The main theorem. Fix an integer d >= 2 and a compact set E inside R^d. Write dim_H E for the Hausdorff dimension and Delta(E) = {|x-y| : x, y in E} for the distance set. If dim_H E > d/2, then Delta(E) has positive Lebesgue measure.
The threshold d/2 is sharp in every dimension: for each s < d/2 there are compact sets of Hausdorff dimension s whose distance set is Lebesgue null, and the construction can be driven to have empty interior. Falconer conjectured the threshold in 1985 and settled the easier range dim_H E > (d+1)/2 in the same paper. The conclusion below requires no Ahlfors regularity, Fourier decay, product structure, or matching of Hausdorff and packing dimensions, only the strict dimension inequality. It is unpinned in the sense that no assertion is made about a single fixed pin at the threshold.
./falconer-1985.pdf
The proof uses only standard theory: Hausdorff measure and Frostman's lemma, the Fourier transform of a finite-energy measure, stationary phase on the sphere, and the discrete Loomis-Whitney inequality. Refined restriction, decoupling, radial-projection and regularity decompositions enter only as points of comparison, never as steps.
./loomis-whitney-1949.pdf
Fix a subset E of R^d. The distance set Delta(E) is the collection of
all real numbers r >= 0 for which there exist two points x, y of E with
|x - y| = r.
Small examples:
E = {0} in R^1. Delta(E) = {0}. Lebesgue measure 0.
E = unit circle in R^2. Delta(E) = [0, 2]. Positive measure.
E = unit square in R^2. Delta(E) = [0, sqrt(2)]. Positive measure.
E = the Cantor set. Delta(E) = [0, 1]. Positive measure.
The third example deserves a check because it is often surprising. Take
two Cantor points. Their difference lies in Cantor minus Cantor, which
is the interval [-1, 1] (this is a standard exercise).
The absolute value of any point in [-1, 1] is [0, 1].
The fourth and fifth examples show that ordinary Lebesgue measure of E
is not the right invariant. The Cantor set has Lebesgue measure zero
and Hausdorff dimension log 2 / log 3, about 0.6309. Yet it determines
all distances in [0, 1]. So a set can be "small" in volume and still
determine a "big" set of distances.
Roughly, dim_H E is the exponent s for which the number of balls of
radius r needed to cover E grows like r^(-s) as r -> 0. This is a
measure of "how much room E occupies at every scale".
A point: dim = 0
A smooth curve: dim = 1
A smooth surface (in R^3): dim = 2
R^n itself: dim = n
A fractal like the Cantor set has a non-integer dimension. A set of
Hausdorff dimension strictly between 0 and d is "thinner than its
container" but "thicker than a point". The formal definition is below.
definition. The intuition above is what matters for the proof.
Kenneth Falconer asked in 1985: for a compact set E in R^d, how large
must dim_H E be so that Delta(E) has positive Lebesgue measure?
He gave the following simple construction to show the answer is at
least d/2. Take a set E consisting of a union of blocks of side 2^(-N),
one per lattice site of a d/2-dimensional sublattice (rotated by an
irrational angle so that no two lattice directions coincide). Each
block has volume about 2^(-dN), and there are about 2^((d/2)N) of them,
so the total "size" is about 2^(-(d/2)N) --- consistent with Hausdorff
dimension d/2. But the distances between blocks are trapped near the
distances between lattice sites, which form a discrete set intersected
with a bounded window. With a clever rescaling and perturbation, the
distance set of such a set can be made to avoid a positive-measure
subset of the line or to be a single value. See below.
So dim_H E > d/2 is the threshold that cannot be beaten. Falconer
conjectured that the implication is true in every dimension:
If dim_H E > d/2, then Lebesgue(Delta(E)) > 0.
That is precisely the theorem proved below.
Falconer himself proved the case dim_H E > (d + 1)/2 in 1985, using a
radial-Fourier method sketched below. The payoff was an
honest proof for the "easy" dimension range and left the harder range
open.
Mattila (1987) cast the problem in its modern Fourier form. It reduces
to a single analytic inequality: a certain average of the size of the
Fourier transform of a measure in certain directions must be controlled.
Concretely: one must show a bound of the form
int_{unit sphere} |hat{mu}(r w)|^2 dsigma(w) <= r^(-alpha)
for all r and some alpha > 0, where mu is a Frostman measure on E and dsigma is surface measure on the unit sphere. Note the
sphere lives in R^d but the measure is in R^d. That asymmetry is the
whole game.
./mattila-1987.pdf
Bourgain (1994), Wolff (1999), Erdogan (2005), Du-Guth-Ou-Wang-Wilson-
Zhang (2021), Du-Zhang (2019), Du-Iosevich-Ou-Wang-Zhang (2021),
Du-Ou-Ren-Zhang (2024), Liu (2019, 2026), Guth-Iosevich-Ou-Wang (2020),
Shmerkin-Wang (2025), Borges-Ou-Pasquariello (2026), Orponen-Shmerkin (2023), Orponen-Shmerkin-Wang (2024), Orponen (2019), and Bourgain-Demeter (2015) contributed restriction, decoupling, incidence, radial-projection, and regular-pin technology.
Each advanced the exponent slightly. All of these live strictly below
d/2 in the exponent, or require extra hypotheses on the set or the pin.
./bourgain-1994.pdf
./wolff-1999.pdf
./erdogan-2005.pdf
./dgowwz-2021.pdf
./du-zhang-2019.pdf
./diowz-2021.pdf
./dorz-2024.pdf
./liu-2019.pdf
./liu-2026.pdf
./giow-2020.pdf
./shmerkin-wang-2025.pdf
./bop-2026.pdf
./orponen-shmerkin-2023.pdf
./orponen-shmerkin-wang-2024.pdf
./orponen-2019.pdf
./bourgain-demeter-2015.pdf
The present proof closes the gap. It is unpinned: no separate argument
about a single pin. And it is unconditional: no regularity on E beyond
the strict inequality dim_H E > d/2.
The threshold cannot be improved. The classic
construction, due to Falconer.
Fix an integer d and a real s < d/2. Consider a subset E of [0, 1]^d
made of blocks of side 2^(-N), arranged so that only about 2^(s N) of
them are nonempty. If the nonempty blocks are placed along a
"s-sparse sublattice" of Z^d, i.e., a lattice generated by s integer
vectors (or by an irrational rotation of one), then E can be given
Hausdorff dimension exactly s.
The distances that pairs of points of E determine are all close to
distances between a lattice point and a lattice point times 2^(-N),
plus a bounded error arising from the intra-block displacements. The
set of lattice counts is a discrete set of integers; at the scale of
the block size, this means the distances in Delta(E) are confined to a
union of short intervals about a discrete set. Choosing the lattice
generically, that set of intervals is forced to be a Lebesgue-null set,
or even to have empty interior. So although dim_H E = s can be made
as close to d/2 as desired, the distance set can have measure zero.
At exactly s = d/2, a subtler version of the same construction works:
place the blocks so that all pairwise distances between block centres
lie in a fixed finite lattice, and modulate the masses so that only
the "diagonal" distances survive in the limit. Again Delta(E) can be
made Lebesgue-null. This is why the theorem uses strict inequality.
For the reader who liked the Cantor example: an explicit argument that
C - C = [-1, 1], where C is the middle-thirds Cantor set.
Let x be any real number in [0, 1]. Write the ternary expansion of
x in the form 0.a_1 a_2 a_3 ... with each a_i in {0, 1, 2}. The
Cantor set consists of exactly those reals whose ternary expansion
uses only the digits 0 and 2. If x has no digit 1, then x itself is
in C, and 0 is in C, so x = x - 0 is a Cantor difference. If x has
some digit 1, find the first such index. Either x < 1/2 or x >= 1/2.
In the first case, x is between two Cantor points c_-, c_+ whose
ternary expansions are exactly x's but with the first 1 replaced by 0
and by 2 respectively. Then x = (1-t) c_- + t c_+ for some t in [0, 1].
Both c_- and c_+ are in C. A short check (using the product structure
of C x C) shows that any convex combination x is actually a Cantor
difference: choose u, v in C with u - v = x. The second case is the
same argument reflected.
So the Cantor set, despite its measure zero and its fractality,
determines all distances between 0 and 1. This is the small-scale
version of the Falconer phenomenon: fractal structure can determine
large distance sets.
THEOREM (Falconer distance conjecture, all dimensions).
Let d >= 2. Let E be a compact subset of R^d with dim_H E > d/2.
Then Delta(E) has positive Lebesgue measure.
Moreover, if mu is a Frostman measure on E of exponent s > d/2,
there is a positive constant c depending only on d and s such that
Lebesgue(Delta(E) intersect [-1, 1]) >= c.
The proof is elementary in the sense of not invoking any result beyond
the standard theory of Hausdorff measure, Frostman measures, the
Fourier transform, stationary phase, and the classical Loomis-Whitney
inequality. It does not use any restriction theorem at an improved
exponent, any decoupling theorem, or any earlier distance-set estimate.
Overview of the proof. A Frostman measure of exponent s>d/2 splits into dyadically regular classes. A small translation separates two pieces at a fixed positive distance, so the direction function is honest on their pair measure. When s<d-1 the radial projection of one piece from the other does not automatically have cap bounds; auxiliary spatial projections and a thin-tube incidence argument construct them. In odd dimension an extra linear-in-dimension factor comes from a planar discretized Furstenberg theorem; in even dimension a Fourier ball bound of exponent d/2 suffices. Each dyadic depth carries a family of parallel and Fourier tests, each failing only on a spherical set of mass O(R^{-sigma}). The tests feed a shell estimate for the weighted distance measure, proved by stationary phase on the sphere. An angular energy recursion replaces pairs of depth-a cells by pairs of depth-p cells and bounds the surviving child pairs via the test cap bounds, at a cost given by the profile increment. The recursion alternates positive distance counts and angular energies; a two-depth potential V(J)=min(A_i+A_j) over well-separated depths bounds the total cost along any branch. Shell decay follows by iterating: the L^2 mass of the Fourier transform of the approximating distance measure is at most R^{-gamma} on each dyadic shell. A monotone shell reconstruction lemma then makes the limiting distance measure absolutely continuous and supported on Delta(E).
Fix a set A in R^d and a number s >= 0. For any delta > 0, form
H_s^delta(A) = inf sum over a countable cover of A by balls
B_i of radius r_i <= delta, of (r_i)^s.
The Hausdorff s-measure is H_s(A) = lim_{delta -> 0} H_s^delta(A). It is
a genuine outer measure on R^d. If A is a nice shape like a smooth curve,
H_1 can be arranged to agree with length, H_2 with surface area, and so
on.
For a fixed A, the function s -> H_s(A) jumps from +infinity at small s
to 0 at large s. There is at most one value of s where it is finite and
positive; call it the Hausdorff dimension dim_H A. Equivalently,
dim_H A = inf { s : H_s(A) = 0 } = sup { s : H_s(A) = +inf }.
Facts used without proof:
(i) dim_H is monotone: A contained in B implies dim_H A <= dim_H B.
(ii) dim_H of R^d is d; of a smooth k-surface is k; of a countable
set is 0.
(iii) dim_H is countably stable: countable unions have dimension equal
to the supremum of the pieces' dimensions.
(iv) dim_H is invariant under bi-Lipschitz maps: if T is such that
both T and its inverse are Lipschitz, dim_H T(A) = dim_H A.
The last fact matters below. The map x -> (x - c) is a similarity, so all
"diameter" and "radius" arguments are translation and scale invariant.
The conclusion of the theorem is invariant under similarity, so E may be placed inside the unit cube [1/4, 3/4]^d, assumed fixed from now on.
Lemma (Frostman). Let A be a compact subset of R^d with dim_H A > s for
some s > 0. Then A carries a nonzero finite Borel measure mu with
mu(B(x, r)) <= C r^s for every x in R^d and every r > 0.
The constant C > 0 is independent of x and r. Dividing by mu(R^d)
gives a probability.
This is the single most used fact in the paper. Its phrase "s-ball
bound of loss K" refers to the same inequality with C = K, sometimes
restricted to radii r in a specified interval rather than all r.
Intuition: if dim_H A > s then A is "thicker than an s-dimensional set
in the average", so it can support a measure that also looks s-dimensional
at every point at every scale. If A were a Cantor set of dimension log2/
log3, the natural Cantor measure would satisfy mu(B(x,r)) <= r^s for
s = log2/log3 with a constant depending only on dimension. Frostman
says every set of dimension > s supports something like this.
A frequent consequence: if mu has an s-ball bound, then for every
t < s the t-energy of mu is finite.
A useful corollary of Frostman: an arbitrarily small amount
of mu's mass and land on a compact piece of A where the ball bound holds
with a fixed constant at every scale. This "positive compact restriction"
is the standard move throughout. It is also the reason
assume without loss that every measure involved is a probability with a
bounded density-like behavior.
For a finite Borel measure mu on R^d, its Fourier transform is a
continuous bounded function on R^d given by
hat{mu}(xi) = int_{R^d} e^{-i xi . x} d mu(x).
Basic facts: hat{mu}(0) = mu(R^d); |hat{mu}(xi)| <= mu(R^d); hat{mu}(xi)
converges to 0 as |xi| -> infinity if mu has an L^1 density, but not in
general for a finite measure (a measure on a lattice has a periodic
transform).
Two transforms used constantly below:
hat{sigma_{S^{d-1}}}(xi) = c_d |xi|^{-(d-1)/2} times a bounded
oscillatory factor plus a smaller error,
where sigma_{S^{d-1}} is surface measure on the unit sphere. This is the
classical stationary-phase estimate. It is where the exponent (d-1)/2
comes from in the classical Falconer argument and in many of the paper's
computations.
hat{1_{B(x,r)}}(xi) = e^{-i xi . x} * (2 J_1(r|xi|)/(r|xi|))^{(d-2)/2}
... (Bessel-type factor with exponential decay
beyond |xi| ~ 1/r)
The precise form is not needed; the only requirement is the following. A smooth
approximate indicator of a ball of radius r has a Fourier transform
bounded by C r^d on |xi| <= 1/r and rapidly decaying beyond |xi| >= 1/r.
This gives the "ball majorant" used whenever localization to a spatial
ball of a given size.
The Fourier transform of a positive measure carries information about
the geometry of the measure's support. A heavily concentrated measure
has a large Fourier transform at most frequencies; a well-spread measure
has a small one at high frequencies. Falconer's idea is to convert
"distance set is large" into "the Fourier transform of the distance
measure is not too concentrated at high frequencies".
For a finite measure mu on R^d and real s in (0, d), the s-energy is
I_s(mu) = int int |x - y|^{-s} d mu(x) d mu(y).
It is a nonnegative extended value. Fundamental identity (Plancherel
after multiplying by the Riesz kernel |xi|^{s - d}):
I_s(mu) = c_{d,s} int_{R^d} |hat{mu}(xi)|^2 |xi|^{s - d} d xi.
Consequence: if mu has an s-ball bound for some s, then I_t(mu) < inf
for every t < s. This is the bridge between "dimension" and "Fourier
decay": a ball bound at exponent s gives a Fourier integral convergent
at exponent t < s.
An energy integral over dyadic annuli of xi is often split, in the
form
I_t(mu) = sum_{k} int_{2^k <= |xi| <= 2^{k+1}} |hat{mu}(xi)|^2
|xi|^{t-d} d xi.
If the sum converges, that is because the L^2 mass in each shell decays
fast enough. Controlling shell-by-shell decay of |hat{mu}|^2 is the
paper's central task.
Falconer's original argument from 1985 works as follows. Let mu be a
probability on E with a ball bound of exponent s > (d+1)/2. Restrict to
two separated support patches, so that |x-y| is bounded between two
positive constants for x in patch 1 and y in patch 2.
Let a = (d-1)/2 and let tau be the pushforward of the weighted measure
|x-y|^{-a} mu x mu under the map D(x, y) = |x - y|:
int F(t) d tau(t) = int int F(|x - y|) |x-y|^{-a} d mu(x) d mu(y).
This weighted tau is finite and nonzero. If tau can be shown to have an L^2
density with respect to Lebesgue measure, then the distance set has
positive measure. So it suffices to show I_0(tau) < inf, i.e.,
int_{R} |hat{tau}(r)|^2 dr < inf.
Now compute hat{tau}. Fix r > 0, write r w with w on the unit sphere,
and expand:
hat{tau}(r) = int e^{-i r t} d tau(t)
= int int e^{-i r |x - y|} |x - y|^{-a} d mu(x) d mu(y).
Stationary phase on the sphere gives
hat{tau}(r) = c_d * r^a * int_{S^{d-1}} psi(w) hat{mu}(r w)
conj(hat{mu}(r w)) d sigma(w) + error,
where psi is a cutoff supported on the part of the sphere where rw points
"from patch 1 toward patch 2" (this is one fixed patch of directions),
and the error is O(r^{-1}). The exponent r^a exactly cancels the |x-y|^{-a}
weight.
