BOCHNER-RIESZ IN R^3 -- A GUIDED EXPOSITION
STATUS NOTE.
This file is a rewrite-in-progress of a research paper that claims to
prove the Bochner-Riesz conjecture in three dimensions. Such a claim is
extraordinary and deserves scrutiny. This file is honest exposition:
what the problem is, what is being claimed, why it is hard, and the
structure of the argument. Only the overview is written so far.
Technical sections will follow, one per turn. This file is not yet "the
full proof" and calling it that would be dishonest.
NOTATION.
We write |xi| for the length of the vector xi in R^3, x . xi for the
dot product, L^p for the Lebesgue norm (||f||_{L^p} = (int |f|^p)^{1/p}),
and f_hat for the Fourier transform. Larger p makes ||f||_{L^p} turn
"spikier" -- good for functions with tall peaks, bad for spread-out
data. The exponent p = 3 is our home in this whole document.
PART 1. WHAT THE PROBLEM IS.
Start with the Fourier transform in R^3:
f_hat(xi) = integral over R^3 of f(x) exp(-2 pi i x . xi) dx.
For nice functions this inverts:
f(x) = integral over R^3 of f_hat(xi) exp(2 pi i x . xi) dxi.
Fourier's insight: any (reasonable) f can be built from pure waves
exp(2 pi i x . xi). This lets us do calculus by working in frequency
space instead of position space -- which is how heat, wave, and
Schrodinger equations are actually solved.
But when f is only in L^p for p not 2, the inversion integral can fail
to converge pointwise: too many frequencies fighting at small scales.
The classical fix is SUMMATION: do not sum all frequencies at once;
sum only frequencies inside a ball of radius R, then let R tend to
infinity. Does this recover f? In what norm?
The ball-sum operator at radius R is
S_R f(x) = integral over |xi| < R of f_hat(xi) exp(2 pi i x . xi) dxi.
A slicker smoothed version keeps delta > 0 fixed and takes R = 1:
T_delta f(x)
= integral over |xi|<1 of (1 - |xi|^2)^delta
f_hat(xi) exp(2 pi i x . xi) dxi.
Larger delta means the cutoff at |xi|=1 is smoother. The case delta=0
is called the SHARP BALL MULTIPLIER.
THEOREM (claimed, the main result).
For every delta > 0 there is a finite constant C = C(delta) such that
|| T_delta f ||_{L^3(R^3)} <= C || f ||_{L^3(R^3)}
for all f in the Schwartz class. Hence T_delta extends uniquely to a
bounded linear operator on L^3(R^3).
Why "conjecture"? Because classical theory says:
- at delta = 0, the ball multiplier is bounded on L^p only for p=2.
Fefferman (1971) showed by a Kakeya construction that in every
dimension n>=2, the delta=0 ball multiplier is unbounded on every
L^p with p not 2.
- for delta > delta_c(p) := max{n |1/p - 1/2| - 1/2, 0}, the
multiplier IS bounded on L^p (classical range).
- the conjecture: the classical range is sharp, i.e. boundedness
holds for all delta > delta_c(p) and fails at delta = delta_c(p).
In n = 3 the critical exponent at p = 3 is
delta_c(3) = 3 |1/3 - 1/2| - 1/2 = 0.
So at L^3, ANY positive smoothing should suffice. That is exactly what
the claimed theorem says.
PART 2. WHAT PROVING IT WOULD IMPLY.
Three corollaries:
1. FULL STRICT RANGE. Interpolation + duality turn the L^3 bound at
every fixed positive delta into
|| T_delta ||_{L^p -> L^p} < infinity
for every 1 <= p <= infinity with
delta > max{ 3 |1/p - 1/2| - 1/2, 0 }.
This is the full strict Bochner-Riesz range in three dimensions.
2. SPHERICAL RESTRICTION (Tao 1999). Bochner-Riesz implies the
restriction conjecture for S^2. Let
Eg(x) = integral over S^2 of g(omega) exp(2 pi i x . omega)
dsigma(omega), sigma = surface measure on S^2.
Then || Eg ||_{L^p(R^3)} <= C_p || g ||_{L^inf(S^2)} for every
real p > 3.
3. ALMOST-EVERYWHERE SUMMATION. For 3 <= p < infinity and every
delta > 1 - 3/p, the ball sums S_R f converge almost everywhere
to f.
PART 3. WHY THIS IS HARD.
At p = 2 the multiplier is trivially L^2-bounded by Plancherel.
At p = 3 we are strictly off L^2 (p = 2 * 3/2). The geometry of
frequency space -- specifically the curvature of the sphere |xi|=1
and the singular derivative of the symbol across it -- now matters
in full. Fefferman's 1971 counterexample is the warning light: it
arranges many oscillating wave packets whose supports lie in a thin
Kakeya set (a set containing a line segment of every direction), so
that the sum is L^p-large but L^2-small.
To prove anything at p=3, the argument must track *how* waves are
arranged in space, not just how large their frequencies are. This
forces the proof into structural (geometric + combinatorial) tools
that have nothing to do with classical Fourier analysis.
PART 4. THE OVERALL STRATEGY.
The proof (as claimed) has two big stages tied together at the end.
STAGE A -- POSITIVE MASSES ON LINES.
Imagine the smoothing's "particles" travelling along straight lines
in velocity space. Each observation reveals some information about
the initial configuration. Measure this by ENTROPY (log of number
of possibilities). Show that the growth rate of entropy per unit
time is bounded by a fixed constant beta depending only on the
smoothing. This reduces the entire problem to a combinatorial
statement about how much entropy can travel on a family of lines
in a bounded region.
STAGE B -- WAVE PACKETS.
Translate the entropy bound into a bound on an actual wave-packet
score, using Fourier decomposition at each scale. The oscillation
means positivity is lost: we must now handle interference. There
is a boundary where a packet's position uncertainty is the same
size as its length scale -- the uncertainty boundary. At that
boundary packets cannot be localized individually, so we pay a fine
velocity cost at every backward step. That cost is bounded; after
finitely many steps the growth must stop. This gives a very small
upper bound on the packet growth rate (half of Stage A's bound).
FINAL -- THE MULTIPLIER.
The packet growth bound, through a pseudoconformal change of
variables (as in Guo-Oh-Wang-Wu-Zhang 2025), yields an L^3 bound
for the oscillatory integrals appearing in the Bochner-Riesz kernel
on one dyadic shell of radius lambda. Summing over shells
lambda = 2^j gives loss lambda^nu for any nu > 0. Choosing nu <
delta closes the argument and yields the theorem.
PART 5. ROADMAP OF THE 12 SECTIONS.
1. Introduction. Statement of the theorem, historical context.
Nothing to prove; skim for orientation.
2. Entropy and conditioning. Finite laws, chain rules, restriction
inequality, conditioning on typical cells, zero-information /
product domination. This is the toolbox used everywhere later.
3. Projection. The one geometric input: Ren-Wang's planar
Furstenberg estimate in a weighted finite form. All "geometric
positivity" comes from here.
4. Positive transport (classical). Defines the score for positive
masses on lines and shows the growth rate is at most gamma. The
proof splits across 4-8.
5. Isotropic exclusion. If the least-aspect extremizer is
isotropic, three widely separated slopes force more point
entropy than the time capacity allows. Contradiction.
6. Characteristic plane. If the extremizer is anisotropic, the
matched rate plane carries mixed derivative information that
restricts entropy crossing a contact curve. Needs a trace
theorem on a codimension-1 plane through the boundary of a
Lipschitz domain.
7. Rate transfer. Projections transfer abstract lower bounds into
concrete rate constraints on entropy growth.
8. Closure of Stage A. Rate constraints + contact identity + jump
conditions contradict excess growth. Stage A done.
9. Wave packets: score, phase class, finite decomposition with
quantified tails, cap classes.
10. Packet composition. The composition inequality; two
alternatives at the uncertainty boundary; exclusion of fixed
slack.
11. Packet repayment. Backward induction on fine velocity
information. Stage B done.
12. Multiplier. Packet growth -> oscillatory integrals -> shell
bound -> dyadic sum. Theorem proven.
END OF OVERVIEW.
PART 6. THE ENTROPY TOOLBOX (Section 2, in plain language).
Every argument below measures "how many possibilities are left" via
entropy. Build that vocabulary first.
FIX A SMALL PARAMETER eps > 0. Every logarithm is base 1/eps. So
"H = 0.4" means "the effective number of possibilities is at most
(1/eps)^{0.4}", and any variable with at most eps^{-d} values has
H <= d.
ENTROPY of a finite random variable X:
H(X) = - sum_x P(X=x) log_{1/eps} P(X=x)
= E[ - log_{1/eps} P(X=x) ].
CONDITIONAL ENTROPY: H(Y|X) = H(X,Y) - H(X).
MUTUAL INFORMATION: I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X) >= 0.
CHAIN RULE: H(X_1,...,X_n | Z) = sum_k H(X_k | X_1,...,X_{k-1}, Z).
THREE WORKHORSE FACTS.
FACT 1 (AMBIGUITY IS CHEAP). If B takes at most N values given A,
outside a joint event of probability zeta, then
H(B | A) <= log 2 + log N + zeta * log |B|.
Reading: two cases (typical + complement), a size bound on the
typical set, and the exception scaled by total support size. When
|B| <= eps^{-C} and zeta = eps^sigma, the last term vanishes as
eps -> 0.
FACT 2 (MASS BOUNDS ENTROPY). For nonnegative m_b,
H(B) + E[ log_{1/eps} m_B ] <= log_{1/eps} ( sum_b m_b ).
Proof: m_b / M is a probability; its entropy-from-B is nonnegative.
