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PART 16. A GUIDED TOUR OF THE SOURCE FILES
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The original paper is a LaTeX manuscript split across several files. This part
says what each file does, so a curious reader can go straight to the part they
care about. The "original" text file is a concatenation of all of them; the
file names below are the actual .tex names.
paper.tex
The document assembly. It sets the title ("Endpoint pointwise convergence for
the Schrodinger equation in higher dimensions", author OpenAI, dated
September 24, 2026), the abstract, and the table of contents, and it pulls in
the section files in order: introduction, packets, frame, fractal,
transverse, sparse, weak, endpoint. The bibliography follows.
preamble.tex
All the LaTeX setup: theorem environments, the notation macros (\R, \Z,
\norm, \abs, \dd, and so on), the package list (amsmath, tikz, hyperref,
cleveref, natbib), and the dimension bookkeeping. Read this only to learn the
notation macros.
tooling-driver.tex
A build wrapper (it sets pdf metadata and inputs paper.tex). Nothing
mathematical.
introduction.tex (Section 1)
The problem, the definitions (Fourier transform, H^s norm, Gaussian
regularization U_a, Lebesgue point f*), the statement of the main theorem,
a full history of the Carleson problem with all references, and the roadmap
of the three principal steps (frame estimate; geometric induction; common
time tree). This is the best entry point to the paper.
packets.tex (Section 2)
The packet dictionary and the packet inequalities:
- packet conventions (scale N, frequency label v, native cell T, energy E);
- weighted endpoint multilinear Kakeya (the Guth/BCT theorem);
- He's robust hyperplane projection theorem;
- the BUFFERED refined decoupling lemma (the repair of the Singh-Singh-
Parmar collar gap, proved here in every dimension);
- weighted estimates with arbitrary auxiliary weights B_b;
- local linear and local multilinear estimates;
- sampling preserves the envelope exponent; weighted transport through a
hierarchy;
- the NEAR-CAUCHY theorem (the sharp N-lossless point count) and its
polynomial starting bound.
frame.tex (Section 3)
The transverse frame estimate f:frame in k dimensions, the reversal of it
(the thing that would happen if the bound failed), the reduction to exact
packets, the finite scale trees and selection lemma, the amplitude-ratio
PROFILES, the smoothing argument for the subpower case, the positive-path
direction law, the angular pruning lemma, ball inflation, the binary slope
rigidity lemma, midpoint closure, the location of the full-slope interval,
the subpower left-limit lemma, and the final incidence contradiction. This
is the single most technical section.
fractal.tex (Section 4)
Compact arrays and admissible measures; the Du-Zhang fractal input in squared
form; the weighted fractal norm with occupancy gain; exact Fourier layers;
the power-loss norm for compact arrays; coarse decoupling inside a strip and
exact transport powers; and the codimension-two strip saving (the real power
gain that the whole endpoint rests on). This is the section that actually
produces the endpoint exponent n/(n+1) in the fractal norm.
transverse.tex (Section 5)
The transverse saving in every fixed dimension: the impossibility statement
t:transverse (no countersequence with high mass and transverse packets); the
multilinear estimates with polynomial transversality dependence; the pruned
incident-tube model; the information profiles F, G_i; the single common
profile lemma; the robust planar projection block test; the finite
alternation lemma; and the growing-degree partitioning contradiction
("no full-to-empty block"). This is where the information calculus is first
used in the paper proper.
sparse.tex (Section 6)
The sparse estimate s:sparse and its proof by the broad/narrow iteration:
the local functional T_beta; the geometric alternative (broad vs narrow);
the broad gain via the transverse saving; narrow transport and the exact
cancellation; stopping; the terminal square-root-scale gain from Du-Zhang;
and the summation with the chosen order of parameters. This is the engine
that works when D is a positive power of L.
weak.tex (Section 7)
The lossless weak estimate w:weak, with the unit density chi and the
sparsity D. It contains the order of choices, the easy ranges, and the
reduction to the active range, then pulls in the three geometric section
files below. This is the longest logical section.
angular.tex (Section 7, first input)
The angular induction lemma w:angular: the narrow-set estimate covering both
the codimension-two strip and the quadratic sublevel set (the second
fundamental form detector), with its geometric iteration, the transport
functional, the recombination over macro cubes, and the terminal
induction/gain calculation. This is the "curvature detector."
wall.tex (Section 7, second input)
Heavy bins and the reduction to broad heavy incidences; tube coordinates and
the incidence-mass bound; polynomial partitioning into a wall; the regular
graph pieces with derivative bounds; the line tests; and the affine plate
covering proposition w:geometry. This is where a curved wall is proved flat.
