A summation rule for the four-coupling beta function of four-dimensional nonplanar Euclidean scalar $φ^{4}$ theory under factorial coefficient bounds, and a domain on which the summed beta function is well defined

Let $\mathbf{c}=(λ,α,μ,ν)\in\mathbb{R}^{4}$ denote the four running couplings of four-dimensional nonplanar Euclidean scalar $φ^{4}$ theory, and let $B(\mathbf{c})=\sum_{p\ge2}\sum_{|\mathbf{m}|=p}β(\mathbf{m})\mathbf{c}^{\mathbf{m}}$ be its formal beta-function series. Assuming the factorial coeffi