An Erdős-Straus divisor–lattice construction for 4/n

We present a deterministic greedy algorithm on the divisor lattice of Ln = lcm (1, 2, . . . , n)
that, for every integer n ≥ 2, produces an Egyptian fraction expansion 4/n = 1/Xn+1/Yn+1/Zn ; with Xn, Yn, Zn ∈ Z>0.
The construction uses only elementary divisor arithmetic and the harmonic sum Hn = ∑ 1/k, 1≤k≤n without invoking any external results on the Erdős–Straus conjecture. The method yields an explicit, canonical triple (Xn, Yn, Zn) for each n, and provides a natural counting of the number of such representations obtainable from the divisor lattice.