The Orbifolds of Permutation-Type as Physical String Systems at Multiples of $\mathbf{c}=\mathbf{26}$ V. Cyclic Permutation Orbifolds

I consider the ${\mathbb{Z}}_{\lambda },$ $\lambda$ prime free-bosonic permutation orbifolds as interacting physical string systems at $\widehat{c}=26\lambda$. As a first step, I introduce twisted tree diagrams which confirm at the interacting level that the physical spectrum of each twisted sector is equivalent to that of an ordinary $c=26$ closed string. The untwisted sectors are surprisingly more difficult to understand, and there are subtleties in the sewing of the loops, but I am able to propose provisional forms for the full modular-invariant cosmological constants and one-loop diagrams with insertions.