Proof. For (i), split the integral at distance one and sum dyadic annuli. The contribution at distance at most \(2^{-j}\) is bounded by \(C2^{-j(s-t)}\), up to a constant depending on the total mass.

For (ii), let \(U_t^\mu(x)=\int |x-y|^{-t}\,\mathop{}\!\mathrm d\mu(y)\). This is finite for \(\mu\)-almost every \(x\), so a sufficiently large level \(M\) discards arbitrarily little mass. Choose a compact subset \(K\subset\{U_t^\mu\le M\}\) losing at most the remaining prescribed mass. If \(K\cap B(z,r)\ne\varnothing\), choose \(x\) in this intersection. Then \[\mu|_K(B(z,r)) \le\mu(B(x,2r))\le (2r)^t U_t^\mu(x)\le M(2r)^t.\] The same bound is immediate if the intersection is empty.

For (iii), rotational invariance and the integrability of the negative \(t\)-moment of the length of an \(m\)-dimensional projection give \[\int_{G(d,m)}|P_V z|^{-t}\,\mathop{}\!\mathrm dV\le C_{d,m,t}|z|^{-t} \quad(z\ne0).\] For \(m<d\), this follows by integrating the density of the first \(m\) coordinates of a uniform unit vector near zero: the radial integral there is bounded by a constant times \(\int_0^1 r^{m-1-t}\,\mathop{}\!\mathrm dr<\infty\). For \(m=d\), the identity projection makes the assertion immediate. Integrating in \((x,y)\) and applying Tonelli proves the claim. ◻

Proof. If the conclusion fails, for each \(j\) choose \(R_j\ge j\) and \(X_j\in\mathcal C(1/j,R_j)\) failing \(\mathcal G(1/j,R_j)\). The assumed subsequence satisfies \(\mathcal G(c,R_j)\) for some fixed \(c>0\). For large \(j\), \(1/j<c\), so monotonicity implies \(\mathcal G(1/j,R_j)\), a contradiction. ◻

Proof. Restrict the coordinate images to fixed bounded charts and normalize the two laws. Their masses are at least the incidence mass, so this costs only subpower factors. Let \(I\) be the resulting incidence mass, sample \(n=\lceil\delta^{-b}\rceil\) independent direction labels \(\omega_1,\ldots,\omega_n\) from \(\nu\), and write \(v_i=v(\omega_i)\). There is a subpower quantity \(T\to\infty\) such that, with failure probability smaller than any prescribed power of \(\delta\), the sample satisfies \[ \#\{i:v_i\in B(v_0,r)\}\le Tnr^b \qquad(\delta\le r\le1). \tag{7}\] To verify this, use a fixed collection of dyadic grids in the direction chart. There are only polynomially many relevant cells. Their expected counts are at most \(Knr^b\), where \(K=\delta^{-o(1)}\), and \(nr^b\ge1\). Choose \(T\) larger than a fixed large multiple of \(K+\log(1/\delta)\). For a binomial count \(Y\), the bound \[\Pr(Y\ge j)\le (e\mathbb E Y/j)^j\] and a union bound prove Equation (7), including arbitrary balls by fixed grid comparison. Increasing \(T\) by another subpower factor makes the failure probability smaller than \(I/4\).

The expected empirical incidence mass equals \(I\). Since it is bounded by one, there exists a sample satisfying Equation (7) and having empirical incidence mass at least \(I/2\). The set \(X_0\) of point labels incident to at least \(In/4\) sampled direction labels has \(\mu\)-mass at least \(I/4\): its complement contributes at most \(I/4\) to the empirical incidence integral. Normalize \(\mu|_{X_0}\) and sample \(m=\lceil\delta^{-f}\rceil\) independent point labels. If their coordinates are \(x_1,\ldots,x_m\), the same argument gives a sample with \[\#\{j:x_j\in B(x,r)\}\le T'mr^f, \qquad T'=\delta^{-o(1)},\qquad \delta\le r\le1.\] In particular, the multiplicities in \(\delta\)-balls are subpower. Color a spatial \(\delta\)-grid with finitely many colors so that cells with the same color are \(\delta\)-separated. Choose a color with the most occupied cells and one sampled label from each of them. Their point coordinates form a separated set \(P\) with \(|P|_\delta\ge m\delta^{o(1)}\). The displayed local count makes \(P\) a \((\delta,f,\delta^{-o(1)})\)-set. Each \(x\in P\) retains the list of at least \(In/4\) good sampled direction labels belonging to its chosen point label.

For \(x\in P\), take the lines exactly through \(x\) in the coordinates \(v_i\) of those listed labels, and let \(\mathcal T_x\) be their \(\delta\)-tubes. Equation (7) bounds the multiplicity in a \(\delta\)-ball of directions by \(CT\), and the count in an \(r\)-ball by \(Tnr^b\). Consequently \[|\mathcal T_x|_\delta\ge cIn/T, \qquad |\mathcal T_x\cap B(\ell,r)|_\delta\le CTnr^b.\] Here the direction component of the line metric supplies the lower distance comparison; the reverse comparison holds because \(x\) is bounded. The normalized loss is therefore at most \(CT^2/I=\delta^{-o(1)}\). In particular, \(\mathcal T_x\) is a nonempty \((\delta,b,\delta^{-o(1)})\)-set of line parameters even when different sampled labels have the same direction coordinate.

For each sampled label \(\omega_i\), its prescribed strips contain the coordinates of all its incident sampled point labels. Moving the central line of such a strip to pass through one of these points changes its intercept by \(O(\delta)\). Thus each original strip accounts for only a bounded number of \(\delta\)-cells in line-parameter space. Altogether the union of our line families has covering count at most \[n\delta^{-b-o(1)}=\delta^{-2b-o(1)}.\] This contradicts Theorem 7 for small \(\delta\). The count treats each original sampled direction label separately; it never asks for one common fiber cover after coincident coordinates are identified. ◻

Proof. The one-vector estimate follows from spherical coordinates. For the second estimate, singular-value decomposition implies that the band conditions constrain the projections of orthonormal singular directions \(u_i\) by \(C_j\delta/d_i\). Keep only the constraints whose right-hand sides are less than a sufficiently small dimensional constant, and order these right-hand sides increasingly. By rotational invariance, one can fix \(V^\perp\) and reveal successive vectors of a random orthonormal frame. Conditional on the earlier constrained vectors, projection onto \(V^\perp\), restricted to their orthogonal complement, has rank \(h\) and nonzero singular values bounded below. Indeed the sum of squared lengths of the earlier projections is less than \(1/2\) if the dimensional constant was chosen small enough. The next vector is uniform on the unit sphere of that complement. Its probability of having projection of length at most \(r\) is at most \(C_n r^h\). The ambient dimension at this last conditional step is at least \(n-j+1\ge h+1\), including the limiting case of a one-dimensional kernel. Multiplying the conditional estimates proves \[\Pr\{w_i\text{ all in the band}\} \le C_{n,j}\prod_i\min(1,\delta/d_i)^h,\] which is equivalent to Equation (8). ◻

Proof. We first choose the fiber covers measurably, enlarging their width by a fixed dimensional factor while preserving each actual direction \(w\). Let \(M_\delta=\delta^{-q-o(1)}\) be their common size bound. Use a finite spatial \(\delta\)-grid in a fixed bounded region containing one point on every axis whose tube meets the source region. Rounding such a point to the grid and keeping the direction \(w\) enlarges the width by only \(C_n\delta\). Candidate covers are the finitely many lists of at most \(\lfloor M_\delta\rfloor\) grid anchors. For each candidate \(J\), the function \[w\longmapsto \mu\left(E_w\setminus\bigcup_{a\in J}T(w,a)\right)\] is measurable by Fubini, where \(T(w,a)\) is the enlarged tube parallel to \(w\) through \(a\). Choose the first candidate for which this function is zero. Coverage up to a \(\mu\)-null set suffices, since every point slice \(\mu_{V,z}\) is absolutely continuous with respect to \(\mu\). For a relation measurable only in the completed product sigma-algebra, first choose a Borel subset agreeing with the relation modulo a product-null set, and discard the \(\nu\)-null set of directions on which its fiber differs by positive \(\mu\)-mass. Every direction slice is absolutely continuous with respect to \(\nu\), so this replacement also changes no sliced incidence. Writing \(J(w)\) for the selected candidate, now replace \(E\) by \[E\cap\left\{(x,w):x\in\bigcup_{a\in J(w)}T(w,a)\right\},\] with the exceptional null directions removed. This loses zero original or sliced incidence and makes the selected covers contain every fiber of the restricted relation.

Intersecting the relation with the two bands does not change its sliced incidence. The expected sliced incidence is comparable from below to the original one. In fact, integrating the point-band indicator in \(z\) gives a constant times \(\delta^h\), and the one-vector Haar estimate supplies another factor comparable to \(\delta^h\). These cancel the two slice normalizations.

Write \(M(V)=\|\nu_V\|\) and \(L(V,z)=\|\mu_{V,z}\|\). Uniformly in \(V\), \[ \int L(V,z)\,dz\le C. \tag{9}\] The two-vector estimate in a projective chart gives the direction-pair kernel \[ \delta^{-2h}\Pr\{w,w'\text{ both within }\delta\text{ of }V\} \le C(|w-w'|+\delta)^{-h}. \tag{10}\] For points, integrate both band indicators in \(z\) first. This yields at most \(C\delta^h\) times the indicator \(|P_{V^\perp}(x-x')|\le2\delta\). Haar averaging therefore gives the analogous normalized kernel \(C(|x-x'|+\delta)^{-h}\). Dyadic annulus summation, using \(q,t>h\), proves \[\begin{align*} \mathbb E_V M(V)^2+\mathbb E_V\int L(V,z)^2\,dz &\le\delta^{-o(1)},\tag{11}\\ \mathbb E_V\iint_{|w-w'|\le r}d\nu_V(w)d\nu_V(w') &\le\delta^{-o(1)}r^{q-h},\tag{12}\\ \mathbb E_V\int\iint_{|x-x'|\le r} d\mu_{V,z}(x)d\mu_{V,z}(x')\,dz &\le\delta^{-o(1)}r^{t-h} \tag{13}\end{align*}\] for \(\delta\le r\le1\). For example, the shells below \(r\) form a geometric sum with successive exponents \(q-h>0\) in Equation (12); the finest shell uses the original ball estimate at radius \(\delta\).

The truncations below first bound direction-slice mass, then enforce both ball bounds, and finally bound the active tube count. Take a subpower number \(K\to\infty\) absorbing all constants, all input losses, the dyadic radius sums in Equations (12) and (13), and the inverse original incidence mass. Enlarge it so the sliced incidence is at least \(K^{-1}\) and the right-hand side of Equation (11) is at most \(K\). First remove planes with \(M(V)>K^3\). By Equations (9) and (11), their incidence contribution is at most \(CK^{-2}\).

In the remaining point slices, remove points at which \[\mu_{V,z}(B(x,Cr))>K^6r^{t-h}\] at some dyadic radius \(r\in[\delta,1]\). Markov’s inequality and Equation (13) bound the expected deleted point mass by \(CK^{-5}\). Since \(M(V)\le K^3\), its incidence cost is at most \(CK^{-2}\). The restricted point measure has the asserted ball bound at every center: any ball meeting the retained set can be enlarged by a fixed factor around a retained point. It also has mass \(O(K^6)\), by covering the bounded support with a fixed number of unit balls.

Similarly remove direction-slice points for which \(\nu_V(B(w,Cr))>K^3r^{q-h}\) at some dyadic radius. Their expected mass is at most \(CK^{-2}\) by Equation (12); Equation (9) bounds the corresponding integrated incidence loss by the same quantity. No independence between direction mass and point mass is used here.

Finally, for a direction \(w\) in the band about \(V\), let \(J(w)\) be its selected list of grid anchors. A bounded tube through \(a\), parallel to \(w\), can meet the source region in the band about \(V+z\) only if \(|P_{V^\perp}a-z|\le C'\delta\): its transverse projection within the bounded source region has diameter \(O(\delta)\). Thus the measurable upper count \[N(V,z,w)=\sum_{a\in J(w)} \mathbf 1_{\{|P_{V^\perp}a-z|\le C'\delta\}}\] dominates the number of active covering tubes and satisfies \[\int N(V,z,w)\,dz \le C M_\delta\delta^h\le K\delta^{h-q}.\] Delete direction/\(z\) pairs for which \(N(V,z,w)>K^{12}\delta^{h-q}\). Their \(z\)-volume for each \(V,w\) is at most \(K^{-11}\). The truncated point mass is \(O(K^6)\) and the truncated direction mass is at most \(K^3\), so this costs at most \(CK^{-2}\) in integrated incidence. Delete the corresponding incidences but keep the marginal restrictions already constructed. At large \(K\) the surviving integrated incidence is at least \((2K)^{-1}\).

