent cap gives transverse Schwartz decay on scale \(R^{1/2}\). For larger \(|T|\), the decay width and constants grow only polynomially in \(1+|T|/R\), which is harmless against \(\psi(T/R)\). Consequently, \[ \|P_v\|_4^4\le C|d_v|^4R^2 =C|d_v|^4\ell^4. \tag{62}\] We next pass from the discrete child samples to a spacetime integral. Write \(F=\sum_vP_v\). In the normalized variables, the child cutoff is a fixed smooth frequency multiplier. All spatial and temporal frequencies of \(F\) lie in a fixed compact set. There is therefore a Schwartz kernel \(H\) on spacetime, with uniform bounds, whose Fourier transform implements the child multiplier on that set. An active child index \(b\) has normalized sample location \[p_b=(X_b,T_b)=\left( \frac{z_b-(t_{I_b}-t_-)U_{\nu_+}}{w_+}, \frac{t_{I_b}-t_-}{r_+}\right).\] At these retained coordinates, up to a unimodular factor, \[\psi(T_b/R)c_b=(H*F)(p_b).\] The time coordinates are integers from \(0\) to \(R-1\); at each time, the spatial coordinates lie in a translate of \(L_{\mathrm f}^{-1}\mathbb Z^2\). Thus the retained samples have bounded multiplicity at unit resolution. Hölder’s inequality and summation over these samples give \[\sum_b|c_b|^4 \le C\int |F(q)|^4 \sum_b|H(p_b-q)|\,dq.\] The summed kernel is uniformly bounded and decays rapidly with the distance to the active sample set. Truncate this distance at \(m^\beta\), where \(\beta>0\) will be chosen small. The omitted integral is bounded by an arbitrarily large negative power times \(R^2\sum_v|d_v|^4\): use the polynomial number of input packets, (62), and the triangle inequality. Cover the retained neighborhood by cubes of side \(\ell\) and denote their union by \(Y\). We have reduced the desired estimate to a bound for \(\int_Y|F|^4\).

We check carefully that the degree hypothesis controls the tube multiplicity on these cubes. Choose \(\beta<\min(\eta,\alpha)/10\). Every cube in \(Y\) has an associated active sample within \(O(\ell+m^\beta)\) of all its points. An enlarged parent tube meeting that cube has, at the time of this sample, transverse distance at most \[ C\bigl(\ell R^{\delta_0}+\ell+m^\beta\bigr) \tag{63}\] from the sample: the normalized tube velocities are bounded. Choose \(0<\delta_0<1/8\) with \(3\alpha\delta_0<\eta/10\). In the stated ratio window, \(R^{\delta_0}\le m^{3\alpha\delta_0} \le m^{\eta/10}\). Then (63) is less than \(m^\eta\ell\) for sufficiently large \(m\), which is exactly the allowed parent-beam radius in child units. Angular overlap gives a fixed multiple of \(\theta_+\), also less than the permitted \(m^\eta\theta_+\). Each such tube is therefore one of the at most \(k\) parents counted at the associated sample. This uses the stated adjacency radius, without replacing it by a larger power.

The sample times satisfy \(0\le T_b<R\), so the cube union \(Y\) lies in \[-C(\ell+m^\beta)\le T\le R+C(\ell+m^\beta).\] In the stated ratio window, \((\ell+m^\beta)/R\le m^{-\alpha/2}+m^{\beta-\alpha}\to0\). Thus \(Y\) is contained in \(\{|T|\le2R\}\) for sufficiently large \(m\), uniformly in that window, as required by Theorem 49.

Partition space into boxes of side \(R\), aligned with the cube grid, over this time slab. In each box retain the packets whose centerlines meet a fixed enlargement of it. Their time-zero centers lie in a box of side \(O(R)\) because the normalized velocities are bounded. The other packets have rapidly decaying tails there. Admissibility gives a polynomial number of boxes meeting \(Y\), so choose the integration-by-parts order after this fixed complexity to make the total discarded contribution negligible. Each relevant tube is charged in only boundedly many boxes, since its length and displacement in the slab are \(O(R)\).

Write \(P_v=\psi(T/R)\widetilde P_v\), where \(\widetilde P_v\) is the underlying canonical extension packet. On the selected cubes, \[\left|\sum_vP_v\right|^4 \le \|\psi\|_\infty^4 \left|\sum_v\widetilde P_v\right|^4.\] Apply the coefficient-cost estimate of Theorem 49 to these canonical packets in each box. Their position centers lie in a box of side \(O(R)\), their coefficients remain arbitrary complex numbers, and their counted-tube multiplicity is at most \(k\). The common cutoff and bounded box multiplicity therefore give \[\sum_{\substack{b:\text{fixed child angle}\\ I_b\subset I_-}}|c_b|^4 \le m^{o(1)} k\,\ell^4 \sum_{\substack{v\in I_-:\text{window overlaps}\ \nu_+}} |d_v|^4.\] All tail errors can be absorbed into this bound, since \(k\ge1\). Since \(R\le m^{3\alpha}\) in the ratio window, the decoupling loss can be made smaller than any prescribed positive power of \(m\) uniformly there. For a prescribed final error power, choose first the error allowed by Theorem 49, then its localization power \(\delta_0\) and the sampling power \(\beta\) small enough to satisfy the preceding constraints. Their constants are fixed before taking \(m\to\infty\).

Each parent window overlaps only boundedly many child windows. Sum over child angles and parent bins and multiply by \(r_+w_+^2\). The scale identity \[r_+w_+^2\ell^4=r_-w_-^2\] proves (61). ◻

Proof. Fix a family satisfying the additional support restriction. For a terminal bin \(I\), let \(\mathcal R_I\) be the initial indices touching at least one retained terminal index in \(I\). Every matrix entry from an initial index outside \(\mathcal R_I\) to a retained index in \(I\) is a locality tail. Lemma 47 and the polynomial coordinate count show that replacing the input by \(d|_{\mathcal R_I}\) changes the output in \(I\) by a negligible error. Apply the squared-sum contraction of Lemma 48 to the full transition matrix on this restricted input. Summing over \(I\) counts each initial index at most \(m^{2\sigma+o(1)}\) times. Therefore \[r w^2\sum_b |c_b|^2 \leq m^{-2(1-\sigma)+o(1)}\delta^2\sum_v|d_v|^2.\] The same lemma gives \(K(c_I)\leq m^{4+o(1)}K(d)\) in every bin. Since \(A(c_I)\leq G(c_I)\), multiplication by the square root of this capacity bound proves amplification at most \(m^{2\sigma+o(1)}\).