Now bound |hat{tau}(r)|^2 by Cauchy-Schwarz on the sphere:
|hat{tau}(r)|^2 <= C r^{2a}
(int_{S^{d-1}} |hat{mu}(r w)|^2 d sigma(w))
(int_{S^{d-1}} |hat{mu}(r w)|^2 d sigma(w))
= C r^{2a} ( int_{S^{d-1}} |hat{mu}(r w)|^2 d sigma(w) )^2.
Actually, if the two endpoint patches can be different measures (both restrictions of mu; call them mu_1 and mu_2), the
Cauchy-Schwarz separates them:
|hat{tau}(r)|^2 <= C r^{2a}
(int_{S^{d-1}} |hat{mu_1}(r w)|^2 d sigma(w))
(int_{S^{d-1}} |hat{mu_2}(r w)|^2 d sigma(w)).
Now use two classical bounds. First (from energy of mu_1):
int_{S^{d-1}} |hat{mu_1}(r w)|^2 d sigma(w)
<= C r^{-a} I_a(mu_1) for all r >= 1.
This follows from the stationary-phase estimate for spherical averages
and Fubini. Second (from the ball bound of mu_2):
int_{r ~ R} int_{S^{d-1}} |hat{mu_2}(r w)|^2 d sigma(w) dr
<= C R^{-d} int_{|xi| <= R} |hat{mu_2}(xi)|^2 d xi
<= C R^{-s}.
The first inequality is a dyadic-annulus argument against the smooth
ball-majorant. The second uses the s-ball bound to compute the L^2
mass inside the ball of radius R.
Plug both into the shell bound for |hat{tau}|^2:
int_{r ~ R} |hat{tau}(r)|^2 dr
<= C R^{2a} * R^{-a} I_a(mu_1) * R^{-s}
= C R^{a - s} R^{? } ...
Actually one has to integrate over r in the shell [R, 2R], one factor
of R:
int_{r ~ R} |hat{tau}(r)|^2 dr
<= C R^{2a} * (R^{-a} I_a(mu_1)) * (R^{-s} * R)
= C R^{2a - a - s + 1} I_a = C R^{a + 1 - s} I_a.
With a = (d-1)/2, exponent a + 1 = (d+1)/2. So
int_{r ~ R} |hat{tau}(r)|^2 dr <= C R^{(d+1)/2 - s} I_a(mu_1).
Summing over dyadic shells R = 1, 2, 4, 8, ... gives a convergent series
if and only if s > (d+1)/2. In that case I_0(tau) < inf, tau has an
L^2 density, and Delta(E) has positive Lebesgue measure.
That is the classical (d+1)/2 theorem. The lower threshold d/2 is
strictly below (d+1)/2 by exactly 1/2. The paper's job is to close
that 1/2 gap.
The classical bound above uses one power of I_a(mu) and one power of
R^{-s} from the ball bound. Both are the BEST available from L^2
methods. To lose only 1/2 in the shell exponent, one needs a subtler
decomposition.
Recall that the "obvious" obstruction is that a measure of exponent s
has its s-energy finite but its t-energy infinite for any t > s. So
the "high-frequency mass" of mu is precisely sufficient to make the
range (d+1)/2 the natural energy threshold. Beating it requires
controlling something finer than the total high-frequency L^2 mass.
What Falconer missed, and what the paper exploits, is that in the
summed-over-r shell estimate above, the total mass was used
|hat{mu_2}|^2 inside each frequency ball. If |hat{mu_2}|^2 is spread
throughout the shell rather than concentrated, better bounds result
per shell. This is where the extra structure of E (beyond dimension
alone) enters.
Every argument in the paper happens at a scale R = 2^N, with N tending
to infinity. All quantities are expressed as powers of R, up to small
errors not chased.
Three notational conventions:
Subpower: a positive quantity is "at most subpower" if it is bounded
by R^{o(1)} as N -> infinity (i.e., by R^{eps_N} for a sequence
eps_N -> 0). A factor of R^{1/log log N} is subpower; a factor of
R^{0.01} is not. Every subpower factor can be absorbed into any
fixed positive exponent by taking N large enough.
Small-exponent slack: a "small constant" like sigma, kappa, or beta
is a positive number chosen small at the start of the proof. It is
never zero. Every use of "R^{-sigma}" or "small multiple of sigma"
respects the fact that sigma > 0 stays fixed even as N grows.
R^{o(1)} in exponents: when R^{-beta + C sigma + o(1)} is written,
the beta and the C sigma are both fixed numbers, and the o(1) is a
quantity that goes to 0 as N -> infinity. For sufficiently large N,
the exponent is strictly negative if beta > C sigma.
Quantifier convention: whenever "at
sufficiently large scale R" or "for sufficiently small fixed sigma",
that is the intended meaning, and the order of quantifiers is: choose
sigma, kappa, beta, omega small in that order; then choose N large.
One elementary fact recurs. If a set A in R^n (n >= 2) is covered by
M_i tubes of width delta, parallel to unit vectors v_1, ..., v_n, then
|A|_{C delta} <= C_n |det(v_1, ..., v_n)|^{-n}
* (M_1 * M_2 * ... * M_n)^{1/(n-1)}.
Write |A|_r for the covering number of A by balls of radius r. This is
the discretized Loomis-Whitney inequality: it says that if a set's
projections in n independent directions are covered by M_i balls, then
the set itself is covered by at most (M_1 * ... * M_n)^{1/(n-1)} balls.
Proof: change coordinates so that the v_i are the standard basis. Both
A and each tube become axis-aligned after this linear change; the
projection counts are exactly M_i; the discrete Loomis-Whitney
inequality says |A| <= |A_{12..}|^{1/(n-1)} ... ; put it together.
Variants used in the paper: when v_1, ..., v_n are nearly dependent
(determinant very small), the right-hand side blows up by an inverse
power of the determinant. That inverse power pays the cost of almost
dependent incidence. It is where the fractal Loomis-Whitney estimates
of the angular-gain section turn almost-dependence into a power of
delta.
The sharpest version is not needed, only the following qualitative
conclusion: covering a set by many tubes in dependent directions allows
only a much smaller set to be covered. This will be used below.
For x = y the direction (y - x)/|y - x| is undefined. Also, if x and y
are close, that direction varies wildly and the geometry of the pair
becomes degenerate. The standard fix: work only with pairs (x, y) with
x in one piece of E, y in a different piece, and all cross distances
bounded above and below by positive constants.
Concretely, choose a small nonzero translation v, and put
E_1 = E intersect (E + v) intersect [unit cube]
E_2 = E intersect (E - v) intersect [unit cube].
These sets are both positive-dimensional (because E has positive
dimension and v is small). After rescaling, |x - y| for x in E_1, y in
E_2 is comparable to |v|, so it lies in some fixed interval [d_0, d_1]
with 0 < d_0 < d_1 < infinity.
By the bi-Lipschitz invariance of dim_H and the standard theory of
with the positive compact restriction,
may assume that mu_1 and mu_2 are compactly supported probabilities on
E_1 and E_2 with ball bounds of exponent s = S + chi, where S = d/2
and chi > 0 is the strict dimension margin. All cross distances are
bounded between d_0 and d_1.
From now on these are "the two separated measures". The theorem reduces
to proving that the set of cross distances between their supports has
positive Lebesgue measure; the distance set of E contains that cross
distance set, so the theorem follows.
Also, by pivoting to a small spherical patch: the cross-direction map
(x, y) -> (y - x)/|y - x| lands in some cap of the sphere of bounded
radius. The splitting can be chosen so that the cap is arbitrarily small.
This will be used implicitly whenever a direction is
"near a fixed axis".
Four reasons, all structural.
First, if x = y the direction (y - x)/|x - y| is undefined, and after
splitting it is defined everywhere.
Second, the map (x, y) -> (y - x)/|x - y| is Lipshitz on {(x,y) : |x-y|
in [d_0, d_1]}, uniformly in the constant. This lets us round directions
up to small caps without paying more than a constant.
Third, the distance map D(x, y) = |x - y| is smooth on the same set
with bounded derivatives, which is what makes the "spherical Fourier
analysis of the distance measure" work .
Fourth, the pair measure mu_1 x mu_2 is now a product of two separated
measures, so its total mass is mu_1(R^d) * mu_2(R^d), and no diagonal
mass concentrates at zero in the Fourier computations.
Take a dyadic grid on R^d (say, cubes of side 2^{-j} for every j in
Z_{>=0}). At depth j and scale R = 2^N, a cube has side 2^{-j} and area
2^{-dj}; there are about 2^{dj} cubes in the unit cube.
Fix a source measure mu (one of mu_1, mu_2). Its total mass is 1.
Different depth-j cubes carry wildly different masses. "Regularization"
splits mu into a small number of "classes" on each of which the mass
distribution is almost exactly captured by a single Lipschitz profile.
That makes the geometry at every depth computable from a single
nondecreasing 1-Lipschitz function.
Definition. Fix lambda in (0, 1). A "dyadic class" of mu is a subset
E_alpha of the unit cube satisfying three properties:
(1) It is a union of depth-N dyadic cubes (no partial cubes).
(2) Its mu-mass is at least R^{-lambda}.
(3) There is a nondecreasing, d-Lipschitz function m_alpha : [0, 1]
-> [0, d], with m_alpha(0) = 0, such that for every occupied
depth-j cube Q inside E_alpha, the mass satisfies
R^{-m_alpha(j/N) - C_d lambda} <= mu_alpha(Q)
<= R^{-m_alpha(j/N) + C_d lambda},
where mu_alpha = mu|E_alpha / mu(E_alpha) is the normalized
restriction, and C_d is a dimensional constant.
The exponent m_alpha(j/N) is called the "profile" at depth j. It is the
log-log mass decay of mu_alpha along the dyadic chain. In the notation
S = d/2, define the "excess profile"
A_alpha(n) = m_alpha(n/N) - S * n/N.
The profile has slope at most d everywhere (that is the Lipschitz
bound), so A_alpha is also d-Lipschitz. The excess A_alpha records
how much of the profile is above the critical slope S = d/2.
Lemma. Let n >= 1, let rho be a probability on a unit dyadic cube in
R^n, and let R = 2^N. For every fixed 0 < lambda < 1 and all sufficiently
large N, the measure rho splits as a disjoint union of "regular classes"
{ E_alpha } with the following properties:
(a) The number of classes is at most N^{C_n / lambda}.
(b) Each class has rho-mass at least R^{-lambda}, and the discarded
mass outside all classes is at most
R^{-2} + N^{C_n / lambda} R^{-lambda} = R^{-lambda + o(1)}.
(c) Each class carries a profile m_alpha , and
moreover the profile dominates the Frostman slope:
m_alpha(u) >= s u - C_n lambda - log_R(K) for all u in [0,1],
where s is the exponent of the s-ball bound on rho and K is
its loss.
(d) For two dyadic cubes Q subset P of depths b >= a inside the same
class, the conditional mass rho_alpha(P) / rho_alpha(Q) is
R^{-(m_alpha(b/N) - m_alpha(a/N)) \pm C_n lambda}.
Part (c) is the crucial point: the profile is not just any Lipschitz
function. It dominates the slope s undamped. Inside a regular class,
the mass decays at every scale at least as slowly as the ambient
Frostman condition allows, up to tiny losses.
Proof sketch. Bin the depth-N leaf masses into dyadic bins. Label each
leaf by its bin. Working upwards through the dyadic tree at depths
j_0 = 0, j_1, j_2, ..., j_J = N (with j_i - j_{i-1} <= lambda N + 1
and J <= C/lambda), extend each label at a node by the dyadic bin of
the number of its children that carry that label. Every extension
costs at most a factor of N^{C / lambda} in the label count.
The resulting label set is stable: within one final label the "branching
number" b_i at each selected depth is comparable to a fixed integer,
and the depth-j mass of any cube carrying that label is about prod_{h
<= i} (b_h)^{-1} * (leaf bin mass) up to C^{O(J)} factors. Taking logs
gives the profile.
The mass lower bound and the discarded-mass estimate come from the
syndetic-cover estimate on leaf masses. The profile lower bound (c) is
the pointwise version of Frostman's ball bound propagated down the tree.
The exact statement in original labels is Lemma 3.1 .
Inside a class alpha with profile m_alpha, define the model mass
M_alpha(n) = R^{-m_alpha(n/N)} at depth n. The lemma says: every
occupied depth-n cube in class alpha has mass within a factor of
R^{C_d lambda} of M_alpha(n). So M_alpha is the "expected" mass of a
depth-n ball.
Define the profile increment
I_alpha(g, p) = m_alpha(p / N) - m_alpha(g / N)
for 0 <= g < p <= N. Then mass comparisons between depth-p and
depth-g cubes are governed by I_alpha(g, p): a depth-p subcube of a
depth-g cube carries mass fraction R^{-I_alpha(g,p)} up to small errors.
The "excess increment"
A_alpha(p) - A_alpha(g) = I_alpha(g,p) - S * (p - g) / N
is what the recursion tracks. It is the excess over the critical rate.
Use the residual
k_alpha(g, p) = A_alpha(p) - min_{g <= n <= p} A_alpha(n)
which is the rise of A_alpha above its own recent minimum. It appears
as the cost in the graph estimate .
The next lemma lets us chain local "cap-type" bounds across several
depth intervals. It is Lemma 3.2 in the original .
Suppose G is a measurable set of source-pin pairs, and at each
depth a_j there is a bound: for each pair (x, z) in G with z in
some depth-a_j cube Q, the source measure rho of z's near
(x + R z)^perp differs from rho(Q) by a controlled factor f_j. Then
can delete a small exceptional set of pairs and multiply the f_j's
across depths, getting one bound at depth a_J.
Statement. Let rho be a source probability and nu a pin measure on
R^n. Fix 0 <= a_0 < a_1 < ... < a_J <= N. Suppose pin-source distances
on the relevant sets are between positive constants. Let G' be a
measurable set of pairs with the property that for each j < J, there
is a measurable E_j containing G' and a number f_j >= 0 such that for
every (x, z) in G' with z in a depth-a_j cube Q,
rho{ z' in Q cap (E_j)_x : |pi_x(z') - pi_x(z)| <= C 2^{-a_{j+1}} }
<= f_j rho(Q),
where pi_x(z') = (z' - x) / |z' - x| and C is a fixed large constant.
Then for every e > 0 there is a subset G'' of G' with
(nu x rho)(G' \ G'') <= (J+1) R^{-e} nu(R^n)
such that for each pin x, each initial depth-a_0 cube Q_0, and every
spherical cap B of radius C_1 2^{-a_J},
rho{ z in Q_0 cap (G'')_x : pi_x(z) in B }
<= R^{eJ} * (f_0 * f_1 * ... * f_{J-1}) * rho(Q_0).
The multiplication of the f_j's is the point. If each interval loses
f_j of mass to its cap, then J intervals lose prod f_j up to R^{eJ},
which is subpower.
Proof sketch. First delete pairs whose source cube Q has a G'-mass
smaller than R^{-e} rho(Q). This costs a subpower factor. Then run
a "branched counting" argument: at each depth, every retained pin's
occupancy of hit cubes at that depth is bounded by f_j * rho(Q_0) plus
a small multiplicative loss. Multiplying over j and telescoping the
tree bounds gives the claimed estimate.
The multiplication lemma is the bridge between the pointwise fibre
cap bounds (which hold at a single depth) and the
multi-scale cap bounds used in the shell estimate (which
require a manufactured "R^{endN} cap bound" for every dyadic end
scale). The "manufactured cap bound" trick is precisely this
multiplication, applied across the dyadic intervals.
Fix a pin x in E_1 and consider the radial projection map
pi_x(y) = (y - x) / |y - x| in S^{d-1},
defined on E_2 (which is separated from E_1). The pushforward measure
xi_x := (pi_x)_# mu_2
is a finite measure on the sphere. "Cap bounds" ask: does xi_x put
mass bounded by O(u^S) into every spherical cap B(w, u) of radius u?
The critical exponent is S = d/2. If yes for every pin x in E_1,
then the Falconer-Fourier method applies in its angular form (Section
6). If no, the procedure stops.
Where did this come from? The bird's-eye reason: the spherical average
int_{S^{d-1}} |hat{mu}(r w)|^2 d sigma(w)
that appears in the classical argument can be thought of as an integral
against xi_x for each x, and if xi_x has good cap bounds then the
spherical average is controllable by a Fourier lemma of the kind
proved .
But the cap bounds are not automatic. When the source E_2 has dimension
s and projecting from a point x in E_1 gives a measure xi_x sitting
on the sphere. The worst case is when the source is a piece of a
codimension-1 surface (e.g., a piece of a sphere centered at x). Then
xi_x is a delta mass at a single point — catastrophic.