Projection changes each direction by \(O(\delta)\). The points of one band whose coordinates \(p_{V,z}(x)\) lie in a ball of radius \(r\ge\delta\) lie in a ball of radius \(Cr\) in the original space; the same assertion holds for directions after unit normalization. These preimage bounds give the asserted ball bounds on the coordinate pushforwards. For each original label \(w\), every active tube in its own selected list becomes a tube of width \(C'\delta\) parallel to \(v_V(w)\). This proves item 3 for each original direction label. ◻

Proof. Let \(L\) be the matrix with columns \(v_i\). Then \(\|L\|\le C_m\) and \(\|L^{-1}\|\le C_m\Delta^{-1}\). Let \(S\subset\mathbb Z^m\) index the \(\delta\)-grid cubes meeting \(L^{-1}A\). The image of such a cube has diameter at most \(C_m\delta\). Each original tube parallel to \(v_i\) becomes a tube parallel to the \(i\)th coordinate axis, of width at most \(C_{m,C}\delta\Delta^{-1}\). Therefore \[|\operatorname{pr}_{\widehat i}S| \le C_{m,C}\Delta^{-(m-1)}M_i.\] The discrete Loomis–Whitney inequality (Loomis and Whitney 1949) \(|S|^{m-1}\le\prod_i|\operatorname{pr}_{\widehat i}S|\) proves the claim. For completeness, if \(X\) is uniform on \(S\), the entropy chain rule gives \[H(X_{\widehat i})\ge \sum_{j\ne i}H(X_j\mid X_1,\ldots,X_{j-1}).\] Summing in \(i\) yields \((m-1)\log|S|\le \sum_i\log|\operatorname{pr}_{\widehat i}S|\), as required. ◻

Proof. We first prove a single-scale assertion at \(u=2^{-N}\), with reciprocal scale \(R=2^N\): for any fixed \(\ell_1<(k-1)^2/k\), sufficiently small fixed losses in the hypotheses permit deletion of \(O(R^{-c_1})\) pair mass so that every remaining fiber has cap mass at most \(R^{-\ell_1}\) at radius \(u\). All choices made below are fixed independently of \(R\).

Apply Lemma 4 to the source measure with a small parameter \(\tau>0\). It suffices to treat one normalized source class \(\rho\), retaining the original pin law \(\lambda\). The discarded source mass is a fixed negative power of \(R\), and the classes will be recombined with their original weights. Write \(m\) for the class profile. Its comparison errors are \(R^{C_k\tau}\), it is \(k\)-Lipschitz, and \[m(1)\ge k-1-C_k\tau-\epsilon.\] The constant absorbs the normalization of a class of mass at least \(R^{-\tau}\). At depths \(a<b\), an occupied depth-\(b\) child of an occupied depth-\(a\) cell \(Q\) has conditional \(\rho\)-mass at most \[ R^{C_k\tau}D^{-1},\qquad D=R^{m(b/N)-m(a/N)}. \tag{14}\]

Divide the depth interval into \(J\) nearly equal blocks, with \(J\) a large fixed integer, and ignore the first two blocks. In every later block \([a,b]\), remove the pairs whose pin is within \(R^{-1/(4J)}\) of the center \(z_Q\) of the source cell \(Q\) at depth \(a\). Summing with the source-cell weights shows that this costs at most \(C_JR^{-(k-1)/(4J)+\epsilon}\) pair mass. For the other pairs the center direction \[v(x,Q)=\frac{z_Q-x}{|z_Q-x|}\] differs from the direction to any point of \(Q\) by \(O(2^{-a}R^{1/(4J)})\). In particular the transverse displacement inside \(Q\) caused by freezing this direction is \[ O(2^{-2a}R^{1/(4J)})=o(2^{-b}); \tag{15}\] here \(a\ge2N/J+O(1)\) and \(b-a=N/J+O(1)\).

Fix a small \(e'>0\). A pair \((x,z)\), \(z\in Q\), is troublesome if the full conditional source mass in the width-\(C2^{-b}\) tube about the line through \(x,z\) exceeds \[ F=R^{e'}D^{-(k-1)/k}. \tag{16}\] The fixed width \(C\) may be chosen as large as any subsequent application requires. If \(F\ge1\), there are no troublesome pairs. Otherwise, Equation (15) and a grid in \(v(x,Q)^\perp\) show that all troublesome sources for this pin lie in at most \(C/F\) parallel tubes of width \(C'2^{-b}\). Indeed, each relevant projected grid cell has a fixed enlargement of conditional mass greater than \(F\); these enlargements have bounded overlap.

For this fixed \(Q\), let \(B_x\) be its troublesome source set when the pin survives the near-pin deletion, and let \(B_x=\varnothing\) otherwise. Put \(\rho_Q=\rho|_Q/\rho(Q)\) and \[I_Q=\int\rho_Q(B_x)\,d\lambda(x).\] We claim that \(I_Q\le R^{-c_2}\), uniformly in \(Q\). Suppose instead that \(I_Q\ge R^{-c_2}\). Jensen’s inequality, applied to the pin-incidence mass at each source, gives \[ \int \rho_Q\Bigl(\bigcap_{i=1}^k B_{x_i}\Bigr) \,d\lambda(x_1)\cdots d\lambda(x_k) \ge I_Q^k. \tag{17}\] Tuples containing a removed pin contribute zero to this integral. For a tuple of surviving pins, if \(|\det(v(x_1,Q),\ldots,v(x_k,Q))|<R^{-\eta}\), one of the successive Gram–Schmidt distances is less than \(R^{-\eta/(k-1)}\). Conditional on the preceding pins, the corresponding pin lies in a \(C R^{-\eta/(k-1)}\)-neighborhood of an affine hyperplane through \(z_Q\). Thus the product measure of these tuples is at most \[C_kR^{\epsilon-\eta\zeta/(k-1)}.\] Choose \(\eta\) small compared with \(e'\), and then choose \(c_2,\epsilon\) so small that this is \(o(R^{-kc_2})\). There is a tuple of determinant at least \(R^{-\eta}\) whose common source set has conditional mass at least \(\tfrac12R^{-kc_2}\). By Equation (14), it needs at least \[c D R^{-kc_2-C_k\tau}\] balls of radius comparable to \(2^{-b}\). On the other hand, Lemma 11, applied to its \(k\) tube covers, bounds this number by \[C D R^{k\eta-e'k/(k-1)}.\] The two estimates are contradictory if \[k\eta+kc_2+C_k\tau<\frac{ke'}{k-1}.\] These parameter requirements are compatible: first choose \(\eta\), then \(c_2\), and finally \(\tau,\epsilon\). This proves the claim. Deleting troublesome pairs and summing over \(Q\) costs at most \(R^{-c_2}\) per block, because the weights \(\rho(Q)\) sum to one.

Apply the endpoint density pruning of Lemma 5, with an additional loss \(R^{e''}\) per block, to the surviving pair set. Its physical-tube version applies even though pins need not be separated from sources by a fixed constant. To check this point directly, fix a parent \(Q\) and consider its depth-\(b\) children meeting one final cap of radius \(2^{-N}\). Every point of such a child is within \(C2^{-b}\) of the target line. Choose one retained witness in one hit child of \(Q\). Its line differs from the target line by transverse distance \(O(2^{-N})\) in the bounded source region. Thus the pre-density-deletion survivors in all hit children of this parent satisfy the same local test at width \(C'2^{-b}\). In each child their mass is at least \(R^{-e''}\) times the full child mass. The one local fraction bound therefore charges the sum of the full masses of hit children by \(R^{e''}F\) times the full parent mass. It uses the survivor set before the final density deletion; no renewed density claim after that deletion is needed.

Multiplying these inequalities telescopes the profile increments. Sum the resulting bounds over the occupied initial cells at the end of the two ignored blocks. Their masses sum to one, so this introduces no factor equal to the number of cells. The ignored growth is at most \(2k/J+o(1)\). Consequently every final cap has normalized class mass at most \[R^{J(e'+e'')+o(1)} R^{-\frac{k-1}{k}(k-1-2k/J-C_k\tau-\epsilon)}.\] Choose \(J\) large depending on \((k-1)^2/k-\ell_1\), then \(e',e''\) small, and then the remaining losses as above. This proves the single-scale assertion. All deletion exponents are positive fixed numbers. Recombining classes with their original weights preserves the cap estimate and the power deletion; the negligible discarded class mass is included in the deletion.

Now choose \(\ell<\ell_1<(k-1)^2/k\). It is enough to consider \(0<e<1\). Apply the single-scale assertion at dyadic \(u\) between \(\delta\) and \(\delta^{e/(2k)}\). At these scales the input loss \(\delta^{-\epsilon}\) is at most \(u^{-2k\epsilon/e}\), so choosing \(\epsilon\) sufficiently small makes all the single-scale hypotheses uniformly valid. The sum of the deletions is at most \[C\log(1/\delta)\,\delta^{ec_1/(2k)},\] which is a positive power of \(\delta\) after decreasing its exponent. Intersect the surviving pair sets. Dyadic comparison gives the desired cap estimate for \(r\le\delta^{e/(2k)}\); the slack between \(\ell\) and \(\ell_1\) absorbs fixed constants. For larger \(r\), \(\delta^{-e}r^\ell\ge1\) at small scales, so the estimate is trivial. This completes the proof. ◻

Proof of Theorem 6. We rule out any sequence with subpower input losses, subpower incidence mass from below, and cover size \(\delta^{-q-o(1)}\). This implies the stated fixed-power result: if no positive \(\epsilon,c\) worked, choose counterexamples with \(\epsilon=c=1/j\) at scales \(\delta_j<1/j\) tending to zero. They would form precisely such a sequence. This observation will also justify all uses of the gain with sufficiently small fixed losses.

By choosing one of finitely many small projective charts, we can assume all relevant directions lie in a single chart and that incidence still has subpower mass. If \(q>k-1\), apply Lemma 10 with \(h=k-1\). Select a slice with subpower incidence and normalize its marginal restrictions; their masses are bounded above by subpower and below by the incidence divided by the other marginal mass. The coordinate maps take values in a plane, with \[b=q-h\in(0,\tfrac12],\qquad f=t-h\in(\tfrac12,1),\qquad b<f,\] and tube count at most \(\delta^{-b-o(1)}\). Apply Lemma 8 with the original sliced points and directions as labels, and with coordinate maps \(p_{V,z}\) and \(v_V\). Item 3 of Lemma 10 gives the ball bounds on their coordinate pushforwards and the separate cover for each original direction label. The labeled incidence relation still has subpower mass, so these data contradict the planar obstruction.

At \(q=k-1\), slicing to the plane would leave direction exponent zero, outside the range of Lemma 8. We therefore put \(h=k-2\), so the selected slices have dimension three, and let \[g=t-h-\tfrac32>0.\] Choose once and for all \(0<a<g/4\). We will find in one slice three directions of determinant at least \(c\delta^a\) with a common incident point set of mass at least subpower. The resulting loss in Lemma 11 is \(O(\delta^{-3a})\), which the strict gap \(g\) will absorb.

Parameterize our chart by \(p\in\mathbb R^k\), with unit direction \(w(p)=(1,p)/\sqrt{1+|p|^2}\). Restricting and normalizing the chart costs only a subpower factor; denote the resulting law in \(p\) by \(\lambda\). In the rest of the endpoint proof the direction law supplied to slicing is \(w_\#\lambda\), and the incidence relation is restricted to this chart. The slice law of the chart labels is \[d\lambda_V(p)=\delta^{-h} \mathbf 1_{\{|P_{V^\perp}w(p)|\le\delta\}}\,d\lambda(p), \qquad \nu_V=w_\#\lambda_V.\] Equation (6) becomes the affine-hyperplane bound in the \(p\) coordinates, because an affine hyperplane there is the intersection of a linear hyperplane with the chart. The ball exponent remains \(k-1\). Choose \[h<\ell<\frac{(k-1)^2}{k}=h+\frac1k.\] Apply Lemma 12 to this chart law, with a small fixed output loss \(\delta^{-e}\). Its hypotheses hold at sufficiently small scales for every fixed input-loss threshold. This deletes a pair set of original product mass \(O(\delta^{c_e})\) and gives, for every anchor \(p\), an \(\ell\)-cap bound \(\delta^{-e}r^\ell\) on the undeleted displacement directions. Also delete pairs at distance less than \(\delta^{b_0}\), where \(b_0>0\) will be small. Denote the union of these two exceptional pair relations by \(\mathcal E_{\mathrm{pair}}\), and call an anchored pair retained when it lies outside this union.

We first record how these deletions transfer to normalized slice-pair mass. Equation (10) shows that for every measurable chart-pair relation \(\mathcal E\) with \((\lambda\otimes\lambda)(\mathcal E)\le\delta^c\), and every fixed \(\theta>0\), \[ \mathbb E_V(\lambda_V\otimes\lambda_V)(\mathcal E) \le C\delta^{c-\theta h} +\delta^{\theta(q-h)-o(1)}. \tag{18}\] For \(0<\theta\le1\), the first term treats distances greater than \(\delta^\theta\), and the second treats smaller distances by the ball bound and annular summation. For \(\theta\ge1\), the kernel is at most \(C\delta^{-h}\), so the whole expectation is at most \(C\delta^{c-h}\le C\delta^{c-\theta h}\). The same annulus calculation bounds the expected mass of the close pairs by \(\delta^{b_0(q-h)-o(1)}\). We leave \(\theta\) free until \(e\), and hence the weak-radial deletion exponent \(c_e\), has been fixed.

