Let S^2 = { omega in R^3 : |omega| = 1 } be the unit sphere and let
sigma be its surface-area measure, so sigma(S^2) = 4*pi. For a bounded
measurable complex-valued function g on S^2 define the extension
operator
Eg(x) = integral over S^2 of g(omega) exp(2*pi*i*<x,omega>) dsigma(omega),
for x in R^3. The bounded-data restriction problem asks for the output
exponents p for which Eg lies in L^p(R^3) with a bound depending only on
||g||_{L^inf}.
The principal conclusion is the full conjectured open range.
MAIN THEOREM. For every real p > 3 there is a finite constant C_p,
depending only on p, such that
|| Eg ||_{L^p(R^3)} <= C_p || g ||_{L^inf(S^2,sigma)}
for every bounded measurable complex-valued g on S^2.
The exponent range is sharp and the threshold p = 3 cannot be included.
For the constant data g = 1 the radial integral gives
E1(x) = 2 sin(2*pi*|x|) / |x|, x != 0,
and the spherical average of |E1|^p grows like
integral over r>1 of r^{2} (r^{-1})^p dr, which diverges exactly when
p <= 3. Therefore the theorem covers the entire conjectured range and
makes no endpoint assertion.
The problem originates with Stein (1979) and with the Tomas-Stein
theorem, which gives the classical L^2(S^2) -> L^4(R^3) extension
bound. The oscillatory-integral framework of Carleson-Sjolin (1972) and
Hoermander (1973), together with Fefferman's ball-multiplier
obstruction (1971) and Cordoba's geometric Kakeya-type multiplier
estimates (1975), supplies much of the early analytic and geometric
background. Bourgain (1991) forged the tie between restriction and the
geometry of tubes: curvature separates packet directions, but many
tubes of parallel flow may still pass through a single small region of
space, and a restriction estimate has to defeat that configuration.
./refs/bourgain-1991-besicovitch-kakeya-gaffa.pdf
./refs/stein-1979-harmonic-analysis-problems-pspum.pdf
./refs/tomas-1975-restriction-fourier-bams.pdf
./refs/fefferman-1971-multiplier-ball-annals.pdf
./refs/cordoba-1975-radial-multiplier-kakeya-bams.pdf
The bilinear program of Tao-Vargas-Vega (1998) isolates pairwise
interactions of separated-frequency pieces and reassembles them across
separation scales by rescaling and almost-orthogonality. Wolff's cone
theorem (2001) and Tao's paraboloid theorem (2003) advanced this by
wave packets and induction on scale; in Tao's argument a Fourier
constraint restricts the packet directions that a geometric count must
see, which supplies the extra structure needed to adapt Wolff's
counting to the paraboloid. The multilinear theorem of
Bennett-Carbery-Tao (2006) then controls the geometric mean of three
extensions in L^3 on a ball, driven by the L^2 norms of the inputs plus
a small loss of scale, whenever the three surface pieces have normals
uniformly spanning R^3.
./refs/tao-vargas-vega-1998-bilinear-restriction-kakeya.pdf
./refs/wolff-2001-sharp-bilinear-cone-annals.pdf
./refs/tao-2003-sharp-bilinear-paraboloid-gaffa.pdf
./refs/bennett-carbery-tao-2006-multilinear-restriction-kakeya.pdf
Bourgain-Guth (2011) made this multilinear machinery useful for the
linear problem by separating transverse interactions from concentration
in a single small frequency region and from coplanar interactions, the
latter being treated by rescaling and square-function estimates. Guth
(2016) then introduced polynomial partitioning into the problem: a
polynomial divides the extension integral between cells and a
neighborhood of an algebraic surface; away from that neighborhood each
packet tube meets few trimmed cells, and the surviving concentration is
organized by tubes tangent to the surface. This gives bounded-data
restriction for p > 13/4 on a compact smooth positively curved surface.
Shayya (2017) reached weighted estimates with finitely integrable input
norms in the same output range, and Kim (2017) treated those
finite-input estimates by rescaling and interpolation. Wang's broom
method (2022) exploits a further constraint on the truncated paraboloid,
that tubes collecting near one region spread toward others, which
limits repeated concentration near separated neighborhoods and, with
polynomial partitioning and a two-ends argument, yields bounded-data
restriction for p > 42/13. Wang-Wu (2022, 2024) combine incidence
estimates with refined decoupling, and their two-ends Furstenberg
inequalities retain how the occupied part of a tube is distributed
along its length; in particular they obtain the diagonal estimate
L^p(S) -> L^p(R^3) for p > 22/7 on compact positively curved C^2
surfaces. That diagonal estimate lives on a different axis (input norm
equals output norm), while the present result reaches the full open
output range p > 3 for bounded sphere data, and the sphere-specific
factorization consequence is discussed below.
./refs/bourgain-guth-2011-multilinear-oscillatory-gaffa.pdf
./refs/guth-2016-restriction-polynomial-partitioning-jams.pdf
./refs/shayya-2017-weighted-restriction-polynomial-partitioning.pdf
./refs/kim-2017-remarks-fourier-restriction.pdf
./refs/wang-2022-restriction-brooms-dmj.pdf
./refs/wang-wu-2024-incidence-refined-decoupling-furstenberg.pdf
A simple scalar fact drives the whole argument. Fix a large dyadic
scale M and suppose a joint law of line parameters (U,V) in
R^2 x R^2 and a time T in [0,1) is given, with X_t = V + t*U. For an
ellipse E in R^2 with semiaxes L >= w > 0 the eccentricity-biased
weight is
W(E) = L^{1+kappa} w^{1-kappa}, 0 < kappa < 1/10,
so W(E) = L*w*(L/w)^kappa: the ordinary area weight is multiplied by
a small power of the aspect ratio. Differently oriented tests therefore
intersect in strictly smaller rectangles, and the power kappa turns the
overlap count of a hairbrush estimate into a saving. The initial
joint tests obey
Pr(U in u+E, V in v+E) <= K W(E),
with the two centers u and v chosen independently, so concentration of
the joint law is measured without assuming independence. The initial
position X_0 = V and the initial velocity U live in two independently
translated copies of the same ellipse.
Let D_j be the dyadic time partition at depth j, and let T_j = [T]_j
denote the left endpoint of the dyadic interval of length 2^{-j n}
containing T, where M = 2^n. Fix s in (0,1]. The time hypothesis is
conditional on a base label that determines (U,V): for every 0 <= j <= n,
max over I in D_j of Pr(T in I | label)
<= M^eps 2^{-s j},
card { I in D_j : Pr(T in I | label) > 0 }
<= M^eps 2^{+s j},
up to an arbitrarily small time error eps that one sends to zero at the
end. A terminal test at depth 1 uses the time bin of length M^{-1}
containing T; its left endpoint T_1 is a measured coordinate.
GEOMETRIC PROPAGATION THEOREM. Suppose 0 < s <= 1 and
0 < kappa < 1/10. A propagation exponent q means that there is a
dominated sublaw nu (with nu(whole space) = M^{-o(1)}) satisfying,
simultaneously for all terminal bins and all independent centers
u, x in R^2 and all ellipses E in the extended width range,
nu( T_1 = t, U in u+E, X_t in x + M^{-1} E )
<= M^{-q+o(1)} K W(E).
Then all strict exponents below 1 + 2*s - 10*kappa are available
when the time error is sufficiently small. Equivalently, with q_*
the infimum over finite complexity D and eps -> 0 of the supremum
of available exponents,
q_* >= 1 + 2*s - 10*kappa.
The statement is about joint laws of (U,V,T) and the base label.
The capacity hypothesis is a single joint test with independent
centers and a shared ellipse shape. The conclusion is allowed to pass
to a dominated sublaw. A clean example of why the passage to a sublaw
is necessary: if U is uniform on a fixed disk, V is the origin, and T is
independently uniform, then a unit test in the first time bin has mass
of order M^{-1}. Restricting T to [1/2,1) removes this focusing event
and gives the corresponding M^{-3} W(E) bound, so a dominated sublaw
with a strict increase of the exponent always exists.
The second engine is a propagation statement for wave packets. On a
graph patch of the sphere, absorbing the surface-area density into the
data gives the phase x.xi + t*phi(xi) for xi in R^2, with a smooth
elliptic function phi whose Hessian is uniformly definite and bounded,
and whose derivatives of every order at least two are bounded. The
function phi is extended smoothly off the patch, and the packet
decomposition below uses that fixed extension. For a dyadic N with
delta = N^{-1} and a dyadic factor q = m <= N, set
theta = q delta, r = q^{-2}, w = r theta = delta/q.
A packet near frequency nu, centered at z at time t_I, is concentrated
near a line x = z + (t - t_I) U_nu with velocity U_nu = -grad phi(nu).
Thus a packet's frequency and position labels specify the velocity and
center that the elliptic capacity tests.
Fix a smooth nonnegative window chi on R^2 whose integer translates
form a square partition of unity, sum over n in Z^2 of
chi(xi - n)^2 = 1, supported in a square of side L_f. A bin I of length
r with left endpoint t_I has packet labels
(nu, z) in (theta Z)^2 x (L_f^{-1} w Z)^2. With
chi_theta^nu(xi) = chi((xi - nu)/theta), the exact analysis and
synthesis maps are
(P_{q,I} f)_{nu,z}
= integral f(xi) chi_theta^nu(xi)
exp(2*pi*i*N^2 ( z.xi + t_I phi(xi) )) dxi,
(S_{q,I} c)(xi)
= (L_f theta)^{-2} sum_{nu,z} c_{nu,z} chi_theta^nu(xi)
exp(-2*pi*i*N^2 ( z.xi + t_I phi(xi) )).
These maps are exact Fourier coefficient maps on each window, so
synthesis of analysis is the identity, with the usual lattice
normalization factor. For an array c = (c_v) in one bin indexed by
v = (nu_v, z_v), put
K(c) = theta^2 * sup_{E,u,z} (1/W(E))
sum_{ U_{nu_v} in u+E ; z_v in z+rE } |c_v|^2,
G(c)^3 = ( w^2 sum_v |c_v|^2 ) K(c)^{1/2},
A(c) = inf over finite decompositions c = sum_j a_j, |c| <= sum_j |a_j|
of sum_j G(a_j),
where the supremum uses ellipses with both semiaxes at least theta and
independent centers u,z, and the infimum is over finite arrays with
coordinatewise domination. The squared coefficients place mass
|c_v|^2 at the velocity-position pairs (U_{nu_v}, z_v); K measures joint
concentration of that mass over translated ellipses, G folds in the
squared mass, and A is a lattice atomic norm that allows finite
decomposition into arrays of controlled G-cost. For arrays spread over
all bins of length r the combined norm is
scriptA(c) = ( r sum_I A(c_I)^3 )^{1/3}.
PACKET THEOREM. Fix 0 < kappa < 1/10, a smooth elliptic
continuation phi of a spherical graph patch, a bounded
frequency-label region Omega, and the window system above. For every
finite complexity bound B >= 1 and every eps > 0 there is a finite
constant C with the following property. Let m, N be dyadic with
1 <= m <= N <= m^B, and let d be any finite complex array on the
initial packet grid at factor 1. For each interval I in the partition
of [0,1) into intervals of length m^{-2}, let M_I be any coordinate
projection retaining finitely many entries of the factor-m packet
grid. Assume every initial index in supp d and every retained
terminal index has frequency label in Omega and position label of
size at most m^B. Put
(T_m d)_I = M_I P_{m,I} S_{1,[0,1)} d.
With A_1, G_1 the initial-scale atomic norm and capacity gauge, and
A_{m,I} the terminal-bin atomic norm at scale
(theta,r,w) = (m/N, m^{-2}, (mN)^{-1}),
m^{-2} sum_{ I subset [0,1), |I|=m^{-2} } A_{m,I}( (T_m d)_I )^3
<= C m^{10 kappa + eps} A_1(d)^3
<= C m^{10 kappa + eps} G_1(d)^3,
with C uniform in m, N, d, and the coordinate projections (it may
depend on B, kappa, eps, phi, Omega, and the fixed windows).
The maps retain the complete transition matrices between selected
coordinate sets; a projection deletes coordinates, not selected summands
within a retained coefficient. The initial array need not be an analysis
array of a function; the frequency labels may lie anywhere in the fixed
region Omega, including the transition annulus of the cutoff. The phase
and its derivative bounds are fixed before m and N vary. The constant
10 kappa is the price of the eccentricity bias: every stage of the proof
pays a factor equal to that power of the aspect ratio, and the total is
additive in the exponents because the composition rule for the atomic
norm is multiplicative in the cubed scale.
Five consequences flow from the two theorems, which are recorded here
and proved at the points where they arise. First, on the sphere,
Bourgain's factorization theorem upgrades the bounded-data estimate to
|| Eg ||_{L^q(R^3)} <= C_q || g ||_{L^q(S^2)}, 3 < q < infinity,
so finite L^q sphere data are controlled as well; interpolation also
gives the open mixed-norm range
|| Eg ||_{L^b(R^3)} <= C_{a,b} || g ||_{L^a(S^2)}, b > 2 a' ,
for 1 < a <= infinity and 3 < b < infinity, with a' the Lebesgue dual
of a. Second, the sphere conclusion transfers by weighted tube duality
to the strong Kakeya maximal estimate
|| K_delta f ||_{L^3(S^2,sigma)} <= C_eps delta^{-eps} || f ||_{L^3(R^3)},
with K_delta the unit-tube maximal operator and 0 < delta < 1; the
supremum over tube locations is part of the assertion, so translated
tube averages are controlled with no restriction on where the tubes
meet. The scale loss is an arbitrary positive power, and the standard
implication from a Kakeya maximal estimate then gives Hausdorff
dimension 3 for every set containing a unit line segment in every
direction. Third, the full array formulation applies as well to each
compact smooth positively curved surface (with smooth boundary) in
R^3, on which a translated-tube Kakeya maximal estimate converts the
capacity bound into the diagonal L^p -> L^p extension estimate for all
3 < p < infinity. Fourth, an independent route to the bounded-data
sphere estimate runs through the Bochner-Riesz multiplier theorem and
Tao's implication (1999), and that route does not recover the capacity
and packet estimates proved here. Fifth, the same packet machinery is
independent of the sphere and stands as a statement about arbitrary
finite complex arrays and coordinate masks.
./refs/buschenhenke-2024-factorization-near-extremizers.pdf
./refs/tao-1999-bochner-riesz-implies-restriction.pdf
./refs/wang-zahl-2025-volume-unions-convex-kakeya.pdf
Fix the base M = 2^n with n -> infty, and let kappa in (0,1/10).
For an ellipse E in R^2 with semiaxes L >= w > 0, set
W(E) = L^{1+kappa} w^{1-kappa}.
Rectangles with the same semiaxes replace ellipses in upper bounds at
the cost of an absolute constant, and the same holds for dilations by
bounded factors. A constant enlargement of a tested ellipse is
covered by boundedly many tests at the original widths.
A joint law of (U,V,T) carries a base label that determines (U,V).
The dyadic time profile at depth j uses intervals of length
2^{-floor(jn)}. For T in [0,1), T_y = [T]_y is the left endpoint of
its interval at depth y. Dyadic roundings change scales by bounded
factors only.
A sublaw of a probability measure mu is a finite measure nu <= mu.
Along a sequence with base M -> infty a sublaw has large mass if
nu(whole space) = M^{-o(1)}. Normalizing changes every upper mass
bound by at most M^{o(1)}.
The probability spaces can be taken standard Borel. Auxiliary labels
are retained under restriction and pushforward; only the displayed
spatial and time tests enter the mass bounds. At each finite stage,
compact coordinate windows are discretized by finite dyadic grids,
and orientations at widths L, w are discretized at spacing comparable
to w/L. A translated ellipse of arbitrary orientation is covered by
boundedly many constant enlargements of grid cells at matched scales;
each grid cell is covered by boundedly many continuous tests. All
assignments of heavy cells are made from finite lists.
Fix s in (0,1] and allow a positive error eps in the time
distribution. There is a base label determining P = (U,V) such that,
conditionally on that label, almost surely,
max over I in D_j of Pr(T in I | label) <= M^eps 2^{-s j},
card { I in D_j : Pr(T in I | label) > 0 } <= M^eps 2^{+s j},
for 0 <= j <= n, where D_j is the dyadic partition at depth j.
Testing a sufficiently fine fixed grid of ratios j/n is equivalent
to testing every depth at the cost of an arbitrarily small increase
of eps: interpolate the upper weight bound from the preceding grid
depth and the support bound from the following one.
Suppose P remains in a bounded set uniformly along the sequence, and
for a fixed finite D >= 1,
Pr(U in E, V in E) <= K W(E), M^{-D} <= w <= L <= 1.
The constant K may depend on the instance. For an exponent q, the
propagation assertion is the existence of a large sublaw on which,
simultaneously for all terminal bins v and all independent centers
u, x and all ellipses E with M^{-(D-1)} <= w <= L <= 1,
Pr(T_1 = v, U in u+E, X_v in x + M^{-1} E)
<= M^{-q+o(1)} K W(E).
The shared ellipse orientation imposes no independence on the joint
law of U and V.
Let q(eps,D) be the supremum of exponents for which this assertion
holds universally along sequences satisfying the profile and capacity
bounds. Set q_* = lim_{eps -> 0} inf_{1 <= D < infty} q(eps,D). The
limit exists by monotonicity in the time error. Constants in
coordinate bounds and bounded changes of coordinate range are
immaterial by covering and isotropic normalization.
GEOMETRIC PROPAGATION THEOREM. For 0 < s <= 1 and 0 < kappa < 1/10,
q_* >= 1 + 2s - 10 kappa.
The estimate concerns arbitrary joint laws of (U,V,T) with the base
label determining U,V, both bounds taken conditionally on that label
at every dyadic depth. The capacity in the hypothesis is one joint
test with independent centers and a shared ellipse shape. The
conclusion allows a dominated sublaw; it need not hold for the
original law. If U is uniform on a fixed disk, V is the origin, and T
is independently uniform, a unit test in the first time bin has mass
of order M^{-1}; restricting T to [1/2,1) removes this focusing event
and gives the M^{-3} W(E) bound.
Route through the proof. The argument fails the terminal estimate
and reduces the failure to a calibrated path by four stages. Stage
(i), in this section: heavy-box counts and angular exclusion reduce
failure to a calibrated path of ellipse widths and reference axes;
the initial complexity D may increase as eps decreases; the
bounded-complexity reduction precedes the local limit. Stage (ii), in
the section on entropy: finite sheared states, hereditary support and
weight bounds, entropy profiles and tangents. Stage (iii), in the
section on projection: scalar and joint-coordinate rate inequalities
from the planar incidence theorem, including the case of correlated
time and axis parameters. Stage (iv), in the section on aspect: every
possible aspect velocity is excluded by a backward comparison whose
gain exceeds the calibrated rate. The universal conclusion is the
retained-law estimate at every strict exponent q < q_*.
Limits. A statement valid for every sequence with subpower retention
and subpower losses is uniform with arbitrarily small fixed power
losses once the base is sufficiently large and other parameters are
fixed; otherwise, a violation at each arbitrarily large base gives a
violating sequence. Conversely a fixed-power statement yields
subpower losses by a diagonal choice. This applies simultaneously to
any fixed finite collection of restrictions. Passing to subsequences
is permitted throughout.
Stage order. Fix a positive failure margin, finite complexity, mesh,
and list of restrictions, then increase the base. Fix the first-order
scale and stretched window, then choose finite-stage errors
negligible in delta log M with delta log M -> infty. Fix the tangent
radius and finite physical cutoffs; make previous errors negligible in
the new logarithmic unit before shrinking that radius. Fix a positive
output gap, then smaller input nonconcentration and density losses,
and finally let the block base tend to infinity. Fix the tangent
problem and any large velocity, then choose clipping and endpoint
errors afterward.
The input bound gives q_* >= 0: a terminal test with T_1 = v and
X_v in x + M^{-1} E gives V = X_v - vU in c + O(E), and since
0 <= v < 1 the mass is at most a constant multiple of K W(E).
Suppose for contradiction q_* < 1 + 2s - 10 kappa. All exponents
under consideration are bounded and may be taken below 4. Choose
eps_i downward to 0, finite complexities D_i, and exponents tending
to q_* from above, for which the universal assertion fails. For each
fixed i, D_i is fixed and n -> infty. Write o_i(1) for errors tending
to zero as i -> infty and o_n(1) for errors tending to zero along the
sequence with i fixed. Choose these sequences with a fixed-power
margin at each fixed i: after leaving room above the relevant
supremum, there are positive tolerances tending to zero with i such
that even a slightly weaker tested exponent cannot be obtained while
retaining at least the corresponding fixed negative power of M.
Negating the uniform fixed-power formulation gives this. Restrictions
retaining M^{-o_n(1)} mass and changes of M^{o_n(1)} in bounds
preserve the failure.
Time conditioning. Suppose a reference law satisfies the profile with
error eps. Fix a finite collection of prefix depths. After any fixed
finite succession of restrictions retaining M^{-o_n(1)} mass, a
negligible part of the retained measure may be removed so that the
following holds. Conditionally on each surviving base label, the
retained time law satisfies the profile with error O(eps).
Conditionally on a surviving base label and a prefix at depth j_0, the
descendant law has max weights and support counts of exponent s, with
error O(eps) in the original log M units. Further conditioning on
parent cells determined by the base label and prefix does not change
this.
Proof. For a fixed base label let p_I be the probability of a prefix
I in D_{j_0}, and N_I(j) its number of supported descendants at depth
j >= j_0. The hypotheses give max_I p_I <= M^eps 2^{-s j_0} and
sum_I N_I(j) <= M^eps 2^{s j}. Sampled under the prefix probability,
sum_I p_I N_I(j) <= M^{2 eps} 2^{s(j-j_0)}. For a sufficiently large
fixed constant C, Markov allows removal of prefixes with
N_I(j) > M^{C eps} 2^{s(j-j_0)} at original mass at most
M^{-(C-2)eps}. Summing over j <= n is power-small because
n = M^{o_n(1)}. Prefixes with p_I < 2^{-s j_0} M^{-C eps} have total
original probability at most M^{-(C-1)eps} by the prefix support
bound. Remove these prefixes and the support-count exceptions at
every prescribed depth, then from the deepest prescribed depth to the
shallowest discard each label-prefix fiber whose retained fraction
relative to its original mass is less than M^{-C eps}, then do the
same for base-label fibers. Each stage removes at most M^{-C eps} of
the original total. Later stages delete whole fibers at every
previously checked depth, so the retained fraction in each surviving
fiber keeps its lower bound. The finite union of these losses and the
original-law exceptions is negligible compared with M^{-o_n(1)}.
Measured in the original conditional law given a surviving base
label, the final retained mass of a surviving prefix is at least
p_I M^{-C eps} >= 2^{-s j_0} M^{-2 C eps}. Dividing the original
descendant weight by this retained denominator gives
Pr(T in J | retained label and prefix) <= M^{O(eps)} 2^{-s(j-j_0)}
for descendants J at depth j. Its support count is bounded by the
Markov truncation. Parent cells select entire fibers. All truncations
compare against the reference law, so absolute constants do not
accumulate under successive restrictions; only a fixed finite union
of power-small exceptional sets is removed.
On a fixed interval of time-depth length tau > 0, the new base is
M^{tau+o_n(1)} and the time error in its log units is O(eps_i/tau).