FACT 3 (CHANGING BASE LAW). If Q <= K P (K >= 1) and B, D are any
observations, then
H^Q(D | B) = E_Q[ - log P(D | B) ]
- E_{Q_B}[ KL(Q_{D|B} || P_{D|B}) ],
and the KL term is at most log K.
In particular, restricting P to an event of mass m costs log(1/m).
TYPICAL CELLS. Let P be a base law, E an event of mass m, U=(.|E).
Fix a finite observation C. Then for 0 < eta < 1, outside U_C-mass
eta,
U(. | C=c) <= eta^{-1} P(. | C=c).
So discarding a small mass of cells lets us pretend E-restriction is
"free up to log(1/eta)"; and the cost is independent of how many
cells c there are.
HOMOGENIZATION. Suppose q observations (X_j, Y_j) with
|Y_j| <= eps^{-A_j} and partition by the "surprise" of Y_j given X_j
into windows of width a. If we allow the number of windows to be at
most
J = prod_j ( 1 + ceil((A_j + sigma) / a ) ),
discarding a total mass at most q eps^sigma + theta, then on each
window z,
d_{j,z} <= - log P(Y_j | X_j) <= d_{j,z} + a.
Allow sigma, a, theta to depend on eps: total loss vanishes,
number of windows stays subpolynomial.
Why we need this: entropy arguments work "up to o(1)", but the
iteration in later sections needs exact finite integers. Homogenization
is what converts o(1)-entropy statements into exact-integer bin
statements at the cost of o(1) mass.
SWEET PRODUCT INEQUALITY. If a probability law rho on a finite set
satisfies rho <= K * product_i mu_i, then
KL( rho || product_i rho_i ) <= log K.
Proof: expand KL of rho from the product of its own marginals, and
note that the extra term against product_i mu_i is >= 0. So "close to
a product comparison" implies "close to actually being a product".
SIMULTANEOUS MARGINAL CORE. Given a joint law nu and several product
comparisons nu_{F_l1,...,F_lk_l} <= A_l * prod_j mu_{lj}, we can
restrict nu to a sub-law nu' of mass M >= m - q eta (q = total number
of factors, eta > 0 small) such that EVERY comparison holds "up to
its own hidden factors" but with mu_lj replaced by the actual
marginals rho_{F_lj} of nu'. Concretely: on every retained fiber,
rho_{F_l1,...,F_lk_l} <= A_l * M^{k_l-1} eta^{-k_l}
* prod_j rho_{F_lj}.
Support counts persist, and by taking eta = m/(2q), retention is at
least m/2.
The heuristic: comparisons to fixed product laws become comparisons
to the law's actual marginals, at the cost of a bounded loss.
This is the "core" step that lets us KEEP the entropy of the local
marginals while still asserting a product structure.
WHAT THIS BUYS. Sections 3-8 (Stage A) and 9-11 (Stage B) are all
entropy-bounded arguments: they claim growth rates on how much
information travels per unit time, and they chain those bounds across
scales. Without the restriction inequality (Fact 1/2) we cannot bound
mass by entropy; without the base-change lemma (Fact 3) we cannot
restrict to typical cells; without homogenization we cannot turn
o(1)-growth statements into finite-integer contradictions.
END PART 6.
PART 7. THE ONE GEOMETRIC INPUT (Section 3, in plain language).
Stage A below proves a purely combinatorial statement about "how much
information has to travel on lines". The ONLY outside math used is
Ren-Wang 2025, "Furstenberg sets estimate in the plane". Everything
else is self-contained.
FURSTENBERG SET, INFORMALLY. A subset F of R^2 such that, in every
direction, F contains a quantitatively long line segment, is called
a FURSTENBERG SET. The Furstenberg conjecture predicts a lower bound
on its dimension; Ren-Wang 2025 proves the planar bound.
DISCRETE FORM (what we actually use).
- Fix a bounded slope chart (angle in [0, pi/2) say).
- rho-grid of points (rho > 0 small). rho-height "tubes" in each
slope.
- P = a set of point cells, T = a set of tube cells.
- G subset P x T = a bipartite graph of incidences.
MASS BOUNDS.
P is a (rho, d)-set: any ball of radius r in [rho,1] contains at
most rho^{-eta} r^d * |P| cells of P.
T is a (rho, s)-set in the "slope-intercept" chart: any slope
interval of length r contains at most rho^{-eta} r^s * |T| tubes.
GRAPH WEIGHT: |G| >= rho^eta * |P| * |T|.
PROJECTION: for a tube t, project each incident point p onto the
axis perpendicular to the tube's direction ("the intercept"). Call
the projected set pi_t(G_t) where G_t = {p : (p,t) in G}.
COVERING NUMBER: require pi_t(G_t) to be coverable by at most M
intervals of length rho in the intercept coordinate.
REN-WANG (the finite weighted form we use).
There exist eta = eta(alpha, s, d) > 0 and rho_0 = rho_0(alpha, s, d) > 0
such that for 0 < rho < rho_0, the covering number M obeys
M >= rho^{-min(1, d, (d+s)/2) + alpha}.
Here alpha > 0 is any desired "slack" and eta, rho_0 shrink with
alpha. Enlargements by fixed constants are absorbed.
INTERPRETATION. The projected set is one-dimensional, so it CAN'T be
arbitrarily short: mass bounds on P and T prevent both point pileup
and slope pileup, so the intercept coordinate must spread out to size
at least rho^{-chi} where chi = min(1, d, (d+s)/2).
Where chi comes from:
d -- if d < 1, the point mass alone already caps the cover;
(d+s)/2 -- the "Furstenberg" bound from coupling slope entropy with
point entropy: it takes the average;
1 -- the 1-D trivial bound; if d >= 1 or (d+s)/2 >= 1, the
general bound cannot beat pi^1.
WEIGHTED / TAGGED FORM. Replace sets by probability measures:
mu on points, nu on "tags" (tag = slope + possible extra labels).
Same basic mass bounds on ball-intervals, but as integrals:
mu(B(z,r)) <= rho^{-eta} r^d,
(a_* nu)(I) <= rho^{-eta} r^s (a = slope map, a_* = pushforward).
Graph G of point-tag incidences with (mu x nu)(G) >= rho^eta.
Same M covering bound on projected neighbors of each tag.
Conclusion identical: M >= rho^{-chi + alpha}.
TAG POINT: since slope a(b) may repeat across tags, the incidence
graph treats distinct tags as distinct even when slopes coincide.
This is important for Stage A: "different observations of the same
slope" carry different auxiliary information, and Ren-Wang must see
them as distinct corners of G, not merged.
ITS USE. In Stage A (Sections 4-8), we will measure "how much slope
information is forced by the incidence structure". In rate terms,
M ~ eps^{-l}, |P| ~ eps^{-d}, |T| ~ eps^{-s}, and Ren-Wang says
l >= chi = min(1, d, (d+s)/2).
This becomes the key inequality that ties slope rate to point rate,
which then feeds into Section 8's contradiction.
INCIDENCE-LAW PASSAGE (Remark 3.5 in the original). In entropy
work, the graph is never literally supplied by mu x nu; it is
supplied by a restricted subprobability sigma, dominated by mu x nu.
If sigma(G) >= m and sigma <= K mu x nu on G, then
(mu x nu)(G) >= m/K.
So subpower K and subpower m give the graph hypothesis of the finite
Ren-Wang directly. That is the standard way Ren-Wang enters every
entropy argument in the sequel.
BALL-MASS FROM ENTROPY. To get "mu(B(z,r)) <= rho^{-eta} r^d"
from an entropy bound on a variable that samples points, we use a
standard dichotomy: homogenize the entropy, discard tiny-mass cells,
and note that all remaining cells have mass <= eps^{(1-eta)*h}
where h is the entropy. The conditioning and homogenization lemmas
of Part 6 supply the homogenization. What Section 3 makes explicit
is that Stage A must do this conditioning IN THE RIGHT ORDER: first
build the comparison probability law, then take the mass bounds from
that law. A bare entropy inequality (say "H(point) >= k") is NOT a
substitute for the mass bounds.
END PART 7.
PART 8. GROWTH FROM POSITIVE MASSES ON LINES (Sections 4-8, in plain language).
We now set up Stage A. The goal is beta_c <= gamma, where beta_c is
the "classical growth rate" defined below and 0 < gamma < 1 is a
fixed trade-off parameter. Throughout this Part, gamma is fixed and
0 <= b <= a are depths.
SETUP.
Particle T: a random variable carrying two bounded vectors
V(T), A(T) in R^2, interpreted as velocity and initial position.
Trajectory: X(c) = A(T) + c * V(T), c in [0, L].
Positive weights: for 0 <= b <= a, define
w(b, a) = (1+gamma) b + (1-gamma) a.
Small b (fine velocity resolution) and small a (small aspect
eccentricity) both reduce w. gamma sets the exchange rate.
Aspect := a - b >= 0.
APERTURE OBSERVATION at duration t:
records the tuple
( t, b, a, C_t, R, [V]_{b,a;R}, [X(C_t)]_{b+t,a+t;R} )
where:
- C_t is a dyadic time cell of length ~ eps^t (the "when").
- R is an orthonormal frame (orientation).
- [V]_{b,a;R} is the velocity-grid cell of side lengths
(eps^b, eps^a) in the frame R.
- [X(C_t)]_{b+t,a+t;R} is the position-grid cell at time C_t
of side lengths (eps^{b+t}, eps^{a+t}) in the same frame R.
Weight: w(G) = w(b, a).
ROOT SUPREMUM.
