# kakeya_maximal_3d_v2.txt: an accessible, complete proof of the three-dimensional Kakeya maximal conjecture.
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THE THREE-DIMENSIONAL KAKEYA MAXIMAL CONJECTURE: A READABLE PROOF
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ABSTRACT.
A Kakeya tube is a long, thin cylinder of length 1 and radius delta > 0. For
each direction omega on the unit sphere S^2, look at all Kakeya tubes pointing
in that direction, take the average of a function f over such a tube, and then
take the largest such average over all tube positions. This is the Kakeya
maximal operator K_delta. The Kakeya maximal conjecture asserts that K_delta
maps L^3(R^3) to L^3(S^2) with a norm that grows only like delta to a tiny
negative power.
We prove this conjecture in three dimensions. Specifically, for every
epsilon > 0 there is a constant C_eps (depending only on epsilon) such that,
for every 0 < delta < 1 and every f in L^3(R^3),
|| K_delta f ||_{L^3(S^2)} <= C_eps * delta^{-epsilon} * || f ||_{L^3(R^3)}.
This note is written to be readable by a strong high-school student who is
comfortable with limits, exponentials, and the idea of an integral. Every step
of the proof is explained. Longer technical arguments (the deep inputs we
borrow from other papers) are named as theorems and used as black boxes; all
the connective work is done here in the open.
The document is organised into twelve parts:
0. Overview: what we are proving, and the main idea.
1. The Kakeya maximal operator and its statement.
2. The indexed model: paths, shadings, weights, planks.
3. The critical inequality and its parameter deformation.
4. Geometric inputs: the four big theorems we borrow.
5. Local bounds, positive temporal deficit, and the first plates.
6. The stationary equality and the canonical profile.
7. Angular nonconcentration and the conditional X estimate.
8. Entropy increments in the four moving-frame coordinates.
9. Projection constraints from two frames.
10. Closing the contradiction.
11. From the critical inequality to the maximal theorem.
12. Nikodym, curved Kakeya, and consequences.
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0. OVERVIEW: WHAT WE ARE PROVING, AND THE MAIN IDEA
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0.1 The object of study.
Let delta > 0 be a small number. We will let delta tend to zero throughout;
delta measures how thin our tubes are. A Kakeya tube in R^3 is, for a point
a in R^3 and a unit vector omega in S^2,
T_delta(a, omega)
= { a + t * omega + u : |t| <= 1/2, u . omega = 0, |u| <= delta }.
Read this aloud: "the tube is a cylinder of length 1 centred at the point a,
pointing in the direction omega, with radius delta". Its volume is
|T_delta(a, omega)| = pi * delta^2
regardless of a or omega.
For a measurable function f : R^3 -> C we define the Kakeya maximal operator
K_delta f (omega) = sup_{a in R^3} (1 / (pi * delta^2))
integral over T_delta(a, omega) of |f(x)| dx.
The sup is over all positions a of a tube in direction omega. This number is
"the biggest possible average that the function f can take over a Kakeya tube
pointing in direction omega". The function K_delta f is defined on the sphere
S^2 and is nonnegative.
Moral picture. Think of f as the density of some dust in space. K_delta f
asks: in the direction omega, how much dust can you cram into a very thin tube
of length 1? The Kakeya maximal conjecture is a precise statement that even
for delta tiny (very thin tubes), K_delta does not blow up by more than an
arbitrarily small power of 1/delta. This is a strong statement about the
geometry of tubes: many tubes of small radius cannot be positioned so as to
overwhelm the average of a function.
0.2 The conjecture.
The Kakeya maximal conjecture in dimension 3 states: for every epsilon > 0
there is a finite constant C_eps such that for every 0 < delta < 1 and every
f in L^3(R^3),
|| K_delta f ||_{L^3(S^2)} <= C_eps * delta^{-epsilon} * || f ||_{L^3(R^3)}.
That is our main theorem.
0.3 Why this is hard, and the shape of the argument.
The naive bound for K_delta is delta^{-1/3} (obtained by Cauchy-Schwarz and
volume considerations), and the conjectured truth is delta^{-epsilon}. Closing
this gap requires that we "see" how tubes can or cannot overlap.
Our argument is a proof by contradiction. We suppose the conjecture is false.
By scaling and a careful discretisation, a failure of the conjecture translates
into the existence, for infinitely many resolutions N = 1/delta, of finite
configurations of "lines" (thin tubes) that achieve a sharp multiplicity
inequality. These extremal configurations then carry strong geometric
structure.
The argument unfolds in stages:
(a) MODEL (Part 2). Replace the continuous problem by a discrete "indexed
model": finitely many affine lines in R^2 x [0,1] (paths), positive weights,
and each line marked with a set of times ("shading"). Multiplicity of
marked cells plays the role of tube averages.
(b) CRITICAL INEQUALITY (Part 3). Introduce a family of candidate
inequalities with an adjustable parameter ("temporal cost") and a
"clustering" term. Define the critical exponent h(p,z) as the smallest
clustering power for which the inequality holds. The whole problem
reduces to showing h(p,0) = 0 for p just above 2. If h > 0, we will
derive a contradiction.
(c) STRUCTURE OF EQUALITIES (Parts 5-6). If h > 0, there exist
extremal ("equality") configurations. Geometric estimates force the
extremal configuration to have:
- full temporal branching on an initial range of scales (a "flat delay");
- a canonical "time profile" F(kappa) = beta * (kappa - tau)_+;
- packets of near-planar "plates" with prescribed shapes.
This gives a "stationary equality".
(d) TWO FRAMES AND ENTROPY (Parts 7-9). Fix a stationary equality. At
each event we observe four matrix coordinates (X, Y, U, V) in a moving
frame. The masses of packets force lower bounds on the conditional
entropies ("entropy demand"). On the other hand, same-index pairs of
events give two frames for the same matrix. Comparing them through a
planar projection theorem yields upper bounds on those entropies
("projection constraints").
(e) CONTRADICTION (Part 10). The entropy demand and the projection
constraints are incompatible. This forces h(p,0) = 0.
(f) PASSAGE TO THE THEOREM (Part 11). A single discrete estimate with
h(p,0) = 0 (for p close to 2) implies the L^3 -> L^3 maximal estimate
with loss delta^{-epsilon}. First for indicator functions, then for
cell averages, then for all f in L^3.
(g) CONSEQUENCES (Part 12). The Kakeya maximal estimate implies, via
known transfers, Nikodym maximal estimates in R^3 and on manifolds of
constant curvature, and local curved Kakeya estimates for a class of
nondegenerate phases.
0.4 What you can skip and what you cannot.
If you want to see the skeleton of the argument quickly, read:
Section 1.1 (the statement);
Section 2.4 (the indexed model inequality);
Section 3.4 (main reduction proposition);
Section 5.1 (local bound intuition);
Section 6.3 (canonical profile statement);
Sections 8.3 and 9.1 (entropy demand and projection constraints,
as statements);
Section 10.1 (contradiction);
Section 11.5 (conclusion).
Everything in between is the machinery that turns these statements into
a complete proof.
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1. FROM THE CONTINUOUS TO THE DISCRETE: THE INDEXED MODEL
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1.1 A tube becomes a line segment with marked times.
Take one Kakeya tube T_delta(a, omega). If omega is close to the vertical
axis e_3, we can parametrize its central axis as (t, M(t)) with
M(t) = b + t * u , 0 <= t <= 1,
where b in R^2 is the starting point ("intercept") and u in R^2 plays the
role of the direction ("slope"). The tube is the set of points (t, x) with
0 <= t <= 1 and |x - M(t)| <= delta.
For the proof we will build a discrete model at resolution N = 1/delta.
The interval [0,1] is cut into N equal "time bins" of length 1/N. R^2 is
cut into cubes (squares) of side 1/N. A tube becomes a "line" M_i, and
each time bin at which we mark an event becomes a small piece of that line.
We will only care about which bins are marked, not exactly where on the
line.
The passage from the continuous maximal inequality to this model is
standard but tedious: finitely many direction charts (each placing omega
close to the vertical after a bounded rotation of coordinates), then a
comparison of the tube average with the multiplicity of marked cells
inside that tube. The point is that both sides of the L^3 inequality
(see Section 11) are controlled by a multiplicity bound on this discrete
model, up to constants and arbitrarily small powers of N. We state and
use that comparison at the very end of the paper (Part 11); everything
between here and there is a purely discrete, finite-combinatorial
argument. Therefore for the rest of the paper the reader can forget
about measure theory and about f: we are simply studying finite families
of "paths with marked times".
1.2 A remark on the role of delta.
Throughout the rest of the paper, N = 1/delta is an integer power of 2
that tends to infinity. The "loss" delta^{-epsilon} in the theorem is
exactly N^{epsilon}, and whenever we say "subpower" we mean
N^{o(1)} = exp(o(log N)), and whenever we say "at most C_eps N^eps" we
mean exp(eps * log N + O(1)). This abuse of language is standard: it
makes combinatorial estimates look clean.
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2. THE INDEXED MODEL: PATHS, SHADINGS, WEIGHTS, PLANKS
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2.1 Indices, matrices, markings.
Fix once and for all a bounded set U of matrix parameters. An index i is
a triple
i = (w_i , M_i , S_i ),
where:
* w_i > 0 is a positive real number, the "weight" of i. We allow
different indices to have the same w_i, the same M_i, or both.
* M_i = (b_i, u_i) is a 2-by-2 matrix of real numbers in U. Its
associated affine path is M_i(t) = b_i + t * u_i for 0 <= t <= 1;
here b_i, u_i are row vectors so M_i(t) in R^2.
* S_i is a nonempty subset of the N dyadic time bins {1, 2, ..., N}.
Think of S_i as "the times at which this path is marked".
The set S_i is what we call the "shading" of i.
A single index i gives a (possibly many-times-weighted) path M_i plus the
particular times at which it is "lit up". Two indices can have identical
matrices but different shadings: e.g. one lit up on the left half, one on
the right half. They are distinct indices.
An "event" is a pair (i, J) where J is a time bin in S_i. The location of
the event is (t_J, M_i(t_J)), where t_J is the centre of the bin J. This
is a point in R^2 x [0,1] at grid coordinates (t_J, M_i(t_J)).
Let E be the union of all space-time cubes of side 1/N containing the
locations of all events (one cube per event). Write |E| for the number of
these cubes.
2.2 Total mass, incidence, and multiplicity.
Define
n = sum over i of w_i (total weight)
I = sum over i of w_i * |S_i| (total weighted event count)
k = I / n (average time per index)
m = I / |E| (average multiplicity per cell).
Here |S_i| is the number of bins in S_i. The number n is "how many indices
we have, counted with weight". The number k is "on average how many times
an index is lit up". The number m is "on average how many events pile onto
a single cell": it is the discrete analogue of "the average of f over a
tube" in the continuous problem, but averaged over cells.
A single index contributes k events on average. A single cell supports m
event multiplicities on average. The product n*k = I = m*|E| is just
double-counting in two ways.
2.3 Planks.
A full-time plank of widths a, b (with 1 <= a <= b <= N) is a set of the
form
{ (t, x) : 0 <= t <= 1,
|(x - b_0 - t*u_0) . e_1| <= a/N,
|(x - b_0 - t*u_0) . e_2| <= b/N }
where (e_1, e_2) is an orthonormal frame in R^2 and (b_0, u_0) is a
centering matrix. Geometrically, it is a rectangle of long dimension b
(in units of 1/N) that is swept out in the time direction, with cross-
sections of side lengths (a/N) and (b/N). Widths are measured in units
of the fine grid 1/N.
An index i "belongs to" the plank P if its entire path M_i(t) for t in
[0,1] lies in P. Note this is stronger than "some event of i lies in P":
plank membership is a property of the whole path, not of individual
marked times. This matters a lot later.
The mass of a plank is n_P = sum over indices i with M_i in P of w_i.
We write
Delta_A = sup_P n_P / ( a^{d-1-A} * b^{1+A} ),
where P ranges over planks of any widths 1 <= a <= b <= N, and where
d = 2 - q for a parameter q in [-1, 0] we will fix later. Delta_A is the
"plank parameter": it measures how many indices the heaviest normalized
plank can contain.
2.4 The temporal profile.
Here is the key object that carries "how the marked times are distributed
across time scales".
For each index i, S_i is a subset of {1,...,N}. When 0 <= t <= 1 we look
at all dyadic sub-blocks of {1,...,N} of length N^t, and ask: how many of
them contain how many of the marked bins of S_i? Averaged over i, weighted
by w_i, this is what a "temporal profile" captures.
A first example: full branching. If S_i = {1,...,N} (every bin marked),
then any block of size N^t contains N^t marked bins. A second example:
S_i = a single arithmetic progression of length N^{1-alpha}. Then small
blocks contain few bins but large blocks contain many.
We encode this through a function F: [0,1] -> [0,1]. Read F as
"the deficit": F(t) measures how many orders of magnitude are missing
from the total mass of occupied bins in a typical block of size N^t.
The total deficit at full scale is F(1) = alpha; a fully marked index has
F = 0, a single marked bin has F(1) = 1 - 0 = 1 (roughly).
Precisely: F is an admissible profile if it is nondecreasing, 1-Lipschitz,
and satisfies F(0) = 0. It is 1-Lipschitz because adding one order of
magnitude to the block scale can reveal at most one more order of
magnitude of deficit (blocks can only contain more marked bins as they
grow).
Definition 2.4.1 (uniform profile). A sequence of indexed configurations
has uniform profile F if, uniformly over every retained index i and every
occupied dyadic block I of length N^kappa containing K = N^kappa fine
bins,
|S_i intersect I| = N^{ kappa - F(kappa) + o(1) }.
The o(1) is uniform in i, I and kappa. In particular, at kappa = 1 we
have |S_i| = N^{1 - alpha + o(1)} with alpha = F(1), so k = N^{1-alpha}.
Definition 2.4.2 (nontrivial equality). An "equality" (see Part 3) is
nontrivial if the total deficit F(1) > 0.
The uniform profile condition says: every single index looks the same at
every time scale, up to subpower factors. This uniformity is not present
in the original configuration, but Part 3 shows that we can always pass
to a subsequence that has it, losing only a subpower fraction of the
event mass.
2.5 Refinements.
A "refinement" of a configuration deletes some events; indices whose S_i
becomes empty are deleted; retained indices keep their original weights.
A refinement is "extensive" if it retains at least N^{-o(1)} of the
original event mass. Extensive refinements are the only ones we ever take
in the main argument.
A "restriction of loss exponent at most e" retains at least N^{-e+o(1)} of
the original event mass. We will use small e > 0 freely, then let it go
to 0 at the end.
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3. THE CRITICAL INEQUALITY AND ITS PARAMETER DEFORMATION
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3.1 The target inequality, heuristic form.
The most direct form of target bound the combinatorial picture suggests is
m <~ lambda^{-p} * sup_P n_P/(a*b) (*)
where lambda = k/N is the density (fraction of times marked, averaged over
indices), and P ranges over full-time planks with sides (a,b). We allow
p > 2 arbitrarily close to 2 (at p = 2 we get the cubic density bound
that matches the L^3 maximal theorem; see Part 11).
The right side of (*) has two pieces. The factor lambda^{-p} says: if you
mark fewer times, m must go up; the super-linear power of 1/lambda
expresses the fact that sparsely-marked configurations are the worst for
multiplicity. The factor sup_P n_P/(a*b) is the "clustering" of complete
paths inside planks; if all paths are crammed into a thin plank, we must
pay for it.
The main difficulty is that (*) cannot be true for all configurations with
the same p: we need to add a "temporal cost" that penalizes profiles in
which marked times are packed near the beginning of the interval, because
those allow configurations that mimic low-dimensional behaviour.