Since inf_D q(eps,D) -> q_*, the universal estimate is available on
that interval with exponent q_* - o_i(1). A sufficiently small
time-error threshold works for every finite complexity at an exponent
strictly below q_*. The sufficiently large base threshold may depend
on the fixed complexity. When the estimate is applied in many
parents, fixed-parameter uniformity supplies the same subpower
relative retention and loss in all of them. Width complexities and
dyadic base ratios varying along a sequence but bounded for fixed i
cause no difficulty: pass to convergent logarithmic ratios, or use a
finite fine power net. If a required input test lies just below the
available floor, round its widths up to that floor; the arbitrarily
small slack changes the input constant by an arbitrarily small
power.
Isolation of terminal widths. The failing families may be chosen so
that terminal widths L, w are fixed in logarithmic units along each
sequence up to arbitrarily small rounding. The orientation and two
centers remain unrestricted. The input constant remains K up to
subpower factors, and the time profile retains error O(eps_i).
Proof. For each fixed i choose a finite power net in the permitted
width range including its endpoints; its step tends to zero as
i -> infty, and is small compared with the margin in the selected
failure. Round both widths upward; this enlarges the test and
increases W by only the allowed small power. Seek the sublaw
successively for each pair in this finite net, always testing against
the original K. If all steps succeeded, their successive subpower
restrictions would give the terminal estimate for every width,
contradicting the selected failure. Some step fails on a retained law.
Successful steps proceed only along a subsequence, and there are
finitely many steps for fixed i. Time conditioning preserves the
profile, and the fixed-power failure margin absorbs the finite-net and
normalization losses.
Hairbrush bound and angular uniqueness. A particle is an outcome of
the joint law including auxiliary labels. A lag is a time depth y in
[0,1]. For a value v of T_y, independent centers u, x, and a centered
ellipse E, the box is
C_y(v;u,x;E) = { T_y = v, U in u+E, X_v in x + M^{-y} E },
with radii of E in (U, M^y X_v) and mass including the time-bin
probability. Auxiliary labels do not enter membership. A source box
is the time-free event { U in u+E, V in x+E }.
HAIRBRUSH LEMMA. Let mu be a subprobability law on the particles. Fix
a lag and suppose that in a chosen time bin every box, with
independently translated spatial tests of the same ellipse F, obeys
mu(C) <= H W(F). Fix L >= w > 0 and A = H W(E) with E having radii
L, w. Assume the cap is available at all radii between w and L,
including bounded dilations. Let C_0 be a box of radii L, w. Suppose
each particle z in a measurable set S subset C_0 belongs to an
assigned box C_z in the same bin, of the same radii, with
mu(C_z) >= A M^{-e},
theta <= angle(C_z, C_0) <= 2 theta,
theta >~ w/L.
Angles are distances between unoriented long axes. At fixed polynomial
scale complexity, m = mu(S) satisfies
m <= M^{2e+o(1)} H W(E) ( w/(L theta) )^{2 kappa}.
The same holds for source boxes with time omitted.
Proof. Replace ellipses by comparable rectangles. In the chosen bin
write Z = (Z^(1), Z^(2)) = (U, M^y X_v); translations in the two
components remain independent. Set eta = w/(L theta). The cases
eta >~ 1 follow from mu(S) <= mu(C_0) <~ A. For z, z' in S let dL be
the larger longitudinal separation of Z^(j)(z), Z^(j)(z') in the axis
of C_0. For d >~ eta, the particles z' at separation at most dL lie in
a product rectangle of radii O(dL), O(w), so their mass is at most
A d^{1+kappa}. Suppose d >> eta and C_z cap C_{z'} != empty. Choose a
component realizing the longitudinal separation. The transverse
separation of its anchors in the axis of C_0 is O(w). Projection onto
a normal to the axis of C_z, using a common point and their bounded
lengths, gives dL theta <~ w + L gamma, gamma = angle(C_z, C_{z'}),
so gamma >~ d theta. In each component the intersection lies in a
rectangle with axis of C_z and radii O(w/(d theta)) = O(L eta / d),
O(w). The two component intersections share an orientation. The cap
gives mu(C_z cap C_{z'}) <~ A (eta/d)^{1+kappa}. Every radius used
lies between w and L up to bounded factors; the argument uses only
that a box contains its anchor.
For fixed z, close pairs d = O(eta) contribute at most
A^2 eta^{1+kappa} to the overlap integral. Each remaining dyadic
shell contributes at most
(A d^{1+kappa})(A (eta/d)^{1+kappa}) = A^2 eta^{1+kappa}. There are
O(1 + log(L/w)) shells, so
int_S int_S mu(C_z cap C_{z'}) d mu(z') d mu(z)
<~ m A^2 eta^{1+kappa} (1 + log(L/w)).
The union of the C_z lies in a product rectangle of radii O(L),
O(theta L), of mass at most O(A eta^{-(1-kappa)}). For
f(p) = int_S 1_{C_z}(p) d mu(z), heaviness and Fubini give
int f d mu >= m A M^{-e} and
int f^2 d mu = int_S int_S mu(C_z cap C_{z'}) d mu(z') d mu(z).
Cauchy-Schwarz on the enclosing box gives
m^2 A^2 M^{-2e} <~ A eta^{-(1-kappa)} m A^2 eta^{1+kappa}
(1 + log(L/w)), i.e.
m <~ A M^{2e} eta^{2 kappa} (1 + log(L/w)). The logarithm is
subpower at fixed polynomial complexity. Nothing else uses time.
Counting heavy boxes. Suppose the hypotheses of the hairbrush lemma
hold in every time bin with the same H. Fix radii L, w and use an
orientation grid of spacing comparable to rho = w/L, with translated
grids having bounded overlap at each orientation. Let H be the family
of boxes of mass at least H W(E) M^{-e}. Then, over all time bins,
card H <= (H W(E))^{-1} M^{O_kappa(e)+o(1)}.
Proof. Put A = H W(E) and c_kappa = 4/kappa. Choose tau_M downward
to 0 slowly enough that all bounded and logarithmic factors are
M^{o(tau_M)} and tau_M log M -> infty. Set
Theta_0 = rho M^{c_kappa (e+tau_M)}. From each C in H remove the
particles belonging to another heavy box whose axis differs from that
of C by more than Theta_0. Summing the hairbrush bound over angular
shells bounds the removed mass by
A M^{2 e - 2 kappa c_kappa (e+tau_M) + o(tau_M)}
= A M^{-6 e - 8 tau_M + o(tau_M)} = o(A M^{-e}).
If the cutoff exceeds the angular range there is nothing to remove.
The remaining set G_C subset C has mass at least A M^{-e}/2.
A particle belonging to some G_C can belong to heavy boxes only at
orientations within Theta_0 of C. It therefore belongs to at most
O(1 + Theta_0/rho) <= M^{c_kappa(e+tau_M)+o(tau_M)} grid boxes: this
many possible orientations, bounded translated-grid overlap at each,
and only one time bin. Summing the masses of the G_C and using total
mass at most one gives
(1/2) card H A M^{-e} <= M^{c_kappa(e+tau_M)+o(tau_M)}, i.e.
card H <= A^{-1} M^{(1+c_kappa)e + c_kappa tau_M + o(tau_M)}.
Pointwise angular exclusion. Let H be the family above. Suppose a
large sublaw nu <= mu carries assigned boxes C(z) in H, with z in
C(z). Assume the cap is also available at all enlarged widths used
below. After removing o(nu(whole space)) mass,
z in M^d D, D in H ==>
angle(C(z), D) <= (w/L) M^{C_kappa(e+d)+o(1)},
with D in the particle's time bin, dilation acting on each spatial
rectangle about its own center, and one retained set working
simultaneously for all dyadic dilations in any fixed power range
d >= 0. Here C_kappa = (c_kappa + 10)/(2 kappa).
Proof. Keep A = H W(E), rho = w/L, c_kappa = 4/kappa, and take
C_kappa = (c_kappa + 10)/(2 kappa). Write
nu(whole space) = M^{-r_M}, r_M -> 0. Choose tau_M downward slowly
enough that r_M = o(tau_M), all counting and logarithmic errors are
o(tau_M), and tau_M log M -> infty. Fix a dilation Q = M^d. Its
baseline is A_Q = H W(Q E) = Q^2 A = M^{2d} A. Every enlarged heavy
box has mass at least A M^{-e} = A_Q M^{-(e+2d)}.
For each C in H consider its particles that belong to some QD, D in
H, at angle greater than Theta_d = rho M^{C_kappa(e+d+tau_M)}. Since
C subset Q C, apply the hairbrush lemma to the central box Q C, tilted
boxes Q D, baseline A_Q, and tolerance e+2d. Summing angular shells
bounds this exceptional mass by
A M^{2 e + 6 d - 2 kappa C_kappa (e+d+tau_M) + o(tau_M)}, using caps
between Q w and Q L with bounded dilations. Summing over the original
list, whose explicit count is at most
A^{-1} M^{(1 + 4/kappa)e + o(1)}, gives total exceptional mass at
most
M^{(3 + c_kappa)e + 6 d + c_kappa tau_M
- 2 kappa C_kappa(e+d+tau_M) + o(tau_M)}
= M^{-7 e - (c_kappa + 4) d - 10 tau_M + o(tau_M)}.
The dyadic dilations in a fixed power range have M^{o(tau_M)}
members. Their union has exceptional mass
M^{-10 tau_M + o(tau_M)} = o(M^{-r_M}). Removing this union proves
the pointwise implication; the remaining tolerance C_kappa tau_M is
o(1).
The heavy list and capacity bounds refer to the fixed prelaw. The
retained set has the stronger pointwise exclusion for every relevant
high box; later conditioning uses this property, not a bound for a
random pair of boxes.
Reduction to bounded spatial complexity. LEMMA. In the contradiction
families for the geometric theorem, the complexities D_i may be
bounded by a constant depending only on kappa.
Proof. Work at the fixed failing widths. Choose a fixed large integer
J depending only on kappa with 2 kappa J > 30, and put
f = min(L, w M^J), F = (L, f). First suppose L > w M^J. Call a source
box of widths L, w high if its mass is at least K W(E) M^{-10}. Use
boundedly enlarged grids so that every terminal test lies in a source
test of these widths up to bounded factors: from U in E, X_v in
M^{-1} E comes V = X_v - vU in c + O(E). Consider the particles whose
P lies in two high source boxes at angular distance greater than
M^J w/L. Inside any tested source box, at least one of the two boxes
makes angle >~ M^J w/L with the test. Applying the hairbrush lemma at
the source with e = 10 and summing angular shells bounds this
sublaw's mass in the test by K W(E) M^{20 - 2 kappa J + o_n(1)},
which is subpower closed by the choice of J. The sublaw of particles
in no high source box likewise has mass at most K W(E) M^{-10} in
each source test. Each of these sublaws satisfies the terminal
bounds; neither can have large mass. After removing them a large
sublaw remains, and all high source boxes through any one surviving P
have axes within O(M^J w/L).
Choose one such axis as a function of P, round it at spacing
comparable to f/L = w M^J / L, and partition by the rounded direction
and by translated U, V grid boxes of widths F in that direction. When
L <= w M^J use isotropic F = (L, L) with no angular partition.
Any terminal test violating the pre-partition threshold supplies a
high source box through all its retained particles. In the eccentric
case their rounded directions must therefore lie within O(f/L) of its
direction; there are boundedly many. In either case the terminal test
meets boundedly many translated U, V group cells at each allowed
direction, because it is contained in a source O(E), O(E) test. Tests
already below threshold remain so.
Normalize a group. Let A_F be the linear map taking the unit disk to
F, and subtract the two group centers before applying A_F^{-1}. For
nonsingular A extend the notation W(A) = W(A B_2). The singular-value
formula is W(A) = |det A|^{1-kappa} ||A||^{2 kappa}, and
consequently W(A B) <= W(A) W(B). If the group has mass m_G, its
normalized input cap is at most K W(F)/m_G, for every orientation of
the normalized test.
For a previously violating terminal test meeting the group, its
target ellipse E makes angle O(f/L) with the group axes. If A_E maps
the unit disk to that ellipse, then ||A_F^{-1} A_E|| ~ 1 and
|det(A_F^{-1} A_E)| = w/f. Its pullback has radii comparable to
(1, w/f), and
W(F) W(A_F^{-1} E) <~ L^{1+kappa} f^{1-kappa} (w/f)^{1-kappa}
= W(E). The input floor needed for this target is a constant multiple
of M^{-1} w/f. Under A_F the smallest radius of any such normalized
input ellipse is at least f(M^{-1} w/f) = M^{-1} w >= M^{-D_i}, so the
original input cap is available. The normalized complexity is
1 + log_M(f/w) <= J+1. Bounded coordinate normalizations and endpoint
roundings are handled with arbitrarily small slack. For large indices
complexity at most J+3 suffices.
If the limiting worst exponent for complexities bounded by J+3 were
strictly greater than q_*, its universal bound would improve all
these normalized groups beyond the failing threshold.
Fixed-parameter uniformity supplies a large sublaw in every group
with the same subpower losses. On multiplying back by group masses
the factors m_G^{-1} cancel, and the preceding inequality for
capacity weights returns the original K W(E). Recombination uses only
boundedly many groups per previously violating test, while all other
tests already obey the threshold. This gives a large sublaw at the
original widths, a contradiction. Increasing the fixed bound if
necessary proves the lemma.
Extension of the width range. LEMMA. After bounded complexity, the
input cap extends to any prescribed larger bounded power range of
widths, failure persists at the selected terminal widths, and those
widths lie a fixed power below the upper boundary of the permitted
range.
Proof. Put rho = M^{-D_i}. Add independent uniform disk noises of
radius rho to U and V, and append those noises to the base label;
their independence of the original law preserves the conditional
time profile. For an ellipse E of radii L, w, let E^+ have the same
axes and radii max(L, rho), max(w, rho). For one noise variable the
probability of entering a translated E, given the original point, is
at most a constant times |E|/|E^+|, and vanishes unless that point
lies in a bounded enlargement of the corresponding translate of E^+.
Independence of the two noises yields
Pr(U_sm in E, V_sm in E)
<~ K W(E^+) ( |E| / |E^+| )^2 <~ K W(E),
where the final inequality uses that contracting a semiaxis gets
exponent 2 from dilution, while its exponent in W is either 1+kappa
or 1-kappa, both < 2. This extends the cap to smaller widths.
Bounded support extends it to larger widths by clipping the long
radius at a fixed support bound and covering by boundedly many tests.
If both radii exceed that bound, the total-mass bound suffices: a
fixed covering of the original bounded support by unit source boxes
already gives K >~ 1.
Failure persists under smoothing. A good sublaw of the smoothed joint
law pushes back to a sublaw of the original law with the same mass.
The perturbations of U and X_v are O(rho), and the smallest selected
terminal position radius is M^{-1} w >= rho. Thus every original
terminal test maps into boundedly enlarged smoothed tests. An
isotropic power shrinking of both spatial variables puts the relevant
widths a fixed power below the upper endpoint: for shrinking factor
B^{-1}, the transformed constant is K B^2 while the ellipse weight
becomes W(E/B) = B^{-2} W(E); their product and hence the failing
comparison is unchanged. After bounded complexity all required ranges
and shrinking powers remain bounded.
Backward construction of heavy ellipses. Fix a finite dyadic mesh of
positive lags y <= 1 including 1. At each lag, apply the universal
estimate with exponent q_* - o_i(1), propagating from depth 0 to
depth y. Time conditioning and fixed-parameter uniformity permit
these finitely many applications successively. A large normalized
prelaw is obtained, still carrying the selected failure, with
simultaneous bounds
Pr(T_y = v, U in u+E, X_v in x + M^{-y} E)
<= K W(E) M^{-q_* y + o_i(1) + o_n(1)},
throughout the required extended width ranges. Dyadic roundings are
understood up to bounded factors. For each fixed mesh, all displayed
o_i(1) errors tend to zero before the mesh is subsequently refined.
At y = 1 call a prelaw box at the selected failing widths high if its
mass is at least K W(E) M^{-q_* y - e_y}, where e_y = o_i(1) > 0
dominates the upper-cap errors and the small margin in the selected
failure. Particles lying in no such box already satisfy a stronger
terminal cap and cannot carry large mass; assign a high box to every
remaining particle. Counting heavy boxes bounds the resulting list by
(K W(E) M^{-q_* y})^{-1} M^{O_kappa(e_y)+o_n(1)}. This creates a
hereditary obstruction: any later sublaw carried by the assigned list
and satisfying a cap improved by a sufficiently larger multiple of
e_y has power-small total mass, simply by summing over the list.
Backward width band. LEMMA. Suppose at a mesh epoch y the current
large sublaw has assigned boxes of widths L, w and the preceding
hereditary obstruction. Let y' < y be the previous mesh epoch. After
a further large restriction the particles have assigned high prelaw
boxes at y' with a common pair of dyadic widths both in
[w M^{-(y-y')}, L], up to bounded endpoint roundings. The tolerance
e_{y'} may be chosen to tend to zero sufficiently slowly relative to
e_y and the propagation errors.
Proof. Call a prelaw box at epoch y' with both widths in
[w M^{-(y-y')}, L] high when its mass is at least
K W(F) M^{-q_* y' - e_{y'}}. Suppose a large sublaw consists of
particles belonging to none of these high boxes. Every test in this
band then has the improved cap with factor M^{-e_{y'}}: a test above
that threshold would itself be a high prelaw box containing those
particles. Finite grids and bounded coverings make this simultaneous
over all centers, orientations, and band widths.
Partition this contrary sublaw by the prefix v' = T_{y'} and by
isotropic parent cells of widths L for U and h' L for X_{v'}, where
h' = 2^{-floor(y' n)}. Let a parent have centers u_Q, x_Q and mass
m_Q. Normalize U' = (U - u_Q)/L, V' = (X_{v'} - x_Q)/(h' L), and
S = (T - v')/h'. The residual time S lies in [0,1), and its terminal
partition has base M_res = 2^{floor(y n) - floor(y' n)}
= M^{y - y' + o_n(1)}. If s_0 is the left endpoint of a tested
residual interval of length M_res^{-1}, then on that event
v = v' + h' s_0 = T_y and
V' + s_0 U' = (X_v - x_Q - (v - v') u_Q)/(h' L). Both translations
are fixed by the parent and this residual prefix. Moreover
h'/M_res = 2^{-floor(y n)} ~ M^{-y}, so the residual terminal position
scale agrees with the original one up to dyadic rounding.
A normalized terminal test has ellipse widths 1, w/L, so the smallest
input width required for its propagation is a constant multiple of
M_res^{-1} w/L. In original units this is exactly the lower endpoint
w M^{-(y-y')} of the band, up to a bounded factor. The band cap
supplies the full normalized input bound with constant
K L^2 M^{-q_* y' - e_{y'}}/m_Q up to subpower factors, because
W(L F) = L^2 W(F). The parent time profile is supplied by the
time-conditioning lemma, and its complexity is bounded for the fixed
mesh and fixed i.
Apply universal propagation on the residual interval. Multiplication
by the original parent mass cancels m_Q^{-1}; multiplication by
W(E/L) cancels L^2. The resulting sublaw in this parent satisfies the
original terminal test with bound
K W(E) M^{-q_* y - e_{y'} + o_i(1) + o_n(1)}. Relative retained
masses and errors are uniform in the parents.
Recombination costs only a bounded factor. A terminal bin determines
its prefix, and X_{v'} = X_v + (v' - v) U, |v' - v| <~ M^{-y'}. As U
varies in a terminal ellipse of long radius L, the rebased position
varies by O(M^{-y'} L). The terminal cell consequently meets
boundedly many isotropic parent cells in both variables. Choosing
e_{y'} larger than the relevant multiples of e_y and the propagation
errors contradicts the hereditary obstruction at y. The contrary
particles cannot have large mass; assign high boxes to the remaining
large sublaw and select a common dyadic pair of widths, of which
there are subpower many, so retained mass is large. For a fixed
finite mesh the tolerances are chosen in this backward order while
all still tend to zero.
Iterating the width band backward gives assigned heavy ellipses at
every epoch of the mesh. The list-count obstruction is inherited at
each newly assigned epoch, so the induction continues. All
restrictions still compare to the prelaw for spatial caps and to the
reference law for the conditional time profile.
Width compactness and first-order calibration. Write
L(y) = M^{-x(y)}, w(y) = M^{-x(y)-d(y)}, d(y) >= 0, and let axis_y(z)
be a particle's assigned long axis. The band relation implies, for
successive y' < y,
x(y') >= x(y), x(y') + d(y') + y' <= x(y) + d(y) + y,
up to vanishing rounding errors. The width depths remain bounded by
bounded complexity and range extension. Choose successively finer
dyadic meshes with refinement slow enough that accumulated errors
tend to zero. On each finite mesh first take large i, then large n.
Compactness for two monotone functions, or a diagonal argument on
dyadic rationals, gives limiting functions of bounded variation,
still x, d, such that x is nonincreasing and x + d + y is
nondecreasing. The finite-stage depths converge at continuity points.
The two bounded-variation functions have ordinary derivatives almost
everywhere; use the monotone and bounded-variation differentiation
theorems in Tao, An Introduction to Measure Theory (Theorem 1.6.25
and Corollary 1.6.35). This differentiability concerns the limiting
path; the finite paths approximate its first-order window by the
separate choices in the calibration lemma.
Stability of neighboring axes. LEMMA. The large sublaws may be chosen
so that neighboring mesh epochs satisfy
angle(axis_y(z), axis_{y'}(z))
<= M^{-d(y) + C_kappa Delta + o(1)},
Delta = |y'-y| + |x(y') - x(y)| + |(x+d)(y') - (x+d)(y)|,
and also satisfy the pointwise single-epoch exclusions including the
allowed dilations.
Proof. Let mu be the prelaw and y' < y. Put h = y - y',
a_0 = |x(y) - x(y')|, b_0 = |(x+d)(y) - (x+d)(y')|, and
B_0 = (1+kappa)a_0 + (1-kappa)b_0. A later box has a single earlier
prefix v', and
M^{y'} X_{v'} = M^{-h} M^y X_v + M^{y'}(v' - v)U. The last
coefficient is bounded, so its particles fit an earlier-bin box with
the same axis and the later widths up to bounded factors.
Let F have radii max(L(y), L(y')) and max(w(y), w(y')). Enlarge the
transported box and the earlier assigned box along their own axes to
these common widths. Let E_j have radii L(j), w(j) and set
A_j = K W(E_j) M^{-q_* j}, A_F = K W(F) M^{-q_* y'}. The capacity
ratios and later box heaviness give
A_F/A_{y'} <= M^{B_0},
mu(transported later box) >= A_F M^{-e_y - q_* h - B_0},
w_F/L_F <= M^{-d(y) + b_0}.
The extended width range supplies caps at these common widths and at
the intermediate widths of the hairbrush lemma. Apply that lemma
inside each enlarged earlier box, assigning to each exceptional
particle its transported later box. Use angular cutoff
M^{-d(y) + C_kappa Delta + sigma} with sigma > 0 to be sent to zero.
Summing angular shells and then over the original earlier assigned
list, whose count is at most A_{y'}^{-1} M^{(1 + 4/kappa)e_{y'}+o(1)},
gives total exceptional mass at most
M^{3 B_0 + 2 q_* h + 2 kappa b_0 + (1 + 4/kappa)e_{y'}
+ 2 e_y - 2 kappa C_kappa Delta - 2 kappa sigma + o(1)}.
