The Kakeya maximal conjecture in three dimensions

1. Direction separation controls full-time planks

In a bounded line chart write $M_i(t)=b_i+tu_i$. An index has a positive weight $\omega_i$ and marked time bins $S_i$. Repeated traces with different indices or shadings are not identified. Distinguish total weight $n=\sum_i\omega_i$, incidence $I=\sum_i\omega_i|S_i|$, mean marked count $k=I/n$, and raw multiplicity $m=I/\#E$ on occupied cells. A plank of widths $a,b$ counts the weight $n_B$ of indices whose entire time trace it contains; a single marked point in the plank does not establish membership.

For the maximal reduction, return to unit weights and $c/N$-separated slopes. Comparing a plank's two endpoint sections places its slopes in an $O(a/N)\times O(b/N)$ rectangle. Thus $n_B\lesssim ab$, and $\Delta_0=\sup_B n_B/(ab)\lesssim1$ persists for every subfamily.

Definition 2.3; (2.1)–(2.8) · pp. 6–8 Proposition 10.2; (10.19)–(10.23) · pp. 91–92

2. A uniform discrete bound and the indicator estimate

Assume $h(p,0)=0$. If each line has $\gtrsim\max\{1,\lambda N\}$ marked bins, then for every $\xi>0$,

If no uniform constant existed, regularize a violating sequence with $\theta=\max\{N^{-1},\lambda\}=N^{-\alpha+o(1)}$. Incidence retention forces $F(1)=\alpha$. At $q=0$ the temporal cost is exactly $p\alpha$, so the violating multiplicity $N^{p\alpha+\xi-o(1)}$ contradicts the critical bound $N^{p\alpha+o(1)}$. This is the passage from asymptotic exponent notation to a constant uniform over all finite configurations.

For the auxiliary operator $K_N$ on cylinders of length 4 and radius $4/N$, direction charts and grouping centers into unit cubes give

For $\lambda\geq N^{-1}$ this implies $\lambda^3\sigma\{K_N1_E>\lambda\}\lesssim N^{p-2+\xi}|E|$. The smaller-density range is handled separately using the finite surface measure and $|E|\geq N^{-3}$ for a nonempty grid set. Integrating the distribution gives $\|K_N1_E\|_3^3\lesssim N^{p-2+\xi}(1+\log N)|E|$.

Proposition 10.2; (10.19)–(10.23) · pp. 91–92

3. Cell averages, level sets, and an arbitrary power loss

Normalize $\|f\|_3=1$ and choose $N^{-1}\leq\delta<2N^{-1}$. The cell average $F_N$ of $|f|$ on each $N^{-1}$-cube satisfies $0\leq F_N\leq N$ and $\|F_N\|_3\leq1$. A fixed cylinder enlargement gives $K_\delta f\leq64K_NF_N$. Bound the part $F_N<N^{-1}$ directly and split the remaining values into $O(\log N)$ dyadic levels. The indicator estimate and Hölder for the finite sum give

For a fixed $\varepsilon$, choose once a $p$ with $h(p,0)=0$ and $(p-2)/3<\varepsilon/2$, then set $\kappa=\varepsilon/2$. Restoring homogeneity proves the full statement. The source also establishes measurability: continuity of tube integrals in their parameters allows the supremum over centers to be restricted to a countable dense set.

Applying the final theorem to indicators yields $|\bigcup_TY(T)|\gtrsim_\varepsilon\delta^\varepsilon\lambda^3\sum_T|T|$ for direction-separated shadings, uniformly over $0<\lambda\leq1$. This final cubic density power is distinct from the intermediate $p+1>3$.

Proposition 10.2; (10.24)–(10.25) · p. 93 Proof of Theorem 1.1 · p. 94

1. Weighted set input and local planiness

The set input requires convex clustering $\sum_{T\subset K}|T|\leq\delta^{-\eta_{\mathrm{in}}}|K|$ and mean shading density $\lambda\geq\delta^{\eta_{\mathrm{in}}}$. It is not applied as an arbitrary-density estimate. Lemma 3.2 samples according to weights, controls finitely many box tests, and selects representatives of repeated tubes. Lemma 3.3 converts full-time plank bounds into convex clustering; disjoint small balls at marked times relate cell counts to shading volume.

An excessive local packet mass would contradict criticality after rescaling. The multilinear input's $\theta^{-1/2}$ cost and cap counts become powers of a sufficiently short block scale and are absorbed by positive slack. Blocks with three transverse rich directions can therefore be discarded, leaving, within each short block, a common direction plane along the reference index's retained times. The planar Furstenberg estimate fills plates while preserving the original shading's Frostman conditions.

[GWZ26, Theorem 1.1, p. 2] Theorem 3.1; Lemmas 3.2–3.3 · pp. 18–21 [Gut10, Theorem 1.3, print p. 264 / PDF p. 2] Proposition 4.1; Lemma 4.3 · pp. 24–28 [RW25, Theorem 4.1, p. 9] Theorem 3.4; Corollary 3.5 · pp. 21–22

2. The temporal profile and compatible packets across scales

Replacing the temporal deficit by total density alone would lose the cost after cutting a time block. The manuscript retains $F$, obtains an initial flat interval by plate filling, and compares an outer scale cut with an inner cut after plate alignment. A minimum-delay choice produces

together with stationary packets preserving the weights of their original indices at every scale. The strict bound $\beta<1$ uses the later angular nonconcentration conclusion of Lemma 6.2. Stationarity means that repeated cuts reproduce the normalized profile and packet shape.

