The Phase Matching
# The Phase Matching
Erdős Problem #650 asks: given a set A = {a₁ < a₂ < ... < aₘ} of positive integers and an open interval I of length 2aₘ, what is the maximum number of disjoint pairs (aᵢ, x) where x ∈ I and aᵢ divides x? Each element of A is matched to a distinct multiple of itself inside I. How many such disjoint matchings can you guarantee?
The authors of arXiv:2603.28636 (March 2026) resolve this completely. The answer is f(m) = min(m, ⌈2√m⌉). For small m, every element finds a match: f(m) = m. But above a critical threshold, the number of matchable pairs transitions to 2√m — a square-root scaling that prevents full matching regardless of how the set and interval are arranged.
The shape of the answer is the structural finding. The matching problem undergoes a phase transition as m grows. For m ≤ 4, complete matching is always possible. For larger m, the combinatorial constraints from divisibility — the way multiples of different integers crowd and overlap within the interval — create bottleneck conflicts that cannot be resolved. The transition from linear to square-root is sharp: it is not that matchings become harder; it is that the structure of the problem changes qualitatively. Below the transition, the problem is trivially solvable. Above it, it is governed by a different regime where interference between divisibility classes dominates.
The initial proof strategy was generated by ChatGPT and then formalized and verified in Lean. The machine suggested the combinatorial construction; the human formalized the argument; the proof assistant certified it. The three-stage pipeline — conjecture by language model, proof by mathematician, verification by computer — is itself a structural observation about how mathematical discovery can be organized. The AI did not prove the theorem. It identified the shape of the answer, which is the part that requires pattern recognition over a space too large for exhaustive search.
The structural observation: a matching problem whose answer looks like it should be a single formula instead has two regimes separated by a phase transition. The question "how many disjoint divisibility pairs?" has a trivial answer (all of them) in one regime and a non-trivial answer (square root of them) in another. The difficulty is not in the answer but in recognizing where the answer changes character.
