Douglas' conjecture asks whether the normalized logarithms of Euler characteristics of moduli spaces of stable representations of the m-Kronecker quiver converge to a continuous function of the slope b/a. That function should vanish at the endpoints of the interval [λ-, λ+] and be positive in its interior. The conjecture is motivated by black hole entropy counting.

A known counting formula expresses mE_m(a,b) as a count of colored gradings on lattice paths. The author constructs maps between such sets of gradings, using shell constructions and run-edits (insertion and deletion of vertical edges), to obtain product estimates on cardinalities and exponential growth rates. The limit is proved first for rational slopes and then extended to irrational slopes by comparison. Separate lower and upper estimates are developed near slope one for the 3-Kronecker quiver.

Douglas' conjecture is proved for every m ≥ 3: a continuous limit function F_m exists on I_m, vanishing at the endpoints and positive in between. A log-concavity conjecture for Euler characteristics near slope one of the 3-Kronecker quiver is also stated.