{
  "schema_version": 1,
  "problem_number": "OWR-4990373-005",
  "title": "Finite Counterexamples to the Monotonicity Conjecture of Amir, Baldasso and Beilin for Median and Majority Dynamics",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Amir, Baldasso and Beilin conjectured (Oberwolfach Report 4/2021, Conjecture 1; Stoch. Proc. Appl. 155 (2023), Conjecture 3.1) that in median dynamics on any locally finite graph (i.i.d. uniform opinions in [0,1], independent rate-one Poisson clocks, a ringing vertex takes the median of its neighbours' opinions, its own opinion being added when its degree is even) the probability that the opinion at a fixed vertex lies in [0,α] is non-decreasing in time for every α > 1/2. Equivalently, in majority dynamics started from i.i.d. Bernoulli(p) opinions, f_p(t) = P_p[ξ_t(x) = 1] is non-decreasing for every p ≥ 1/2. They proved this for complete graphs and for Z. We show that the conjecture is false as stated. On a connected graph with six vertices and eight edges, all of odd degree, we determine f_p at one vertex for every p: with r = p(1−p), f_p(t) − f_p(∞) = r(2p−1) e^{−t} [ (9/16) r t − (4+r)/4 − (7/24) r e^{−2t} + r (71/200 + 7t/80) e^{−4t} ]. Hence, for every p ∈ (1/2,1), f_p(t) > f_p(∞) for all t > 16/(9r) + 26/27, and f_p is eventually strictly decreasing; the conjecture fails at this vertex for every α ∈ (1/2,1). The formula is obtained by a finite computation in exact rational arithmetic over the 64 configurations of the graph, and one instance is certified separately by rational bounds which do not use it. On a connected 3-regular graph on ten vertices, an exact closed form shows that f_p(t) > f_p(∞) for all large t, for every p ∈ (1/2,1), with a larger overshoot (f(3) − f(∞) > 3.47·10^−3 at p = 7/10). All degrees in both examples are odd, so neither depends on the rule for ties at even degree. For infinite graphs we obtain only disconnected counterexamples. The conjecture for vertex-transitive graphs, in particular for Z^d (d ≥ 2) and for regular trees, and for connected infinite graphs is not settled, and we have no certified counterexample among regular graphs of degree at least four. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "keywords": [
    "median dynamics",
    "median process",
    "majority dynamics",
    "marginal distributions",
    "monotonicity in time",
    "counterexample",
    "finite graphs",
    "cubic graphs",
    "continuous-time Markov chains",
    "uniformization",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-4990373-005",
    "math.PR",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-4990373-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-4990373-005/paper.pdf?v=7a5d74972106",
  "doi": "10.5281/zenodo.23496532",
  "zenodo_record_url": "https://zenodo.org/records/23496532",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The counterexamples are finite graphs with odd degrees; the vertex-transitive versions (Z^d, regular trees) and connected infinite graphs remain open.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}