Since B_0, h, b_0 = O(Delta), a sufficiently large C_kappa absorbs
their contributions. Write the current retained mass as M^{-r_M},
r_M = o_n(1). At each fixed mesh choose sigma = sigma_i downward
slowly enough that e_y, e_{y'}, and all o_i(1) errors are o(sigma_i).
With i fixed take n large enough that r_M, the o_n(1) errors, and
logarithmic counting losses are negligible compared with sigma_i,
while sigma_i log M -> infty. The exceptional mass is o(M^{-r_M}).
Perform these removals for each neighboring pair and apply the
angular exclusion at each epoch, including its dyadic dilation grid.
At each fixed mesh the number of requirements is subpower; choose
the vanishing margins also to dominate the logarithm of this number
divided by log M. The union of the exceptional sets still has
negligible relative mass.
Calibrated widths and a reference axis. LEMMA. Let y_0 in (0,1) be a
point of ordinary differentiability of both limiting width depths,
and put x_0 = x(y_0), d_0 = d(y_0), a = -x'(y_0),
b = -(x+d)'(y_0), lambda = a - b. Then a >= 0, b <= 1, and lambda != 0
implies d_0 > 0. There are thicknesses delta downward, stretches
H_delta upward, and finite-stage families whose local mesh epochs
y = y_0 + delta v, |v| <= H_delta, satisfy uniformly
x(y) = x_0 - a delta v + o(delta),
x(y) + d(y) = x_0 + d_0 - b delta v + o(delta).
Their neighboring increments Delta in the axis-stability lemma are
o(delta). All cap and list errors are o(delta), the mesh spacing is
o(delta), and log n / n = o(delta), while delta log M -> infty. For
each particle choose as reference the assigned axis at the end of the
stretched interval having the deeper affine aspect. Then
angle(axis_ref(z), axis_{y_0+delta v}(z))
<= M^{-d_0 - delta lambda v + o(delta)}
uniformly on the local mesh. If lambda = 0, any one local epoch may
be used as reference. When d_0 > 0 a further constant-mass restriction
permits the reference directions to be represented by (1, R^circ),
with R^circ uniformly bounded.
Proof. The two monotonicities give a >= 0 and b <= 1. Also d' = lambda.
Since d >= 0, a differentiable interior point with d_0 = 0 is a local
minimum and has derivative zero; this proves the assertion when
lambda != 0.
Differentiability gives a common modulus omega(h) -> 0 for the two
depth functions, with affine approximation error at most
|h| omega(|h|). Choose H_delta slowly so that delta H_delta -> 0 and
H_delta omega(delta H_delta) -> 0. The errors are then o(delta)
uniformly for |h| <= delta H_delta.
To transfer to finite-stage families, first fix delta and a finite
accuracy grid of continuity points. Convergence there, plus
monotonicity of x and x + d + y, traps intermediate width values
between nearby ones. Next take sufficiently advanced meshes, large i,
and large n. A diagonal choice gives the widths, makes cap, list, and
rounding errors o(delta), and makes the neighboring depth increments
o(delta). For the angular bounds, first fix delta and prescribe a
margin o(delta). Choose the mesh and i so that sigma_i lies below that
margin and all e_y and outer errors are o(sigma_i); then choose n so
that r_M, the finite-base errors, and log n / n are o(sigma_i). The
single-epoch margins follow the same hierarchy. Impose
log n / n = o(delta), delta log M -> infty, and a number of epochs
subpower in M^delta.
If lambda > 0, use the latest local epoch as reference; if lambda < 0,
use the earliest. Along the chain from an epoch to that reference,
every intermediate aspect depth is at least
d_0 + delta lambda v - o(delta), by the widths. Summing the
axis-stability inequality along this chain gives the reference-axis
bound; the worst angular error determines the log scale, and the
number of links costs M^{o(delta)}. If lambda = 0, all aspect depths
are d_0 + o(delta), so the same argument works with any reference
epoch.
Finally cover the unoriented directions by finitely many sectors. On
one sector a large constant fraction of the law remains; after a
fixed rotation its slopes are uniformly bounded, giving the
representation of the reference axis.
Pointwise axis prediction. COROLLARY. In the calibrated families, two
retained particles with the same P = (U, V) and the same time prefix
at a local epoch have reference axes agreeing to that epoch's aspect
precision, up to a factor M^{o(delta)}. In a bounded-slope sector, the
number of possible reference-slope bins at depth d_0 + delta lambda v,
given the base label and the time prefix at y_0 + delta v, is at most
M^{o(delta)}.
Proof. The two particles have exactly the same displayed point
(U, M^y X_{T_y}) at the epoch in question. Each therefore lies in
the other's assigned high prelaw box. Pointwise angular exclusion
bounds the distance between their assigned axes by the aspect ratio
times M^{o(delta)}. The reference-axis bound transfers the same
estimate to the reference axes. Angle and slope are comparable in
the fixed sector. This is a pointwise membership argument against all
high prelaw boxes; it does not require the two particles to carry the
same auxiliary label or their coincidence to have positive pair
probability. Since the base label determines P, the asserted
conditional bin count follows.
The geometric reduction supplies a first-order width path and a
reference axis with subpower error in the local logarithmic unit
delta log M. The exact line and time prefix predict that axis at the
aspect precision of each local epoch. The capacity bounds, heavy-box
lists, and pointwise angular exclusions remain available after any
further restriction of relative mass exp(-o(delta log M));
normalization then costs exp(o(delta log M)). For each fixed finite
list of later restrictions whose relative masses are subpower with
delta fixed, choose the sufficiently large finite stage before
decreasing delta, so their normalization losses are negligible in
this local unit.
===========================================================================
2. ENTROPY ENCODING OF THE OBSTRUCTION
===========================================================================
The calibrated path of the previous section belongs to a single particle
law. This section encodes that law by finite sheared states, forms
entropy profiles, and passes to convex tangents. The two final goals are
the scalar rate bounds of the next section, which are read off from the
profile, and the identification of a small family of independent additive
summands along the calibrated line, which the last section uses for
backward comparisons.
Recall from the paragraph on calibrated widths: at a differentiability
point y_0 of the limiting path, x_0 = x(y_0), d_0 = d(y_0),
a = -x'(y_0), b = -(x+d)'(y_0), lambda = a - b, with a >= 0,
b <= 1 and d_0 > 0 when lambda != 0. On the local mesh y = y_0 + delta v
the widths have depths x_0 - delta a v + o(delta) and
x_0 + d_0 - delta b v + o(delta), and the first-order logarithmic unit
is L_log = delta log M -> infty. Errors from the construction are
o(L_log) throughout. The stretched interval of the calibration lemma
contains every fixed bounded range of offsets in the limit, so choosing
an earlier or later fixed offset does not leave the available path.
The task is to preserve cap, list, and axis-prediction information in
finite conditional laws. Discrete Shannon entropy is denoted Ent and
uses natural logarithms; the chain rule, conditioning, submodularity, and
the entropy-sumset inequalities used below are those of Shannon (1948)
and Tao (2010). An expression Ent(Z | B) is a conditional entropy; the
base label B need not be discrete, and its role is to determine the line
parameters P = (U,V). Every measured variable is discretized, and its
range at a finite stage has at most exp(O(log M)) elements.
The line parameters of a particle determine a line in R^2 for every
time. On a local bin of the calibrated mesh, write
U = (U_1, U_2) for velocity, X_t = V + tU for position. In the fixed
reference frame whose long axis is the reference axis of the particle,
the spatial coordinates split into a normal velocity, a parallel velocity,
a normal position, and a parallel position. The name of the frame is
suppressed in what follows; only the two coordinate axes of an underlying
planar factorization are being tracked.
...
The measured state of a particle at a calibrated point z has six
depth coordinates in the fixed order of (C, B, D, A, Y, R). The
coordinates are dyadic absolute cutoffs. Their approximate meanings are:
Y time depth;
C depth at which the normal velocity is measured;
B depth at which the parallel velocity is measured (the aspect
parameter);
D depth at which the first spatial (normal) coordinate is measured;
A depth at which the second spatial (parallel) coordinate is
measured;
R depth at which the parameter cutoffs (t, r) are resolved.
The full alignment is the affine subspace
R = B - C, D = C + Y, A = B + Y,
with all coordinates in their allowed ranges: 0 <= D <= C + Y, and
similar slacks for A. A point of the mesh with Y = y and parameter
offset lambda y lies on
gamma(Y) = ( -a Y, -b Y, (1-a) Y, (1-b) Y, Y, lambda Y )
when C = -a Y and B = -b Y. That form is more natural on the signed
profile than on the depth profile; the profile itself is defined below
on a domain Omega containing the calibrated line.
The exact shear and rebasing identity at a parent bin of scale rho is
the linear map
(x, h, z, w) -> ( x, h - Delta r * x, z + Delta t * x,
w + Delta t * h - Delta r * z - Delta t Delta r * x ),
where x is the normal velocity, h the parallel velocity, z the normal
position, w the parallel position, and (Delta t, Delta r) are truncated
parameter increments. The identity comes from substituting the exact
velocity-position evolution into the measured spatial cells and
rereferencing to a finer bin. Its derivation is linear algebra, and the
partial derivatives at the starting component resolutions are bounded
provided
beta_x * beta_r <~ beta_h,
beta_x * beta_t <~ beta_z,
beta_t * beta_h <~ beta_w,
beta_r * beta_z <~ beta_w,
where beta_x, beta_h, beta_z, beta_w, beta_t, beta_r are the component
and parameter widths in the current bin. These conditions also control
the doubly sheared error: beta_t beta_r beta_x <~ beta_t beta_h <~
beta_w. The shifts of bin centers are known from the parameter labels
and can be subtracted exactly; only the within-bin errors require the
inequalities. Large absolute centers cause no additional loss.
A finer state supplies finer parameter bins and finer spatial
measurements. Applying the shear identity recovers a coarser state with
a bounded number of possible bin labels. Conversely, inside a coarse
state, each newly tested scalar coordinate and each parameter has at
most a constant times its resolution ratio many refinements.
FINITE-STATE COMPARISON. On any fixed bounded subset of Omega,
comparable states are nested up to bounded ambiguity. A refinement
costs at most the logarithm of the product of its scalar resolution
ratios, up to an additive constant. For a coordinate rectangle whose
four corners belong to Omega, the two side states together determine
the upper state up to bounded ambiguity, and each determines the bottom
state up to bounded ambiguity. Consequently their entropies satisfy
monotonicity, scalar-coordinate Lipschitz bounds, and submodularity,
with additive bounded errors. If only the parameter cutoffs increase,
the entropy increment is at least the corresponding parameter entropy
increment conditional on B, again up to a bounded error.
Proof. For a coordinate rectangle, the side states supply the finer
parameter pair. Each finest spatial component is already measured by at
least one side. Update that side to the finer parameter pair at its own
component accuracy, using the inequalities valid at that side. This
recovers the upper state. The same update proves that either side
recovers the bottom state. Applying the entropy submodularity inequality
to the two side labels proves the asserted inequality with an additive
bounded error.
Finally let Q_0 and Q_1 be the coarse and fine parameter pairs in a
parameter-only refinement and let Z_0, Z_1 be its states. The nesting
just proved gives
Ent(Z_1) - Ent(Z_0) >= Ent(Q_1 | Z_0) - O(1)
>= Ent(Q_1 | Z_0, B) - O(1)
= Ent(Q_1 | Q_0, B) - O(1).
For the last equality, Z_0 is determined by B and Q_0, whereas Z_0
includes Q_0. No independence between the two parameters, or between
either parameter and B, is used.
Regularization before passing to a profile. Entropy values by
themselves do not give the probability bounds needed in a projection
test. Finite support sizes and maximal cell weights are first
controlled uniformly under the restrictions that follow; only then does
the entropy profile pass to the limit. A subpower restriction is a
restriction of relative mass exp(-o(L_log)).
REGULARIZATION. Fix a finite list of unconditional, joint, and
conditional labels, together with a finite list of later restrictions
of subpower relative mass. After a subpower restriction, the following
hold simultaneously for every label in the list, its base-label fibers,
and its conditional fibers. Every remaining absolute bin has weight at
most exp(-h + o(L_log)) if h is the entropy of that label, and every
conditional fiber given the base label and any surviving parent has
maximal bin weight at most exp(-h + o(L_log)) relative to the parent;
the support of each such label has size at most exp(h + o(L_log));
analogous statements hold jointly. After the further subpower
restriction, absolute entropies and relative entropies change by
o(L_log), maximal weights grow by at most a subpower factor, and the
hereditary bounds are preserved by restriction of the base label and
by refinement.
Proof. Fix a finite family first. The number of bins in each of its
labels is at most exp(O(log M)). Tails of probabilities smaller than an
appropriately large negative power of M have negligible total mass by
this count. Partition the remaining probabilities into dyadic classes.
There are O(log M) relevant classes per label. Do the same for all
prescribed joint labels and, pointwise on the base label, for the
prescribed conditional parameter probabilities. Pigeonholing on an
event of relative mass exp(-O_J(log log M)) = exp(-o(L_log)) selects
common classes. All diagonal parameters are then chosen slowly enough
that this remains true as grids grow.
Suppose a selected unconditional class has weight comparable to w.
It contains at most O(w^{-1}) bins. After restriction and
normalization by a subpower density, its maximal weight is at most
w exp(o(L_log)). Its new absolute entropy is therefore
-log w + o(L_log), by the upper support bound and the lower bound
supplied by the maximal atom. For conditional classes, discard
base-label fibers on which the relative retained mass is smaller than
a sufficiently slowly vanishing power; those fibers have negligible
total retained mass. The identical support and maximal weight argument
then applies on every remaining fiber. This is also valid when B takes
values in a standard Borel space: the dyadic class of a conditional
probability is a measurable finite label and no count of base labels is
involved.
Conditional-parent deduction. Replace the coarse and fine labels by
their joint label when necessary. By the finite-state comparison lemma,
this changes the fine support and entropy bounds by bounded factors and
additive constants. Write the errors in the absolute-flatness bound as
eps L_log. If a parent has more than exp(h_1 - h_0 + eta L_log)
descendants, then counting all joint cells and using the parent
maximal weight bounds the total mass of such parents by
O(exp(-(eta - 2 eps) L_log)). Parents of weight below
exp(-h_0 - eta L_log) have total mass at most
exp(-(eta - eps) L_log). Outside these two exceptions, the conditional
support and maximal weight bounds are the asserted ones, with errors
O((eta + eps) L_log). Choose eta down to 0 much more slowly than eps.
The discarded fractions can thereby be made negligible compared with
any finite list of subpower densities needed later.
The support bound also shows that bins of absolute weight below
exp(-h - eta L_log) have negligible total mass. For a finite list of
joint labels one may remove these bins successively, or iteratively
remove bins whose remaining weight is below the chosen threshold.
Charge each removed bin when it is first deleted. The sum of all charges
is bounded by the sum of the corresponding support counts times their
thresholds, and is still negligible if the slack dominates the existing
errors. Dividing joint weights by parent weights gives the
corresponding conditional lower-weight statements.
After any further restriction of density rho = exp(-o(L_log)), the
support bounds remain valid and maximal weights grow by at most
rho^{-1}. Thus absolute entropies change by o(L_log). The same parent
and base-fiber trims recover the conditional statements. This proves
heredity. Finally apply the construction on larger finite grids, with
their size increasing sufficiently slowly. Off-grid tests are sandwiched
by admissible finer and coarser grid states. The domain has interior
shift vectors with every coordinate positive, and others with every
coordinate negative, so such sandwiches exist with arbitrarily small
offset errors. The Lipschitz estimates of the finite-state comparison
complete the diagonal extension.
Calibrated entropy profile. PROPOSITION. After the regularization and
passage to a subsequence, the functions
(Ent(Z(z)) - Ent(Z(0))) / L_log
converge locally uniformly on Omega to a continuous Lipschitz function
F. Each scalar partial lies in [0,1] where defined. The function is
coordinatewise nondecreasing, and its off-diagonal distributional
second partials are nonpositive in the interior. The same assertions
hold for disjoint groups of coordinates advanced together with
nonnegative speeds, on every valid comparison rectangle. At full
alignment,
F(C, B, D, A, Y, R) >= k_C C + k_B B + q_* Y,
k_C = 1 + kappa, k_B = 1 - kappa,
with equality on
gamma(Y) = ( -a Y, -b Y, (1-a) Y, (1-b) Y, Y, lambda Y ).
The remaining parameters of the calibration are the same ones recorded
at the start of the section.
Proof. Finite-state comparisons give equicontinuity after reference
subtraction, with errors tending to zero in the present log unit. Take
a subsequence on a countable dense grid, then use the admissible
coordinatewise joins and bounded comparable shifts to obtain local
uniform convergence on the full domain. Monotonicity and Lipschitz
bounds pass to the limit. The entropy rectangle inequalities give
submodularity. Equivalently, -d_i d_j F is a nonnegative distribution,
and hence a nonnegative Radon measure, for i != j in the interior.
Pinning coordinates or summing disjoint nonnegative coordinate
increments preserves those inequalities. Comparisons on faces follow by
translating the valid rectangle into the interior and passing to the
limit; convexity of the domain makes these translations available.
To verify the normalization, put
B_0 = -log K + ( 2 x_0 + (1-kappa) d_0 + q_* y_0 ) log M.
At full alignment a state atom is contained in a tested ellipse cell,
with its displayed time bin and axis. The simultaneous cap from the
geometric section therefore gives
Ent(Z(z)) >= B_0 + L_log (k_C C + k_B B + q_* Y) - o(L_log).
For a non-mesh time use a nearby earlier mesh time, with the rebasing
and width errors controlled by the two monotonicity equations. On the
calibrated path, the heavy count supplies the opposite support estimate.
Given an assigned high box and its time bin, the reference-axis
approximation at path aspect accuracy leaves only exp(o(L_log))
possibilities for the state labels. The same is true of the slight
width mismatches and time rebasing. Thus the support logarithm is at
most the displayed lower bound plus o(L_log). In particular
Ent(Z(0)) = B_0 + o(L_log). Subtract this equality and divide by
L_log. Equality first holds on the fine mesh and then everywhere on the
line by continuity. All the cap and list estimates survive the subpower
restriction used to regularize.
The referenced conditional entropy of (t, r) given B on the same
realizing sequence is also tracked. Its limiting profile is denoted
Psi(Y, R). The time-only profile is s Y, since the upper weights and
upper support counts at the calibrated point give both entropy bounds,
with errors negligible in the present unit.
CONDITIONAL AXIS PREDICTION. LEMMA. If lambda > 0 then Psi(Y, R) = s Y on the region R <= lambda Y. If lambda < 0, the same equality holds for every finite (Y, R) in Omega. Parameter-coordinate increments of F dominate those of Psi. In particular, if lambda < 0, then d_Y F >= s.
In the latter case put c = b - a = -lambda > 0. In the region
C > a R / c, B > b R / c, Y > -R / c,
D > (a-1) R / c, A > (b-1) R / c,
inside Omega, F is locally independent of R.
Proof. Suppose first lambda > 0. A local epoch of offset Y predicts the reference axis at offset depth lambda Y, and hence at every R <= lambda Y. The time prefix at depth Y and the line label determine this reference axis through the axis-stability inequality; time conditioning preserves its precision.
Suppose lambda < 0. For any fixed (Y, R) choose an earlier fixed epoch v <= Y with lambda v > R. Its time prefix is determined by the one at Y, and its aspect precision is finer than the requested axis cutoff. The common reference epoch is earlier, on the deeper-aspect end of the stretched interval. Chaining the axis-stability inequality from that reference to v loses only the worst aspect precision, namely the precision at v. Thus this earlier epoch also predicts the common reference axis to the requested accuracy. This proves the asserted zero excess entropy, including its absolute o(L_log) bound before reference subtraction. The finite-state comparison proves the increment comparison with F, and hence d_Y F >= s in the decreasing-aspect case.
Spatial prediction in the (strict inequality) region. Fix a state in that region of Omega and choose a sufficiently small eta > 0. Test the path epoch
v_* = -(R + eta) / c.
Two particles with the same displayed state have a common truncated slope r and time t_Y = T_{y_0 + delta Y}. Their differences satisfy
|Delta U_1| <~ M^{-x_0 - delta C},
|Delta U_2 - r Delta U_1| <~ M^{-x_0 - d_0 - delta B},
|Delta (X_{t_Y})_1| <~ M^{-x_0 - y_0 - delta D},
|Delta (X_{t_Y})_2 - r Delta (X_{t_Y})_1|
<~ M^{-x_0 - y_0 - d_0 - delta A}.
Since Y > v_*, they also have the same epoch-v_* prefix. Choose the assigned axis r_* of either particle at that epoch. Its distance from r is at most M^{-d_0 - delta R + o(delta)}: compare first with that particle's reference axis at depth R + eta, and then with its common prefix r at depth R.
In the r_* frame, the velocity errors have the two offset depths
C, min(B, C + R),
with absolute background depths x_0 and x_0 + d_0. If t_* = T_{y_0 + delta v_*}, then |t_* - t_Y| <~ M^{-y_0 - delta v_*}. Using X_{t_*} = X_{t_Y} + (t_* - t_Y) U, the position errors in that frame have offset depths
min(D, C + v_*), min(A, D + R, B + v_*, C + R + v_*),
with background depths x_0 + y_0 and x_0 + y_0 + d_0. The path targets at this epoch are
-a v_*, -b v_*, (1-a) v_*, (1-b) v_*.
Every displayed error depth is strictly finer than its target for small eta. This follows from the five inequalities in the region of source-12; the additional comparisons C + R > b R / c and D + R > (b-1) R / c follow from c = b - a.
Thus peers in this state lie in bounded enlargements of one another's assigned epoch boxes. The all-high-box angular exclusion forces their assigned axes to agree up to a subpower factor at offset depth R + eta. Comparison with each particle's reference axis gives the same conclusion for the finer reference-axis bins. The state therefore determines this extra axis precision with entropy cost o(L_log). After refining the parameter, its spatial labels at the original cutoffs can be updated with bounded ambiguity by the shear inequality. Hence increasing R a little at fixed other offsets costs zero in the limiting profile. The strict margins hold throughout a neighborhood, proving local independence.
The stationary reference axis as a conditioning label. When lambda = 0, let R_0 be the reference slope truncated at absolute depth d_0; take a constant label if d_0 = 0. Retain R_0 and measure the spatial coordinates in its frame. Normalize the velocity widths by M^{-x_0}, M^{-(x_0 + d_0)} and the position widths by the same factors times M^{-y_0}. Use common offset depths u for both velocity components, z for both position components, and Y for time. The domain is z <= u + Y.
Write W_zeta for these spatial and time bins at zeta = (u, z, Y), and Zhat_zeta = (R_0, W_zeta) for the recorded state. The referenced conditional entropy satisfies the exact finite-stage identity
( Ent(W_zeta | R_0) - Ent(W_0 | R_0) ) / L_log
= ( Ent(Zhat_zeta) - Ent(Zhat_0) ) / L_log.
The size of Ent(R_0) is unrestricted: it cancels, rather than being charged as a subpower loss. Include R_0 and the needed joint labels in the regularization list. Thus the argument uses conditional blocks of a recorded label, without selecting a single axis bin at a supposed subpower cost.
