An Exact Logarithmic-State Algorithm for Recursive Tight-Cycle Constructions

The classical recursive construction for tight-cycle-free three-uniform hypergraphs has optimal edge count \(R(n)\)\({}=\max_{0\leq m<n}\{m\binom{n-m}{2}\)\({}+R(m)\}\). We give an exact algorithm for a single input \(n\) that visits only \(O(\log(n+2))\) states. A square-completed cubic potential shows that every maximizing split is one of the two integers surrounding \(((2n\)\({}+1)\sqrt{3}\)\({}-(2n\)\({}+3))\)\(/4\). Their descendants occupy at most three consecutive integers per depth and halve in size. The same identity gives an explicit linear error bound and \(R(n)\)\({}=(\sqrt{3}\)\(/3\)\({}-1\)\(/2)n^3\)\({}+(\sqrt{3}\)\({}-2)n^2\)\(/2\)\({}+O(n)\). Integer square roots suffice for exact evaluation and reconstruction of an optimal block sequence. Nearest-integer rounding alone fails at \(n=106\). The construction, recurrence, leading density and finite directed-star interpretation are classical and credited. The single-forbidden-tight-five-cycle Turan density conjecture is not resolved. This is an unrefereed preprint; no absolute priority is claimed.