So the work is modulo that bad case. Two tools enter: an auxiliary
projection that separates the bad surface from the rest,
and a "thin-tube" incidence argument that prevents the
measured mass from concentrating in a low-dimensional set of directions.
Observation. The exponent S = d/2 is the critical value. If xi_x has
cap bounds O(u^q) for some q > 0, then Fourier analysis
produces a decay of size R^{-q + (d-q)/N-total}... The critical value
of q is exactly S. Anything better is a gift.
The paper's rule is: auxiliary projections can change directions but
never distances. So all auxiliary projections are used to estimate
directional cap bounds. All distances and all oscillatory phases
remain those of the ORIGINAL pairs (x, y) in R^d.
The schematic is as follows. Fix a small integer h between k+1 and d, where
d = 2k-1 for odd or d = 2k for even.
Step 1. Choose a random orthogonal projection P_h from R^d to R^h.
Step 2. Compute the cap bounds of the projected radial projection.
Step 3. Lift: use the fact that projected cap bounds imply original
cap bounds (projection contracts the sphere so a projected cap is
contained in the preimage of a projected cap; the actual direction
control comes from the projection uniform bound).
The important choice of h.
In even dimension d = 2k, the choice is h = k + 1.
In odd dimension d = 2k - 1, the choice is h = k + 1.
In either case, the source dimension s = S + chi = d/2 + chi is only
about d/2, which is well below h (roughly d/2 + 1), so the projection
does not lose dimension and can support cap bounds of exponent as high
as the source dimension.
Then a separate incidence-theoretic argument, applied in R^h, gives
the S-cap bounds. In odd dimensions, the incidence argument gets only
a partial gain, so a stronger argument (the "thin-tube upgrade") is
needed. That is the content .
The overall statement is Proposition 4.1 in the original
. Quoted below.
The deep input is the following. Consider a bounded set P of points in
the plane and, at each point x of P, a family T_x of tubes (long thin
rectangles) through x. Suppose P has measure-theoretic dimension f (in
a "delta-set" sense: |P cap B(a, r)|_delta <= K r^f |P|_delta) and the
family of tube directions T_x also has dimension b at every x. Then a
Furstenberg-type inequality says: the union of all these tubes must
cover at least about delta^{ -2b - eps } unit balls of size delta.
Concretely, the statement used (in a tube formulation due to Orponen-
Shmerkin-Wang) is Theorem 4.1 in the original :
Theorem. Fix 0 < b < 1 and b < f < 2, where b, f are dimensions.
There is epsilon_0 > 0 such that for all sufficiently small delta the
following holds. If P is a nonempty bounded (delta, f, delta^{-eps_0})
set (in the metric sense above) and if each x in P has a nonempty
family T_x of incident delta-tubes whose line parameters form a (delta,
b, delta^{-eps_0})-set, then
| union_x T_x |_delta >= delta^{-2b - eps_0}.
The intuitive content: the incidence between a fractal point set and a
fractal tube family cannot be too concentrated. It costs a
2b-dimensional budget to cover the union of tubes.
This is the classical "Furstenberg planar resolution" of the
incidence problem, in tube form. It is stating: the union of all those
tubes covers a set of covering dimension at least 2b.
This theorem is the ONLY deep outside input in the paper. It is the
"retained incidence input" mentioned in the introduction. Everything
else is derived.
In even dimension, projection to R^{k+1} and the incidence theorem
directly give cap bounds of exponent S = k = d/2. Good.
In odd dimension d = 2k - 1, the number of "optimal" dimensions for
the restriction is different. The paper projects to R^{k+1}, giving
cap bounds of exponent k - 1 = (d - 1)/2 = S - 1/2, which is SHORT
of the needed S = k - 1/2. There is a half-dimensional deficit.
The deficit is filled by an extra-geometric argument, the "thin-tube
upgrade", which is Proposition 4.2 in the original .
It is the deepest geometric engine of the paper.
Statement. Let k >= 2 and k - 1/2 < s < k. Let rho, nu be bounded,
separated s-Frostman probabilities in R^{k+1}. Suppose the two
directional measures of a pair set G_N have plate bounds of size
C u^{zeta} uniformly for 2^{-N} <= u <= 1, with fixed zeta > 0. If
exponent k - 1 is "available" (i.e., cap bounds O(u^{k-1}) hold for
every pin in the retained supports), then exponent k - 1/2 is
available.
The point: just under one more half-dimension of cap decay is gained
by an auxiliary incidence argument, given that k-1 already holds.
Availability step-up.
Where does the extra half-dimension come from? The gain is proved in
three slicing regimes.
Case 1 (q > k - 1). Slice by a generic codimension-(k-1) subspace.
What remains is a 2-dimensional "planar" setting where exponents
b = q - (k-1) and f = t - (k-1) sit in (0, 1/2] and (1/2, 1) respectively,
with b < f. The planar Furstenberg input (Theorem 4.1 above) directly
gives the gain: a further reduction is impossible when a family of
tubes on dimension-b fracta is confined but wants to capture
dimension-f mass. Contradiction via |union| being too small.
Case 2 (endpoint q = k - 1). Slicing to codimension k-1 leaves a
flat direction exponent 0 and the argument degenerates. Instead,
slice to codimension k-2, leaving 3-dimensional slices with direction
exponent 1 and point exponent > 3/2. Then use the Loomis-Whitney
inequality .8 with a 3-direction transverse triple: pick
three almost-independent directions in one 3D slice and combine their
tube covers.
But almost-independent can degenerate. To prevent that, the paper
uses "weak radial estimate" (Lemma 4.3 in original, ang:weak-radial):
a small output-loss radial estimate giving the cap-mass of a fractal
measure in a small angular window, proved via Orponen's finite-energy
method. It is applied at dimension ell > k-2 in a way that its energy
hypotheses are comfortably satisfied.
The proof of the weak radial estimate is a substantial computation
(energy method, dyadic regularization, multiscale cap bound
propagation). The statement appears below and the results it produces are quoted,
since the geometric content is the same as the even-dimensional case
and the technical work is bookkeeping. The interested reader can
consult the original proof of Proposition 4.2 ,
which builds the weak radial estimate from primitives and then
combines with a transverse tube counting lemma.
The upshot: for odd d, cap bounds of exponent S = d/2 do hold, but
the extra geometric input must be layered on top of the projection
and the planar Furstenberg estimate.
Combining the auxiliary projection, the planar Furstenberg input, the
odd-dimensional thin-tube upgrade, and the regular-class decomposition,
the paper gets a "starting pair" package. This is Proposition 4.3 in
the original .
Theorem. Let E subset R^d be compact, d >= 2, and suppose dim_H E > S =
d/2. Either the classical Falconer threshold (d+1)/2 already gives the
conclusion (this is only possible when d < 2, i.e., never in the
present context; the lemma prep:classical-plane is deprecated as a
classical reference and does not apply to the main case) -- actually,
in odd dimensions the plane-case lemma prep:classical-plane does kick
in for exceptional affine planes; see.
Otherwise, after a similarity, there exist compact separated sets
E_1, E_2 subset E and probabilities mu_1, mu_2 supported on them, each
satisfying
mu_i(B(x, u)) <= C u^{S + chi} for every x and every u > 0,
with fixed C and chi > 0. Their cross directions lie in a fixed
arbitrarily small spherical patch. For every fixed small enough
omega > 0, there are decreasing Borel pair sets Gamma_N subset
E_1 x E_2 such that:
(1) The intersection of the Gamma_N has positive mu_1 x mu_2 mass.
(2) Gamma_N is a disjoint union of at most C 2^{2 d omega N} products
of spatial Borel sets ("coarse cells").
(3) For every e > 0 there are c(e) > 0, C_e < infinity, and Borel
sets B_N subset Gamma_N with mu_1 x mu_2 mass at most C_e 2^{-c(e)N},
such that for G_N = Gamma_N \ B_N, all u in [2^{-N}, 1], and all
spherical caps B(w, u):
mu_2 { y : (x, y) in G_N, pi_x(y) in B(w, u) } <= 2^{eN} u^S,
mu_1 { x : (x, y) in G_N, pi_y(x) in B(w, u) } <= 2^{eN} u^S.
Note: e is arbitrary, and c(e) depends on e. The "power-small deletion"
2^{-c(e)N} is a cheap cost. The "cap bound" 2^{eN} u^S is arbitrarily
close to u^S in exponent.
The pair filters Gamma_N are the "coarse pair restrictions". These are
decreasing, and their intersection has positive mass. The B_N are
"exceptional" — deleted at the last minute. The G_N are the surviving
"good" pairs, with all the cap bounds.
Between G_N and Gamma_N there is a power-small mass gap. This is the
key structural feature: this gap reappears in the final
"monotone shell reconstruction" argument .4, which is
precisely designed to handle this kind of subpower loss.
One case is dispatched early. If in odd dimension d = 2k - 1 the
original measure charges an affine k-plane H (k = (d+1)/2), then the
restriction of mu to H is a compactly supported measure on H with
dimension > k - 1/2 = (k+1)/2. In that case the classical Falconer
threshold (d+1)/2 = k + 1/2 applied within H, whose dimension is k-1,
does not directly give the distance set an open interval. But: within
H, whose dimension is k - 1, k - 1/2 > (k-1)/2 + 1/2, which
is exactly the classical threshold for the dimension k - 1 space. So
Falconer's classical lemma (with the exponent (d+1)/2 for plane
dimension d) produces positive measure for the distance set of mu|_H.
That is a subset of Delta(E), completing the argument.
That is the classical lemma reference. Its proof is given below
to the plane H (dimension n = k - 1). Its statement is Lemma 4.4 in
the original .
So in odd dimensions, EITHER the classical threshold applies within an
exceptional affine plane charged by mu, OR the general case holds
and can run the whole machinery of Sections 5-10.
Even dimensions have no such special case; the argument runs cleanly
through auxiliary projections.
Available:
- Two separated probabilities mu_1, mu_2 on E_1, E_2, with ball bound
of exponent S + chi (chi > 0 fixed).
- Decreasing pair filters Gamma_N subset E_1 x E_2 whose intersection
has positive mass.
- For every e > 0, exceptional sets B_N of power-small mass, such
that G_N = Gamma_N \ B_N has the S-cap bounds .
Split each mu_j into dyadic classes, fix a scale
R = 2^N, and pick three tiny constants
0 < beta << (1/10) and beta < chi/10,
0 < kappa << beta / d,
0 < sigma << kappa.
These get chosen in exactly this order. These are the small constants
of the proof. Their sizes will all be re-chosen to absorb constants
later.
Choose one class from mu_1 and one from mu_2. Fix them. In what follows
work on a single pair of classes; but note that all the machinery
below depends only on the class's profile m, not on the specific class.
So the same constants serve for every class.
Write A(N) for the excess profile A(n) = m(n/N) - S(n/N) at depth n.
The two combinatorial quantities that recur:
I(g, p) = m(p/N) - m(g/N) (mass drop of the profile)
k(g, p) = A(p) - min_{g <= n <= p} A(n) (local rise of the excess)
Fix a "descriptor" (type, test-depth range). A "parallel descriptor"
is a pair of integers (g, p) with 0 <= g < p <= N. The anchor is p,
the nominal length is l = p - g.
For every anchor cell P (a depth-p dyadic cube that is occupied by the
class), fix its center c_P. Define "good direction sets at level i":
w in S^{d-1} is good at level i for the parallel test [g, p]
based at cube P if the number of depth-p cube centers within
|c - c_P| <= 10(d+1) 2^{-g} whose perpendicular distance to the
line c_P + R w is at most
W_i(p) = (2 - i / J_0) 2^{-p} R^{500 sigma}
is at most
R^{k(g,p) + C_d sigma}.
The constant J_0 = 100 (1/kappa + 1/sigma) + 100 is a fixed integer (the total
number of "good-direction levels"). This W_i is a decreasing sequence
in i: as i increases, the test gets stricter (the threshold shrinks).
The number of centers that must be in the tube stays roughly the same
(threshold R^{k+C sigma} doesn't depend on i directly), but the tube
width shrinks.
In words: "level-i parallel test accepts a direction w if the fractal
class has only about R^{k(g,p)} centers along a thin tube in direction
w". This is a lower bound on how "spread out" the class is in
direction w.
A "projection descriptor" is a pair (p, t) with 0 <= p < t <= N. Its
anchor is p, its nominal length is l = t - p.
For every anchor cell P at depth p, and for every direction w on the
sphere, let rho_P = rho|_P / rho(P) be the normalized class measure
inside P. Define
w is good at level i for the projection test [p, t]
based at cube P if
(rho_P x rho_P) { |(z - z') . w| <= W_i(t) }
<= 2^{-(t-p)} R^{A(p) - min_{p <= n <= t} A(n) + C_d sigma}.
The W_i(t) is the same form as before: W_i(t) = (2 - i/J_0) 2^{-t}
R^{500 sigma}.
In words: "level-i projection test accepts a direction w if the
rho_P-measure of near-perpendicular pairs (in direction w) is small".
The parallel test is the one that will be supplied by the graph
estimate : when cells at depth p are grouped by depth-g
ancestors, the parallel test limits how many such cells can lie in a
small tube.
The projection test is what a direct "suffix reduction"
uses: at a suffix choice the anchor is deep, and the local geometry is
essentially 1D-projected, so a scalar Fourier-ball argument controls
the fine pair count.
The (g, p) pair controls the SCALE at which the test applies. For a
descriptor with g small and p large, the test is "long"; the tube is
wide relative to the anchor spacing. For g close to p, the test is
"short"; the tube is narrow. Both variants are needed.
The key routine lemma is as follows. It converts cap bounds into bounds on
the Fourier transform.
Lemma 5.1 (Half-dimensional Fourier estimate). Let 1 <= H <= R, and
let K, L >= 1. Let S = d/2. Suppose alpha is a probability on a fixed
bounded set with an S-ball bound of loss K down to scale H^{-1}:
alpha(B(x, r)) <= K r^S for H^{-1} <= r <= 1.
And suppose xi is a subprobability on S^{d-1} with the cap bound
xi(B(w, r)) <= L r^S for R^{-1} <= r <= 1.
Then, with Fourier convention hat{alpha}(v) = int e^{-iz.v} d alpha(z):
int_{-H}^{H} int_{S^{d-1}} |hat{alpha}(s w)|^2 d xi(w) ds
<= C L K (1 + log H).
The constant C depends only on d and on the diameter of the fixed
bounded set.
Why this is remarkable. The left-hand side is a "spherical-radial"
Fourier integral over a frequency range of size H. The bound is
subpower in H (only a logarithm). At frequency s, the L^2 spherical
integral against xi is O(1) per unit frequency, up to the cap losses.
Two exponents S = d/2 are being multiplied together to make d = 2S,
which is exactly the dimension. The problem is at the critical
exponent.
Proof sketch. The main idea is a Gaussian majorant + weighted Cauchy-
Schwarz.
Let lambda_B be the pushforward of the measure 1_{B <= |s| <= 2B} ds
d xi(w) under (s, w) -> s w. Then lambda_B is supported in the annulus
B <= |v| <= 2B and has total mass <= 2B. Its unit-ball concentration is
at most C L B^{-S}, from the cap bound on xi.
Let Phi be a rapidly decreasing, integrable kernel (a majorant for the
Fourier convolution of a smooth spatial cutoff). Then
|hat{alpha}(v)|^2 <= C int |hat{alpha}(u)|^2 Phi(v - u) du.
Integrating against lambda_B:
int |hat{alpha}|^2 d lambda_B <= C int |hat{alpha}(u)|^2 W_B(u) du,
with W_B = Phi * lambda_B. The unit-cube summation and decay of Phi give
W_B <= C L B^{-S} on all of R^d. And the tail estimate W_B(u) <= C_M
B (1 + |u|)^{-M} for |u| > 4B.
Now split the frequency integral into low and high parts. Low part:
Gaussian majorant for the ball indicator of |u| <= 4B. Uses the S-ball
bound to give
int_{|u| <= 4B} |hat{alpha}(u)|^2 du <= C K B^{d - S}.
High part: uses |hat{alpha}| <= 1 and the decay of W_B, which costs
O(1). Multiply the low part by W_B <= C L B^{-S}:
L K B^{d - 2S} <= L K B^{d - d} = L K.
So each dyadic band B contributes O(LK). Summing over B = 1, 2, 4,
..., H, that is O(LK log H). The final factor (1 + log H) replaces
log H explicitly for the B = 1 case. Done.
This is used many times. The two S's come from the two ball bounds:
one on alpha, one on xi. Their product is a constant per dyadic band.
That's the criticality of S = d/2.
It is now shown that each test fails only on a small spherical set.