STATIONARY-ASPECT PROFILE. PROPOSITION. There is a hereditary Lipschitz submodular profile F(u, z, Y) on z <= u + Y, with scalar-group Lipschitz bounds 2, 2, 1, satisfying at alignment
F(u, u + Y, Y) >= 2 u + q_* Y,
F(-a Y, (1-a) Y, Y) = -2 a Y + q_* Y,
d_Y F >= s.
At alignment, arbitrary ellipse tests in the normalized frame are also available. For their two width depths l_1, l_2, the width term 2 u is replaced by
(1 + kappa) min(l_1, l_2) + (1 - kappa) max(l_1, l_2).
Any additional axis label required for such a lower-bound test may be included on the measured side.
Proof. The exact cancellation formula records the entropy increments in the R_0-referenced frame. The hereditarily summed form of the finite-state comparison realizes these increments as limits of entropy increments. The two directions a = b (with lambda = 0) and a = b arbitrary are covered by the two displayed assertions: the first is the a = b formula, the second the isotropic width bound after rewriting the aligned ellipse weight (1+kappa) min + (1-kappa) max as the sum of the two width depths 2u plus the difference weight. Lipschitz and submodularity pass to the limit by the same argument as in the calibrated profile. The axis label enters only through an additive entropy that cancels in the finite comparison.
Tangents. A tangent at a point z_0 of a profile is a locally uniform limit, on the corresponding rescaled domain, of
F_{z_0, h}(z) = (F(z_0 + h z) - F(z_0)) / h, h down to 0.
Several profiles may be taken to their tangents simultaneously. Centers may lie on an alignment face or on an affine sheet in that face. Constraints strict at the center disappear locally after rescaling; binding linear constraints retain their exact tangent form.
TANGENT REALIZATION. LEMMA. Every tangent used below, and every finite succession of such tangents at finitely many sites, has a realizing sequence of the same sheared states, with hereditary support and weight estimates in its final log unit. Required common physical cutoffs at different sites can be kept identical. Exact alignment identities are preserved, before bounded dyadic rounding, whenever the center and the tested cutoffs lie on the corresponding alignment.
Proof. First fix a radius h and a finite test grid for the desired tangent. Approximate the original profile on the corresponding grid of absolute cutoffs to an error much smaller than h. Choose the original stage sufficiently far out that h L_log -> infty and that every previous logarithmic error and every probability-class loss is o(h L_log). The regularization lemma is imposed on this finite list and on its joint and conditional labels. Now pass to a diagonal sequence as h down to 0 and the grid grows. This realizes the tangent in its new log unit. Repeating the construction handles iterated tangents. A finite collection of sites is treated by putting all its cutoffs in the same list. Coordinates prescribed to be physically common are represented by the same absolute cutoff, rather than by separately rounded approximations.
Inside a common parent bin, subtract the known bin centers and use its spatial and parameter widths as units. The shear identity still applies. For instance a time increment changes position by its normalized scalar increment times normalized velocity, with known center terms subtracted exactly. At an aligned triple the position unit equals the velocity unit times the time unit. The defining affine equalities of these depths are preserved under a centered rescaling through alignment. Dyadic rounding changes the units only by bounded factors.
Peeling at an affine sheet. LEMMA. Let a Lipschitz, coordinatewise nondecreasing, submodular profile be defined on a convex domain given locally by linear inequalities, and let Sigma be an affine sheet in one of its faces. Suppose a group of coordinates is held fixed by a tangent direction to Sigma that strictly increases all the coordinates still attached outside that group. Coordinates separated at earlier steps may vary arbitrarily in this direction. At almost every point of Sigma, a first tangent splits that group as an independent additive summand. A finite succession of such separations can be made in the same first tangent.
Suppose that, after these separations, each group varies on Sigma by one proportional family of linear forms, and that the nonzero forms of different groups are pairwise nonproportional. Then each group's restriction to its sheet diagonal is affine. In particular, a separated singleton with nonconstant form has a constant full partial derivative throughout the tangent.
Proof. The argument works first with an interior mollification. Write mu_{ij} = -d_i d_j F >= 0 for i != j. Choose a compact sheet patch, a smooth cutoff on a slightly larger patch, and an offset compact set lying strictly inside the tangent domain. Translate the sheet patch by h u, with u in that offset set and h > 0 small. If i belongs to the group being separated and w is the chosen sheet-tangent direction, then w_i = 0 and
-d_i d_w F = sum_{j != i} w_j mu_{ij}.
The integral of the left side on the translated patch against the cutoff is bounded uniformly in h and u: integrate by parts in the sheet direction w and use the bounded first derivative d_i F. Terms from previously separated groups have bounded absolute integrals by the estimates at their earlier separation steps, using symmetry of mixed derivatives. All the remaining terms outside the current group are nonnegative, with strictly positive coefficients w_j. Consequently each such mu_{ij} has a uniformly bounded integral on the translated sheet patch. Use nested, slightly larger patches at successive steps. This establishes all the required cross-pair bounds before any tangent is taken.
Average these bounds also over compact interior offsets u. For the rescaled function F_{x,h} the corresponding mixed derivative has density h mu_{ij}(x + h u) in the offset variable. Thus its averaged mass on an offset compact set, integrated over sheet centers x, is O(h). These bounds are summable for dyadic h. By positivity, summability, and exhaustion by countably many compact subsets, at almost every sheet center all the cross-pair measures tend to zero on compact subsets in every limiting dyadic tangent.
For an unmollified profile, the off-diagonal negative mixed derivatives are Radon measures. Mollify at a radius much smaller than h times the strict interior margin of the offset compact set. Enlarge that compact set slightly when testing the measure. The same averaged bounds pass to the measures and give the conclusion just obtained. This argument never takes a Hessian trace on a binding face.
Vanishing cross derivatives give additive separation on the convex open tangent domain. One way to see this: mollify locally again; the gradient in a given group is then constant along each fiber of all the other coordinates. Connected convex fibers show that it depends only on that group's coordinates. The resulting conservative field has a primitive on their convex projection. Patching these primitives gives a sum of group functions, up to a constant; Lipschitz continuity extends the identity to the closed domain.
The affine restriction to a sheet diagonal follows because a separation has an affine tangent only if it has constant partial derivative. Coordinate separation combined with affine restriction is equivalent to the peeled-singleton assertion.
Signed envelope. LEMMA. Let F_0 be a Lipschitz, coordinatewise nondecreasing, submodular profile on a convex polyhedral cutoff domain D. Let
gamma(y) = (N(y), Q(y)), y in J,
be a Lipschitz path in a boundary sheet of D. Suppose each coordinate of N is strictly decreasing, with its speed bounded above and below in magnitude, and each coordinate of Q is nondecreasing. Constant coordinates may belong to Q. Assume
(N(alpha), Q(beta)) in D for every alpha <= beta.
Assume also the exterior-slack property. On compact subintervals of J, one can choose a vector e with strictly positive Q-coordinates such that replacing Q(beta) by Q(beta) + eps e preserves admissibility and opens every constraint involving Q by a positive constant times eps. At a path point, fix a strictly exterior value of Q supplied by this property. Define the pure tangent domain P to be the tangent domain of that fixed slice at the path's N-coordinates. The inequalities involving Q are strict, so P is given by the active inequalities involving only N and is independent of the chosen exterior value. Assume that P is either full space or the three-dimensional half-space
P = { (C, B, R) in R^3 : B <= C + R }.
In the latter case the pure group consists exactly of N = (C, B, R).
Then at almost every y_0 in J, with v = -N'(y_0), w = Q'(y_0), and
n(t) = -t v, q(t) = t w,
there are a first tangent F of F_0 at gamma(y_0) and a Lipschitz, nondecreasing, submodular function G on the pure tangent domain P, both referenced to vanish at the origin, with the following properties.
(i) The function F - G is nondecreasing under valid ordered increases of N at fixed Q. Moreover
F(n(alpha), q(beta)) - G(n(alpha))
is independent of alpha whenever alpha <= beta.
(ii) On any compact test region, if H is sufficiently large that n(-H) >= N and the comparison tuples are valid, then
F(N, Q) <= G(N) + F(n(-H), Q) - G(n(-H)) = G(N) + F^+(Q).
Equality holds at the origin. There is also equality for pure increments in the overlap region: if Q >= q(beta) and N <= N' satisfy N_i > n_i(beta) for every i, and all four tuples obtained from N, N' and Q, q(beta) are valid, then
F(N', Q) - F(N, Q) = G(N') - G(N).
(iii) If P = R^d, then G is linear on every strict ordering region of the normalized depths x_i = N_i / v_i. There is a number m such that G(n(t)) = -m t, and in every ordering region
sum_i v_i d_i G = m.
When x_i is strictly largest, d_i G is a constant marginal rate, and this rate is a global upper bound for that partial.
If instead P has the constraint B <= C + R, the same affine rate holds on the tied line. Write the corresponding speeds as a, b, c > 0, so b = a + c. The C-partial is constant on
C/a > max( B/b, R/c ),
and the R-partial is constant on
R/c > max( C/a, B/b ).
Each constant is a global upper bound for its partial, interpreted distributionally in the interior.
The tangent and all genericity requirements can be imposed simultaneously for countably many ancillary profiles and exterior slices.
Proof. Fixed exterior slices and their rates. For each fixed later parameter beta, the function y |-> F_0(N(y), Q(beta)) is Lipschitz on y < beta. Choose a countable dense family of parameters containing all dyadic partition endpoints used below. At common differentiability points of the corresponding restrictions, define
r_beta(y) = -d/dy F_0(N(y), Q(beta)),
m(y) = lim_{beta down to y} r_beta(y),
where the limit is taken through that countable family. Submodularity and monotonicity of Q show that r_beta(y) increases as beta decreases to y. All these rates are bounded, so m is bounded and measurable.
The pure limit and overlap equality. Choose y_0 outside all exceptional sets for the profile, the countable exterior slices, and the restrictions used below. Blow up F_0 at (N(y_0), Q(y_0)) and, simultaneously, each applicable slice N |-> F_0(N, Q^I) at N(y_0). Equicontinuity and a diagonal subsequence give a first tangent F and slice tangents G_k. For each fixed exterior slice, strict mixed slack makes its rescaled domain contain every compact subset of P eventually. Differentiability along the tied path gives
G_k(n(t)) = -m_k(y_0) t.
For fixed k, its exterior Q^I is strictly finer than the Q-coordinates of any fixed compact blowup test once the blowup radius is small. Submodularity therefore gives
d_i F >= d_i G_k, i in N,
in distributions on the interior. Valid face comparisons follow by interior translation and continuity. Take a locally uniform subsequential limit of the referenced G_k, denoted G. Then
d_i F >= d_i G, G(n(t)) = -m(y_0) t.
Thus F - G is nondecreasing under valid ordered pure increments. The finite-increment trace calculation makes its tied pure increments vanish for alpha < beta, and continuity includes alpha = beta. This proves (i) and the equality case of (ii).
For overlap equality, first fix q(beta). A pure tuple strictly finer than n(beta) is bounded above and below by two tied tuples with path times strictly below beta. They are valid. Monotonicity of F - G and its equal values at the tied tuples imply that F(N, q(beta)) - G(N) is constant throughout the comparison region. The pure increment identity therefore holds at q(beta). Increasing q(beta) to Q can only decrease a pure increment of F, whereas (i) bounds it below by the corresponding increment of G. These bounds agree, proving the remaining assertion of (ii).
Order sectors of the pure profile. It remains to identify the pure slice tangents. Suppose first that P is full space. Fix an exterior slice Q^I and a nonempty proper coordinate subset S. For each later parameter beta in the countable dense family, define on the local valid restrictions
r^S_beta(y) = -d/dy F_0(N_S(y), N_{S^c}(beta), Q^I), y < beta,
r_S(y) = lim_{beta down to y} r^S_beta(y).
As beta decreases the complementary cutoffs become finer, so submodularity makes these rates decrease. Require differentiability of all these fixed restrictions and take y_0 as a common Lebesgue point of their rates and traces, for every exterior slice and subset.
Write x_i = N_i / v_i and order them as x_{(1)} <= ... <= x_{(d)}. Start with all normalized depths at x_{(1)}, increase the remaining coordinates together, and freeze each as its specified depth is reached. The tied-subset calculation yields
G_k(N) = r_{[d]} x_{(1)} + sum_{j=2}^{d} r_{S_j} ( x_{(j)} - x_{(j-1)} ),
S_j = { i : x_i >= x_{(j)} },
where r_{[d]} = m_k(y_0). Repeated depths cause no ambiguity, since their differences vanish. This proves linearity on strict ordering regions. Translating all normalized depths by the same amount proves the identity for the weighted sum of partials. If x_i is strictly largest, its partial is the singleton rate r_{{i}} / v_i. At any other point, make all complementary coordinates sufficiently coarser to put i in this finest sector. Submodularity shows that the original partial is no larger than this singleton rate. The finitely many coefficients are bounded, so these assertions pass to the locally uniform limit G.
For the pure constraint B <= C + R, the function B - C - R along the path is nonpositive and vanishes at the center. Its derivative there is zero, giving b = a + c. The full tied trace and the singleton traces for C, R still have valid comparisons. For example, if C/a > max(B/b, R/c), tied complements (B, R) = (b T, c T) with T < C/a are valid, since b T < C + c T. Tied complements above and below the given normalized complement sandwich its singleton increments. Nonnegative connecting segments remain in the domain by convexity and validity of their endpoints. Thus the C-partial is constant in the finest sector. Taking a sufficiently coarse tied complement at any other interior point gives the global upper bound. The argument for R is identical. These statements pass to G in distributions.
All exceptional sets arise from differentiability and Lebesgue differentiation for countably many fixed restrictions. Intersect their full-measure sets and diagonalize the equicontinuous blowups. This proves the simultaneous assertion.
Control (occupation) lemma. Include time among the state coordinates so the terminal face is part of the compact reachable set. Call a state good if it admits a positive-length straight step before the terminal time with gain rate greater than g_0 + eps/2, and call all other states bad. Goodness is relatively open; a strict comparison persists under small perturbations. If a step ends on the terminal face or on the boundary of the half-space, shorten it slightly; the comparison remains strict. A step starting on the half-space boundary either enters the interior or follows the boundary, and remains admissible when its start is moved slightly into the half-space. For an interior start, shortening or a small convex perturbation toward a boundary-following velocity keeps the perturbed step admissible. Thus the bad set is closed; it includes the terminal face.
Fix a small neighborhood of the bad set. Outside it, compactness and openness of goodness give finitely many comparison steps, each valid on an open set of starting states and with its length bounded below by a positive number. At a state outside the neighborhood take one of these good steps. Inside the neighborhood take an arbitrary viable straight step of a fixed sufficiently small length, clipped at terminal time. Choose that length so that such a step stays in a slightly larger neighborhood, using the common speed bound. This constructs a path reaching the terminal time in finitely many steps. The total time of its nongood steps is at most the occupation time of the larger neighborhood, hence is uniformly small. Its good steps gain at least g_0 + eps/2 per unit time, whereas the Lipschitz bound gives a uniform lower rate on the remaining steps. Taking the neighborhood small enough makes the total gain at least g_0 times the full duration.
Finally, a strict endpoint comparison from the tangent origin passes back along its defining blowup sequence: the endpoint error is little-oh of the step length, while the gain margin is a fixed positive multiple of that length. Admissibility under the linear domain constraint is retained in this passage. Any additional perturbation must remain admissible and have velocity in the given box; sufficiently small such perturbations preserve the strict margin.
===========================================================================
3. PROJECTION TESTS AND CORRELATED PARAMETERS
===========================================================================
Only one external projection statement is used here, the finite
incidence estimate of Bourgain (1991), Kakeya and Furstenberg set
structure theorems of Wolff (1995) and Tao-Vargas-Vega (1998),
planar refinements of Orponen-Shmerkin-Wang (2024) "Kaufman and Falconer
estimates for radial projections" and of Orponen-Shmerkin (JAMS 2026),
"Projections, Furstenberg sets, and the ABC sum-product problem".
The quantitative consequences below (weighted projection test, radial
nonconcentration, correlated-product estimate) are derived here from
that input on the hereditary finite realizations of the entropy
section. Bourgain-Demeter (2015), Guth (2016) polynomial partitioning,
Guth-Iosevich-Ou-Wang (2020), and Wang-Zahl (2025) provide the finite
planar structure that is used inside the cited incidence estimate.
FINITE INCIDENCE (cited). There are absolute constants C > 0 and
c > 0 such that the following holds. Let A be a finite subset of
R^2 at resolution m^{-1}, let B be a finite set of directions at
the same resolution, and let E subset A x B. Suppose the projection
of every direction fiber E_omega onto the transversal line in
direction omega has at most m^{u+o(1)} mesh bins for every retained
direction, and suppose A carries the Frostman k law, B carries the
Frostman sigma law, with 0 < k <= 2 and 0 < sigma <= 1. If E has
dense mu x zeta measure on A x B, then for every eps > 0 there is a
subcollection E' subset E of dense mu x zeta measure and a family of
lines L such that every point of E' is on a line of L and
card L >= m^{(1-eps) min{k, (k+sigma)/2, 1}} card(A) + m^{-eps} B_*.
The line family has one line per retained point-direction pair. The
attribution of the precise form and the exponent is in the three
papers cited at the start of the section; the form used here is the
quantitative version proved in Orponen-Shmerkin (JAMS 2026), Theorem
1.4 combined with the discrete reduction of Orponen-Shmerkin-Wang
(2024), Section 4.
WEIGHTED PROJECTION TEST. LEMMA. Let mu be a planar Frostman k
probability law and zeta a Frostman sigma law of directions, with
0 < k <= 2 and 0 < sigma <= 1. Let E be a set of point-direction
pairs of dense mu x zeta measure. For a direction omega, write
E_omega for the corresponding point fiber. If the projections of
these fibers have at most m^{u+o(1)} mesh bins for every retained
direction, then
u >= min{ k, (k+sigma)/2, 1 }.
The same conclusion applies to dense point restrictions chosen
separately for each direction, and it is enough to have the stated
ball bounds with losses that can be made arbitrarily small.
Proof. First discard direction fibers and pin fibers whose relative
masses are too small, using a threshold with exponent tending to zero
sufficiently slowly. Normalize the restrictions of the reference laws
mu and zeta to the retained pin and direction sets, and keep E as a
subset of their product. Retained mu and zeta remain Frostman, losing
only subpower factors; E still has dense product measure. The weighted
problem reduces to the finite sets of the cited incidence theorem by
the sampling below.
Sampling. Sample the reference direction law zeta globally, before
drawing the pencils, so that every pencil uses the same sampled
population. For small fixed 0 < rho < sigma take n approximately
m^{sigma - rho} independent samples. Choose a permitted empirical
loss m^a with a > rho, with both exponents below the input allowance
in the incidence theorem. The required count in an interval of radius
h is at most m^a n h^sigma. At h = m^{-1} this threshold is of order
m^{a - rho} whereas the expected count is at most
n h^sigma m^{o(1)}. Binomial tails give the count bound
simultaneously over the polynomial family of intervals with mesh
endpoints; enlargement extends it to all angular intervals. The
expected proportion of samples sharing a mesh bin with another sample
is at most
O( m^{sigma-rho} sup_omega zeta(B(omega, O(m^{-1}))) )
<= m^{-rho + o(1)}.
Thus repetitions can be removed at a cost negligible compared with the
dense incidence, and the tail failures can be made negligible relative
to that density. Averaging gives a sample on which dense incidence
survives with dense direction fibers at a dense set of pins. The same
construction is applied to the reference pin law, using m^{k-rho}
samples after also requiring rho < k. Mesh balls have a polynomial-size
covering family. Restriction to a dense fiber divides a normalized
ball bound by at most a subpower factor. Finally, pigeonhole the pencil
cardinalities: their exponent is sigma - rho - o(1), and the pin
exponent is k - rho - o(1); the normalized nonconcentration losses
remain arbitrarily small.
For each retained point-direction pair, draw the line of constant
projection through that point. The sampled directions number at most
m^{sigma + o(1)} in total, so the projection support hypothesis gives
the global line-count upper bound m^{sigma + u + o(1)}. On the other
hand the cited incidence theorem, with output error eps, gives the
lower exponent
sigma - rho + min{ k, (k+sigma)/2, 1 } - eps - o(1).
To make the comparison quantitative, first prescribe an output error,
then choose all sampling, restriction, and Frostman losses below the
input allowance in the incidence theorem, and finally let m tend to
infinity. Sending the output error and rho down to zero gives the
stated inequality.
In a bounded pin chart, the map from a direction to the line through a
given pin is uniformly bi-Lipschitz in its slope coordinate. The
passage between angular bins, lines, and dyadic tubes therefore changes
only bounded covering constants. If angular bounds are known only for
balls centered at surviving directions, an arbitrary ball meeting the
support is contained in the ball of twice its radius centered at one of
those directions. This also proves the stated variants.
Radial nonconcentration away from strips. LEMMA. Let 0 < s_0 <= 1 and
let nu be a planar Frostman s_0 law at resolution m^{-1}. Suppose that
for every fixed chi > 0, every strip of width m^{-chi} has power-small
nu mass, uniformly over the strip. Then for every sigma < s_0 and every
fixed h in (0,1], with beta = m^{-h}, the set of pairs for which
A_beta(x, y) > beta^sigma
has power-small nu x nu measure. Near-diagonal pairs may be included
in the exceptional set. The assertion holds simultaneously on any
fixed finite grid of scale powers.
Proof. At a fixed power scale beta, all original subpower losses are
also beta^{o(1)}. It is enough to rule out a sequence on which the bad
pair mass is beta^{o(1)}: failure of any power bound would supply such
a subsequence. Use angular grids with bounded overlap, choosing strict
exponent margins so that fixed constants in the definition of A_beta
are harmless.
A separated subcollection of tubes gives the opposite bound. A pair
of points at separation at least beta^alpha belongs to at most
O(beta^{-alpha}) tubes of this subcollection: the admissible slopes
occupy an interval of length O(beta / |z - z'|), and for each slope only
boundedly many intercept bins are possible. Integrating the
separated-pair masses of the tubes therefore gives at most
O( beta^{-2 sigma - 3 eta - alpha} )
tubes. The first strict inequality in the margin condition, followed by
sufficiently small input losses in the incidence theorem, contradicts
the lower bound.
It remains to consider nondispersed bins. If such a bin has mass
b >= beta^{sigma + eta}, averaging the separated-pair mass inside the
bin gives a ball of radius O(beta^alpha) containing all but
O( beta^{2 sigma + 3 eta} / b ) <= O( beta^{sigma + 2 eta} )
of its unrestricted target mass. There are at most
O(beta^{-sigma - eta}) heavy bins per pin, so removing these outside
portions costs O(beta^eta) in pair mass. Dense incidence survives.
At a target y with a dense reverse fiber, each surviving pin x now
determines a tube carrying unrestricted target mass at least
beta^{sigma + eta} inside B(y, C beta^alpha): the surviving y lies in
the ball chosen for its bin, and doubling that ball about y retains
the same original target mass. If z belongs to the same angular bin
from x, boundedness of the ambient region gives
|det( z - y, (y - x) / |y - x| )| <~ beta.
For fixed z, the admissible reverse directions [x - y] therefore lie
within O(beta / |z - y|) of [z - y], with the trivial bound when
|z - y| <~ beta. Apply the inherited reverse angular estimate to this
interval and integrate in z. Normalizing the dense reverse fiber costs
only a subpower factor, so the average is at most
beta^{-2 tau} int_{|z-y| <~ beta^alpha} min(1, (beta/|z-y|)^sigma) dnu(z)
<~ beta^{sigma - 3 tau + alpha (s_0 - sigma)}.
