# Verification report — OWR-4990373-005 (Amir–Baldasso–Beilin: monotonicity in time of the marginals of median and majority dynamics)

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete negative answer (counterexample)** to the question as stated; every statement
it makes is proved, two of the main ones by finite exact computations, and the scope is as follows.
- **Settled.** Conjecture 1 of Amir's abstract in Oberwolfach Report 4/2021 (= Conjectures 3.1 and 3.2 of Amir,
  Baldasso and Beilin, Stoch. Proc. Appl. 155 (2023)) is false: on the connected 6-vertex graph B (all degrees odd),
  at the vertex x = 0, f_p(t) = P_p[xi_t(x) = 1] overshoots its limit and is eventually strictly decreasing for every
  p in (1/2, 1) (Theorem 3.1, Corollary 3.2); the same holds on the connected planar cubic graph C10 (Theorem 4.1).
  So t -> mu_t(x)([0, alpha]) is not non-decreasing for any alpha in (1/2, 1), and t -> mu_t(x)([0, alpha)) is not
  non-increasing for any alpha in (0, 1/2). No tie rule enters (all degrees odd).
- **Method.** Theorems 3.1 and 4.1 and Proposition 5.1 are finite exact computations (closed forms over the 2^n
  configurations in exact rational arithmetic, with the polynomial dependence on p); Proposition 3.3 and Theorem 4.1(a)
  are exact rational certificates by uniformization that do not use the closed forms. The overshoot on B is small
  (9.389e-7 at p = 61/100, t0 = 223/25) but certified exactly; on C10 it is 3.474e-3 at p = 7/10, t = 3.
- **Not new.** The model and its conventions, the median-to-majority reduction, the proved cases K_N and Z and the
  quenched counterexample on K_{3,m} are due to Amir, Baldasso and Beilin; the bound f_p(t) >= p (p >= 1/2) is due to
  Mossel, Neeman and Tamuz (used only as a consistency check); the energy argument and uniformization are standard.
- **Open (not addressed).** Vertex-transitive graphs, in particular Z^d (d >= 2) and regular trees; connected infinite
  graphs (only disconnected infinite counterexamples follow); regular graphs of degree >= 4 (no certified example).
  Conjecture 2 of the report (= ABB Conjectures 3.7/3.8) is a different question.

The note is unrefereed.

## Statement checked
- **Primary source.** G. Amir (joint work with R. Baldasso and N. Beilin), "Majority dynamics, median dynamics and
  percolation", Oberwolfach Report 4/2021 (Spatial Networks and Percolation), Oberwolfach Rep. 18 (2021), no. 1,
  pp. 171–174, doi:10.4171/OWR/2021/4 (Crossref: report pp. 149–224, issued 2022-03-14).
  - Read in full in the open PDF of the report (text extraction). Section 2 of the abstract defines median dynamics
    (i.i.d. uniform opinions, rate-1 clocks, median of the neighbours, own opinion added at even degree; coin flips
    mentioned as an alternative).
  - Conjecture 1 is on p. 172 (Conjecture 2 on p. 173). As printed it says that for any G, x and alpha > 1/2 "the
    function t -> mu_t(x) is monotone non-decreasing": the set to which mu_t(x) is applied is missing. Conjecture 2
    is written with mu_t(x)([0, alpha]); the abstract refers to [2] = arXiv:1911.08613 for this part and says the
    conjectures hold on Z and K_n.
- **Source paper.** G. Amir, R. Baldasso, N. Beilin, "Majority dynamics and the median process: Connections,
  convergence and some new conjectures", Stoch. Proc. Appl. 155 (2023) 437–458, doi:10.1016/j.spa.2022.10.015;
  read in the arXiv version arXiv:1911.08613v2 (2 January 2023), whose arXiv record carries the journal reference.
  The journal version was not read; all numbers refer to v2.
  - Section 2: "locally-finite (possibly infinite) graph"; generator (2.2)–(2.3); own vertex added to N(x) at even
    degree; finite G gives a finite Markov chain; Prop. 2.1 (well defined for bounded degree); Prop. 2.2 via (2.8).
  - Section 3: (3.1) mu^x_t(alpha) = mu^x_t([0, alpha)); Conjecture 3.1 (alpha <= 1/2, non-increasing) and the majority
    form Conjecture 3.2 (p <= 1/2); Prop. 3.3 (Mossel–Neeman–Tamuz); Prop. 3.6 (quenched version fails on K_{3,m},
    presented as refuting only that stronger version); Conjectures 3.7/3.8; Props. 3.10 (K_N) and 3.11 (Z, with
    P[xi_t(0) = 1] = 3p^2 - 2p^3 - e^{-t} p(1-p)(2p-1)).
  - Section 5: for median dynamics with coin flips the authors believe Conjectures 3.1, 3.7 and 4.2 should also hold.
- **Corpus record.** ulamai/UnsolvedMath, OWR-4990373-005 (dataset v1.8.0, label `open`); its text copies the OWR
  garble.