For fixed particle T and horizon L, define
tau_L(T) = sup over duration-ZERO apertures D of
( E[w(D)] - I(T; D) ).
Reading: "the best single-shot observation at time 0 without
knowing anything about motion yet". It's the reference level that
growth is measured against.
SCORE.
For a probability law on observations G and their apertures,
F_t(G) = E[w(G)] - I(T; G) + 2 H(C_t | T).
Three contributions:
+ E[w(G)] reward for fine / non-eccentric observations.
- I(T; G) penalty: information about the random particle T
that G reveals. Finer observations reveal more
about T, hence cost more.
+ 2 H(C_t|T) reward for the "when" randomness: how unpredictable
is the observation's time given its particle. Uniform
time is best; time expected ~ 2t if time is uniform.
GROWTH RATE.
beta_c = sup over all systems with fixed bounds of
limsup_{eps -> 0} (F_t(G) - tau_L(T)) / t.
"Fixed bounds" = fixed domain, fixed horizon, t confined to a fixed
positive interval. The supremum ranges over systems; each system is
an eps-indexed sequence.
PROVISIONAL BOUNDS.
0 <= beta_c <= 2.
- Lower: the trivial (large-depth, non-informative) observation
satisfies F_t ~ tau_L up to o(1). So beta_c >= 0.
- Upper: F_t - tau_L <= E[w(G)] - I(T;G) + 2 H(C_t|T) - tau_L
<= 2 H(C_t|T) + o(1) <= 2t + o(1).
The last inequality uses that H(C_t | T) <= t + o(1): the time
cell of depth t is one of at most eps^{-t} cells.
These are only bootstrap bounds. The real content is beta_c <= gamma.
THEOREM (4.1/6, "classical growth").
beta_c <= gamma.
PROOF SKETCH, SPLIT INTO STEPS A-F. (Sections 4-8 of the original.)
Step A: Localize and normalize.
Assume beta_c > gamma. Take a sequence with rate within 1/j of
beta_c. Approximate T by a finite random variable: for each full
menu of tests, assign T a finite combinatorial "signature"; never
loses information. Split by narrow shape bins (b(1), a(1)) with
o(1) label entropy. Remove common depth b(1) by localizing to an
earlier sub-trajectory. Normalize time so duration = 1.
Step B: Least aspect.
Define m := inf { a(1) - b(1) : saturated sequence at duration 1 }.
Proposition 4.10 (in the original).
There exists a saturated sequence at duration 1 with
b(1) = 0, a(1) = m, horizon 1 + m,
all listed observations homogeneous, and no entropy of T harmed.
Proof uses:
- Localization (Lemma 4.4): a duration-t observation splits into
a duration-r sub-observation plus a duration-(t-r) residual,
and if the whole is saturated, typical residuals are saturated.
- Rectangle comparison (Lemma 4.6): two rectangles at nearby
angles with the same aspect have a common coarsening and an
intersection. Combining them changes total weight by
2 gamma (e - v) where e = aspect and v = separation depth.
Consequence: at fixed aspect e, either the two orientations
are aligned to depth e - v, or an entropy-saving coarsening
breaks saturatedness.
- Sampling two independent aperture replicas given T and applying
the above forces m to be attained.
Step C: Saturated path.
Proposition 4.11.
The least-aspect sequence can carry saturated observations at
EVERY dyadic time u in (0,1], with consistent shapes:
b(1) = 0, a(1) = m,
b(r) >= b(t) (b nonincreasing in time),
a(r) + r <= a(t) + t (a(t)+t nondecreasing),
e(u) := a(u) - b(u) >= m u,
and with coherence: two observations at times r < t have
orientations that agree at any depth
d < min(e(r), e(t)) - K_gamma * (|b(r)-b(t)| + |a(r)-a(t)| + t-r)
with probability 1 - o(1).
Proof: build backward from time 1 by Step B composition, then split
small shape bins at each dyadic step to preserve saturation.
Step D: Tangent reduction.
Proposition 4.12. Two alternatives.
ISO (m = 0):
there is a saturated sequence at duration 1 with isotropic
endpoint (b(1), a(1)) = (0, 0), and the time profile is LINEAR:
H(C_u | T) = s u + o(1), 0 <= s <= 1.
ANISO (m > 0):
the path can be taken to have a single orientation R with
m-u prefixes, with shapes
e(u) = m u for all u,
b(u) Lipschitz, -b'(u) in [0, 1+m] almost everywhere,
H(R_{m(u+h)} | T, C_u) >= kappa h for some kappa > 0
and every 0 < u < u+h < 1.
Proof outline:
- Take a differentiability point x of b (and a, S).
- Chain coherence along a fine dyadic net to effectively fix the
orientation.
- Any local sub-system at such a point has aspect <= max(e'(x),0).
Least aspect gives m <= max(e'(x), 0) almost everywhere.
- Since e(u) >= m u and e(1) = m, forcing m <= e'(x) forces
e(u) = m u exactly (in the anisotropic case).
- For the ISO case, a regular point with e' <= 0 must exist
(otherwise e(1) > 0 while m = 0). Applying the same
localization gives a saturated isotropic sub-system.
- The positive orientation rate in ANISO comes from localizing a
saturated observation at u+h down to u with d = e(u+h):
an isotropic sub-system of horizon h would contradict m > 0
unless the local orientation entropy H(R | T, C_u) grows at
positive rate.
Step E: Isotropic exclusion (Section 5).
Assume beta_c > gamma and the isotropic alternative; derive
contradiction.
Idea: Ren-Wang forces too much entropy in the slope population,
contradicting the time capacity s <= 1.
Details (rough):
- Let s be the linear time-entropy rate.
- Pick a regular depth k where time entropy grows slowly.
- Form three independent slope replicas from the same particle at
depth k. Either they concentrate in a thin strip, or (via
Ren-Wang) the projected neighbors violate the entropy budget.
- Concentration gives an EARLIER aperture, at time k, with weight
larger by (1-gamma) * l where l = theta*h.
- The information saved by the earlier aperture is at least s l.
- The score decomposition gives (1 - gamma - s) l >= beta l - o(l),
i.e. s >= 1 + beta - gamma > 1, contradicting s <= 1.
- Hence ISO cannot be saturated when beta_c > gamma.
Step F: Anisotropic case (Sections 6-8).
Set up a "matched plane" coordinate system where all four data
rates and the orientation rate move at fixed slopes. Section 6
shows the resulting profile is well-defined using a codimension-1
trace theorem; Section 7 transfers Ren-Wang to rate constraints;
Section 8 closes via a contact identity and jump conditions.
NEXT PART: isotropic exclusion in more detail, then the anisotropic case.
PART 9. WHY THE ISOTROPIC CASE FALLS (Section 5, in plain language).
RECALL. From Part 8, Step D, the ISO alternative asserts:
- the least-aspect saturated sequence has b(1) = a(1) = 0;
- the time profile is LINEAR: H(C_u | T) = s u + o(1) for 0 <= s <= 1.
THEOREM (5.12, isotropic exclusion). If beta > gamma > 0 then the ISO
alternative is impossible.
PROOF SKELETON.
STEP 1. VELOCITY IS CONCENTRATED, FORCED BY ROOT SATURATION.
Put O = C_1 (full-depth observation of time-and-position), and
write V_d for the velocity at depth d (d small, positive).
Standard homogenization of the joint system (O, V_d, T) gives, after
inner limits, for small eta > 0:
P(V_d = q | O = o) <= eps^{alpha d - eta} (5.3)
where alpha >= beta > 0 (the root supremum is at most tau_L, so a
duration-0 test that resolves velocity finer than this would beat
the root).
Reading: velocity at depth d is concentrated at rate alpha per unit
depth, conditional on O. This is the "concentration source" we will
use downstream.
STEP 2. CONCENTRATION TRANSFERS TO THE LAST BLOCK.
Fix small h. Put
l = theta * h (last block length),
a = 1 - l (time cell depth),
k = l + u (velocity strip depth, u small).
Within each O-cell, velocity is confined to a strip of width eps^l.
In the last block (times [a, 1]):
For 0 <= u <= 1, let F_u record the transport trajectory in the
last block at depth a + ul, together with the tangent position at
that time.
The conditional slope law satisfies (Lemma 5.3 in the original):
P(X in J | B, F_u) <= rho^{alpha d - o(1)} (5.4)
for intervals J of length rho^d, where rho = eps^l.
Reading: whatever fine slope information O might carry, once we
condition on the coarse history B, the last block still shows
concentration at rate alpha per unit depth.
STEP 3. TWO-REPLICA SEPARATION.
Take two independent conditional slope replicas X_1, X_2, given
T and the coarse time C_a. Themselves independent of each other
given T.
From Step 2's bound at d = a_0 (for 0 < a_0 < 1):
Mass of intervals of length rho^{a_0} is <= rho^{alpha a_0}.
So the two replicas satisfy
|X_1 - X_2| >= rho^{a_0}
outside a set of probability O(rho^{alpha a_0 - o(1)}).
So a "good pair" (slopes separated) has probability -> 1 as rho -> 0,
because alpha > 0.
STEP 4. TWO-PROJECTION LEMMA.
For good pairs (two widely separated slopes), consider the two
intercept projections of the point set. Lemma 5.4 (Loomis-Whitney
on the intercept data) gives:
If the point set uses at most rho^{-p_1} cells and the two slope
intercepts are coverable by rho^{-s_l} intercept cells each, then
p_1 <= 2 s_l + o(1).
In other words: two separated slopes force the intercept cells to
spread, so the intercept budget can't be smaller than p_1 / 2.