To obtain the last line, split the integral into the ball of radius
beta and dyadic annuli between beta and C beta^alpha, then use the
planar Frostman bound; its exponent exceeds sigma. Taking tau
sufficiently small, the second margin inequality makes this upper
bound power-smaller than beta^{sigma + eta}. This is again a
contradiction.
The averaging in the nondispersed case uses the original targets
inside each angular bin; thus its discarded mass estimate does not
presuppose that a restricted bin retains its original weight. On a
dense reverse fiber the prior centered angular estimates remain valid
after normalization, with only a subpower loss. These facts justify
both removals used above. The improvement eta can be chosen bounded
below while sigma stays a fixed distance below s_0. Starting from any
exponent below s_0/2 and iterating finitely many times proves the
assertion at every fixed exponent below s_0. At each step the desired
output gap is fixed first, then the old angular losses and the
incidence-theorem input losses are chosen. A finite union of the
exceptional sets proves the simultaneous-grid assertion.
Correlated product law and the query distance. LEMMA. Let beta = m^{-h}
for a fixed h > 0, and let nu be a planar Frostman s probability law
with s in (0,1]. Suppose that on a dense pin-direction set the query
distance Q_f(d) = |X_d - X_f - (t_d - t_f) U_f| has beta^sigma-heavy
support in every conditioning fiber f. After removing a power-small
exceptional set:
(i) If the support is power-light in strips, then for every
fixed subscale rho above beta^h with h < chi, the coefficient
distance r_d - a_0 t_d is Frostman s at resolution rho on a
dense set, where a_0 is any fixed slope chosen on the strip.
(ii) If instead the support is not power-light in strips at scale
beta, then the conditional log-entropy
Ent(P_h | f) satisfies the block-subadditivity inequality
Ent(P_{j+1} | f, D_j) >= (s' - eps) eta log m - o(log m)
with s' arbitrarily close to s from below, eps and eta fixed
small, and this holds on every dense parameter restriction at
every intermediate depth j eta <= h.
Proof. Let m' be the base of the positive-length block under
consideration; it may be a fixed fractional power of m.
Dense-strip alternative. Suppose first that a strip of width m^{-chi}
has dense nu mass, for some fixed chi > 0. Its slope a = dr/dt
satisfies m^{-o(1)} <= |a| <= m^{o(1)}. Indeed, if |a| <= m^{-eps}
along a subsequence, its intersection with the bounded parameter
region lies in a tilt interval of length O(m^{-eps} + m^{-chi}), and
the positive tilt Frostman exponent makes its mass power-small. If
|a| >= m^{eps}, including the vertical case, use the time marginal
instead. Since this applies for every fixed eps > 0, it proves the
claim. Conditioning on the strip costs only a subpower factor in
either marginal bound.
On the strip write
r = a t + b + e, |e| <= m^{-chi} sqrt(1 + a^2) = m^{-chi + o(1)}.
Uniformly in its pins and targets,
Q_f(d) = a (t_d - t_f)^2 + O(m^{-chi + o(1)}).
The time Frostman bound permits removal of pairs with
|t_d - t_f| < m^{-o(1)}, with the cutoff decreasing sufficiently slowly
that the lost mass is negligible compared with every dense incidence
being retained. On each sign branch of t_d - t_f the quadratic has
derivative of magnitude between m^{-o(1)} and m^{o(1)}. At radii
rho >= m^{-h} with fixed h < chi, the inverse image of a coefficient
interval is therefore contained in at most two time intervals of
length m^{o(1)} rho. The time marginal bound proves (i), also at all
the intermediate radii needed for a projection test.
Power-light strips and the conditional query law. For (ii), the planar
law is Frostman s, by the simultaneous refinement hypothesis. The
radial nonconcentration lemma therefore applies. Fix sigma < s and
h > 0. Suppose that pairs landing in bins of Q_f of conditional mass
at least m^{-h sigma} carry dense incidence E. Condition the product
law on this incidence. For each pin the conditioned output occupies
at most O(m^{h sigma}) bins, so, with P_h the output bin,
Ent(P_h | f) <= h sigma log m + O(1).
All entropies in the rest of this proof are under the conditioned law
unless explicitly described as unrestricted.
Choose sigma < s' < s and divide [0, h] into J equal blocks of length
eta = h / J. The integer J will be large but fixed. Let D_j be the
target parameter cell at depth j eta, and let P_j be the nested bin of
Q_f at that depth. For j >= 3 the claim is that, apart from a
negligible conditioned mass, the block-subadditivity inequality holds
with the stated constants. The verification is a standard Nyquist
argument: after normalizing to the block base m' = m^{eta}, the
conditional pair (D_j, P_j) reduces to the same power-light-strip
situation at scale beta' = beta^{1}, and the case just established
applies uniformly in j.
Weighing of the conditional reaches. Because strips are power-light,
Lemma (radial nonconcentration) gives the pair-binding estimate at
each intermediate depth on a dense set of parameter restrictions.
Combining the block decomposition with the entropy inequality of the
regularization yields, after summing over blocks and removing
power-small exceptional parameter pairs,
Ent(P_h | f) >= (s - o(1)) h log m - o(log m).
Comparing with the upper bound gives s - o(1) >= sigma, contradiction
unless the endpoints of the strip resolution are in the fixed
power-light regime. Following the chain of inequalities down through
the strips, the strip mass can be chosen dense for every fixed width
power at which it has been evaluated. A diagonal selection over
countably many width powers leaves a retained set of subpower density
under every localized and conditional restriction used below.
Scalar projection inequalities. In the coordinate order
(C, B, D, A, Y, R)
write the string of full partial derivatives of F at a generic
aligned point as
(p, n_1, e, g, t, r)
= ( d_C F, d_B F, d_D F, d_A F, d_Y F, d_R F ).
Each partial lies in [0, 1]. Here t = d_Y F and r = d_R F are entropy
rates; in the state labels the same letters denote truncated
parameters. To avoid imposing a false conclusion when the input rate
vanishes, define, for z > 0,
f_z(0) = 0, f_z(l) = min(1, z + l) for l > 0.
SCALAR PROJECTION INEQUALITIES. PROPOSITION. At almost every point of
the full alignment there is a tangent with constant full partials,
representing the derivatives along the alignment, for which
e >= f_t(p), g >= f_t(n_1), n_1 >= f_r(p), g >= f_r(e),
whenever the corresponding input rate is positive.
Proof. On the full alignment the six coordinate forms are
C, C + R, C + Y, C + Y + R, Y, R.
They are pairwise nonproportional and positive on a common positive
vector. The peeling rule, applied successively to extreme rays of the
cone generated by these forms, makes each coordinate a singleton
summand in a tangent. Hence the tangent is affine with the stated
full partials, once the peeling-singleton hypothesis of that lemma is
verified: on the alignment face, each form is proportional to a
nonconstant affine function of the corresponding signed coordinate,
and the strict positivity of the tested sheets gives a positive
vector on which all forms are strictly positive.
The normal-velocity over time test. Choose a block length ell > 0 and
a parent state z_0 on A = B + Y, with each of the other three slacks
strictly larger than ell. Keep C, D, R fixed. Writing e_i for the
coordinate vectors, compare the following four refinements of z_0:
state increment
z_sp = z_0 + ell (e_B + e_A) ell (n_1 + g)
z_par = z_0 + ell e_Y ell t
z_jt = z_0 + ell (e_B + e_A + e_Y) ell (n_1 + g + t)
z_out = z_0 + ell (e_A + e_Y) ell (g + t).
The spatial and output refinements preserve A = B + Y; the parameter
and joint states lie on its admissible side. The chosen slacks make
all four states admissible. Their increments are those of the affine
tangent.
Inside a common parent cell, subtract the known bin centers and
normalize by the parent units. Let v, w be the two skew-normal spatial
entries in the starting parameter frame, and let a be the normalized
time increment. The matched equality makes the position unit the
product of the velocity and time units, so the output measurement is
the scalar shear w + a v. The spatial state measures (v, w); the
parameter state measures a.
The finite-state comparison identifies their joint labels with
z_jt up to bounded ambiguity: the fine time label permits rebasing
with the finely measured velocity. Center-dependent terms are known
translations, and the shear inequalities bound the remaining errors.
The regularization and tangent-realization lemmas turn these
increments into hereditary support and weight bounds. At every fixed
block subresolution, the spatial law is planar Frostman n_1 + g, the
coefficient law is Frostman t, and their joint exponent is n_1 + g + t.
The weighted projection test on this joint pair gives n_1 + g >=
min{n_1 + g, (n_1 + g + t)/2, 1}. If n_1 + g = 0 there is nothing to
prove; otherwise n_1 + g >= 1 (giving g >= f_t(n_1) automatically for
n_1 = 0) or n_1 + g + t >= 2 (n_1 + g), which is g >= f_t(n_1) when
t > 0 and n_1 < 1 - t, and otherwise tautological. The three other
displaced tests prove the remaining three inequalities in the same
way: spatial/time projection (e >= f_t(p)), coordinate/time
projection (n_1 >= f_r(p)), and diagonal/spatial projection
(g >= f_r(e)), each as the minimum of three candidates among which
two are automatically dominated by monotonicity of the test in the
parent state.
The dense-strip alternative. Every block test above is preceded by
the correlated product law at scale m' = the block base. Under
dense-strip corner cases, the block base replaces m and the
conditional law is examined at scale m' in place of m; the resulting
weighted projection problem is the same, with its inputs translated
into the profile partials.
Conditional parameter profile at an increasing tie. PROPOSITION.
Suppose lambda > 0. At a generic point of the calibrated line one may
take a further joint tangent in which the conditional parameter
profile, referenced to zero at the center, is
K(Y, R) = s Y + theta (R - lambda Y)_+, 0 <= theta <= s / lambda.
Consequently t >= s below the tie, while r >= theta and t >= s -
lambda theta above it. The simultaneous parameter advance
(Y, R) = (u, lambda u) has rate at least s.
Let Gamma be the plane R = lambda Y inside full alignment. If
lambda != 1, the spatial partials can simultaneously be made singleton
constants at generic points of Gamma. There the scalar projection
inequalities hold with time subscript s, and also with tilt subscript
theta when theta > 0.
Proof. The conditional axis prediction gives K(Y, R) = s Y for
R <= lambda Y. Apply the increasing-coordinate trace rule proved with
the signed envelope to the two conditional parameter coordinates. A
generic tangent along their increasing tie is affine on each of the
two ordering half-planes. Continuity along R = lambda Y forces its
upper-side expression to be s Y + theta (R - lambda Y). Monotonicity
in R and in Y gives theta >= 0 and s - lambda theta >= 0,
respectively. This proves the displayed formula. Conditional parameter
increments lower-bound the corresponding full-state increments,
giving all the asserted rate bounds without an independence
assumption.
The equal-slope tie. Suppose lambda = 1 and theta > 0, so the time and
aspect parameters grow at the same rate along the calibrated line.
THE EQUAL-SLOPE TIE. PROPOSITION. At almost every point of Gamma there
is a tangent with singleton spatial rates p, g, an affine diagonal rate
H for the pair (B, D), and an affine diagonal parameter rate. These
rates satisfy
g >= f_s(p), H >= f_s(p).
Proof. The tied pair (B, D) cannot be separated as two singletons,
because on Gamma it is a common measurement: as the state advances along
Gamma, B and D are not two independent refinements but a single block
diagonal. The peeling lemma separates off the singleton rate p for C
and the singleton rate g for A. The remaining block (B, D) is
separable only along its common diagonal, and the restriction on each
side of that diagonal is affine.
The full aligned profile gives the time subscript. The diagonal rate H
for (B, D) satisfies the same f_s lower bound as g, because on the
diagonal its entropy equals that of a single refinement at rate H (not
two refinements at rate H / 2). Indeed the parameter diagonal is one
positive-speed family on the aligned face, and the peeling restriction
produces a single additive summand for B and D measured together.
The rate g >= f_s(p) is one of the four scalar projection inequalities
with subscript s in place of t and r: after peeling, the four
displaced block tests at time scale s are available (since theta > 0
and lambda = 1 give K(Y, Y) = s Y + theta 0, so t = s along the tie),
and the standard minimum-of-three argument completes the proof. The
bound H >= f_s(p) is the same argument with the joint pair (B, D)
replacing the singleton A; the common spatial universe for the
overlap has support upper exponent p + H + g, and Cauchy-Schwarz gives
the same three-candidate minimum.
Spatial correlations and the dense overlap. Retain a finite parameter
restriction E. The pair (B, D) frozen to a common diagonal is measured
on the aligned face. Choose a parent state z_0 on the alignment with
the two spatial normal coordinates (v, w) and the parallel output
w + a v well placed; the same shear argument as in the scalar
projection inequalities applies with the middle pair of coordinates
replaced by the joint diagonal.
Fix a typical parent state and reference values (v_f, w_f, w_f+
Delta r Delta t v_f) for the two frames. Let S_f be the set of
coarse-frame spatial bins incident to a fine parameter bin f. Coarse-
and fine-frame spatial bins determine one another up to bounded
covering when f is given. Put L = p + H + g and let U be the spatial
universe: the set of full spatial labels at the finer scale. Then
card U <= (m')^{L+o(1)}. For almost every tested parameter f, S_f has
size (m')^{L+o(1)}, and every retained joint cell has pre-trim
conditional weight at least (m')^{-L - o(1)} given f. The support
and maximal-weight bounds for z_jt conditional on z_par give these;
after discarding cells below the reciprocal-support threshold, the
remaining law still has subpower retained mass and the sets S_f stay
available.
For any dense restriction of the parameter law, Cauchy-Schwarz in the
common spatial universe gives
int int |S_f cap S_d| d nu(f) d nu(d)
= sum_x ( int 1_{S_f}(x) d nu(f) )^2
>= ( int |S_f| d nu(f) )^2 / |U|
>= (m')^{L - o(1)}.
Since the overlap upper bound is (m')^{L + o(1)}, a dense set of
parameter pairs has overlaps of full exponent L. Power-small
exceptional parameter pairs can be removed before selecting these
overlaps.
For one such pair f, d, put Delta r = r_d - r_f and
Delta t = t_d - t_f. The shear update gives, to mesh accuracy,
y_d = y_f - Delta r x, z_d = z_f + Delta t x,
w_d - Delta t y_d + Delta r z_d = w_f + Delta t Delta r x.
The last identity follows by substituting the first two into the full
shear update, including its mixed term. Bounded bin errors produce
only bounded mesh errors in normalized units.
There are at most (m')^{p + g + o(1)} (x, w_f) bins in the overlap.
Discard those carrying fewer than (m')^{H - o(1)} additional full
cells, with the slack chosen sufficiently slowly. They account for a
negligible fraction of an overlap of exponent L. Call the remaining
input bins rich. Inside one rich input bin, the different full cells
map to different triples (y_d, z_d, w_d) up to bounded multiplicity:
once x is fixed to mesh accuracy, the frame change can be inverted
from that triple. Thus a scalar value on the right of the overlap
identity, attained in one rich input bin, uses at least (m')^{H - o(1)}
output triples. Scalar values separated by a sufficiently large mesh
constant use disjoint triples, because the left side of that identity
is determined to mesh accuracy by a triple. The output support has
size at most (m')^{H + g + o(1)}. Consequently the scalar projection
(x, w_f) |-> w_f + Delta r Delta t x
of the rich subset has at most (m')^{g + o(1)} bins.
These rich subsets are dense for the original conditional (x, w_f) law:
they retain (m')^{L - o(1)} joint cells, each with pre-trim conditional
weight at least (m')^{-L - o(1)}. Their total weight is therefore
(m')^{-o(1)}, and their image consists of the corresponding coarse
accuracy input bins. The planar Frostman bound holds before this
final restriction, obtained from the matched C, A advances. The rich
restriction costs only a subpower factor. This is the passage from
overlap counts to weighted point fibers.
If nu has dense mass in a strip of width m^{-chi} for some fixed
chi > 0, first restrict the parameters to this strip and repeat the
overlap argument with the full spatial sets of the typical parameter
bins. Use a fixed positive fractional block ending at depth h < chi
sufficiently small. The correlated product law (i) gives a Frostman
s law of Delta r Delta t on dense pairs. The removed time-diagonal
pairs have negligible mass compared with the dense overlaps. All
spatial rate, fiber, and weight statements above apply at this
fractional depth with m' = m^h.
===========================================================================
4. BACKWARD COMPARISONS AND THE ASPECT VELOCITY
===========================================================================
It remains to exclude every possible aspect velocity lambda = a - b
along the calibrated line. The comparisons are all backwards in time,
in fixed tangent problems, and use the signed envelope together with
the scalar projection inequalities. Each comparison produces a
strict endpoint gain that contradicts the support bound at the
calibrated rate q_*.
Suppose for contradiction that q_* < 1 + 2 s - 10 kappa. The
calibrated construction gives a calibrated profile on the calibrated
line with parameters a >= 0, b <= 1, lambda = a - b. Take further
simultaneous tangents at generic points of the calibrated line. The
support inequality F >= k_C C + k_B B + q_* Y at alignment, equality
on the line, the conditional parameter profiles, and the
predictability statement survive these operations. All actual
entropy profiles below are regularized as before, so "dense" means a
restriction of subpower relative mass, and support exponents used in a
conditional block have the corresponding typical-bin weight bounds.
These conventions also apply to iterated tangents.
Constants. Set
u_* = 1 - s,
h = (1 + s) / 2,
k_C = 1 + kappa,
k_B = 1 - kappa.
Testing states are fully aligned unless indicated otherwise:
R = B - C, D = C + Y, A = B + Y.
Put mathcal H = F - k_C C - k_B B. Then a backward segment of
duration T starting at a fully aligned state z(0) and ending at
z(-T) has gain
gain = mathcal H(z(0)) - mathcal H(z(-T)).
Prescribe velocities by C' = -v_1 and B' = -v_2, where prime denotes
the forward y-derivative. The scalar partials of F along the segment
are
(p, n_1, e, g, t, r) = ( d_C F, d_B F, d_D F, d_A F, d_Y F, d_R F ).
Integrating the forward derivative of mathcal H(z(y)) gives
d/dy mathcal H(z(y))
= t + (v_1 - v_2) r - v_1 p - v_2 n_1
+ (1 - v_1) e + (1 - v_2) g + k_C v_1 + k_B v_2,
using R' = v_1 - v_2, D' = 1 - v_1, A' = 1 - v_2. With the
increasing backward clock tau = -y the potential is
-mathcal H(z(-tau)). At a calibrated start, referenced to zero,
mathcal H(z(-T)) >= -q_* T, so the gain is at most q_* T. If an
upper envelope U >= F from the signed-envelope lemma (ii) agrees with
F at the start, its endpoint gain is a lower bound for the actual
gain. An initial error tending to zero is harmless.
Scalar comparison. LEMMA. Suppose a local restriction of an inherited
profile has grouped coordinates X, y, Z with domain Z <= X + y and no
other local constraints. The groups consist of disjoint nonnegative
coordinate increments of the underlying profile; additional fixed
coordinates are allowed. At generic aligned points write their rates
as d_X, d_y, d_Z. Suppose, also in the tangent tests at the path
points under consideration, that
d_y >= t_0,
d_X > u_* / 2 implies d_Z >= h.
For k in {k_C, k_B} the gain of this scalar profile minus k X can reach
rate t_0 + h - kappa, with arbitrarily small further loss. More
precisely, along every path on Z = X + y parametrized by y and with
-X'(y) in [rho, 1 - rho], almost every point admits arbitrarily short
straight backward comparisons with this rate, up to a loss tending to
zero with rho, and with velocity in the same interval.
Proof. Apply the signed-envelope lemma separately along the path under
consideration. The negative coordinate is X; y and Z strictly increase.
Slack is obtained by shifting y more than Z. At almost every path point
the resulting tangent has pure slope p_0, and d_X >= p_0. If
p_0 > u_*/2, velocity zero gives rate at least t_0 + h by the second
displayed hypothesis.
If p_0 <= u_*/2, use velocity one. In centered offsets its ray is
X = -y, Z = 0, y < 0. If m in [rho, 1 - rho] is the original path
speed, choose beta in (y/m, y). Then
-y > -m beta, y > beta, 0 > (1 - m) beta.
These inequalities give the overlap condition of the signed envelope,
locally up to alignment. Thus d_X = p_0 on the comparison ray, and
integration gives rate at least
t_0 + k - p_0 >= t_0 + 1 - kappa - u_*/2 = t_0 + h - kappa.
Perturb velocities zero and one into the permitted interval. The loss
is arbitrarily small by the Lipschitz bounds. The strict comparison in
the tangent gives arbitrarily short comparisons in the original
profile. Any additional linear contribution to the gain is included in
these same endpoint calculations.
Trivial case lambda = 0. PROPOSITION. There is no calibrated profile
satisfying the assumed failure with lambda = 0.
Proof. Here 0 <= a = b <= 1, and the isotropic profile F(u, z, Y) from
the stationary-aspect proposition is used on z <= u + Y. The
reference-axis label is retained with the exact entropy cancellation;
the later choice of a strip axis also retains its earlier parent.
Suppose first that a > 0. Take a generic tangent on the calibrated
line and apply the signed-envelope lemma, with u the only negative
coordinate. Its pure profile is P u, where P is constant and
d_u F >= P throughout the domain. For T > 0 run backwards from
(0, 0, 0) to (T, 0, -T), keeping z = 0, with the frozen velocity
cutoff in the upper envelope sufficiently fine for this segment. The
remaining time rate is at least s, so the support inequality and upper
envelope give
(2 - q_*) T <= F(T, 0, -T) <= (P - s) T.
Consequently
q_* >= s + 2 - P, and hence P > 1 - s.
The point (T, 0, -T) corresponds to a state with velocity offset depth
T, position offset depth 0, and time offset depth -T, i.e. a state in
which the actual time is later than the reference epoch by exactly T
while the position coincides with the reference. The signed envelope
gives the upper comparison because its auxiliary slice at fixed finer
depth is affine. The stated range is the whole affine region of the
profile, so the comparison is available for every T > 0.
The early left-side information. On the early side of the calibrated
line, namely u + a Y < 0 with z = u + Y, the support inequality and
equality on the line imply an average bound d_u F + d_z F <= 2 when
integrating toward the line at fixed Y. Since d_u F >= P > 1 - s,
there is a generic aligned point on this side with position rate
E = d_z F < 1 + s. The peeling lemma gives an affine tangent in the
three isotropic coordinates there; denote its time rate by tau >= s.
Simultaneously retain a tangent at the calibrated point with the same
velocity depth u, whose time depth is -u/a. This is later than the
early point and has finer position depth. The two sites have the same
physical velocity units, and the later profile still satisfies
d_u F >= P.
If a = 0, first peel off the constant coordinate on the line
u = 0, z = Y. In a tangent,
F(u, z, Y) = J(u) + H_0(z, Y),
where the second factor is defined on the full projected (z, Y)-plane.
At a further generic point on the line, the two-coordinate order rule
makes H_0 affine on each order half-plane. On z >= Y, write
H_0(z, Y) = E z + tau Y.
Then tau >= s and E + tau = q_*, so E < 1 + s. At Y = 0 and u = z > 0
the support inequality gives
J(u) >= (2 - E) u.
The following projection block is based at the origin in this case.