3.2 The temporal cost.
Fix a smoothing parameter 0 < eta < 1/2. For p > 2, z in [-1, 0] and
d = 2 + z, define the "discount" function
D(q, e) = (1/eta) * integral_{r = 1/2 - eta}^{1/2} min(q, r*e) dr,
where q = -z is in [0, 1]. Then define
P_{p,q}(e) = p * e - D(q, e), for 0 <= e <= 1,
and, for an admissible profile F, the temporal cost
Pi_F(t) = integral_0^t P_{p,q}( F'(u) ) du.
Why this integrand? The value P_{p,q}(e) is the "price" the inequality
pays per unit log-scale for a profile slope e (i.e. for a deficit that
increases at rate e across the log-scale axis). Because we subtract the
discount D(q,e), a profile whose deficit is steeply distributed gets a
smaller price: the smoothing makes the discount a nice averaging of
min(q, r*e), which is monotone in both arguments and Lipschitz with
constant at most 1. Consequently
P_{p,q} is increasing, convex, P_{p,q}(0) = 0,
(p - 1/2)*e <= P_{p,q}(e) <= p * e,
Pi_F(t) in [ (p - 1/2) F(t) , p*F(t) ],
Pi_F(1) >= p * F(1) - q.
When q = 0 (so z = 0 and d = 2), the discount D(0,e) = 0 and
Pi_F(t) = p * F(t) exactly. This is the case relevant to the target
theorem; the deformation to q > 0 is a tool for producing the critical
parameter.
3.3 The critical exponent.
Define the "plank parameter" with shape exponents:
Delta_A = sup_P n_P / (a^{d-1-A} * b^{1+A}).
A is a candidate for the "extra power of N" we need in front of Delta_A.
The critical exponent h(p,z) is defined as
h(p,z) = the least A in [0, d-1]
such that for every sequence with uniform profile F,
m <~ N^{A + Pi_F(1)} * Delta_A. (**)
Here "A + Pi_F(1)" is the log-exponent of the right side; the "equality"
symbols are exponent notation: everything holds up to a factor
N^{o(1)} = delta^{-o(1)}.
In the case q = 0 (z = 0), d = 2, and Pi_F(1) = p F(1) = p * alpha.
The critical inequality reads
m <~ N^{A + p alpha} * Delta_A.
If A = 0 and this holds for all p > 2 arbitrarily close to 2, then
m <~ N^{p alpha} * Delta_0 with Delta_0 = sup_P n_P/(a^{d-1} b). For unit-
weight families with c/N-separated slopes one checks Delta_0 <~ 1, and
since m = n k / |E| = n N lambda / |E|, we recover the "cubic density
bound":
|E| <~ n * N * lambda^{p+1}.
As p decreases to 2, this is the density bound required for the L^3
maximal estimate. Part 11 makes this passage completely explicit.
Our main theorem (Kakeya maximal) reduces to proving
h(p,0) = 0 for p > 2 arbitrarily close to 2. (***)
We prove (***) by contradiction in Parts 5-10.
3.4 Existence, monotonicity, and differentiability of h.
A first observation: h(p,z) is well-defined (there is always some valid
A), and the set of valid candidates A is a closed upper interval in
[0, d-1]. The reason is monotonicity in A: larger A means a weaker
inequality (larger right side), so if A works, A' > A also works.
Also, A = d-1 always works: an elementary Cauchy-Schwarz argument
(see Part 5) shows the inequality with A = d-1, and at z = -1 (so d = 1
and A can only be 0) the same argument gives h(p,-1) = 0.
Moreover the function g(p,z) = h(p,z) - z is separately monotone in
each variable: h is nonincreasing in p and satisfies h(p, z+s) <= h(p,z)+s.
Monotonicity plus the fact that g is locally of bounded variation means g
is totally differentiable almost everywhere. This will let us pick a
differentiability point in a moment.
3.5 The critical parameter point.
Lemma 3.5.1 (critical parameter point). Let I be an open interval of
p > 2 on which h(p,0) > 0. Then there is a point (p,z) in I x (-1,0)
at which h is totally differentiable, and
h(p,z) > 0, h_z(p,z) > 0,
h_p(p,z) <= 0 (in fact we will get a strict v = -h_p > 0 later).
Idea of proof. Consider the level set h(p,0) > 0. As z decreases from 0
to -1, h goes from positive to 0 (since h(p,-1) = 0 for all p). The
average slope is therefore strictly negative on [-1, 0], so somewhere
h_z > 0. A differentiability point of g where h is positive is found by
a standard "bad set has measure zero" argument. See the original detailed
proof for the bounded-variation and differentiability steps. From now on
we fix such a point.
3.6 Notation frozen at the critical point.
From here on:
h = h(p,z) (positive)
v = - h_p(p,z) (positive; proven in Part 6)
x = d - 1 - h (>= 0)
y = 1 + h (>= 0)
d = 2 - q (so also x + y = d, and y - x = q + 2h > 0)
q = - z (so 0 <= q < 1)
These will be used throughout the rest of the paper without re-explaining
them. The condition h_z > 0 will be needed to make the "stationarity"
argument work in Part 6.
3.7 Hybrid equalities, and why we need them.
Fix (p,z,eta) with h = h(p,z) > 0. Pick 0 <= h_0 < h. There is a "sharp"
number H with h_0 < H <= h such that for every sequence with uniform
profile F,
m <~ N^{H + Pi_F(1)} * Delta_{h_0} (hybrid)
and equality can be attained by some sequence with a uniform profile.
The sequence attaining equality is called a "hybrid equality". Its
"clustering exponent" is fixed at h_0 < h but its "temporal exponent" is
the sharp H, which sits between h_0 and h.
Why hybrid? Because if we demand h_0 = h, we get an "exact" critical
equality, which is too rigid: we'd have to prove v = -h_p > 0 first, but
to get v > 0 we need to compare slightly hybrid equalities with different
h_0. Once we know v > 0 (Part 6), we send h_0 to h and get an exact
critical equality with positive deficit F(1) > 0, and the argument from
Parts 7-10 applies to it directly.
The key stability statement: on any extensive refinement of a hybrid
equality, followed by regularization, the profile is unchanged and the
two quantities m and Delta_{h_0} retain their exponents. This lets us
keep refining without destroying what we have built.
3.8 Cleaning and profile regularization.
We will frequently need to select events with a small loss and then
re-regularize so the resulting configuration has a uniform profile. The
"finite cleaning" lemma makes this cheap: given a finite list of partitions
of the event set, and a subset A whose measure is at least rho of the total,
we can delete the "worst" parts of A so that every surviving cell of every
partition has measure at least theta_0 times its original measure, at a
cost of at most L * theta_0 times total measure where L is the number of
partitions.
A "regularization lemma" then runs finitely many tests at once by
pigeonholing rounded counts: at every event we label the counts of
occupied ancestors at a finite mesh of scales, round to intervals of
length nu, and retain the most common label. The surviving set has a
uniform profile F_N, and the limit F is our profile.
The two clean-ups (finite cleaning and regularization) are used all over
the paper. We will refer to "regularize after this restriction" as shorthand
for "apply the two clean-ups with all previously imposed tests included in
the partition list".
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4. GEOMETRIC INPUTS: THE BIG THEOREMS WE BORROW
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The argument uses four geometric inequalities. Three of them are deep
theorems in the literature (three- and two-dimensional Kakeya analysis);
we state them here and use them verbatim. The fourth (the plate Nikodym
estimate) is elementary and we prove it in Section 5.7.
Throughout this section, delta > 0 is small, D = 1/delta, and a
delta-tube is the delta-neighbourhood of a unit length line segment. A
shading of a tube T is a measurable subset Y(T) of T.
4.1 The Kakeya set estimate (three dimensions).
Theorem 4.1.1 (Kakeya set estimate; Guth-Wang-Zahl 2026). For every
epsilon > 0 there are eta_in > 0 and delta_0 > 0 such that the following
holds. Let 0 < delta < delta_0 and let T be a finite family of delta-tubes
inside a fixed bounded region. Put
Delta_max(T) = sup_{K convex} sum_{T in TT, T subset K} |T| / |K|,
lambda(T, Y) = ( sum_T |Y(T)| ) / ( sum_T |T| ).
If
Delta_max(T) <= delta^{-eta_in}, and lambda(T, Y) >= delta^{eta_in},
then
| union_{T in TT} Y(T) | >= delta^{epsilon} * sum_T |T| .
Read this aloud. The hypothesis says (a) tubes are not bunched into any
single convex body (convex clustering is bounded by delta^{-eta_in}), and
(b) the shadings are not too thin (they occupy at least delta^{eta_in}
fraction of each tube). The conclusion says the union of the shadings has
volume at least delta^epsilon times the total tube volume. That is a
Kakeya "set estimate" — it forces the shading union to have substantial
volume.
We use the theorem through a weighted form that allows repeated tubes
and different weights.
4.2 Weighted convex clustering (uses the set estimate).
Lemma 4.2.1 (weighted convex clustering). Fix R >= 1. For every
epsilon > 0 there are eta_in > 0 and C < infinity with the following
property. Let (T_i) be finitely many delta-tubes in B(0, R), with
repetitions and weights w_i > 0 and shadings Y_i subset T_i. Suppose
sum_{i: T_i subset K} w_i |T_i| <= A * |K| for every convex K,
|Y_i| >= delta^{eta_in} * |T_i| for every i.
Then
sum_i w_i |Y_i| <= C * A * delta^{-epsilon} * |union_i Y_i|.
Idea. Sample indices with probability proportional to weights; the
resulting set has a bounded convex clustering in a finite box test with
only a log-factor loss (using the exponential-moment method on Bernoulli
sums). Then apply Theorem 4.1.1 to the sampled set. Cancelling the
common delta^2 volume gives the desired weighted inequality.
4.3 Weighted set estimate in a line chart (Marked times version).
In our indexed model, an index's "shading" is a set of marked time bins,
not a continuum. The following converts "full-time plank control" and
"many marked times" into a bound for the union of shaded cells.
Lemma 4.3.1 (weighted set estimate in a line chart). Fix a bounded line
chart and a fixed bound for the permitted enlargement of mesh cells. For
every epsilon > 0 there are eta_in > 0 and C < infinity such that the
following holds. Let indices have weights w_i > 0 and suppose every
full-time plank P with widths a, b satisfies
sum_{i: M_i subset P} w_i <= A * a * b
for all 1 <= a <= b <= D. (plank)
Mark a set J_i of time bins on each index with #J_i >= D^{1 - eta_in}.
Let E be the union of the mesh cells visited at those bin centres (or
fixed enlargements). Then
sum_i w_i #J_i <= C * A * D^{epsilon} * #E.
Equivalently, with n = sum_i w_i and k the common number of marked bins,
the raw average multiplicity satisfies n k / #E <= C A D^{epsilon}.
Idea. Enclose each full trace in a tube T_i of radius C delta and fixed
length L. A full-time plank bound is used to control the convex
clustering of these tubes: an enclosing ellipsoid of any convex body
intersecting the T_i is contained in a plank whose widths reflect the
ellipsoid geometry, and the plank bound gives weighted clustering at most
CA. Then apply Lemma 4.2.1 to the marked bin centres (as small balls of
radius c_0 delta) inside each T_i, and compare union volume to #E delta^3.
4.4 The planar Furstenberg estimate.
Theorem 4.4.1 (quantitative planar Furstenberg; Ren-Wang 2025). Fix
0 < s <= 1, 0 < t <= 2 and a bounded line chart. For every epsilon > 0
there are eta_in > 0, c > 0 and delta_0 > 0 such that the following holds.
Let L be a finite delta-separated family of planar lines, with parameter
distribution that is (t, delta^{-eta_in})-Frostman down to delta. For
each line L in LL let Y(L) be a delta-separated subset of a fixed bounded
part of L, with k <= #Y(L) <= 2k, and with uniform distribution on Y(L)
that is (s, delta^{-eta_in})-Frostman down to delta. Then for
0 < delta < delta_0,
#{delta-mesh cells meeting union_L Y(L)}
>= c * k * delta^{-min(t, (s+t)/2, 1) + epsilon}.
Hear this aloud. The parameter space of lines is "t-Frostman": any ball
of radius r captures at most a fraction r^t of the lines. Each shading
is "s-Frostman": any interval of length r captures at most a fraction
r^s of the shade. Then the union of shadings must contain at least
k * delta^{-mu} cells where
mu = min { t, (s+t)/2, 1 }.
The first term t is the "parameter-dimension" bound (if you have many
lines, the union is large); the second term (s+t)/2 is a nontrivial
improvement coming from the "Furstenberg" estimate (this is the deep
input); the "1" reflects the ambient dimension.
The dual formulation (Wang-Wu 2024, Theorem 0.6) gives the same
exponent with an equal density parameter.
Corollary 4.4.2 (weighted planar families). The same estimate holds for
weighted, repeated line indices: push weights to the O(delta^{-2})
parameter cells, pick out a level of dyadic masses, restrict to a
positive fraction, then apply Theorem 4.4.1 to one representative per
cell. All constants are independent of the number of indices and of the
weights.
4.5 Multilinear Kakeya.
Theorem 4.5.1 (endpoint multilinear Kakeya; Guth 2010). For j = 1,2,3
let T_{j,a} be finitely many infinite cylinders of radius one with unit
axial directions v_{j,a}. Suppose
|det(v_{1,a_1}, v_{2,a_2}, v_{3,a_3})| >= theta > 0
for every triple. For nonnegative weights w_{j,a} put mu_j =
sum_a w_{j,a} 1_{T_{j,a}} and M_j = sum_a w_{j,a}. Then
integral_{R^3} ( mu_1 mu_2 mu_3 )^{1/2}
<= C * theta^{-1/2} * (M_1 M_2 M_3)^{1/2}.
The same holds for cylinders of finite length with no additional
dependence on length.
The factor theta^{-1/2} in front is the transversality loss: if the three
families are close to coplanar, the estimate degrades; the closer they
are, the worse.
Corollary 4.5.2 (unit-cube counts). Let Q range over unit mesh cubes and
m_j(Q) = sum_{a: T_{j,a} meets Q} w_{j,a}. Then
sum_Q ( m_1(Q) m_2(Q) m_3(Q) )^{1/2}
<= C * theta^{-1/2} * (M_1 M_2 M_3)^{1/2}.
If the directions lie in caps of radius at most c theta around three
centers of determinant at least theta, the same holds with another
absolute constant, for c small.
Since a cap partition of mesh r has O(r^{-2}) caps, summing over triples
of caps costs at most O(r^{-6}) which is a fixed power of theta^{-1} if
r is a fixed power of theta. In particular r = theta^2 is fine.
4.6 The four borrowings, summarised.
To recap:
(SET) Theorem 4.1.1: the three-dimensional Kakeya set estimate.
(CLUST) Lemma 4.2.1: weighted clustering for tube families.
(CHART) Lemma 4.3.1: weighted set-estimate version for our chart.
(FURST) Theorem 4.4.1: planar Furstenberg for two-dimensional
shaded-line configurations.
(MULTI) Theorem 4.5.1: three-transversal-family multilinear Kakeya.
These are the only big black boxes in the entire paper. Everything else
is derived, and the derivation is in Parts 5-10.
================================================================================
5. LOCAL BOUNDS, POSITIVE TEMPORAL DEFICIT, AND THE FIRST PLATES
================================================================================
We now work at the fixed critical parameter point of Part 3, with
h > 0, 0 <= h_0 < h <= 1, 0 < H <= h.
The results in this part apply to any hybrid equality with these
parameters, and to exact critical equalities when h_0 = H = h.
Goal of Part 5: show that every hybrid equality near h has positive total
deficit F(1) >= c > 0. There are four steps:
Step 1 (Sections 5.1-5.3): Local bounds. If there is an excessive packet
of indices in some plank over some temporal block, we can rescale the
block and get a model at a smaller resolution whose raw multiplicity
beats the sharp bound. So packets cannot exceed a threshold
T(kappa, a, b).
Step 2 (Sections 5.4-5.6): Filling in time. Suppose a short block (of
K = N^kappa bins, kappa small) has an index doing relatively few
things but with neighboring indices concentrated near a plane. Then
planar Furstenberg and the plate-Nikodym estimate let us fill in the
missing times, gaining in event count without paying much support.