The classes of admissible families increase with \(\sigma\). Consequently \(\gamma_\sigma\) is nondecreasing and at most \(\gamma\); also \(\gamma_1=\gamma\), since there are only \(m^2\) terminal time bins. The set on which equality holds is therefore an upper interval, possibly without its lower endpoint. Its infimum satisfies \(s\geq\gamma/2>0\), and the other assertions about \(\gamma_\sigma\) follow. Equality at \(\sigma=s\) is not needed.

To obtain comparable initial amplitudes, first normalize the largest one. The polynomial coordinate counts and the polynomial comparison between coordinate norms and the packet norms allow amplitudes below a sufficiently small fixed negative power of \(m\) to be discarded. The remaining amplitudes lie in \(O(\log m)\) dyadic classes. Writing the input as their sum and using linearity followed by the triangle inequality for the output norm, one class retains at least an inverse-logarithmic fraction of the output norm. Restriction to that class can only decrease the initial raw gauge. Its cubed amplification therefore loses a subpower factor. Rescaling its amplitudes gives the interval \([1,2]\). The support restriction is hereditary under this operation. ◻

Proof. Select dyadic classes for all the indicated positive statistics. For each time bin touched by a remaining parent \(v\), select one remaining low child \(b\). Its intermediate neighborhood contributes at least a constant times \(s_j'\) to \(D_j(v)\). These contributions for distinct time bins are disjoint, because each suffix has a unique child time bin. Thus the number of touched bins is at most \(C d_j/s_j'\).

If (69) failed by a fixed positive power of \(m\) along a subsequence, the bound \(C d_j/s_j'\) would, after rebasing the parent interval, be at most \(\ell_j^{2\sigma+o(1)}\) for some fixed \(\sigma<s\). Here \(\log_m\ell_j\to1/J\), and the low radius remains a fixed positive power in the \(\ell_j\) units within each family. Thus the selected parent and child supports satisfy the hypothesis of (65). Its strict improvement for the full transition contradicts the retained amplification, proving (69).

For the pruning, begin at level \(J-1\) and proceed backwards. At level \(j\), delete every parent touching fewer than \(\ell_j^{2s}m^{-a}\) distinct child time bins in the current child mask, initially with a fixed \(a>0\). The difference made to the output is a diagram whose transition at this level starts from the deleted parent set and ends in that current child mask. That transition has the strict sparsity improvement just described. A telescoping decomposition of the successive mask changes contains only \(J\) such error diagrams. Each has a fixed exponent gap, and the triangle inequality shows that their total is negligible compared with a sufficiently accurate near-extremal diagram.

We may consequently let the deficit \(a=a_i\downarrow0\) slowly enough that its positive exponent gap still dominates the amplification errors in the \(i\)-th family. This is a diagonal choice: first use the gap for each fixed deficit, and then take the initial accuracies and the family bases sufficiently far out. No quantitative continuity of \(\gamma_\sigma\) at \(s\) is required. The resulting pruned graph retains amplification \(m^{\gamma-o_i(1)}\).

Each surviving parent now has at least \(\ell_j^{2s}m^{-a_i}\) child time bins with surviving low children. Choose one child in each such bin and iterate. The descendant bins associated with distinct child time bins are disjoint. Induction therefore gives at least \[\prod_{h=j}^{J-1}\ell_h^{2s}m^{-a_i} =(m/m_j)^{2s}m^{-(J-j)a_i}\] terminal bins from every surviving level-\(j\) vertex.

Fix a root and one reachable vertex in each of its reachable level-\(j\) time bins. The just constructed terminal descendants are disjoint for these different time bins, and all are accessible from the root in the original high graph. In that graph this root reaches at most \(m^{2(s+\epsilon_i)}\) terminal time bins. Dividing this bound by the lower descendant count gives (70). If later masks remove some descendants, the original pruned descendants remain valid witnesses in this counting argument; later masks can only decrease the set of reachable prefixes. ◻

Proof. Proceed forwards. At transition \(j\), count the low parents of each child in the already selected parent mask, discard the zero-count class by the locality observation, and select a positive dyadic class retaining amplification \(m^{\gamma-o_i(1)}\). Subsequent forward selections do not change any already fixed parent layer. Thus the asserted incoming degrees hold in the final graph. Induction from the roots gives the bounds \(k\) and \(2^Jk\) for the number of incoming paths to each leaf. The bound for a specified root–leaf pair follows from Lemma 54, because the radius powers tend to zero.

It remains to choose the leaves with the stronger norm property. On the finite coordinate space, the lattice norm \(\mathcal A\) has a nonnegative dual functional \(\lambda\) of dual norm one satisfying \[\sum_b\lambda_b|c_b|=\mathcal A(c).\] One may take the absolute values of a norming functional: the lattice property preserves its dual norm. Polynomial norm comparisons and coordinate counts permit tiny coordinate amplitudes to be discarded with negligible norm, hence negligible pairing with \(\lambda\). Discarding additionally those contributions \(\lambda_b|c_b|\) smaller than a sufficiently large negative power of \(m\) times \(\mathcal A(c)\) loses negligible total pairing, by the polynomial coordinate count. Partition the remaining leaves jointly into dyadic classes for these two quantities. There are only a power of \(\log m\) such classes. Some class \(\mathcal B\) satisfies \[\sum_{b\in\mathcal B}\lambda_b|c_b| \geq (\log m)^{-C}\mathcal A(c), \qquad |c_b|^2\asymp h_1k\quad(b\in\mathcal B)\] for a suitable \(h_1>0\). Keep this class as a final leaf mask. Incoming path counts for its leaves are unaffected.

If \(\mathcal B'\subset\mathcal B\) occupies a fraction \(f\), comparability of \(\lambda_b|c_b|\) gives \[\mathcal A(c|_{\mathcal B'}) \geq \sum_{b\in\mathcal B'}\lambda_b|c_b| \geq \tfrac12 f\,(\log m)^{-C}\mathcal A(c).\] For \(f=m^{-o_i(1)}\), cubing this inequality loses only a vanishing exponent. The functional need not norm any of the later restricted outputs. ◻

Proof. Let \(H\) be the original high path count and \(L\) the final low path count. By Lemma 55, \(L\geq m^{-\alpha_i}H\), where \(\alpha_i=o_i(1)\). Write \(T^{(j)}\) for the level-\(j\) time interval containing the terminal time. For a vertex \(v\), let \(P_H(v)\) and \(P_L(v)\) be its high and final low prefix counts, and let \(D_L(v)\) be its final low suffix count. Thus \(P_L(v)\leq P_H(v)\) and \(D_L(v)\leq D_j(v)\).

