  This article is the third of four that completely characterize a solution
space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial
differential equations (PDEs) in $2N$ variables that arises in conformal field
theory (CFT) and multiple Schramm-Lowner evolution (SLE). The system comprises
$2N$ null-state equations and three conformal Ward identities that govern CFT
correlation functions of $2N$ one-leg boundary operators. In the previous two
articles (parts I and II), 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.
  Extending these results, we prove in this article that
$\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ entirely consists of (real-valued)
solutions constructed with the CFT Coulomb gas (contour integral) formalism. In
order to prove this claim, we show that a certain set of $C_N$ such solutions
is linearly independent. Because the formulas for these solutions are
complicated, we prove linear independence indirectly. We use the linear
injective map of lemma 15 in part I to send each solution of the mentioned set
to a vector in $\mathbb{R}^{C_N}$, whose components we find as inner products
of elements in a Temperley-Lieb algebra. We gather these vectors together as
columns of a symmetric $C_N$ by $C_N$ matrix, with the form of a meander
matrix. If the determinant of this matrix does not vanish, then the set of
$C_N$ Coulomb gas solutions is linearly independent. And if this determinant
does vanish, then we construct an alternative set of $C_N$ Coulomb gas
solutions and follow a similar procedure to show that this set is linearly
independent. The latter situation is closely related to CFT minimal models.
