# P2914: A rigorous stochastic-dominance counterexample in one-dimensional coalescence - ID: `P2914` - Reference: `coalescence-stochastic-dominance-counterexample` - Page: https://theoremdb.org/statements/P2914 - Record maturity: Reviewed problem with recorded work ## Problem In the one-dimensional red-blue coalescence process, let \(U\) be uniform on \([1,1.01]\), let a red interval have length \(R=U\), and let a blue interval have length \(B=U-1\) when \(U<1.0008\) and \(B=U\) otherwise. Prove that blue wins almost surely under every complete legal coalescence order, even though \(R\) strictly stochastically dominates \(B\). ### Problem setup - **Remark.** The initial state is a bi-infinite alternating sequence of red and blue intervals whose lengths are independent with the stated color laws. A legal move recolors an interval that is shorter than each of its two opposite-color neighbors, then merges the resulting three adjacent intervals. - **Definition.** A coalescence order is complete if every move that remains legal is eventually performed. Blue wins if every fixed point of the line eventually lies in a blue interval. ### What counts as a solution - Give a rigorous bound establishing the finite probability inequality required by the published coalescence criterion for the stated distribution, then derive almost-sure blue victory for every complete legal order. - All computer-assisted probability bounds must use exact or outward-rounded arithmetic, publish the event definition and parameter vector, and include replayable code or a finite certificate. ## Status UNKNOWN as of 2026-07-31. The published paper reports this distribution as a counterexample with overwhelming simulation confidence. Its Claim 2.3 depends on a finite probability inequality verified numerically, and the authors explain that an analytic finite calculation could convert it into a theorem. Give a rigorous bound establishing the finite probability inequality required by the published coalescence criterion for the stated distribution, then derive almost-sure blue victory for every complete legal order. [1](#reference-1) ## Work ### Evidence for the current status **Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-07-31. The published paper reports this distribution as a counterexample with overwhelming simulation confidence. Its Claim 2.3 depends on a finite probability inequality verified numerically, and the authors explain that an analytic finite calculation could convert it into a theorem. Give a rigorous bound establishing the finite probability inequality required by the published coalescence criterion for the stated distribution, then derive almost-sure blue victory for every complete legal order. UNKNOWN as of 2026-07-31. The published paper reports this distribution as a counterexample with overwhelming simulation confidence. Its Claim 2.3 depends on a finite probability inequality verified numerically, and the authors explain that an analytic finite calculation could convert it into a theorem. A complete resolution must satisfy this condition: Give a rigorous bound establishing the finite probability inequality required by the published coalescence criterion for the stated distribution, then derive almost-sure blue victory for every complete legal order. ### Background and intake notes The paper already supplies the infinite-process reduction. Tables for the finite distribution, interval enclosures, and rare-event decompositions can be reused to replace its Monte Carlo step with a proof. - Original intake status: UNKNOWN as of 2026-07-27. The published paper reports this distribution as a counterexample with overwhelming simulation confidence. Its Claim 2.3 depends on a finite probability inequality verified numerically, and the authors explain that an analytic finite calculation could convert it into a theorem. - The MathOverflow question, answer, and all comments were checked on 2026-07-27. The answer links the later paper and identifies stochastic dominance as one of the original conjectures disproved computationally. - The published Transactions of the AMS article and arXiv:1610.07430 were checked. For the stated distribution they reduce blue victory to a finite estimate q(n0,r)<0.058 with n0=2,000,000 and report 987 successes in 1000 trials rather than a proof of that estimate. - The paper's simulation gives a p-value below 10^{-12} against the adverse threshold, which makes the example a strong candidate for interval arithmetic, exact convolution, or a concentration bound. Statistical confidence alone is outside the acceptance test. - A local corpus search for coalescence, stochastic dominance, red-blue intervals, and the numerical distribution found no duplicate. - Recorded example: For every t, Pr(R>t)>=Pr(B>t), with strict inequality for some t, because B either equals U or is shifted down by one. The claimed winner therefore runs against the interval-length stochastic order. ### Open directions - **Route 1** (reported): Give a rigorous bound establishing the finite probability inequality required by the published coalescence criterion for the stated distribution, then derive almost-sure blue victory for every complete legal order. [1](#reference-1) ### Computational notes - The published experiment reports 987 successful trials out of 1000 for the finite criterion at n0=2,000,000. The record treats this as evidence and asks for a certified replacement. ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `coalescence-stochastic-dominance-counterexample`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. MathOverflow: Alternating colors on a line, infinitely often or converge?. Question 56048, its answer, and every visible comment were checked on 2026-07-27; the numerical distribution and proof gap come from the paper linked in the answer. Question 56048, its answer, and every visible comment were checked on 2026-07-27; the numerical distribution and proof gap come from the paper linked in the answer. https://mathoverflow.net/questions/56048/alternating-colors-on-a-line-infinitely-often-or-converge - Also cited at See dataset.references[0] for the exact external source and locator. - Also cited at Editorial research route recorded 2026-07-31 - forum; reference source; checked 2026-07-31 - Source use: citation_only - Source used to formulate or check the problem record. - Source used to assess the problem's recorded status. - For A rigorous stochastic-dominance counterexample in one-dimensional coalescence: UNKNOWN as of 2026-07-27. The published paper reports this distribution as a counterexample with overwhelming simulation confidence. Its Claim 2.3 depends on a finite probability inequality verified numerically, and the authors explain that an analytic finite calculation could convert it into a theorem. - Source named by the research packet. 2. Paul Balister, Béla Bollobás, Jonathan Lee, and Bhargav Narayanan, “Coalescence on the real line”. Transactions of the American Mathematical Society 371(3) (2018), 1583-1619. DOI 10.1090/tran/7391. Full journal article relevant to A rigorous stochastic-dominance counterexample in one-dimensional coalescence. https://doi.org/10.1090/tran/7391 - scholarly_publication; reference source; arXiv:1610.07430, checked 2026-07-31; checked 2026-07-31 - Open copy: https://arxiv.org/abs/1610.07430 - Source use: citation_only - Source used to assess the problem's recorded status. - For A rigorous stochastic-dominance counterexample in one-dimensional coalescence: UNKNOWN as of 2026-07-27. The published paper reports this distribution as a counterexample with overwhelming simulation confidence. Its Claim 2.3 depends on a finite probability inequality verified numerically, and the authors explain that an analytic finite calculation could convert it into a theorem.