Integers divisible by exactly one of three moduli and cube billiard words with deleted edge hits

For pairwise coprime integers a < b < c, list the integers divisible by exactly one of a, b, c in increasing order and record which modulus each is a multiple of. The resulting periodic word w(a,b,c) over {A, B, C} is the cutting sequence of the segment from the origin to (bc, ac, ab) in the cubic lattice, with every crossing of a lattice edge deleted. For cube billiard words in totally irrational directions, which never meet an edge, the number of factors of length m is exactly m² + m + 1. We show how deleting edge hits interacts with this count. First, once c > 3a(m−1), the length-m factors of w(a,b,c) are exactly those of the two-moduli word u(a,b) with at most one defect (a reduction theorem). Second, for a ≥ 3 and c > 3as with s = ⌊(b+1)/a⌋, the factor complexity of w(a,b,c) equals m² + m + 1 for every m ≤ s and is strictly larger at m = s+1; the first new factor is BA^(s−1)B, produced by a single deleted multiple of a. We also determine the two-letter factors for all c > b and show that the n-th term lies within five positions of its density estimate. The threshold statement holds for every triple with a ≥ 3 and abc ≤ 120,000, by exhaustive computation in the range c ≤ 3as not covered by the theorem, and we conjecture that it holds for all c > b. Files: the paper (PDF) and Oberoi_2026_exactly_one_three_moduli_code.zip, which contains the Python scripts (standard library only) used for every computation in the paper, including an independent brute-force check (exhaustive_check.py).