We next bound triples that are close to linearly dependent. For two retained anchored pairs \((p,p'),(p,p'')\), set \[\alpha=\left| \frac{p'-p}{|p'-p|}\wedge\frac{p''-p}{|p''-p|}\right|.\] In a bounded chart, \[ |w(p)\wedge w(p')\wedge w(p'')| \asymp |p'-p|\,|p''-p|\,\alpha. \tag{19}\] To see this, put \(u=p'-p\) and \(v=p''-p\) and subtract the first unnormalized column from the other two. The orthogonal decomposition of exterior powers gives the identity \[|(1,p)\wedge(0,u)\wedge(0,v)|^2 =|u\wedge v|^2+|p\wedge u\wedge v|^2.\] It supplies the lower bound, while boundedness of \(p\) supplies the upper bound. Normalizing the three columns changes only fixed constants. Consequently a wedge smaller than \(2\delta^a\) implies \(\alpha\le C\delta^{a-2b_0}\).

For \((w_1,w_2,w_3)=(w(p),w(p'),w(p''))\), the largest singular value is comparable to one. Their first two columns span area at least \(c\delta^{b_0}\), so the second singular value is at least \(c\delta^{b_0}\); by Equation (19), the third is at least \(c\delta^{2b_0}\alpha\). Lemma 9 therefore bounds their normalized expected slicing weight by \[ \delta^{-3h}\Pr_V\{\max_{1\le i\le3}|P_{V^\perp}w_i|\le\delta\} \le C\delta^{-3hb_0}(\alpha+\delta)^{-h}. \tag{20}\] This includes singular values below \(\delta\), since the kernel is truncated there. At fixed \(p,p'\), the weak radial estimate and the two antipodal caps give \[\lambda\{p'':(p,p'')\text{ is retained},\ \alpha\le r\} \le C\delta^{-e}r^\ell \qquad(\delta\le r\le1).\] Let \(\mathcal T_\delta\) be the chart triples whose two anchored pairs are retained and whose original wedge has size less than \(2\delta^a\). Summing dyadic annuli in Equation (20), up to \(C\delta^{a-2b_0}\), gives \[ \mathbb E_V\lambda_V^{\otimes3}(\mathcal T_\delta) \le C\delta^{(a-2b_0)(\ell-h)-e-3hb_0}. \tag{21}\] Since \(\ell>h\), first choose \(b_0>0\) sufficiently small depending on \(a,\ell,h\), and then \(e>0\) sufficiently small, so that this exponent is positive. These choices also ensure \(a-2b_0>0\). With \(e\) fixed, the weak radial estimate fixes \(c_e>0\). Now choose \(\theta>0\) small enough that \(c_e-\theta h>0\). Since \(q-h=1\), the second exponent in Equation (18) is also positive. Together with the close-pair bound, this proves that \(\mathcal E_{\mathrm{pair}}\) has power-small expected slice-pair mass.

Perform the restrictions in Lemma 10, and write \(\widetilde\lambda_V=(w^{-1})_\#\widetilde\nu_V\) for the restricted law of chart labels. It is dominated by \(\lambda_V\) and has the same uniform subpower mass bound as \(\widetilde\nu_V\). Define \[F(V,z,x)=\widetilde\lambda_V \{p:(x,w(p))\in\widetilde E_{V,z}\}.\] Its integrated first moment is the retained sliced incidence and is at least subpower, while the total integrated point mass is bounded by a fixed constant. Hölder’s inequality consequently yields \[ \mathbb E_V\int\int F(V,z,x)^3 \,d\widetilde\mu_{V,z}(x)\,dz \ge\delta^{o(1)}. \tag{22}\] Dropping the point-incidence indicators bounds the contributions of bad triples and exceptional anchored pairs by the left-hand sides below. Equation (9) and \(\widetilde\lambda_V\le\lambda_V\) give \[\begin{align*} \mathbb E_V\int \|\widetilde\mu_{V,z}\| \widetilde\lambda_V^{\otimes3}(\mathcal T_\delta)\,dz &\le C\mathbb E_V\lambda_V^{\otimes3}(\mathcal T_\delta),\\ \mathbb E_V\int \|\widetilde\mu_{V,z}\|\,\|\widetilde\lambda_V\| (\widetilde\lambda_V\otimes\widetilde\lambda_V) (\mathcal E_{\mathrm{pair}})\,dz &\le C\sup_V\|\widetilde\lambda_V\|\, \mathbb E_V(\lambda_V\otimes\lambda_V) (\mathcal E_{\mathrm{pair}}). \end{align*}\] Both quantities are power-small by Equation (21) and the preceding bound for \(\mathcal E_{\mathrm{pair}}\); the latter also covers the two possible anchored pairs after changing the constant. No independence between the point and direction slices is used. Hence a subpower amount of the triple incidence in Equation (22) has original wedge at least \(2\delta^a\). Projection to \(V\) and unit normalization change each vector by \(O(\delta)\), so their determinant in \(V\) is at least \(\delta^a\) for small \(\delta\).

The total parameter mass for integrating \((V,z)\) and three restricted chart labels is at most subpower, since \(\|\widetilde\lambda_V\|\) has a uniform subpower bound and the \(z\)-domain is bounded. It follows that some slice and some such triple have a common incident point set of sliced mass at least subpower. Let \(A\) be its projected point coordinates. The pushforward of its restricted point measure has the same mass and inherits the point-coordinate ball bound, which implies \[|A|_\delta\ge\delta^{-(t-h)+o(1)}.\] For each of the three original direction labels, the projected common point set has its own cover by \(\delta^{-(q-h)-o(1)}=\delta^{-1-o(1)}\) tubes in that label’s projected direction. By Lemma 11, \[|A|_\delta\le\delta^{-3/2-3a-o(1)}.\] This contradicts \(3a<g=t-h-3/2\). When \(k=2\) we have \(h=0\); all slicing weights equal one and the same proof applies, with the \(z\)-integral interpreted as integration over the singleton zero-dimensional space. The endpoint and the theorem are proved. ◻

Proof. Choose a fixed enlarged width \(C'>C\) large enough for the spatial anchor rounding used in Lemma 10. Let \(\epsilon,c\) be supplied by Theorem 6 for width \(C'\delta\), and choose \(\epsilon_1\le\epsilon/2\) and \(0<c_1<\min(\epsilon/2,c/3)\). In a fixed bounded spatial \(\delta\)-grid consider all candidate lists \(J\) of at most \(\lfloor\delta^{-q-c_1}\rfloor\) anchors. Keep the actual direction \(w\) in every tube, and define \[B_\delta^+ =\left\{w:\max_J \mu\left(\bigcup_{a\in J}T_{C'}(w,a)\right) \ge\delta^{c_1}\right\}.\] The candidate masses are measurable, so this is a Borel set. Rounding a point on each axis in an original bad cover gives a candidate at the enlarged width with at least the same captured mass. Consequently \(B_\delta\subset B_\delta^+\); equality is not needed. For each \(w\in B_\delta^+\) choose the first candidate attaining the threshold. Its measurable incidence relation has mass at least \(\delta^{c_1}\nu(B_\delta^+)\) and uses at most \(\delta^{-q-c_1}\le\delta^{-q-\epsilon}\) tubes per fiber. Theorem 6 bounds its incidence by \(\delta^c\), hence \(\nu(B_\delta^+)\le\delta^{c-c_1}\le\delta^{c_1}\). ◻

Proof. Put \(g(u)=m(u)-qu\) and \(F(u)=\min_{u\le v\le1}g(v)\). The function \(F\) is nondecreasing and Lipschitz: if \(u<v\), a minimizer for \(F(u)\) either lies in \([v,1]\), or comparison with \(g(v)\) bounds \(F(v)-F(u)\) by \(\mathop{\mathrm{Lip}}(g)(v-u)\). Moreover, \(F(0)=0\) and \(F(1)\ge s-q>t-q\). Absolute continuity gives an interior differentiability point \(A\) at which \(F'(A)>t-q\). At this point \(F(A)=g(A)\), since a strict gap would force the minimum to be attained a fixed distance to the right and would make \(F\) locally constant. There are contact points \(B>A\) arbitrarily close to \(A\): otherwise \(F\) would be constant immediately to the right of \(A\). Choose such a \(B<\min(2A,1)\) so close that \(F(w)-F(A)\ge(t-q)(w-A)\) on \([A,B]\). Using \(g(w)\ge F(w)\) proves Equation (24). The first inequality follows from \(m(w)\ge sw\), and the last follows from \(g(w)\ge F(B)=g(B)\) for \(w\ge B\). ◻

Proof. It is enough to prove the conclusion in one orientation and intersect the two resulting pair sets. In this orientation regularize only the source law, using Lemma 4. We first establish the following uniform assertion for an individual normalized class: sufficiently small losses in its profile comparisons and in the input cap bound imply a fixed positive gain and a power bound for the deleted pair mass. During this class assertion, \(\rho\) denotes the normalized class. Restrict \(G\) to pairs whose source belongs to that class, and discard the null pairs outside \(\mathop{\mathrm{supp}}\nu\times\mathop{\mathrm{supp}}\rho\). Deleted pair mass is measured by \(\nu\otimes\rho\). Only the reverse laws \((\pi_z)_\#(\nu|_{G^z})\) are used below. They remain unnormalized restrictions of the original pin law, so normalizing a source class does not alter their cap or plate constants.

We prove this assertion by contradiction, which also fixes its quantifiers. If it failed, there would be scales \(R=\delta^{-1}=2^M\to\infty\), class-comparison losses tending to zero, input cap losses tending to zero, and counterexamples to every gain and deletion exponent tending to zero. The class profiles are equi-Lipschitz. Pass to a uniformly convergent subsequence with limit \(m\), where \(m(0)=0\) and \(m(u)\ge su\), by the lower profile bound in Lemma 4 and the vanishing regularization losses. All normalized cell masses and conditional cell comparisons are then \(R^{o(1)}\) times those given by \(m\).

Fix \(t\in(k-\tfrac12,s)\) and choose \(A,B\) by Lemma 15. Use the four depths \(0,a=\lfloor AM\rfloor,b=\lfloor BM\rfloor,M\). Rounding changes only \(R^{o(1)}\) factors. We establish local radial fiber estimates on the three intervals. For an occupied source cell \(Q\) at the initial depth of an interval \([u,v]\), write \(\rho_Q=\rho|_Q/\rho(Q)\) and define \[\mathcal R_{Q,v}(x,z) =\rho_Q\{z'\in Q: |\pi_x(z')-\pi_x(z)|\le C_{\mathrm{rad}}R^{-v}\}, \qquad x\in\mathop{\mathrm{supp}}\nu,\quad z\in Q\cap\mathop{\mathrm{supp}}\rho.\] Here \(C_{\mathrm{rad}}\) is a sufficiently large fixed width constant. The comparison source \(z'\) always ranges over the full conditional class law in \(Q\). We will bound this quantity only at retained witness pairs \((x,z)\); no individual pair fiber is normalized. The function \(\mathcal R_{Q,v}\) is Borel by parameterized integration.

First consider \([u,v]=[0,A]\) or \([B,1]\), and a source cell \(Q\) at depth \(uM\). For every \(u\le w\le v\), the profile comparisons and Equations (23) and (25) give \[ \rho_Q(B(z,CR^{-w}))\le R^{-q(w-u)+o(1)}. \tag{26}\] At a fixed witness \(z\in Q\), integrate pins only on \(G^z\). Bounded separation implies \[|\pi_x(z)-\pi_x(z')|\le C_{\mathrm{rad}}R^{-v} \quad\Longrightarrow\quad \pi_z(x)\text{ lies in two caps of radius } C'R^{-v}/|z-z'|.\] For \(\tfrac12R^{-w}<|z-z'|\le R^{-w}\), the product of the cap bound and Equation (26) is at most \(R^{-q(v-u)+o(1)}\); equivalently one may sum over all intermediate dyadic distance annuli. The innermost ball \(|z-z'|\lesssim R^{-v}\) has the same bound by Equation (26). Hence \[ \int_{G^z}\mathcal R_{Q,v}(x,z)\,d\nu(x) \le R^{-q(v-u)+o(1)}. \tag{27}\] Fubini and Markov therefore delete only \(R^{-e'+o(1)}\) pair mass, for any fixed \(e'>0\), and leave \[\mathcal R_{Q,v}(x,z)\le R^{-q(v-u)+e'}\] at every retained witness \((x,z)\) in this cell. Summing over cells uses the cell weights \(\rho(Q)\).

For the middle interval put \(D=R^{-(B-A)}\). After rescaling an \(a\)-cell \(Q\) to bounded size, its conditional probability has a \(t\)-ball bound down to scale \(D\), with loss \(D^{-o(1)}\), by Equation (24). For the unchanged reverse laws, \(B-A>0\) is fixed along this subsequence, so their \(R^{o(1)}\) cap loss is \(D^{-o(1)}\) on the required radii. At \(q=k-1\), their fixed plate constant also fits the permitted small \(D\)-power loss for small \(D\). Apply Corollary 13 with a sufficiently large fixed tube-width constant. It supplies a fixed \(c_1>0\) and a common Borel set \(\mathcal B_Q\) containing all directions for which a union of at most \(D^{-q-c_1}\) parallel tubes of width \(C'D\) carries at least \(D^{c_1}\) conditional source mass. For every \(z\in Q\), \[ ((\pi_z)_\#(\nu|_{G^z}))(\mathcal B_Q)\le D^{c_1}. \tag{28}\] The directional law in this statement is a subprobability and has not been renormalized. The set \(\mathcal B_Q\) depends on \(Q\) and its conditional source law, not on \(z\). If necessary decrease \(c_1\) so that the same exponent bounds the exceptional mass. Constant-width enlargements and gridding are part of this Corollary.