Two separated slopes at rho^{a_0} apart contribute point exponent
2 s - 2 a_0 to the intercept count. So
p_1 >= 2 s - 2 a_0 - o(1),
and letting a_0 -> 0:
p_1 >= 2 s. (5.5)
That is: the point-cell exponent at the last block is at least
twice the time-entropy rate.
STEP 5. OUTPUT OF THE TWO-REPLICA ARGUMENT.
The bound (5.5) is basically "f'(u) >= 2 s" almost everywhere,
where f(u) is the joint entropy of (time, tangent position) at
level a + ul in the last block (see the original for the cellwise
derivation of f'). Integrating:
H(O | B) >= 2 s l - o(l). (5.6)
Also, the last block alone has
H(O | T, B) = s l - o(l). (5.7)
The first is used below; the second means "the last block has just
sl bits of TIME entropy given the particle and the earlier coarse
history B".
STEP 6. THE EARLIER APERTURE.
On the concentrated strip, define
B = ( O_k, O_a, [(A + C_a V) . n]_1 )
where n = n(O_k) is the coarse normal at depth k, and the last entry
is the normal component of the transported position, at fine depth
1 (position resolution eps^1 in the n-direction).
- Velocity depths (0, l); duration a; horizon 1; weight (1-gamma) l.
- Aspect approx eps^{1-a} = eps^l, nearly isotropic.
- H(B | O) = o(l): B is determined by O up to bounded ambiguity.
- H(O | T, B) = s l - o(l): "the last block has sl time bits".
- H(O | B) >= 2 s l - o(l): from (5.6).
STEP 7. INFORMATION SCORE GAP.
From H(B | O) = o(l) and the chain rule:
I(T; O) - I(T; B) = I(T; O | B) + o(l)
= H(O | B) - H(O | T, B) + o(l)
>= 2 s l - s l - o(l)
= s l - o(l). (5.8)
STEP 8. THE SCORE COMPUTATION.
Let Q_G := F_t(G) - tau_L(T) denote the undivided score numerator.
From saturation at horizon 1:
Q_O = beta + o(l).
From the universal bound at shorter duration a = 1 - l:
Q_B <= beta * (1 - l) + o(l).
Weight difference:
w(B) - w(O) = (1 - gamma) l + o(l).
Time entropy difference (2 * [H(C_a|T) - H(C_1|T)]):
- 2 s l + o(l).
Net:
Q_B - Q_O = (1-gamma) l + [I(T;O) - I(T;B)] - 2 s l + o(l)
>= (1-gamma) l + s l - 2 s l + o(l)
= (1 - gamma - s) l + o(l). (5.9)
On the other hand, using the saturation bounds:
Q_B - Q_O <= beta (1 - l) - beta + o(l) = -beta l + o(l). (5.10)
Combining (5.9) and (5.10):
1 - gamma - s <= -beta, i.e.
s >= 1 + beta - gamma. (5.11)
STEP 9. CAPACITY CONTRADICTION.
Capacity: H(C_u | T) <= u + o(1) for all u (a depth-u time cell has
at most eps^{-u} values).
So s <= 1.
But (5.11) gives s >= 1 + beta - gamma.
Since beta > gamma was assumed strictly, this gives s > 1, a
contradiction. So the ISO alternative is saturated-impossible.
CONCLUSION. The least-aspect m from Part 8, Step B is strictly
positive. We enter the ANISO case (Sections 6-8).
WHY THIS IS SUBTLE.
- The proof uses Ren-Wang exactly ONCE, in Step 4 via the two-
projection lemma. Concentration from Step 2 forces two replicas
of the slope to be well-separated. Widely separated slopes then
constrain how small the intercept projections can be. This is
the point where Ren-Wang's "Furstenberg sets carry dimension" is
applied.
- The specific factors 1 s l and 2 s l come from the isotropic
structure: time entropy is 1 bit per unit depth, but joint
time-position entropy is 2 bits per unit depth (two dimensions
of tangency, one of time). The information gained by going from
B (some coarse data + fine normal) to O (everything) reflects
this difference.
- The estimate "H(O|B) >= 2 s l" is what carries the REN-WANG
NOVELTY. It says: even though time alone would carry sl bits,
the geometric constraint of packing separated slopes forces the
pair to carry twice as much.
NEXT PART: the anisotropic case (Sections 6-8). That's where the
matched plane lives.
PART 10. THE MATCHED PLANE AND ITS TRACES (Section 6, in plain
language).
RECALL. We are in the ANISO case: m > 0. From Part 8, Step D:
- the saturated path has b'(u) in [-1-m, 0] almost everywhere;
- a(1) - b(1) = m;
- the orientation at "m-u" prefix carries kappa h bits of entropy
per level of u, conditional on (T, C_u).
GOAL. Rule out ANISO when beta > gamma.
OBSTRUCTION. The natural "rates of information flow" along the
saturated path are not smooth in u: they can shock, switch signs,
etc. Section 8 will want to write down a linear relation between
these rates at a boundary point ("contact point"), and we can only
do that if we have some control over the behavior of the profile
NEAR the boundary.
SOLUTION. Build an abstract "matched plane" where the four geometric
data coordinates and the depth variable live on a single
(n+1)-dimensional domain, and prove a trace theorem: the directional
derivatives of the limiting profile have strong limits as we
approach the boundary of the admissible cone.
MATCHED DOMAIN.
Fix:
- n >= 2 (the number of data coordinates in the application;
n = 4 works: velocity-x, velocity-y, position-x, position-y);
- distinct reals c_1 < ... < c_n (matched speeds);
- an order graph E: pairs (j,k) with j < k;
- an open interval I in the depth coordinate.
The matched domain is
D = { (L, v) in R^n x I : L_k - L_j <= (c_k - c_j) v for (j,k) in E }.
The trace plane (the matched direction) is
Pi(i, u) = (i + c_1 u, ..., i + c_n u, u).
The invariant cone is
K = { xi in R^{n+1} : xi_k - xi_j <= (c_k - c_j) xi_v for (j,k) in E }.
The tangent vectors
e = (1, ..., 1, 0) and t = (c_1, ..., c_n, 1)
span the plane and are tangent to every contact face of D.
THE OBJECT WE STUDY: a "limiting profile" f(L, v). In the application,
f stands for (1/ell) log of the mass of particles whose information
gains along the n directions have rates L_j at depth v. Its
directional derivatives f_j and f_v are the local information rates.
THE TRACE THEOREM (Theorem 6.1).
Suppose f is locally Lipschitz on D and, in the distributional
sense on the interior, satisfies
0 <= f_j <= d_j, 0 <= f_v <= d_v,
f_jk <= 0 (j != k), f_jv <= 0.
Then there exist unique locally bounded fields a_1, ..., a_n, D
on R x I, with 0 <= a_j <= d_j and 0 <= D <= d_v, such that:
1. STRONG TRACES. For almost every interior offset eta in K^o,
the restrictions of the ambient gradients to Pi(.) + eta exist
a.e. As eta -> 0 through such offsets, they converge strongly
in L^1_loc(di du) to (a_1, ..., a_n, D). The convergence is
QUANTITATIVE:
|| f_alpha(Pi(.) + eta) - A_alpha ||_{L^1(Q)}
<= C_Q |eta| + omega_alpha,Q'(C_Q |eta|),
where A = (a_1,...,a_n,D) and omega is the L^1 modulus of A.
2. The fields satisfy 0 <= a_j <= d_j, 0 <= D <= d_v.
3. They are continuous in (i, u).
WHERE THE TRACES COME FROM.
The four "monotonicity / concavity" inequalities are exactly the
distributional statements of a partial-order monotonicity. If
L <= L' componentwise and u >= u', the chain rule combined with
f_jk <= 0, f_jv <= 0 gives
f(L', u) - f(L, u) - [f(L', u') - f(L, u')] <= 0.
So f is SUBMODULAR in the L-coordinates with respect to the rank
ordering given by the c_j and the graph E.
Submodularity + the four first-derivative bounds imply that f is BV
on compact subsets of D. Standard BV/trace theory
(via dyadic-cube sums over the domain boundary) then produces the
traces, and the L^1 convergence rate follows from the modulus
continuity of the profiles on the compact Q' containing Q.
Sections 6.2-6.5 of the original give the full argument: the box
comparison, the mixed-sign transfer, and the trace limit.
WHY THIS IS THE RIGHT TOOL FOR SECTION 8.
The score identity at a "contact point" -- a point where the
saturated extremizer is tight against the boundary of the
admissible region -- is a linear relation among the trace rates
(a_1, ..., a_n, D). The trace theorem makes this identity writable
even when the extremizer is non-smooth at the contact point.
The continuity (property 3) is essential: Section 8 needs to
choose a contact point based on the saturated rate vector, then
continuously move the boundary parameter, and the traces have to
move continuously as well.
NEXT PART: Sections 7-8 -- rate transfer, contact identity, and the
final closure of the classical growth theorem.
PART 11. RATE TRANSFER AND CLOSURE (Sections 7-8, in plain
language).
GOAL (final theorem of Stage A). beta_c <= gamma.
CONTEXT. From Parts 8-9, we are in the ANISO case: m > 0. From
Part 10 we have a trace theorem on the matched domain D: the
directional derivatives of the limiting profile have strong
boundary traces (a_1, ..., a_n, D).
SECTION 7: RATE TRANSFER.
The "rate data" at a saturated point consists of:
- the trace vector (a_1, ..., a_n, D) at the boundary;
- the four coordinate capacities d_j and d_v;
- a "shape parameter" p (in [m, 1], in the application)
measuring where the extremal test sits within its velocity strip.
Saturation of beta_c > gamma gives certain rate equalities; capacity
gives certain upper bounds. Section 7 compares them.
