Exact value of the critical threshold Tcrit in the three-throw mystery-balls process

Determine the exact value of the critical threshold Tcrit for the three-throw "mystery balls" process used to analyze sparse polynomial coefficient recovery, where t balls are thrown independently and uniformly at random into r = ⌊τ t⌋ boxes for each of three throws and balls in private boxes are iteratively removed across rounds; specifically, prove the precise value of τcrit that separates the regime where all balls are removed with high probability from the regime where a positive fraction remains, and verify or refute the numerical conjecture τcrit ≈ 0.407265.

Background

The paper analyzes a probabilistic "game of mystery balls" that models the coefficient-recovery step in sparse polynomial multiplication. In each round, for each of three independent throws, t balls are placed uniformly at random into r = ⌊τ t⌋ boxes; balls that land alone in a box for any throw are removed, and the process repeats. The authors show empirically a phase transition in τ: for large enough τ the process removes all balls (“win”), while for small τ a positive fraction remains (“lose”).

The analysis introduces a critical threshold Tcrit governing this phase change. While extensive numerical experiments support a specific value, the paper explicitly states this as a conjecture. Establishing the exact threshold is important for rigorously justifying the constants in the algorithmic complexity bounds that depend on τ > Tcrit.