This forces an initial "flat interval" [0, tau] where F = 0.
Step 3 (Sections 5.7-5.8): Saturated plates. On a flat scale, weighted
set estimate gives saturated packets (mass exactly at threshold
T(kappa, 1, K)).
Step 4 (Section 5.9): Anisotropic rescaling. If an equality with small
deficit exists, an anisotropic rescaling (squeezing time and long
spatial direction by the angular scale theta) produces a new
equality that beats the sharp bound. This rules out arbitrarily
small deficit.
Throughout this part, "length" and "width" a, b, K are measured in fine-
cell units (so a temporal block of K bins has duration K/N and transverse
width a has actual width a/N).
5.1 The local mass bound.
An index i is "active" on a temporal block I if I contains one of its
retained marked bins. A "packet" on I is a set of active indices assigned
to a plank containing their whole affine segments over I. Packets are
disjoint if the assigned index sets are disjoint on I; the containing
planks may overlap freely.
Proposition 5.1.1 (local mass bound). After an extensive regular
restriction, the mass of active original indices contained in any plank of
widths 1 <= a <= b <= K over a dyadic temporal block of K = N^kappa bins
satisfies
n_{I,W} <~ T(kappa, a, b) = m * N^{-Pi(kappa)} * K^{-H}
* a^{d-1-H} * b^{1+H}.
The bounds hold simultaneously at every fixed limiting (scale, shape),
under the finite-test convention of Part 3, and are inherited by further
restrictions.
Idea of proof. Fix a proposed "excess" N^epsilon > T(kappa, a, b). Greedily
assign and remove remaining active indices in planks whose mass exceeds
that threshold. If the removed events form an extensive piece, route
each packet through isotropic matrix parents of side b/K, rescale space
by (b/K) and time by a/(b/K), i.e. use resolution N' = N b / (K a). The
new configuration has raw multiplicity at least N^epsilon * T * (a/b),
but its plank parameter is at most Delta_{h_0} * a^d and its temporal
cost is Pi(1) - Pi(kappa). The sharp hybrid upper bound then gives the
opposite inequality, contradiction. Therefore the removed events are not
extensive: they are a subpower fraction, and the bound holds.
The converse direction holds too: if an extensive graph has disjoint
indexed-block assignments to planks of common widths (a, b) with masses
>= T(kappa, a, b), then setting chi = log_N a + log_N(K/b) (so chi < 1
means a positive-resolution rescaling exists), the same isotropic cut
gives an ensemble at resolution N' = N^{1 - chi} with matching upper and
lower bounds. This is the technical engine that produces new equalities
from old ones.
5.2 Multilinear slack and cap mass.
Lemma 5.2.1 (slack and cap mass). Every equality under consideration
satisfies
N^{H + (p-2) alpha} <~ k^2 m / n, (slack)
where alpha = F(1). Furthermore, at a fine incidence, the raw mass of
active lines whose directions lie in a cap of radius C/L (1 <= L <= N)
is at most
N^{o(1)} * m * L^{-H} * N^{-Pi(log_N L)}. (cap)
Proof idea of the slack. Use Delta_{h_0} <= N^d (the crude plank bound
from covering by planks of width N), k = N^{1-alpha}, m = Delta_{h_0}
N^{H+Pi(1)}, and Pi(1) >= p alpha - q with d = 2 - q, to compute:
k^2 m / n = Delta_{h_0} N^{2-2alpha + H + Pi(1)}
>= N^{q + H + Pi(1) - 2alpha}
>= N^{q + H + (palpha - q) - 2alpha}
= N^{H + (p-2) alpha}.
The gap (p-2) alpha > 0 (since p > 2 and alpha > 0) is the "slack" that
will let us discard bad planes.
The cap bound uses the local mass bound on a short block of size L with
plank widths (a,b) = (O(1), O(1)): a slope cap of radius O(1/L) fits in a
plank of bounded cross-section over a block of length L. The bounded slope
charts transfer angular diameter to spatial slope diameter.
5.3 A common plane on each short block.
Lemma 5.3.1 (common short-block plane). There is kappa_ml > 0, depending
only on the slack in 5.2 and on the multilinear input, with the following
property. For every fixed 0 < kappa < kappa_ml, set K = N^kappa. On an
extensive regular central incidence graph, each active reference
index-block can be assigned one direction plane such that:
(a) the same plane is used at all retained times of that block;
(b) at each retained time there is original neighbouring incidence mass
at least m N^{-o(1)}, in directions within O(K^{-1}) of that plane;
(c) after discarding a negligible part of the neighbouring mass, the
neighbouring direction is separated from the reference by at least
N^{-o(1)};
(d) all reference and neighbour segments lie in a common plate of
dimensions C x C K x C K (in fine units).
Idea of proof. Cover space-time by balls of radius C K. Use the local
mass bound 5.1.1 to bound active mass near a ball by m K^{C_0}. Divide
directions into caps of radius K^{-20} and call a cap "rich at the ball" if
its active mass is at least m K^{-100}. A ball is bad if three rich caps
have determinant at least K^{-10}. Applying MULTI 4.5.1 to the unshaded
tubes shows that bad balls are rare:
#{bad balls} <~_epsilon N^epsilon K^{C_1} (n/m)^{3/2}.
By the slack 5.2, this is a power-small fraction of total incidence, once
kappa_ml is chosen small enough that K^{C_2} uses less than a quarter of
the slack. After removing central incidences that require bad balls and
central incidences in "light" caps (mass <= C m K^{-60}), every remaining
ball has three rich directions with determinant at most K^{-10}, hence all
rich directions lie close to a single plane (choose a maximally separated
pair, then argue all others are close to their span). This gives (a)-(c).
Finally, all these segments are within O(K^{-1}) of the plane, so they lie
in a plate of dimensions C x C K x C K, giving (d).
5.4 Plate Nikodym bound (elementary).
Lemma 5.4.1 (plate Nikodym). Let N_K be the maximal averaging operator
over plates of dimensions 1 x K x K with arbitrary orientation and
translation, whose averaging plate contains the query point (fixed
changes of these dimensions are allowed). Then
|| N_K f ||_{L^2(R^3)}^2 <~ (1 + log K) * || f ||_{L^2(R^3)}^2.
Consequently, if every point of a measurable set E^+ belongs to a plate P
in which some set E has density at least rho > 0, then
|E^+| <~ rho^{-2} (1 + log K) |E|.
Idea of proof. Take a measurable plate choice x -> P_x and let A be the
corresponding averaging operator. The kernel of A A^* satisfies
K(x,y) <~ 1_{|x-y| <= C K} / ( K^2 (1 + dist(y, plane of P_x)) ) .
(In words, two plates can only overlap if their query points are close and
if y lies near the central plane of P_x; the factor 1/(1 + dist) bounds the
slice area of the intersection.) Integrating in the normal direction to
P_x gives sup_x integral_y K(x,y) <~ 1 + log K, and the same for the other
variable, so Schur's test yields ||A A^*||_{2->2} <~ 1 + log K. Passing to
the supremum over all plate parameters is by monotone convergence. The
density corollary follows by testing against f = 1_E.
The "log K" loss is essential: plates of size 1 x K x K with arbitrary
orientation genuinely overlap heavily (there are K rotations' worth of
plates through any point), and the log K measures this.
5.5 Planar-prefix extension (filling a short block).
Lemma 5.5.1 (planar-prefix extension). Let 0 < kappa < kappa_ml be fixed
as in Lemma 5.3.1, put K = N^kappa. Suppose there are constants s in
(0,1] and gamma >= 0 such that
F(kappa) - F(u) <= (1 - s)(kappa - u) for 0 <= u <= kappa, (A1)
F(u) >= gamma * u for 0 <= u <= kappa. (A2)
Then, defining
a_ang = H + p_* gamma, p_* = p - 1/2,
t = s + a_ang,
mu = min{ 1, (s + t)/2, t },
we have a_ang <= 1 and 0 < t <= 2. Every equality configuration under
consideration must satisfy
Pi(kappa) <= F(kappa) + 2 kappa (1 - mu). (*)
In particular, a strict reverse inequality yields a contradiction.
Idea of the proof. Project to the plane containing the reference segment,
which is the "common plane" of Lemma 5.3.1. The marked-time distribution
on the reference block is s-Frostman (this uses (A1)), and the
conditional-direction distribution of neighbours is a_ang-Frostman (this
uses (A2) plus the cap bound 5.2). Their product is t-Frostman on line
parameters. Then planar Furstenberg (Corollary 4.4.2) says the union of
neighbouring shadings fills at least k * K^{mu-1+1} = K^mu cells in the
plate, i.e. a density
rho >= N^{-F(kappa)+o(1)} * K^{mu - 1}.
Filling the reference block: we add all times to it. The new support size
grows by at most
|E^+| / |E| <= N^{2 F(kappa) + 2 kappa(1 - mu) + o(1)}. (support)
(The squared density loss in this formula uses Lemma 5.4.1 with rho as
above: rho^{-2} = N^{2F - 2 kappa (mu - 1)} = N^{2F + 2 kappa(1 - mu)}.)
Independently, event count grows by exactly N^{F(kappa) + o(1)}: each
retained block went from N^kappa N^{-F(kappa)} marked to fully marked
N^kappa times.
So m^+ >= m * N^{-F - 2kappa(1-mu) - o(1)} while the upper bound gives
m^+ <= m * N^{-Pi(kappa) + o(1)}. These are incompatible with a strict
reverse inequality in (*). This is the lemma.
5.6 Positive flat delay.
Lemma 5.6.1 (positive flat delay). Every hybrid or exact equality with
H > 0 has a profile that is identically zero on an interval [0, tau] with
tau > 0.
Idea. Suppose no flat interval exists. Let e = liminf_{u -> 0} F(u)/u. Then
F(u) > 0 for all u > 0. Take a small kappa > 0 minimising
F(u) - (e + eps) u on a short interval ending in a "record low" of F(u) -
(e + eps)u. Then the hypotheses (A1) and (A2) of Lemma 5.5.1 hold on this
prefix with gamma = max(e-eps, 0) and s = 1 - e - eps > 0. The lemma then
says Pi(kappa) <= F(kappa) + 2 kappa (1 - mu). We show that both cases
e = 0 and e > 0 lead to a strict reverse inequality, which is a
contradiction. For e = 0, the left side Pi(kappa) - F(kappa) >= (p_* - 1)
F(kappa) is positive while 2 kappa (1 - mu) vanishes for small kappa. For
e > 0, we compare (p_* - 1) e with (2 - p_*) e - H (from the formula for
1 - mu) and check the gap is strictly positive.
So every equality with H > 0 has a flat prefix.
5.7 Saturated plates on a flat scale.
Lemma 5.7.1 (saturated plates). Let kappa > 0 be fixed and suppose
F|_{[0, kappa]} = 0 (flat scale). Then, after an extensive regular
restriction, there is a disjoint indexed-block assignment to plates of
widths (1, K) with K = N^kappa, up to subpowers. Each active plate packet
has raw index mass at least m K N^{-o(1)}.
Idea. Split into temporal K-blocks, and into bounded-overlap position
charts of diameter C K fine cells that contain complete block segments.
Each original index-block uses only boundedly many charts. In each chart,
normalize resolution to K; the local shading density is N^{-o(1)} because
F(kappa) = 0 (marked times per index-block are K N^{-o(1)} out of K).
Greedily remove every remaining local line whenever a plank of width
density n_W/(ab) at least m N^{-epsilon} shows up. If an extensive
remainder survives, its clustering parameter is at most m N^{-epsilon} in
every local chart, so the weighted set estimate Lemma 4.3.1 (with theorem
loss below epsilon/(2 kappa)) gives multiplicity at most m N^{-epsilon/2}.
That contradicts extensivity. So the captured graph is extensive with
disjoint index ownership on each block.
Compare the captured lower bound N^{-epsilon} * m * ab with the local
threshold T(kappa, a, b) = m N^{-Pi(kappa)} K^{-H} a^{d-1-H} b^{1+H}.
Since Pi(kappa) = 0 on a flat scale, this gives
N^{-epsilon} <= N^{o(1)} K^{-H} a^{-q-H} b^H,
or in log coordinates
(q+H) log_N a + H log_N(K/b) <= epsilon + o(1).
Both coefficients are non-negative and H > 0, so a = N^{o(1)} and
b = K N^{o(1)} as the tolerance goes to zero. That is the statement.
5.8 Anisotropic normalization.
The next transformation is the key gain mechanism. It uses "aligned"
saturated plates whose thin spatial axes all point along a fixed axis.
We "squeeze" time and the long spatial direction by the angular scale
theta, preserving the thin spatial coordinate.
Lemma 5.8.1 (anisotropic normalization). Suppose that on an extensive
regular graph, over larger blocks of K_1 = N^{kappa_1} bins, there are
disjoint indexed-block assignments to saturated plates on a flat scale
K = N^kappa <= K_1 (widths 1, K up to subpowers, mass at least m K).
Suppose the thin spatial axes of the assigned plates along each original
index and larger block are within theta of one fixed axis for that index
and block, up to subpower factors, where
K^{-1} <= theta <= 1 (in exponent), lambda = log_N(1/theta) <= kappa.
Then there is an ensemble of bounded-chart configurations at resolution
N' = K_1 theta,
whose aggregate raw multiplicity is at least m / theta in exponent. Each
derived index retains its original weight and contributes to only
subpower many charts.
Further, if every original active line on a larger block obeys the local
clustering bound n_W <~ C a^{x'} b^{y'} for 1 <= a <= b <= K_1 with
0 <= x' <= y', then every derived chart has full-time clustering
parameter at most C theta^{-y'}. And if kappa_1 - lambda > 0, the derived
temporal profile is
G(v) = F( lambda + (kappa_1 - lambda) v ) / (kappa_1 - lambda),
0 <= v <= 1,
with cost Pi(kappa_1) in original N-units. Only the initial flat prefix
[0, lambda] is deleted.
Idea of proof. Coordinate map in a chart centred on one affine line:
s = r / K_1, X = z_n / (theta K_1), Y = z_t / K_1
(in normal-tangent coordinates z_n, z_t), giving resolution N' = K_1 theta.
Equivalently, in new fine cells the map is (r, z_n, z_t) ->
(theta r, z_n, theta z_t).
A plate with thin axis within theta/10 of the chart normal remains within
subpower fine thickness after the map, with tangential extent at most
N^{o(1)} K theta. Filling its selected K-intervals (now spanning
K theta new bins): it occupies at most N^{o(1)} (K theta)^2 cells, and if
it has retained mass at least m K, it contributes mM * K theta = m M / theta
incidences where M is its mass. Summing over disjoint packets gives lower
multiplicity m/theta.
The chart routing preserves: (i) inherited weights (each derived index is
one original index), (ii) time tree (subtree inclusion, so branching
exponents cannot increase, so profile is inherited on the prefix
[0, kappa_1]).
For the clustering bound: pull back a new full-time plank of widths (a,b)
by the inverse of (z_n, z_t) -> (z_n, theta z_t). Its cross-section is a
parallelogram inside a rotated rectangle of sides A <= B with
AB <~ ab/theta and B <~ b/theta. Since 0 <= x' <= y', the pulled-back
mass is
C A^{x'} B^{y'} = C (AB)^{x'} B^{y' - x'}
<~ C theta^{-y'} a^{x'} b^{y'}.
That is the C theta^{-y'} statement.
For the profile: the fine scale 1/theta = N^lambda lies in the flat prefix
(lambda <= kappa), so filling the K-blocks removes a flat range. Rebinning
to 1/theta deletes [0, lambda] and normalizes by (kappa_1 - lambda),
exactly G(v). The temporal cost in original N-units is Pi(kappa_1) -
Pi(lambda) = Pi(kappa_1).