Choose \(\beta_i\downarrow0\) sufficiently slowly that \(\beta_i-\alpha_i\) dominates the other errors and \(m^{\alpha_i-\beta_i}\to0\) within each family. Call a remaining level-\(j\) vertex bad if \[D_L(v)<m^{-\beta_i}D_j(v).\] The number of final low paths through bad vertices at this level is bounded by \[\sum_{v\ {\rm bad}}P_L(v)D_L(v) \leq m^{-\beta_i}\sum_v P_H(v)D_j(v) =m^{-\beta_i}H.\] In the final sum, \(v\) ranges over all vertices of the original high graph at that level. Its equality to \(H\) holds because every high path visits exactly one such vertex. Summing over \(J+1\) levels shows that the event \(\mathcal G\) that every visited vertex is good has probability \(1-o(1)\) under uniform final low path counting.

Consider this unrestricted uniform low law, conditional on a complete vertex prefix ending at a good \(v\) at level \(j\). For a prescribed child time bin \(I\), its next-bin probability is \[ \frac{\displaystyle \sum_{\substack{b\ {\rm low\ child\ of}\ v\\I_b=I}}D_L(b)} {D_L(v)} \leq m^{\beta_i+o_i(1)}\frac{s_j'}{d_j} \leq m^{o_i(1)}\ell_j^{-2s}. \tag{71}\] If the numerator is nonzero, choose one remaining child \(b\) in that bin. Its intermediate neighborhood contains every other low child there, and the original high suffix sum on the neighborhood is comparable to \(s_j'\). This bounds the numerator. Goodness and \(D_j(v)\asymp d_j\) bound the denominator, and (69) gives the final inequality.

To iterate (71), kill a path as soon as it visits a bad vertex. Fix a nested sequence of time bins through level \(j\). Conditional on each surviving vertex prefix, the next prescribed-bin probability is bounded by (71). Averaging over the vertex prefixes having the same time prefix preserves that bound. Induction, with the process killed at every step, therefore gives, for every root \(v\) and level-\(j\) bin \(I\), \[ \Pr(\mathcal G,\ T^{(j)}=I\mid v) \leq m^{o_i(1)}\prod_{h<j}\ell_h^{-2s} =m^{o_i(1)}m_j^{-2s}. \tag{72}\] If the root is bad the left side is zero. In writing \(\mathcal G\) on the left, one may first require survival only through level \(j\), for which the induction applies directly, and then impose survival at later levels. This can only decrease the probability. Multiple vertex prefixes with the same time prefix introduce no additional count.

Discard roots for which \(\Pr(\mathcal G\mid v)<1/2\). Their total probability under the original low root law is at most \(2\Pr(\mathcal G^c)=o(1)\). Keep the good paths on the other roots. Dividing (72) by their conditional survival probability costs at most two, so the maximum probability of a level-\(j\) time bin, conditional on each retained root, is \(m^{-2s\log_m m_j+o_i(1)}\). The support bound at that level is (70).

Between consecutive level depths, monotonicity of probabilities uses the preceding level for the maximum weight, and monotonicity of supports uses the following level for the number of occupied bins. Since the gap in base-\(M=m^2\) depths is \(1/J+o(1)\), these give (1) at every dyadic depth with error at most \(s/J+o_i(1)\). The dyadic roundings change only bounded factors.

The retained measure is still uniform on a subset of paths. Its positive root weights have polynomial ratios, because the numbers of paths are positive integers with polynomial upper bounds. Select a dyadic class of root weights carrying an inverse-logarithmic fraction of the measure, and keep all retained paths from those roots. If their number is \(N_0\), the new root weights are comparable to \(1/N_0\). This selection of whole root fibers does not change the root-conditional time laws and retains a fraction \(m^{-o_i(1)}\) of all final low paths. ◻

Proof. Put the uniform law of Lemma 58 on its path subset, take the root as base label, and assign it line parameters \((U_{\nu_v},z_v)\). The root weights are comparable to \(1/N_0\), and \(1\leq|d_v|\leq2\). The input capacity definition therefore gives the density constant \[K_0\lesssim \frac{K(d)\delta^{-2}}{N_0}\] for tests with radii at least \(\delta\).

We fix the width range before applying geometry. Choose a polynomial cutoff \(H\geq m^2\delta\) bounding the norms of all tested velocities and all spatial coordinates after division by \(r\), including the root-line positions and terminal position labels. In the axes of a rectangular test with half-widths \(e_j\), each coordinate interval meets the bounded support in a set contained in an interval of half-width \(\min(e_j,H)\). Choose the two centers independently, using the same clipped half-widths and axes. The resulting joint test covers the original test on the support, and its weight is no larger, up to the fixed ellipse–rectangle comparison. Thus one polynomial width range controls the full terminal capacity supremum.

Choose \(B=m^A\) larger than this cutoff, the upper width bound in the statement, and the root coordinate bounds, with a fixed power margin. This margin also covers the endpoint enlargements. With \(M=m^2\), we have \(B\geq M\delta\). Rescale both line coordinates by \(B^{-1}\). The new input floor is \[\delta'=\delta/B=M^{-D_m},\qquad D_m=\tfrac12(\log_mN+A)\geq1.\] The density constant becomes \(K_0B^2\), since \(W\) is homogeneous of degree two. The output weight of a test is \(W(F/B)=B^{-2}W(F)\), so these factors cancel on returning to the old coordinates. The new output floor \(M\delta'\) corresponds exactly to \(M\delta\) in the old coordinates. This also covers \(m=N\), for which the unnormalized floor would have given the inadmissible value \(D=1/2\). To make the complexity fixed, take a subsequence with \(D_m\to D_0\), and choose \(D_*=D_0+a\), where \(0<a<\zeta/16\). Eventually \(0<D_*-D_m<2a\). A radius at least \(M^{-D_*}\) but below the available input floor \(M^{-D_m}\) can therefore be rounded up by a factor at most \(M^{2a}\). Doing this to both radii increases \(W\) by at most \(M^{4a}=m^{8a}\). Thus the input cap at the fixed complexity \(D_*\) costs at most this additional factor. The output floor for \(D_*\) is smaller than the available output floor, so every required output test is still allowed.