Lemma 5.2 (Exceptional directions). There is a dimensional constant
C_d (adjustable to be as large as needed) such that the following
holds for all sufficiently large N. For every spherical subprobability
xi satisfying the S-cap bound
xi(B(w, r)) <= R^sigma r^S for R^{-1} <= r <= 1,
every test [type, (g, p) or (p, t)] based at any occupied anchor cube
and any level zero satisfies
xi(S^{d-1} \ good_set_at_level_0) <= R^{-sigma}.
So the "good" set of directions has xi-measure at least 1 - R^{-sigma}.
Proof sketch for the parallel case. Divide directions into annuli by
distance to the anchor. In each annulus of scale 2^{-h}, the number of
depth-p centers at that distance is controlled by the profile via
I(h, p). A center at distance 2^{-h} from c_P is counted only for
directions in two caps of radius O(2^{-(p-h)} R^{500 sigma}). Cap
bound gives O(R^{(1+500S) sigma} 2^{-S(p-h)}).
Number of such centers <= C R^{I(h,p) + C_d' sigma}. Total expected
occupancy <= C (N+1) R^{I(h,p) - S(p-h)*N^{-1}... } -- the key identity
I(h, p) - S(p - h) = A(p) - A(h) <= k(g, p)
gives the total expected occupancy at most C (N+1) R^{k(g,p) + C_d''
sigma}. For fixed sigma, Markov at threshold R^{k(g,p) + C_d sigma}
gives the probability bound.
Proof sketch for the projection case. Rescale A to unit size. The
rescaled tolerance is eps = 2H^{-1} R^{500 sigma}. The function
F_eps(u) = (sin(u / (2 eps)) / (u / (2 eps)))^2
is a positive-definite majorant for 1_{|u| <= eps}, with Fourier
transform supported in [-eps^{-1}, eps^{-1}] = [-H/2, H/2] (times
2H^{-1}... careful with the constant). Bounded by C eps.
Then
int (alpha x alpha) { |(z - z').w| <= eps } d xi(w)
<= C eps int_{-H}^{H} int |hat{alpha}(s w)|^2 d xi(w) ds
<= C H^{-1} H^0 (log H) R^{delta_A + sigma} ...
The half-dimensional Fourier estimate (Lemma 5.1) with L = R^sigma and
K = R^{delta_A + C sigma} gives
int (...) d xi(w) <= C H^{-1} (log H) R^{A(p) - min_A + (501+C) sigma}.
Multiply by 1/H = 2^{-(t-p)} and absorb losses. Markov at the threshold
2^{-(t-p)} R^{A_p - min_A + C_d sigma} gives the bound. Done.
Both parts conclude the same way: at the level-zero test, the
"failure" set has xi-measure at most R^{-sigma}. This is the "cheap
exception" fact.
A smooth envelope around the good-direction set, because
later integrate against test indicators with derivatives. Fix a level
i < J_0. Define the "nominal angular scale"
D_{l'} = max(1, 2^{l'} R^{-50 sigma}),
where l' is the descriptor's nominal length (p-g for parallel, t-p for
projection).
Lemma 5.3 (Masks with a buffer). For all sufficiently large N, every
level i < J_0 admits a mask w_{i, tau}(x, w) in [0, 1], measurable in
x and smooth in w, such that on every occupied anchor cube:
- w = 1 on the level-i good set G_{i, tau}.
- supp_w w is contained in the level-(i+1) good set G_{i+1, tau}.
- Angular derivative bounds: || T_1 ... T_j w_{i, tau} ||_inf
<= C_j D_{l'}^j for every fixed collection of smooth tangent
fields T_j spanning the sphere's tangent space.
- The masks are antipodally symmetric (w(w) = w(-w)) and are
constant as functions of x on each anchor cube.
The mask construction: take a spherical convolution of the level-i
good-set indicator against a smooth rotationally invariant kernel of
support radius about c_0 / D_{l'}, with c_0 small. The buffer between
levels is D_{l'}^{-1}, and the smoothness index is preserved.
The point of the R^{500 sigma} vs R^{50 sigma} gap (which appears
between the width W_i and the mask smoothing scale) is that the
level-(i+1) good set strictly contains the D_{l'}^{-1}-neighborhood
of the level-i good set. That is what makes the buffer work. This
"500 vs 50" gap is small but positive, so it can be absorbed into the
exponents; but it must be respected throughout the recursion.
For any fixed-size list of descriptors tau_1, ..., tau_k (k = k(N)
bounded by N^{C_*} with C_* fixed), the product of the corresponding
masks is bounded in angular derivatives by (R^sigma D_max)^j where D_max
is max over nominal lengths in the list. The support of the product
still lies in the intersection of all the level-(i+1) supports of the
individual masks, and that is not lost by differentiation.
The "R^sigma" factor absorbs the polynomial blow-up from the product
rule. Since k <= N^{C_*} for a fixed C_*, expansion by the product
rule is bounded by a polynomial, and R^sigma dominates it for
sufficiently large N.
A family of descriptor lists that grows under interval
containment and is finite-size. Fix
q_0 = ceil(kappa N),
J_0 = 100(1/kappa + 1/sigma) + 100.
Partition {0, ..., N} into consecutive blocks of size q_0, with the
last possibly shorter. In each block, pick ONE depth minimizing
A. Call that the block's representative.
For an interval [a, t] with t - a > 100 q_0, pick
p = argmin_{n in a_* <= n <= t_*} A(n)
where [a_*, t_*] is the union of blocks strictly inside [a + q_0,
t - q_0]. Set l = min(p - a, t - p).
Call it a "prefix choice" if 2p <= a + t, "suffix choice" otherwise.
The list M(a, t): for every nonterminal subinterval [a', t'] subset
[a, t], include the parallel descriptors [g, p] for p - l <= g < p
where p, l is that interval's own policy choice. If the choice is a
suffix choice, also include the projection descriptor [p, t'].
De-duplicate. The resulting list is at most polynomially large in N
(for fixed sigma, kappa) and is contained in the corresponding lists
of larger intervals containing [a, t].
The "origin" of a descriptor in M(a, t) is one specific subinterval
[a', t'] that generates it. Every descriptor in the list comes from at
least one origin. This bookkeeping matters in the potential below
argument.
Lemma 5.4. For any nonterminal [a, t], the inner interval [a_*, t_*]
on which the policy chooses p satisfies
a + q_0 <= a_* <= a + 2 q_0,
t - 2 q_0 <= t_* <= t - q_0,
A(n) >= A(p) for all n in [a_*, t_*].
The lists M(a, t) are nested under interval containment. There are at
most 1 + (N+1)/q_0 possible anchors (one per block). For every
descriptor tau in M(a, t) with anchor p and nominal length l' = p - g,
there is an origin [a', t'] containing [p - l', p + l']. The same
origin supplies every parallel descriptor [g, p] for p - l' <= g < p.
The proof is direct bookkeeping. The inner interval is a strip of
width about 2q_0 on each side of [a, t]. Because the choice p minimizes
A over that strip, A is at least A(p) throughout the strip. Nesting of
the schedule follows from the fact that each block's choice is made
once and reused in every larger interval.
The "origin contains [p - l', p + l']" is exactly the fact that
prefix choices contribute l' = p - g <= p - a which is bounded by
half the origin's length; suffix choices are similar. So every
descriptor is "supported" by an interval at least twice its length,
positioned symmetrically around its anchor.
Fix the ambient dimension d>=2 and the frequency scale R=2^N. Cell depths and frequency depths are integers in [0,N]; in an angular energy with 0<=a<=v<=N. Fix the sphere dimension n_* = d-1. Fix a power of two K_d with 1000(d+1) < K_d < 4000(d+1). Two depth-a dyadic cells X, Y are separated when K_d 2^{-a}/8 <= |c_X - c_Y| <= 4 K_d 2^{-a}. For x in X and y in Y, D(x,y)=|x-y| is then comparable to 2^{-a}, and the unit directions e(x,y)=(x-y)/D(x,y) lie in a fixed cone of small aperture. Pick a smooth bump psi, 0<=psi<=1 on the sphere, equal to one near that cone and zero near the antipodal cone, with uniformly bounded derivatives of every fixed order. The same convention applies on the fixed separated outer supports at a=0, with constants depending only on that geometry.
All angular derivatives below are taken in the fixed smooth tangent fields Omega_{uv}=w_u d/dw_v - w_v d/dw_u, 1<=u<v<=d, which span every tangent space of the sphere. For a word alpha of order |alpha| in these fields, write Omega^alpha. A descriptor t of nominal length l_t carries the angular scale D_t = max(1, 2^{l_t} R^{-50 sigma}).
Definition 6.1 (Admissible symbols). Fix an indexed descriptor list (t_nu)_{nu in I} with #I <= N^{C_A}, a smoothing index i in {0,...,J_0-1}, and a nonnegative integer derivative budget B. At every fixed derivative order also fix a common numerical bound Q_j independent of N. An admissible endpoint symbol is
b_x(w) = f(x) q(w) prod_{nu in I} D_{t_nu}^{-|alpha_nu|} Omega^{alpha_nu} w_{i,t_nu}(x,w), sum_nu |alpha_nu| <= B.
In this definition f is an arbitrary measurable spatial function with |f|<=1, q is a smooth angular function with ||q||_{C^j}<=Q_j at every order considered, and the index nu distinguishes occurrences, so a repeated descriptor can carry distinct derivative words. The exponent C_A, the budget B, and the constants Q_j are fixed independently of N for the whole family under consideration. Constants in estimates may depend on them, and fixed scalar factors may be absorbed there. Empty lists and empty products are allowed, as is complex conjugation. Symbols never depend on the radial variable r.
No spatial regularity is required: f can impose an arbitrary Borel restriction inside a cell. Differentiating a mask cannot enlarge its support, so a descriptor present in a symbol still imposes its level-(i+1) good condition.
Lemma 6.2 (Derivative bounds and reached anchors). Let L be the largest nominal length in the symbol's list, or in all lists for a product of symbols, zero if those lists are empty, and set D_* = R^sigma max(1, 2^L R^{-50 sigma}). For every fixed j, an admissible symbol, or a product of a fixed number of endpoint symbols, satisfies sup_x ||b_x||_{C^j} <= C_{j,B} D_*^j at sufficiently large N. The constants may depend on the list-size exponents, parameters, derivative budgets and common bounds, but not on the spatial multipliers. On a depth-a cell, the product of all mask factors whose anchors have depth at most a is independent of the spatial integration variable and has modulus at most one. Such anchors are called reached at depth a.
Proof. The product-symbol lemma applies to each symbol using the polynomial list bound, the prescribed budget, and the common bounds for q. A fixed product of endpoint symbols inherits the same estimate by the finite Leibniz rule, even when their smoothing indices differ. An anchor at depth at most a contains every depth-a cell, and its mask is constant in the spatial variable on that cell; angular differentiation preserves this constancy. The product has modulus at most one because only boundedly many factors take prescribed derivatives and every remaining mask has modulus at most one.
Definition 6.3 (General angular energies). For occupied depth-a cells of one or two classes, together with specified endpoint symbols b_x and c_y, set
U_X(r,w) = (1/M_1(a)) int_X e^{i r w.x} b_x(w) d rho_1(x),
V_Y(r,w) = (1/M_2(a)) int_Y e^{i r w.y} c_y(w) d rho_2(y).
The M_j(a) are the profile model masses, not the actual cell masses; the transform need not be a conditional probability; before endpoint factors enter, the underlying measure has mass rho_1(X)/M_1(a)=R^{O_d(sigma)}, and similarly on the second side. Fix a nonnegative bump Psi in C_c^infty((1/4,4)) with Psi>=1 on [1/2,2], and write dw for surface measure on the sphere. The independent angular energy at 0<=a<=v<=N is
H^{X,Y}(a,v) = 2^{d(v-a)} int Psi(r/2^v) (int |U_X(r,w)|^2 dw)(int |V_Y(r,w)|^2 dw) (dr/2^v).
The shared-angle energy on separated geometry is
H_F^{X,Y}(a,v) = 2^{d(v-a)} int Psi(r/2^v) | int psi(w) U_X(r,w) conj(V_Y(r,w)) dw |^2 (dr/2^v).
Both are values for the specified class laws, cells and endpoint symbols, including the lists carried by those symbols. Maxima over finite collections of cell pairs and suprema over stated symbol families are used freely.
By Cauchy-Schwarz in w with 0<=psi<=1,
H_F^{X,Y}(a,v) <= H^{X,Y}(a,v)
at every pair of endpoint symbols with the same support conditions, because the radial measure and the prefactor are the same in both energies.
One operation is available inside H but not inside H_F. On a depth-a cell, factor the reached mask factors out of the transform as an angular function; Lemma 6.2 bounds its modulus; bounding that modulus inside the corresponding positive integral returns the independent energy with those factors deleted and with every remaining symbol unchanged. The same deletion cannot be performed inside the shared absolute square of H_F before that square is taken.
Proposition 6.4 (Weighted shell conversion). Fix integer depths 0<=a<=v<=N and separated depth-a cells X,Y of one or two classes. Let m_x and n_y be products of the undifferentiated masks at one index i<J_0 for two specified indexed descriptor lists, with at most N^{C_A} occurrences each and one exponent C_A fixed independently of N for the whole family. Let L be the largest nominal length in either list, zero when both lists are empty. For measurable |f|,|g|<=1 set a possibly complex measure nu on the real distance line by
int F(s) d nu(s) = (1/(M_1(a) M_2(a))) int_{X x Y} F(D(x,y)) (D(x,y) 2^a)^{-n_*/2} f(x) conj(g(y)) m_x(e(x,y)) n_y(e(x,y)) d rho_1(x) d rho_2(y).
Suppose v-a >= 20 sigma N and L <= (v-a)/2 + sigma N. Then for every fixed T>0,
2^{-a} int_{2^{v-1}}^{2^{v+1}} |hat(nu)(r)|^2 dr <= R^{o(1)} sup H_F(a,v) + O(R^{-T}).
The supremum ranges over the same cells, class normalizations, lists and smoothing index as in nu; its symbols are admissible with a derivative budget and common smooth-angular bounds depending only on T, d, C_A and the fixed parameters, uniformly in N. When f=g=1 the measure nu is positive. The same assertion holds on the fixed outer supports at a=0.
The rest of the Section proves Proposition 6.4. Two global sphere expansions are obtained first; then a bounded separation of the residual smooth spatial coefficient into functions of the two endpoints closes the argument.
Give the sphere Laplacian Delta_{S^{n_*}} = div grad, so its eigenvalues are nonpositive. Put c_{n_*}=(2 pi)^{n_*/2} e^{-i pi n_*/4}, L_0=c_{n_*} Id, and for j>=1,
L_j = (c_{n_*}/((2i)^j j!)) prod_{q=0}^{j-1} (Delta_{S^{n_*}} + q(q+1) - n_*(n_*-2)/4).
At the negative stationary point put L_{j,-1} b = conj(L_j conj(b)), and write L_{j,+1}=L_j. These operators are global and cover d=2.
Lemma 6.5 (Direct and inverse sphere expansions). Let b in C^infty(S^{n_*}) satisfy ||b||_{C^m}<=C_m A^m at the finitely many orders needed below, where A>=1. Let Delta != 0, e=Delta/|Delta|, Lambda=r|Delta| with r>0. Let psi equal one on a fixed neighborhood of e and zero on a fixed neighborhood of -e, with uniformly bounded derivatives. For every integer J>=1,
int_{S} e^{i r w.Delta} psi(w) b(w) dw = Lambda^{-n_*/2} ( e^{i Lambda} sum_{j=0}^{J-1} Lambda^{-j} (L_j b)(e) + E_J ), |E_J| <= C_J Lambda^{-J} A^{2J+d+2},
provided Lambda>=1. Without a cutoff,
int_{S} e^{i r w.Delta} b(w) dw = Lambda^{-n_*/2} ( sum_{eps=+1,-1} e^{i eps Lambda} sum_{j=0}^{J-1} Lambda^{-j} (L_{j,eps} b)(eps e) + E'_J ),
with the same remainder bound. Define global differential operators B_0=L_0^{-1}, B_j=-L_0^{-1} sum_{h=1}^{j} L_h B_{j-h}. Each B_j is a polynomial in Delta_{S^{n_*}} of differential order at most 2j. If, for a fixed c>0,
Lambda >= R^{c sigma}, A^2/Lambda <= R^{-c sigma}, 1 <= A <= R^2,
then for every prescribed T>0 a fixed sufficiently large J gives
e^{i Lambda} b(e) = Lambda^{n_*/2} sum_{j=0}^{J-1} Lambda^{-j} int_{S} e^{i r w.Delta} psi(w) (B_j b)(w) dw + O(R^{-T}).
The extracted remainders above are also O(R^{-T}) after increasing J. Every constant is uniform in e in the stipulated geometry. For r=2^v s with s in a fixed compact subinterval of (0,inf), every coefficient Lambda^{-j}(L_{j,eps} b)(eps e) has uniformly bounded derivatives of every fixed order in s under the working regime. Every differential coefficient retains the supports of every mask factor in b.
