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This article is the last of four that completely characterize a solutionspace $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partialdifferential equations (PDEs) in $2N$ variables that arises in conformal fieldtheory (CFT) and multiple Schramm-Loewner evolution (SLE). The system comprises$2N$ null-state equations and three conformal Ward identities that govern CFTcorrelation functions of $2N$ one-leg boundary operators. In the first twoarticles, we use methods of analysis and linear algebra to prove that$\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. Building onthese results in the third article, we prove that $\dim\mathcal{S}_N=C_N$ and$\mathcal{S}_N$ is spanned by (real-valued) solutions constructed with theCoulomb gas (contour integral) formalism of CFT.  In this article, we use these results to prove some facts concerning thesolution space $\mathcal{S}_N$. First, we show that each of its elements equalsa sum of at most two distinct Frobenius series in powers of the differencebetween two adjacent points (unless $8/\kappa$ is odd, in which case alogarithmic term may appear). This establishes an important element in theoperator product expansion (OPE) for one-leg boundary operators, assumed inCFT. We also identify particular elements of $\mathcal{S}_N$, which we callconnectivity weights, and exploit their special properties to conjecture aformula for the probability that the curves of a multiple-SLE$_\kappa$ processjoin in a particular connectivity. Finally, we propose a reason for why the"exceptional speeds" (certain $\kappa$ values that appeared in the analysis ofthe Coulomb gas solutions in the third article) and the minimal models of CFTare connected.

A solution space for a system of null-state partial differential equations 4