The definition of the geometric exponent as a small-error limit of an infimum over finite complexities is used here: the sufficiently small time-error threshold may be chosen for all finite \(D_*\), although the large-base threshold and subpower losses can depend on that fixed complexity. Thus the preceding choice of \(J\), followed by \(i\) and then the family base, is valid.

Apply Theorem 5, with base \(M=m^2\), to obtain a dominated sublaw of subpower mass. The difference between the chosen geometric exponent and \(1+2s-10\kappa-\zeta\) absorbs the arbitrarily small fixed rounding loss just described. After normalization, its line-test probabilities satisfy \[ \Pr\bigl(T_1=t,\ U\in F,\ X_t\in rF\bigr) \leq \frac{K(d)\delta^{-2}}{N_0}\,W(F)\, m^{-2(1+2s-10\kappa-\zeta)+o_i(1)} . \tag{74}\] As throughout, translations in the two tests are independent.

There are at most \(N_0m^{2s+o_i(1)}\) possible root–terminal-time pairs. Discard pairs whose probability in this normalized law is less than \[N_0^{-1}m^{-2s-\tau_i},\] where \(\tau_i=o_i(1)\) tends to zero slowly enough to dominate the support error. This discards \(o(1)\) mass. Take \(\mathcal P\) to be the path atoms in the support of the law after this deletion. Each represented root in a specified terminal bin has pair probability at least this threshold. Dividing (74) by the threshold proves (73), with all normalization factors absorbed in \(m^{o_i(1)}\).

Let \(L\) again be the final low path count. The starting uniform law for geometry is on a subset of at least \(m^{-o_i(1)}L\) paths. Its dominated geometric sublaw retains subpower mass. After normalization, the weight of any path atom is consequently at most \(m^{o_i(1)}/L\). The remaining support therefore represents at least \(m^{-o_i(1)}L\) paths, even though their new weights need not be equal.

Every original final leaf has between \(k\) and \(2^Jk\) incoming low paths. Choose a threshold \(k m^{-\upsilon_i}\), with \(\upsilon_i=o_i(1)\) sufficiently slow, and keep the leaves with at least this many represented incoming paths. The represented paths going to the other leaves are at most \(k m^{-\upsilon_i}\) times the original leaf count, which is negligible compared with the total represented path count. Since no leaf has more than \(2^Jk\) incoming paths, the kept leaves occupy a fraction \(m^{-o_i(1)}\) of all final leaves. Lemma 57 ensures that this last output restriction retains amplification \(m^{\gamma-o_i(1)}\). ◻

Proof. Consider an output test \(E,rE\), where the ordered radii of \(E\) are \(e_1\geq e_2\geq\theta=m\delta\). Let \(F\) have the same axes and radii \[f_j=\max(e_j,\delta/r).\] By Lemma 54, the initial root of any represented path ending in this output test belongs to the corresponding line test \(F,rF\), enlarged by \(m^{o_i(1)}\). Indeed the angular endpoint error is at most \(m^{o_i(1)}\theta\), and the position error from the root line is at most \(m^{o_i(1)}\delta\). The definition of \(F\) covers both errors. Its radii satisfy the floor required in (73). The harmless power enlargement changes its weight and the resulting count only by \(m^{o_i(1)}\).

For a fixed represented root at this time, the terminal angular grid supplies at most \(m^{o_i(1)}\) possible leaf angles. The leaf positions lie both in the spatial test \(rE\) and in an enlarged root beam of radius \(m^{o_i(1)}\delta\). Lattice counting bounds their number by \[m^{o_i(1)}w^{-2}\prod_{j=1}^2\min(re_j,\delta).\] No boundary correction is needed beyond a constant factor: both \(re_j\) and \(\delta\) are at least the position mesh \(w\). For any root and leaf, the number of paths is \(m^{o_i(1)}\), by Lemma 57.

Each kept leaf has \(|c_b|^2\lesssim h_1k\) and at least \(k m^{-o_i(1)}\) represented incoming paths. Its squared coefficient is therefore bounded by \(h_1m^{o_i(1)}\) times that incoming count. Summing represented paths in the test, then using (73), gives \[\begin{align*} K(c_I|_{\mathrm{kept}}) &\leq m^{o_i(1)} \sup_E h_1\theta^2W(E)^{-1} w^{-2}\prod_{j=1}^2\min(re_j,\delta)\\ &\hspace{25mm}\cdot K(d)\delta^{-2}W(F) m^{-2(1+s-10\kappa-\zeta)} . \end{align*}\] For the products of semiaxes, \[\prod_{j=1}^2\min(re_j,\delta) =\delta^2\,\frac{e_1e_2}{f_1f_2}, \qquad \frac{W(F)}{f_1f_2} =\left(\frac{f_1}{f_2}\right)^\kappa \leq\left(\frac{e_1}{e_2}\right)^\kappa =\frac{W(E)}{e_1e_2}.\] The inequality uses the reduction of aspect ratio when small radii are raised to a common floor. Finally \(\theta^2w^{-2}=r^{-2}=m^4\). Substitution proves (75). ◻

Proof. For the first bound, fix a terminal time bin \(I\) and let \(\mathcal R_I\) be the roots having an original high path to that bin. Roots outside this set give a negligible error by the locality observation, applied to the original high graph with output restricted to \(I\). Along the unique sequence of time ancestors of \(I\), the successive exact masked transitions satisfy the squared-sum bound of Lemma 42, with the parent-bin rebasing of Lemma 46. The transition at level \(j\) has squared operator norm at most \(\ell_j^2\), and masks are orthogonal coordinate projections. With the scale normalizations this gives \[w^2\sum_{b\in I}|c_b|^2 \lesssim \delta^2\#\mathcal R_I\] apart from negligible error. Summation with weight \(r=m^{-2}\) counts each root at most \(m^{2(s+\epsilon_i)}\) times by the original high time-support bound. This proves the first term in (76).