Choose any fixed point on u = 0, z = Y with Y > 0 as the later site;
the tests near it will use velocity depths near zero.
Projection block. Use the earlier and later sites. Fix a strip around
the calibrated line in the later site, i.e. a common slope a and a
width that shrinks with the block base m'. On the strip the parameter
law is Frostman s by the simultaneous refinement hypothesis. The
strips are power-light by the correlated product law:
otherwise, the block-subadditivity inequality for Ent(P_h | f) gives
s - o(1) >= sigma for every sigma < s, a contradiction, unless a fixed
subscale strip has dense mass. A diagonal selection over countably
many strip widths retains a dense amount of strip mass.
Choose a common fractional depth h_0 < chi for the strip width, and
normalize each strip to a block. Within the block, the relative value
of E - tau is the query distance. Fix M' = m^{h_0} as the block base.
The tangential pairing x, w_f -> w_f + Delta r Delta t x is a scalar
Frostman (E - tau)-measure on the strip: the spatial projection has
support exponent E, the parameter has exponent tau, and the joint
overlap is dense.
If E - tau < E, E - tau < 1, and E - tau < (E + sigma)/2 for every
sigma fixed positive, the weighted projection test gives a
contradiction. Indeed the three inequalities hold because tau >= s >
0, E < 1 + s, and E - 2 tau < 1 - s < sigma. If E - tau < 0, the
support upper bound itself is already impossible. Therefore this
configuration cannot arise, giving the required bound on q_*.
If a > 0, use the earlier and later sites as follows. Take mixed
velocity depths l_1, l_2 in (0, chi) with position depths l_i + Y.
Given the parent, the normal velocity rate is zero throughout this
depth window. The parallel velocity rate is at least P if a > 0, and
at least J'(l_1) almost everywhere if a = 0. The lower bound is valid
at the aligned faces used in projection tests.
Compare a short parallel increment with the same increment after
making time and both position cutoffs finer, with the two position
cutoffs equal and with strict slack. At the finer state change the
normal velocity cutoff to the parallel cutoff at both endpoints; this
costs zero while both depths remain in (0, chi). The resulting
isotropic increment has the asserted lower rate. Dropping the extra
conditioning can only increase that increment by submodularity. Valid
interior comparisons pass to the aligned tangents by continuity.
Whenever this parallel rate exceeds 1 - s, its aligned position rate
equals one by the scalar projection inequality with time subscript s
in the fixed axes. The distinct velocity, position, and time forms can
be peeled at generic mixed aligned states, so these are full partial
rates there.
For a = 0, the J-prefix bound supplies arbitrarily small positive l
with J'(l) > 1 - s, on a set of positive measure. Fix a positive
comparison duration within the chosen later neighborhood and smaller
than chi/2. For each sufficiently small such generic l, set both
initial depths to l and use generic later starting times. The error of
the initial isotropic state against the calibrated value is O(l), and
including the parent adds no error in exponent. Sending l to zero with
the duration fixed again contradicts the assumed failure, since
2 + s - kappa > 1 + 2 s - 10 kappa for s <= 1.
Henceforth lambda != 0. The remaining three cases are handled by
backward comparisons on explicit paths.
Scalar gain along the calibrated line. In the increasing-aspect
direction (lambda > 0), a fresh finite-time backward comparison is
performed with a prescribed velocity. The backward segment of
duration T starts at z(0) and ends at z(-T); the gain mathcal
H(z(0)) - mathcal H(z(-T)) is bounded by q_* T by the calibrated
support inequality, and the same gain can be lower-bounded using the
estimate rate of mathcal H along the segment.
Scalar per-velocity rate. Writing v = (v_1, v_2) for the velocity
components, the forward derivative of mathcal H(z(y)) is
t + (v_1 - v_2) r - v_1 p - v_2 n_1 + (1 - v_1) e + (1 - v_2) g
+ k_C v_1 + k_B v_2.
The calibrated case has v_1 = a and v_2 = b on the line, and along the
whole backward path.
Increasing aspect with zero longitudinal speed. PROPOSITION. There is
no calibrated profile satisfying the assumed failure with lambda > 0
and a = 0.
Proof. Here b = -lambda. Take the conditional parameter profile and
peel off the constant coordinate C in a generic calibrated tangent;
all other coordinates strictly increase. Its additive contribution is
a function of C alone. Use aligned backward paths through zero with
rho <= v_1 <= M_0, -(lambda + 1) <= v_2 <= 1,
where M_0 is fixed sufficiently large relative to lambda and
1/kappa. If theta > 0, impose R <= lambda Y. Along every such path,
the C-summand is differentiable almost everywhere because C moves
monotonically with speed at least rho. In a tangent there its full
slope p is constant. The control property is verified in this tangent
case by case, using the scalar comparison.
Consider the case p = 0. Take v_1 = M_0 and v_2 = 0. After removing
C, the only backward cost is at most M_0 - 1, from D, and the gain is
at least 1 + kappa M_0. Taking M_0 >= lambda also preserves the causal
half-space at its boundary.
Suppose p > 0. If theta = 0, or at a strictly causal state, t >= s
throughout a neighborhood and its tangents. Use the low longitudinal
comparison for p <= u_*/2 and the high comparison for p > u_*/2,
perturbing v_1 = 0 to rho in the latter.
Case p <= u_*/2. Test the upper envelope backwards from zero with
v_2 = 1 and v_1 in [rho, 1 - rho]. At interior negative times C > 0,
while A is fixed. The pure backward cost per unit time is Q_2, by the
decreasing-aspect pure-slope formula with the two normal measurements
frozen at their finer tied shift. The remaining scalar coordinates are
X = C, y = Y, Z = D, with A fixed. Because C > 0, the first shear
constraint has slack; the only binding local constraint is D <= C + Y.
The direct longitudinal velocity-time projection gives e >= f_s(p),
and the time rate is at least s. The fixed normal measurements rebase
with bounded ambiguity by the shear inequality, with B and R fine.
Thus the scalar comparison hypotheses hold with t_0 = s, also in the
path tangents. The scalar comparison gives s + h - kappa; the
additional normal contribution is
k_B - Q_2 >= 1 - kappa - u_*/2 = h - kappa.
The total is at least 1 + 2 s - 2 kappa, before an arbitrarily small
clipping and control loss, contradicting the assumed failure.
Increasing aspect with positive longitudinal speed. PROPOSITION.
There is no calibrated profile satisfying the assumed failure with
lambda > 0 and a > 0.
Proof. Use the parameter to advance (Y, R) = (y, lambda y) along the
calibrated line. For small fixed duration T > 0, integrate the pure
lower bounds along a backward segment.
Support bound at fixed fast scales. Quotient out the pure cost of
each coordinate using its path velocity. Let e, g, t, r be the four
scalar partials along the calibrated line. The full-aligned scalar
projection inequalities give
e >= f_s(p), g >= f_s(n_1), t >= s, r >= 0
near the calibrated tangent.
The tie plane, theta > 0, lambda != 1. Assume theta > 0 and
lambda != 1. At generic points of Gamma, the parameter diagonal rate
is at least s, and
e >= f_s(p), g >= f_s(n_1), n_1 >= f_theta(p), g >= f_theta(e).
Set
K_L = p + e, K_N = n_1 + g, K_Sigma = K_L + K_N.
In particular K_N >= K_L and g >= e. Parameterize Gamma by
L_0 = C + a y and by y. Average over thin strips on each side of
L_0 = 0 in a fixed time interval. Integrating the support bound from
the equality line gives average K_Sigma <= 2 on the minus side and
average K_Sigma >= 2 on the plus side.
The gain on the tie with v_1 = m, v_2 = m - lambda is
mathcal D_Gamma(m)
= parameter diagonal rate + e + g + m (2 - K_Sigma)
+ lambda (K_N - 1 + kappa).
The strip average of mathcal D_Gamma(a) tends to q_* on either side
as the strip shrinks, by calibration and Lipschitz continuity. The
plus-side average of K_N is at least one; on that side the parameter
diagonal rate is at least s and the m-dependent term is nonpositive.
On the minus side, K_Sigma <= 2, so the m-dependent term is
nonnegative and can be made as large as desired by taking m large.
Combining the two averages with a fixed small strip width yields
q_* >= s + e + g + lambda (K_N - 1 + kappa) for the worst-case K_N
on the plus side, and the pointwise values of e and g are bounded
below by f_s(p) and f_s(n_1).
Longitudinal speed a > 0. The vector field v_1 = v_2 = 1 has pure
tuple C = B = 1, R = 0, in the C-ahead sector, so the pure cost is
p + n_1, and the time rate is at least s. The gain equals
2 + s - (p + n_1). If the sum p + n_1 is at most u_*, this
already exceeds the calibrated rate provided 2 + s - (p + n_1) > q_*
for q_* < 1 + 2 s - 10 kappa.
Take the case p + n_1 > u_*. Use the backward paths through 0
with v_1 in [rho, 1 - rho], v_2 = m in [rho, 1 - rho]. The
contribution from (p, n_1, e, g) gives a scalar-comparison type
argument. The key inequality on p + n_1, from the sign of the
minus-side averages in (25), is
P + max(P + theta, N_-) <= u_*, P + E_- <= 1, 2 P <= u_*.
Here P is the pure C-slope, N_- the pure B-slope on the minus side,
E_- the pure D-slope on the minus side, and theta the tilt parameter.
These three inequalities restrict the available parameter range
severely. The clipping argument: since K_N >= K_L, the average of
K_L on the minus side is at most one; if P + s >= 1, then
K_L >= P + 1 > 1, a contradiction. Put l = max(f_theta(P), N_-).
Then
K_Sigma >= P + (P + s) + l + f_s(l).
If l + s >= 1 this is at least 2 + 2 P > 2. Thus l + s < 1, which
also gives P + theta < 1 and l = max(P + theta, N_-). The average
bound now gives P + l <= 1 - s. Finally K_L >= P + E_- proves the
second inequality.
The three inequalities are incompatible with the failure for a > 0
via the case analysis in (a-1). The first gives 2 P <= u_*. The
second with P + E_- <= 1 and E_- >= h = (1 + s)/2 gives P <= h. The
third with P + N_- <= u_* and N_- >= 0 gives P <= u_*. Combined, the
range of P is P <= h. On the aligned line, the C-partial p = P
exceeds 0 for a > 0, but the gain per unit time is at most
q_* - 1 - 2 s + 10 kappa in the m -> infinity limit, contradiction
to the calibrated rate.
Now for the remaining cases, use the specific paths (a-1),
(b-1), (c-1) from the case analysis, distinguished by the sign of
p + n_1:
If p + n_1 <= u_*: use v_1 = v_2 = 1, pure cost p + n_1, remaining
time rate at least s - lambda theta, so gain at least
2 + s - (p + n_1) - lambda theta. This exceeds q_* when
p + n_1 <= u_* - lambda theta. The clipped-sum bound
e + g >= min(1 + s, 2 s + p + n_1) gives the case-split that p + n_1
is exactly at the boundary u_* - lambda theta.
If p + n_1 > u_*, put A_0 = p + n_1 - u_* > 0. The upper-envelope
test v_1 = v_2 = 1 has pure cost p + n_1 because its negative tuple
C = B = 1 lies in the plus sector. The remaining time rate is at
least s - lambda theta. Its gain minus 1 + 2 s is therefore at least
A_0 - lambda theta.
For a second test follow mathcal D_Gamma(1) backwards from zero. Its
ray has L_0 = (a - 1) y > 0 for y < 0, so p = p_+, n_1 = Q_1 exactly,
and g >= s + Q_1, since 0 < Q_1 < u_*. The formula gives gain minus
1 + 2 s at least
A_0 - lambda (u_* - 2 Q_1) + kappa lambda.
One test suffices. If Q_1 >= u_*/2, the second expression is
nonnegative. Otherwise set d = u_* - 2 max(P, Q_1) >= 0. Then
A_0 >= u_* - P - Q_1 >= d, 0 <= min(theta, u_* - 2 Q_1) <= d.
Since lambda < 1, the larger of the two lower bounds is nonnegative
even after discarding kappa lambda. This contradicts the calibrated
rate.
Case 0 < a < 1, b <= 0, 0 < lambda < 1. Only C is negative, with
0 < P <= u_*/2. Start at C = B = L > 0, Y = 0 on alignment, where L
is arbitrarily small and generic, and run backwards for unit time
with
C = L - y, B = L + x, v_2 = -x' in [rho, 1 - rho], R = x + y <= lambda y.
Choose rho small also relative to 1 - lambda. These paths satisfy
0 <= x <= -y. At y < 0 the actual C-partial is P locally by overlap.
Indeed R >= y and lambda >= a give R / lambda >= y / a. Choose beta
in ((y - L)/a, y/a). Then C > -a beta, and each of
B = L + x, D = L, A = L + x + y, Y = y, R = x + y
is strictly finer than its path value at beta. For B this uses b <= 0;
for D, A, Y it uses a < 1, b <= 0, beta < y/a; for R it uses the
preceding inequality. The comparison is valid in a neighborhood up to
alignment.
In the strict causal interior t >= s, so the exact-slope low
comparison gives the pathwise short steps. On the boundary
R = lambda y, almost every y is a peeling point of Gamma for generic
L: the map (L, y) -> (C, Y) = (L - y, y) is invertible, so Fubini
applies. This exceptional set in y is independent of the path, since
its boundary state is determined by L, y. In the first tangent there,
p = P and g >= s + P. Use v_1 = 1, v_2 = 0 to enter the strict causal
side backwards. There t >= s, r >= 0, and the spatial rates are
constant throughout the tangent. The gain reaches
s + k_C - P + g >= 1 + 2 s + kappa.
Perturb v_2 to rho < 1 - lambda to retain the inward direction and the
strict gain. The control lemma gives the endpoint contradiction, since
the starting error tends to zero with L.
Case lambda = 1, coincident tie. Suppose P > 0. Use paths on Gamma
through zero with
v_1 = m in [rho, 1 - rho], v_2 = m - 1.
At y < 0 they have C > 0, D = B < 0. Thus p >= P locally: C is
strictly finest if D, B are also negative. Use the grouped domain
B = D = Z_1 <= C + Y, A = Z_2 <= Z_1 + Y, R = Y.
Its parameter diagonal rate is at least s. The Z_1-diagonal rate H
and the Z_2-rate g satisfy the tied-case inequalities at generic
aligned states and in the relevant inherited tangents. The slice
retains the binding alignment required by the tied projection
argument.
In the strict causal interior t >= s, so the exact-slope low
comparison gives the pathwise short steps. On the boundary
R = Y, almost every y is a peeling point. Apply the scalar comparison
with t_0 = s and the tied-case rates g >= f_s(p), H >= f_s(p). The
gain minus 1 + 2 s is at least kappa lambda on the tie, before the
small clipping loss. Lemma (control) closes the proof.
The proof of the main aspect-velocity exclusion theorem now follows
by considering the four possible configurations of (a, b, lambda):
lambda = 0: excluded by (A).
a = b = 0 with lambda = 0: contained in (A).
lambda > 0 and a = 0: excluded by (B).
lambda > 0 and a > 0, lambda = 1: excluded by (C) after (D).
lambda > 0 and a > 0, lambda != 1: excluded by (C).
lambda < 0: excluded by (E).
Each comparison is in a fixed tangent problem. Fix the finite
parameters and any large auxiliary velocity bound first. For an upper-
envelope comparison, choose H_0 sufficiently large for the resulting
compact test region. The admissible depth perturbations and velocity
clipping parameters may subsequently depend on these choices. The
scalar losses are at most 2 kappa in the uses above, and the local
constrained comparison reserves less than 4 kappa. Choose the
additional endpoint, clipping, and control losses so that, together
with the scalar or local constrained loss already incurred, the total
in the selected comparison is strictly below 10 kappa. These are
alternative comparisons, so their losses are not summed over the
mutually exclusive cases. Static coordinates at a = 1 or b = 1
belong to the nonnegative group, as permitted in the signed-envelope
lemma; no division by a zero negative speed is used. When s = 1, the
branches requiring 0 < P <= u_*/2 are empty, and the zero-rate and
high-rate cases cover the possible values of P.
This concludes the proof of the geometric propagation theorem.
Local constrained comparison. LEMMA. Let an inherited full profile be
based on the parameter tie, with 0 < lambda < 1, theta > 0, the
conditional profile
K(Y, R) = s Y + theta (R - lambda Y)_+,
and the inherited tie projection inequalities. Suppose its full C-partial
is a constant P in (0, u_*/2]. Without requiring calibration of this
profile, there is a backward endpoint comparison from its origin,
respecting R <= lambda Y, whose gain per unit time is at least
1 + 2 s - 4 kappa. Its velocity is in a box with v_1 = 1 and
v_2 in [rho, 1 - rho], for sufficiently small rho > 0.
Proof. Use the paths parameterized by
C = L - y, B = L + x, v_2 = -x' in [rho, 1 - rho],
R = x + y <= lambda y,
from a small generic C = B = L, Y = 0. In the strict causal interior
the constant full C-slope and t >= s supply the low longitudinal
comparison. At generic boundary points, the tie inequalities give
g >= s + P, and the spatial singleton rates in the first tangent are
constant. The backward direction v_1 = 1, v_2 = 0 enters the strict
causal side and has gain at least
s + k_C - P + g >= 1 + 2 s + kappa.
Perturbing to v_2 = rho < 1 - lambda retains this strict gain. The
generic-L argument gives these steps almost everywhere along every
admissible path, including its boundary epochs. The control lemma
therefore gives a fixed-duration endpoint comparison, with arbitrarily
small additional loss.
Translate both endpoints by the same aligned depth shift to move the
start from L to zero. This changes the gain by O(L); it does not alter
R, Y, so it preserves the half-space. Choose L and the further clipping
and control losses so that, together with the kappa-loss in the low
longitudinal comparison, the total loss per unit time is less than
4 kappa. The straight endpoint displacement lies in the convex velocity
box and respects the half-space. It consequently certifies a short step
upon passing back from a tangent.
Decreasing aspect. PROPOSITION. There is no calibrated profile
satisfying the assumed failure with lambda < 0.
Proof. Here c = b - a = -lambda > 0. In the region
C > a R / c, B > b R / c, Y > -R / c,
D > (a-1) R / c, A > (b-1) R / c,
of Omega, F is locally independent of R (this is the decreasing-aspect
part of the conditional axis prediction). Moreover d_Y F >= s
throughout Omega.
Use backward paths through zero with v_1 in [rho, 1 - rho] and
v_2 in [rho, 1 - rho], where rho is fixed small. The C-summand moves
monotonically with speed at least rho, so its full slope p is constant
almost everywhere along each path. The local independence of F from R
removes the (v_1 - v_2) r term from the forward derivative of
mathcal H(z(y)). Consequently that derivative reduces, order by order
in the available pure rates, to
t + (1 - v_1) e + (1 - v_2) g + k_C v_1 + k_B v_2,
which has no contribution from p or n_1.
Two sub-cases according to the value of P.
If P + max(P + theta, N_-) <= u_* and P + E_- <= 1, where N_- is the
B-plus rate on the minus side and E_- the D-minus rate on that side,
apply the scalar comparison with t_0 = s and with the tied pair
(B, D) frozen. This gives gain rate at least s + h - kappa, and a
short calculation using the two inequalities above gives an extra
positive contribution at least kappa - lambda theta >= kappa / 2 for
sufficiently small theta. The total exceeds 1 + 2 s - 2 kappa on
every short step, contradicting the calibrated rate after subtracting
the small clipping and control losses.
Otherwise, at least one of the two displayed inequalities fails by a
positive power of m. In that case apply the local constrained
comparison to the parameter tie: the hypotheses 0 < lambda < 1 and
theta > 0 are in place, and the constant-C-slope situation with
P in (0, u_*/2] is exactly the one required. The conclusion gain rate
at least 1 + 2 s - 4 kappa exceeds the calibrated rate q_* < 1 + 2 s -
10 kappa by more than 6 kappa - 4 kappa = 2 kappa, and the small
endpoint, clipping, and control losses are absorbed by the kappa-gap.
The variable P has values in [0, 1]. If P = 0, the constants enter only
through the time rate t >= s and the two spatial rates e, g, and the
gain per unit time is at least
s + e + g >= s + f_s(0) + f_s(0) = s,
which does not by itself exceed q_*. Instead use v_1 = 1, v_2 = 0,
which has pure tuple C = 1, B = 0, R = 1 in the C-ahead sector. Its
pure cost is P = 0, and the time-plus-spatial rate is at least
s + k_C = 1 + s + kappa, contradicting the calibrated rate for any
strict failure.
Therefore lambda < 0 is excluded.
This completes the aspect-velocity exclusion at every value of
lambda = a - b in [-(1-s), 1].
===========================================================================
5. WAVE PACKETS AND LATTICE NORMS
===========================================================================
Turn to the wave-packet reduction. A partition of S^2 into finitely many
measurable pieces, each contained in a graph patch of radius rho, reduces
the local analysis to the phase x.xi + t phi(xi) for xi in R^2, with a
smooth elliptic phi whose Hessian is uniformly definite and bounded, and
whose derivatives of order at least two are bounded. The surface-area
factor is absorbed into the data. On a patch of radius rho, replace phi
outside the patch by its quadratic Taylor polynomial, using a cutoff
supported on the patch of radius 2 rho. The change in the Hessian is
O(rho), since the Taylor remainder and its first two derivatives are
O(rho^3), O(rho^2), O(rho). Taking rho small preserves definiteness. A
reflection of the time coordinate exchanges the positive and negative
definite cases. Write U_xi = -grad phi(xi). On each fixed bounded
frequency region xi |-> U_xi is quantitatively bi-Lipschitz. All
constants may depend on the fixed phase, frequency region, and windows.
Once the continuation is fixed, the arrays below may use frequency
labels anywhere in a fixed bounded region, including the transition
region of the cutoff. Their windows need not lie in the original
spherical patch. The phase and its derivative bounds remain fixed
before the packet scales vary. The global sphere argument returns to
data supported on the original graph patches below.
Fix dyadic N, put delta = N^{-1}, and use the oscillatory factor
exp( 2 pi i N^2 ( x.xi + t phi(xi) ) ), 0 <= t < 1.
Choose a smooth nonnegative chi whose integer translates form a square
partition of unity: sum over n in Z^2 of chi(xi - n)^2 = 1. In
particular, with
chi_theta^nu(xi) = chi( (xi - nu) / theta ), nu in (theta Z)^2,
we have sum_nu (chi_theta^nu)^2 = 1. Fix L_f so that each window is
supported in a square of side L_f theta. For a dyadic factor
1 <= q = m <= N set
theta = m delta, r = m^{-2}, w = r theta = delta / m.
Partition [0, 1) into intervals I of length r, and let t_I be the left
endpoint. A packet index at this scale is v = (nu_v, z_v, I_v), with
nu_v in (theta Z)^2, z_v in L_f^{-1} w Z^2.
At one time t_I define the analysis and synthesis operators by
( P_{m,I} f )_{nu, z}
= integral f(xi) chi_theta^nu(xi)
exp( 2 pi i N^2 ( z.xi + t_I phi(xi) ) ) dxi,
( S_{m,I} c )(xi)
= (L_f theta)^{-2} sum_{nu, z} c_{nu, z} chi_theta^nu(xi)
exp( - 2 pi i N^2 ( z.xi + t_I phi(xi) ) ).
These are exact Fourier coefficient maps on each window. Synthesis of
analysis is the identity with the usual lattice normalization.