Delete the pairs in Equation (28), at cost \(O(D^{c_1})\) by Fubini. For a pin still represented in \(Q\), at least one retained witness has a good direction. Let \(c_Q\) be the cell center and \(w_{x,Q}=\pi_x(c_Q)\). All witness directions differ, up to sign, from \(w_{x,Q}\) by \(O(R^{-A})\). Since \[ R^{-A}=o(D),\qquad R^{-2A}=o(R^{-B}), \tag{29}\] a tube cover at \(w_{x,Q}\) with a slightly smaller fixed width would also be a forbidden cover at that witness direction. Thus the conditional source law is robust at the center direction, uniformly over every represented pin. The directions may be regarded as unoriented for this assertion.

For \(z,z'\in Q\), Taylor expansion of \(y\mapsto\pi_x(y)\) on the separated supports gives \[|\pi_x(z)-\pi_x(z')|\le C_{\mathrm{rad}}R^{-B} \quad\Longrightarrow\quad |P_{w_{x,Q}^{\perp}}(z-z')| \le C''(R^{-B}+R^{-2A})\le C'''R^{-B}.\] Grid the perpendicular projection at this scale. For each pin \(x\), let \[\mathcal H_{x,Q} =\{z\in Q\cap\mathop{\mathrm{supp}}\rho:\mathcal R_{Q,B}(x,z)>D^{q+c_1/2}\}.\] If \(z\in\mathcal H_{x,Q}\), some one of a bounded number of neighboring projected grid cells has full \(\rho_Q\)-mass at least a fixed multiple of \(D^{q+c_1/2}\). There are at most \(C D^{-q-c_1/2}\) such heavy cells. Enlarge each by its bounded number of neighboring grid cells. Thus \(\mathcal H_{x,Q}\) lies in at most that many tubes, of a fixed width permitted by the preceding robustness assertion. For small \(D\) this is less than \(D^{-q-c_1}\). For every represented pin, robustness therefore gives \(\rho_Q(\mathcal H_{x,Q})\le D^{c_1}\).

From the pair relation surviving the bad-direction deletion, delete the pairs \((x,z)\) with \(z\in\mathcal H_{x,Q}\). If a pin has no surviving witness in \(Q\), this deletion has zero fiber mass. Otherwise the preceding bound applies. Integrating over the full pin law, and then summing with the cell weights \(\rho(Q)\), costs \(O(D^{c_1})\). This deletion intersects the first surviving Borel relation with the threshold event for \(\mathcal R_{Q,B}\); it is Borel and needs no measurable choice of the witness used in the robustness argument. Every remaining witness satisfies \(\mathcal R_{Q,B}(x,z)\le D^{q+c_1/2}\).

Denote the three local factors, in interval order, by \[f_0=R^{-qA+e'},\qquad f_1=R^{-(q+c_1/2)(B-A)},\qquad f_2=R^{-q(1-B)+e'}.\] Let \(G'\) be the common Borel pair relation left in the class-restricted \(G\) after the two outer Markov deletions and the two middle deletions, before endpoint density pruning. At each \((x,z)\in G'\) in the initial cell \(Q\) of the corresponding interval \([u,v]\), the local estimate is \[\rho\{z'\in Q: |\pi_x(z')-\pi_x(z)|\le C_{\mathrm{rad}}R^{-v}\} =\rho(Q)\mathcal R_{Q,v}(x,z)\le f_j\rho(Q).\] Thus Lemma 5 applies at the four endpoint depths with this \(G'\) and with \(E_0=E_1=E_2=\mathop{\mathrm{supp}}\nu\times\mathop{\mathrm{supp}}\rho\): the comparison sources are the full class law, while the estimates are required only at witnesses in \(G'\). The fixed choice of \(C_{\mathrm{rad}}\) includes the enlargement from rounding the endpoint depths. Use density threshold \(R^{-e''}\). The Lemma deletes \(O(R^{-e''})\) pair mass and multiplies the factors with at most \(R^{3e''}\) loss; its density test concerns \(G'\) before this last deletion. Choose \(e',e''>0\) so that \[2e'+3e''<\tfrac14c_1(B-A).\] The resulting cap mass is at most \(R^{-q-c_1(B-A)/4}\), and every deletion has a fixed positive power bound. This contradicts the selected counterexamples. In particular, although \(A,B\) arose from a limiting profile, the contradiction proves uniform positive constants for sufficiently small fixed comparison losses.

Choose the regularization parameter small enough for this uniform class assertion, and then choose \(\varepsilon_0\) small enough for its input cap loss. Discarded classes have power-small total mass. Apply the assertion to every remaining class and recombine with its original source weight. The weights sum to at most one, so both cap estimates and deletion estimates sum without a factor equal to the number of classes. Finally repeat with the two original measures interchanged and intersect the outputs. Decreasing \(c\) absorbs fixed constants and proves the Lemma. ◻

Proof. Availability is downward closed. It is also closed under increasing limits inside \([k-1,k-\tfrac12]\): for a desired loss \(e\), choose an available \(q\) within \(e/2\) of the supremum, apply its estimate with loss \(e/2\), and use \(u\ge2^{-N}\).

Suppose that an available exponent \(q<k-\tfrac12\) is given. Let \(c,\varepsilon_0\) be the constants in Lemma 16, and choose \(q'>q\) with \(q'\le q+c/2\) and \(q'\le k-\tfrac12\). For any desired loss \(e>0\), choose \(e_1>0\) so small that \(2e_1q'<e\). First use availability of \(q\) with input loss \(\varepsilon<e_1\varepsilon_0/2\). For every integer \(M\in[e_1N,N]\), that input loss is at most \(2^{\varepsilon_0M}\), so Lemma 16 applies at \(\delta=2^{-M}\) to the same remaining pair set. Intersect the outputs over these \(M\). Their combined deletion is at most \[C\sum_{M\ge e_1N}2^{-cM}=O(2^{-ce_1N}),\] in addition to the original power-small deletion. At these radii the output is stronger than the requested \(2^{eN}u^{q'}\) bound. At larger radii that bound is trivial from mass at most one, for all large \(N\), by the choice of \(e_1\). Dyadic enlargement is absorbed by the remaining loss. Hence \(q'\) is available. Applying this to the supremum of the available exponents rules out a supremum below \(k-\tfrac12\); closure gives the endpoint itself. ◻

Proof. Choose positive normalized restrictions \(\rho_1,\rho_2\) to sufficiently small separated compact patches. Put \(a=(n-1)/2\) and \[\tau=D_\#\bigl[D^{-a}(\rho_1\otimes\rho_2)\bigr], \qquad D(x,y)=|x-y|.\] This is a nonzero finite positive measure on the distance line. The elementary sphere-kernel estimate \[|\widehat{\sigma_{S^{n-1}}}(\xi)|\lesssim(1+|\xi|)^{-a}\] follows by a partition into the two stationary patches and their complement, using stationary phase on the former and integration by parts on the latter. Consequently \[ \int_{S^{n-1}}|\widehat{\rho_1}(rw)|^2\,d\sigma(w) \lesssim r^{-a}I_a(\rho_1),\qquad r\ge1. \tag{30}\] The energy is finite since \(a<s\). The ball bound and a smooth Fourier ball majorant give \[ \int_{r\asymp R}\int_{S^{n-1}} |\widehat{\rho_2}(rw)|^2\,d\sigma(w)\,\frac{dr}{R} \lesssim R^{-n}\int_{|\xi|\lesssim R}|\widehat{\rho_2}(\xi)|^2d\xi \lesssim R^{-s}. \tag{31}\] For the last inequality, the inverse transform of the majorant is \(O_L(R^n(1+R|x-y|)^{-L})\); integrating by dyadic annuli against the \(s\)-ball bound proves the claim.

Choose a smooth angular cutoff equal to one on all the cross directions and zero near their antipodes. The leading spherical stationary-phase term, uniformly on the separated patches, gives, up to the harmless Fourier-sign choice, \[\widehat\tau(r)=c_n^{-1}r^a \int_{S^{n-1}}\psi(w)\widehat{\rho_1}(rw) \overline{\widehat{\rho_2}(rw)}\,d\sigma(w) +O(r^{-1}),\qquad c_n\ne0.\] Indeed the error before multiplication by \(r^a\) is \(O(r^{-a-1})\), because distances stay in a compact subset of \((0,\infty)\). Cauchy–Schwarz, Equations (30) and (31), and \(2a+1=n\) imply \[\int_{r\asymp R}|\widehat\tau(r)|^2\,dr \lesssim R^{n-a-s}+R^{-1} =R^{(n+1)/2-s}+R^{-1}.\] The dyadic sum converges. The negative frequencies are conjugates, and low frequencies are bounded. Plancherel therefore gives an \(L^2\) density for \(\tau\), proving the conclusion. This is the classical threshold argument of Falconer (Falconer 1985). ◻

Proof. Choose \(S<s_0<\dim_HE\) and use Frostman’s Lemma to obtain an \(s_0\)-ball probability. Fix \(u>S\) below \(s_0\) and all the target dimensions. Lemma 2 gives finite \(u\)-energy for almost every successive projection. To make all the projected ball bounds hold on the same restriction, use the potential proof of that Lemma: for each projected law \(\eta\), retain a set on which \(U_u\eta\) is bounded, and pull this set back to the original space. Each discarded preimage can have arbitrarily small mass, so the intersection of these finitely many restrictions has positive mass. An inner compact restriction has the desired ball bound in every space. Decrease the exponent slightly if necessary to obtain one common \(S+\chi\).

For the last assertion, almost every \((k+2)\)-tuple of original points is affinely independent: sample successively and use zero mass on every span of dimension at most \(k\). For each fixed such tuple, almost every projection to \(\mathbb R^{k+1}\) preserves its independence. Fubini makes this true for almost every tuple for almost every projection, simultaneously with energy preservation. If a projected hyperplane had mass \(a>0\), the dependent tuples would have mass at least \(a^{k+2}\), a contradiction. Subsequent restrictions preserve this property. ◻

Proof. Put \[u=h-\epsilon,\qquad p=1+\frac{\epsilon}{4h}, \qquad t=h-\frac{\epsilon}{2}.\] Then \(1<p<2\) and \(up<t<h(2-p)\). Initially suppose that the source is smooth. For \(w\in S^h\), let \(P_w\) be projection to \(w^\perp\), let \(f_w\) be the density of \((P_w)_\#\alpha\), and put \(\eta_w=(P_w)_\#\beta\). The radial density is \[F(x,w)=\int_0^\infty\alpha(x+rw)r^h\,dr \lesssim f_w(P_wx),\] because the supports lie in a fixed bounded region.

If \(g\ge0\) and \(\|g\|_{L^{p/(p-1)}(\eta_w)}=1\), Hölder on the product space gives \[I_u(g\eta_w)\le I_{up}(\eta_w)^{1/p} \lesssim I_t(\eta_w)^{1/p}.\] The second inequality uses bounded support and \(up<t\). The energy identity \(I_u(\eta)=c_{h,u}\int|\widehat\eta(\xi)|^2|\xi|^{u-h}\,d\xi\) follows by writing the Riesz kernel as an integral of Gaussian kernels; see (Mattila 1995, Lemma 12.12) for this identity with our Fourier convention. Fourier duality and Cauchy–Schwarz with reciprocal weights therefore imply \[ \|f_w\|_{L^p(\eta_w)} \lesssim I_t(\eta_w)^{1/(2p)} \left(\int_{w^\perp}|\widehat\alpha(\xi)|^2 |\xi|^{h-u}\,d\xi\right)^{1/2}. \tag{32}\] The pairing can first be justified with bounded smooth truncations, then passed to the limit by the same energy bound.

Write \(A(w)\) for the Fourier integral in Equation (32) and \(B(w)=I_t(\eta_w)\). Polar coordinates and rotational invariance give \[\int_{S^h}A(w)\,d\sigma(w) =C\int_{\mathbb R^{h+1}}|\widehat\alpha(\xi)|^2 |\xi|^{h-u-1}\,d\xi =C'I_{h+\epsilon}(\alpha).\] Set \(r_0=1/(2-p)\). The one-vector estimate \(\sigma\{w:|P_wz|\le a|z|\}\lesssim a^h\), together with \(tr_0<h\), shows that \[\big\||P_wz|^{-t}\big\|_{L^{r_0}(d\sigma)}\lesssim|z|^{-t}.\] Minkowski gives \(\|B\|_{L^{r_0}}\lesssim I_t(\beta)\). Raise Equation (32) to the \(p\)th power and apply Hölder with exponents \(2/p\) and \(2/(2-p)\) to obtain \[ \int\!\int F(x,w)^p\,d\sigma(w)\,d\beta(x) \lesssim I_{h+\epsilon}(\alpha)^{p/2}I_t(\beta)^{1/2}. \tag{33}\]

For a general source, let \(\alpha_\varepsilon\) be its convolution with a nonnegative smooth approximate identity of scale \(\varepsilon\) and sufficiently small support. Separation persists, and the Fourier energy formula gives a uniform energy bound. The joint measures \[\Lambda_\varepsilon =\bigl[(x,y)\mapsto(x,\pi_x(y))\bigr]_\# (\beta\otimes\alpha_\varepsilon)\] have uniformly bounded densities in \(L^p(\beta\otimes\sigma)\) by Equation (33). Separation gives weak convergence to the analogous joint measure \(\Lambda\). Duality against continuous functions, followed by \(L^{p'}\) density (Folland 1999, Theorem 6.15 and Proposition 7.9), shows that \(\Lambda\) has an \(L^p\) density \(F\). Disintegration, checked on a countable determining family of continuous functions on \(S^h\), identifies \(F(x,\cdot)\) as the radial density for \(\beta\)-almost every \(x\). A Borel version supplies joint measurability.