For the second bound, apply Proposition 50 at each actual masked transition. Lemma 57 bounds its low-parent multiplicity by \(2k_j\). Use Remark 52 with adjacency power \(\eta_i=\eta_i^{\mathrm{lo}}\): within each family the localization and sampling powers are chosen below this fixed positive power. Thus the full masked transition has the required multiplicity bound, and the product of its localization, coloring, and decoupling losses over the fixed \(J\) transitions is \(m^{o_i(1)}\). The factor \(2^J\) is absorbed in the same loss. Successive applications of (61) yield \[r w^2\sum_{b\ {\rm kept}}|c_b|^4 \leq k\,m^{o_i(1)} \delta^2\sum_{v\in\mathcal R}|d_v|^4 \lesssim k\,m^{o_i(1)}\delta^2\#\mathcal R .\] Since \(|c_b|^2\asymp h_1k\) on the kept leaves, division by \(h_1k\) proves the second term in (76). ◻

Completion of the proof of Theorem 45. For each terminal bin, use its entire kept coefficient array as a single atom in the definition of \(A\). Equations (75) and (76) imply \[\begin{align*} \mathcal A(c|_{\mathrm{kept}})^3 &\leq r\sum_I G(c_I|_{\mathrm{kept}})^3\\ &\leq m^{10\kappa+\zeta+o_i(1)} \delta^2\#\mathcal R\,K(d)^{1/2}\\ &\hspace{10mm}\cdot m^{1-s}h_1^{1/2} \min\bigl(m^{-2(1-s)},h_1^{-1}\bigr). \end{align*}\] For every positive \(h_1\), \[h_1^{1/2}\min\bigl(m^{-2(1-s)},h_1^{-1}\bigr) \leq m^{-(1-s)}.\] Also \(1\leq|d_v|\leq2\) gives \(G(d)^3\asymp\delta^2\#\mathcal R\,K(d)^{1/2}\). The cubed amplification of the kept output is consequently at most \(m^{10\kappa+\zeta+o_i(1)}\).

By Lemmas 57 and 59, this same output has amplification at least \(m^{\gamma-o_i(1)}\). Our choice \(\zeta<\gamma-10\kappa\), followed by sufficiently large \(i\) and then sufficiently large \(m\) within that family, is a contradiction. Thus the assumption \(\gamma>10\kappa\) is impossible, which proves the propagation bound and its uniform finite-family formulation. ◻

Proof. Choose a dyadic \(N\) with \(4R\le N^2<16R\), and put \(\delta=N^{-1}\). For a ball with center \(b_B\in\mathbb R^3\), set \(a_B=b_B-(0,0,N^2/2)\). The coordinates \(y=a_B+N^2(z,t)\) place that ball inside \(\{|z|\le 1/4,\ 1/4\le t\le3/4\}\). We shall also translate the physical sampling lattice by \[u\in Q=[0,L_{\mathrm f}^{-1})^2\times[0,1).\] For each such \(u\), define \[f_{B,u}(\xi)=f(\xi) e^{2\pi i(a_B+u)\cdot(\xi,\phi(\xi))}.\] These functions have the same \(L^\infty\) and \(L^2\) bounds as \(f\). All estimates that follow are uniform in \(u\) and in the ball locations. Those locations enter only through this modulation of the input; the retained packet coordinates stay in the same bounded normalized region.

At factor \(1\), retain the analysis coefficients of \(f_{B,u}\) with angular windows meeting the fixed frequency support and positions \[z\in L_{\mathrm f}^{-1}\delta\mathbb Z^2,\qquad |z|\le Z_0,\] where \(Z_0\) is a sufficiently large fixed constant. Denote the resulting finite array by \(d^{B,u}\). For each angle \(\nu\), Fourier-series Parseval gives \[ \sum_z|d_{\nu,z}^{B,u}|^2 \le (L_{\mathrm f}\delta)^2 \int |f(\xi)\chi_\delta^\nu(\xi)|^2\,d\xi \le C\delta^4. \tag{79}\] The velocity centers \(U_\nu=-\nabla\phi(\nu)\) are separated by at least \(c\delta\), since the Hessian of the extended phase is uniformly definite. An ellipse \(E\) whose radii satisfy \(L\ge w\ge\delta\) thus contains at most \(C|E|\delta^{-2}\) such centers: its \(O(\delta)\) enlargement has area comparable to \(|E|\). Ignoring the positional restriction in the definition of capacity, we obtain from (51) and (79) \[ K(d^{B,u}) \le C\delta^4\sup_E\frac{|E|}{W(E)} \le C\delta^4, \tag{80}\] because \(|E|/W(E)\) is a constant multiple of \((w/L)^\kappa\le1\).

We next prove the estimate that saves the number of balls: \[ \sum_{B\in\mathcal B}\sum_v|d_v^{B,u}|^2 \le C\delta^2. \tag{81}\] Let \(T_B\) be the retained analysis operator acting on the unmodulated function \(f\), so that \(T_Bf=d^{B,u}\), and let \(T=(T_B)_{B\in\mathcal B}\). The frame identity shows that each \(T_B\) has operator norm at most \(L_{\mathrm f}\delta\). Hence the diagonal blocks of \(TT^*\) have operator norm at most \(C\delta^2\).

An entry in an off-diagonal block is, up to complex conjugation, an integral of the form \[ \int A(\xi)e^{2\pi i(q\cdot\xi+s\phi(\xi))}\,d\xi, \qquad (q,s)=a_B-a_{B'}+N^2(z-z',0), \tag{82}\] where \(A\) is a product of two \(\delta\)-scale windows supported in a fixed compact set. The common translation \(u\) cancels. The position offsets have size \(O(N^2)=O(R)\), so, for \(B\ne B'\) and sufficiently small \(\mu_0\), \[H:=|(q,s)|\ge c(R/\mu)^{C_0}.\] The amplitude and its derivatives have bounds polynomial in \(R\). There is a fixed exponent \(a\ge0\), independent of \(\kappa\), \(\epsilon\), and the ball locations, for which \[ \left|\int A(\xi)e^{2\pi i(q\cdot\xi+s\phi(\xi))}\,d\xi\right| \le C R^aH^{-1/2}. \tag{83}\] For completeness, write \(\Psi(\xi)=q\cdot\xi+s\phi(\xi)\). If \(|s|\) is sufficiently small compared with \(H\), then \(|\nabla\Psi|\ge cH\) on the fixed support and integration by parts proves (83). Otherwise \(|s|\ge cH\). Uniform definiteness of the Hessian implies \[|\nabla\Psi(\xi)-\nabla\Psi(\xi')| \ge cH|\xi-\xi'|.\] Consequently the set \(|\nabla\Psi|\le2H^{3/4}\) has area at most \(CH^{-1/2}\). Insert a smooth cutoff to this set. Its contribution is bounded by \(CH^{-1/2}\|A\|_\infty\). On the complementary region, integrate by parts with the vector field \(\nabla\Psi/|\nabla\Psi|^2\). This field has size \(O(H^{-3/4})\) and divergence \(O(H^{-1/2})\); the derivative of the cutoff has size \(O(H^{1/4})\). One integration by parts therefore gives \(CR^aH^{-1/2}\), after increasing the fixed exponent \(a\) to cover the amplitude derivatives. This proves (83).