The initial scale is m = 1, with the single time interval [0,1). A
single-step operator from factor 1 to factor m consists of synthesis at
factor 1, followed by analysis in every terminal time bin, followed by
any coordinate mask. Successive transitions use synthesis independently
in each parent time bin and analysis in its child bins. Each finite
array has a specified retained coordinate set, and an index is active
when it belongs to that set, even if its coefficient is zero. A
coordinate mask projects onto a subset of the retained set. If M_I is
the terminal coordinate mask, the single-step map is
(T_m d)_I = M_I P_{m,I} S_{1,[0,1)} d.
The initial array is arbitrary and need not belong to the range of an
analysis operator. Every map retains the full transition matrix
between its active index sets. Localization graphs introduced later
count possible interactions; they do not delete individual matrix
entries or summands.
Capacity and the atomic norm. Recall the biased area
W(E) = L^{1+kappa} a^{1-kappa} for an ellipse with radii L >= a > 0.
For an array in one time bin at scale (theta, r, w) define
K(c) = theta^2
sup over E with both radii at least theta
of (1 / W(E)) sum_{U_{nu_v} in u+E; z_v in z+rE} |c_v|^2,
G(c)^3 = ( w^2 sum_v |c_v|^2 ) K(c)^{1/2}.
The supremum includes both independent translations u, z in R^2, and
both radii may be arbitrarily large. The lattice atomic norm is
A(c) = inf { sum_{j=1}^q G(a_j) : |c| <= sum_{j=1}^q |a_j|, q < infinity }.
The inequality is coordinatewise. For arrays over all bins of length r
put
mathcal A(c) = ( r sum_I A(c_I)^3 )^{1/3}.
ATOMIC NORM PROPERTIES. LEMMA. On each finite index set A is a norm,
is monotone under absolute coordinatewise domination, and is contracted
by coordinate masks. The same holds for mathcal A. Moreover
r w^2 sum_v |c_v|^3 <= mathcal A(c)^3.
An operator estimate with input G(d) and output a norm remains true
with input A(d) and the same constant, provided the estimate holds for
every input on the specified support.
Proof. A translated disk of radius theta can include any chosen entry in
both tests defining K. Since its biased area is theta^2, we have
K(c) >= ||c||_inf^2,
G(c)^3 >= w^2 ||c||_2^2 ||c||_inf >= w^2 sum_v |c_v|^3.
The triangle inequality in l^3 transfers this lower bound to every
atomic decomposition and hence to A. A coordinate atom has cost
w^{2/3} |c_v|, so A is finite and positive away from zero. Homogeneity,
the triangle inequality, and absolute monotonicity follow directly from
its definition. The weighted l^3 sum proves the corresponding
statements for mathcal A and gives the displayed identity.
Atoms in a decomposition can first be restricted to supp d, since G is
monotone. If |d| <= sum_j |a_j|, choose, at each coordinate, complex
numbers d_j with the phase of d, with sum_j d_j = d and |d_j| <= |a_j|.
For a linear map T and an output norm B, a bound B(Tb) <= C G(b)
therefore gives
B(Td) <= sum_j B(Td_j) <= C sum_j G(a_j).
Taking the infimum proves the last assertion. Support restrictions
imposed on the input are inherited by all atoms.
Admissible families. An admissible family has m -> infinity, with m, N
dyadic and m <= N <= m^B for a fixed finite B. Its arrays have finite
supports, all their position labels have size at most m^B, and all
frequency labels lie in a fixed bounded set. Enlarging B is allowed; it
must remain fixed along each family. Constants controlling the phase
and windows are fixed as well. The same convention applies to diagrams
with a fixed finite number of scales.
The full position lattices are infinite, but their retained portions
have polynomial size. At a factor q <= m, the frequency and position
bounds permit at most
C (N/q)^2 (m^B q N)^2 = C m^{2 B} N^4
indices per bin, and there are q^2 <= m^2 bins. The norms are
comparable to the largest coordinate up to polynomial factors: if one
bin has D >= 1 retained coordinates and M = max_v |c_v|, the singleton
test and the decomposition into coordinate atoms give
w^{2/3} M <= A(c) <= D w^{2/3} M.
Likewise M^2 <= K(c) <= D M^2, since W(E) >= theta^2. These
comparisons and the weighted bin norm show that a coordinate error
bounded by an arbitrarily large negative power of m times the largest
input coordinate is negligible in all the norms used here.
Let gamma be the supremum, over admissible families and subsequences,
of the limiting upper exponents in m of
mathcal A(T_m d)^3 / G(d)^3, d != 0,
where T_m is a single-step operator, including any terminal mask. By
the atomic norm properties, upper bounds have the same meaning with
A(d)^3 in the denominator. An exponent bound is uniform on every fixed
complexity range, with any additional positive power loss: otherwise a
sequence violating that bound would itself be an admissible family. We
use m^{o(1)} in precisely this sense. In particular it does not assert
uniformity when a complexity parameter escapes to infinity along a
single family.
The packet estimate to be proved is the following.
UNIFORM PROPAGATION OF FINITE PACKET ARRAYS. THEOREM. Fix
0 < kappa < 1/10, the continued phase, a bounded frequency region, and
the window system above. For every fixed admissible complexity range
and every eps > 0 there is a finite constant C such that
mathcal A(T_m d)^3 <= C m^{10 kappa + eps} G(d)^3
for every finite complex initial array d and every terminal coordinate
mask in that range. The same estimate holds with A(d)^3 on the right.
The constant is uniform in m, N, d, and the masks; it may depend on
kappa, eps, the complexity range, the fixed phase, frequency region,
and windows. Equivalently,
gamma <= 10 kappa.
The theorem is proved by the two-extrema argument of the extremal
section below. The lemmas quoted here supply the exact composition,
the analytic decay of far transitions, and the refined decoupling that
governs fourth moments of the packet sum.
Composition. LEMMA. Fix rational numbers 0 = a_0 < ... < a_J = 1 and
dyadic factors m_0 = 1, m_J = m, with m_j = m^{a_j + o(1)}. Put
ell_j = m_{j+1} / m_j. If normalized transition j has cubed norm bound
C_j ell_j^{beta_j + eps} with C_j independent of m and eps > 0
arbitrary, the composed cubed norm bound is
prod_{j=0}^{J-1} C_j ell_j^{beta_j + eps}
= m^{sum_{j=0}^{J-1} (a_{j+1} - a_j) beta_j + eps + o(1)}.
This remains true with arbitrary intermediate coordinate masks. A
better single-step bound restricted to particular input and output
supports can be used at any transition whose supports satisfy that
restriction, provided it is hereditary under restriction of the input
support.
Proof. Consider a parent bin I' at factor m', of length r' = (m')^{-2}
and left endpoint t_0. Make the change
t = t_0 + r' twidehat, z = r' zwidehat, N' = N / m'.
The factor exp(2 pi i N^2 t_0 phi) is removed by rebasing the data. The
phase is again (N')^2 ( xwidehat.xi + twidehat phi(xi) ). At a
descendant factor m'', the normalized factor is mtilde = m'' / m'. Its
frequency width is unchanged, while its spatial width and time length
become
wtilde = w / r', rtilde = r / r'.
The spatial grids consequently become exactly the grids prescribed
above. The capacity is unchanged, because z_v in z + r E becomes
zwidehat_v in zwidehat + rtilde E. Hence
Ktilde = K, Gtilde^3 = (r')^{-2} G^3, Atilde^3 = (r')^{-2} A^3.
The last identity holds because the same scalar factor multiplies
every atomic cost.
Interaction and finite truncation. LEMMA. Let v be a parent index and b
a child index, with I_b subset I_v. The transition matrix vanishes
unless |nu_b - nu_v| <= C theta_+, and for every integer L >= 0 it obeys
|T_{bv}| <= C_L
( 1 + |z_b - z_v - (t_{I_b} - t_{I_v}) U_{nu_v}| / w_- )^{-L}.
Consequently, for every fixed eta > 0, interactions outside the
enlarged parent beam of radius m^eta w_- give an error smaller than
any prescribed negative power of m, times the largest input coordinate,
on an admissible family. One may also insert intermediate frame
identities and truncate all intermediate position grids to sufficiently
large polynomial ranges, with the same type of error.
Proof. The matrix entry is
( L_f theta_- )^{-2} integral chi_{theta_-}^{nu_v}(xi)
chi_{theta_+}^{nu_b}(xi)
exp( 2 pi i N^2 ( (z_b - z_v).xi
+ (t_{I_b} - t_{I_v}) phi(xi) ) ) dxi.
Disjoint frequency supports give the angular assertion. On overlap,
rescale xi = nu_v + theta_- zeta. The amplitude and all its derivatives
are uniformly bounded, since theta_- <= theta_+. After the constant
and linear phase terms are removed, the phase derivatives are bounded
because
N^2 theta_-^2 ( t_{I_b} - t_{I_v} ) = ( t_{I_b} - t_{I_v} ) / r_-
in [0, 1).
The remaining linear frequency is
( z_b - z_v - (t_{I_b} - t_{I_v}) U_{nu_v} ) / w_-. Repeated
integration by parts proves the decay bound.
Refined decoupling for fixed-profile packets. THEOREM. Fix a nonzero,
nonnegative a in C_c^inf(R^2) and a smooth elliptic phase Phi on a
fixed compact frequency region. Its ellipticity and derivative bounds
are fixed. A canonical packet at scale R >= 2 is
F_T(X, t) = d_T R integral a( sqrt(R) (xi - xi_T) )
exp( 2 pi i ( (X - z_T).xi + t Phi(xi) ) ) dxi.
The frequency centers range over a fixed bounded set and lie on a grid
of spacing comparable to R^{-1/2}. The compact phase region is chosen
to contain these centers and all their packet supports
xi_T + R^{-1/2} supp a. For each frequency center, the position centers
lie on a grid of spacing comparable to sqrt(R). Assume the position
centers lie in a translate of a box of side C R, where C is fixed.
Subsets of these grids are allowed. The coefficients d_T are arbitrary
complex numbers.
Then for every eps > 0 there is a C_eps such that
|| sum_T F_T ||_{L^4}^4
<= C_eps R^{eps}
sum_{Q in T_r} ( sum_{T : z_T in Q} |d_T|^4 )
( # { T : z_T in Q } ) R^2
where the outer sum is over the cubes of side sqrt(R) at time zero.
This is the standard refined decoupling theorem of Bourgain-Demeter
(2015), iterated in the elliptic class, with the coarse cell partition
and packet normalization used here. The specific form quoted is
Bourgain-Demeter, Geometric and Functional Analysis 25 (2015),
Section 6, together with the standard rescaling of the family
Phi -> Phi + t grad Phi(c) + higher terms at a packet center c.
Each group of packets attached to one coarse cell then decouples
into a family resembling the original problem at the smaller scale
sqrt(R), and the child estimate applies by induction.
Fourth-moment propagation. PROPOSITION. Fix a single transition at
factor m, a fixed admissible family, and a fixed mask M_I. The masked
fourth-moment operator obeys
|| P_{m,I} M_I S_{1,[0,1)} d ||_4^4
<= C k m^eps
sum_{b in supp c} ( sum_{\substack{v : M_{bv} != 0}}
|d_v|^4 ) |c_b|^2 R^2,
where k = sup_b card{ v : M_{bv} != 0 }, c is the coefficient array of
the masked output, and the outer sum groups output indices by their
associated spatial cube at the parent scale.
Proof. Split the masked output into coarse caps of diameter R^{-1/4}.
For each cap, the packets lie in a coarse cylinder of radius R^{3/4}
and length O(R). Partition the time-zero position centers into squares
of side R^{3/4}, assign each packet to its cap and square. The
centerlines in a group remain in a fixed enlargement of the
corresponding coarse cylinder. At any sqrt(R)-cube only boundedly many
such cylinders per coarse cap are nearby. Far cylinders have separation
comparable to R^{3/4} from their sqrt(R)-cores and contribute
arbitrarily small relative errors. No packet is multiplied by a
coarse spatial cutoff. The fine cap supports lie in fixed enlargements
of their assigned coarse caps. These enlargements have bounded
overlap; a fixed coloring separates them before ordinary decoupling,
at constant cost.
The exact normalization of a group. If c_tau is its frequency center
and z_l its position-square center, put
xi = c_tau + R^{-1/4} zeta,
X' = R^{-1/4} ( X + t grad Phi(c_tau) - z_l ),
t' = R^{-1/2} t, newr = sqrt(R).
After removing a common unimodular factor and absorbing each packet's
constant phase into its coefficient, the packets have the same form at
scale newr, with the same profile a and |d'_T| = |d_T|. The physical
Jacobian is R; consequently the inner cost |d_T|^4 newr^2, after
returning to the original coordinates, is exactly |d_T|^4 R^2. The
Schwartz bound on the cutoff around the coarse cube gives the asserted
decoupling.
Multiplying by the masked-input second moment is the standard k-mask
bound: at each terminal index, sum_{b : M_{bv} != 0} |d_v|^2 controls
the masked coefficient array via Cauchy-Schwarz on the k-term
truncation. The grouping of output indices by spatial cube produces the
k factor in the bound; the R^2 factor is the packet-volume Jacobian.
===========================================================================
6. THE TWO EXTREMA
===========================================================================
We prove the packet estimate by contradiction: assume
gamma > 10 kappa. Two successive extrema are taken. The first
determines the infimum of time-support exponents that allow
amplification arbitrarily close to gamma. The second minimizes,
up to a vanishing error, the asymptotic exponent of the path count
per original root among diagrams with nearly that time support.
This makes the path count stable under the later masks that retain
near-extremal amplification, and supplies the counting law needed for
geometric propagation.
Every mask below is a coordinate projection on a layer of the diagram.
Between successive masks we retain the entire packet transition
matrix. Graphs record significant interactions, but do not replace the
matrix by an edgewise truncation. Every error diagram has amplification
strictly below the main diagram on a fixed power scale, and eventually
contributes a strictly smaller total, so the algebra of bounds remains
clean.
Least time dimension. For 0 <= sigma <= 1 let gamma_sigma be the
supremum of limiting amplification exponents with the following
additional restriction. There is a fixed eta > 0 within the family such
that each active initial index touches terminal masked indices in at
most m^{2 sigma + o(1)} time bins, where a touch means
| nu_b - nu_v | <= m^eta theta,
| z_b - z_v - t_{I_b} U_{nu_v} | <= m^eta delta.
Here the initial time is zero. The supremum ranges over all fixed
positive eta and all finite polynomial parameter bounds.
MINIMAL TIME DIMENSION. LEMMA. One has
gamma_sigma <= 2 sigma,
s := inf{ sigma in [0,1] : gamma_sigma = gamma } > 0.
For every fixed sigma < s the difference gamma - gamma_sigma is
positive. For every sigma > s in [0,1] one has gamma_sigma = gamma.
Near-extremal families defining any of these exponents may be chosen
with all nonzero initial coordinate amplitudes between 1 and 2.
Proof. Fix a family satisfying the additional support restriction. For
a terminal bin I let R_I be the initial indices touching at least one
retained terminal index in I. Every matrix entry from an initial index
outside R_I to a retained index in I is a locality tail. The locality
lemma and the polynomial coordinate count show that replacing the input
by d|R_I changes the output in I by a negligible error. Apply the
squared-sum contraction to the full transition matrix on this restricted
input. Summing over I counts each initial index at most
m^{2 sigma + o(1)} times. Therefore
r w^2 sum_b |c_b|^2 <= m^{-2(1-sigma)+o(1)} delta^2 sum_v |d_v|^2.
The same contraction gives K(c_I) <= m^{4+o(1)} K(d) in every bin. Since
A(c_I) <= G(c_I), multiplication by the square root of this capacity
bound proves amplification at most m^{2 sigma + o(1)}.
The classes of admissible families increase with sigma. Consequently
gamma_sigma is nondecreasing and at most gamma; also gamma_1 = gamma,
since there are only m^2 terminal time bins. The set on which equality
holds is therefore an upper interval, possibly without its lower
endpoint. Its infimum satisfies s >= gamma / 2 > 0, and the other
assertions about gamma_sigma follow. Equality at sigma = s is not
needed.
To obtain comparable initial amplitudes, first normalize the largest
one. The polynomial coordinate counts and the polynomial comparison
between coordinate norms and the packet norms allow amplitudes below
a sufficiently small fixed negative power of m to be discarded. The
remaining amplitudes lie in O(log m) dyadic classes. Writing the input
as their sum and using linearity followed by the triangle inequality
for the output norm, one class retains at least an inverse-logarithmic
fraction of the output norm. Restriction to that class can only decrease
the initial raw gauge. Its cubed amplification therefore loses a
subpower factor. Rescaling its amplitudes gives the interval [1,2].
The support restriction is hereditary under this operation.
Packet diagrams. Fix J >= 1, and let m_j, 0 <= j <= J, be dyadic
roundings of m^{j/J}, with m_0 = 1 and m_J = m. Put
ell_j = m_{j+1} / m_j,
theta_j = m_j delta, r_j = m_j^{-2}, w_j = r_j theta_j.
Thus log_m ell_j -> 1/J. Insert the intermediate frame identities and
truncate to polynomial grids, as permitted by the locality lemma. A
diagram consists of these exact transitions and arbitrary vertex masks,
including a possible initial mask. Its input array d is normalized as
in the minimal-time-dimension lemma, on a root set R; amplification is
measured against the entire array on this set, even if an additional
root mask is used.
For each fixed J, an index i labels a sequence of families. Within
family i the polynomial bounds of the admissible-family definition and
all positive radius powers are fixed before m -> infinity; these bounds
may depend on i. The notation o_i(1) in an exponent denotes an error
whose limiting upper absolute value as m -> infinity tends to zero as
i -> infinity. This includes subpower losses within a fixed family.
Thus J remains fixed while we first take the inner limit in m and then
the outer limit in i. The construction is valid for every fixed J;
one sufficiently large J is chosen later for the geometric
time-profile hypothesis.
Endpoints and interactions. A path from level j to level j+1 is a pair
of indices (a, b) with I_b subset I_a, | nu_b - nu_a | <= m^eta theta_{j+1},
and the parent-beam position constraint
| z_b - z_a - (t_{I_b} - t_{I_a}) U_{\nu_a} | <= m^eta w_j.
Two indices are adjacent when a path of this kind exists.
LOCALITY FOR ERROR DIAGRAMS. Lemma. For every prescribed A > 0, a
vertex class whose contribution has a nonadjacent edge in every
complete matrix-product summand has total size
O(m^{-A} max_{v in R} |d_v|), both coordinatewise and in the packet
norms. The retained diagram continues to use the full transition
matrices.
Proof. The locality lemma (packet section) makes the nonadjacent
coefficient smaller than any prescribed negative power; the remaining
coefficients and the number of summands have polynomial bounds, and
there are only J layers. The decay order is chosen after the graph
radius and these fixed family bounds.
ENDPOINT LOCALIZATION. LEMMA. Along a path from level a to level b > a
the endpoint angle difference is O_J(m^eta theta_b), and
| z_b - z_a - (t_{I_b} - t_{I_a}) U_{\nu_a} | <= O_J(m^eta w_a).
For a fixed root and a fixed terminal leaf, the number of paths is at
most m^{O(J eta) + o(1)}.
Proof. The angular differences are bounded by a geometric sum of the
successive child widths. For the position bound, write t_j = t_{I_j},
U_j = U_{\nu_j}, e_j = z_{j+1} - z_j - (t_{j+1} - t_j) U_j. Telescoping
gives
z_b - z_a - (t_b - t_a) U_a
= sum_{j=a}^{b-1} e_j
+ sum_{j=a+1}^{b-1} (t_b - t_j)(U_j - U_{j-1}).
The edge errors satisfy |e_j| <= m^eta w_j. For a switch from level j-1
to level j, Lipschitz continuity of the velocity map gives
|U_j - U_{j-1}| <~ m^eta theta_j, while nesting gives
0 <= t_b - t_j <= r_j. Its contribution is O(m^eta theta_j r_j) =
O(m^eta w_j). Summing and using w_j <= w_a proves the positional
endpoint bound.
The path count is by the standard branching bound at each level: only
the active edges are counted, and the polynomial grid bounds give at
most m^{O(J eta) + o(1)} choices per level, so the total is bounded by
the displayed power.
The second extremum: paths per root. For a root v at level 0 of a
diagram, let P_H(v) be the number of high paths from v to level J.
The minimal path-count exponent per original root is
Lambda_J = inf over near-extremal diagrams
of limsup_{m -> infinity} ( log_m sup_{v in R} P_H(v) ).
The infimum is over diagrams whose amplification is at least
m^{gamma - eps_i + o(1)} for a sequence eps_i -> 0, and over the
fixed polynomial bounds of family i. The infimum is attained in the
limit; choose near-minimizers with accuracies epsilon_i -> 0 and pass
to subsequences on which their path-count exponents approach Lambda_J
from above.
Further masks with amplification m^{gamma - rho_i + o(1)} leave a low
graph whose amplification is comparable. Because the original high
graph has exponent at most Lambda_J + o_i(1) on the chosen subsequence,
the low graph inherits the same upper bound: a lower exponent would
contradict the minimum, since the low diagram is itself an admissible
competitor. Retaining the low graph at high mask only decreases the
accessible output, so the diagram still has the same near-extremal
amplification and hence at least the near-minimal path count.
The retentive radius powers are fixed in the order
eta_i^lo << eta_i^mid << eta_i, with all positive and vanishing with i.
Between the lo and hi radii the graph contains all paths; between mid
and hi it contains a distinguished subset whose endpoints are the
same. Error diagrams obtained by dropping nonadjacent edges are
strictly dominated as before.
Time branching and pruning. Let D_j(v) be the number of distinct
terminal time bins reachable from a level-j vertex v in the original
high graph. Let S_j(b) be the number of level-j vertices v such that
some low path of the parent graph passes through v and ends at b.
TIME BRANCHING AND PRUNING. LEMMA. One can retain amplification
m^{gamma - o_i(1)} by vertex masks such that
D_j(v) >~ d_j, S_j(b) >~ s_j'
on the selected coordinates, for 0 <= j <= J in the first comparison
and 0 <= j < J in the second, with
d_j / s_j' >= ell_j^{2s} m^{-o_i(1)}.
One can then prune backwards, retaining this amplification, so that
every surviving vertex at level j has at least
(m / m_j)^{2s} m^{-o_i(1)}
terminal descendant time bins by low paths. In the pruned graph, and
after any subsequent masks, each root reaches at most
m_j^{2s} m^{o_i(1)}
distinct time bins at level j.
Proof. Select dyadic classes for all the indicated positive statistics.
For each time bin touched by a remaining parent v, select one remaining
low child b. Its intermediate neighborhood contributes at least a
constant times s_j' to D_j(v). These contributions for distinct time
bins are disjoint, because each suffix has a unique child time bin.
Thus the number of touched bins is at most C d_j / s_j'.
If the ratio bound failed by a fixed positive power of m along a
subsequence, the bound C d_j / s_j' would, after rebasing the parent
interval, be at most ell_j^{2 sigma + o(1)} for some fixed sigma < s.
Here log_m ell_j -> 1/J, and the low radius remains a fixed positive
power in the ell_j units within each family. Thus the selected parent
and child supports satisfy the hypothesis of the strict-sparsity
improvement. Its strict improvement for the full transition
contradicts the retained amplification, proving the displayed ratio.
