For every stochastic transition matrix M=(m_ij)_i,j∈1,2, every alpha_1,alpha_2∈[0,1), and every C_1,C_2>0, the two-site moving-target search problem has an optimal deterministic stationary threshold selector. Thus, for some p^∈[0,1], site 2 is selected when p p^; at p=p^ either minimizing action may be selected. If an interval of beliefs consists entirely of ties, the threshold may be placed anywhere in that interval with ties resolved monotonically.

Ross’s conjecture

Ross’s conjecture asks whether an optimal search strategy always has a threshold form in the two-site moving-target model. The conjecture concerns the infinite-horizon problem and has also motivated finite-horizon and endpoint extensions.

Known results

MacPhee and Jordan proved the conjecture when the transition matrix satisfies $\det P \le 0$ .

August 2026 claimed proof

Yunpeng Li’s preprint claims an optimal deterministic stationary threshold selector for every classical two-site stochastic transition matrix with both overlook probabilities less than $1$ , thereby covering the previously unresolved regime $\det P>0$ . It also claims finite-horizon threshold results and extensions to cases with overlook probability $1$ . The preprint reports that ChatGPT (GPT-5.6 Sol) assisted with strategy brainstorming, literature searches, checks, and drafting; Li states that the mathematics was independently verified. No independent verification, referee report, counterexample, withdrawal, or retraction was found.

Current status (as of August 2026): Ross’s conjecture has a claimed complete proof covering the classical range and extensions, but its correctness remains unverified, so independent confirmation is still open.

Ross’s conjecture

Primary source

Additional references

Progress summary

Solutions 0