[#P2914] A rigorous stochastic-dominance counterexample in one-dimensional coalescence
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 \(B\).
1Context
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.
2Problem setup
Definition 1 (The initial state). 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 2 (A coalescence order). 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.
Remark 1. 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.
3What 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.
1Status
Current status (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.[1]
1Packet records
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
Notes and companion material
Original intake 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.
- 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 1. 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.
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.
How the 2 records connect
ProblemA rigorous stochastic-dominance counterexample in one-dimensional coalescence
2See also
How to cite
TheoremDB contributors, “A rigorous stochastic-dominance counterexample in one-dimensional coalescence,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/coalescence-stochastic-dominance-counterexampleThis page as plain text: coalescence-stochastic-dominance-counterexample.md
This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net[1], current as of July 31, 2026.
1References
- Packet source. 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. ↗forum · discovery source · checked 2026-07-31Source use: original summary.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.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.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.
- 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. Status evidence identified in the source record and checked at the linked publication. ↗ open copy ↗journal article · primary source · arXiv:1610.07430, checked 2026-07-31 · checked 2026-07-31Source use: original summary.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.Also cited at Full journal article relevant to A rigorous stochastic-dominance counterexample in one-dimensional coalescence.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.
An original CC0 reformulation motivated by the cited MathOverflow thread and its linked coalescence paper; no source prose was copied.