THE RATE-TRANSFER LEMMA (informal). At every contact point (i, u)
of the trace plane, the linear functional of the trace data that
appears in the score satisfies
F(p) = 1 + s + o(1),
where p is the shape parameter, s is the time entropy profile
(H(C_u|T) = s u + o(1)) from the ISO profile, and F is a specific
linear combination of a_1, ..., a_n, D with coefficients depending on p.
Two structural facts built in:
- F is a nonnegative combination of the trace rates (since it comes
from product information at an admissible test);
- 0 <= a_j <= d_j and 0 <= D <= d_v (from the trace theorem).
SECTION 8: CLOSURE.
Set up. Choose a contact point (i, u) with the extremal rate vector,
and apply the trace theorem to get the traces (a_1, ..., a_n, D).
Algebraic strict gap (Lemma 8.2). Under the assumption beta > gamma
> 0 and the capacity bound s <= 1:
2 s - beta - gamma * m < 1 + s.
This is the "algebraic threshold". It's strictly below 1+s.
The contact identity (Lemma 8.1). At the contact point,
score = a combination of trace data = F(p) . (1 + o(1)),
and the strict gap says F(p) < 1 + s.
Lemma 8.4 (closure-positive-x). On the "minus side" of the
contact face (the side with B <= 0), a specific trace-derived
coordinate x satisfies x = x^- >= gamma > 0. Consequence: F is
nondecreasing in the shape parameter p on that side.
Putting it together.
- The tuple lives on the minus side (choose B <= 0 there).
- From Lemma 8.4, F is nondecreasing along the p-parameter.
- Therefore:
if p >= p_*: F(p) >= F(p_*) >= 1 + s (using p_* definition);
if p = 1: F(p) >= F(0) >= 1 + s.
In both cases, F(p) >= 1 + s.
- But the contact identity + strict gap gave F(p) < 1 + s.
- Contradiction.
CONCLUSION. No matched-clock least-aspect extremizer can exist under
beta > gamma. Since any saturated extremizer at beta > gamma would
produce one, beta_c > gamma is impossible, and Theorem (4.1) is
proved: beta_c <= gamma.
CASE DETAILS.
- m < 1, m = 1, m > 1 are covered by the "algebraic lemma",
which verifies the algebraic threshold in each range.
- At intermediate values p = 1 or p = m, one component of F lacks
a strong trace, but its coefficient in F is zero, so the argument
is unaffected.
- The continuity of the traces (Part 10, property 3) lets the
contact point be varied continuously, so the case split
p >= p_* vs p < p_* is exhaustive.
WHAT THIS BUYS. Theorem (4.1) beta_c <= gamma is the ONLY outside
input to Stage B. Once accepted, we can prove the packet growth
bound using exactly this kind of "hard classical sub-argument".
NEXT PART: Stage B. Packet scores, phase classes, and the packet
growth theorem (Section 9).
PART 12. STAGE B: PACKETS AND PHASE CLASSES (Section 9, in plain
language).
WHAT CHANGES. Stage A worked with POSITIVE masses m_O on sites. Stage
B works with squared amplitudes m_O = ||^2 -- the "square of
a coherent sum". Positivity is only partial: |sum|^2 involves sums
of oscillating terms, and interference can cancel or amplify. The
score has to be designed carefully.
PHASE CLASS. Let c in [0,1] denote longitudinal time, v in R^2 the
input velocity, x in R^2 the observation position. A wave packet
has a phase function Phi(c, x, v). Requirements:
(i) d_v Phi(c, x, v) = zeta(v, x - cv) for some function zeta.
(ii) zeta is defined for every p in R^2 and on all bounded v-patches.
(iii) d_p zeta(v, p) is symmetric and uniformly positive definite:
a_0 I <= d_p zeta(v, p) <= a_1 I, for all (v, p),
(or its negative; the sign can be flipped).
(iv) Derivatives of d_p zeta of order j are bounded by A_Phi^{j+1} (j!)^3.
Fixed bounds a_0^{-1}, a_1, A_Phi.
The class is closed under "rescaling": after translating to an
earlier anchor, subtracting a phase, and dilating to bounded velocity
neighborhood, the resulting phase is again in the same class (with
changed constants).
WAVE PACKETS. A wave packet is a quantum
g_T(v) = rho^{-1} eta((v - V_T)/rho) * exp(-i lambda Phi(C_r, X_T, v)),
where:
- (V_T, X_T) are the velocity and (trajectory-transported) position
of the packet at anchor time C_r;
- rho is the packet's velocity width in input space;
- lambda is the frequency parameter of the phase.
The AMPLITUDE m_O(f) is the modulus-squared pairing
m_O(f) = ||^2
in the canonical form, or up to a bounded factor in the site-dependent
adaptive form.
LOG-SUM IDENTITY. Let m_O >= 0 be the packet masses at positions O,
with at least one positive. Let Ent(O) denote the entropy of the
maximizing observation law (with weighted masses m_O^{3/2}).
Then:
sup_over_laws_pi { Ent_pi(O) + (3/2) E_pi[log_{1/eps} m_O] }
= log_{1/eps}( sum_O m_O^{3/2} ).
Reading: if you allocate your probability inversely proportional to
m_O^{3/2}, the weighted sum of entropy and (3/2)*log-mass equals the
log_{1/eps} of the cubic packet sum. This is the deep link between
entropy and L^{3/2} norms.
PACKET SCORE (the analogue of F_t from Stage A).
Fix a horizon t and a small depth parameter Planck (the "Planck
depth" of the packet system, tied to lambda by eps = lambda^{-1}
and Planck = t = L = 1 below).
For a probability law on observation sites O and their apertures D:
P_t(O, D) = E[w(D)] + Ent(O|D) + (3/2) E[log_{1/eps} m_O].
This mirrors the classical score but with:
- Ent(O|D) instead of -I(T;G);
- (3/2) E[logeps m_O] as the packet-mass analogue of log-sum.
Root supremum:
K_0 = sup over duration-ZERO apertures (b = a = 0) of
[ Ent(O) + (3/2) E[logeps m_O] ].
(Root apertures are "the best nothing-to-see tests".)
GROWTH RATE.
beta_q = sup over all packet systems of limsup_{eps -> 0}
(P_t - K_0) / t.
THEOREM (9.1, packet growth). beta_q <= gamma/2.
Equivalently: for fixed bounded data and every nu > 0, there is
eps_0 > 0 such that
P_t <= K_0 + (gamma/2 + nu) t for 0 < eps < eps_0.
Notice the factor 1/2: Stage B "buys back" half of the classical
budget by exploiting that packet masses are squared amplitudes
rather than plain positive masses. This gain is the engine of the
entire argument.
END OF PART 12.
NEXT PART: Section 10 -- packet composition at a time cut. This is
where cross-aperture decompositions and packet classes are set up.
PART 13. PACKET COMPOSITION AT A TIME CUT (Section 10, in plain
language).
SETTING. Fix a packet system with input f, endpoint time t, endpoint
depths (b, a). We want to compare its score P_t with an "earlier"
score at time r < t, plus the score of the local residual. This is
how beta_q <= gamma/2 will be proved: by slowly oozing entropy
backwards across a fixed number of cuts, and finding that the growth
cannot continue with fixed slack.
WHY THE REDUCTION IS NOT OBVIOUS. Wave packets are not positive
masses; they interfere. We cannot simply "split by packet". The
mechanisms that make it work:
(1) FINITE FOURIER DECOMPOSITION. Localize in velocity to one
localizer J_0 = (C_r, V_b*, (X + (C_r - C_t) V)_{r+b*}).
After rescaling, decompose the rescaled input into finitely
many Fourier atoms g_T with quantified tails. Only finitely
many atoms interact with any given endpoint test, up to a
controlled tail.
(2) GREEDY CAP CLASSES. Bin atoms by mass; assign each atom a
"cap" E(T) (a small velocity-position rectangle). Every class
has an upper cap count plus its weight bounded by a single
number g.
(3) LOCAL ROOT COMPARISON. Local Bessel (analysis + synthesis)
bounds the local root supremum K_0^loc from above by the
"candidate value" A_J = log N + (3/2) log m + (1/2) g (where
N = class size, m = mass bin, g = cap parameter).
(4) FOUR-DEFICIT SPLIT. After localizing, we have four natural
"deficits" d_0, d_1, d_2, d_3. Their mean sum is bounded by
the total growth gap. Markov's inequality picks a sublaw on
which all four are small simultaneously.
THE FINITE ERROR BUDGET. All these steps lose a small amount:
e_pkt = C (eta + xi + logeps S)
+ C_{data, eta, xi, A} (1 + log(2+ell)) / ell,
where:
- eta is the "packet enlargement exponent" (transport window);
- xi is the "coefficient-cutoff bin mesh";
- S bounds the number of auxiliary tags at each localizer;
- the last term absorbs the finite analytic truncation errors.
Order of limits: fix eta, xi; take eps -> 0; then eta, xi -> 0.
THE TWO BACKWARD ALTERNATIVES (this is the crucial structural fact).
Given a duration-one saturation array at rate beta_q (from
Definition 10.4), and a saturation gap at a fixed further cut,
EITHER:
(A) FORCED ARRAY. There is a fixed nu > 0 and kappa > 0 such
that, at every saturation array in a fixed outer subsequence,
the actual root aperture scores with b >= u_0 - nu are at
least kappa below K_0. Consequence: at the chosen cut, the
aperture cannot be forced to be degenerate. The earlier
aperture is uniformly eccentric: a - b >= nu/2.
(B) NO FORCED ARRAY. Then exact ISOTROPIC boundary tests
(a = b = u_t) saturate the local cut. This is the case where
the local score propagates without forced eccentricity.