There are \(O(N^2)\) retained angular labels and \(O(N^2)\) retained position labels per ball. Thus each off-diagonal row of \(TT^*\) has at most \(C\mu^{-8}R^2\) entries. By (83), its absolute row sum is at most \[C\mu^{-8}R^{a+2}(R/\mu)^{-C_0/2}.\] Choose \(C_0\) so large that \(C_0/2\ge a+3\) and \(C_0/2\ge8\). This row sum is then \(O(R^{-1})=O(\delta^2)\). The same holds for columns, since the matrix is Hermitian. The Schur bound, together with the diagonal block bound, proves \(\|TT^*\|\le C\delta^2\) and hence (81). This choice of \(C_0\) is made once and does not depend on the power loss \(\epsilon\).

Apply the single-step packet map from factor \(1\) to factor \(N\) to \(d^{B,u}\). Retain only the terminal angle \(\nu_0\) and positions in a fixed bounded region containing \(|z|\le1\), and denote the output by \(c^{B,u}\). At this terminal factor, \[\theta=1,\qquad r=w=N^{-2},\qquad rw^2=N^{-6}.\] If the initial positions had not been truncated, the frame identity would give exactly the samples \[F_0\bigl(a_B+u+N^2(z,t_I)\bigr).\] The omitted positions contribute \(O_M(N^{-M})\), uniformly on the retained terminal region, for every fixed \(M\). Indeed, choose \(Z_0\) larger than a fixed bound for the retained terminal positions and the velocities on the frequency patch. If \(|z'|>Z_0\), then the phase in the synthesis–analysis matrix has, on a parent \(\delta\)-cap, nonvanishing gradient of size at least \(cN^2(1+|z'|)\). Rescaling that cap and integrating by parts gives a matrix bound \(C_L(1+N|z'|)^{-L}\) for every \(L\). There are \(O(N^2)\) relevant parent angular labels, and their position mesh is comparable to \(N^{-1}\). Thus, for \(L>1\), the squared matrix bounds sum to \[\sum_{\nu\text{ relevant}}\sum_{z':\,|z'|>Z_0}(1+N|z'|)^{-2L} \le C_LN^{4-2L}.\] Write \(\widetilde d=\mathsf P_{1,[0,1)}f_{B,u}\) for the full initial analysis array. The frame identity gives \(\sum_v|\widetilde d_v|^2\le C\delta^2\), so Cauchy–Schwarz bounds each omitted contribution by \(C_LN^{1-L}\). Taking \(L\ge M+1\) proves the claimed error, using only smoothness of the windows and the phase.

The supports just described belong to one fixed polynomial complexity range. Theorem 45, (53), and (80) therefore give, for every fixed \(0<\kappa<1/10\) and \(\eta>0\), \[\begin{align*} N^{-6}\sum_v|c_v^{B,u}|^3 &\le \mathcal A(c^{B,u})^3 \le C_{\kappa,\eta}N^{10\kappa+\eta}G(d^{B,u})^3 \\ &\le C_{\kappa,\eta}N^{10\kappa+\eta} \delta^4\sum_v|d_v^{B,u}|^2. \tag{84}\end{align*}\] In the last line we used \(G(d)^3=\delta^2(\sum|d_v|^2)K(d)^{1/2}\) at the initial factor. Summing (84) and using (81), the physical sample sum is at most \[C_{\kappa,\eta}N^6N^{10\kappa+\eta}\delta^4\delta^2 =C_{\kappa,\eta}N^{10\kappa+\eta}.\] The sampling errors are harmless: there are at most \(C\mu^{-8}N^6\le CN^{22}\) samples, and \(M\) may be chosen arbitrarily large. The inequality \(|a+b|^3\le4(|a|^3+|b|^3)\) absorbs their total cubed contribution.

The terminal physical lattice is \(\Lambda=L_{\mathrm f}^{-1}\mathbb Z^2\times\mathbb Z\), with fundamental cell \(Q\). The retained lattice points include each \(\lambda\in\Lambda\) for which \(\lambda+Q\) meets \(B-a_B\), since the ball lies in the interior normalized region fixed above. Consequently, \[\sum_{B\in\mathcal B}\int_B|F_0(y)|^3\,dy \le\int_Q\sum_{B\in\mathcal B}\sum_{v\text{ retained}} |F_0(a_B+u+N^2(z_v,t_{I_v}))|^3\,du.\] Apply the uniform sampled bound to the right-hand side. Integration over \(Q\) tiles physical space by \(\Lambda\), so it introduces no \(N\)-dependent density factor. Finally choose \(\kappa\) and \(\eta\) sufficiently small that \((10\kappa+\eta)/2\le\epsilon\). Since \(N^2\asymp R\), this proves (78). ◻

Proof. The convolution of surface measure with itself has density \[ (\sigma*\sigma)(y)=\frac{2\pi}{|y|} \mathbf1_{\{0<|y|<2\}}. \tag{86}\] To verify the constant, use rotational invariance and fix \(\omega\in S^2\). Under surface area measure in the second variable, \(s=\omega\cdot\nu\) has density \(2\pi\) on \([-1,1]\), and \(|\omega+\nu|=(2+2s)^{1/2}\). Thus, for a radial test function \(q\), \[\int q(|y|)\,d(\sigma*\sigma)(y) =8\pi^2\int_0^2q(r)r\,dr,\] which is exactly the integral against the radial density in (86). In particular, \[\|\sigma*\sigma\|_2^2 =\int_0^2\frac{4\pi^2}{r^2}\,4\pi r^2\,dr =32\pi^3.\] Let \(\lambda=h\sigma\), a finite complex measure. Its convolution satisfies the total-variation domination \[|\lambda*\lambda|\le\|h\|_\infty^2(\sigma*\sigma),\] so it has an \(L^2\) density of squared norm at most \(32\pi^3\|h\|_\infty^4\). The function \((Eh)^2\) is the Fourier transform, with the positive phase convention, of this convolution. Plancherel proves (85). Finally, differentiating the defining integral is justified by boundedness of the frequency support and gives, for example, \[\|Eh\|_\infty\le4\pi\|h\|_\infty, \qquad \|\nabla Eh\|_\infty\le8\pi^2\|h\|_\infty.\] ◻