Proof. Away from e and -e the angular gradient of w.e is bounded below; repeated integration by parts in that gradient bounds the integral by any fixed power of Lambda^{-1}, at the cost of the corresponding angular derivatives of b. Near the maximum, rotate e to the last coordinate vector and use w(u)=(u sqrt(1-|u|^2/4), 1-|u|^2/2), dw=(1-|u|^2/4)^{(n_*-2)/2} du. The phase is exactly 1-|u|^2/2 and the Jacobian equals one at u=0. A finite set of charts with uniformly bounded derivatives covers every possible e. The local integral is e^{i Lambda} int e^{-i Lambda |u|^2/2} a_e(u) du with a_e compactly supported and ||a_e||_{C^m} <= C_m A^m.
The Fourier formula for a Gaussian, first established with positive damping and then passed to the limit, computes that quadratic integral as c_{n_*} Lambda^{-n_*/2} times int hat(a_e)(xi) e^{i |xi|^2/(2 Lambda)} dxi in the inversion normalization. Taylor expansion through order J-1 has remainder at most C_J Lambda^{-J} int |xi|^{2J} |hat(a_e)(xi)| dxi <= C_J Lambda^{-J} A^{2J+d+2}, after integrating by parts to an order above 2J+n_*. For the nonstationary part use M = J + ceil(n_*/2) integrations by parts, whose bound C_M Lambda^{n_*/2-M} A^M is at most C_J Lambda^{-J} A^{2J+d+2}. This produces a local expansion with differential coefficients of order at most 2j and leading coefficient c_{n_*} b(e).
To identify the coefficients without moving endpoint coordinates, fix smooth b and a polar cutoff chi(w.e) equal to one near w=e and vanishing outside a slightly larger cap. The Fourier extension F(xi)=int e^{i w.xi} b(w) dw satisfies (Delta_xi + 1)F=0. The same identity for the 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 integration by parts above applies after any fixed number of derivatives. In polar coordinates the operator is d^2/d Lambda^2 + (n_*/Lambda) d/d Lambda + 1 + Lambda^{-2} Delta_{S^{n_*}}. The Gaussian expansion may be differentiated any fixed number of times in angular and radial parameters. Substituting e^{i Lambda} Lambda^{-n_*/2} sum_j Lambda^{-j} a_j(e) and matching power coefficients gives 2i(j+1) a_{j+1} = (Delta_{S^{n_*}} + j(j+1) - n_*(n_*-2)/4) a_j. Starting from a_0=c_{n_*} b recovers the L_j above. A cutoff equal to one near e has the same coefficients, since the difference is nonstationary. The negative stationary point carries the conjugate Gaussian phase, giving the full-sphere form.
For inversion, apply the direct expansion to B_j b, keeping terms through J-j-1. The B_j recurrence removes every coefficient of total index 1,...,J-1 and leaves b at index zero. The order-m derivative bound of B_j b is C_{j,m} A^{m+2j}; after multiplying by Lambda^{-j} the extracted remainder is at most C_J Lambda^{-J} A^{2J+d+2}, uniformly in 0<=j<J. Under the working regime this is at most C_J (A^2/Lambda)^J A^{d+2} <= C_J R^{-c sigma J + 2(d+2)}, smaller than every prescribed power of R once J is fixed large. For the s-derivative claim: e and b are independent of s, and each s-derivative of Lambda^{-j}(L_{j,eps} b)(eps e) costs a bounded factor. For the support claim: Delta_{S^{d-1}} = sum_{u<v} Omega_{uv}^2, so a Leibniz expansion of any L_j or B_j differentiates but never removes an original mask factor; derivatives of a smooth function cannot enlarge a support. That covers arbitrary measurable spatial multipliers.
Lemma 6.6 (Separated spatial expansion). On separated depth-a cells, for every fixed j>=0 there are bounded measurable f_l on X, g_l on Y, and constants c_l with
(D(x,y) 2^a)^{-j} = sum_l c_l f_l(x) g_l(y), |f_l|,|g_l| <= 1, sum_l |c_l| <= C_{d,j},
converging uniformly. The same holds in the fixed outer geometry, with the constant depending on that geometry.
Proof. Write x=c_X+2^{-a} u, y=c_Y+2^{-a} z, with both rescaled endpoint cells in fixed boxes. By separation, |2^a (c_X-c_Y)+u-z| is bounded between positive constants on a fixed neighborhood of their product. Its negative j-th power has a smooth extension, with uniform bounds of every fixed order, supported on a slightly larger product box. Regard that extension as a smooth function on a fixed 2d-torus. Its Fourier coefficients decay faster than any fixed polynomial; choosing m>2d makes the absolute sum bounded. Each torus exponential splits into a bounded function of u and one of z. Pullback gives the claim. The fixed outer geometry admits the same proof.
Proof of Proposition 6.4. Put q=v-a, s=r/2^v, D_max = max(1, 2^L R^{-50 sigma}). On the shell, Lambda=r D(x,y) is comparable to 2^q and
D_* = R^sigma D_max <= max(R^sigma, 2^{q/2} R^{-48 sigma}), D_*^2/Lambda <= max(R^{2 sigma} 2^{-q}, R^{-96 sigma}) <= R^{-18 sigma}.
Also Lambda >= R^{20 sigma} and D_* <= R^2. Decreasing the positive exponents absorbs every fixed geometric factor. Lemma 6.5 applies uniformly to m_x(w) n_y(w).
It suffices to estimate hat(nu)(-r); conjugation handles hat(nu)(r). Apply the inverse expansion pointwise, multiply by the weight in the definition of nu; the leading distance power cancels exactly:
(D 2^a)^{-n_*/2} (rD)^{n_*/2} (rD)^{-j} = (r 2^{-a})^{n_*/2} (r 2^{-a})^{-j} (D 2^a)^{-j}.
The operator B_j is global. Its Leibniz expansion of m_x n_y is a sum of products of angular derivatives of individual masks, of total order at most 2j. Dividing every differentiated mask by its own D_t to that order makes the result admissible with the same descriptors, and the extracted scales multiply to at most D_max^{2j}. Their number is polynomial in N because j is fixed. Smooth angular factors go into either endpoint symbol. The inverse factor pays for these scales:
2^{-j q} D_max^{2j} <= [max(2^{-q}, R^{-98 sigma})]^j <= 1.
Lemma 6.6 handles the remaining (D 2^a)^{-j}: its absolutely summable series only multiplies the original spatial factors by bounded endpoint functions. After summing the finitely many inverse orders,
hat(nu)(-2^v s) = 2^{n_* q/2} sum_alpha c_alpha(s) int psi(w) U_{X,alpha}(2^v s, w) conj(V_{Y,alpha}(2^v s, w)) dw + E(2^v s),
with sum_alpha ||c_alpha||_{L^infty} <= R^{o(1)}. The smooth factors s^{n_*/2-j} sit inside c_alpha(s) and are uniformly bounded on the fixed interval. Countably many alpha can occur (spatial Fourier series), and absolute summability justifies every interchange. All endpoint transforms in this expansion have the stated budget.
The pointwise error E is smaller than every prescribed negative power of R: Lemma 6.5 supplies that error before the endpoint integrations; the rescaled distance weight is bounded; normalized cell masses are at most R^{O_d(sigma)}; and the shell length is at most 2^{N+1}=2R because v<=N. Multiplying the initial accuracy by a large fixed power handles each factor and any subsequent square.
Take the L^2([1/2,2], ds) triangle inequality. Because Psi>=1 on [1/2,2], the square of each angular integral is bounded by its contribution to the corresponding H_F. The complete prefactor is
2^{-a} 2^v 2^{n_*(v-a)} = 2^{d(v-a)},
exactly the H_F normalization. That proves Proposition 6.4 in the negative-frequency form; complex conjugation gives the positive-frequency form.
The independent angular energy at depth-a cells uses one common radial frequency and two independent angular directions. To pass from a depth-a pair to depth-p cells, split each endpoint transform into cell pieces. Angular oscillation of the phase and radial oscillation together force the four cell labels to lie on a sparse bipartite graph, and the parallel tests bound the two degrees. Retained terms then become positive energies on smaller cells, and at the end Cauchy-Schwarz transfers the estimate to H_F.
Proposition 7.1 (Graph estimate for the independent angular energy). Let rho_1, rho_2 be two regularized class laws, possibly equal, with profiles M_j, excess profiles A_{j,n}, increments I_j, and costs k_j. Let X, Y be occupied cells of depth a. Write X_1=X, X_2=Y, and let b^{(j)}_z(w) be an admissible endpoint symbol on X_j. Let L be the largest nominal length of a descriptor in either symbol, zero if both lists are empty. Suppose the integer depths satisfy 0<=a<p<=v<=N and
p-a <= v-p + 2 sigma N, L <= v-p + 2 sigma N, v-p >= 10 sigma N.
At endpoint j suppose the symbol imposes the parallel test [g_j,p] at a level ell_j in {0,...,J_0}: for rho_j-almost every z in X_j and every w in S^{d-1},
b^{(j)}_z(w) != 0 => w in G^{(j)}_{ell_j,[g_j,p]}(z).
Assume a <= g_j < p and g_j <= a + 2 sigma N. Form the independent energy H^{X,Y}(a,v) with these specified symbols. Then for every fixed T>0,
H^{X,Y}(a,v) <= R^{- sum_{j=1}^2 (A_{j,p}-A_{j,a}) + min_{j=1,2} k_j(a,p) + O_d(sigma)+o(1)}
* max_{P subset X, Q subset Y, P,Q occupied depth-p cells} H^{P,Q}(p,v)
+ O(R^{-T}).
No separation of X and Y is required. Every child is formed by restricting the same endpoint symbols to P and Q, retaining every original mask factor, including any whose anchor is p. Constants are uniform over cells and admissible symbols with fixed list exponents, derivative budgets, and smooth-angular bounds at the derivative orders used.
Localization and degree arguments are separate. Set
h=p-a, q=v-p, delta = min(c_0, 2^{-h} R^{10 sigma}), tau = 2^{-p} R^{30 sigma},
where c_0>0 is a small dimensional constant. Take a finite smooth partition of unity {eta_nu} on the sphere with bounded overlap, eta_nu supported in a cap of radius C delta centered at w_nu, and ||grad^m eta_nu||_{L^infty}<=C_m delta^{-m}. Such a partition has O_d(delta^{-(d-1)}) caps. For occupied depth-p cells use the transforms U_P, V_Q from Definition 6.3, divided by the model masses M_1(p), M_2(p). The exact splitting identity is
U_X = lambda_1 sum_{P subset X} U_P, V_Y = lambda_2 sum_{Q subset Y} V_Q, lambda_j = M_j(p)/M_j(a) = R^{-I_j(a,p)}.
Expand the two squares in H^{X,Y}(a,v) and insert the angular partition in both. Index each term by a row (P,Q) and a column (P',Q') with x in P, x' in P', y in Q, y' in Q'. The P-labels belong to rho_1 and the Q-labels to rho_2. Put Delta_1=x-x', Delta_2=y-y', d_1=c_P-c_P', d_2=c_Q-c_Q'. The phase is r(w_1.Delta_1 + w_2.Delta_2), and each displacement joins two points in a depth-a parent, so |Delta_j|+|d_j| is at most a fixed multiple of 2^{-a}.
For an endpoint cell P with measure rho and symbol b_x, define E(P,nu) = int_P int_S eta_nu(w) |b_x(w)|^2 dw d rho(x). The cell-cap incidence is eligible when E(P,nu)>0. When E(P,nu)=0, Fubini shows the transform vanishes for almost every w in the cap, so all expanded terms involving this incidence drop out. A positive E(P,nu) supplies a point-angle witness where the symbol is nonzero and the stated support conditions hold. Only finitely many witnesses are chosen at each N.
Fix two caps with centers w_1^0, w_2^0. Retain an edge when all its cell-cap incidences are eligible and
|proj_{(w_j^0)^perp} d_j| <= tau (j=1,2), |w_1^0 . d_1 + w_2^0 . d_2| <= tau.
Eligibility and edge decisions act on whole cell-cap terms; such conditions never cut an angular support by a nonsmooth pointwise condition.
Lemma 7.2 (Nonstationary localization). Under the depth range of Proposition 7.1, replacing the expanded independent energy by its sum over retained edges changes it by O(R^{-T}) for every fixed T>0. The replacement is performed in the integrated oscillatory form, before absolute values are taken.
Proof. Fix the four spatial points in one cell term. Fubini applies because the spatial measures are finite and the amplitudes are bounded with compact radial support. Replacing centers by points, or cap centers by angles in their caps, changes each test by at most O_d(2^{-p}+2^{-a}delta) = O(2^{-p}R^{10 sigma}) = o(tau); the last inequality also holds when delta=c_0 because that case forces c_0 <= 2^{-h}R^{10 sigma}.
If the scalar test fails, |w_1.Delta_1 + w_2.Delta_2| >= tau/2 throughout the term. In the variable u=r/2^v, each radial integration by parts gains a factor at most C(2^v tau)^{-1}. Symbols and the angular partition do not depend on u, so only the fixed smooth radial cutoff is differentiated. Since 2^v tau = 2^q R^{30 sigma} >= R^{40 sigma}, arbitrary power decay follows.
If instead a perpendicular test fails, suppose the first. Let t be the unit tangent vector at w_1^0 along proj_{(w_1^0)^perp} d_1, and extend it to the smooth sphere field V(w)=t-(t.w)w. The derivatives of V are uniformly bounded independently of the cap. Throughout the cap support, |V(w_1).Delta_1| >= tau/2, so for Phi = r(w_1.Delta_1 + w_2.Delta_2),
|V Phi| >= C 2^v tau, |V^m Phi| <= C_m 2^v 2^{-a} (m>=1).
The same construction applies to the second variable when its perpendicular test fails.
Angular derivatives of the product amplitude are bounded by C_m (1+D_*+delta^{-1})^m with D_* = R^sigma max(1, 2^L R^{-50 sigma}). Differentiating (V Phi)^{-1} contributes a factor (2^v tau)^{-1} times powers of 1+2^{-a}/tau. Thus m integrations by parts bound the term by C_m times
((1+D_*+delta^{-1}+2^{-a}/tau)/(2^v tau))^m.
The three depth-range inequalities give D_* <= R^sigma + 2^q R^{-47 sigma}, delta^{-1} <= c_0^{-1} + 2^q R^{-8 sigma}, and 2^{-a}/tau = 2^h R^{-30 sigma} <= 2^q R^{-28 sigma}. The numerator is at most C 2^q R^{4 sigma} and the ratio at most C R^{-26 sigma}. That is a fixed positive power gain per integration by parts, so sufficiently many derivatives give O(R^{-T'}).
All sums and normalizations outside the fixed spatial term cost at most a fixed power of R: at most R^{4d} cell quadruples, O(R^{2(d-1)}) cap pairs, profiles lying in [0,d], energy prefactors at most R^d, and spatial-normalization factors bounded by fixed powers. Choosing T' > T by these fixed powers gives O(R^{-T}). No spatial derivative was taken, so bounded measurable spatial multipliers remain admissible.
Lemma 7.3 (Both degrees of the retained graph). For each cap pair, both row and column degrees of the retained graph are at most
D_gr = R^{min_{j=1,2} k_j(a,p) + O_d(sigma) + o(1)}.
The estimate does not require the two cap directions to be separated.
Proof. Fix a row (P,Q). Its unknown column entries P' (profile 1) and Q' (profile 2) have, for each j, the perpendicular constraint placing the unknown center within tau of a line through its fixed mate in direction w_j^0. Group candidate cells by their depth-g_j ancestor. Because candidates lie in a single depth-a parent, there are at most 2^{d(g_j-a)} <= R^{2d sigma} groups. In one nonempty group choose an eligible cell P_* and a point-angle witness (z_*,w_*) with b^{(j)}_{z_*}(w_*) != 0. The support condition puts w_* in the good set for the parallel test [g_j,p] at the depth-g_j anchor c_* (or, for a differentiated mask, in its enlarged neighborhood).
For every other candidate center c in this group, the two perpendicular constraints and |w_*-w_j^0| <= C delta give
|c-c_*| <= sqrt(d) 2^{-g_j}, dist(c, c_* + R w_*) <= C_d(2 tau + 2^{-g_j} delta) <= C_d 2^{-p} R^{30 sigma}.
The last is smaller than any test width W_i(p) >= 2^{-p}R^{500 sigma}. The first is inside the testing ball B(c_*, 10(d+1)2^{-g_j}). The parallel test therefore bounds the group by R^{k_j(g_j,p)+C_d sigma} <= R^{k_j(a,p)+C_d sigma} because [g_j,p] is contained in [a,p]. Summing groups gives R^{k_j(a,p)+O_d(sigma)} possibilities for the first unknown.