Either alternative contributes to the final closure: (A) uses the
classical bound in an eccentric local window; (B) uses the
isotropic classical bound directly. Both force bounded growth.
WHY THE ALTERNATIVES EXHAUST THE PROBLEM.
The point of the split is that a saturating endpoint test either
has to look at the boundary of the aperture space (case B, where
the local root comparison becomes "free") or it has to stay inside
(case A, where the eccentricty a - b is bounded below). The two
cases have different classical bounds to apply, but both fit the
same argumentative structure. That is the "closure" that the
four-deficit split delivers.
WHAT IS ACTUALLY PROVED IN SECTION 10 (list of main statements).
Lemma 10.1 (Localizing labels). J_0 above is determined by
(endpoint observation, aperture) up to bounded ambiguity,
and by (particle, coarse time) up to a bounded number
of choices.
Lemma 10.2 (Finite packet decomposition with quantified tails).
For fixed data and depths, finitely many Fourier atoms
capture the endpoint pairings up to C_A eps^A ||f||_2.
Lemma 10.3 (Greedy cap classes). Each mass bin can be partitioned
into classes with cap-count parameter g, assigned caps
E(T), and per-class size N_all, satisfying
Cap(F) + w(F) <= g + e_pkt
for every root cap F, and
log_{1/eps} #{T : E(T) = E} >= g - w(E) - e_pkt
for the assigned cap.
Proposition 10.4 (Local root comparison). With A_J as above, the
local root supremum satisfies
K_0^loc <= A_J + e_pkt.
Proposition 10.5 (Universal modulus). P_t <= K_J + beta_q (t-r) + e_pkt
for every localizer J acting on a residual subsystem.
Proposition 10.6 (Four-deficit split). On a typical sublabor with
all four deficits small, we have
P_t <= Ent(J) + b_* + E[P_J] + e_pkt,
and the mean of the four deficits equals
K_0 + beta_q t - Ent(J) - b_* - E[P_J] + e_pkt.
Markov's inequality selects a sublaw on which all four
deficits are small relative to their mean gain.
Proposition 10.7 (Boundary alternatives and finite chains).
Under beta_q > gamma/2, there exist q successive
backward cuts, with h > 0 fixed, satisfying either
(A) or (B) above at every cut.
WHY THIS MATTERS. The four-deficit split is what lets us conclude
the growth constraint. Without it, "typical" defers to "expected"
which defers to "average over the whole universe", losing control of
the errors. The finite cap classes make the defers finite.
END PART 13.
NEXT PART: Section 11 (packet repayment) -- the backward induction
on fine velocity information that closes Stage B.
PART 14. PACKET REPAYMENT (Section 11, in plain language).
WHERE WE ARE. Section 10 established the two-alternative split at each
backward cut (eccentric-forced vs isotropic-boundary). Section 11
closes it: it proves Theorem 9.1, beta_q <= gamma/2.
FIVE INGREDIENTS.
1. FINE-VELOCITY LABEL W_f. At each backward cut we carry a label W_f
whose meaning depends on the alternative:
- ISOTROPIC: W_f records two velocity coordinates at the earlier
cut's depth.
- ELONGATED: W_f records only the velocity component along the
aperture normal n_f(P_f) at the endpoint. Coherent with the
aperture's normal as carried forward.
2. INCOMING CREDIT k_f (integer-like). Defined by
sup_w P(W_f = w | P_f = p) <= eps^{k_f - alpha_f}
for every p with positive support. Small k_f = coarse (few bits
per cell); large k_f = fine. The alpha_f = o(1) tracks accumulated
error.
CAPACITY: over a cut of depth h,
k_f <= d h + C xi + o(1),
where d = 1 (elongated) or d = 2 (isotropic), and xi is the small
bin mesh from Section 9.
3. RETENTION LEMMA (11.1). Suppose the current law satisfies the
credit bound with credit k_f. Let A be an event of probability
p_A. Then outside a sub-event of conditional mass eps^zeta, the
restricted law satisfies
P(W_f = w | P_f = p, A) <= eps^{k_f - alpha_f - zeta - logeps(1/p_A)}.
So restricting costs logeps(1/p_A) of credit. This is the exact
bookkeeping that survives every score-split in Section 10: the
mass before splitting is preserved, so the credit adjusts by the
conditioning probability alone. It is "credit economics".
4. PREDICTION LEMMA (11.2, elongated case). On the buffered
sub-population pi^buf, suppose two independent copies
(O,D), (O',D') of apertures at the same P are separated by
angle >= eps^{4h} with probability >= 1/4. Then there is a
deterministic normal n(P), defined on the atom-partition cells,
with
angle(n(D), n(P)) <= C eps^{4h}
on a part of pi^buf mass >= 3/4.
Proof idea: replace each aperture by its isotropic coarsenings at
widths a and b. Both are valid classical apertures. Apply the
classical bound (Theorem 4.1) to both. The classical growth is
(1+gamma) e in weight, where e = a - b is the aspect. But the
total information I(P; D, D') is more than the "one aperture"
information when the two apertures are far apart in angle. The
intersection of the two projected strips (in the velocity x
position product) is small: it has size bounded by eps^{4h}.
Hence I(P; D, D') is forced below I(P;D) + I(P;D'|D) by an
amount ~ 2 gamma e - small, unless a fixed predicting normal
n(P) collapses the entropy. The prediction makes the elongated
"continuation normal" coherent with the packet class.
5. REPAYMENT INEQUALITY (Proposition 11.3). For a prepared cut with
incoming credit k_f at the endpoint and outgoing credit k at the
earlier cut:
P_t <= B_r + gamma h + (1/2)(k - k_f) + C eta.
Here:
- P_t is the endpoint packet score.
- B_r is the "isotropic boundary score" at the earlier cut r:
the score of the exact isotropic aperture at width u_r, from
the trace theorem + the local classical bound.
- gamma h is the per-step classical outlay from Theorem 4.1.
- (1/2)(k - k_f) is the credit adjustment: going backward, the
earlier law has finer velocity observation than the endpoint
(since W_f is measured at the earlier depth), so k < k_f;
the term is NEGATIVE and helps.
CLOSURE (Section 11.4). Apply the repayment inequality q times
backward from r = 1 to r = 0 with step h and qh = 1:
P_1 <= B_0 + q gamma h + (1/2)(k_q - k_0) + q C eta.
Capacity: k_q - k_0 <= d qh = d. Combined with the eccentricity
condition (elongated case)
C_*(h + eta) < gamma e_0,
which forces the credit adjustment to pay back exactly half of the
classical outlay at each step (because e_0 is bounded below by
nu/2 from Prop 10.7), we get
P_1 <= B_0 + (gamma/2) * 1 + o(1).
Saturation: P_1 >= K_0 + beta_q - o(1). Root bound: B_0 <= K_0 + o(1).
Therefore
beta_q <= gamma/2 + o(1).
WHY THE FACTOR 1/2.
- The classical argument is stated for POSITIVE masses (the m_O in
Section 4). The packet score uses m_O = ||^2, the SQUARE
of an amplitude.
- The credit repayment exploits this in two places:
(a) The synthesis Gram matrix of packets satisfies
|| <= C eps^h * sum over rapid kernel
(Lemma 10.3 / strip coherence). The eps^h factor is the
amplitude sqrt(|mass|) instead of |mass|.
(b) When we replace a classical pairing by a packet pairing, the
budget T transforms as "one half T" because m_O is already
a squared modulus. Hence the scale factor 1/2.
- Net effect over q steps at unit total time: the packet budget
shrinks by exactly 1/2 relative to the classical budget.
WHAT IS PROVED IN SECTION 11 (list).
Lemma 11.1 (Retention of credit under conditioning).
Lemma 11.2 (Orientation prediction from isotropic replacements).
Proposition 11.3 (Repayment inequality).
Theorem 11.4 (Packet growth: beta_q <= gamma/2, for both
alternatives).
HOW THE TWO ALTERNATIVES SPLIT.
ISOTROPIC case: apply the repayment inequality directly with d = 2;
the isotropic boundary score B_r comes from the trace theorem.
ELONGATED case: d = 1; the aspect e_0 pays for the credit at rate
epsilon^{4h}, giving the prediction lemma, and the repayment
inequality closes.
END PART 14.
NEXT PART: Section 12 (multiplier). Uses Theorem 11.4 (beta_q <=
gamma/2) to bound the Bochner-Riesz operator norm on one dyadic
shell, then sums to close the theorem.
PART 15. FROM PACKETS TO BOCHNER-RIESZ (Section 12, part A).
GOAL. Turn packet growth (beta_q <= gamma/2) into an operator norm
bound for Bochner-Riesz on one dyadic shell, then sum over shells.
THE BOCHNER-RIESZ KERNEL.
The multiplier is m_delta(xi) = (1 - |xi|^2)_+^delta, delta > 0.
Its kernel (with the standard convention) is
K_delta(x) = integral over {|xi|<1} of
exp(2 pi i x . xi) (1 - |xi|^2)^delta dxi.
Lemma 15.1 (endpoint expansion).
K_delta is bounded on R^3. For R >= 1, at x = R e_3:
K_delta(R e_3) = exp(2 pi i R) b_+(R) + exp(-2 pi i R) b_-(R),
|d^j b_+-/dR^j| <= C_{delta,j} R^{-2-delta-j}.
In particular K_delta is in L^1(R^3) whenever delta > 1.
Proof. Integrate over disks perpendicular to e_3 first. This
gives the exact formula
K_delta(R e_3) = (pi/(delta+1)) *
integral over z in [-1,1] of
exp(2 pi i R z) (1 - z^2)^{1+delta} dz.
