The orbifolds of permutation type as physical string systems at multiples of c=26 V. Cyclic permutation orbifolds

I consider the Zλ,λ prime free-bosonic permutation orbifolds as interacting physical string systems at ĉ=26λ. 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.