Repeat with \(\alpha\) and \(\beta\) interchanged. Choose positive compact subsets on which the relevant radial norms are bounded and on which the density identities hold. Restricting the sources only decreases positive radial measures; normalization costs a fixed constant. The two uniform bounds therefore hold simultaneously. For each resulting radial kernel \(K_x\), choose the density version \[\widetilde F(x,w)=\limsup_{n\to\infty} \frac{K_x(B(w,2^{-n}))}{\sigma(B(w,2^{-n}))}.\] This is jointly Borel by parameterized integration. At every retained pin the preceding domination gives \(K_x\ll\sigma\), so spherical differentiation identifies this version as its density almost everywhere in \(w\), with the stated \(L^p\) bound. Here one may view both measures as ambient Radon measures and apply (Simon 2014, chap. 1, Theorems 3.23–3.24). The compact good-pin sets contain the supports of the restricted laws; hence this construction introduces no new exceptional pins. Apply the same choice in the reverse orientation. ◻

Proof. Use the jointly Borel densities \(F(x,w)\) and \(H(y,w)\) of the two radial laws. With \(T=R^{e/2}\), delete pairs for which \[F(x,\pi_x(y))>T\quad\text{or}\quad H(y,\pi_y(x))>T.\] Their product mass is \(O(T^{1-p})\), since \(\int_{F>T}F\le T^{1-p}\int F^p\), and similarly for \(H\). Each remaining directional density is bounded by \(T\), so the spherical cap-volume bound gives \(CTu^h\le R^eu^h\) for large \(R\).

Let \(P\) be the fixed projection. On all actual cross directions, \(|Pw|\ge c>0\), because projected separation is bounded below and original distances are bounded above. The map \[\Phi(w)=Pw/|Pw|\] is Lipschitz on this region and satisfies \(\Phi(\pi_x(y))=\pi_{Px}(Py)\). The image of any original cap, intersected with the actual directions, is contained in a projected cap of comparable radius. Pushforward of the source measure gives the asserted bound at each original pin. This uses no injectivity of \(P\), and works in both orientations. ◻

Proof. Put \(a=s-(k-1)>1/2\). Covering a bounded portion of an affine \((k-1)\)-plane by \(O(r^{-(k-1)})\) balls gives \[ \rho(L_r)\le C r^a \tag{34}\] uniformly over such planes and \(0<r\le1\). This also bounds neighborhoods of all lower-dimensional affine subspaces. We shall fix a sufficiently small \(b>0\) and then a sufficiently large integer \(j_0\).

Call \(H\) heavy at width \(r\) if \(\rho(H_r)\ge r^b\). We first prove that its parameter set has an \(r\)-net of size \[ O(r^{-C_0b}), \tag{35}\] where \(C_0\) is independent of sufficiently small \(b\). Restrict to the compact family of hyperplanes meeting a fixed bounded neighborhood of the supports, and use the usual metric given by normal direction, modulo sign, and offset. Fix \(L>2/a\). Within \(H_r\), choose \(k+1\) points successively, each at distance at least \(r^{Lb}\) from the affine span of its predecessors. Equation (34) bounds each forbidden set by \(Cr^{Lab}\le r^b/2\) when \(r\) is sufficiently small. Thus the product measure of admissible tuples is at least \(c r^{(k+1)b}\), and every such tuple has \(k\)-simplex volume at least \(c_k r^{kLb}\).

If a fixed admissible tuple belongs to the width-\(r\) plates of two hyperplanes, elementary linear algebra bounds the parameter distance between these hyperplanes by \(Cr^{1-kLb}\). Indeed the matrix of the \(k\) difference vectors has bounded operator norm and smallest nonzero singular value at least a constant times its \(k\)-volume; both normals have scalar products \(O(r)\) with every difference. The offset is then controlled by either first vertex. The hyperplane parameter space has dimension \(k+1\), so an \(r\)-separated list contains at most \(Cr^{-k(k+1)Lb}\) hyperplanes compatible with this tuple. Integrating tuple incidences over any \(r\)-separated list of heavy hyperplanes proves Equation (35), with, for example, \(C_0=(k+1)(1+kL)\) after increasing fixed constants. A maximal separated list is the desired net. Its plates enlarged to width \(C_kr\) contain every original width-\(r\) heavy plate in the bounded region, as well as every grid cell of side \(r\) meeting one.

For each \(j\ge j_0\), put \(r_j=2^{-j}\) and delete every pair of depth-\(j\) cells which both meet some heavy width-\(r_j\) plate. These are finite unions of products of Borel cells. Let \(E_j\) be the deleted pair set. With \(J_N=\lfloor\omega N\rfloor\), define \[ \Gamma_N=(\mathop{\mathrm{supp}}\rho_1\times\mathop{\mathrm{supp}}\rho_2) \setminus\bigcup_{j_0\le j\le J_N}E_j. \tag{36}\] The sets decrease, and nested grids give the stated cell partition. We prove positive limiting mass by bounding new losses over all \(j\).

At a fixed width \(r\), use the enlarged plates of its net. Two such plates with normal angle greater than \(\sqrt r\), if they intersect the bounded region, have their intersection within an \(O(\sqrt r)\)-neighborhood of an affine \((k-1)\)-plane. This follows by solving the two normal equations: the smallest singular value of the pair of unit normals is comparable to their angle. Equations (34) and (35) therefore show that the union of all these intersections has \(\rho_1\)-mass at most \[ C r^{a/2-2C_0b}. \tag{37}\] Outside this exceptional set, all listed plates containing a fixed first endpoint have pairwise normal angle at most \(\sqrt r\). Their bounded portions lie in one plate of width \(C\sqrt r\): the common endpoint controls the offsets as well as the normals. Consequently all second endpoints lost at this scale lie in that single plate.

For \(j>4j_0\), set \(j'=\lfloor j/3\rfloor\). Taking \(j_0\) large ensures \(j'\ge j_0\) and \(C2^{-j/2}<2^{-j'}\). The single plate just obtained is contained in a width-\(2^{-j'}\) plate containing the first endpoint. If this enclosing plate is heavy at that earlier width, every pair under consideration was already deleted by \(E_{j'}\): it contains the actual endpoints, so both of their depth-\(j'\) cells meet it. If it is light, all newly affected second endpoints have total \(\rho_2\)-mass at most \(2^{-bj'}\). It follows from Equation (37) that the new product mass lost at scale \(j\) is at most \[ C2^{-j(a/2-2C_0b)}+2^{-b\lfloor j/3\rfloor}. \tag{38}\]

Treat the initial scales \(j_0\le j\le4j_0\) together. Their combined net has at most \(C_b2^{4C_0bj_0}\) members. Enlarge them all to width \(C2^{-j_0}\) and repeat the intersection argument at angle \(2^{-j_0/2}\). The exceptional first-endpoint mass is at most \[C_b2^{-(a/2-8C_0b)j_0}.\] For every other first endpoint, all lost second endpoints lie in one width-\(C2^{-j_0/2}\) hyperplane plate. The hypothesis of zero hyperplane masses implies \[ \sup_H\rho(H_\varepsilon)\longrightarrow0 \qquad(\varepsilon\downarrow0). \tag{39}\] Indeed a contrary sequence of hyperplanes has a convergent subsequence in the compact parameter space; its shrinking plates would force positive mass on the limiting hyperplane. Thus the total initial loss tends to zero as \(j_0\to\infty\).

Choose \(b>0\) so small that \(8C_0b<a/4\) and \(kLb<1/2\). The sum of Equation (38) over \(j>4j_0\) also tends to zero as \(j_0\to\infty\). Choose \(j_0\) so large that the combined loss is less than \(1/2\). This proves positive limiting mass.

Finally fix a pin \(x\). Directions in a linear-hyperplane \(u\)-neighborhood correspond to sources in a width-\(Cu\) affine plate through \(x\). At an available comparable exclusion scale, a heavy plate leaves no retained pairs, while a light plate has source mass at most its width to the power \(b\). At radii below the last exclusion scale use the last scale instead. The finitely many radii above the first scale cost a fixed constant. Thus both fibers obey \[\lambda\{w:\mathop{\mathrm{dist}}(w,V)\le u\} \le C\max(u,2^{-J_N})^b \le C_\omega u^{\omega b}, \qquad 2^{-N}\le u\le1.\] Dyadic rounding and fixed width constants change only \(C_\omega\). We may take \(\zeta=\omega b\), or any smaller positive exponent. ◻

Proof. Use Lemma 19 to choose an initial Frostman probability of exponent greater than \(S\). In odd dimension \(d=2k-1\), if it charges an affine \(k\)-plane, its normalized positive restriction to that plane still has this exponent, and \[S=k-\tfrac12\ge(k+1)/2\qquad(k\ge2).\] Lemma 18, applied isometrically within the plane, proves the first alternative. These are original distances. Otherwise project generically to \(\mathbb R^{k+1}\), preserving energies above \(S\) and giving zero mass to all hyperplanes.

In both parities project further, if needed, to \(\mathbb R^{h+1}\), where \(h=\lfloor S\rfloor\). All target dimensions are greater than \(S\). In even dimension this is the only auxiliary projection. Use simultaneous potential restrictions in Lemma 19 to obtain ball exponents above \(S\) in all spaces. Choose two distinct support locations whose smallest projected images are distinct; this is possible since that projected law is nonatomic. Restrict to small compact positive-measure neighborhoods of these locations. Their supports are separated in every required space, and their original cross directions lie in a fixed small patch. Normalize and decrease the common exponent to \(S+\chi\) if necessary.

Choose \(0<\epsilon<\min(\chi,h,1)\) small enough that both laws in \(\mathbb R^{h+1}\) have finite \(I_{h+\epsilon}\) energy. Apply Lemma 20 and pull its positive compact restrictions back to the original measures. Inner regularity permits positive compact preimages; further restriction preserves all established bounds. These are the final \(\mu_1,\mu_2\).

If \(d\) is even, \(h=S\). Take \(\Gamma_N=E_1\times E_2\) and apply Lemma 21, lifting to the original space. This includes \(d=2\), where the smallest projection is the identity.

If \(d=2k-1\), write \(\mu_i'\) for the laws in \(\mathbb R^{k+1}\). Their exponent may be decreased to an \(s\in(k-\tfrac12,k)\). They give zero mass to every affine hyperplane. Apply Lemma 22 with the prescribed \(\omega\). The integer cap bounds obtained in \(\mathbb R^k\) from Lemma 21 lift to \(\mathbb R^{k+1}\), and persist on restriction to the coarse filters. Thus \(k-1\) is available there. Proposition 17 gives availability of \(k-\tfrac12=S\). Lift the resulting cap bounds and the exceptional pair sets to the original space.

Pullback preserves product masses and decreasing intersections. The auxiliary finest-depth cell partition has at most \(C2^{(k+1)\omega N}\) cells on each side. Its retained cell-pair atoms are disjoint products. As \(k+1\le d\), their number is at most \(C2^{2d\omega N}\). Pullbacks of the individual cells are spatial Borel sets, and pullback preserves disjointness, so these products form a partition of \(\Gamma_N\). In even dimensions the entire product is already a one-piece partition. This proves every part of the Proposition. ◻

Proof. The first two assertions follow from \(m(0)=0\) and the fact that the slopes of \(m\) lie in \([0,d]\). The lower bound follows from Equation (2); the fixed Frostman constant is absorbed into \(R^{C_d\sigma}\) for sufficiently large \(N\). The occupied-cell mass assertion is Equation (1). A ball of radius \(C2^{-g}\) meets \(O_{d,C}(1)\) depth-\(g\) cubes. Their full masses are at most \(M(g)R^{C_d\sigma}\) each, whereas each occupied depth-\(p\) cube has mass at least \(M(p)R^{-C_d\sigma}\). Summation gives Equation (41), with a larger \(C_d\).

