Take the integers up to x whose prime factors all come from a restricted set Q of primes with relative density δ. Count those that have a divisor in the interval (y, 2y]. How does this count depend on δ?
There is a phase transition at δ = 1/log 4. Below this critical density, the count exhibits one asymptotic regime. Above it, a qualitatively different one. At the threshold itself, the behavior is explicitly determined.
This extends Ford's 2008 result, which covered only δ = 1 (the unrestricted case, all primes available). Schlitt's finding is that the arithmetic structure of the multiplication table changes character at a specific logarithmic constant — not at δ = 0 or δ = 1, but at 1/log 4 ≈ 0.721.
The constant 1/log 4 is not arbitrary. It emerges from the interaction between the density of available primes and the divisibility structure they generate. Below this density, there are too few primes to produce the rich divisibility patterns that characterize unrestricted integers. Above it, the restricted set behaves qualitatively like the full set of primes.
A sharp threshold in a problem about multiplication, governed by a specific constant from the logarithmic structure of the integers. The phase transition is in the arithmetic, not the analysis.