This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical 4+1-dimensional hyperbolic Yang–Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time.

Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary topological class at optimal regularity in the third paper, here we perform a blow-up analysis that shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence of a non-trivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.

Proving a conjecture of Talagrand, a fractional version of the ``expectation-threshold" conjecture of Kalai and the second author, we show that $p_c (\mathcal{F}) = O(q_f(\mathcal{F})\mathrm{log}\ \ell(\mathcal{F}))$ for any increasing family $\mathcal{F}$ on a finite set $X$, where $p_c(\mathcal{F})$ and $q_f(\mathcal{F})$ are the threshold and ``fractional expectation-threshold" of $\mathcal{F}$, and $\ell(\mathcal{F})$ is the maximum size of a minimal member of $\mathcal{F}$. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (Johansson--Kahn--Vu), bounded degree spanning trees (Montgomery), and bounded degree graphs (new). We also resolve (and vastly extend) the ``axial" version of the random multi-dimensional assignment problem (earlier considered by Martin--Mézard--Rivoire and Frieze--Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdős--Rado ``Sunflower Conjecture."

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r\in [0,1]$ so that there is an inscribed rectangle in $\gamma$ of aspect ratio $\mathrm{tan}(r\cdot \pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in $\mathbb{R}\times \mathbb{R}P^3$. We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in $S^1$ to prove that $1/3$ is a sharp lower bound on the probability that a Möbius strip filling the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.

For every smooth Jordan curve $\gamma$ and rectangle $R$ in the Euclidean plane, we show that there exists a rectangle similar to $R$ whose vertices lie on $\gamma$. The proof relies on the theorem of Shevchishin and Nemirovski that the Klein bottle does not admit a smooth Lagrangian embedding in $\mathbb{C}^2$.

We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $\mathrm{U}(n) \times \mathrm{U}(n+1)$ in the stablecase.