flattening.tex (Section 7, third input)
Assigning packets to plates; the dyadic inverse width h; the density bands;
the two energy budgets (per time bin and global density-weighted); the frame
cutoff verification; the tuple bound (the frame estimate in k = n-1); and
the summation that produces the final lambda^2 M_Y bound and closes both
inductions. This is where the geometry meets the frame estimate.
endpoint.tex (Section 8)
The transition from frequency-local maximal bounds to the actual theorem:
compact layers; the short-time estimate; the dual square sum on one time
tree; the local L^1 endpoint maximal estimate; the Gaussian local-integral
lemma; and the final representative construction, including the careful
ordering of limits. This is where the main theorem is actually proved.
forced_profiles.tex, time_tree.tex
TikZ figures. The first shows the two forced shapes of the amplitude-ratio
profiles (the "power case" and the "subpower case"). The second shows the
heavier-child path in one common dyadic time tree. They are worth looking at
while reading frame.tex and endpoint.tex respectively.
references.bib
The bibliography. It is unusually complete for a paper of this kind and is
the place to find the exact statements of every external input.
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PART 17. GLOSSARY
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Almost everywhere (a.e.):
Everywhere except on a set of measure zero.
Ball parameter / growth (m):
A measure mu has ball parameter m if mu(B(z,r)) <= m r^n for all centers z
and all r >= 1. It says the measure does not concentrate more than n-
dimensional Lebesgue measure would.
Compact array / packet:
A piece of a wave localized in both space and frequency. A scale-N packet
with frequency label v lives in its native unit cube under the map
X_{N,v}(x,t) = ((x - 2vt)/N, t/N^2).
Decoupling:
A family of inequalities (Bourgain-Demeter) saying that a wave with separated
frequency pieces has L^p size comparable to the l^2 sum of the sizes of the
pieces, for p up to the decoupling exponent q_k = 2(k+2)/k.
Density parameter (chi):
The local mass of the measure, mu(B(z,1)) <= m chi. It interpolates between
a spread-out measure (small chi) and a concentrated one (chi near 1).
Endpoint:
The exact value s = s_n of the exponent, not a value strictly larger. The
difficulty is that no epsilon slack is available.
Fractal norm / weighted fractal estimate:
The sum over cubes of mu(cube)^beta sup_cube |field|^2, controlled by the
input energy. beta = 2/(n+1) is the fractal exponent giving the endpoint.
Frame estimate:
The transverse saving in k = n-1 dimensions that says k+1 transversally
independent big packet fields cannot concentrate. Its proof is the
amplitude-profile/incidence argument of Part 12.
Gaussian regularization (U_a):
The Schrodinger evolution with the extra Fourier factor e^{-a|xi|^2}. It
makes the evolution of general L^2 data well defined; the theorem is about
removing it (a -> 0).
Heavy bin:
A 1/R frequency bin whose weighted coefficient energy at a macro cube is at
least a threshold W = X^{-P_0} m L^{n+1}/M_Y. Heavy bins are what the
geometric induction tracks.
H^s (Sobolev space):
Functions whose frequencies are damped by |xi|^s; equivalently the norm
integral (1+|xi|^2)^s |fhat|^2 dxi. Higher s means smoother. The endpoint
here is s_n = n/(2(n+1)).
Lebesgue point / f*:
A point x where the average of f over a ball around x settles to a definite
value f*(x). Almost every point is a Lebesgue point.
Multilinear Kakeya:
The theorem bounding the overlap of d families of tubes that are pairwise
transverse. In this paper it gives the frame estimate its geometric content.
Near-Cauchy theorem:
The sharp point count |Y| <= C R^{q_k+eps} sum s_T^2 for points where the
field is a fraction N^{k/2}/R of its trivial Cauchy size. It is proved by
the scale-separation/occupancy contradiction of Part 8.5.
Polynomial partitioning:
The Guth-Katz theorem giving a low-degree polynomial whose sign regions
split any finite point set evenly. Its zero set is the "wall."
Projection theorem:
He's discretized projection theorem: a fat fractal set has a projection of
comparably large size in some direction. It is used as the final
contradiction device.
Sparse estimate / sparsity (D):
The gain D^{-c_s} obtained when the sparsity parameter D (a measure of how
spread out the measure is relative to packet tubes) is a positive power of L.
Strip saving:
The genuine power gain S^{n/(n+1) - c_0 alpha} when the input frequency
support lies in a thin codimension-two strip of width S^{-alpha}. This is
the technical engine that closes the endpoint.
Time tree:
One dyadic decomposition of the time interval [0,1), shared by all
frequencies. The "heavier-child" path and the convexity-deficit telescoping
make the frequency sum converge without an extra logarithm.