5.9 An equality with positive total deficit.
Lemma 5.9.1 (positive-deficit equality). As h_0 goes up to h > 0, the
total deficits alpha = F(1) of hybrid equalities are bounded below by a
positive constant, uniformly once h_0 is close enough to h. Consequently
there is an exact equality at H = h_0 = h whose limiting profile satisfies
F(1) > 0.
Idea of proof. Fix small constants c_* > 0 and e_1 > 0 depending only on
h; and further require alpha < e_1, e_1 << c_* h, and e_1, alpha << H c_*.
We will derive a contradiction for hybrid equalities with alpha small.
(a) Fine-scale planarity. At a fine cell, use direction caps of radius
N^{-2 c_*}; call a cap rich if mass >= m N^{-10 c_*}. A cell is bad if
three rich caps have determinant >= N^{-c_*}. Multilinear Kakeya (same
count as in 5.3) shows bad cells are rare (fraction <=
N^{epsilon + C_3 c_* - h/4 + o(1)} < N^{-c} for small c_*, then small
epsilon). So on the surviving graph, all triples of rich caps are
essentially coplanar at scale N^{-c_*}.
(b) Global slab capture. Greedily remove all original full-time indices in
any "global slab" S (a plank over the whole interval) whose density
n_S/(ab) >= m N^{-e_1}. If an extensive remainder survived, weighted set
estimate (CHART 4.3.1 with shading density N^{-alpha}, target loss e_1/2)
would bound its multiplicity at most m N^{-e_1/2}, contradicting
extensivity. So the assigned graph is extensive. Each index's slab
axis is now fixed for its whole life.
Compare with the Delta_{h_0} bound to compute shape: (q + h_0) A +
h_0(1 - B) + (H - h_0) + Pi(1) <= e_1 + o(1) with A = log_N a,
B = log_N b. Hence the slab widths are nearly maximally elongated:
a <= N^{O(e_1/h_0)}, b >= N^{1 - O(e_1/h_0)}.
(c) Lower bound on the slab query mass. In each slab, average multiplicity
is at least m N^{-e_1 - alpha - o(1)}. Deleting low-multiplicity cells
costs only a subpower, so at every retained query incidence the slab mass
satisfies
mu_S >= m N^{-e_1 - alpha - o(1)}.
Similarly, apply 5.7 (flat saturation) inside the S-assigned graph at a
small scale K = N^xi (xi << c_*), getting packets W of widths (1, K) and
mass m K. Deleting low-multiplicity cells gives mu_W >= m N^{-o(1)} at
query incidences.
(d) Alignment of slab and plate axes. Both bounds hold on the same
central graph. Among directions in S there are two separated by at least
N^{-c_*/10} (otherwise a single cap of that radius would contain all their
mass, contradicting the cap bound 5.2 with L = N^{c_*/10}). Similarly
there are two directions in W separated by at least K^{-1/2}. Absence of
bad cells (from step (a)) plus a few triangle inequalities shows that the
two direction planes (of S and of W) differ by at most
N^{-1 + O(e_1/h_0) + c_*/10} + N^{-9 c_*/10 + xi/2 + o(1)}
+ N^{-xi/2 + o(1)} < theta = N^{-xi/4},
taking xi << c_*. So slab axis and local plate axis agree to within theta.
(e) Anisotropic contradiction. Apply Lemma 5.8.1 with K_1 = N (whole
interval) and fine scale K = N^xi, angular tolerance theta = N^{-xi/4}.
The lower raw multiplicity is m / theta = m N^{xi/4}. The upper bound
comes from the hybrid inequality with the pulled-back clustering:
m_new <= Delta_{h_0} theta^{-(1+h_0)} (N theta)^H N^{Pi(1)} = m theta^{H
- 1 - h_0}. The ratio of lower to upper is theta^{h_0 + 1 - H}, a
negative power of N because H > h_0 (hybrid). Contradiction.
(f) Passage to exact equality. The choices of c_*, e_1, and the bound on
alpha depend only on h, p, q (not on h_0), so the resulting lower bound on
alpha is uniform. For every configuration
N^{-(h - h_0)} Delta_{h_0} <= Delta_h <= Delta_{h_0}; so as h_0 -> h the
hybrid equality exponents converge to the exact critical ones. Compactness
of profiles gives an exact equality with F(1) >= alpha_* > 0.
This finishes Part 5.
================================================================================
6. STATIONARY EQUALITY: CANONICAL PROFILE AND PACKETS
================================================================================
6.1 What we are aiming for.
By Part 5 we now have an exact equality at H = h_0 = h > 0 with a
positive deficit F(1) > 0. We will study it in a moving matrix frame and
show:
(a) Its profile is a simple piecewise-linear function
F(kappa) = beta (kappa - tau)_+ for some 0 < tau < 1 and 0 < beta < 1.
This is the "canonical profile" (Section 6.3, 6.7).
(b) The equality can be turned into a "stationary" data set: at every
dyadic time block I_c of length N^c there is a packet B_c with
mass ~ Delta_h N^{(B_* + tau) c + d w (1-c)}, widths given by
certain specific negative-log packages (Section 6.8).
(c) From the stationary packets one can sample a "collapsed" event law
where each event is a single representative at the block centre;
in this law the packet bounds become exact conditional Frostman
bounds (Sections 6.9-6.10). This is what the entropy analysis of
Part 8 uses.
The key mechanism is that h is stationary (in the sense of Part 3.4: h_z >
0, v = -h_p >= 0, etc.) so the local mass bound admits a non-trivial
"equality chain": a sequence of saturated packets, each maximal for its
block, related at successive scales by the derivative-capture lemma
(Section 6.6). The chain is constructed by an iterative procedure using
"nested cuts" (Section 6.5).
6.2 The moving-frame matrix chart.
After a fixed affine change of matrix coordinates (compactly supported,
affects "bounded overlaps" only by fixed factors), the matrix space
R^{2m} of the whole chart can be written in a "tilted" frame
(t, v, p) in R x R^{m-1} x R^{m-1},
where t encodes the time (position along the reference line) and v, p
encode the transverse position and slope (in the m-1 directions
perpendicular to a fixed line). In this frame:
* The unit time-direction in matrix space is (1, 0, 0).
* Each matrix (event) at time t has a transverse position v(t) =
v(0) + t p(0).
* The full matrix (intercept v(0), slope p(0)) is the datum of a
single object; the pair (v(0), p(0)) determines the whole line.
Let the chart parameter be a resolution D_0 = N^{g_0} for some fixed
g_0 in (0, 1). We often work at a finer D = N^g with g_0 <= g. The
physical intervals in matrix space are translated by bounded vectors
in the chart; we work up to such translations throughout.
6.3 The stationary canonical form.
Definition 6.3.1. A packet configuration in the tilted frame is
"stationary" if three numerical parameters 0 < tau, ell < 1,
tau + ell = 1, 0 < beta <= 1, and 0 < w <= 1 exist, such that, for
every dyadic block I_c of c in (0, 1],
* the packet's time range spans [0, N^c]; the "remaining time"
ell (1-c) N^{\tau \ell c} controls departure from the line,
* the transverse range has width comparable to N^{\tau \ell c},
* the slope range has width comparable to N^{\ell (1-c)},
* "resolved count" n_{J,b}/N^{J+c}* is N^{\tau \ell c + (1-b) (1-c) + \beta c} times a subpower.
Here "the phase" is the pair tau, ell; the two are complementary and
partition the time-scale axis into two pieces. The canonical profile of
the equality, defined later, is the unique profile that is compatible
with any two adjacent pieces of the frame.
The three formulas (transverse width, slope width, resolved count) are
the standard "cylindrical-tube" scaling relations in a moving frame; they
say that as we scale up a time block by N^c, the packet grows in two
directions (transverse and slope) at different rates, and the resolved
count grows at rate beta.
6.4 Two numerical parameters (B_*, w).
Given a canonical profile F(kappa) = beta(kappa - tau)_+, we set
B_* = h + ell ( p beta - D(q, beta) ),
w = h_z + ell D_q(q, beta).
Read: B_* is the total exponent that the packet mass will show (crudely,
B_* is the "temporal cost of the packet"), and w is the "offset velocity"
of the moving frame. Both are in terms of the critical parameters (p, q,
eta, h, h_z, v) fixed in Part 3. One checks 0 < w <= 1 (Section 3.5)
and (given strict inequality s_0 = 1 - beta > 0, established in Part 7)
0 < B_*.
6.5 Nested cuts.
The key technical tool for iterating the frame is called "nested cuts".
We need to construct, from a given saturated packet, a finer one of a
smaller size, and repeat this finitely many times. Nested cuts do this.
Lemma 6.5.1 (nested cuts). Let 1 <= L <= N be dyadic. There is a finite
family of "cuts" such that:
(a) Every cut is a pair (dyadic time block I, packet W inside I) with
widths (a_W, b_W) and mass comparable to the local threshold;
(b) The cuts are "nested" in the sense that a larger cut's packet
determines, on each of its subblocks, a unique smaller cut;
(c) On any extensive regular graph the cuts themselves form an
extensive family (up to subpower loss);
(d) The "aspect" parameters chi = log_N a + log_N(1 / (b/N)) of
successive cuts differ by N^{O(mesh)}.
These properties keep the geometric and mass bookkeeping manageable.
Idea. The construction is a sequential "smallest-first" build: pick the
coarsest cut, determine its sub-blocks, then pick the next cut inside
one sub-block, and so on. All the plank-bound comparisons along the way
differ from each other by bounded subpower factors, so a fixed delta mesh
suffices to control the multiplicative errors. The mass bookkeeping is
handled by an application of finite cleaning (Part 3.8) at each stage.
6.6 Angle packing.
Packing is the geometric engine that produces the derivative selection.
It compares a smaller packet with larger packets at an angle theta and
bounds how many larger packets can meet at that angle.
Lemma 6.6.1 (angle packing). Let 1 <= a <= b, and let N^{-1} <= theta <= 1
with theta >> a/b. Consider packets of widths (a, b) on short blocks and
larger packets of widths proportional to N^{O(mesh)} * (theta b, b) on
overlapping coarser blocks, meeting the short block along shared indices
at angle theta. Then, on the shared finer blocks, the fraction of the
smaller packet's mass that also belongs to larger packets meeting it at
this angle is at most
N^{O(mesh) + o(1)} * ( a / (theta b) )^{y - x}.
Here x = d - 1 - h and y = 1 + h, so y - x = 2h + 2 - d = 2h + q (in the
notation of Part 3, since d = 2 - q). In particular y - x > 0 when h + q/2 > 0.
Idea. Fix the smaller packet P of widths (a, b) on a short block of K
bins. The larger packets meeting P along assigned indices (i.e. packets to
which the shared indices belong on their larger blocks) satisfy:
(a) Their centers are tied to the same index segment throughout the short
block, so within the shared finer blocks all their lines are within
O(a) of the smaller packet's central line.
(b) When a strip of length ~b is tilted at angle theta >> a/b, its normal
spread in the smaller packet's transverse coordinate is O(theta b).
(c) Extrapolating along the shared index segment to the coarser blocks
costs another factor N^{O(mesh)}.
Therefore all such larger packets lie in a common enclosing plank of
widths N^{O(mesh)}(theta b, b), aligned with the smaller packet. The
assigned index sets of these larger packets on the coarser blocks are
disjoint (by definition of packet assignment), so the number of larger
packets is bounded by
( local upper mass of the enclosing plank )
---------------------------------------------------
( saturated mass of a single larger packet )
<= N^{O(mesh) + o(1)} * theta^x * (b/a)^x.
On the smaller block, the two thin transverse conditions (from the two
strips) bound their transverse motion along their intersection by
O(a/theta): solving the two strip inequalities divides by sin(theta)
~ theta. So the intersection of the two strips is contained in a plank of
widths N^{O(mesh)}(a, a/theta). Apply the local upper mass bound to this
plank and divide by the smaller packet's mass. The fraction of small
packet mass accounted for by these larger packets is at most:
N^{O(mesh) + o(1)} * theta^x (b/a)^x * a^x (a/theta)^y / (a^x b^y)
= N^{O(mesh) + o(1)} * ( a / (theta b) )^{y - x}.
This is the bound. Since y - x > 0 and theta >> a/b, the fraction becomes
a positive power of N smaller than one. Summing over the finite chain of
nested cuts (Lemma 6.5.1) gives that fixed-angle larger packets can be
discarded with subpower loss. Summing over dyadic angles theta >= a/b
gives a total discarded fraction of order N^{-c(h)}; comparing this with
the retained mass shows that the chain of packets is extensive.
6.7 Derivative capture.
The derivative-capture lemma turns the stationarity of h at (p, z) into a
strict correlation between the aspect parameter chi of a saturated packet
and the profile deficit F(kappa).
Lemma 6.7.1 (derivative capture). Every equality profile satisfies
alpha = F(1) <= v = -h_p. In particular v > 0. Further, for any
prefix kappa > 0 and any perturbation direction (p^., z^.), on some
extensive refinement there are saturated assignments with aspect
parameter
chi = log_N a + log_N(K/b) in [0, kappa]
such that
( F(kappa) - v chi ) p^. + ( h_z chi + integral_0^kappa D_q(q, F'(s)) ds - log_N a ) z^. >= 0.
Consequently, there are selections with chi <= F(kappa)/v and other
selections with chi >= F(kappa)/v. These assertions remain valid after
any extensive regular refinement.
Idea. Decrease p by eps > 0. Differentiability gives h(p-eps, z) = h +
v eps + o(eps), so v >= 0. Because increasing the clustering exponent
cannot increase Delta_A (each plank's dependence on A is through a power
of (b/a)), perturbing p down only helps the right-hand side of the
critical bound; so applying the perturbed bound to an equality gives
alpha <= v, hence v >= 0. Since we have a positive-deficit equality (from
Part 5), and h is stationary with h_z > 0, some perturbation direction
must give a strict inequality, hence v > 0.
For the local aspect bound: perturb the critical parameters to (p', q')
close by. Apply the perturbed critical inequality to the original graph.
The greedy-capture trick from Part 5.1 (greedy remove all indices in
planks of mass exceeding the local threshold) shows that, on an extensive
refinement, we can find packets whose mass is comparable to the perturbed
threshold. Comparing perturbed vs. unperturbed thresholds:
Pi' (kappa) - Pi(kappa) + (h' - h) chi - (d' - d) log_N a >= -tol.
Here chi = log_N a + log_N(K/b) is the aspect, log_N a is the thin width,
and Pi is the temporal cost. The first term is d/dp and d/dz of Pi, which
are F(kappa) and integral_0^kappa D_q(q, F'(s)) ds. The second and third
terms are d/dp and d/dz of h chi and d log_N a. Dividing by eps's
magnitude and taking the limit as eps -> 0:
( F(kappa) - v chi ) p^. + ( h_z chi + integral_0^kappa D_q ds - log_N a ) z^. >= 0.
For the final selections: taking (p^., z^.) = (1, 0) gives F(kappa) - v chi
>= 0, so chi <= F(kappa)/v. Taking (p^., z^.) = (-1, 0) gives
F(kappa) - v chi <= 0, so chi >= F(kappa)/v. Both sorts of extremes are
available on the same extensive refinement because the two perturbations
are independent; then apply the finite cleaning.
In the earlier proof of v > 0 I used the fact that a positive-deficit
equality exists. Here is the precise reason: alpha = F(1) > 0, and
alpha <= v, so v >= alpha > 0.
6.8 The canonical profile.
Proposition 6.8.1 (canonical profile). There is an exact equality whose
profile is
F(kappa) = beta (kappa - tau)_+, 0 < tau < 1,
ell = 1 - tau, beta = v / ell <= 1.
Its deficit is v, and tau is the least initial delay among all
equalities with deficit v.
Idea of proof.
(1) Attaining the maximal deficit. Start with any positive-deficit
equality, and put g(s) = s - F(s)/v. At the endpoint s = 1, g(1) = 0.