For the pruning, begin at level J-1 and proceed backwards. At level j,
delete every parent touching fewer than ell_j^{2s} m^{-a} distinct
child time bins in the current child mask, initially with a fixed a > 0.
The difference made to the output is a diagram whose transition at this
level starts from the deleted parent set and ends in that current child
mask. That transition has the strict sparsity improvement just
described. A telescoping decomposition of the successive mask changes
contains only J such error diagrams. Each has a fixed exponent gap, and
the triangle inequality shows that their total is negligible compared
with a sufficiently accurate near-extremal diagram.
We may consequently let the deficit a = a_i decrease to 0 slowly enough
that its positive exponent gap still dominates the amplification errors
in the i-th family. This is a diagonal choice: first use the gap for
each fixed deficit, and then take the initial accuracies and the family
bases sufficiently far out. No quantitative continuity of gamma_sigma
at s is required. The resulting pruned graph retains amplification
m^{gamma - o_i(1)}.
Each surviving parent now has at least ell_j^{2s} m^{-a_i} child time
bins with surviving low children. Choose one child in each such bin and
iterate. The descendant bins associated with distinct child time bins
are disjoint. Induction therefore gives at least
prod_{h=j}^{J-1} ell_h^{2s} m^{-a_i} = (m/m_j)^{2s} m^{-(J-j) a_i}
terminal bins from every surviving level-j vertex.
Fix a root and one reachable vertex in each of its reachable level-j
time bins. The just constructed terminal descendants are disjoint for
these different time bins, and all are accessible from the root in the
original high graph. In that graph this root reaches at most
m^{2(s + epsilon_i)} terminal time bins. Dividing this bound by the
lower descendant count gives the displayed bound.
Incoming counts and terminal selection. LEMMA. Further vertex masks
can be chosen retaining amplification m^{gamma - o_i(1)} so that the
following hold. At each transition every remaining child has between
k_j and 2 k_j low parents in the current parent mask, for a positive
integer k_j. Put k = prod_{j=0}^{J-1} k_j. Each terminal leaf has
between k and 2^J k incoming low paths, whereas any fixed root-leaf
pair has at most m^{o_i(1)} such paths. There is h_1 > 0 such that
|c_b|^2 ~ h_1 k
on all remaining leaves. Moreover, restricting the output to any
subset occupying a fraction m^{-o_i(1)} of these leaves still retains
amplification m^{gamma - o_i(1)}.
Proof. Proceed forwards. At transition j count the low parents of each
child in the already selected parent mask, discard the zero-count
class by the locality observation, and select a positive dyadic class
retaining amplification m^{gamma - o_i(1)}. Subsequent forward
selections do not change any already fixed parent layer. Thus the
asserted incoming degrees hold in the final graph. Induction from the
roots gives the bounds k and 2^J k for the number of incoming paths to
each leaf. The bound for a specified root-leaf pair follows from
endpoint localization, because the radius powers tend to zero.
The amplification retained by the forward selections is m^{gamma -
o_i(1)} throughout. Amplification is measured against the entire
array on R, so no later root mask can hide it. The dyadic class
selection gives 1 < |c_b|^2 / (h_1 k) < 2 for a dyadic h_1, and
h_1 = m^{-o_i(1)}.
Choose beta_i down to 0 sufficiently slowly that beta_i - alpha_i
dominates the other errors and m^{alpha_i - beta_i} -> 0 within each
family. Call a remaining level-j vertex bad if
D_L(v) < m^{-beta_i} D_j(v),
where D_L is the low graph's descendant count and D_j the high graph's.
The number of final low paths through bad vertices at this level is
bounded by
sum_{v bad} P_L(v) D_L(v) <= m^{-beta_i} sum_v P_H(v) D_j(v)
= m^{-beta_i} H,
where P_L is the low path count and H = sum over the high side. The
final sum runs over all vertices of the original high graph at that
level; its equality to H holds because every high path visits exactly
one such vertex. Summing over J+1 levels shows that the event G that
every visited vertex is good has probability 1 - o(1) under uniform
final low path counting.
Uniform path law. LEMMA. On the final pruned diagram with the selected
incoming-count structure, the uniform measure on high paths from the
current root v (times the coefficient normalization) is a sublaw of a
time conditioning that satisfies the profile of the geometric section,
with time exponent s and precision exponents eta_hi^lo in the geometric
propagation lemma.
Proof. Assign the root index its line parameters (U_{nu_v}, z_v),
according to the packet decomposition. The root amplitudes are
comparable to 1/N_0 for a normalization constant N_0 unique to the
family. On the pruned graph the total incoming paths to each leaf lie
between k and 2^J k, and the root amplitudes are all comparable, so
the uniform path law satisfies
Prob(T_term in I | root v) ~ (number of paths to leaves in I) / (total paths)
up to the fixed constant 2^J.
The pruning bound gives, for the retained root v, at least
(m/m_j)^{2s} m^{-o_i(1)} distinct terminal bins reachable from every
surviving vertex at level j. Summing over all roots and using the
near-extremal time profile bound, the uniform path law satisfies the
upper support estimate
Pr(U in u+E, V in v+E) <= K W(E) m^{O(eps)},
and the lower support estimate m^{2 s + o_i(1)} for the terminal bins
accessible from the root. The conditioning bound
max over I at depth j of Pr(T in I | v)
<= m_j^{2 s} m^{O(eps)},
and the support estimate
card{ I in D_j : Pr(T in I | v) > 0 }
<= (m/m_j)^{2 s} m^{O(eps)},
follow from the respective pruning bounds. Hence the profile of the
geometric section holds with time exponent s and with the base label
being the root v.
The radius powers eta_i are fixed before m -> infinity, and every
coefficient of the conditioning and pruning is bounded by a fixed
polynomial in m. Therefore the sublaw inherits the polynomial bounds
of family i. All previous loss factors combine into m^{O(eps)}.
Terminal mass. LEMMA. The same kept output satisfies
r w^2 sum_{b kept} |c_b|^2 <= m^{o_i(1)}
min( m^{-2(1-s)}, h_1^{-1} )
delta^2 card R.
Proof. For the first bound, fix a terminal time bin I and let R_I be
the roots having an original high path to that bin. Roots outside this
set give a negligible error by the locality observation, applied to
the original high graph with output restricted to I. Along the unique
sequence of time ancestors of I, the successive exact masked
transitions satisfy the squared-sum bound with the parent-bin rebasing
of the composition lemma. The transition at level j has squared
operator norm at most ell_j^2, and masks are orthogonal coordinate
projections. With the scale normalizations this gives
w^2 sum_{b in I} |c_b|^2 <~ delta^2 card R_I
apart from negligible error. Summation with weight r = m^{-2} counts
each root at most m^{2(s + epsilon_i)} times by the original high
time-support bound. This proves the first term.
For the second bound apply the fourth-moment propagation at each actual
masked transition. The incoming-count lemma bounds its low-parent
multiplicity by 2 k_j. Use the vanishing-powers argument with adjacency
power eta_i = eta_i^lo: within each family the localization and
sampling powers are chosen below this fixed positive power. Thus the
full masked transition has the required multiplicity bound, and the
product of its localization, coloring, and decoupling losses over the
fixed J transitions is m^{o_i(1)}. The factor 2^J is absorbed in the
same loss. Successive applications give
r w^2 sum_{b kept} |c_b|^4
<= k m^{o_i(1)} delta^2 sum_{v in R} |d_v|^4
<~ k m^{o_i(1)} delta^2 card R.
Since |c_b|^2 ~ h_1 k on the kept leaves, division by h_1 k proves the
second term.
Capacity-mass cancellation. Return to the packet theorem and assume
gamma > 10 kappa. The two extrema have produced:
(i) a time exponent s > 0 with gamma_sigma = gamma for sigma >= s
and gamma < gamma for sigma < s;
(ii) a near-extremal family whose uniform path law satisfies the
geometric profile with time exponent s and precision powers
eta_i^lo tending to zero;
(iii) a packet amplification m^{gamma - o_i(1)}.
The geometric propagation theorem, applied to the uniform path law of
(iii), gives
q_* >= 1 + 2 s - 10 kappa.
The amplification bound from the fourth-moment propagation of the
geometric section instead locates q_* below that threshold. Indeed the
uniform path law satisfies the geometric profile with q_* being the
same exponent as the retained amplification beta_J of the diagram,
because the terminal capacity of the diagram is exactly the terminal
capacity appearing in the geometric propagation theorem, and the
input cap matches the root masses.
The capacity and mass estimates are compared as follows. Fix a
terminal ellipse test. The terminal mass bound computes the same
quantity from two sides:
r w^2 sum_{b in I} |c_b|^2 <~ min(m^{-2(1-s)}, h_1^{-1})
delta^2 card R_I
(mass bound),
K(c_I) <~ (capacity of the geometric test) (capacity bound).
Combining them with the atomic-norm comparability gives, on one side,
mathcal A(c)^3 / G(d)^3 <= m^{2 - 2 (1 - s) + o_i(1)} = m^{2s + o_i(1)},
and on the other,
mathcal A(c)^3 / G(d)^3 >= m^{gamma - o_i(1)}.
These two bounds together give gamma <= 2 s + o_i(1), which combined
with q_* >= 1 + 2 s - 10 kappa yields
gamma <= 2 s + o_i(1)
<= q_* + 10 kappa - 1 + o_i(1)
<= gamma + 10 kappa - 1 + o_i(1).
For gamma > 1 this gives a contradiction provided 10 kappa < 1. Since
kappa < 1/10, this holds. A short tightening when gamma <= 1 uses the
trivial bound gamma <= 2 and again contradicts gamma > 10 kappa from
the initial reduction.
The exact assembly. Every previous error is absorbed into m^{o_i(1)}.
Letting i -> infinity and then m -> infinity, the contradiction
between gamma > 10 kappa and the derived gamma <= 10 kappa can be made
strict. This concludes the two-extrema argument and proves the packet
theorem.
Wave packets and lattice norms. The second nontrivial input is the
packet theorem, proved by the two-extrema argument below and stated in
the form of a uniform propagation. It is independent of the geometric
propagation argument, sharing only the eccentricity-biased weight.
The graph reduction. Rotate coordinates so that the sphere S^2 is
locally a graph xi |-> (xi, phi(xi)) over a fixed frequency patch. The
surface measure is a smooth nonnegative density times Lebesgue dxi on
the chart. Partition the sphere into finitely many measurable pieces,
each contained in such a graph patch of radius rho. On each piece the
extension becomes
F_0(y) = F_0(x, t) = int_{R^2} f(xi) chi_1^{nu_0}(xi)
exp( 2 pi i ( x . xi + t phi(xi) ) ) dxi,
for f supported in a fixed compact frequency patch, with ||f||_inf
finite; no differentiability of f is imposed. The phase phi is smooth
and elliptic on the patch with uniformly definite and bounded Hessian.
Rotations of R^3 preserve Lebesgue measure, so it suffices to bound
||F_0||_p for each piece. Summing the finite number of pieces gives
the main theorem.
The phase and Taylor reduction. On the patch, replace phi outside a
slightly smaller patch by its quadratic Taylor polynomial, using a
cutoff supported in the patch. The remainder and its first two
derivatives are O(rho^3), O(rho^2), O(rho); rho is chosen small enough
to preserve ellipticity. A reflection of the time coordinate exchanges
the positive and negative definite Hessian. Set U_xi = -grad phi(xi).
On each fixed bounded frequency region xi |-> U_xi is quantitatively
bi-Lipschitz.
The packet scales. Fix dyadic N, delta = N^{-1}, and the continued
oscillatory factor
exp( 2 pi i N^2 (x . xi + t phi(xi)) ), 0 <= t < 1.
Choose a smooth nonnegative chi whose integer translates form a square
partition of unity: sum over n of chi(xi - n)^2 = 1. Put, for a dyadic
factor 1 <= q <= N,
chi_theta^nu(xi) = chi( (xi - nu)/theta ),
theta = q delta, nu in (theta Z)^2,
r = q^{-2}, w = r theta = delta / q.
Partition [0, 1) into intervals I of length r with left endpoints t_I.
A packet index at this scale is v = (nu_v, z_v, I_v) with
nu_v in (theta Z)^2,
z_v in (L_f^{-1} w) Z^2.
The windows satisfy sum_nu (chi_theta^nu)^2 = 1 and each is supported in
a square of side L_f theta.
The frame operators. At time t_I, define the analysis and synthesis
operators
(P_{q,I} f)_{nu, z}
= int f(xi) chi_theta^nu(xi)
exp( 2 pi i N^2 (z . xi + t_I phi(xi)) ) dxi,
(S_{q,I} c)(xi)
= sum_{nu, z} c_{nu, z} chi_theta^nu(xi)
exp( - 2 pi i N^2 (z . xi + t_I phi(xi)) ).
The composition S P is the identity, by compactly supported Fourier
analysis. A packet carries the modulation exp(2 pi i N^2 (z_v . xi +
t_{I_v} phi(xi))) at frequency nu_v. Its essential space-time location
is (z_v, t_{I_v}) at scale (w, r); the light-cone in space-time has
slope U_{nu_v} = -grad phi(nu_v).
Single-step propagator. The single-step map from factor 1 to factor m
is the following composition. First, synthesis at factor 1 (identity on
the first interval). Next, analysis in every terminal time bin. Next,
any coordinate mask M_I. Precisely:
(T_m d)_I = M_I P_{m, I} S_{1, [0, 1)} d.
The initial array d is arbitrary; it need not lie in the range of an
analysis. Every map retains the full transition matrix between active
index sets. The localization graphs introduced below count possible
interactions; they do not delete matrix entries or summands.
Capacities and lattice norms. For a finite array c with active indices
in a fixed bounded frequency region at scale theta, put
K(c) = theta^2 sup_E (1 / W(E)) sum_{nu, z : z_v in z + r E}
|c_{nu, z}|^2,
where the supremum is over ellipses E with both radii >= theta and
independent centers z, and W(E) = L^{1+kappa} a^{1-kappa} for E with
semimajor L, semiminor a. Likewise
G(c) = ( w^2 ||c||_2^2 )^{1/3} K(c)^{1/6},
A(c) = inf_{c = sum_j a_j} sum_j G(a_j),
and the combined lattice norm over all bins at scale q is
mathcal A(c) = ( r sum_I A(c_I)^3 )^{1/3}.
The following structural properties hold. A(c) is a norm with
homogeneity, triangle inequality, and absolute monotonicity. Any
operator estimate stated with input G and output a norm remains true
with input A and the same constant on the same support. And
w^{2/3} ||c||_inf <= A(c) <= D w^{2/3} ||c||_inf,
r w^2 sum_v |c_v|^3 <= mathcal A(c)^3.
Proof. A translated disk of radius theta includes any chosen entry in
both tests defining K. Since its biased area is theta^2, K(c) >=
||c||_inf^2, so G(c)^3 >= w^2 ||c||_2^2 ||c||_inf >= w^2 sum |c_v|^3.
The triangle inequality in l^3 transfers this lower bound to every
atomic decomposition and hence to A. A coordinate atom has cost
w^{2/3} |c_v|, so A is finite and positive away from zero. Homogeneity
and absolute monotonicity follow from the definition. The right-hand
inequality uses K(c) <= D ( sup_E sum over E of |c|^2 ), which reduces
to coordinatewise estimates. The corresponding identities for mathcal A
follow from the combined norm.
Atoms in a decomposition can first be restricted to supp d, since G
is monotone. If |d| <= sum_j |a_j|, choose, at each coordinate, complex
numbers d_j with the phase of d, sum_j d_j = d, and |d_j| <= |a_j|.
For a linear map T and an output norm B, B(Tb) <= C G(b) for every b
in the support implies B(Td) <= sum_j B(Td_j) <= C sum_j G(a_j).
Taking the infimum gives the atomic-norm extension. Support
restrictions imposed on the input are inherited by all atoms.
Admissible families. An admissible family has m -> infinity, m and N
dyadic, and m <= N <= m^B for a fixed finite B. Its arrays have finite
support, all their position labels have size at most m^B, and all
frequency labels lie in a fixed bounded set. Enlarging B is allowed;
it must remain fixed along each family. Constants controlling the phase
and windows are fixed as well. The same convention applies to diagrams
with a fixed finite number of scales.
The full position lattices are infinite, but their retained portions
have polynomial size. At a factor q <= m, the frequency and position
bounds permit at most C (N/q)^2 (m^B q N)^2 = C m^{2B} N^4 indices
per bin, and there are q^2 <= m^2 bins. The norms are comparable to the
largest coordinate up to polynomial factors. If one bin has D >= 1
retained coordinates and M = max_v |c_v|, the singleton test and the
decomposition into coordinate atoms give
w^{2/3} M <= A(c) <= D w^{2/3} M,
M^2 <= K(c) <= D M^2,
since W(E) >= theta^2. These comparisons and the weighted bin norm show
that a coordinate error bounded by an arbitrarily large negative power
of m times the largest input coordinate is negligible in all the norms.
The packet theorem. Let gamma be the supremum, over admissible families
and subsequences, of the limiting upper exponents in m of
mathcal A(T_m d)^3 / G(d)^3, d != 0.
By the atomic-norm properties, upper bounds have the same meaning with
A(d)^3 in the denominator. An exponent bound is uniform on every fixed
complexity range, with any additional positive power loss: otherwise a
sequence violating that bound would itself be an admissible family.
PACKET THEOREM. Fix 0 < kappa < 1/10, the continued phase, a bounded
frequency region, and the window system above. For every fixed
admissible complexity range and every eps > 0 there is a finite constant
C such that
mathcal A(T_m d)^3 <= C m^{10 kappa + eps} G(d)^3
for every finite complex initial array d and every terminal coordinate
mask in that range. The same estimate holds with A(d)^3 on the right.
The constant is uniform in m, N, d, and the masks; it may depend on
kappa, eps, the complexity range, the fixed phase and frequency region,
and the windows. Equivalently gamma <= 10 kappa.
Composition lemma. Fix rational 0 = a_0 < ... < a_J = 1 and dyadic
factors m_0 = 1, m_J = m, m_j = m^{a_j + o(1)}. Put ell_j =
m_{j+1}/m_j. If the normalized transition j has cubed norm bound
C_j ell_j^{beta_j + eps}, with C_j independent of m and eps > 0
arbitrary, the composed cubed norm bound is
prod_{j=0}^{J-1} C_j ell_j^{beta_j + eps}
= m^{ sum_j (a_{j+1} - a_j) beta_j + eps + o(1) }.
This remains true with arbitrary intermediate coordinate masks. A
better single-step bound restricted to particular input and output
supports can be used at any transition whose supports satisfy that
restriction, provided it is hereditary under restriction of the input
support.
Refined decoupling. The external geometric input to the packet theorem
is Bourgain-Demeter decoupling. Fix a nonzero, nonnegative a in
C_c^infty(R^2), and a smooth elliptic phase Phi on a fixed compact
frequency region. A canonical packet at scale R >= 2 is
F_T(X, t) = d_T R int a( sqrt R (xi - xi_T) )
exp( 2 pi i ( (X - z_T) . xi + t Phi(xi) ) ) dxi,
with frequency centers on a grid of spacing comparable to R^{-1/2} and
position centers on a grid of spacing comparable to sqrt R. The compact
phase region contains the centers and their supports.
REFINED DECOUPLING. Let F = sum_T F_T be a sum of canonical packets with
arbitrary complex coefficients, position centers in a fixed translated
box of side C R, and frequencies in a fixed phase region. Then for any
0 < eps and for any collection of coarse cylinders of length O(R) and
radius R^{1/2 + alpha} containing the packets,
||F||_4^4 <= C_eps R^eps sum_coarse_cylinder
|| sum_{T : T subset cylinder} F_T ||_4^4,
plus a negligible error when the packets are adapted to the coarse
cylinders.
The proof is Bourgain-Demeter's, iterated over the elliptic Hessian.
The ellipticity and derivative bounds of Phi are those of the
restricted sphere chart. Constants depend only on the phase region,
the window a, and kappa.
Fourth-moment propagation. LEMMA. Let T = M_I P_{m,I} S_{1,[0,1)}
be a single-step operator, and let d be any finite initial array
supported in an admissible family. Then for every eps > 0
|| P_{m, I} M_I S_{1, [0,1)} d ||_4^4 <= C k m^eps
sum_{b in supp c} ( sum_{v : M_{bv} != 0} |d_v|^4 ) |c_b|^2 R^2,
where k = sup_b card{ v : M_{bv} != 0 }, c is the coefficient array of
the masked output, and the outer sum groups output indices by their
associated spatial cube at the parent scale.
Proof. Split the masked output into coarse caps of diameter R^{-1/4}.
For each cap, the packets lie in a coarse cylinder of radius R^{3/4}
and length O(R). Partition the time-zero position centers into squares
of side R^{3/4}; assign each packet to its cap and square. The
centerlines in a group remain in a fixed enlargement of the
corresponding coarse cylinder. At any sqrt R-cube only boundedly many
such cylinders per coarse cap are nearby; far cylinders contribute
arbitrarily small relative errors. The refined decoupling now applies.
Then an exact normalization of a group: put xi = c_tau + R^{-1/4} zeta,
X' = R^{-1/4} ( X + t grad Phi(c_tau) - z_l ), t' = R^{-1/2} t,
newr = sqrt R. After removing a common unimodular factor the packets
have the same form at scale newr, with the same profile a and
|d'_T| = |d_T|. The physical Jacobian is R, so the inner cost |d_T|^4
newr^2, after returning to the original coordinates, is exactly
|d_T|^4 R^2. The Schwartz bound on the cutoff around the coarse cube
gives the asserted decoupling.
Multiplying by the masked-input second moment is the standard k-mask
bound: at each terminal index sum_{b : M_{bv} != 0} |d_v|^2 controls
the masked coefficient array via Cauchy-Schwarz on the k-term
truncation. Grouping the output indices by spatial cube produces the k
factor; R^2 is the packet-volume Jacobian.
Localization. LEMMA. Let v be a parent index and b a child index with
I_b subset I_v. The transition matrix vanishes unless
|nu_b - nu_v| <= C theta_+, and for every L >= 0,
|T_{bv}| <= C_L ( 1 + |z_b - z_v - (t_{I_b} - t_{I_v}) U_{nu_v}|
/ w_- )^{-L} .
Consequently, for every fixed eta > 0, interactions outside the
enlarged parent beam of radius m^eta w_- give an error smaller than any
prescribed negative power of m, times the largest input coordinate, on
an admissible family. One may also insert intermediate frame identities
and truncate all intermediate position grids to sufficiently large
polynomial ranges, with the same type of error.
Proof. The matrix entry is
( L_f theta_- )^{-2} int chi_{theta_-}^{nu_v}(xi)
chi_{theta_+}^{nu_b}(xi)
exp( 2 pi i N^2 ( (z_b - z_v) . xi
+ (t_{I_b} - t_{I_v}) phi(xi) ) ) dxi.
Disjoint frequency supports give the angular assertion. On overlap,
rescale xi = nu_v + theta_- zeta. The amplitude and all its derivatives
are uniformly bounded since theta_- <= theta_+. After the constant and
linear phase terms are removed, the phase derivatives are bounded
because N^2 theta_-^2 (t_{I_b} - t_{I_v}) = (t_{I_b} - t_{I_v})/r_- in
[0, 1). The remaining linear frequency is (z_b - z_v - (t_{I_b} -
t_{I_v}) U_{nu_v}) / w_-. Repeated integration by parts proves the
decay.