For the last assertion, if \(p\le n\le t\), a physical ball of radius \(2^{-n}\) meets a bounded number of depth-\(n\) cubes, and division by \(\rho(P)\) gives an upper bound \[R^{-[m(n/N)-m(p/N)]+C_d\sigma} =2^{-S(n-p)}R^{A_p-A_n+C_d\sigma}.\] Dyadic comparison of radii yields Equation (42). Only depths through \(t\) are used, so no mass estimate below \(H^{-1}\) is involved. ◻

Proof. The interval \(|s|\le1\) costs at most \(2\). For a dyadic \(B\ge1\), let \(\lambda_B\) be the pushforward of \[\mathbf 1_{\{B\le|s|\le\min(2B,H)\}}\,ds\,d\xi(w)\] under \((s,w)\mapsto sw\). It is supported in the annulus \(B\le|v|\le2B\) and has total mass at most \(2B\). The preimage of a unit frequency ball has radial length \(O(1)\) and lies in at most two spherical caps of radius \(O(B^{-1})\). Since \(B\le H\le R\), the cap bound applies. A fixed covering handles radii differing from \(B^{-1}\) by fixed constants, and the case \(B=O(1)\) uses the total mass bound. Thus \[ \sup_v\lambda_B(B(v,1))\le C LB^{-S}. \tag{46}\]

Translation of \(\alpha\) changes only the phase of its transform, so assume its support is in a fixed ball. Choose a smooth compactly supported spatial function equal to one on that ball. Fourier convolution and the weighted Cauchy–Schwarz inequality give a nonnegative, integrable, rapidly decreasing kernel \(\Phi\), independent of \(B\), for which \[|\widehat\alpha(v)|^2 \le C\int_{\mathbb R^d}|\widehat\alpha(u)|^2\Phi(v-u)\,du.\] Indeed one may take a rapidly decreasing majorant for the absolute value of the cutoff’s Fourier transform. Integrating against \(\lambda_B\) and writing \(W_B=\Phi*\lambda_B\), we obtain \[\int|\widehat\alpha(v)|^2\,d\lambda_B(v) \le C\int|\widehat\alpha(u)|^2W_B(u)\,du.\] Summing Equation (46) over unit cubes against the decay of \(\Phi\) gives \(W_B\le CLB^{-S}\). On \(|u|>4B\), the support and total mass of \(\lambda_B\) give, for every fixed \(M\), \[ W_B(u)\le C_M B(1+|u|)^{-M}. \tag{47}\]

A Gaussian majorant and its positive Fourier transform show that \[\begin{split} \int_{|u|\le4B}|\widehat\alpha(u)|^2\,du &\le CB^d\iint e^{-cB^2|z-z'|^2}\,d\alpha(z)\,d\alpha(z')\\ &\le CKB^{d-S}. \end{split}\] For completeness, the last bound follows by splitting the \(z'\) integral into the ball of radius \(B^{-1}\) and annuli with radii \(2^jB^{-1}\). Every radius is at least \(H^{-1}\). Up to radius one use the assumed ball bound; above radius one use total mass at most one, which is bounded by \(Kr^S\). The resulting sum is bounded by \(CKB^{-S}\sum_{j\ge0}2^{jS}e^{-c'4^j}\). The part with \(|u|\le4B\) therefore costs at most \[CLK B^{d-2S}=CLK.\] For the complement, use \(|\widehat\alpha|\le1\) and Equation (47), choosing \(M>d+1\). Its contribution is at most \(C B^{d+1-M}\le C\), which is absorbed by \(CLK\). Summing over the \(O(1+\log H)\) dyadic shells proves Equation (45). ◻

Proof. First take a parallel descriptor. The anchor centre itself costs at most one in the count. Divide the other centres into annuli of distance comparable to \(2^{-h}\) from \(c_P\), where \(g-O_d(1)\le h\le p+O_d(1)\). A given centre in this annulus is counted only for directions in two caps of radius \[O(2^{-(p-h)}R^{500\sigma}).\] For radii at most one, the angular mass is at most \(C R^{(1+500S)\sigma}2^{-S(p-h)}\). For larger radii the same bound, after adjusting the constant, follows from the subprobability property. All small radii used are at least a fixed multiple of \(R^{-1}\); changing this multiple costs only a fixed cap covering. The number of centres is bounded by \(C R^{I(h,p)+C_d'\sigma}\) by Equation (41). If an endpoint annulus has \(h\) outside \([g,p]\), replace it by the nearer endpoint, which changes scales only by a dimensional constant. The identity \[I(h,p)-S(p-h)/N=A_p-A_h\le k(g,p)\] then gives expected count at most \[C(N+1)R^{k(g,p)+C_d''\sigma}.\] Here \(C_d''\) is dimensional, including the factor \(1+500S\). Choose \(C_d>C_d''+3\). For fixed \(\sigma\) and sufficiently large \(N\), Markov’s inequality at threshold Equation (43) gives Equation (48).

For a projection descriptor, rescale \(P\) to unit size and let \(\alpha\) denote its conditional law. Set \(H=2^{t-p}\) and \(\delta_A=A_p-\min_{p\le n\le t}A_n\ge0\). By Lemma 25, its \(S\)-ball loss is \(K=R^{\delta_A+C_d'\sigma}\). The rescaled tolerance of the level-zero test is \(\varepsilon=2H^{-1}R^{500\sigma}\). A constant multiple of \[F_\varepsilon(u)= \left(\frac{\sin(u/(2\varepsilon))}{u/(2\varepsilon)}\right)^2\] majorizes \(\mathbf 1_{[-\varepsilon,\varepsilon]}\). Its Fourier transform is nonnegative, supported in \([-\varepsilon^{-1},\varepsilon^{-1}]\subseteq[-H,H]\), and bounded by \(C\varepsilon\). Fourier inversion and Tonelli’s Theorem thus bound the mean collision mass by \[\begin{split} \int (\alpha\otimes\alpha) \{|(z-z')\cdot w|\le\varepsilon\}\,d\xi(w) &\le C\varepsilon \int_{-H}^H\int|\widehat\alpha(sw)|^2\,d\xi(w)\,ds\\ &\le C H^{-1}(1+\log H) R^{\delta_A+(501+C_d')\sigma}. \end{split}\] The last inequality is Lemma 27 with \(L=R^\sigma\). Increasing the same dimensional \(C_d\) so that \(C_d>504+C_d'\) absorbs constants and \(1+\log H\) and gives Equation (48) by Markov’s inequality. This argument also covers large tolerances; alternatively, if the threshold is at least one, the test holds for every direction. The choices of \(C_d\) depend only on previously fixed dimensional constants, not on any small parameter. ◻

Proof. Consider directions \(w,w'\) at distance at most \(cD_{l'}^{-1}\). In a parallel test \([g,p]\), changing direction changes the distance of a tested centre to the line by at most \[C_d2^{-g}|w-w'| \le C_dc2^{-p}R^{50\sigma}.\] Here \(l'=p-g\); if \(D_{l'}=1\), use \(2^{l'}\le R^{50\sigma}\). In a projection test \([p,t]\) the change in the tested scalar displacement is at most \[C_d2^{-p}|w-w'| \le C_dc2^{-t}R^{50\sigma},\] by the same argument with \(l'=t-p\). By contrast, the consecutive width gap at the relevant fine depth \(n\) is \[W_i(n)-W_{i+1}(n)=J_0^{-1}2^{-n}R^{500\sigma}.\] For fixed parameters and sufficiently large \(N\), each perturbation above is strictly less than half this gap. Every configuration counted at width \(W_{i+1}\) in direction \(w'\) was therefore counted at width \(W_i\) in direction \(w\). If \(w\) is good at level \(i\), then \(w'\) is good at level \(i+1\). The strict remaining margin also gives the asserted inclusion for the closure of a smaller neighbourhood.

To construct the mask, take \(r=c_0D_{l'}^{-1}\) with \(c_0\) sufficiently small, and the indicator of the spherical \(2r\)-neighbourhood of \(G_{i,\mathfrak t}\). Convolve it on the sphere with a nonnegative rotationally invariant smooth kernel of integral one, supported where the chordal distance is less than \(r\). Such a kernel has angular derivative \(L^1\) bounds \(C_jr^{-j}\), by rescaling in a fixed finite collection of spherical charts. The convolution equals one on \(G_{i,\mathfrak t}\) and is supported in the closure of its \(3r\)-neighbourhood, contained in \(G_{i+1,\mathfrak t}\). The derivative bounds in Equation (49) follow. Both tests and the kernel are invariant under \(w\mapsto-w\), so the mask is even. The construction is made separately for each of the finitely many anchor cubes; no spatial differentiability is required. ◻

Proof. An undifferentiated mask has norm at most one. Only a bounded number of factors are prescribed derivatives, and their normalized derivatives have bounds \(C_jD_{l'}^j\) by Lemma 29, uniformly over the allowed indices. At order \(j\), the product rule gives at most a polynomial in \(N\) terms, by the assumed list-size bound and because \(j\) is fixed. Each term is bounded by a constant times \(\max(1,2^LR^{-50\sigma})^j\). For fixed \(\sigma>0\), its polynomial multiplicity is at most \(R^{\sigma j}\) for sufficiently large \(N\), giving Equation (50). The order-zero bound has no such multiplicity. Finally, the support of a derivative of a smooth function is contained in the support of the function. Thus none of the differentiations removes an imposed good-direction condition. ◻

Proof. The eligible blocks are consecutive. The first starts less than one block length after \(a+q_0\), and the last ends less than one block length before \(t-q_0\). Nonterminality ensures that there are eligible blocks. For an index in their union, its own block representative has no larger profile value, and the selected \(p\) has no larger value than that representative. This proves the first assertions.

Each origin’s choice is made independently of the interval in whose list it is being included. Hence an origin contributing to a smaller interval contributes the same descriptors to every larger one, proving nesting. Anchors are block representatives, of which there are at most \(1+(N+1)/q_0\). A descriptor is specified by its type, anchor, and one other integer depth. Their number is therefore at most \(2(N+1)(1+(N+1)/q_0)\).

A parallel descriptor coming from \([a',t']\) has \(l'=p-g\le l\). A projection descriptor occurs only for a suffix choice and then has \(l'=t'-p=l\le p-a'\). These observations prove Equation (51); the list of parallel descriptors supplied by the same origin contains all the asserted ones. Deduplication preserves the conclusion for each membership: by definition, at least one of the origins that generated a descriptor in \(\mathcal M(a,t)\) lies inside \([a,t]\). That origin may be chosen for this membership independently of any choice made for another list. ◻

Proof. Apply Lemma 30 to each endpoint symbol, using the polynomial list bound, the prescribed derivative budget and the common bounds for \(q\). A fixed product of endpoint symbols then satisfies the same estimate by the finite product rule, even if their smoothing indices differ. For the last assertion, an anchor cell at depth at most \(a\) contains the depth-\(a\) cell, and its mask is constant in the spatial variable on that cell. Angular differentiation preserves this constancy. The product has bounded modulus because only boundedly many factors were prescribed derivatives and all other masks have modulus at most one. ◻

Proof. We first obtain a local expansion with a controlled remainder. Away from \(e\) and \(-e\), the angular gradient of \(w\cdot e\) is bounded below. Repeated integration by parts with this gradient bounds the integral there by any chosen power of \(\Lambda^{-1}\), paying at most the corresponding angular derivatives of \(b\). Near the maximum, rotate \(e\) to the last coordinate vector and use \[w(u)=\bigl(u\sqrt{1-|u|^2/4},1-|u|^2/2\bigr),\qquad dw=(1-|u|^2/4)^{(n_*-2)/2}\,du.\] The phase is exactly \(w\cdot e=1-|u|^2/2\) and the Jacobian equals one at \(u=0\). These charts and their derivatives can be chosen uniformly on finitely many patches of possible \(e\). Thus the local integral is \(e^{i\Lambda}\int e^{-i\Lambda|u|^2/2}a_e(u)\,du\), with \(a_e\) compactly supported in a fixed coordinate ball and \(\|a_e\|_{C^m}\lesssim_m A^m\).

The Fourier formula for a Gaussian, obtained first with positive damping and then by taking its limit, expresses the quadratic integral \(\int e^{-i\Lambda|u|^2/2}a_e(u)\,du\) as \(c_{n_*}\Lambda^{-n_*/2}\) times the integral of \(\widehat a_e(\xi)e^{i|\xi|^2/(2\Lambda)}\) with the Fourier inversion normalization. Taylor expansion through order \(J-1\) has remainder at most \[C_J\Lambda^{-J} \int |\xi|^{2J}|\widehat a_e(\xi)|\,d\xi \le C_J\Lambda^{-J}A^{2J+d+2}.\] The last inequality follows by integrating by parts in the transform to an order greater than \(2J+n_*\); all supports have fixed diameter. For the nonstationary portion take \(M=J+\lceil n_*/2\rceil\) integrations by parts. After extracting \(\Lambda^{-n_*/2}\), its bound is \(C_M\Lambda^{n_*/2-M}A^M\le C_J\Lambda^{-J}A^{2J+d+2}\). We have proved Equation (64) with differential coefficients of order at most \(2j\) and leading coefficient \(c_{n_*}b(e)\).