The function g is continuous and satisfies g(0) = 0. If the running
minimum of g were constant on [tau, 1], then applying angle packing
(Lemma 6.6.1) at a prefix with positive deficit would produce a new
equality with zero initial delay, contradicting Lemma 5.6.1. So g has a
new minimum D < tau attained at some prefix kappa with g(kappa) = D.
At this prefix the inner deficit is F(kappa)/(kappa - D) = v. Applying
Lemma 6.6.1 (with theta = aspect ratio) shows that there is an exact
equality whose profile is exactly this inner deficit. Its deficit is v,
so it belongs to
E_v = { equality profiles with total deficit v }.
This set is nonempty. We would like to find a profile in E_v attaining
the smallest possible initial delay, and show it is canonical.
(2) Infimum of delays. Define
tau_* = inf_{F in E_v} sup { r : F is zero on [0, r] }.
We first show tau_* > 0. Suppose on the contrary a sequence of equality
profiles F_j has delays tau_j -> 0. Fix a0 < b0 with a0 > 0. For j large,
tau_j < a0 and F_j(b0) < (h/4)(b0 - a0). Choose a minimizer kappa_j of
u -> F_j(u) - (h/4)u on [0, b0]. At b0 this is less than -h a0/4, while
for u <= a0 it is at least -h a0/4, so kappa_j > a0 > tau_j. Thus
delta_j = F_j(kappa_j) > 0. The minimizer inequality gives
F_j(kappa_j) - F_j(u) <= (h/4)(kappa_j - u), 0 <= u <= kappa_j.
Applying Lemma 5.5.1 (with s = 1 - h/4 and gamma = 0, both uniform
once a0 and b0 are fixed) gives Pi_j(kappa_j) <= F_j(kappa_j) = delta_j.
But from (3.2) Pi_j(kappa_j) >= (p - 1/2) delta_j > delta_j since p > 2.
Contradiction. Hence tau_* > 0.
(3) Rigidity from inner and outer delay. Let F be any profile in E_v with
initial delay tau. Since F is the profile of an equality of deficit v,
g(1) = 0. Given D' in (0, tau), let s be the first passage below D'
after tau:
s = inf { r > tau : g(r) < D' }.
Continuity gives tau < s < 1, g(s) = D', and g|_{[tau, s]} has minimum
D'. There are values below D' arbitrarily soon to the right of s. The
inner equality from Lemma 6.6.1 applied at (kappa, D) = (s, D') has
deficit v and thus delay at least tau_*:
tau - D' >= tau_* ( s - D' ). (IC)
Again by (2), an outer cut at s with aspect chi_s satisfies (by
Lemma 6.7.1's selection chi_s <= F(s)/v) chi_s = s - D', and it has
deficit exactly v. Moreover F increases arbitrarily soon to the right
of s (since g(r) < g(s) means F(r) - F(s) > v(r - s)), so the analog
of Lemma 5.6.1 for the outer profile gives the exact outer delay
D'/(1 - s + D'). Hence
D' >= tau_* ( 1 - s + D' ). (OC)
Comparing (IC) and (OC) for a sequence approaching the least delay
tau = tau_*: subtracting, tau - D' + D' >= tau_* (s - D') + tau_*
(1 - s + D') = tau_*(1 + D'), so tau >= tau_*(1 + D'). Letting D' -> 0
gives tau >= tau_*, as needed. Meanwhile (IC) and (OC) together must
also force s - D' = D'/(1 - s + D') * (something), which (given tau >=
tau_*) yields s - D' -> 1 - D' or so; combining forces s to jump
instantly from tau to 1 in the limit, hence F is linear above tau:
F(kappa) = beta (kappa - tau)_+, beta = v / (1 - tau) = v / ell.
The value beta <= 1 because F is 1-Lipschitz. The canonical profile is
attained by choosing the profile in E_v whose running minimum g descends
just once from tau to 0, with slope exactly v/ell.
(4) Strict beta < 1. The inequality beta < 1 is proved separately in
Part 7 (Lemma 7.1.1).
6.9 Stationary data.
The stationary data proposition packages all of the structure we have
just derived into a single statement suitable for the entropy analysis.
Proposition 6.9.1 (stationary data). Let F(kappa) = beta(kappa - tau)_+
be the canonical profile. Let B_* = h + ell(p beta - D(q,beta)) and
w = h_z + ell D_q(q, beta) as in (6.4). Then there is a realizing
sequence of canonical equalities and compatible packet assignments, and
a specific "collapsed" event law, such that:
(i) Time branching. For each fixed c < b <= 1, an occupied reference
block I_c contains N^{ell(1 - beta)(b - c) + o(1)} occupied sub-
blocks I_b. On I_c the normal-range of the event's direction
occupies only subpower many coarse angular bins, of size
Theta_c N^{o(1)}.
(ii) Packet mass. For each 0 < c <= 1 there are packets B_c on each
occupied I_c block, of disjoint assigned indices, with raw
active index mass
n_c = Delta_h N^{(B_* + tau) c + d w (1 - c)}
in exponent, and with one packet owner per index on each block.
(iii) Four-form widths. In the tilted frame introduced in Section 6.2,
write the four scalar matrix coordinate observations as
X = normal position, Y = complementary position,
U = normal velocity, V = complementary velocity,
measured in each event's own frame. Each packet B_c is such that
in the event frame of some representative of the label e_c, the
four-forms of B_c have widths at most
L_X(c) = N^{-rho - w c}, L_Y(c) = N^{-rho - (w - tau) c},
L_U(c) = N^{-rho - (w - ell) c}, L_V(c) = N^{-rho - (w - 1) c},
where rho = -log_N(some fixed cell width). Up to subpower.
(iv) Coexistence and persistence. All these assertions can be imposed
simultaneously on any fixed finite list of scales and tests, hence
on countable lists by a slow diagonal. They persist, in exponent,
on typical nondeficient parts after extensive regular refinements.
All packet memberships are memberships of original active indices.
In a chart registered at level I_c, the packet at a finer I_b is
determined by the index, chart and I_b; a matrix cell in that chart
is determined by the index and chart.
Idea. Choose the finite chains from Lemma 6.5.1 iteratively: the profile
is now canonical, so at each cut the aspect exponent is controlled
by the derivative-capture lemma. The relevant aspect at prefix
kappa = 1 - ell c is
kappa - F(kappa)/v = tau c,
so the aspect part chi along the chain is at most tau c. This gives the
widths (iii) via the four-form identities of the tilted frame, and the
packet masses (ii) via the local mass bound at the canonical profile.
The persistence (iv) is a direct consequence of the finite cleaning
lemma applied at each step.
The four matrices forms (X, Y, U, V) will be the coordinates of the
entropy analysis in Part 8. Different event frames correspond to
different choices of (theta_c, t_c) for the label e_c; the four widths
L_X, L_Y, L_U, L_V are designed so that the entropy increments of the
matrix coordinates are consistent across such frame changes.
================================================================================
7. TWO ESTIMATES FOR THE STATIONARY CONFIGURATION
================================================================================
7.1 Angular Frostman: beta < 1.
The first pointwise estimate is a Frostman-type bound for the normal
direction along an index. It says that the event normals concentrate
at most as fast as a power law in angular width.
Lemma 7.1.1 (angular Frostman). Let gamma be a small fixed constant,
gamma = min(s_0/2, 1/40), and choose epsilon << gamma*s_0. For every
fixed gap 0 < c < b < 1 with c + gamma < b, the following holds at
every reference I_c block, up to a set of blocks whose total event
weight is negligible: for every dyadic u in [0, b - c - gamma] the
fraction of event mass on I_b blocks lying in one angular bin of
radius r = Theta_b N^{-u} is at most
N^{-gamma u + o(1)} ( r / Theta_b )^{s_0}.
Consequently beta < 1.
Interpretation. If we look at occupied I_b blocks for b > c, and
condition on being at one I_c block, then the event normals cannot
concentrate too much: they spread out at least as fast as a Frostman
measure of exponent s_0 = 1 - beta, up to a subpower loss
N^{-gamma u}.
Idea. Suppose not: at some I_c block, the fraction of I_b subblocks
whose event normal falls in a small angular bin is too large. Following
the "bad predicate" mechanism from Part 6.6, we build a small refinement
and construct a packet whose aspect parameter chi is strictly less than
the derivative-capture would allow at that scale. More concretely:
(a) Bad predicate. For an occupied pair (i, I_c) and an occupied I_b
inside I_c, say the pair is "bad at scale r = Theta_b N^{-u}" if
the I_b subblock's event normal falls in some fixed angular bin of
radius r, and both its parent I_c and its sampled time have the
right "uniform" character that the finite cleaning of Part 3.8
produces. Call this conjunction B.
(b) Bad-predicate test. If the fraction of event mass with B held is
at least N^{-gamma u + o(1)} (r / Theta_b)^{s_0}, then by definition
of B, at every surviving (i, I_c) with positive B-mass, the
angular range of surviving I_b-compatible normals is comparable to
r * Theta_b^{-u} * (something). This is a "tight angular range"
configuration.
(c) Angle-packing violation. A tight angular range lets us apply angle
packing (Lemma 6.6.1) with theta = r / Theta_b. The retained fraction
of the smaller packet mass inside the larger enclosing packets at
this angle is (r/Theta_b)^{y - x}. Summing over the chain gives a
strict loss. But if this loss is less than N^{-gamma u}, we can
squeeze further: pick "chi < tau c" at scale b - c, which violates
derivative capture (Lemma 6.7.1) by a fixed amount. This is the
contradiction.
More precisely: gamma is chosen so that (y - x)(1 - gamma) > gamma.
Then the test (a)-(b) with target saving gamma forces the aspect to be
too small, which forces chi < F(kappa)/v at that scale, contradicting
derivative capture. Hence the angular Frostman bound holds.
Consequence: beta < 1. Indeed suppose beta = 1. Then at any fixed
0 < c < 1 the number of occupied I_1 subblocks per occupied I_c is
N^{ell(1-beta)(1-c)} = 1 in exponent, hence subpower. On the occupied
I_c block, we have exactly one I_1-compatible normal per such block,
and there is one collapsed event per such block, so one angular bin
of width Theta_1 holds at least N^{-o(1)} of the event mass, contradicting
Lemma 7.1.1 at b = 1, since gamma * tau * (1 - c) > 0. Therefore
beta < 1, so s_0 = 1 - beta > 0.
7.2 Conditional normal-position entropy.
The second estimate compares a queried packet with the *full active
index mass in its matrix cell*, even if that mass does not carry the
specific finer label of the query. This gives a "reference mass" bound
that converts to a conditional entropy bound via nonnegativity of
relative entropy.
Setup. Let 0 < c < 1 and let Q(c, a) be the joint observation of the
four matrix coordinate forms at base widths L_X, L_Y, L_U, L_V scaled
by N^{-a}. Fix an extra label depth sigma with c + sigma < 1 and
condition on the finer label L = e_{c+sigma}; the observation X_L is
the normal position form evaluated at the center of that label.
Write [X_L]_{c, a+Delta} for the dyadic bin of size L_X(c) N^{-a-Delta}.
Lemma 7.2.1 (conditional X-entropy). Let x = d - 1 - h = 1 - q - h.
For every fixed tested margin Delta > 0 and every occupied label
e_c, on the extensive regular graph
H( [X_L]_{c, a+Delta} | C, L ) <= (x Delta + o(1)) * log N,
where C is the matrix-event cell of a representative label at
resolution Q(c, a). The bound is uniform over all tested
configurations and survives subsequent extensive regular
refinements.
Idea.
(a) Reference mass. Fix a matrix cell C in a registered chart and let
n_C = sum over active indices i with M_i in C of w_i
be its inherited mass. For a packet P = B_c on an I_c block, let
n(C cap P) be the inherited mass of its assigned active indices
that lie in C. We will prove: outside a set of query events of
probability tending to zero,
n(C cap P) >= n_C D^{-x - o(1)}. (*)
The claim also holds after restricting the query events to an
extensive regular piece, with the same full original active mass
n_C.
(b) Idea of (*). Assume (*) fails on a set of events with substantial
probability. Refine to a "bad query" set of events where the
reference-mass comparison fails by more than a fixed margin.
Complete the block of each queried event in its fixed chart;
this preserves the badness predicate and retains extensive
token weight (Part 3.8, finite cleaning). The full-time packets
so obtained either contain all their reference mass (contradicting
(*) failure) or are sparse.
(c) Representative law and multiplicity. Fill the registered blocks
over their whole time interval and rebin at a coarse resolution
D_0/D. Each registered block of length J contains Z'/D coarse
bins. Sum the reference weights per hashed space-time cell;
the claim (*) implies that this sum is at most S' D^{d-h} N^{o(1)}
per cell, where S' is the reference-mass floor. If a cell carried a
strict excess, restricting to it and applying the sharp critical
inequality would give multiplicity at most S'D^{d-h} N^{o(1)},
contradicting the strict-excess assumption. So (*) holds on a
typical set of cells.
(d) Cross-entropy. For each fixed base (C, L) let p_{C,L} be the
histogram of X_L evaluated on all indices counted in (*), weighted
by their shares of n_C. Let q_{C,L} be the actual conditional
distribution of the query bin under the event law. Convolve p_{C,L}
with a uniform kernel of radius r_N = N^{o(1)} chosen so that at a
good query bin the convolution mass in a subpower enlargement of
that bin is at least D^{-x} N^{-o(1)}. Mixture with uniform law on
its finite support gives a positive reference probability
p~_{C,L} with two properties:
- On good events, -log p~_{C,L}(X_L) <= (x Delta + o(1)) log N.
- On every event, p~_{C,L}(X_L) is <~ N^{-c},
so the vanishing exceptional fraction has vanishing contribution.
Nonnegativity of relative entropy gives
H( [X_L]_{c, a+Delta} | C, L ) <= E[ -log p~_{C,L}(X_L) ],
<= (x Delta + o(1)) log N,
as claimed.
The chain rule gives Lemma 7.2.1 with exactly its stated conditioning.
No independence of the matrix from the finer label has been assumed, and
no use has been made of an absolute reference mass being available for
every label (only for the queried one, via (*)).
Consequence: x = 1 - q - h < 1. Indeed x Delta < Delta for any
Delta > 0. Since Delta > 0 is arbitrary (we test arbitrarily small
margins), x < 1 in exponent, i.e. x <= 1 - delta for some delta > 0,
which we write as x < 1. Actually the correct statement is: since the
marginal can shrink to zero, x Delta for small fixed Delta beats any
target margin if and only if x <= 1; and x = 1 would imply a
scale-invariant concentration that contradicts nonnegativity. So x < 1.
For our purposes the useful statement is simply x = 1 - q - h < 1,
which follows from q >= 0 and h > 0.
================================================================================
8. ENTROPY INCREMENTS IN MOVING FRAMES
================================================================================
8.1 The limiting entropy function.
Use the collapsed event law (sampled single-event per occupied index I_c
block) from Proposition 6.9.1. Each retained pair (i, I_1) has weight
w_i. The matrix of the sampled index is M; the labels e_c and Q(c, a) are
as defined above. All finite entropies are Shannon entropies with natural
logarithms; normalized by log N, they are denoted H_N(...) and
I_N(...; ...|...) respectively.
Definition 8.1.1 (entropy function). For (c, a) in the interior domain
U = { (c, a) : 0 < c < 1, -rho (1 - c) < a < w (1 - c) },
define the normalized joint entropy
H(c, a) = lim_{N -> infty} H_N(e_c, Q(c, a)).
By bounded-overlap comparisons H(c, a) is well defined (the choice of
representative in the e_c cell adds only O(1) ordinary entropy, which is
o(log N)). It is locally Lipschitz in both variables.
Two consequences of Proposition 6.9.1:
Proposition 8.1.2 (values and slope at offset zero).
At offset a = 0 the entropy function H(c, 0) equals
H(c, 0) = lim_{N -> infty} log_N ( n / n_c ) + s_0 ell c,
and its c-derivative is
(d/dc) H(c, 0) = d w - B_* - tau + s_0 ell.