In an event's normal frame $\vartheta$, use $X=M_i(t)_1+\vartheta M_i(t)_2$, $Y=M_i(t)_2$, $U=u_{i,1}+\vartheta u_{i,2}$, and $V=u_{i,2}$. Their scale speeds are $(0,\tau,\ell,1)$. Packet mass lower bounds become a demand on conditional coordinate entropy.

Propositions 5.4, 5.6 · pp. 44–48 Lemma 6.2 · pp. 51–54 Proposition 7.4; Lemmas 7.5–7.6 · pp. 65–68

3. Two events on one index are not independent

The two-frame transformation is

Time and normal differences give scalar projections of coordinate pairs. When the middle speeds tie, $\tau=\ell=1/2$, the joint $Y,U$ rate must be retained. After fixing these coordinates, the term $\Delta t\Delta\vartheta\,V$ in $X'$ requires product-slope pinning.

The manuscript compares the actual same-index joint law with the product of its conditional matrix and label marginals. If the recorded support fell below the required projection exponent, its hit probability under the product law would be $D^{-c}$, while the actual law hits with probability one because both events use the same matrix. Data processing for relative entropy would require mutual information $\gtrsim\log D$, contradicting the short-gap bound $o(\log D)$ in Lemma 8.5. Bias introduced by pinning is charged through domination bounds for both marginals.

Proposition 8.1 · pp. 68–69 Lemmas 8.4–8.7; (8.27) · pp. 73–78 Sections 9.1–9.2 · pp. 79–85

4. Closing the contradiction with the prescribed loss and limit order

For distinct speeds, the conditional rates $a_1,y_1,u_1,v_1$ satisfy, among other constraints, $a_1\geq y_1$, $u_1\geq v_1$, and $a_1<1\Rightarrow u_1\leq(a_1-s_0)_+$, where $s_0=1-\beta>0$. Five cases, determined by speed ties and the parameter regime, give a positive gap between these constraints and packet demand. Localization to $v_1>0$ cannot use weak convergence alone; Lemma 7.5 supplies strong $L^1$ traces or subsequential compactness.

Restrictions retain $N^{-o(1)}$ of incidence. Finite tests, scales, and positive gaps are fixed before increasing resolution. The smoothing width $\eta$ is fixed at the start in terms of $p_{\min}-2$, not readjusted after selecting $h,\tau,\beta$. With this order, the contradiction excludes a positive critical exponent and supplies the analytic reduction in the preceding section.

Proposition 10.1 · pp. 86–90

1. Obtain the full range and transfer to Euclidean Nikodym

The external source uses $\delta$-neighborhoods of unit segments with normalization $\delta^{-2}$. Covering such a neighborhood by three manuscript cylinders gives $\widetilde K_\delta f\leq3\pi K_\delta f$. Also $\|\widetilde K_\delta f\|_\infty\leq\delta^{-2}\|f\|_1$. Interpolation with $1/p=1-2\theta/3$ and $1/q=\theta/3$ gives $q=2p'$ and exponent $1-3/p-\eta\theta$. The loss $\eta>0$ is arbitrary; including both endpoints gives $K(3)$. Proposition 2.3 of GLX25 transfers this full range to the Euclidean Nikodym operator.

Corollary 11.1 · pp. 94–95 [GLX25, Proposition 2.3; (6)–(9), pp. 13–14]

2. Fix the local geometry on a constant-curvature manifold

Apply the same proposition's geodesic-straightening diffeomorphism on a fixed compact chart. Bounded bi-Lipschitz and volume-comparison constants compare tube averages on $U\Subset V$; the choice of $\delta_0$ keeps the input inside $V$. Start with a smaller loss to absorb the transfer's arbitrarily small additional loss. Setting $p=q=3$ yields the local endpoint.

[GLX25, Proposition 2.3; (6)–(9), pp. 13–14] Corollary 11.1 · pp. 94–95

1. Preserve the phase hypotheses and operator normalization

Theorem 1.12 of GLX25 transforms the phase locally into $x\cdot y+t h(y)+q(y)+f(t)$ with nondegenerate $\nabla_y^2h$. Terms depending only on $y$ cancel in the gradient difference, and terms depending only on $(x,t)$ do not contribute to $\nabla_y$. The curves become straight lines $x=\omega-t\nabla h(y)$. Proposition 4.1 makes the ordinary and curved $K(3)$ assertions equivalent, giving the same $p,q$ range and $\delta^{1-3/p-\varepsilon}$ power. Spatial and direction-parameter comparison constants may depend on the fixed phase and chart, which also determine $\delta_{\mathrm{curv}}$. The $p=q=3$ conclusion is included.

[GLX25, Definitions 1.7, 1.10–1.11; Theorem 1.12, pp. 3–5] [GLX25, Proposition 4.1; Definition 4.5, p. 19] Corollary 11.2 · pp. 95–96