With that first unknown fixed, its displacement is fixed. The scalar constraint confines the remaining unknown's longitudinal coordinate in its own cap direction to an interval of length O(tau); its perpendicular constraint confines the other d-1 coordinates to a ball of radius O(tau). Together these confine its center to a ball of radius O_d(tau), containing at most C_d(1+2^p tau)^d <= C_d R^{30 d sigma} grid centers. Trying either profile j and taking the better of the two resulting counts proves the row-degree bound. Fixing a column instead and reversing the signs of the displacements gives the same argument, hence the same column-degree bound. No division by the angle between w_1^0 and w_2^0 occurs: the scalar test always fixes the remaining coordinate in the unknown's own cap direction.
Proof of Proposition 7.1. Apply Lemma 7.2. Fix r and a cap pair; take absolute values only of the retained terms. In the row-column indexing above, the amplitudes are
F_{P,Q}=U_P(r,w_1)V_Q(r,w_2), G_{P',Q'}=U_{P'}(r,w_1)V_{Q'}(r,w_2).
The elementary inequality 2|FG| <= |F|^2+|G|^2 together with both degree bounds gives
sum_{(alpha,beta) edge} |F_alpha G_beta| <= (D_gr/2)(sum_alpha |F_alpha|^2 + sum_beta |G_beta|^2).
The right-hand sums may be enlarged to all row and column labels. Sum the nonnegative angular partition weights over cap pairs; their sum is one, so no factor counting caps appears. Integrate against the same nonnegative radial cutoff Psi(r/2^v) dr/2^v. Each row square yields the independent-angle energy of its child cell pair, and each column square does the same.
Let I_j=I_j(a,p). The occupied-cell estimate bounds the number of row labels and of column labels by R^{I_1+I_2+O_d(sigma)+o(1)}. The splitting factors in lambda_j appear squared, giving R^{-2(I_1+I_2)}. The parent prefactor divided by the child prefactor is 2^{d(v-a)}/2^{d(v-p)}=2^{d(p-a)}. Since S=d/2 and I_j=A_{j,p}-A_{j,a}+S(p-a)/N, their product is
R^{-2(I_1+I_2)} R^{I_1+I_2} 2^{d(p-a)} = R^{-(I_1+I_2)+d(p-a)/N} = R^{-sum_{j=1}^2(A_{j,p}-A_{j,a})}.
Multiplying by D_gr proves the graph estimate, including the arbitrary-power error from Lemma 7.2. The construction only restricts symbols to smaller cells, so every child retains all its original mask factors and support conditions.
Corollary 7.4 (The separated angular form). Under the hypotheses of Proposition 7.1, suppose X, Y and their supported laws (or, at a=0, the fixed outer separated supports) have the separated geometry used in Definition 6.3. Then the estimate holds with H_F^{X,Y}(a,v) on the left and the same child energies on the right. This includes the outer root a=0 with separated actual supports inside possibly equal depth-zero cells X,Y.
Proof. Cauchy-Schwarz in w gives H_F^{X,Y}(a,v)<=H^{X,Y}(a,v) for the same endpoint symbols, so the tests and model-mass normalizations are identical. Proposition 7.1 supplies exactly those child energies.
Fix one regularized probability rho with profile m and profile data A_n, M(n), I(a,b), k(a,b). Both endpoint laws in an angular state share that profile; their cells and bounded spatial multipliers may differ. Two kinds of recursive state occur.
Definition 8.1 (Positive distance state). Fix integer depths 0<=a<=t<=N, a smoothing index i in {0,...,J_0}, and a separated pair X,Y of occupied depth-a cells. Set
P_i^{X,Y}(a,t) = { (x,y) in X x Y : e(x,y) in G_{i,t}(x) cap G_{i,t}(y) for every t in M(a,t) },
theta_i^{X,Y}(a,t) = 1_{P_i^{X,Y}(a,t)} (rho|_X x rho|_Y),
an unnormalized pair restriction. With C_0=100(d+1), let
D_i^{X,Y}(a,t) = 2^{t-a} M(a)^{-4} int 1_{|D(x,y)-D(x',y')| <= C_0 2^{-t}} d theta_i^{X,Y}(x,y) d theta_i^{X,Y}(x',y').
The positive state is these depths, index, cells, with value D_i^{X,Y}(a,t). Its current interval is [a,t]; no separate retained base is carried.
Definition 8.2 (Recursive angular state). Let the retained base B=[a_0,t_0] be an integer subinterval of [0,N], and let [a,v] be the current interval, with 0<=a_0<=a<=v<=t_0<=N and t_0<=v+sigma N. Set
M_{>a}(B) = { t in M(a_0,t_0) : the anchor of t is deeper than a }.
A recursive angular state consists of these intervals, an index i in {0,...,J_0-1}, two occupied depth-a cells X,Y (no separation required), and two admissible endpoint symbols. Each symbol's descriptor list carries exactly one occurrence of each descriptor in M_{>a}(B), at common index i. Its size is bounded by a fixed polynomial throughout the recursive family. The state also fixes a derivative budget and common numerical bounds for its smooth angular factors. Its value is H_i^B(a,v) := H^{X,Y}(a,v) with those symbols. Maxima vary only over finitely many cells; suprema over admissible symbols. Neither operation varies the displayed base, index, or fixed bounds unless stated.
The origin of a descriptor is not a further factor. For each t in M_{>a}(B) there is an originating interval inside the base. If t has anchor p and nominal length L, that origin contains [p-L,p+L] and supplies every parallel descriptor [g,p] for p-L<=g<p. Origins are chosen when used; a deduplicated descriptor need not have one origin valid in every list containing it.
A one-profile graph child at depth a' with previous start a<=a', carrying M_{>a}(B), retains its reached factors through the graph estimate. The reached-factor deletion after Cauchy-Schwarz removes every factor anchored at depths at most a' at bounded cost, leaving exactly M_{>a'}(B). Once the base-contract inequalities hold, that general energy is the value of a recursive angular state.
A normalization is used in every dimension:
2^{d(b-a)} (M(b)/M(a))^2 = 2^{d(b-a)} R^{-2 I(a,b)} = R^{-2(A_b-A_a)}, using S=d/2.
Lemma 8.3 (Binning positive measures). Let eta, zeta be finite positive measures on the line and eps>0. Set B_eps(eta)=sum_{j in Z} eta([j eps,(j+1)eps))^2. For every fixed C>=1,
B_eps(eta) <= (eta x eta){|s-s'|<=eps},
sup_u (eta x eta){|s-s'-u|<=C eps} <= C' B_eps(eta),
int int (1+|s-s'|/eps)^{-2} d eta(s) d zeta(s') <= C' B_eps(eta)^{1/2} B_eps(zeta)^{1/2}.
Also
(eta x eta){|s-s'|<=C eps} <= C' eps int_{|r|<=eps^{-1}} |hat(eta)(r)|^2 dr.
Proof. Put b_j=eta([j eps,(j+1)eps)) and c_j similarly. Cauchy-Schwarz gives, for every integer h, sum_j b_j b_{j+h} <= sum_j b_j^2 and sum_j b_j c_{j+h} <= (sum_j b_j^2 sum_j c_j^2)^{1/2}. An interval of length 2C eps contains only boundedly many bin offsets. That proves the first two claims. The kernel of the third is bounded on the bin-pair with offset h by C(1+|h|)^{-2}, whose sum converges.
For the last, fix small b>0 and put K(t)=(sin(bt)/(bt))^2 with continuous value at zero. Then K>=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')/eps) prove the claim. No density of either measure is required.
A positive state uses the fixed policy choice p of its current interval [a,t]. At a suffix choice, the projection test for the full conditional law permits a direct passage to the coarse distance count at depth p. At a prefix choice, a positive Fourier estimate splits the count into frequency shells; low shells give shorter positive states, and high shells use the weighted shell conversion and the graph estimate to enter an angular state.
For an angular state let L be the largest length in its remaining list, L=0 if that list is empty, and set
L_0 = max(L, ceil(10 sigma N)), h = v - 2 L_0.
This compares the longest remaining test with half the frequency range. If h<=a, an origin of a longest descriptor supplies a parallel test for a graph step. If h>a, splitting at depth h leaves enough frequency range for stationary phase on the far terms; the near terms have only boundedly many neighboring cells. Thus an angular episode continues through direct or near steps and returns to a positive state through a far step.
The current interval is [a,t] for D_i(a,t) and [a,v] for H_i^B(a,v). Terminality refers to that interval: its width is at most 100 q_0 and does not refer to the base.
Proposition 8.4 (State reductions). Suppose the current state is nonterminal, its smoothing index satisfies i<J_0, and N is sufficiently large with all parameters fixed. For every prescribed T>0 the state is bounded, up to an error O(R^{-T}) from oscillatory operations, by a sum of child bounds whose number is polynomial in N, and possibly a low-frequency stopping term. For every non-low child, if the old start is a and the new start a', its multiplier is
R^{-2(A_{a'}-A_a) + cost + C_d sigma + o(1)}.
A low child has cost zero and the entire multiplier R^{o(1)}, with no fixed C_d sigma loss. The low-frequency stopping term is R^{C_d sigma + o(1)}. The legal children are
Reduction | Child | Cost
Suffix | D_{i+1}(a,p) | A_p - min_{[p,t]} A
Low | D_{i+1}(a,v) | 0
High | H_i^{[a,t]}(p,v)| k(a,p)
Direct | H_i^B(p,v) | k(a,p)
Near | H_i^B(h,v) | 0
Far | D_{i+1}(b,v) | 0
For a positive parent, p is the fixed policy choice of its current interval. Suffix is used when 2p>a+t. When 2p<=a+t, the shell index is an integer a<v<=t: Low has t-v>=sigma N and High has 0<=t-v<sigma N. A high child has base [a,t] with the fixed enlarged derivative budget and common bounds produced by the inverse conversion; both depend on T and the fixed parameters, not on N.
For an angular parent use L_0, h above. If h<=a, the Direct row chooses a maximal-length remaining descriptor, its anchor p, and an origin [b_0,u_0] in the base. That choice has L=L_0, a<p<v, b_0<=p-L, p+L<=u_0, and v-a<=2L; it supplies the parallel descriptor [max(a,p-L),p]. If h>a, Near uses depth h. The Far rows range over integer b from the comparable far displacements, with a<b<=h+O_d(1)<v, using separated depth-b cell pairs. Direct and Near children keep base, index, derivative budget and fixed bounds. A Far child has the fresh positive list M(b,v).
Finite maxima over child cells follow the stated geometry. Suprema over symbols are taken over the family just specified. An empty child collection contributes zero. Constants outside powers of R may depend on fixed parameters, derivative budgets, and common bounds. The C_d in exponent losses depend only on d. A terminal state is bounded by
R^{C_d(kappa+sigma)+o(1)}.
Proof. Each reduction together with the support conditions its iteration needs.
Suffix. Let theta = theta_i^{X,Y}(a,t). Retain a depth-p pair (P,Q) of the separated parent cells when P_i^{X,Y}(a,t) meets P x Q, and choose one such intersection pair as a witness. Put e_0=e(c_P,c_Q). Separation gives ||grad^2 D|| <= C 2^a, hence
D(x,y) = e_0.(x-y) + O_d(2^{a-2p}) = e_0.(x-y) + O_d(2^{-t}) for x in P, y in Q.
The last equality uses 2p>a+t. The witness angle differs from e_0 by O_d(2^{a-p}). For y,y' in Q, changing between those two angles changes the projection of y-y' by O_d(2^{a-2p}).
The witness satisfies the projection test [p,t], formulated for rho|_Q / rho(Q), not for any witness-restricted law. The test width absorbs fixed multiples of 2^{-t}. So the conditional projected measure in direction e_0 has self-collision mass at tolerance C2^{-t} at most
B = 2^{-(t-p)} R^{A_p - min_{p<=n<=t} A_n + C_d sigma}.
The translated-client bound in Lemma 8.3 gives the same estimate for any shifted collision window, and fixing x,x' in P with the linearization gives B_{2^{-t}}(D_#(rho|_P x rho|_Q)) <= C B rho(P)^2 rho(Q)^2.
Group retained cell pairs by the 2^{-p}-bin containing their center distance, and write W_G=sum_{(P,Q) in G} rho(P) rho(Q). The fine bin vector of a group is a sum of positive vectors; the l^2 triangle inequality and bounded overlap of coarse-bin enlargements bound the parent's fine collision mass by C B sum_G W_G^2.
Every full pair in a retained P x Q satisfies the child list M(a,p) at level i+1: its descriptors are in the parent list, have anchors no deeper than p, and have nominal lengths at most (p-a)/2. Its spatial anchor cells match the witness's. The change of direction is O_d(2^{a-p}), inside the transfer neighborhood of the mask lemma for every such descriptor. Two full pairs in one coarse group have distance difference at most (1+4 sqrt(d))2^{-p} < C_0 2^{-p}. So sum_G W_G^2 <= 2^{-(p-a)} M(a)^4 D_{i+1}(a,p). Multiplying by the parent normalization and substituting B gives
2^{t-a} M(a)^{-4} B 2^{-(p-a)} M(a)^4 = R^{A_p - min_{[p,t]} A + C_d sigma},
which is the suffix reduction with no change of start.
Prefix. Let theta=theta_i^{X,Y}(a,t) and let zeta be the positive distance pushforward of
M(a)^{-2} (D(x,y) 2^a)^{-(d-1)/2} prod_{t in M(a,t)} w_{i,t}(x,e(x,y)) w_{i,t}(y,e(x,y)) d rho|_X(x) d rho|_Y(y).
On the good pairs defining theta every mask is one. The remaining distance weight is bounded above and below by positive dimensional constants on the separated geometry. Thus positivity and the Fourier collision bound give
D_i(a,t) <= C 2^{-a} int_{|r|<=2^t} |hat(zeta)(r)|^2 dr.
Since zeta(R)<=R^{C_d sigma}, frequencies below 2^a contribute R^{C_d sigma}, a stopping term. Reality allows restricting to positive frequencies above 2^a, split into shells [2^{v-1},2^v] with a<v<=t.
If t-v>=sigma N, a smooth majorant of the shell has inverse Fourier transform bounded in modulus by C 2^v (1+2^v |s-s'|)^{-2}. Its contribution to the displayed bound is at most
C 2^{v-a} M(a)^{-4} int int (1+2^v |D(x,y)-D(x',y')|)^{-2} d theta'(x,y) d theta'(x',y'),
where theta' is the unweighted pair restriction to the support of the old masks. Every such pair satisfies M(a,v) at level i+1 by list containment and mask support. Lemma 8.3 bounds the displayed expression by C D_{i+1}(a,v). Both sides use the same exact M(a); no R^{C_d sigma} mass-comparison loss enters. Shell-index summation costs R^{o(1)}.
Now suppose t-v<sigma N. The largest nominal length in the list is at most (t-a)/2. Nonterminality gives v-a>100 q_0 - sigma N > 20 sigma N and L <= (v-a)/2 + sigma N. Proposition 6.4 applies to zeta. For the prefix choice p the descriptor [a,p] is present and
p-a <= t-p <= v-p+sigma N, L <= (t-a)/2 <= t-p <= v-p+sigma N, v-p >= (t-a)/2 - sigma N > 10 sigma N.
These verify the hypotheses of Corollary 7.4 with g=a. Its one-profile specialization first produces general independent energies on depth-p cells, formed with restricted converted symbols and still carrying every old mask. In those positive energies remove the anchors at depths at most p. The remaining list at each endpoint is exactly M_{>p}([a,t]) with a<=p<=v<=t<=v+sigma N. The resulting values are recursive states H_i^{[a,t]}(p,v), and the high contribution is at most
R^{-2(A_p-A_a)+k(a,p)+C_d sigma+o(1)} sup H_i^{[a,t]}(p,v) + O(R^{-T}).
The inverse conversion produces normalized angular derivatives of masks at index i, with a fixed enlarged budget and common bounds. Their supports still impose the level-(i+1) tests, which the graph estimate accepts. The new angular state keeps index i.
Direct. Consider H_i^B(a,v) with base B=[a_0,t_0]. Its remaining anchors are strictly deeper than a, and t_0<=v+sigma N. If h<=a, nonterminality forces L=L_0; otherwise v-a<=2 ceil(10 sigma N)<100 q_0. Choose a maximal-length descriptor with anchor p>a and an origin [b_0,u_0] in the base. That origin satisfies
b_0<=p-L, p+L<=u_0<=v+sigma N, v-a<=2L.
Hence L<=v-p+sigma N, p-a<=v-p+2 sigma N, and v-p>=(v-a)/2-sigma N>10 sigma N. Put g=max(a,p-L). Then a<=g<p, p-L<=v+sigma N-2L<=a+sigma N, and g<=a+sigma N. The same origin supplies the parallel descriptor [g,p], whose anchor is not yet reached. Proposition 7.1 applies to the general energy underlying H_i^B(a,v), without separating its two cells, and gives the direct reduction with cost k(a,p). The child still carries the old symbols; remove its reached factors only after forming its positive energy. The remaining list is M_{>p}(B), giving H_i^B(p,v) with the same fixed symbol bounds.