## Readings
| Reading | Answer | Where |
|---|---|---|
| OWR Conjecture 1 read as: t -> mu_t(x)([0, alpha]) non-decreasing for every alpha > 1/2 (the only reading consistent with Conjecture 2 and with ABB) | false: fails at x = 0 of B and of C10 for every alpha in (1/2, 1) | Cor. 3.2, Thm 4.1 |
| ABB Conjecture 3.1: t -> mu_t(x)([0, alpha)) non-increasing for alpha <= 1/2 | the same statement by the symmetry eta -> 1 - eta; false for every alpha in (0, 1/2) | Lemmas 2.1, 2.2 |
| ABB Conjecture 3.2 (majority form) / f_p non-decreasing for p >= 1/2 | false for every p in (1/2, 1) on B and C10 | Thm 3.1, Thm 4.1 |
| [0, alpha) versus [0, alpha] | immaterial: eta_t(x) is a.s. one of the initial values | Lemma 2.1 |
| Coin rule (zero-temperature Glauber dynamics) / median dynamics with coin flips | B and C10 have only odd degrees, so all rules coincide; the counterexamples apply | Remark 2.5 |
| Even degrees with the ABB rule | an additional example A (degrees 2,4,4,3,3,4) | Prop. 5.1 |
| Restricted to connected graphs / regular graphs | B and C10 are connected; C10 is 3-regular | Sections 3, 4 |
| Infinite graphs | only disconnected (disjoint union of copies of C10); connected infinite case open | Section 6.2 |
| Vertex-transitive graphs, Z^d, regular trees | not addressed; float searches only (evidence) | Section 6.3 |
| Quenched version (fixed update sequence) | already refuted by ABB (Prop. 3.6, K_{3,m}); not the question | Section 1.1 |

## Results in the paper
- **Lemmas 2.1–2.4.** Median form = majority form; symmetry f_{1-p} = 1 - f_p; fixation (every flip lowers the number
  of disagreeing edges for odd degrees and for the ABB rule at even degree, not for the coin rule at even degree) and
  the absorption recursion for f_p(inf); an overshoot over the limit refutes monotonicity.
- **Lemma 2.6.** Backward recursion for g_s(t) = E_s[xi_t(x)]; closed forms in the basis t^{i-1}/(i-1)! e^{-kt} by the
  partial-fraction rule (2); every coefficient of f_p is a polynomial of degree <= n in p (p-independent per-state
  coefficients), so n+1 nodes determine it.
- **Lemma 2.7.** Two-sided rational uniformization bounds (Lambda = n, P = I + Q/Lambda, Poisson truncation with an
  explicit geometric tail, rational bounds for e^a).
- **Theorem 3.1 / Corollary 3.2 (graph B).** f_p(inf) = p + r(2p-1)(1 + 14r/75) = p^2(28p^3 - 70p^2 - 94p + 211)/75 and
  f_p(t) - f_p(inf) = r(2p-1) e^{-t}[(9/16) r t - (4+r)/4 - (7/24) r e^{-2t} + r(71/200 + 7t/80) e^{-4t}], r = p(1-p);
  overshoot for t > 16/(9r) + 26/27; f_p'(0) = r(2p-1) > 0 and f_p' < 0 for large t.
- **Proposition 3.3.** p = 61/100, t0 = 223/25: f(inf) = 41541389119/62500000000; L - f(inf) = 9.3891186962e-7 > 0
  with a two-sided enclosure of width < 3e-65.
- **Theorem 4.1 (graph C10).** (a) p = 7/10: f(inf) = 90794489017687/115200000000000, f(3) >= f(inf) + 3.474e-3.
  (b) 19-term closed form for all p (Table 1, coefficients p + (2p-1)Q_{0,1}(r) and (2p-1)Q_{k,i}(r)), only i = 1, 2 at
  k = 1, c_{1,2} = r^2(2p-1)(80905r^2 + 5355504r + 5556600)/1587600. (c) overshoot for all large t, every p in (1/2, 1).
- **Proposition 5.1 (graph A, ABB rule).** Closed form for all p; leading term (2/3) r^2 (2p-1) t^2 e^{-t}; certified
  instance p = 7/10, t0 = 27/5, f(t0) - f(inf) >= 4.776e-4.
- **Corollary 1.2.** The conjecture fails on finite connected, finite connected cubic and infinite (disconnected)
  cubic graphs, for every level alpha and every tie rule.
- **Section 6.** Heuristic picture (not used in proofs); infinite graphs; open questions and float searches.