We identify these coefficients without making a choice of moving endpoint coordinates. For this identification fix a smooth \(b\) and use a polar cutoff \(\chi(w\cdot e)\) equal to one near \(w=e\) and zero outside a slightly larger fixed cap. The Fourier extension \(F(\xi)=\int e^{iw\cdot\xi}b(w)\,dw\) satisfies \((\Delta_\xi+1)F=0\). The same identity for its polar-cutoff outgoing part has an error decreasing faster than every power of \(|\xi|\): derivatives of the cutoff are supported away from both stationary points, so the preceding integration by parts applies, also after any fixed number of derivatives. In polar coordinates the differential operator is \[\partial_\Lambda^2+\frac{n_*}{\Lambda}\partial_\Lambda +1+\Lambda^{-2}\Delta_{S^{n_*}}.\] The Gaussian expansion can be differentiated any fixed number of times in its angular and radial parameters: differentiate the compactly supported coordinate amplitudes and Gaussian formula, and take more terms to bound the differentiated remainder. We may therefore substitute the local outgoing expansion \(e^{i\Lambda}\Lambda^{-n_*/2} \sum_{j\ge0}\Lambda^{-j}a_j(e)\). Uniqueness of its power coefficients gives \[2i(j+1)a_{j+1} =\left(\Delta_{S^{n_*}}+j(j+1) -\frac{n_*(n_*-2)}4\right)a_j.\] Starting from \(a_0=c_{n_*}b\), this is exactly Equation (63). A cutoff equal to one near \(e\) has the same coefficients, since the difference is nonstationary. The minimum has the conjugate Gaussian phase, proving Equation (65).

For inversion, apply Equation (64) to \(B_jb\), expanding its series through index \(J-j-1\). Equation (66) cancels every coefficient of total index \(1,\ldots,J-1\), and its index-zero coefficient is \(b\). The derivative bound for \(B_jb\) at order \(m\) is \(C_{j,m}A^{m+2j}\). After multiplication by \(\Lambda^{-j}\) its extracted remainder is therefore at most \(C_J\Lambda^{-J}A^{2J+d+2}\), uniformly in \(0\le j<J\). Under Equation (67) this is at most \[C_J(A^2/\Lambda)^J A^{d+2} \le C_J R^{-c\sigma J+2(d+2)}.\] Taking \(J\) large proves all the asserted arbitrary-power estimates.

Finally, \(e\) and \(b\) are independent of \(s\), and each \(s\) derivative of \(\Lambda^{-j}(L_{j,\epsilon}b)(\epsilon e)\) costs only a fixed constant times \(\Lambda^{-j}A^{2j}\), which is bounded. The same elementary Taylor remainder argument can accommodate any prescribed finite number of radial derivatives by increasing the truncation order. In the present applications \(\Lambda\lesssim R^2\), so even unstripped radial derivatives of the error can be made an arbitrary negative power by increasing that order once more. For the support assertion, use \[\Delta_{S^{d-1}}=\sum_{u<v}\Omega_{uv}^2.\] Leibniz expansion of any \(L_j\) or \(B_j\) differentiates, but never removes, each original mask factor. Derivatives of a smooth function cannot enlarge its support. This proves the assertion, including for arbitrary measurable spatial multipliers. ◻

Proof. Write \(x=c_X+2^{-a}u\), \(y=c_Y+2^{-a}z\). Both rescaled endpoint cells lie in fixed boxes. By Equation (52), \(|2^a(c_X-c_Y)+u-z|\) is bounded above and below by positive constants on a fixed neighborhood of their product. Consequently its negative \(j\)th power has a smooth extension, with uniform bounds of every fixed order, supported in a slightly larger product box. Regard this extension as a smooth function on a fixed torus of dimension \(2d\). Its Fourier series has coefficients bounded by \(C_m(1+|k|+|l|)^{-m}\). Choosing \(m>2d\) makes their absolute sum bounded. Each Fourier exponential splits into one function of \(u\) and one of \(z\), each of modulus one. Pulling them back proves the claim. The proof on the fixed separated outer supports is identical. ◻

Proof of Proposition 35. Put \(q=v-a\), \(s=r/2^v\), and \(D_{\max}=\max(1,2^L R^{-50\sigma})\). On the shell in question, \(\Lambda=rD(x,y)\asymp2^q\) and \[D_*=R^\sigma D_{\max} \le\max(R^\sigma,2^{q/2}R^{-48\sigma}),\qquad \frac{D_*^2}{\Lambda} \lesssim\max(R^{2\sigma}2^{-q},R^{-96\sigma}) \lesssim R^{-18\sigma}.\] Moreover \(\Lambda\gtrsim R^{20\sigma}\) and \(D_*\le R^2\). Fixed geometric constants can be absorbed by slightly decreasing these positive exponents. Lemma 36 therefore applies uniformly to the endpoint product \(m_x(w)n_y(w)\).

We first estimate \(\widehat\nu(-r)\), which has the outgoing phase \(e^{irD}\). This suffices also for \(\widehat\nu(r)\): conjugating the spatial multipliers conjugates \(\nu\), and \(\widehat{\overline\nu}(-r)=\overline{\widehat\nu(r)}\). Use Equation (68) pointwise for each \((x,y)\) and multiply by the weight in Equation (60). The leading distance power cancels exactly: \[ (D2^a)^{-n_*/2}(rD)^{n_*/2}(rD)^{-j} = (r2^{-a})^{n_*/2}(r2^{-a})^{-j}(D2^a)^{-j}. \tag{70}\] The operator \(B_j\) is global. Its Leibniz expansion on \(m_xn_y\) is a sum of products of angular derivatives of the individual masks, of total order at most \(2j\). Divide every differentiated mask by its own scale \(D_{l_{\mathfrak t}}\) to the corresponding order. The resulting factors are admissible, retain every descriptor, and their extracted scales have product at most \(D_{\max}^{2j}\). Their number is polynomial in \(N\), since \(j\) is fixed. Smooth angular factors, if produced, can be included in either endpoint symbol.

The inverse factor pays for all these scales, since \[ 2^{-jq}D_{\max}^{2j} \le\bigl[\max(2^{-q},R^{-98\sigma})\bigr]^j\le1. \tag{71}\] Apply Lemma 37 to the remaining factor \((D2^a)^{-j}\). Its absolutely summable expansion only multiplies the original arbitrary spatial factors by bounded endpoint functions. After summing the finitely many inverse orders, we obtain \[ \widehat\nu(-2^vs) =2^{n_*q/2}\sum_\alpha c_\alpha(s) \int\psi(w)U_{X,\alpha}(2^vs,w) \overline{V_{Y,\alpha}(2^vs,w)}\,dw +\mathcal E(2^vs), \tag{72}\] where \[\sum_\alpha\|c_\alpha\|_{L^\infty([1/2,2])} \le R^{o(1)}.\] The factors \(s^{n_*/2-j}\) are included in \(c_\alpha(s)\) and are uniformly bounded on this interval. The index set can be countable because of the spatial Fourier series; its absolute summability justifies all interchanges. All endpoint transforms in Equation (72) have the form Equation (56) with the stated enlarged fixed budget.

The pointwise error \(\mathcal E\) is smaller than any prescribed negative power of \(R\). Indeed Lemma 36 supplies such an error before the endpoint integrations. The rescaled distance weight is bounded, and the normalized cell masses are \(\rho_j(X)/M_j(a)\le R^{O_d(\sigma)}\). Increasing the initial accuracy absorbs these factors and any subsequent square. The shell length is at most \(2^{N+1}=2R\), since \(v\le N\), and is absorbed by the same choice.

Apply the triangle inequality in \(L^2([1/2,2],ds)\) to Equation (72). Since \(\Psi\ge1\) there, the square of each angular integral is bounded by its contribution to the corresponding \(\mathsf H_F\) energy. The complete prefactor is \[ 2^{-a}\,2^v\,2^{n_*(v-a)}=2^{d(v-a)}. \tag{73}\] This is precisely the normalization in Equation (58), proving Equation (62). Complex conjugation gives the positive-frequency formulation stated in the Proposition. ◻

Proof. Put \(b_j=\eta([j\varepsilon,(j+1)\varepsilon))\) and define \(c_j\) using \(\zeta\). Cauchy–Schwarz gives, for every integer \(h\), \[\sum_jb_jb_{j+h}\le\sum_jb_j^2, \qquad \sum_jb_jc_{j+h}\le \left(\sum_jb_j^2\sum_jc_j^2\right)^{1/2}.\] An interval of fixed length \(2C\varepsilon\), with any center, permits only a bounded number of bin offsets. This proves (93)–(94). The kernel in (95) is bounded on the bin pair with offset \(h\) by \(C(1+|h|)^{-2}\), whose sum is finite.

For the last assertion, choose a fixed sufficiently small \(b>0\) and put \(K(t)=(\sin(bt)/(bt))^2\), with its continuous value at zero. Then \(K\ge c>0\) on \([-C,C]\), and its nonnegative Fourier transform is bounded and supported in \([-1,1]\). Fourier inversion and Fubini, applied to \(K((s-s')/\varepsilon)\), prove (96). These arguments require no density for either measure. ◻

Proof. We prove every reduction, including the support conditions needed for its iteration.

Proof. The index \(0\) qualifies in Equation (111). For \(n\le\beta N/(2d)\), the Lipschitz bound in Equation (110) gives \(A_{j,n}\le\beta/2\), proving the lower bound in Equation (112). For \(n\ge N/4\), the lower profile bound gives \(A_{j,n}\ge\chi/4-C_d\sigma>\beta\) when \(\sigma\) is sufficiently small. This proves its upper bound.

By maximality of \(c\), both \(A_{j,c+1}\) exceed \(\beta\), so \(A_{j,c}>\beta-d/N\). Both profiles exceed \(\beta\) at every later index. For at least one side \(j_*\), we also have \(A_{j_*,c}\le\beta\). Since \(\min_{0\le n\le c}A_{j_*,n}\ge-C_d\sigma\), this gives Equation (114). The width of \([c,N]\) exceeds \(3N/4\); thus it exceeds \(100q_0\) for sufficiently small fixed \(\kappa\) and large \(N\). Every pair entering its potential consists of two indices covered by Equation (113), which proves Equation (115). ◻

Proof. Apply the cap preparation in Proposition 23 with loss \(R^\sigma\). It gives \(\mathcal G_N\subset\Gamma_N\) such that \[(\mu_1\otimes\mu_2)(\Gamma_N\setminus\mathcal G_N) \lesssim R^{-c_*} \quad\hbox{for some }c_*>0.\] For \(\mu_1\)-almost every first endpoint \(x\), the angular subprobability \[\xi_{N,x}=e(x,\cdot)_* \bigl(\mathbf 1_{\mathcal G_N}(x,\cdot)\mu_2\bigr)\] has the required \(S\)-cap bound with loss \(R^\sigma\) down to scale \(R^{-1}\). The change from \(\pi_x(y)\) to \(e(x,y)=-\pi_x(y)\) preserves cap bounds, since the antipodal map sends caps to caps. The analogous assertion holds for \(\mu_2\)-almost every second endpoint with the endpoints reversed. These are bounds for restrictions of the original opposite probability.

Fix \(C_1,C_2\) and one descriptor \(\mathfrak t\) used at the first endpoint. For \(\mu_1\)-almost every \(x\in C_1\), the anchor cell determined by \(\mathfrak t\) and \(x\) is occupied and \(\xi_{N,x}\) has the cap bound above. Let \(F_{C_1,C_2,\mathfrak t,x}\) be the directions failing this level-zero test. For such \(x\), Lemma 28 applies directly to \(\xi_{N,x}\) and gives \[ \mu_2\{y\in C_2:(x,y)\in\mathcal G_N, \ e(x,y)\in F_{C_1,C_2,\mathfrak t,x}\}\le R^{-\sigma}. \tag{121}\] Restricting the set of \(y\) to \(C_2\) only decreases the left side; no normalization by \(\mu_2(C_2)\) occurs. Integrating in \(x\) over \(C_1\) costs at most \(a_{1,C_1}R^{-\sigma}\). Sum first over \(C_1\), whose original masses have sum at most one, and then over \(C_2\) and the descriptors. The latter two counts are \(R^{o(1)}\), by Lemma 4 and Lemma 31. Thus the first-endpoint failure mass is \(R^{-\sigma+o(1)}\). The same argument in the other orientation gives this bound for second-endpoint failures. There is no sum over anchor cells: a point and descriptor already specify one.

The product in Equation (117) equals one if all its level-zero tests hold. Hence \(1-\Theta_N\) on retained class pairs is bounded by the indicator of the union of these failures. Including the original mass outside retained class pairs and \(\Gamma_N\setminus\mathcal G_N\) gives \[\int_{\Gamma_N}(1-\Theta_N)\,d(\mu_1\otimes\mu_2) \lesssim R^{-c_*}+R^{-\sigma+o(1)}.\] The bounded weight \(W\) proves Equation (119), for example with any fixed \(0<\eta<\min(c_*,\sigma)/2\) and sufficiently large \(N\). The set \(\mathcal G_N\) was used only in this estimate: it is not a multiplier in Equation (118).