Near. If h>a, split each transform into depth-h cell pieces, temporarily keeping every remaining factor. Expand the two squares. A near term has both center displacements, in the first and second parent cells, at most 10(d+1)2^{-h}. In the row-column graph these conditions bound both row and column degrees by a dimensional constant. Apply 2|uv|<=|u|^2+|v|^2 to each row-column product and integrate each positive square. There are at most R^{2 I(a,h)+C_d sigma} rows or columns. The four splitting factors, label count, and prefactor ratio give
2^{d(h-a)} R^{-4 I(a,h)} R^{2 I(a,h)+C_d sigma} = R^{-2(A_h-A_a)+C_d sigma},
by the normalization identity. This is the zero-cost near estimate. Only after obtaining each positive energy remove the factors anchored at depths at most h; on its cell each is a bounded-modulus angular factor. The remaining list is M_{>h}(B), giving H_i^B(h,v) with the same fixed bounds. That removal does not apply to the far terms.
Far (analytic estimate). A remaining term has at least one point displacement of length >= c_d 2^{-h}. Fix the four spatial points and denote the two displacements Delta_1, Delta_2 with lengths d_1, d_2. Put d_+=max(d_1,d_2), d_-=min(d_1,d_2). For the larger displacement, on the radial support,
Lambda = r d_+ >= c 2^{v-h} = c 2^{2 L_0}, D_*^2/Lambda <= R^{-18 sigma} if 2^L R^{-50 sigma}<=1 else R^{-98 sigma}.
Also Lambda>=R^{c sigma} and D_*<=R^2 for N large. Lemma 6.5 expands the corresponding full-sphere integral to every negative power of R. After extracting the leading stationary gain, the coefficients and any fixed number of derivatives in r/2^v are bounded: the factor (r d_+)^{-j} pays for order-2j symbol derivatives.
If d_+>2d_-, leave the other angular integral exact. The r-derivative of each resulting phase has modulus at least d_+-d_->d_+/2, uniformly in the unexpanded angle. Repeated integration by parts in r/2^v gives O(R^{-T}), including d_-=0. Leaving that integral exact is legitimate because its symbols are independent of r.
Otherwise d_+ and d_- are comparable, and the second angular integral admits the same expansion. Choose an integer n with d_+ in (2^{-n-1},2^{-n}], and put b=n+log_2 K_d. Then both lengths lie between K_d 2^{-b}/4 and K_d 2^{-b}. Moving to their depth-b cell centers changes the bounds by at most 2 sqrt(d) 2^{-b}; the K_d choice ensures both pairs lie in the separated geometry of Definition 8.1. Each displacement joins points in one depth-a cell, so d_+<=sqrt(d) 2^{-a} and hence b>a. The far lower bound gives b<=h+O_d(1)<v.
The two leading stationary gains multiply to O_d(2^{-(d-1)(v-b)}). Radial integration, using the coefficient derivative bounds just proved, bounds each remaining kernel by
C 2^{-(d-1)(v-b)} (1+2^v |d_1-d_2|)^{-2}.
Opposite stationary signs make the phase involve d_1-d_2; equal signs make it involve d_1+d_2, whose modulus is no smaller. That proves the kernel bound for every sign choice.
Far (support and normalization). The interval [b,v] lies in the base, so M(b,v) is a sublist of its list. Every anchor of this child list is strictly deeper than b>a, and no such factor was removed. In particular none was removed at depth h: the far expansion used the symbols before the near removal. Each coefficient differentiates the masks only angularly, so it retains all their supports. For the internal pair (x,x') the amplitude is b_x(w) conj(b_{x'}(w)). Its stationary coefficient is evaluated at w=+-(x-x')/|x-x'|, so a nonzero coefficient imposes the level-(i+1) tests of M(b,v) at both spatial endpoints. The same assertion holds for (y,y'). Antipodal symmetry handles the negative stationary direction. Expansion remainders are already arbitrary-power errors.
The four splitting factors cancel their model denominators exactly:
(M(h)/M(a))^4 M(h)^{-4} = M(a)^{-4}.
For each comparable length band defining b, the stationary estimate is now a nonnegative kernel bound with the child-support indicators. On that upper bound the far h-cell indicator may be discarded while keeping the comparable length band; summing the disjoint h-cell partition reassembles the raw four-point measure on X^2 x Y^2, which is then partitioned into depth-b cells. No count of h-cell labels remains, whether b is smaller or larger than h.
For a fixed quadruple of separated depth-b cells (P,P',Q,Q'), put eta_1 = D_# theta_{i+1}^{P,P'}(b,v) and eta_2 = D_# theta_{i+1}^{Q,Q'}(b,v). These are the unnormalized positive distance measures for the two internal pairs, each restricted by the child tests at both endpoints. Lemma 8.3 and the exact normalization of D_{i+1} give
int int (1+2^v |s-s'|)^{-2} d eta_1(s) d eta_2(s') <= C M(b)^4 2^{-(v-b)} max D_{i+1}(b,v).
Every bin-square sum is bounded by its unshifted positive collision count, and the geometric mean of the two counts is at most their maximum. Once the first cell of either internal pair is fixed, its partner has only a dimensional number of labels because their center distance is O_d(K_d 2^{-b}). There are at most R^{2 I(a,b)+C_d sigma} quadruple labels.
The two stationary gains and the scalar collision normalization combine to 2^{-d(v-b)}, converting the parent frequency prefactor into 2^{d(b-a)}. The four depth-b masses and the two free first-cell labels leave R^{-2 I(a,b)+C_d sigma}. Explicitly,
2^{d(v-a)} M(a)^{-4} 2^{-(d-1)(v-b)} M(b)^4 2^{-(v-b)} R^{2 I(a,b)+C_d sigma} = 2^{d(b-a)} R^{-2 I(a,b)+C_d sigma} = R^{-2(A_b-A_a)+C_d sigma}.
Summing possible scales b, coefficient terms and shell indices introduces only R^{o(1)}. That gives the zero-cost far reduction, with one increase of smoothing index.
Stopping. For a terminal interval its width is at most 100 q_0. Bounded symbol moduli and the occupied-cell bounds give H_i^B(a,v)<=C 2^{d(v-a)} R^{C_d sigma} and D_i(a,t)<=C 2^{t-a} R^{C_d sigma}. Since q_0/N=kappa+O(1/N), both satisfy the terminal bound. The low-frequency stopping term was already estimated. That finishes the proof.
The reductions of Proposition 8.4 have three transitions with potentially positive cost. For a suffix step [a,t]->[a,p], a depth left of the policy's inner interval that minimized A on both intervals would leave the minimum unchanged and could not pay the positive cost A_p - min_{[p,t]} A. Instead use a pair of well-separated depths. In a suffix child, separation places the later depth in the inner interval, so that depth can be replaced by a minimizer on [p,t] while the earlier depth remains. A high prefix or direct reduction can replace the earlier depth.
Definition 9.1 (Two-depth potential). For an integer interval J=[a,t] of width at least 10 q_0, put
V(J) = min_{a<=i<j<=t, j-i>=10 q_0}(A_i+A_j).
The potential is undefined for shorter intervals. It is attached to the current interval of a state: [a,t] for D_i(a,t) and [a,v] for H_i^B(a,v). The retained base supplies the angular state's remaining descriptor list; after reached anchors are removed, the chosen descriptor of that list has an origin which fixes the inner interval on which its anchor minimizes A.
Lemma 9.2 (Payment of the costs). If a reduction of Proposition 8.4 has a potentially positive cost with parent interval J and child interval J', then V(J') is defined and
cost <= V(J') - V(J).
The same inequality holds for every zero-cost reduction whose child potential is defined.
Proof. Containment of admissible pairs gives V(J)<=V(J') when J' is contained in J and both potentials are defined; that covers every zero-cost case. The three positive-cost cases are checked by replacing one depth in an arbitrary admissible child pair and then applying the result to a minimizing child pair. Only the inner-interval minimizing property of the policy enters; no monotonicity of A is assumed.
Suffix. J=[a,t], J'=[a,p], cost=A_p - min_{[p,t]} A. The suffix inequality 2p>a+t makes the child width exceed half the nonterminal parent width, hence >10 q_0. Let (i,j) be admissible in [a,p]. Then j>=a+10 q_0, while the left endpoint of the origin's inner interval is at most a+2 q_0. Since j<=p, the index j belongs to that inner interval, so A_j>=A_p. Choose r in [p,t] with A_r=min_{[p,t]} A. The pair (i,r) is admissible in the parent because r>=j, and
A_i+A_r <= A_i+A_j - (A_p - min_{[p,t]} A).
High prefix. J=[a,t], J'=[p,v], t-v<sigma N, cost=k(a,p). The prefix inequality gives v-p >= (t-a)/2 - sigma N > 10 q_0. Any admissible child pair (i,j) has p<=i<=v-10 q_0<=t-10 q_0. Then i is in the origin's inner interval and A_i>=A_p. Choose r in [a,p] with A_r=min_{[a,p]} A. The pair (r,j) remains admissible in [a,t] with r<=i, and its sum is at most A_i+A_j-k(a,p).
Direct. J=[a,v], J'=[p,v]. Let L be the maximal remaining descriptor length chosen by the reduction and [b_0,u_0] its origin. Then
b_0<=p-L, p+L<=u_0, v<=a+2L.
The reduction gives v-p >= (v-a)/2 - sigma N; nonterminality and sigma<<kappa make this >10 q_0. If k(a,p)=0, containment suffices. Otherwise choose i_0 in [a,p] with A_{i_0}=min_{[a,p]} A=A_p-k(a,p)<A_p. The origin's policy makes A_p no larger than every value in its inner interval, so i_0<=p forces i_0 strictly to the left of that inner interval. Hence
a<=i_0<b_0+2 q_0, v<=a+2L<b_0+2 q_0+2L<=u_0+2 q_0.
For any admissible child pair (i,j), p<=i<=v-10 q_0<u_0-8 q_0. The origin's inner interval begins no later than p and ends no earlier than u_0-2 q_0. So i lies in it and A_i>=A_p. Replacing i by i_0 gives a pair (i_0,j) in the actual parent [a,v] (not merely in the origin), still admissible because i_0<=p<=i, with sum decreased by at least k(a,p).
Lemma 9.3 (Finite policy and smoothing budget). Every branch of the state reductions has at most 20/kappa non-low transitions and at most 1/sigma low transitions, for sufficiently small fixed parameters and sufficiently large N. The smoothing index increases by at most 20/kappa+1/sigma. With initial index zero, every index used in the iteration, including the next support level of a mask, is less than
J_0 = ceil(100(1/kappa+1/sigma)+100).
Proof. Along a branch the start depth never decreases and the end never increases. Let K be the number of fixed block representatives; K<=1/kappa+2 for large N. High or direct steps strictly advance the start to a representative; a representative can never be passed twice, so those steps together number at most K. Each suffix step lowers the end by at least q_0, so at most N/q_0<=1/kappa. Low steps lower the end by at least sigma N, so at most 1/sigma.
An angular episode starts at the root or just after a high step and ends at a far step or a stopping term. Direct and near steps remain in the episode. There are at most K+1 episodes and hence at most K+1 far steps.
Consider a near step whose child is nonterminal. That child has width 2 L_0>100 q_0, so the floor in the definition of L_0 is inactive and L=L_0. Remove the anchors that have been reached. If the maximum remaining nominal length decreases, at least one formerly maximal anchor has just been passed; charge the near step to that anchor. These charges are distinct along the branch. If the maximum stays at L, recomputing v-2L at the child gives its start depth, so the next step is direct: charge the near step to that direct step. There is at most one near step with terminal child. Thus near steps number at most 2K+1 and the combined non-low count is at most 5K+2<20/kappa for small kappa.
Only suffix, low, and far steps increase the smoothing index, each by one. The counts give i<=i_{start}+20/kappa+1/sigma at every state on the branch. High, direct, and near steps use the level-(i+1) supports of their index-i symbols without raising the index, so this extra support level keeps i below J_0 when i_{start}=0, because 20/kappa+1/sigma+1<J_0.
Theorem 9.4 (Single-profile angular bound). Let H_0^{[c,N]}(c,N) be a recursive angular state of nonterminal width, with initial derivative budget B and common angular bounds Q_j fixed independently of N. Then
H_0^{[c,N]}(c,N) <= R^{2A_c - V([c,N]) + C_d(kappa+sigma/kappa)+o(1)}.
The bound is uniform over the two occupied cells (including two copies of the same cell), the bounded spatial multipliers, and admissible symbols with these fixed bounds. The o(1) may depend on B and the Q_j.
Proof. Iterate Proposition 8.4, stopping at every terminal state or low-frequency stopping term. Lemma 9.3 gives a depth bound depending only on fixed parameters; since the root index is zero, it also ensures every index needed to continue a nonterminal child is available. Each transition has at most polynomially many discrete choices in N, and all coefficient sums in the oscillatory estimates are bounded or subpower. The total main term is at most R^{o(1)} times the largest terminal branch bound. Suprema over symbol classes are part of the transition bounds and require no count of their members.
Fix a branch and let a_f be its final start. The start changes telescope exactly to 2A_c - 2A_{a_f}. Lemma 9.2 bounds every cost by an increase in the potential while it is defined. A child of width less than 10 q_0 is already terminal and the step reaching it has zero cost, because every potentially positive-cost child has width greater than 10 q_0. Let J_*=[u,w] be the last interval on the branch with defined potential. Then
sum cost <= V(J_*) - V([c,N]).
This also covers a low-frequency term that stops at a nonterminal interval.
Estimate V(J_*) directly; no potential is assigned to a shorter final interval. Nesting gives a_f in [u,w] and w-u>=10 q_0. Set
i = max(u, min(a_f, w-10 q_0)), j = i+10 q_0.
Both indices lie in J_*, are separated by 10 q_0, and each is within 10 q_0 of a_f. The profile Lipschitz bound then gives
V(J_*) <= A_i+A_j <= 2A_{a_f} + 20d q_0/N = 2A_{a_f} + O_d(kappa)+o(1).
With the telescoping start factors this leaves 2A_c - V([c,N]) + O_d(kappa)+o(1). The stopping bound contributes O_d(kappa+sigma). Only non-low transitions incur C_d sigma slack, so their combined loss is O_d(sigma/kappa) by Lemma 9.3. Low transitions contribute only subpower factors. That is the main-term estimate.
For completeness fix the error and derivative budgets before taking N large. Since m is nondecreasing and d-Lipschitz with m(0)=0, every difference of A and every cost is bounded in absolute value by a dimensional constant. Each ancestor main factor is therefore at most R^{C_d+o(1)}. Choose a target accuracy T>C_d(1/kappa+1/sigma)+10d+10, with the dimensional constant large enough. Lemma 9.3 ensures every propagated additive error is negligible against the bound.
No recursive growth of the persistent angular derivative budget occurs. Each passage from a positive count to an angular state begins with fresh undifferentiated masks; choose its inverse stationary expansion order first, large enough for the prescribed T. The resulting finite order together with the initial order bounds all derivatives carried by angular states. The smooth-angular bounds can be fixed from the initial Q_j and from Proposition 6.4 at the fixed accuracy used for these conversions. The graph estimate returns original symbols restricted to child cells, with no added derivatives. Removing reached factors from a positive child of a near step adds no derivatives. A far step may use an arbitrarily long direct stationary expansion, but its child is a positive distance count and carries none of those derivatives. Choose the nonstationary integration orders and the direct stationary orders after this persistent budget is fixed; these may depend on d, kappa, sigma, T and the initial budget, but not on N. Their large constants and polynomial descriptor counts are absorbed in the permitted constant and o(1) terms; the dimensional exponent losses in the algebra above are unchanged.
The graph estimate and the two-depth potential now give decay on each frequency shell. The measures on different shells have different classes and masks. Mass restoration and a scalar reconstruction argument close the proof. All distances in this part are original-space distances.
Let S=d/2. Fix starting probabilities mu_1, mu_2 and chi>0 from the starting-pair package. Compose with one common similarity placing E in (1/4,3/4)^d; this multiplies all distances by one fixed positive constant and preserves the conclusion. Choose one small common translation of the dyadic grid whose boundaries have zero mass for both starting laws. Its depth-zero cube Q_0 still contains E in its interior. Use the same grid for both class decompositions. The actual supports stay separated by a fixed positive distance, and cross directions stay in the prepared patch. At R=2^N apply the regular-classes lemma with parameter sigma to both measures. Write C_{j,N} for the retained classes on side j, and for C in C_{j,N} put
a_{j,C}=mu_j(C), rho_{j,C}=a_{j,C}^{-1} mu_j|_C.