Split into z-near-1, z-near-(-1), and a compact interior.
The interior decays faster than any inverse power of R (via
repeated integration by parts in z).
Near z = 1: write t = 1 - z, t in [0, t_0). Then
contribution = exp(2 pi i R) times
integral over t in [0, t_0] of
exp(-2 pi i R t) t^{1+delta} a(t) dt,
with a smooth and supported in [0, t_0). On dyadic shells t ~ h
(h >= 1/R), N integrations by parts give
C_{delta,j,N} R^{-N} h^{2+delta+j-N}.
Sum over shells h = 1/R, 2/R, ..., t_0. For N > 2 + delta + j
the sum telescopes to C_{delta,j} R^{-2-delta-j}. This gives b_+.
The z near -1 side gives b_- with the conjugate symbol.
DYADIC SHELL DECOMPOSITION.
Take a smooth radial partition: bounded piece plus shells
|x| ~ lambda = 2^j, j >= 1.
Bounded piece: integrable, and convolution by an L^1 kernel is
bounded on every L^p (Young's inequality).
Each shell, by Lemma 15.1, is a sum of two kernels
lambda^{-2-delta} exp(+- 2 pi i |x|) a_{lambda,+-}(x/lambda),
with amplitudes supported in a fixed annulus and with all
derivatives bounded uniformly in lambda.
In the convolution operator, substitute x = lambda z, y = lambda y'.
The volume factor lambda^3 cancels the input and output dilations
of the L^p norm. The rescaled operator is
(A_{lambda,+-} f)(z) = integral over y of
exp(+- 2 pi i lambda |z-y|) a_{lambda,+-}(z-y) f(y) dy,
and the shell operator equals lambda^{1-delta} A_{lambda,+-}.
THE OSCILLATORY ESTIMATE (Theorem 15.2).
For every phase class (Section 9), every amplitude class with
bounded derivatives, every lambda >= 2, and every f in L^3(R^2):
||E_lambda f||_{L^3([0,1] x K_x)}
<= C_nu lambda^{-1+nu} ||f||_3,
where
(E_lambda f)(c,x) = integral of
exp(i lambda Phi(c,x,v)) B(c,x,v) f(v) dv.
Proof. Apply Section 9-11 with epsilon = lambda^{-1},
Planck = t = L = 1. Restrict first to |f| <= 1_F for a fixed
bounded ball F. The endpoint packet sum bound gives
sum over O of m_O^{3/2}
<= C_{gamma,eta} lambda^{gamma/2+eta} |F|.
Each output box has volume O(lambda^{-3}). Cube-sum:
||E_lambda f||_3^3
<= C_{gamma,eta} lambda^{-3+gamma/2+eta} |F|,
hence
||E_lambda f||_3
<= C_{gamma,eta} lambda^{-1+gamma/6+eta/3} |F|^{1/3}.
For general f: split into dyadic level sets
2^k < |f| <= 2^{k+1} supported on F_k. Apply the above to each
piece. Holder in k absorbs the factor log(lambda) in the
remaining positive power of lambda, provided
gamma/6 + eta/3 < nu/2.
THE PSEUDOCONFORMAL CHANGE OF VARIABLES (Section 12.2).
Unit-cube partition of (z,y) in R^3 x R^3. The annular support
of a_{lambda,+-} allows only a bounded number of neighbor cubes.
Translate each pair of cubes to a fixed region. A smooth angular
partition reduces the phase to regions where
0 < d_0 <= z_3 - y_3 <= d_1 < infinity.
Substitute, following Guo-Oh-Wang-Wu-Zhang 2025:
s = (z_3 - y_3)^{-1}, Z = s(z_1,z_2), v = (y_1,y_2).
For g(q) = sqrt(1 + |q|^2):
|z - y| = s^{-1} g(Z - s v),
|det d(Z,s)/dz| = s^4.
Both maps and their inverses have bounded derivatives on the
region, uniformly in lambda.
Choose a fixed positive interval [s*, s*+D] containing the
s-range. Put s = s* + D c, Z = D x. Direct differentiation:
Phi(c,x,v) = +- 2 pi (s*+Dc)^{-1} g(D x - (s*+Dc) v),
d_v Phi(c,x,v) = zeta(v, x - c v),
zeta(v,p) = -+ 2 pi grad g(D p - s* v),
d_p zeta(v,p) = -+ 2 pi D Hess g(D p - s* v).
This matches the phase class of Section 9, modulo the modification
of g to have a globally positive definite Hessian (next turn).
PART 15 (continued). Sections 12.3-12.7.
MODIFYING g FOR GLOBAL DEFINITENESS (Section 12.3).
After the pseudoconformal change, the phase involves g(q) = sqrt(1+|q|^2).
The Hessian of g at q=0 is I, but its eigenvalues decay as |q| -> inf
(asymptotically 1/(1+|q|^2)). The phase class of Section 9 requires a
Hessian bounded above and below by positive constants on all of R^2.
We fix this, with Gevrey constants that survive differentiation.
FIX. Define h : [0,infinity) -> [0,infinity) by
h(r) = 0 for r <= R_0,
h(r) = integral from 0 to (r - R_0) of (r - R_0 - u) chi(u) du
for r > R_0,
where chi >= 0 is a smooth bump supported on [0,1]. Then h = h' = h''
= 0 on [0, R_0], and for r > R_0:
h''(r) = chi(r - R_0) >= 0,
h'(r) = integral from 0 to (r - R_0) of chi(u) du >= 0.
Both are bounded above by a fixed constant.
Set g_tilde(q) = g(q) + h(|q|). On |q| <= R_0, g_tilde = g, so the
modification vanishes on every actual argument region of the
applications. Globally, Hess g_tilde decomposes into the tangential
direction (eigenvalue h'(r)/r) and the radial direction (eigenvalue
h''(r)); both are nonnegative. If chi is large enough (fixed constant
depending only on g), the added part dominates the decaying Hessian
of g, giving
c_0 I <= Hess g_tilde <= C_0 I on all of R^2.
GEVREY BOUNDS. On |q| in [R_0, R_0+1], the modification is a
composition of q -> |q| (analytic away from origin) with smooth
scalar functions of |q|; chain rule gives
|d^j Hess g_tilde(q)| <= A_0^{j+1} (j!)^3
for a fixed A_0. On |q| > R_0+1, Hess g_tilde is a polynomial in |q|
plus the radial part of Hess g; same bound. On |q| < R_0, it is the
Hessian of g, which is analytic with Gevrey constant. So the phase
class of Section 9 holds with grad g_tilde.
Under this modification, the phase class is closed under:
- translation to earlier anchors (composing with a shift and phase
subtraction);
- coarsening to bounded velocity neighborhoods;
- differentiation up to any fixed order.
Constants depend only on the fixed data.
SLICE ARGUMENT (Section 12.4).
R^3 = R^2 x R_{y_3}. The annular support of a_{lambda,+-} localizes
to a fixed bounded range of z_3 - y_3, corresponding to a fixed
s-range [s_*, s_* + D]. For each fixed y_3, the operator restricts
to a 2D oscillatory integral in (y_1, y_2) of the form treated by
Theorem 15.2. Constants in Theorem 15.2 are uniform in the parameter
s in [s_*, s_* + D].
Per-slice bound: for each y_3,
||(A_{lambda,+-} f)(., y_3)||_{L^3(R^2_x)}
<= C_nu lambda^{-1+nu} ||f(., y_3)||_{L^3(R^2_x)}.
Minkowski's inequality over y_3:
||A_{lambda,+-} f||_{L^3(R^3)}
= || ||(A_{lambda,+-} f)(., y_3)||_{L^3(R^2_x)} ||
<= C_nu lambda^{-1+nu} || ||f(., y_3)||_{L^3(R^2_x)} ||
= C_nu lambda^{-1+nu} ||f||_{L^3(R^3)}.
(Minkowski applies because the RHS is an L^3 norm in y_3 of a
nonnegative quantity.)
SHELL SUM AND CONCLUSION (Section 12.5).
Each shell operator equals lambda^{1-delta} times A_{lambda,+-}.
Combining with the last display:
|| shell_lambda(K_delta) ||_{L^3 -> L^3}
<= C_{delta,nu} lambda^{-delta+nu}.
Sum over dyadic shells lambda = 2^j, j >= 1:
sum_j lambda^{-delta+nu} <= 1 / (1 - 2^{-delta+nu})
< infinity, for 0 < nu < delta.
Bounded piece of K_delta: L^1 function, convolution is bounded on
L^3 by Young's inequality.
Therefore:
Theorem 15.3. T_delta is bounded on L^3(R^3), for every delta > 0.
FULL STRICT RANGE (Section 12.6).
Extend the L^3 bound to L^p for 1 <= p <= infinity via analytic
interpolation (Stein 1956) between the L^3 branch and an L^infinity
branch at higher orders.
Setup. Fix 3 < p < infinity. Put
theta = 1 - 3/p in (0,1),
c = delta - theta > 0,
z(w) = c + w for w in the closed strip 0 <= Re w <= 1.
Family of multipliers:
m_{z(w)}(xi) = (1 - |xi|^2)_+^{c+w}.
Boundary Re w = 0: T_c is bounded on L^3 by Theorem 15.3.
Boundary Re w = 1: T_{c+1} has kernel K_{c+1}, which lies in
L^1 by Lemma 15.1 (since c+1 > 1). So T_{c+1} is bounded on
L^infinity.
Scalar pairing for Schwartz f, g:
= integral over |xi|<1 of
(1-|xi|^2)^{c+w} f_hat(xi) conj(g_hat(xi)) dxi.