## Computations (programs and outputs in reproducibility/)
- **The exact program of this note** (`check_note.py`, standard library, about 3 s; TOTAL failures: 0). Exact
  rational implementation of Lemmas 2.3, 2.6, 2.7 and checks of: graph data (degrees, connectedness, automorphism
  groups |Aut(B)| = 4, |Aut(C10)| = 32, orbits); energy decrease (B, C10, A, K5, B minus 45) and energy-neutral coin
  flips on A; f(inf) by recursion (three instances); Theorem 3.1 at 7 nodes, by direct expansion and by
  interpolation; Corollary 3.2 arithmetic; the 19 polynomials of Table 1 at 11 nodes, by direct expansion and by
  interpolation, the absence of c_{1,i} for i >= 3, and the quartic/quadratic identity; Proposition 5.1; two-sided
  enclosures for B (width 2.5e-65), C10 (2.7e-55) and A (4e-58); ABB (3.7) for K5; symmetry; FLOAT check of the
  Mossel–Neeman–Tamuz bound; mutation control (B minus edge 45: c_{1,2} < 0, no overshoot, upper bound below f(inf)).
- **Programs with which the results were first obtained** (`original/`): exact certificates for B, C10, A
  (cert_exact.py); all-p proofs for B (cert_B_allp.py) and C10 (cert_allp_interp.py); an independent derivation of
  Theorem 3.1 with p symbolic (closedform_sympy.py); float cross-checks against scipy's expm_multiply (agreement to
  5.7e-15 for B, 8.2e-15 for C10); graph facts; the float searches. On 2026-10-20 the nine programs with a recorded
  output were re-run from the package: all outputs byte-identical.
- **The LaTeX Table 1** was parsed back and compared entry by entry with the polynomials checked by `check_note.py`.
- **Float searches (evidence only).** All-odd-degree connected graphs on 6 and 8 vertices (28 of 13 + 224 with a
  violation); cubic graphs on 8, 10, 12 vertices (0/5, 3/19, 16/85; largest overshoot 8.7e-3; none vertex-transitive);
  all 2, 6, 16, 59 connected 4-regular graphs on 7–10 vertices (one float-only apparent overshoot 7.3e-6 on 10
  vertices, not certified, not claimed); 5-regular: all 3 on 8 vertices and 59 of the 60 on 10 vertices, found by
  random sampling (none); 62 vertex-transitive graphs and three cubic vertex-transitive
  graphs on 18 vertices (none); all connected graphs on <= 7 vertices by a grid test (none on <= 5 vertices).

## Independent verification runs
The results were first obtained with the programs in `original/`. Two independent verification runs, both
AI-assisted, followed on 2026-10-20; each wrote its own programs and examined the written result before the note.
- **First run (statement and proofs).** Compared every statement about the OWR abstract and ABB v2 with the texts;
  re-derived Lemmas 2.1–2.4, the backward recursion, the partial-fraction table (by induction), the uniformization
  bound and its tail, the polynomial-degree argument, Corollary 3.2 and the identity and sign of c_{1,2} for C10.
  Computations of its own: Theorem 3.1 with p symbolic (sympy, partial fractions of the Laplace transforms); a third
  method, truncated Laurent expansions at the poles, for B at 13 rational p, for the 19 coefficients of C10 at
  p = 7/10 and for c_{1,2} of C10 at 13 points; f(inf), own uniformization bounds and 30-digit matrix exponentials for
  A, B and C10; graph facts; consistency with Mossel–Neeman–Tamuz; the derivative sign. Verdict: all claims
  confirmed. Required corrections: the reason given for "at k = 1 only i = 1, 2" in Theorem 4.1(b); the explicit
  scope of the energy argument in Lemma 2.3; keeping the scoping in every public statement.
- **Second run (independent computation and literature).** Its own exact engine (pools with self-inclusion,
  assertion of the energy decrease, exp-polynomials in the monomial basis, coefficients as exact polynomials in p
  without interpolation): closed form of B identical for all p; all 19 polynomials of C10 identical (a mutated
  constant is detected); graph A leading term (2/3) r^2 (2p-1) t^2 e^{-t}. Two-sided rational uniformization
  enclosures: B at (61/100, 223/25) width about 1.5e-65, C10 at (7/10, 3) width about 1.5e-55 (after it found and fixed
  a float leak in its own first version). Monte Carlo of the raw clock process: C10 at p = 0.7, f(3) - f(inf) =
  3.463e-3 ± 9.0e-5 (exact 3.474e-3), about 38 standard errors; B's overshoot is below Monte Carlo resolution.
  Mutation test: B without edge 45 gives f(t0) - f(inf) = -3.02e-5. Scan of all connected graphs on <= 7 vertices.
  Re-ran nine certificate programs: byte-identical outputs. Literature and bibliography (Crossref). Verdict: all
  claims confirmed. Suggestions adopted: Conjecture 1 is on p. 172; DOIs for Damron–Sen and Mossel–Neeman–Tamuz;
  OpenAlex and zbMATH in the novelty paragraph; the two-sided enclosure, the simulation and the mutation test in the
  Verification paragraph; graph A only as an addition; a heuristic picture of the overshoot.