Proof. Take the unit cubes to be half-open, and select such a point in every cube of an arbitrary finite subcollection. A uniform Lipschitz bound shows that \(|F_0|\ge\mu/4\) on the ball of radius \(c\mu\) about each selected point, where \(c>0\) is fixed and small. Each ball contributes at least \(c'\mu^7\) to the integral of \(|F_0|^4\). These balls have bounded overlap, since their radii are bounded and there is one center in each of distinct unit lattice cubes. Lemma 63 bounds the size of the finite subcollection by \(C\mu^{-7}\). Since the same bound holds for every finite subcollection, the whole set of such cubes is finite and has the asserted size. ◻

Proof. Put \(\beta=\alpha/4\), choose an integer \(J>8/\beta\), and set \(D=2C_0+2\). For sufficiently small \(\mu\), we have \(\#X\le\mu^{-8}\). Define \[R_0=\mu^{-1},\qquad R_{j+1}=(R_j/\mu)^D \quad(0\le j<J), \qquad n_j(x)=\#\bigl(X\cap B(x,R_j)\bigr),\] where balls are open and counts include the multiplicities of the indexed list. For every point \(x\) of the list, some \(j<J\) satisfies \[ n_{j+1}(x)\le\mu^{-\beta}n_j(x). \tag{88}\] Otherwise \(n_J(x)>\mu^{-J\beta}n_0(x)\ge\mu^{-J\beta}\) would contradict \(\#X\le\mu^{-8}\).

Assign each point one such index \(j\), and group the points further according to \(2^k\le n_j(x)<2^{k+1}\). There are \(O_\alpha(\log(1/\mu))\) groups. In each group choose a maximal disjoint subcollection of the radius-\(R_j\) balls centered at its points. The triples of the selected balls cover that group.

Join two selected centers by an edge when their distance is less than \(R_{j+1}/2\). For a selected center \(a\), the radius-\(R_j\) balls centered at its neighbors are disjoint and lie in \(B(a,R_{j+1})\), since \(R_j\le R_{j+1}/2\). Each such ball captures at least \(2^k\) indices of the whole list, whereas (88) gives \(n_{j+1}(a)<2^{k+1}\mu^{-\beta}\). The graph therefore has degree at most \(2\mu^{-\beta}\). Greedy coloring splits the selected balls into \(O(\mu^{-\beta})\) colors. Including all the groups, the number of families is at most \[C_\alpha\mu^{-\beta}\log(1/\mu) \le C_\alpha\mu^{-\alpha/2}.\]

Use radius \(\rho=3R_j\) for each covering family. Centers in one color have distance at least \(R_{j+1}/2\), and, for sufficiently small \(\mu\), \[\frac{R_{j+1}}2 =\frac12(R_j/\mu)^{2C_0+2} \ge(6R_j/\mu)^{C_0} =(2\rho/\mu)^{C_0}.\] Finally, write \(R_j=\mu^{-a_j}\), so that \(a_0=1\) and \(a_{j+1}=D(a_j+1)\). The finite number \(J\) depends only on \(\alpha\) and \(C_0\). Enlarging \(C_\alpha\) to at least \(a_{J-1}+1\) gives \(6R_j\le\mu^{-C_\alpha}\) for small \(\mu\), which proves (87). The selected balls are centered at points of the list, so every family has at most \(\#X\) balls. ◻

Proof. Let \(X=(x_{Q'})\) be the finite list supplied by Lemma 64. It suffices to prove \(\#X\le C_\alpha\mu^{-3-\alpha}\), since the level set in (89) is contained in the corresponding unit cubes.

The Fourier transform of \(F_0\), as a tempered distribution, is supported in a fixed compact set. Choose a Schwartz function \(\psi\) whose Fourier transform is \(1\) on a neighborhood of that set. Then \(F_0=\psi*F_0\); the convolution is absolutely convergent and the identity holds pointwise because \(F_0\) is bounded and continuous. Hölder’s inequality yields \[ |F_0(x)|^3 \le\|\psi\|_1^2 \int_{\mathbb R^3}|\psi(x-y)|\,|F_0(y)|^3\,dy. \tag{90}\] Apply Lemma 65, with the separation exponent from Lemma 62, and assign every index of \(X\) to one ball in one covering family. Suppose the assigned covering ball has radius \(\rho\). Truncating the integral in (90) to \(|x-y|\le\rho\) costs at most \[C\|F_0\|_\infty^3 \int_{|z|>\rho}|\psi(z)|\,dz \le C\rho^{-4}\le C\mu^4.\] Since \(|F_0(x)|\ge\mu/2\), this error is absorbed into the left-hand side for sufficiently small \(\mu\). The truncated region lies in the doubled assigned covering ball.

There is a uniform bound \[ \sup_{y\in\mathbb R^3}\sum_{x\in X}|\psi(x-y)|\le C_\psi. \tag{91}\] Indeed, at most one selected point lies in each unit lattice cube, and Schwartz decay bounds the sum by the convergent lattice sum of a sufficiently high inverse power of distance. In particular no pairwise separation assumption on the selected points is needed.

Let \(X_{\mathcal F}\) denote the indices assigned to one covering family \(\mathcal F\), of common radius \(\rho\). Summing the truncated form of (90) and using (91), we obtain \[\mu^3\#X_{\mathcal F} \le C\sum_{B\in\mathcal F}\int_{2B}|F_0(y)|^3\,dy \le C_\epsilon(2\rho)^\epsilon.\] For the last inequality, the doubled balls satisfy all the hypotheses of Lemma 62: their radius is at least \(\mu^{-1}\), their separation is the required one, and their number is at most \(\#X\le\mu^{-8}\) after reducing the threshold for \(\mu\).