Analytic in w on 0 < Re w < 1. On compact subsets of Re w > -c, all
derivatives are bounded by integrable powers of |log(1-|xi|^2)|, so
analyticity and continuity extend to the closed strip.
POLYNOMIAL GROWTH ON BOUNDARIES. The imaginary part of the exponent
contributes a factor exp(|Im w| |log(1-|xi|^2)|) to |m_{z(w)}|.
On the Re w = 0 branch, this factor multiplies the (1-|xi|^2)^c
which is in L^3; the extra factor grows polynomially in |Im w|,
controlled by the beta-integral representation of the kernel (a
standard reduction, see Lemma 12.7 of the original). Similarly for
the Re w = 1 branch with the L^1 kernel. So both boundary families
satisfy the polynomial-in-Im w growth hypothesis of Stein's theorem
(Stein 1956).
STEIN INTERPOLATION. Given boundary operators bounded on L^{p_0}
at Re w = 0 and L^{p_1} at Re w = 1, with polynomial growth in Im w,
the interpolant at w = theta is bounded from L^{p_theta} to itself
with
1/p_theta = (1 - theta)/p_0 + theta/p_1.
Here p_0 = 3, p_1 = infinity, so
1/p_theta = (1 - theta)/3,
i.e. p_theta = 3/(1 - theta). Setting theta = 1 - 3/p gives
1/p_theta = (1 - (1 - 3/p))/3 = 1/p, consistent.
Therefore T_delta = T_{c+theta} is bounded on L^p(R^3) for
every p with 3 < p < infinity and delta > theta = 1 - 3/p.
DUALITY. Taking adjoints, T_delta bounded on L^p is equivalent to
T_delta bounded on L^{p'} = p/(p-1). So the same result holds for
1 < p < 3/2, with delta > 1 - 3/p' = 1 - 3(1-1/p) = 3/p - 2.
COMBINED STRICT RANGE. For all 1 <= p <= infinity:
delta > max{ 3 |1/p - 1/2| - 1/2, 0 }.
At p = 3 the threshold is 0, so T_delta is bounded on L^3 for all
delta > 0.
THEOREM 15.4 (Bochner-Riesz conjecture, R^3; strict range).
For every delta > 0, T_delta is bounded on L^p(R^3) for every p
with delta > max{ 3 |1/p - 1/2| - 1/2, 0 }.
This matches the conjectured sharp range for Bochner-Riesz in
three dimensions.
EXTENSIONS.
(i) Almost-everywhere convergence of S_R f for 3 <= p < infinity
and delta > 1 - 3/p follows from the L^p bounds by standard
maximal-function methods.
(ii) Spherical restriction (Tao 1999). The L^p bounds of Theorem
15.4 imply the restriction conjecture for the sphere S^2:
||Eg||_p <= C_p ||g||_infinity, for all p > 3.
CLOSING COMMENT ON THE ARGUMENT.
The argument above is the claimed proof. Its five pillars are:
- finite entropy calculus (Section 2);
- the Ren-Wang Furstenberg estimate (Section 3);
- the classical positive-mass growth bound beta_c <= gamma
(Sections 4-8, via matched plane + trace theorem + closure);
- the packet growth bound beta_q <= gamma/2 (Sections 9-11, via
finite Fourier decomposition + greedy cap classes + repayment);
- the multiplier step (Section 12, pseudoconformal change +
analytic interpolation).
Each pillar is a substantial piece of mathematics. A reader who
wants to check any single step should go to the corresponding
section of the original paper (bochner_riesz_3d_original.txt).
The point of this companion was to make the OVERALL STRUCTURE
readable to a motivated high-school student, and to identify the
five pillar statements that would need individual verification.
END PART 15.
NEXT PART: Part 16 -- closing note on scope, what remains to verify,
and a reading guide for a high-school reader.
PART 16. CLOSING NOTE ON SCOPE AND READING.
WHAT THIS FILE IS.
A plain-language tour of the claimed proof of the Bochner-Riesz
conjecture in R^3. All 12 sections of the original paper
(bochner_riesz_3d_original.txt) are covered in narrative form.
The goal: make the structure readable to a motivated high-school
student, and identify the five pillar statements a reader would
need to verify individually.
WHAT THIS FILE IS NOT.
A complete rigorous proof. It is not a substitute for the original
(15093 lines). The original contains all the technical detail:
the entropy calculus, the trace theorem, Bessel/Schur estimates,
the four-deficit splits, the exact credit arithmetic, and the
analytic-interpolation family. This companion summarizes. Where
it says "the calculation gives", "the argument shows", or "one
checks that", the actual check is in the original.
THE FIVE PILLARS. Each is a substantial independent piece.
1. Finite entropy calculus (Section 2, Part 6 here).
- entropy-restriction inequality (mass bounded by entropy);
- typical conditioning (base change at bounded cost);
- simultaneous-replacement theorem (product comparison to
actual marginals at fixed cost).
2. Ren-Wang's planar Furstenberg estimate (Section 3, Part 7).
The ONE outside theorem used anywhere in the argument.
Weighted, tagged, and bounded-graph versions are reductions
this pillar must supply. If this pillar fails, the entire
argument (specifically: isotropic exclusion in Section 5)
falls.
3. Classical positive-mass growth beta_c <= gamma
(Sections 4-8, Parts 8-11 here).
- localize, finite signature.
- least-aspect extremizer m.
- saturated path and coherence.
- tangent reduction: ISO (m=0) or ANISO (m>0).
- isotropic exclusion via Ren-Wang (Section 5).
- ANISO exclusion via matched plane, trace theorem,
contact identity, and closure (Sections 6-8).
4. Packet growth beta_q <= gamma/2
(Sections 9-11, Parts 12-14).
- finite Fourier decomposition with quantified tails.
- greedy cap classes.
- local root comparison.
- four-deficit split.
- two boundary alternatives (forced eccentric vs.
isotropic boundary).
- packet repayment by fine velocity credit.
5. Multiplier step (Section 12, Parts 15A-15B).
- Bochner-Riesz kernel expansion.
- pseudoconformal change of variables.
- modified quadratic g_tilde.
- oscillatory estimate (Theorem 15.2).
- slice argument (Section 12.4).
- shell sum (Theorem 15.3).
- analytic interpolation (Theorem 15.4).
WHAT A SKEPTICAL READER SHOULD CHECK FIRST.
- Section 3 (Part 7): does Ren-Wang's discrete theorem really
supply the WEIGHTED, TAGGED, BOUNDED-GRAPH version used? This
is the only place where the argument could break outside the
paper. Everything else is self-contained algebra.
- Section 5 (Part 9): the two-replica argument and the
Loomis-Whitney projection lemma. Do the constants survive the
localization layers when the ensemble has been trimmed to
typical homogeneous parts?
- Section 8 (Part 11): the contact identity and the strict-gap
algebra. Case-by-case checking of the algebraic threshold
(m < 1, m = 1, m > 1, p >= p_*, p < p_*) is the subtle part.
- Section 10 (Part 13): the four-deficit split. Does the
split correctly count errors at every trim, and is the credit
preserved under re-composition?
- Section 12.5 (Part 15B): the slice argument. Does Theorem
15.2 in 2D truly give a bound uniform in s in [s_*, s_*+D]?
If s enters the phase derivatives, the constants could
degenerate. The original asserts uniformity; the check is
direct but nontrivial.
READING GUIDE.
- Curious high-school reader with no harmonic analysis background:
read Part 1 (overview), Part 6 (entropy vocabulary), Part 7
(Ren-Wang statement, skip proofs), Parts 8-9 (classical growth
outline, skip case splits), Parts 12-13 (packet setup), Part 15
(multiplier overview). Skip the technical parts.
- Graduate student in harmonic analysis:
read Parts 6-15 in order. Cross-check each against the
corresponding section of the original. Take any step that
seems skipped as a signal to look at the original.
- Expert:
focus on Parts 9 (isotropic exclusion) and 14 (packet
repayment) -- those are where the novel argumentative
structures live.
WHERE THIS COMPANION MOST LIKELY DIVERGES FROM THE ORIGINAL.
- The trace theorem in Section 6 (Part 10 here) is stated with
less detail than the original. The property list is correct
but the submodularity/box-comparison proof is a sketch. Consult
Section 6.2-6.5 of the original for the full argument.
- The credit repayment algebra in Section 11 (Part 14) is
summarized at the level of principle. The exact cancellation
(why 1/2, why gamma/2) relies on Lemma 10.3's eps^h factor in
the strip coherence bound; the original gives the bookkeeping.
- The 3D-to-2D slice in Section 12.4 (Part 15B) works for fixed
slabs z_3 - y_3 away from the diagonal, with a finite angular
partition and unit-cube cover. The precise covers are in
Section 12.4 of the original.
FINAL HONESTY.
This file is a guided tour. Whether the original proof is correct
is a question for an expert in harmonic analysis. If it is, this
file is a readable gate into it. If it is not, this file inherits
every flaw.
The reason the "full proper entire proof" cannot be reproduced in
a few thousand lines of plain language is that the original is
15093 lines of graduate-level harmonic analysis, and its content
is genuinely irreducible: it is a claimed proof of a famous open
problem. This companion does not claim to be complete. It claims
to be readable, structurally correct in its summary, and to
clearly point at the five pillars that need checking.
If you are the reader this file was written for: start with Part
1, then Part 6, then Part 7. If you want more after that, go to
the original. If you want less, that is honest too; the
Bochner-Riesz conjecture is not a high-school topic, but the
STRUCTURE of a long proof is something worth seeing.
END.