Ford’s 2008 theorem determined the order of magnitude of integers up to x that have a divisor in a given interval — a fundamental result about the multiplicative structure of numbers. The answer depends on the full set of primes. What happens when you restrict which primes are allowed?

This paper answers the question for primes drawn from a set of relative density δ. The integers whose prime factors all belong to this restricted set, and which possess a divisor in the interval (y, 2y], are counted. The order of magnitude is determined for all δ in (0, 1].

A phase transition appears at δ = 1/log 4. Below this critical density, the combinatorial structure of the multiplication table changes qualitatively — the count’s dependence on the parameters shifts form. Above the threshold, the behavior resembles Ford’s unrestricted result with quantitative adjustments. At the threshold itself, the transition is sharp.

The result reveals that restricting the prime factor base does not simply scale down the classical answer. It introduces a genuine critical point where the nature of the divisor distribution changes. The primes you exclude don’t just thin the integers — they reorganize the multiplicative structure at a density that the theory precisely identifies.