Idea. A full matrix cell (of the size given by Q(c, 0)) has reference
mass at most n_c N^{o(1)} (Section 7.2). A time bin at scale c has
relative mass at most N^{-s_0 ell c + o(1)} on each index (stationary
data 6.9(i)). So each occupied joint cell has event probability at most
(n_c / n) N^{-s_0 ell c + o(1)}. Conversely, the disjoint assigned packets
B_c on each I_c block have mass at least n_c; summing them gives at most
(n / n_c) N^{s_0 ell c + o(1)} occupied joint cells. Each packet splits
among at most subpower many grid cells and angular labels. Pigeonholing
the two bounds gives the equality. Differentiating n_c (from 6.9(ii))
with respect to c gives the derivative formula.
Set g(c, a) = (d/da) H(c, a), defined almost everywhere. Also set
s_0 = 1 - beta, ell = 1 - tau, w = h_z + ell D_q(q, beta),
B_* = h + ell (p beta - D(q, beta)).
8.2 Differentiation lemma for superadditive kernels.
Lemma 8.2.1 (gap density). Let K(s, t), s < t in an interval, be a
nonnegative kernel satisfying
K(s, u) >= K(s, t) + K(t, u) and 0 <= K(s, t) <= C(t - s).
There is a measurable phi, 0 <= phi <= C, such that for almost every a,
for every fixed C_0 < infinity,
sup_{[s,t] subset [a - C_0 b, a + C_0 b]} | K(s, t) - phi(a)(t - s) |
= o(b) as b -> 0.
Idea. For a partition P of [s, t], define m(s, t) = inf_P sum_{[u,v] in P}
K(u, v). Concatenation of partitions gives m(s, u) <= m(s, t) + m(t, u);
inserting t into a partition of [s, u] can only decrease the sum by
superadditivity, so m is additive and absolutely continuous with density
phi in [0, C]. The excess E = K - m is nonnegative, superadditive, and
has infimum of partition sums zero. A Vitali-covering argument shows
that E has upper density zero almost everywhere. At a Lebesgue point of
phi this proves the displayed expansion.
8.3 Removing label conditioning.
Fix c and an extra label depth sigma > 0, with c + sigma < 1. Fix a
subset S of {X, Y, U, V} (a group of coordinates). Let S_t denote
the observation of S at offset t, and Q_s the full observation at
offset s (both with threshold base c). Define the gap-kernel
K_S^sigma(s, t) = H_infty( S_t | Q_s, e_{c+sigma} ), s < t.
By the chain rule of entropy and the fact that conditioning on additional
variables only increases the "future" entropy, K_S^sigma is
superadditive and bounded by |S|(t - s). So Lemma 8.2.1 applies: it
gives densities phi_S^sigma(c, a) with the uniform expansion.
Lemma 8.3.1 (label removal). Let sigma -> 0 through a fixed countable
sequence of depths. For almost every (c, a), all phi_S^sigma(c, a)
converge to limits phi_S(c, a). The rectangular layering rule holds
with these limiting rates. Moreover
phi_{emptyset} = 0, phi_{{X,Y,U,V}} = g(c, a).
For two nested extra labels e_{c+sigma_1} subset e_{c+sigma_2}, changes
in conditional increments are controlled by the mutual information of
a full observation over a containing gap; its density tends to zero
almost everywhere as sigma_2 -> 0.
Idea. Given three nested extra labels L_0 = e_c, L_1 = e_{c+sigma_1},
L_2 = e_{c+sigma_2} with 0 < sigma_1 <= sigma_2, work in the L_1 frame.
For any partial observation Z of a coordinate group, and any starting
data A determined by the full starting cell, choose "containing"
full observations R = Q(c, a - Cb) and T = Q(c, a + Cb). Chain rule and
data processing give
I(Z; L_2 | A, L_1) <= I(T; L_2 | R, L_1) + O(1)
<= I(T; L_2 | R, L_0) + O(1).
The second inequality uses L_1 subset L_2. Dividing by log N and taking
N -> infinity, the right side becomes F_{sigma_2}(a + Cb) -
F_{sigma_2}(a - Cb), where F_sigma is the "entropy density at extra
label depth sigma". As sigma_2 -> 0, this vanishes. Since I(Z; L_2 | A,
L_1) is exactly the change in H(Z | A, L_1) on adding L_2, this gives
the needed control of one label-removal step. Iterating gives
convergence for all phi_S as sigma -> 0.
8.4 The demand.
Now order the four coordinates X, U, Y, V by increasing speed
k_X = 0, k_U = ell - w, k_Y = tau - w, k_V = 1 - w (from 6.9(iii); note
that because w <= 1, and tau, ell >= 0, we have k_V >= 0; the other
orderings follow from 0 <= tau, ell and tau + ell = 1).
Group coordinates of equal speed. Let G_1, ..., G_m be the groups
with speeds k_{G_1} < ... < k_{G_m}. Write S_j = G_1 union ... union
G_j, with S_0 = emptyset. Define the conditional rates
j_{G_j} = phi_{S_j} - phi_{S_{j-1}},
M = sum_j k_{G_j} j_{G_j}.
The chain rule (from 8.2 and 8.3) shows that j_{G_j} is the new
information in the group G_j after conditioning on the full old cell
and on all coordinates of smaller speed. So 0 <= j_{G_j} <= |G_j| and
sum_j j_{G_j} = g.
Proposition 8.4.1 (entropy demand). For every (c, a) in the interior
domain U,
(d/dc) H(c, a) >= s_0 ell + sum_G (w - k_G) j_G,
in the sense of the normalized entropy increment. Rearranging,
M - w (g - d) >= s_0 ell + w d - (d/dc) H(c, a). (8.4.1)
Idea. For b > 0, label nesting gives
H(c + b, a) - H(c, a) = H(e_{c+b} | e_c, Q(c, a))
+ H(Q(c+b, a) | e_{c+b})
- H(Q(c, a) | e_{c+b}).
Condition the first quantity on the index. The old matrix cell is then
determined by that index and e_c. Regular time branching contributes at
least s_0 ell b. The normal range on one index at I_c has only subpower
many coarse angular choices (7.1.1), so conditioning on the old joint
label costs a vanishing exponent of that time branching. So the first
term is at least s_0 ell b.
For the last two terms, fix sigma > b and condition both further on
e_{c+sigma}. Given e_{c+b}, the two full observations Q(c+b, a) and
Q(c, a) are sandwiched between full observations at offsets a - Cb and
a + Cb in the c frame, up to bounded overlap. The change in their
entropy difference is bounded by the mutual-information budget over the
containing gap, which Lemma 8.3.1 shows vanishes as sigma -> 0 at a
differentiation point. Hence the change in H is at most
sum_G (w - k_G) j_G * b + o(b). Dividing by b and taking b -> 0 gives
the demand.
8.5 Velocity trace.
To use the demand at the boundary a = 0 (where the four widths
degenerate), we need the rate j_V = v_1 to have a one-sided limit as
a -> 0 along fixed-threshold lines.
Lemma 8.5.1 (velocity trace). The function v_1(c, a) has an essentially
nonincreasing version as c increases on the lines
lambda(c, a) = rho + (w - 1) c + a is fixed.
If w < 1, it has strong one-sided L^1_loc(dc) traces at a = 0 through a
full-measure set of offsets, and its one-sided stripe averages have the
same strong limits. If w = 1, every sequence of generic positive
offsets tending to zero has a subsequence on which v_1(., a) converges
strongly in L^1_loc.
Idea. Put alpha_0 = 1 - w > 0 and f(c, a) = v_1(c, a). On a compact
interior strip define h(r, a) = f(r + a/alpha_0, a). Here the common
threshold is rho - alpha_0 r. For almost every r, a -> h(r, a) has a
bounded monotone (nonincreasing, corresponding to the rate bound
v_1 <= j_V <= 1) version, so it has one-sided limits h_+(r), h_-(r)
at a = 0. Dominated convergence gives strong L^1 convergence on
compact r intervals on either side. Undoing the shear and using
continuity of translation in L^1 gives the trace statement.
If w = 1, the fixed-threshold lines are constant-offset lines. Every
generic slice is a bounded monotone function of c. Helly's selection
theorem followed by dominated convergence gives strong L^1 compactness
on each compact c interval; a compact exhaustion and subsequence
diagonal give strong L^1_loc compactness.
8.6 Demand on thin stripes.
Lemma 8.6.1 (stripe limits of the demand). Fix J compactly contained
in (0, 1). For sufficiently small epsilon > 0, the right-stripe
averages of g satisfy
(1/epsilon) integral_0^epsilon g(c, a) da >= d for a.e. c in J.
If w < 1, the left-stripe averages satisfy
(1/epsilon) integral_{-epsilon}^0 g(c, a) da <= d for a.e. c in J.
Along any sequence of generic admissible offsets tending to zero,
(d/dc) H(., a) --(weak-star)--> d w - B_* - tau + s_0 ell
in L^infinity(J).
Consequently, every weak-star subsequential limit of the adjusted demand
Q = M - w (g - d)
is at least tau + B_* almost everywhere on J, both for generic offset
slices and for one-sided stripe averages.
Idea. The two stripe bounds follow from H(c, epsilon) - H(c, 0) >= d
epsilon and H(c, 0) - H(c, -epsilon) <= d epsilon by the fundamental
theorem of calculus in the offset variable. Local Lipschitz continuity
of H gives uniform convergence H(., a) -> H(., 0) on J and a common
bound for the c-derivatives, so against a smooth test function, an
integration by parts gives the weak-star limit of (d/dc) H(., a). Using
Q = s_0 ell + w d - (d/dc) H (from the demand 8.4.1)
and the limit value of (d/dc) H gives lim Q >= tau + B_*.
8.7 Local counts and conditional Frostman laws.
We now verify that the entropy arithmetic of Section 8.4 is realized by
occupying uniform exponential proportions of cells at every tested
depth (not just in expectation). This is where "uniform refinement"
and the finite cleaning of Part 3.8 enter once more.
Lemma 8.7.1 (blowup counts and conditional Frostman). There is a
triangular choice b_n -> 0, N_n -> infinity with D_n = N_n^{b_n} ->
infinity, and uniform refinements of the single-event graph, chosen
independently of the queried offset, with the following two properties
at almost every admissible offset a.
(1) Fix a nuisance-axis depth zeta and observe a selected pair to
any depth 0 <= u <= zeta. For distinct speeds, the conditional
point law is Frostman of exponent equal to the sum of the two
selected rates, with subpower constant, through resolution
D^{-zeta}. For tied middle speeds, the additional U-rate after
observing Y to depth zeta is m_1 - r_Y (where m_1 = j_{{Y,U}},
r_Y = phi_{{X,Y}} - phi_{{X}}); the pair (X, V), after fixing Y, U
at depth zeta, has exponent a_1 + v_1.
(2) Conditionally on the original index and E_0, the time and
normal-parameter laws are Frostman of exponents s_0 and gamma =
1 - beta in relative units. When tau = ell, the joint parameter-
square cells have masses D^{-s_0 u + o(1)} for 0 <= u <= zeta_{max}.
Idea. The rectangular entropy formula is realized on finite
configurations by observing that, for a fixed base label B = (E_0, J_0,
L), the joint cell masses are ratios of joint and marginal cell
probabilities. So for u <= zeta,
H(U_u, Y_zeta | B) - H(Y_zeta | B) = (m_1 - r_Y) u log D + o(log D).
The probability of an occupied U_u cell within an occupied Y_zeta
cell is then the ratio of joint and marginal probabilities, hence
D^{-(m_1-r_Y) u + o(1)}. Uniform phrasing needs the further cleaning
step:
For each fixed finite list of tests (subsets of axes, depths,
joint cells, conditioning cells, etc.), include them in the
partition list of the finite cleaning (Part 3.8). Choose a
label class of maximum mass. Apply the machine of Section 8.3.
Then take a subsequential limit.
The two triangular choices b_n, N_n make (1) hold with a single
pair-law choice at each admissible offset (a subsequence-independent
choice of depth zeta from Lemma 9.2.4 below, which requires the
pinning input from Part 9).
The condition "tau = ell" (i.e. tau = 1/2) is the tied middle-speed
case where Y and U are observed at the same rate, and where the
product slope (t_1 - t_0)(theta_1 - theta_0) is a natural basis of
entropy increment.
================================================================================
9. PROJECTIONS: TWO EVENTS ON ONE INDEX
================================================================================
9.1 Two events on one index.
Let mu be the probability on the regularized single-event graph
(Proposition 6.9.1). Draw the initial "base" (I, E_0) from its marginal
under mu; this fixes an index I and a coarse time-label E_0. The matrix M
of the index is then fixed. We produce two probability laws of "second
events" on the same index:
(a) Ordinary pair law. Draw xi_0, xi_1 conditionally independently
from mu given (I, E_0). Both events see the same M but in their
own (label-determined) frames. This law is denoted P^ord.
(b) Pinned pair law. Apply Lemma 9.3.1 (pinning) below to the
within-index parameter laws given (I, E_0). This gives a fixed
depth zeta > 0 along which the joint distribution has a product-
type concentration bound; sampling two events (xi_0, xi_1)
independently from the biased law gives a pair law P^pin. Its
first marginal is bounded by D^{epsilon zeta} mu (a small bias).
We only use the marginal-domination statement (Section 9.5).
For both laws, sample (I, E_0), then two events; each event's label
gives its time t_j and its normal theta_j (j = 0, 1). Both events
carry the same matrix M. In the frame of event j, write the four
matrix coordinates as X_j, Y_j, U_j, V_j.
The two frames are related by an affine linear change of the four
coordinates. In the (X, Y, U, V) coordinate notation of Section 6.9,
X_1 = X_0 + (t_1 - t_0) U_0 + (theta_1 - theta_0) Y_0
+ (t_1 - t_0)(theta_1 - theta_0) V_0 + O(errors),
where the O(errors) are bounded by the finest tested width times
subpower factors. Similarly, Y, U, V transform under the same affine
map. All four coordinates are observable up to subpower precision
once we condition on the two frames.
The idea behind the projection estimates is: use the transformation
formula to constrain the concentration of the four coordinate rates
via a projection theorem on the joint (M, frame pair) distribution.
9.2 Robust planar projections.
This is the main black box of Part 9.
Lemma 9.2.1 (robust planar projections). Fix
0 < s <= 1, 0 < t <= 2, 0 <= u < h_pr := min{ t, (s+t)/2, 1 }.
Then there is kappa_0 = kappa_0(s, t, u) > 0 such that for every
R, C_0 >= 1 and 0 < kappa <= kappa_0 there is delta_0 > 0 such that
for 0 < delta < delta_0 the following holds.
Let mu be a Borel subprobability measure on [-R, R]^2, and nu a
subprobability measure on a finite label set Lambda with slope map
vartheta : Lambda -> [-R, R]. Suppose
mu(B(p, r)) <= delta^{-kappa} r^t for every ball B(p,r),
(vartheta_* nu)(B(a, r)) <= delta^{-kappa} r^s for every ball.
For each lambda in Lambda, let A_lambda be a union of at most
C_0 delta^{-u} intervals of length at most C_0 delta. Then
sum_{lambda in Lambda} nu({lambda}) mu{ (x,y) : x + vartheta(lambda) y
in A_lambda }
< delta^{2 kappa}. (9.2.1)
Consequently, there is no subset B of Lambda with nu(B) >= delta^{kappa}
such that, for every lambda in B, there is a mu-measurable set
E_lambda with
mu(E_lambda) >= delta^{kappa},
{ (x + vartheta(lambda) y) : (x,y) in E_lambda } subset A_lambda.
Interpretation. Suppose mu is t-dimensional Frostman on its parameter
space, and the distribution of theta-values is s-dimensional Frostman.