The filters \(\Gamma_N\) decrease and have positive limiting product mass. Continuity of a finite measure along decreasing sets proves Equation (120) and monotonicity of \(\alpha_N\). The positive lower bound for \(W\) shows \(\alpha_\infty(\mathbb R)>0\). Finally, the two original supports are compact subsets of \(E\), and \(D\) is continuous. Every pushforward above is therefore carried by their compact distance image, which is contained in \(\Delta(E)\). ◻

Proof. For the moment fix one normalized class pair and one product \(X_\ell\times Y_\ell\) in the spatial partition of \(\Gamma_N\). The corresponding component measure \(\nu\) is the pushforward of \[W(x,y)\mathbf 1_{X_\ell}(x)\mathbf 1_{Y_\ell}(y) b_{1;C_1,C_2}(x,e(x,y))b_{2;C_1,C_2}(y,e(x,y)) \,d\rho_{1,C_1}(x)\,d\rho_{2,C_2}(y).\] The product indicators are permitted bounded measurable spatial multipliers in the oscillatory estimates. At depth zero take \(X=Y=Q_0\), each with its own normalized class law. These are occupied root cells, with \(M_1(0)=M_2(0)=1\). Their supported laws lie in the fixed separated sets \(E_1,E_2\), so the outer-support geometry of Section 7 applies. The cell-center separation in Equation (52) is the internal-cell convention; the outer root uses the separation of the supported laws. In particular, all root spatial diameters in the graph argument are \(O_d(1)\), while the actual cross distances have a fixed positive lower bound. By Lemma 31, the largest nominal length of an endpoint descriptor is at most \[L\le\max\bigl(c,(N-c)/2\bigr)\le N/2.\] Thus the root shell conversion, Proposition 35, applies with \(a=0\), \(v=N\). Its weight is precisely \(W\); there is no internal normalization of the distance weight at this root. For any prescribed fixed \(T>0\), it bounds the squared Fourier norm on the positive shell by \(R^{o(1)}\sup\mathsf H_F(0,N)+O(R^{-T})\). The supremum retains the root cells, class normalizations, lists, and smoothing index zero; its admissible symbols have the fixed derivative budget and common numerical angular bounds supplied by that Proposition for \(T\).

Apply Corollary 41 at \(p=c\) to each energy in this supremum. The required parallel descriptor is the extra \([0,c]\) on each side, with \(g=0\). All scale conditions hold: \(c>0\) for large \(N\), \(c<N/4\), \(L\le N/2<N-c\), and \(N-c\ge10\sigma N\). Since \(A_{j,0}=0\), the graph factor is \[ R^{-A_{1,c}-A_{2,c}+\min_j k_j(0,c)+O_d(\sigma)+o(1)}. \tag{123}\] The graph estimate produces a general two-profile energy \(\mathsf H(c,N)\) on a pair of occupied \(c\)-cells, still carrying the same endpoint symbols.

On these cells the extra \([0,c]\) factor is spatially constant within each endpoint cell. It, and any of its normalized differentiated versions produced by shell conversion, can be removed at bounded cost from the positive independent angular energies. The remaining descriptor lists are exactly those with base \([c,N]\) and with reached anchors omitted. This removal is made in the general positive energy \(\mathsf H\), where the two angular factors are nonnegative.

To separate the two profiles, write their nonnegative angular energies as \(E_1(r),E_2(r)\) and use the common measure \(d\lambda_N(r)=\Psi(r/2^N)\,dr/2^N\). Cauchy–Schwarz gives the exact normalization \[ \begin{split} 2^{d(N-c)}\int E_1(r)E_2(r)\,d\lambda_N(r) &\le \left(2^{d(N-c)}\int E_1(r)^2\,d\lambda_N(r)\right)^{1/2}\\ &\quad\cdot \left(2^{d(N-c)}\int E_2(r)^2\,d\lambda_N(r)\right)^{1/2}. \end{split} \tag{124}\] Each parenthesis is the independent angular energy formed by duplicating one endpoint law, cell, and symbol. With the unreached descriptors of \(\mathcal M_j(c,N)\), it is the single-profile recursive state \(\mathsf H_0^{[c,N]}(c,N)\) for profile \(j\); denote this value by \(\mathsf H_j(c,N)\). Its symbols have the fixed derivative budget and common numerical angular bounds supplied by shell conversion; restriction to the \(c\)-cells and duplication preserve these bounds. These states impose no separation condition, so Theorem 49 applies to each, including the case of two identical cells. It yields \[\mathsf H_j(c,N) \le R^{2A_{j,c}-\mathcal V_j([c,N]) +O_d(\kappa+\sigma/\kappa)+o(1)}.\] The profile terms in Equation (123) and the geometric mean of these bounds combine to \[ \begin{split} &-A_{1,c}-A_{2,c}+\min_j k_j(0,c) +\frac12\sum_{j=1}^2 \bigl(2A_{j,c}-\mathcal V_j([c,N])\bigr)\\ &\hspace{20mm} =\min_j k_j(0,c)-\frac12 \bigl(\mathcal V_1([c,N])+\mathcal V_2([c,N])\bigr) \le-\beta+C_d\sigma+2d/N, \end{split} \tag{125}\] where the last inequality uses the cost and potential bounds in Lemma 50. Including the stated losses proves uniformly for each component that \[ \int_{2^{N-1}\le r\le2^{N+1}}|\widehat\nu(r)|^2\,dr \lesssim R^{-\beta+C_d(\kappa+\sigma/\kappa)+o(1)}+R^{-T}. \tag{126}\] Here \(T\) can be any fixed sufficiently large number, after choosing the expansion and integration-by-parts orders. The term \(O_d(\sigma)\) has been included in \(O_d(\sigma/\kappa)\), with \(\kappa<1\).

We make the parameter and error choices explicit. First choose \(\beta\) as in Lemma 50. Next choose \(\kappa\) so small that the entry interval is nonterminal and \(C_d\kappa<\beta/16\). Then choose \(\sigma\) so small that all the profile and transition hypotheses hold and \(C_d\sigma/\kappa<\beta/16\). Finally choose \(\omega>0\) so small that \[ 4d\omega<\beta/8. \tag{127}\] The preparation of \(\Gamma_N\) is used with this fixed \(\omega\) and cap loss \(R^\sigma\). Its deletion exponent \(c_*\) may depend on both parameters; only \(c_*>0\) was needed in Lemma 51.

For completeness, the error accuracy is fixed after these parameters and before \(N\) tends to infinity. Theorem 49 bounds the number of steps on a branch by a constant depending on \(\kappa,\sigma\). Choose a fixed local negative-power accuracy large enough to dominate all ancestor factors and all errors on such a branch, leaving the root remainder \(R^{-T}\) in Equation (126), with, say, \(T>10+4d\omega\). The inverse expansion orders are then fixed and determine a finite symbol derivative budget. The direct expansion and integration-by-parts orders are chosen for that budget and accuracy, as in Theorem 49. The graph step and Equation (124) introduce no persistent new derivative orders; the spatial indicators have zero angular derivatives. These choices affect constants and the threshold for \(N\), and do not change the dimensional coefficients in the exponent losses above. We now take \(N\) sufficiently large for all these fixed choices and for the uniform \(o(1)\) losses.

There are at most \(L_N\lesssim R^{2d\omega}\) spatial products in \(\Gamma_N\). Let \(\nu_{C_1,C_2,\ell}\) denote the preceding component. The exact recombination is \[\tau_N=\sum_{C_1,C_2}a_{1,C_1}a_{2,C_2} \sum_{\ell=1}^{L_N}\nu_{C_1,C_2,\ell}.\] Triangle inequality in the Fourier \(L^2\) norm costs no class-count factor, because \[\sum_{C_1,C_2}a_{1,C_1}a_{2,C_2} =\left(\sum_{C_1}a_{1,C_1}\right) \left(\sum_{C_2}a_{2,C_2}\right)\le1.\] It costs at most \(L_N\) for the spatial products. Squaring therefore costs \(R^{4d\omega}\), giving \[ \int_{2^{N-1}\le r\le2^{N+1}} |\widehat{\tau_N}(r)|^2\,dr \lesssim R^{-\beta+C_d(\kappa+\sigma/\kappa)+4d\omega+o(1)} +R^{-T+4d\omega}. \tag{128}\] Our successive choices leave a strictly negative exponent; for example, increasing the starting \(N\) if necessary, the right side is \(O(R^{-\beta/2})\). Each \(\tau_N\) is a real positive measure, so \(\widehat{\tau_N}(-r)=\overline{\widehat{\tau_N}(r)}\). The same bound holds on the negative shell, proving Equation (122) with some \(\gamma>0\). ◻

Proof. Choose \(\vartheta\in C_c^\infty(\mathbb R)\) equal to one on \([-1,1]\) and supported in \([-2,2]\). Let \(Q_N\) be convolution with the inverse Fourier transform of \(\vartheta(2^{-N}r)\), and put \(P_N=Q_N-Q_{N-1}\). The multiplier of \(P_N\) is supported in \(\{2^{N-1}\le|r|\le2^{N+1}\}\). Dilation of the fixed Schwartz kernel gives a constant \(C_\vartheta\) such that, for every finite measure \(\nu\), \[ \|Q_N\nu\|_{L^1}\le C_\vartheta\|\nu\|_{\mathrm{TV}},\qquad \left\|\sum_{N=a}^bP_N\nu\right\|_{L^1} \le2C_\vartheta\|\nu\|_{\mathrm{TV}}\quad(a\le b). \tag{130}\] The second bound follows from telescoping to \(Q_b-Q_{a-1}\); in particular its constant is independent of the number of bands.

With the convention \(\widehat\nu(r)=\int e^{-irt}\,d\nu(t)\), Plancherel (Mattila 1995, equation (12.4)) and the uniformly bounded band multipliers give \[\|P_N\tau_N\|_{L^2} \lesssim \left(\int_{2^{N-1}\le |r|\le2^{N+1}} |\widehat{\tau_N}(r)|^2\,dr\right)^{1/2}.\] Thus \[ G:=\sum_{N\ge K}P_N\tau_N\quad\hbox{converges in }L^2, \qquad H:=\sum_{N\ge K}P_N(\alpha_N-\tau_N) \quad\hbox{converges in }L^1. \tag{131}\] Both series converge absolutely in the indicated Banach space.

It remains to account for the variation of \(\alpha_N\). Set \(\zeta_n=\alpha_n-\alpha_{n+1}\ge0\). Monotonicity and finiteness give \[ \sum_{n\ge K}\|\zeta_n\|_{\mathrm{TV}} =\alpha_K(\mathbb R)-\alpha_\infty(\mathbb R)<\infty, \qquad \alpha_N-\alpha_\infty=\sum_{n\ge N}\zeta_n \quad\hbox{in total variation}. \tag{132}\] For a finite \(M\ge K\), define \(Z_M=\sum_{N=K}^MP_N(\alpha_N-\alpha_\infty)\). Substituting Equation (132) into this finite sum and using the bounded operators from measures to \(L^1\) gives the exact identity \[ Z_M=\sum_{n\ge K} (Q_{\min(n,M)}-Q_{K-1})\zeta_n. \tag{133}\] For each fixed \(M\) the series on the right converges in \(L^1\). Moreover, Equation (130) shows that \[Z:=\sum_{n\ge K}(Q_n-Q_{K-1})\zeta_n\] converges absolutely in \(L^1\). Subtracting the two series cancels all \(n\le M\), and gives \[ \begin{split} \|Z_M-Z\|_{L^1} &\le2C_\vartheta\sum_{n>M}\|\zeta_n\|_{\mathrm{TV}}\\ &=2C_\vartheta \|\alpha_{M+1}-\alpha_\infty\|_{\mathrm{TV}} \longrightarrow0. \end{split} \tag{134}\] This is where monotonicity is used. The original double series has not been rearranged without a convergence justification.

Finally, finite telescoping gives \[ Q_M\alpha_\infty =Q_{K-1}\alpha_\infty +\sum_{N=K}^MP_N\tau_N +\sum_{N=K}^MP_N(\alpha_N-\tau_N)-Z_M. \tag{135}\] The fixed low-frequency term \(Q_{K-1}\alpha_\infty\) is a smooth \(L^1\) function by convolution with a Schwartz kernel. By Equations (131) and (134), the right side converges in \(L^1+L^2\), hence in distributions, to \[f=Q_{K-1}\alpha_\infty+G+H-Z\in L^1_{\mathrm{loc}}(\mathbb R).\] The kernels defining \(Q_M\) form an approximate identity, so the left side of Equation (135) converges in distributions to \(\alpha_\infty\). Therefore \(\alpha_\infty=f\,\mathcal L^1\). Since \(\alpha_\infty\) is a finite positive measure, this locally integrable density is nonnegative almost everywhere and belongs to \(L^1(\mathbb R)\). This proves absolute continuity. ◻

Proof. Choose the parameters as in Proposition 52. Lemma 51 gives decreasing \(\alpha_N\) with nonzero limit \(\alpha_\infty\), and approximants \(\tau_N\) with summable total-variation error. Proposition 52 gives the second summability hypothesis of Lemma 53. That Lemma makes \(\alpha_\infty\) absolutely continuous. By Lemma 51 it is carried by \(\Delta(E)\) and has positive mass, so \(\Delta(E)\) cannot be a Lebesgue null set. ◻

Proof of Theorem 1. Start with a Frostman probability on the given compact set \(E\subset\mathbb R^d\), \(d\ge2\), with exponent strictly larger than \(d/2\). In the odd-dimensional exceptional affine-plane case, Lemma 18 already proves the conclusion using original distances inside that plane. In every other case, Proposition 23 supplies the separated original probabilities and filters used above, and Proposition 54 proves the conclusion. This includes \(d=2\): its cap preparation is the integer case of Lemma 21. The auxiliary projections used in preparation control directions only; the measures in Equation (118) and their nonzero limit are pushforwards by \(|x-y|\) in \(\mathbb R^d\). Thus the argument establishes the asserted positive measure for the original distance set in every dimension \(d\ge2\). ◻