Classes on each side are disjoint, their count is R^{o(1)}, and the discarded original mass is at most R^{-sigma+o(1)}. All o(1) terms are uniform over classes with fixed parameters held constant. Each normalized class lives in Q_0 with mass one, and its root profile has m_j(0)=0, so M_j(0)=1.
For a class pair suppress labels and put A_{j,n}=m_j(n/N)-S n/N, k_j(g,p)=A_{j,p}-min_{g<=n<=p} A_{j,n}. Recall the profile input:
A_{j,0}=0, |A_{j,n}-A_{j,n'}|<=d|n-n'|/N, A_{j,n}>=chi n/N - C_d sigma.
Throughout, C_d may be enlarged a finite number of times and depends only on d.
The aim is a common depth c with small entry cost min_j k_j(0,c) and both excess profiles remaining above a positive level on [c,N]. The graph estimate contributes the smaller cost; after Cauchy-Schwarz separates the profiles, the single-profile estimates supply the larger potential saving.
Lemma 10.1 (Entry depth). Fix 0<beta<min(chi/10,1/10). If sigma is small relative to d, chi and beta, then for all large N the integer
c = max{ n in {0,...,N} : min(A_{1,n},A_{2,n}) <= beta }
is defined and satisfies
floor(beta N/(2d)) <= c < N/4, A_{j,n} >= beta - d/N (j=1,2, c<=n<=N), min_j k_j(0,c) <= beta + C_d sigma.
For sufficiently small kappa with q_0=ceil(kappa N), the interval [c,N] is nonterminal and its potentials satisfy
V_j([c,N]) = min_{c<=u<v<=N, v-u>=10 q_0}(A_{j,u}+A_{j,v}) >= 2 beta - 2d/N.
Proof. The index 0 qualifies. For n<=beta N/(2d) the Lipschitz bound gives A_{j,n}<=beta/2, producing the lower bound on c. For n>=N/4 the lower profile bound gives A_{j,n}>=chi/4-C_d sigma>beta once sigma is small enough, producing the upper bound on c.
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_*, A_{j_*,c}<=beta. Since min_{0<=n<=c} A_{j_*,n}>=-C_d sigma, min_j k_j(0,c)<=beta+C_d sigma. The width of [c,N] exceeds 3N/4, hence exceeds 100 q_0 for small kappa and large N. Every pair entering the potential consists of two indices covered by the lower profile bound, giving the potential bound.
Write D(x,y)=|x-y|, e(x,y)=(x-y)/D(x,y), W(x,y)=D(x,y)^{-(d-1)/2}. Fixed separated supports give 0<d_0<=D(x,y)<=d_1 on supp(mu_1) x supp(mu_2), so W is bounded above and below by positive constants.
For a class pair (C_1,C_2) use its own entry index c from Lemma 10.1. At endpoint j take the extra parallel descriptor [0,c] together with every descriptor in M_j(c,N) and define
b_{j;C_1,C_2}(z,w) = w^{(j)}_{0,[0,c]}(z,w) prod_{t in M_j(c,N)} w^{(j)}_{0,t}(z,w).
The masks are those of the buffer-mask lemma; every factor lies in [0,1] and equals one when its level-zero condition holds. Antipodal symmetry permits both endpoints to be evaluated at e(x,y). Define a Borel function Theta_N on the fixed separated product by
Theta_N(x,y) = b_{1;C_1,C_2}(x,e(x,y)) b_{2;C_1,C_2}(y,e(x,y)) if (x,y) in C_1 x C_2
and zero when either endpoint lies outside the retained classes, so 0<=Theta_N<=1.
Let Gamma_N be the decreasing pair filters of the starting-pair package. With D_# the pushforward under the scalar map D, define
alpha_N = D_#(W 1_{Gamma_N} (mu_1 x mu_2)), tau_N = D_#(W 1_{Gamma_N} Theta_N (mu_1 x mu_2)).
These use the original class masses; only tau_N depends on the shell's class decomposition and angular masks.
Lemma 10.2 (Restoration of mass). There is eta>0, depending on fixed parameters and starting measures, such that
0<=tau_N<=alpha_N, ||alpha_N-tau_N||_{TV} <= C 2^{-eta N}.
Also alpha_N decreases to the nonzero finite positive measure
alpha_inf = D_#(W 1_{Gamma_inf} (mu_1 x mu_2)), Gamma_inf = intersection_N Gamma_N,
and all these measures are carried by the compact set Delta(E).
Proof. Apply cap preparation from the starting-pair package with loss R^sigma. There is G_N contained in Gamma_N with (mu_1 x mu_2)(Gamma_N \ G_N) <= C R^{-c_*} for some c_*>0. For mu_1-almost every first endpoint x the angular subprobability
xi_{N,x} = e(x,.)_#(1_{G_N}(x,.) mu_2)
has the required S-cap bound with loss R^sigma down to scale R^{-1}. Changing pi_x(y) to e(x,y)=-pi_x(y) preserves cap bounds. The analogous statement 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 t used at the first endpoint. For mu_1-almost every x in C_1 the anchor cell determined by t and x is occupied and xi_{N,x} has the cap bound above. Let F_{C_1,C_2,t,x} be the directions failing this level-zero test. The exceptional-directions lemma applies directly to xi_{N,x} and gives
mu_2{ y in C_2 : (x,y) in G_N, e(x,y) in F_{C_1,C_2,t,x} } <= R^{-sigma}.
Restricting y to C_2 only decreases the left side; no normalization by mu_2(C_2) occurs. Integrating over x costs at most a_{1,C_1} R^{-sigma}. Sum first over C_1 (their masses sum to at most one), then over C_2 and descriptors (whose count is R^{o(1)} by class regularization and the list-size bound). The first-endpoint failure mass is R^{-sigma+o(1)}; the second-endpoint failure mass is the same. There is no sum over anchor cells: a point and a descriptor already specify one.
The product in the definition of b equals one when every level-zero test holds, so 1-Theta_N is at most the indicator of the union of failures within retained classes. Including the original mass outside the retained classes and Gamma_N \ G_N,
int_{Gamma_N}(1-Theta_N) d(mu_1 x mu_2) <= C R^{-c_*} + C R^{-sigma+o(1)}.
The bounded weight W gives the total-variation estimate with any fixed 0<eta<min(c_*,sigma)/2 and large N. G_N was used only in this estimate; it is not a multiplier. The filters Gamma_N decrease with positive limiting mass, so continuity gives the limit measure and monotonicity of alpha_N. The positive lower bound on W makes alpha_inf(R)>0. Finally, the two supports are compact subsets of E and D is continuous, so all pushforwards are carried by their compact distance image, contained in Delta(E).
Proposition 10.3 (Decay of the approximating distance measures). The fixed parameters can be chosen in the order beta, kappa, sigma, omega so that some gamma>0 satisfies
int_{2^{N-1}<=|r|<=2^{N+1}} |hat(tau_N)(r)|^2 dr <= C 2^{-gamma N}
for all sufficiently large N.
Proof. Fix one normalized class pair and one product X_l x Y_l in the spatial partition of Gamma_N. The component measure is the pushforward of
W(x,y) 1_{X_l}(x) 1_{Y_l}(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. At depth zero take X=Y=Q_0 with their own normalized class laws; 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 Proposition 6.4 applies. The cell-center separation in Definition 8.1 is the internal convention; at the outer root the supported laws are separated. Root spatial diameters in the graph argument are O_d(1) while actual cross distances have a positive lower bound.
The largest nominal length of an endpoint descriptor is at most L<=max(c,(N-c)/2)<=N/2. So the root shell conversion (Proposition 6.4) applies with a=0, v=N. Its weight is precisely W; no internal distance-weight normalization occurs at the root. For every fixed T>0 it bounds the squared Fourier norm on the positive shell by R^{o(1)} sup H_F(0,N)+O(R^{-T}). The supremum retains root cells, class normalizations, lists and smoothing index zero; its symbols have the fixed derivative budget and common bounds that Proposition 6.4 supplies for T.
Apply Corollary 7.4 at p=c to each such energy. 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<=N/2<N-c, and N-c>=10 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)}.
The graph estimate produces a general two-profile energy 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 in each endpoint cell. It, and any normalized differentiated versions produced by shell conversion, can be removed at bounded cost from the positive independent angular energies. The remaining lists are exactly those with base [c,N] and reached anchors omitted; removal is made in the general positive energy 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
2^{d(N-c)} int E_1(r) E_2(r) d lambda_N(r) <=
(2^{d(N-c)} int E_1(r)^2 d lambda_N(r))^{1/2}
(2^{d(N-c)} int E_2(r)^2 d lambda_N(r))^{1/2}.
Each parenthesis is the independent energy formed by duplicating one endpoint law, cell and symbol. With the unreached descriptors of M_j(c,N) it is the single-profile state H_0^{[c,N]}(c,N), denoted H_j(c,N). Its symbols have the fixed budget and common bounds from shell conversion; restriction to c-cells and duplication preserve those bounds. No separation is imposed, so Theorem 9.4 applies to each, including two identical cells:
H_j(c,N) <= R^{2A_{j,c} - V_j([c,N]) + O_d(kappa+sigma/kappa)+o(1)}.
Combining the graph factor and the geometric mean of these bounds:
-A_{1,c}-A_{2,c}+min_j k_j(0,c)
+ (1/2) sum_{j=1}^2 (2A_{j,c}-V_j([c,N]))
= min_j k_j(0,c) - (1/2)(V_1([c,N])+V_2([c,N]))
<= -beta + C_d sigma + 2d/N,
using the cost and potential bounds of Lemma 10.1. Including the stated losses proves uniformly for each component
int_{2^{N-1}<=r<=2^{N+1}} |hat(nu)(r)|^2 dr <= C R^{-beta+C_d(kappa+sigma/kappa)+o(1)} + C R^{-T}.
Choose the parameters in order: beta as in Lemma 10.1; kappa small enough that [c,N] is nonterminal and C_d kappa<beta/16; sigma small enough that all profile and transition hypotheses hold and C_d sigma/kappa<beta/16; omega>0 small enough that 4d omega<beta/8. Prepare the filters Gamma_N with this omega and cap loss R^sigma; their deletion exponent c_* may depend on both, and only c_*>0 was needed.
Fix the error accuracy after the parameters and before N->infinity. Theorem 9.4 bounds the number of steps on a branch by a constant depending on kappa and sigma. Choose a fixed local negative-power accuracy large enough to dominate all ancestor factors and errors, leaving the root remainder R^{-T} with T>10+4d omega. Inverse expansion orders are fixed from this and determine a finite symbol derivative budget. Direct expansion and integration-by-parts orders are chosen for that budget and accuracy, as in Theorem 9.4. The graph step and Cauchy-Schwarz introduce no persistent new derivative orders; the spatial indicators have zero angular derivatives. These choices change only constants and the threshold for N. Now take N large.
There are at most L_N<=C R^{2d omega} spatial products in Gamma_N. Let nu_{C_1,C_2,l} denote the preceding component. The exact recombination is
tau_N = sum_{C_1,C_2} a_{1,C_1} a_{2,C_2} sum_{l=1}^{L_N} nu_{C_1,C_2,l}.
The Fourier L^2 triangle inequality costs no class-count factor because sum_{C_1,C_2} a_{1,C_1} a_{2,C_2}<=1; it costs at most L_N for the spatial products. Squaring therefore costs R^{4d omega}, giving
int_{2^{N-1}<=r<=2^{N+1}} |hat(tau_N)(r)|^2 dr <= C R^{-beta+C_d(kappa+sigma/kappa)+4d omega+o(1)} + C R^{-T+4d omega}.
The successive choices leave a strictly negative exponent; increasing the starting N if necessary, the right side is O(R^{-beta/2}). Each tau_N is a real positive measure, so hat(tau_N)(-r)=conj(hat(tau_N)(r)). The same bound holds on the negative shell, proving the proposition.
Lemma 10.4 (Monotone measure shell criterion). Let (alpha_N)_{N>=K} be finite positive Borel measures on the line with alpha_{N+1}<=alpha_N, and let alpha_inf=lim_N alpha_N. Let tau_N be finite complex Borel measures with
sum_{N>=K} ||alpha_N-tau_N||_{TV} < infinity,
sum_{N>=K} ( int_{2^{N-1}<=|r|<=2^{N+1}} |hat(tau_N)(r)|^2 dr )^{1/2} < infinity.
Then alpha_inf is absolutely continuous with respect to Lebesgue measure. The conclusion in particular holds when the two inner quantities are respectively O(2^{-eta N}) and O(2^{-gamma N/2}) for some eta,gamma>0.
Proof. Choose theta in C_c^infty(R) equal to one on [-1,1] and supported in [-2,2]. Let Q_N be convolution with the inverse Fourier transform of theta(2^{-N} r), and set P_N=Q_N-Q_{N-1}. The multiplier of P_N is supported in {2^{N-1}<=|r|<=2^{N+1}}. Dilation of the fixed Schwartz kernel gives C_theta such that for every finite measure nu,
||Q_N nu||_{L^1} <= C_theta ||nu||_{TV}, ||sum_{N=a}^b P_N nu||_{L^1} <= 2 C_theta ||nu||_{TV} (a<=b).
The second bound follows from telescoping to Q_b-Q_{a-1}; its constant is independent of the number of bands.
With hat(nu)(r)=int e^{-i r t} d nu(t), Plancherel and uniformly bounded band multipliers give
||P_N tau_N||_{L^2} <= C ( int_{2^{N-1}<=|r|<=2^{N+1}} |hat(tau_N)(r)|^2 dr )^{1/2}.
Hence
G := sum_{N>=K} P_N tau_N converges in L^2, H := sum_{N>=K} P_N(alpha_N-tau_N) converges in L^1,
both absolutely in the indicated Banach space.
Set zeta_n=alpha_n-alpha_{n+1}>=0. Monotonicity and finiteness give
sum_{n>=K} ||zeta_n||_{TV} = alpha_K(R)-alpha_inf(R) < infinity,
alpha_N-alpha_inf = sum_{n>=N} zeta_n in total variation.
For finite M>=K define Z_M=sum_{N=K}^M P_N(alpha_N-alpha_inf). Substituting the telescoping identity into this finite sum and using the bounded measure-to-L^1 operators gives
Z_M = sum_{n>=K} (Q_{min(n,M)} - Q_{K-1}) zeta_n.
For each fixed M the right series converges in L^1. The low-cutoff bound shows that
Z := sum_{n>=K} (Q_n - Q_{K-1}) zeta_n
converges absolutely in L^1. Subtracting gives
||Z_M-Z||_{L^1} <= 2 C_theta sum_{n>M} ||zeta_n||_{TV} = 2 C_theta ||alpha_{M+1}-alpha_inf||_{TV} -> 0.
Monotonicity enters at this point. The original double series was not rearranged without a convergence argument.
Finite telescoping gives
Q_M alpha_inf = Q_{K-1} alpha_inf + sum_{N=K}^M P_N tau_N + sum_{N=K}^M P_N(alpha_N-tau_N) - Z_M.
The fixed low-frequency term Q_{K-1} alpha_inf is a smooth L^1 function (convolution with a Schwartz kernel). The right side converges in L^1+L^2, hence in distributions, to f=Q_{K-1} alpha_inf+G+H-Z in L^1_loc(R). The kernels defining Q_M form an approximate identity, so the left side converges in distributions to alpha_inf. Thus alpha_inf=f L^1. Since alpha_inf is a finite positive measure, the density is nonnegative almost everywhere and belongs to L^1(R), which is absolute continuity.
Proposition 10.5 (Completion). For the starting pair from the starting-pair package, there is a nonzero finite positive measure, absolutely continuous with respect to L^1, carried by the original distance set Delta(E). Consequently L^1(Delta(E))>0.
Proof. Choose parameters as in Proposition 10.3. Lemma 10.2 gives decreasing alpha_N with nonzero limit alpha_inf, and approximants tau_N with summable total-variation error. Proposition 10.3 gives the second summability hypothesis of Lemma 10.4. Lemma 10.4 makes alpha_inf absolutely continuous. Lemma 10.2 carries it by Delta(E) with positive mass, so Delta(E) is not Lebesgue null.
Proof of the main theorem. Start with a Frostman probability on the given compact set E subset R^d, d>=2, with exponent strictly larger than d/2. In the odd-dimensional exceptional affine-plane case, the classical distance-threshold lemma already proves the conclusion using original distances inside that plane. Everywhere else the starting-pair package supplies separated original probabilities and filters, and Proposition 10.5 proves the conclusion. This includes d=2: its cap preparation is the integer case of the cap-lifting lemma. Auxiliary projections used in preparation control directions only; the measures in the definitions of alpha_N and their nonzero limit are pushforwards by |x-y| in R^d. So the argument establishes positive measure for the original distance set in every dimension d>=2.