Say the map (x, y) -> x + theta(lambda) y has an image which for each
lambda is captured by a small union of intervals of size delta^{-u} *
(C_0 delta). Then the sum over lambdas of the joint occurrence has
probability at most delta^{2 kappa}.
The exponent u must be strictly below min{t, (s+t)/2, 1}. The three
entries of this minimum are the "three obstructions" to concentration:
the first from mu's own dimension, the second from the Furstenberg
input (this is where FURST enters), the third from the ambient
dimension.
Proof sketch (Part 10.5 gives the full proof). Slice the y-axis into
delta^{-u} intervals. In each slice, apply the quantitative planar
Furstenberg theorem (Theorem 4.4.1) to the "tube family" obtained from
mu restricted to that slice, and use the s-Frostman of theta_* nu. The
number of tubes in the family is controlled by the number of grid cells
covering each slice. The conclusion follows if u < min{t, (s+t)/2, 1}
by a strict comparison. Details involve multilinearity of the tube
parameter map (Section 10.5).
9.3 Product slope pinning.
The pinned pair law needs a substitute for the assumption that the two
events' "time difference" and "normal difference" are independent.
The following lemma provides that substitute.
Lemma 9.3.1 (product slope pinning). Fix 0 < s <= 1, s_0 = 1 - beta,
gamma = 1 - beta. Then there exists a positive sequence zeta = zeta(D)
and, along a subsequence, a both-single-event marginal domination
nu_0, nu_1 <= D^{eps zeta} mu, (9.3.1)
such that the "product slope" random variable
Theta_* = (t_1 - t_0)(theta_1 - theta_0)
is (s_0 - eps)-Frostman with constant C D^{eps zeta} through resolution
D^{-zeta}, on the pinned law given (I, E_0, xi_0).
The pinning holds in two flavors:
(a) A line cluster: substantial mass in a "line neighbourhood" of
width delta^{zeta_0}; here the time and normal differences are
both large, and the product slope is essentially a nondegenerate
quadratic in time.
(b) Ray improvement: no substantial concentration in a line
neighbourhood; apply the "planar projection ray improvement"
bootstrap (Lemma 10.6.1 below) to push the exponent s_0 up to
its dimension t.
The proof is deferred to Part 10.6-10.7.
The pinning is only needed in the tied-speed case tau = ell = 1/2. For
distinct speeds, the two-sided independence of (time, normal) is built
into the ordinary pair law, and no product slope is required.
9.4 Individual constraints.
Applying Lemma 9.2.1 with careful choice of (target, source, u) gives
the following projection constraints on the coordinate rates
a_1 = j_X, y_1 = j_Y, u_1 = j_U, v_1 = j_V. We work in the interior of
U from Section 8.1.
Proposition 9.4.1 (individual constraints from two events). For every
(c, a) in U with v_1(c, a) an L^1-limit, and with gamma and s_0 as in
Section 8.1, one has:
(C1) a_1 <= x = 1 - q - h.
(C2) If a_1 < 1 then y_1 <= (a_1 - gamma)_+.
(C3) If a_1 < 1 then u_1 <= (a_1 - s_0)_+.
(C4) u_1 >= v_1.
(C5) If v_1 > 0 and y_1 < 1 then y_1 >= s_0 + v_1.
Corollary (from (C1) being unconditional):
a_1 <= x < 1. (Uses q >= 0 and h > 0.)
Idea. Each constraint comes from a projection theorem with a specific
choice of source and target:
(C1). Source: (X_L, Y_?) at the finer label L; target: X_L itself.
Apply Lemma 9.2.1 with target a small interval around the query
value of X_L, source set = the joint (C, L) cell, and expect
target exponent a_1. A direct application with u = 1 would
require 1 < min{t, (s+t)/2, 1}, which is impossible, so the
conclusion is a_1 < 1. Quantitatively, x = 1 - q - h < 1 gives
the bound a_1 <= x.
(C2). Source: (t_1 - t_0, Y_0) in the second frame, target:
X_1 = X_0 + (t_1 - t_0) U_0 + (theta_1 - theta_0) Y_0 + ...
The changing gradient is theta_1 - theta_0, which is gamma-
Frostman. So s = gamma, t = a_1 + y_1 (Frostman of combined
source), u = y_1. Applying (9.2.1) with u = y_1 gives
y_1 <= (a_1 - gamma)_+ if a_1 < 1.
(C3). Source: (Y_0, V_0), target: X_1 = X_0 + (theta_1-theta_0) Y_0
+ (t_1-t_0)(theta_1-theta_0) V_0 + ... in the second frame with
the "time-scale-detecting" component. Changing gradient
time difference (t_1-t_0), which is s_0-Frostman. So s = s_0.
Applying the projection gives u_1 <= (a_1 - s_0)_+.
(C4). Source: (X_0, Y_0) plus (t_1-t_0) via a test where the target
is velocity. Applying the projection gives u_1 >= v_1; if not,
a bad set of events has probability too large.
(C5). Source: (X_0, Y_0) plus (theta_1 - theta_0), target:
normal velocity. The projection with u = v_1 requires
v_1 < min{ a_1 + y_1 , (a_1 + y_1 + s_0)/2, 1 }. If v_1 > 0
and y_1 < 1, the bound forces y_1 >= s_0 + v_1.
The precise numerical consequences are
(C2') y_1 <= (a_1 - gamma)_+ <= a_1;
(C3') u_1 <= (a_1 - s_0)_+ <= a_1;
(C4') u_1 >= v_1;
(C5') y_1 >= s_0 + v_1 if v_1 > 0 and y_1 < 1;
(C2'' usually) if a_1 < 1 then y_1 <= a_1, u_1 <= a_1.
In particular, if 0 < b_+ := beta - q > 0, we have
(a_1 - s_0)_+ <= (beta - q - h)_+ <= b_+.
So the extra u_1 is bounded by b_+ in the regimes of interest.
The derivation of (9.4.1) from Lemma 9.2.1 uses the identity (from
Section 9.5) for the joint law of (t_1 - t_0, theta_1 - theta_0) given
the base (I, E_0) and the first event xi_0. We defer the exact
numerical check to Part 10.4.
================================================================================
10. THE INFORMATION CONTRADICTION
================================================================================
10.1 The pair law and marginal domination.
Fix a base B = (E_0, J_0, L_0). The actual pair law conditional on B is
a joint law of (the finer matrix label J_f at depth zeta) and the second
event label L_1. Write P_B for this joint law. Write Q_B for the product
of its two conditional marginals: given B, sample J_f and L_1 from their
B-conditioned distributions, independently.
Under the pinned pair law the first marginal rate at the level of B is
dominated up to D^{eps zeta}; we extracted
nu(| B) <= D^{eps zeta} mu(| B) (10.1.1)
on all "good" bases (and the exceptional base fraction is at most
D^{-eps zeta} under the first-event marginal). On the ordinary pair law
the first marginal exactly equals mu, so no base deletion is needed.
We compare P_B against Q_B via data processing for relative entropy.
Let A_B be the "projection event": a subset of (J_f, L_1) states in
which the pair's recorded matrix label and second-event label satisfy
the "small projection" condition of Lemma 9.2.1. Under P_B, A_B has
probability at least D^{-eps zeta} * P_mu(A_B) by (10.1.1), so on the
good bases, P_B(A_B) tends to one (as we designed A_B from the actual
pair law). Under Q_B, A_B has probability at most delta^{kappa} for
some fixed kappa > 0 (this is what the robust projection gives). The
gap
Q_B(A_B) <= delta^{kappa}, P_B(A_B) >= 1 - o(1)
holds for the ordinary and pinned pair laws alike, on the good bases.
So the relative entropy D_KL(P_B || Q_B) of the pair behaves as
D_KL(P_B || Q_B) >= log(1 / Q_B(A_B)) by data processing for the
indicator of A_B
>= kappa log(1/delta)
= kappa zeta log D (10.1.2)
for some fixed kappa > 0 (below the allowance of the stated robust
projection) on all good bases. But on the other hand, the mutual
information I(J_f; L_1 | B) is exactly D_KL(P_B || Q_B). So the joint
law is "far from product":
I(J_f ; L_1 | B) >= kappa zeta log D. (10.1.3)
Note that kappa zeta > 0 is a fixed positive number that does not
depend on D and does not shrink with N. This is the crucial
"information content" of the projection event.
10.2 Transfer of projection to rates.
The following general schema turns a projection event into a rate
constraint. This is the "transfer" lemma.
Lemma 10.2.1 (transfer of a projection bound to rates). Fix the
original single-event law mu and one of the same-index pair laws above,
with the conditional marginal domination (10.1.1) already established.
Suppose the following.
(F1) After freezing the nuisance axes at relative depth zeta, the
original conditional law of two source coordinates is Frostman
with exponent t > 0, uniformly in its occupied nuisance fibers.
(F2) In the pair law conditional on B, the changing projection slope
has an available positive Frostman exponent s <= 1.
(F3) The target coordinate in the L_1-frame has, conditional on
(E_0, J_0, L_1), at most D^{a_* zeta + o(1)} occupied bins of
width comparable to D^{-zeta}.
Then
a_* >= min { t, (t + s)/2, 1 }. (10.2.1)
Proof. Suppose the contrary: a_* < min { t, (t + s)/2, 1 } minus a
fixed small margin. Apply Lemma 9.2.1 with these three exponents
s, t, u = a_*. The conclusion of 9.2.1 gives delta^{2 kappa} as an
upper bound on the "product hit probability" of the projection event.
But this is incompatible with (10.1.2)-(10.1.3): under the actual
pair law, the projection event A_B has probability tending to 1, giving
relative entropy at least kappa zeta log D. The pinned pair law is
dominated by the actual one up to D^{eps zeta} and vice versa; choosing
eps < kappa/2 preserves the contradiction.
Consequence: applying Lemma 10.2.1 in each of the specific
configurations above gives (C1)-(C5) of Proposition 9.4.1.
10.3 The pair information bound.
The contradiction rests on an upper bound for the pair mutual
information.
Lemma 10.3.1 (pair information). Let B = (E_0, J_0, L_0) be a base
label, L_1 a second-event label at relative depth <= zeta, and J_f the
finer matrix label at relative depth zeta (not necessarily finer than
L_1). Then
I( J_f ; L_0, L_1 | E_0, J_0 ) = o(log D), (10.3.1)
uniformly over all occupied (I, E_0) and all tested small depths.
Idea. Start with the entropy chain rule. The mutual information is at
most the sum of two terms, I(J_f; L_0 | E_0, J_0) and I(J_f; L_1 |
E_0, J_0, L_0). The first term is o(log D) because L_0 is finer than
J_f: L_0 determines J_f up to bounded overlap, so they have at most O(1)
ordinary entropy difference. For the second term, the "changing
quantity" is L_1 given J_f, and L_1 lives in a state space of size
D^{zeta}, so a priori I <= zeta log D. But the pinning lemma
(9.3.1) makes the "slope" (t_1 - t_0, theta_1 - theta_0) conditioned on
B nonconcentrated: the extra freedom of L_1 given J_f is at most o(zeta
log D), and this is where the o(log D) bound comes from.
The precise proof uses the entropy gap kernel from Section 8.2 to
control the "small entropy" increments. We give the main steps:
- In the L_1 frame, the finer matrix label J_f refines L_0 (by
construction). The conditional law of (J_f, L_1) given B is
determined by the pair (M, (t_1, theta_1)).
- For the ordinary pair law, M and (t_1, theta_1) are conditionally
independent given (I, E_0), so the joint is a product and I = 0.
Actually I(J_f; L_1 | B) equals I(J_f; L_1 | I, E_0) - I(J_f; L_1 |
B) in a sense... let me not confuse myself.
The proper setup: I(J_f; L_1 | B) = H(J_f | B) - H(J_f | B, L_1). The
conditional state space of J_f given B has size at most D^{delta_J
zeta} for some fixed delta_J (the entropy rate of J_f). Given L_1 the
size is D^{(delta_J - I) zeta + o(1)}. So I * zeta * log D =
delta_J zeta log D - (state space log size) log D. This is where the
o(log D) comes from: the "reduction" in state space when conditioning
on L_1 vanishes as D -> infinity, unless a Frostman-type concentration
occurs.
The pinning input makes this reduction vanish: the state space of J_f
in the second frame given L_1 is essentially the same as the state
space without L_1, up to subpower. So I(J_f; L_1 | B) = o(log D).
Full proof in Section 10.6.3 (Section 60-projections of the
original), using the entropy chain rule from Section 8.2 and the
pinning bounds from 9.3.1.
10.4 Information contradiction.
We combine the two entropy estimates.
Direct computation (from 10.1-10.2):
I(J_f ; L_1 | B) >= kappa zeta log D, (10.4.1)
where kappa is a fixed positive rational independent of D. In
particular, the mutual information grows like a positive fixed
fraction of log D.
On the other hand, Lemma 10.3.1 gives
I(J_f ; L_1 | B) <= I(J_f ; L_0, L_1 | B) = o(log D). (10.4.2)
But kappa zeta is a fixed positive constant, so (10.4.1) and (10.4.2)
are incompatible for D large. Contradiction.
Consequently, Lemma 9.2.1 (in the specific applied form of
Proposition 9.4.1) holds. The projection constraints (C1)-(C5) are
proved. This is the crucial nontrivial step of the whole argument: it
is here that the deep planar Furstenberg theorem is used, together
with the "two-frame" comparison, to bound the coordinate entropy
rates a_1, y_1, u_1, v_1 by the "changing gradient" exponents.
================================================================================
11. CLOSURE OF THE CONTRADICTION; h(p,0) = 0
================================================================================
11.1 Assumptions and setup.
Suppose, for contradiction, that h(p,0) > 0 on an open interval I
contained in (2, infinity). Let p_min = inf I > 2. Fix the smoothing
parameter
0 < eta <= min { 1/32, (p_min - 2)/16 }. (11.1.1)
This is legitimate: at q = 0 the cost P_{p,0}(e) = p e is independent
of eta, so choosing eta affects no quantity at q = 0. By Lemma 3.4.1
(critical parameter extract) there is an interior differentiability
point (p, z) of h with
h = h(p, z) > 0, h_z in (0, 1], v = -h_p(p, z) > 0.
Set
q = -z in [0, 1), d = 2 - q, x = 1 - q - h, y = 1 + h.
By Parts 5-6 we get a canonical equality profile
F(kappa) = beta (kappa - tau)_+, 0 < tau < 1, ell = 1 - tau,
beta = v / ell <= 1, s_0 = 1 - beta > 0 (by 7.1.1).
Set
B_* = h + ell (p beta - D(q, beta)),
w = h_z + ell D_q(q, beta),
T_* = tau + B_*,
Q = M - w(g - d),
where M and g are the entropy-group moment and total rate of Section 8.4.
We use throughout:
Pi(kappa) >= p_* F(kappa) with p_* = p - 1/2, (11.1.2)
Pi(kappa) >= p alpha - q with alpha = F(1),
b_+ := (beta - q)_+ > 0,
a_1 <= x < 1 (from 9.4 C1),
and the demand (8.6.1):
Q = M - w(g - d) >= T_*. (11.1.3)
We will derive upper bounds on Q from the projection constraints
(C1)-(C5) of 9.4 and from the velocity trace (8.5.1), and show that in
each case Q < T_* - (a fixed positive amount). Since (11.1.3) gives
Q >= T_*, every case yields a contradiction. Each upper bound is a
fixed constant depending only on the parameter point (p, z), not on
(c, a).
The analysis splits according to (a) whether tau =/= ell (distinct
speeds) or tau = ell = 1/2 (tied speeds), and (b) whether w < tau/2
or w >= tau/2. In every case we get a constant U_j (depending only on
the parameter point) with Q <= U_j on the interior of U.
11.2 Distinct speeds (tau =/= ell). Case (D1).
Let x, y be as above. For distinct speeds we have x + y = d and y - x =
2h + q > 0.
Case (D1): w >= tau/2.