Transverse / transversality:
k+1 frequencies v_1,...,v_{k+1} are delta-transverse when the parallelepiped
of their paraboloid directions (2v_i,1) has volume at least delta. Nonzero
transversality means "genuinely different directions."
Tube moment condition:
The condition that each packet tube carries at most m L^{n+1}/D of the
measure, with D the sparsity. It rules out measure concentrating on a single
packet tube.
Unit density (chi):
See "density parameter."
Wall / plate:
The wall is the zero set of the degree-X^C polynomial partitioning the level
set. The small-Hessian argument shows each piece of the wall is near an
affine hyperplane; that hyperplane is a PLATE, and the frame estimate is
applied on plates in one fewer dimension.
Weak L^p / L^{p,infinity}:
A size condition weaker than L^p: the measure of {|f| > lambda} decays like
lambda^{-p}. The whole geometric part produces a weak L^p maximal bound with
p > 2, and the endpoint sum converts it into the L^1 maximal estimate.
Key exponents (all fixed by the dimension):
s_n = n/(2(n+1)) the endpoint Sobolev exponent;
beta = 2/(n+1) the fractal weight exponent;
e = n^2/(n+1) the scaling exponent in the level;
q_k = 2(k+2)/k the decoupling exponent;
p_k = 2(k+1)/k the multilinear/frame exponent;
gamma_n = 2n/((n+1)(n+2)) the Du-Zhang fractal exponent.
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PART 18. HONEST NOTES ON THE STATUS OF THE RESULT
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This document describes a specific manuscript. It is worth being clear about
what is fully established and what a careful reader should check.
What the paper claims.
The manuscript claims a complete proof that, for every integer n >= 3 and
every f in H^{n/(2(n+1))}(R^n), the free Schrodinger evolution converges
almost everywhere to f as t -> 0+, in the precise sense of Part 5, namely a
single full-measure set, uniform in t, with Gaussian regularization removed
simultaneously for all real times in [0,1). Combined with Bourgain's
counterexample (which shows failure for smaller s) and with Carleson's
one-dimensional result, this would settle the entire Carleson pointwise
convergence problem: convergence holds exactly for s >= n/(2(n+1)) in every
dimension. (For n = 2, the endpoint is claimed in the companion planar paper,
not here; this paper is about n >= 3.)
What is genuinely new here, as described by the paper.
- The buffered refined-decoupling lemma, repaired in all dimensions (fixing
the collar gap flagged by Singh-Singh-Parmar).
- The near-Cauchy theorem with its N-lossless R^{q_k} count.
- The transverse saving in every fixed dimension (Part 12).
- The quadratic-sublevel narrow estimate that detects the second fundamental
form, and the resulting flattening into affine plates (Part 13).
- The two-parameter (scale/sparsity) induction and the two energy budgets
that make the endpoint close.
What depends on external inputs.
- Du-Zhang (2019) fractal/maximal estimate.
- Bourgain-Demeter decoupling (2015).
- Du-Guth-Li-Zhang multilinear refined Strichartz (2018).
- Bennett-Carbery-Tao / Guth multilinear Kakeya, and Tao's multilinear
curved Kakeya.
- He's discretized projection theorem.
- Guth-Katz polynomial partitioning and Wongkew's tubular neighborhood bound.
- The planar companion PSE2026 for the "limiting information calculus" and
"block comparison and telescoping costs" lemmas and the time-tree
mechanism.
These are all cited with explicit theorem locators. A serious verification
would have to confirm that each is applied in its stated range and with the
stated constants, especially the "buffered" repair and the planar lemmas.
How to read the correctness claim.
This guide faithfully summarizes the paper as it is written. It does not
independently re-prove every inequality (that is not something a summary
can do). A proof of this size and technical depth can only really be
certified by sustained expert reading of the actual manuscript, ideally with
independent checks of the external inputs. If you are using this result, cite
the paper and treat its proof as the source of truth, not this guide.
Why the result would matter if correct.
The Carleson pointwise convergence problem is a central question in harmonic
analysis with a long history and many deep partial results. Settling the
endpoint in all dimensions would close the problem completely. It would also
demonstrate that the endpoint version of the Schrodinger maximal estimate
(as opposed to the strict inequality) is true, which is the technically
difficult and previously open case. The proof techniques -- the lossless
packet algebra, the near-Cauchy bound, the transverse frame estimate, the
curvature-driven flattening, and the single time tree -- are likely to be
useful in adjacent endpoint problems as well.
One-line summary.
For n >= 3, every f in H^{n/(2(n+1))}(R^n) has an almost-everywhere
Schrodinger evolution to f as t -> 0+, at the exact Sobolev exponent, on one
time-independent full-measure spatial set, uniformly over the time interval.
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