Fix \(\alpha\) first, and then choose \(\epsilon>0\) so small that \(C_\alpha\epsilon\le\alpha/2\), using the exponent in (87). Thus \[\mu^3\#X_{\mathcal F}\le C_\alpha\mu^{-\alpha/2}.\] There are at most \(C_\alpha\mu^{-\alpha/2}\) families. Each index was assigned exactly once, and summing proves \(\#X\le C_\alpha\mu^{-3-\alpha}\), as required. ◻

Proof of Theorem 1. Fix \(p>3\) and choose \(0<\alpha<\min\{1,p-3\}\). Let \(\mu_*>0\) be smaller than the thresholds required above for this choice of \(\alpha\). By the layer-cake identity and Proposition 66, the contribution of sufficiently small thresholds is bounded by \[p\int_0^{\mu_*}\mu^{p-1} |\{|F_0|>\mu\}|\,d\mu \le C_{p,\alpha}\int_0^{\mu_*}\mu^{p-4-\alpha}\,d\mu <\infty.\] For larger thresholds, the uniform \(L^4\) estimate gives \(|\{|F_0|>\mu\}|\le C\mu^{-4}\), and the level set is empty above the uniform \(L^\infty\) bound. Their contribution is therefore finite as well. We conclude that \(\|F_0\|_p\le C_p\).

There are only finitely many graph pieces in (77). Rotations preserve Lebesgue measure, so the \(L^p\) triangle inequality gives \(\|Eg\|_p\le C_p\) under the normalization \(\|g\|_\infty=1\). Multiplication by \(\|g\|_\infty\) proves the result for arbitrary nonzero bounded measurable complex data. When the essential supremum is zero, the extension is identically zero. All constructions and estimates depend only on the almost-everywhere class of the data, which completes the assertion of the theorem. ◻

Proof of Corollary 3. Sphere factorization, due to Bourgain (Bourgain 1991), upgrades Theorem 1 to \(\|Eg\|_{L^q(\mathbb R^3)}\lesssim_q\|g\|_{L^q(S^2)}\) for every \(q>3\); see (Buschenhenke 2024, secs. 1.1, (1.1)–(1.2)) for the weak-type factorization and interpolation in the sphere case. Fix \(3<q<4\) and put \(r=q/2\). Consider unit tubes \(T_j=T_\delta(x_j,\omega_j)\) in \(\delta\)-separated directions. Choose smooth nonnegative bumps \(b_j\), bounded by \(1\), supported on disjoint caps of radius \(c\delta\) about \(\omega_j\), and equal to \(1\) on the concentric caps of radius \(c\delta/2\). Here \(c>0\) is a fixed small constant. The modulation \[h_j(\zeta)=e^{-2\pi i\delta^{-2}x_j\cdot\zeta}b_j(\zeta)\] translates the extension to the center of \(\delta^{-2}T_j\). We have \(\|h_j\|_q^q\lesssim\delta^2\) and \(|Eh_j|\gtrsim\delta^2\) on that dilated tube. To see the lower bound, write \(y-\delta^{-2}x_j=s\omega_j+v\), where \(|s|\le\delta^{-2}/2\), \(v\perp\omega_j\), and \(|v|\le\delta^{-1}\). On the cap, the phase after removing its value at \(\omega_j\) has magnitude at most \(C(c+c^2)\). Taking \(c\) sufficiently small makes the real part of the remaining integrand a fixed positive fraction of \(b_j\), whose integral is comparable to \(\delta^2\).

For arbitrary weights \(a_j\ge0\) and independent random signs \(\varepsilon_j\), Khintchine’s inequality and the diagonal estimate give \[\delta^{2q-6}\Big\|\sum_j a_j1_{T_j}\Big\|_r^r \lesssim_q \mathbb E\Big\|E\Big(\sum_j\varepsilon_j a_j^{1/2}h_j\Big)\Big\|_q^q \lesssim_q\delta^2\sum_j a_j^r.\] The factor \(\delta^{-6}\) comes from \(y=\delta^{-2}x\), while the last inequality uses the disjoint cap supports. Taking the \(r\)th root gives the weighted tube estimate \[\Big\|\sum_j a_j1_{T_j}\Big\|_{L^r(\mathbb R^3)} \lesssim_q \delta^{12/q-4} \Big(\delta^2\sum_j a_j^r\Big)^{1/r}.\]

Set \(p=r'=q/(q-2)\). For nonnegative \(f\in L^p(\mathbb R^3)\) and \(c_j\ge0\), the normalization \(|T_j|=\pi\delta^2\) gives \[\delta^2\sum_j c_j\frac1{|T_j|}\int_{T_j}f =\frac1\pi\int f\sum_jc_j1_{T_j} \lesssim_q\delta^{12/q-4}\|f\|_p \Big(\delta^2\sum_jc_j^r\Big)^{1/r}.\] Duality for the discrete measure assigning mass \(\delta^2\) to each direction bounds the \(p\)-norm of these tube averages. The bound is uniform in all tube locations, so each \(T_j\) may independently approximate the supremum defining \(K_\delta f(\omega_j)\).

Choose a maximal \(\delta\)-separated net on \(S^2\), with angular cells of area comparable to \(\delta^2\). Every direction in a cell is within \(\delta\) of its center \(\omega_j\). For \(\delta<1/4\), the corresponding tubes satisfy \[T_\delta(a,\omega)\subset T_{2\delta}(a-\omega_j/4,\omega_j) \cup T_{2\delta}(a+\omega_j/4,\omega_j).\] Indeed, the transverse displacement is at most \(3\delta/2\) and the axial displacement at most \(1/2+\delta^2<3/4\). Thus \(K_\delta f(\omega)\le8K_{2\delta}f(\omega_j)\). Sphere packing splits the net into a fixed number of \(2\delta\)-separated families. Apply the discrete bound at width \(2\delta\) to each family and sum over the angular cells. This gives Bourgain’s strong maximal transfer, in the form stated in (Mattila, n.d., Theorem 3.3, pp. 10–11): \[\|K_\delta f\|_{L^p(S^2)} \lesssim_q\delta^{12/q-4}\|f\|_{L^p(\mathbb R^3)}, \qquad p=\frac{q}{q-2}.\] For \(\delta\ge1/4\), the same bound follows directly from Hölder’s inequality. Interpolating with \(\|K_\delta f\|_\infty\le\|f\|_\infty\), at parameter \(p/3\), yields \[\|K_\delta f\|_{L^3(S^2)} \lesssim_q\delta^{-4(q-3)/(3(q-2))}\|f\|_{L^3(\mathbb R^3)}.\] Choosing \(q>3\) sufficiently close to \(3\) proves the claim. ◻