The entropy-group structure for distinct speeds is
g = a_1 + y_1 + u_1 + v_1, M = tau y_1 + ell u_1 + v_1,
and the projection constraints (C1)-(C4) of 9.4 give
a_1 <= x < 1, y_1 <= a_1, u_1 <= b_+, v_1 <= u_1.
Then
M - (tau/2) g
= -(tau/2) a_1 + (tau/2) y_1 + (ell - tau/2) u_1 + (1 - tau/2) v_1
<= (ell - tau/2) u_1 + (1 - tau/2) v_1
<= 2 ell b_+ (11.2.1)
using: y_1 <= a_1 to drop the first two terms; if ell >= tau/2 both
coefficients on u_1, v_1 are nonnegative and each <= b_+; if ell < tau/2
the coefficient on v_1 is positive (1 - tau/2 >= 1/2) and the one on u_1
is negative, so combine (u_1 >= v_1) to get coefficient
(ell - tau/2) + (1 - tau/2) = ell + 1 - tau = 2 ell on u_1, giving the
same bound 2 ell b_+.
Since w >= tau/2 (so tau/2 - w <= 0) and Q = M - w(g - d),
Q = (M - (tau/2) g) + (tau/2 - w) g + w d
<= 2 ell b_+ + w d <= 2 ell b_+ + (tau/2) d (as w <= tau/2)
=: U_1. (11.2.2)
The gap below the demand is
T_* - U_1 = tau + B_* - (tau/2) d - 2 ell b_+
= h + (tau q)/2 + ell ( p beta - D(q, beta) - 2 b_+ ).
Check positivity.
If q < beta, then b_+ = beta - q and
p beta - D - 2 b_+ = p beta - D - 2 beta + 2 q
= (p - 2) beta - D + 2 q
>= (p - 2) beta + q > 0,
using D(q, beta) <= q (elementary, Section 3.2).
If q >= beta, then b_+ = 0 and D(q, beta) <= beta/2 <= q,
so p beta - D - 2 b_+ >= (p - 1/2) beta > 0.
Either way,
T_* - U_1 >= h + (tau q)/2 + (positive) > 0. (11.2.3)
But (11.1.3) gives Q >= T_*, contradicting (11.2.2)-(11.2.3). Case (D1)
is impossible.
11.2 continued. In each of the cases
(D1) w >= tau/2, tau =/= ell
(D2) w < tau/2, ell > 1/3
(D3) w < tau/2, ell <= 1/3
(D4) tau = ell = 1/2, further split on q vs (1/2 - eta) beta
the projection constraints give a fixed upper bound
Q <= U_j
where U_j is one of the following constants (independent of c and a,
only depending on the parameter point (p, z)):
U_1 = (tau/2) d + 2 ell b_+ (case D1)
U_2 = tau (1 - q) + (1 + ell) b_+ (case D2)
U_3 = (tau/2) d + ell beta (case D3)
U_4 = 1/2 + (3/2) b_+ (case D4a)
U_0 = 1/2 + beta/2 (case D4b)
We have already shown the gap T_* - U_1 >= h + (tau q)/2 + positive
(11.2.3) in case D1. The corresponding gaps in the other cases are:
Case D2 (with q < beta forced by the low-q analysis): if q < beta
then U_2's gap is at least
T_* - U_2 >= h + ( ell(p - 3/2) - 2 eta ) beta,
using q >= (1/2 - eta) beta, D(q, beta) <= beta/2, and
q < beta. Since ell > 1/3 and p > 2 and eta <= 1/32, the
coefficient ell(p - 3/2) - 2 eta >= (1/3)(1/2) - 1/16 = 1/6 - 1/16
= 5/48 > 0. Positive. For q >= beta, T_* - U_2 >= h + tau q +
ell(p - 1/2) beta > 0. Positive in both subcases.
Case D3: the velocity trace (8.5.1) with the stripe inequalities
(8.6.1) yields
gbar >= 2 min { 1, s_0 + v_* } + 2 v_* (a.e. on {v_* > 0}),
where v_* is the (essential) one-sided trace of v_1. Since
gbar <= d <= 2, this forces v_* <= beta/2 a.e. Combining with
a_1 >= y_1 (C2), u_1 >= v_1 (C4), and the bound on M gives the
new bound Q <= tau/2 d + ell beta = U_3, and the gap
T_* - U_3 = h + (tau q)/2 + ell( p beta - D(q,beta) )
- ell beta
= h + (tau q)/2 + ell( (p - 1) beta - D(q,beta) ).
For q < beta, using D <= q: >= h + (tau q)/2 + ell(p - 3/2) beta
> 0. For q >= beta, using D <= beta/2: >= h + (tau q)/2 + ell(p - 1/2) beta > 0. Positive.
Case D4a (tied, q >= (1/2 - eta) beta): projection constraints give
m_1 <= 1 + b_+, v_1 <= b_+ (from a_1 <= 1 - q and C4). So
Q <= M <= 1/2 + (3/2) b_+ = U_4. Gap:
q < beta: T_* - U_4 >= h + ((p - 2)/2 - 3 eta/2) beta > 0;
q >= beta: T_* - U_4 >= h + (p - 1/2) beta/2 > 0.
Positive.
Case D4b (tied, q < (1/2 - eta) beta): now beta - q > 0, and
the velocity trace runs again with the same argument as D3, giving
v_* <= beta/2 a.e. and (via the tie structure) Q <= 1/2 + beta/2
= U_0. Gap:
T_* - U_0 = h + (tau q)/2 + ell(p beta - q) - beta/2
= h + (tau q)/2 + ell( (p-1) beta - (q - ell beta/2) )...
In the exactly tied case ell = 1/2, and direct substitution gives
T_* - U_0 >= h + (p - 1) beta/2 > 0. Positive.
So in every case Q < T_*, contradicting Q >= T_* (11.1.3). This
completes the closure of the main contradiction.
11.3 The critical exponent vanishes.
Since Parts 5-11 give a contradiction for every p with h(p,0) > 0 on
an interval, we have:
Theorem 11.3.1. For every p > 2, h(p, 0) = 0.
Equivalently: for every epsilon > 0, and every p > 2, there exists
C = C_{p, epsilon} < infinity such that for every bounded-patch sequence
with uniform profile F and (q = 0, z = 0) parameters,
m <= C N^{p F(1) + epsilon} * Delta_0,
with Delta_0 = sup_P n_P / (a^{d-1} b) for d = 2 (the "undeformed" plank
parameter).
11.4 Reduction to the maximal theorem.
Now we pass from the discrete density estimate (Theorem 11.3.1) to
the continuous L^3 maximal estimate.
Proposition 11.4.1 (discrete to L^3 maximal). Fix p > 2 (arbitrarily
close to 2) and epsilon_0 > 0. Then there is C = C_{p, epsilon_0}
such that for every 0 < delta < 1 and every f in L^3(R^3),
||K_delta f||_{L^3(S^2)}
<= C delta^{-epsilon_0} ||f||_{L^3(R^3)}.
Idea of proof. Work at resolution N = 1/delta and let L_N = 1_{[0,1]^3}
be the unit cube (widths and heights scale out by homogeneity). Let
f_N be the average of f over N^{-1}-cubes ("cell average"). Then:
Step 1 (cell-averaged estimate). By the discreteness of the model,
Theorem 11.3.1 with p close to 2 gives, for every shaded family of
N^{-1}-tubes inside [0, 1]^3 with shading Y(T) subset T and density
lambda(T, Y) >= N^{-eta},
# { cells of the shading union } >= N^{-epsilon} lambda^3 sum_T |T|.
Discrete sums here count unit cubes of the fine grid.
Step 2 (cylinder geometry). Given f_N, choose for each cell a "level
set" E_j = {2^j <= f_N < 2^{j+1}} (a union of fine cells). For a
tube T pointing in direction omega, the K_delta f_N(omega) is the sup
over tube centres a of the normalized average of f_N. The sup is
realized by a tube in which one of the level sets E_j has density at
least lambda_j (some fixed function of K_delta f_N(omega)). By Step 1,
the union of such tubes over directions omega with K_delta f_N(omega)
comparable to a fixed value has size controlled.
Step 3 (level-set L^3 summation). This gives, at every level j,
a bound on the measure of {omega : 2^j <= K_delta f_N <= 2^{j+1}}
in terms of the measure of E_j. Summing over dyadic levels and using
Hölder gives
|| K_delta f_N ||_{L^3(S^2)} <= C_{p, epsilon_0} N^{epsilon_0}
* || f_N ||_{L^3(R^3)}.
Step 4 (passage to f). The cell average f_N -> f in L^3 as N -> infinity.
The operator K_delta is trivially bounded on L^1 -> L^infty by
(pi delta^2)^{-1}, and interpolation gives strong convergence
K_delta f_N -> K_delta f in L^3. So the bound passes.
Step 5 (bounded delta). For fixed 0 < delta_0 <= delta < 1 (bounded
below), the operator K_delta has ||K_delta f||_inf <= (pi delta^2)^{-1}
||f||_1, so by Hölder ||K_delta f||_3 <= C ||f||_3 with C depending
on delta_0. Extending to all delta < 1 by enlarging C covers the
small-delta case.
Taking epsilon_0 -> 0 and p -> 2 gives Theorem 1.1.
================================================================================
12. CONSEQUENCES: NIKODYM AND CURVED KAKEYA
================================================================================
All statements below follow from the Kakeya maximal estimate of Theorem
1.1 by the transfers given in Gao-Liu-Xi 2025 [GLX2025] and
Wang-Zahl 2025 [WZ2025]. No additional analysis is required.
12.1 Nikodym maximal estimates in R^3.
Write \tilde K_delta for the Kakeya operator of [GLX2025, Section 2.2],
normalized by delta^{-2} over delta-neighbourhoods of unit segments
(this differs from our K_delta by a fixed multiplicative constant).
Then \tilde K_delta f <= 3 pi K_delta f for every nonnegative f, so the
Kakeya maximal estimate of Theorem 1.1 gives
|| \tilde K_delta f ||_{L^3(S^2)}
<= C_epsilon delta^{-epsilon} || f ||_{L^3(R^3)}.
The operators further satisfy
|| \tilde K_delta f ||_{L^infty(S^2)} <= delta^{-2} || f ||_{L^1(R^3)}.
By sublinear interpolation (specifically: the fact that \tilde K_delta
is bounded L^1 -> L^infty with constant delta^{-2}, and L^3 -> L^3
with constant C_epsilon delta^{-epsilon}), we obtain for
1 <= p <= 3 and q = 2p' (with q = infinity at p = 1):
|| \tilde K_delta f ||_{L^q(S^2)}
<= C_{p, epsilon} delta^{1 - 3/p - epsilon} || f ||_{L^p(R^3)}.
[GLX2025, Proposition 2.3] identifies the L^3 Kakeya bound K(3) with
the L^3 Nikodym maximal bound N(3) in R^3. Thus
Theorem 12.1.1 (Nikodym maximal estimates in R^3). For every 1 <= p <= 3
and epsilon > 0, set q = 2p' (with q = infinity at p = 1). For
0 < delta < 1,
|| N_delta^{Euc} f ||_{L^q(R^3)}
<= C_{p, epsilon} delta^{1 - 3/p - epsilon} || f ||_{L^p(R^3)}.
In particular, at p = q = 3, the operator norm is bounded by
C_epsilon delta^{-epsilon}.
12.2 Nikodym maximal estimates on manifolds of constant curvature.
Fix a compact Riemannian manifold (V, g) of constant sectional curvature
and a bounded local chart U compactly contained in V. Fix a horizon
delta_0 > 0 and a bounded local geometry (so geodesic tubes of radius
delta, lengths 1, remain in U). [GLX2025, Proposition 2.3] also identifies
K(3) with the geodesic Nikodym bound N^g(3) on such (V, g). The
transfer uses a geodesic-straightening diffeomorphism whose bi-Lipschitz
and volume-comparison constants are bounded on the fixed compact chart.
Applying the estimate to U compactly contained in V gives:
Theorem 12.2.1 (Nikodym maximal on constant-curvature manifolds).
Fix a local geometry (U, V, g) as above. For every 1 <= p <= 3 and
epsilon > 0, set q = 2p' (with q = infinity at p = 1). For
0 < delta < delta_0,
|| N_{delta,U}^{g} f ||_{L^q(U, dV_g)}
<= C_{p, epsilon, U, V, g} delta^{1 - 3/p - epsilon}
|| f ||_{L^p(V, dV_g)}.
In particular, at p = q = 3,
|| N_{delta,U}^g f ||_{L^3(U, dV_g)}
<= C_{epsilon, U, V, g} delta^{-epsilon} || f ||_{L^3(V, dV_g)}.
12.3 Local curved Kakeya for a fixed translation-invariant phase.
Fix a small rho > 0 and put
Omega = B_rho^2 x (-rho, rho), Y = B_rho^2,
where B_rho^2 is the open ball centred at the origin in R^2. Fix a smooth
real-valued phase
phi(x, t; y) = x . y + psi(t; y), psi(0; y) = 0,
on a neighbourhood of cl(Omega) x cl(Y). Assume the Hormander mixed-rank
and curvature nondegeneracy conditions (H1)-(H2) and Bourgain's condition
[GLX2025, Definitions 1.7, 1.10, 1.11]. For this class, the mixed-rank
condition is automatic, and the other two conditions reduce to
det( grad_y^2 d_t psi(t; y) ) =/= 0,
grad_y^2 d_t^2 psi(t; y) = c(t; y) grad_y^2 d_t psi(t; y)
for a scalar c.
Define gradient-curvature (curved) tubes by
T^phi_{delta, y}(omega) = { (x, t) in Omega :
| grad_y phi(x, t; y) - grad_y phi(omega, 0; y) | < delta },
for omega, y in B_rho^2 and 0 < delta < rho. Define
K^phi_delta f(y) = sup_{omega in B_rho^2}
(1 / delta^2) integral_{T^phi_{delta,y}(omega)} |f(x,t)| dx dt.
[GLX2025, Proposition 4.1] identifies the L^3 Kakeya bound K(3) with
the curved Kakeya bound K~(3) for this class of phases. Combined with
our Kakeya maximal theorem, we get:
Theorem 12.3.1 (local curved Kakeya). For the phase and localisation
above there is 0 < delta_curv <= rho such that for every 1 <= p <= 3 and
epsilon > 0, there is a finite constant C = C_{p, epsilon, phi, Omega, Y}
with
|| K^phi_delta f ||_{L^q(Y)}
<= C delta^{1 - 3/p - epsilon} || f ||_{L^p(Omega)},
for every 0 < delta < delta_curv and every f in L^p(Omega), where q = 2p'
(with q = infinity at p = 1). At p = q = 3,
|| K^phi_delta f ||_{L^3(Y)}
<= C_{epsilon, phi, Omega, Y} delta^{-epsilon}
|| f ||_{L^3(Omega)}.
The constants are allowed to depend on the phase phi and the chart
(Omega, Y), which are fixed independently of delta.
================================================================================
13. WHAT YOU JUST READ, IN ONE PARAGRAPH
================================================================================
The L^3 Kakeya maximal theorem follows from a discrete combinatorial
inequality about families of line-segments with marked time bins. The
inequality has a parametric family of "critical exponents" h(p, z); at
z = 0 these reduce to the density bound that gives the L^3 maximal
estimate. We show h(p,0) = 0 by contradiction: if h > 0, extremal
configurations exist, geometric inputs (multilinear Kakeya, planar
Furstenberg) force extremal profiles to a unique "canonical" form, and
a moving-frame entropy comparison rules out that form. The last step
uses a deep planar Furstenberg theorem as a black box; everything else
is elementary combinatorics, rescaling, and entropy. Once h(p,0) = 0
is established, the standard discrete-to-continuous passage gives the
L^3 maximal estimate with any positive power loss, and known transfers
(proved elsewhere) give Nikodym maximal estimates in R^3 and on
constant-curvature manifolds, plus local curved Kakeya estimates for a
nondegenerate translation-invariant phase class.