## Second verification round (2026-10-20, on the text of the note)
A further independent verification run, AI-assisted, checked the text of the note before any further edit, with
programs of its own (reproducibility/independent_run_2/, with README and outputs).
- **Sources.** OWR abstract pp. 171–174 and ABB arXiv v2 read again (Sections 2, 3, 3.1 and 5); corpus record; Crossref
  records of the four DOIs (all agree with the bibliography). The Elsevier version of ABB was not read.
- **Own computations.** Closed forms of B, C10 and A in the monomial basis t^j e^{-kt}, coefficient polynomials formed
  directly in a symbolic p (no interpolation): Theorem 3.1, all constants of (3), (4) and Corollary 3.2 (bracket
  coefficients 9/16, (4+r)/4, 7/24, 71/200, 7/80, rates 1, 3, 5, f(0) = p, threshold 16/(9r) + 26/27, 60206/7137 <
  223/25), all 19 entries of Table 1, both forms of (5), no t^2 e^{-t} term, and Proposition 5.1 are identical. Every
  per-state function of B agrees with scipy expm (6.7e-16). Exact certificates by two methods of its own (rational
  bounds for e^{-t} in its closed forms; uniformization in integer arithmetic): B at (61/100, 223/25) 9.3891186962314e-7
  (both > 9.389e-7); C10 at (7/10, 3) 3.4740871121831e-3; A at (7/10, 27/5) 4.776913650723e-4. Widths of Lemma 2.7 at the
  stated parameters (2.49e-65, 2.69e-55, 4.0e-58); partial fractions (2) symbolically; graph data (networkx and an own
  automorphism search) and edge lists of text, figure, check_note.py and original/cert_exact.py identical. Mutation test:
  B without the edge 45 has no overshoot at (61/100, 223/25) (-3.02e-5) and c_{1,2} = -23 r^2 (2p-1)/16 < 0. Monte Carlo
  of the raw clock process on C10 (2e6 runs): f(3) - f(25) = 3.40e-3 ± 1.6e-4 (exact 3.474e-3; evidence only).
- **Package re-run.** source.zip extracted: check_note.py output identical; the nine certificate programs of original/
  byte-identical; recorded outputs of the float searches read (counts as stated, see the corrected coverage above).
- **Literature.** 9 arXiv API queries, Crossref, the OpenAlex work record (6 citations; the citing list was refused
  with HTTP 429), Semantic Scholar (the same 7 citing works), a zbMATH Open search, one web search: no printed answer.
- **Verdict.** Every statement confirmed; no mathematical error; required corrections of wording only (below, 9–13).

## Corrections made
1. Theorem 4.1(b): the loose reason for "at k = 1 only i = 1, 2 occur" was replaced by the polynomial argument
   (coefficients of degree <= 10 that vanish at the 11 nodes vanish identically); the same principle is stated once
   in Lemma 2.6(c) and used for B (degree <= 6, 7 nodes).
2. Lemma 2.3: the strict energy decrease is stated for odd degrees and for the ABB self-inclusion rule, and its
   failure for the coin rule at even degree is noted; it is used only for B, C10 (odd degrees) and A (ABB rule).
3. Scoping in title, abstract, introduction, Section 6 and "Scope and priority".
4. The page of Conjecture 1: p. 172 of OWR 4/2021 (abstract pp. 171–174).
5. References with DOIs from Crossref: Damron–Sen (PTRF 178 (2020) 25–68) and Mossel–Neeman–Tamuz (AAMAS 28 (2014)
   408–429), exact titles.
6. Novelty paragraph: OpenAlex 6 citing works (same as Semantic Scholar's 7), Crossref count 4, zbMATH review.
7. Verification paragraph and graph A as an addition; heuristic labelled as such; the two-sided enclosure for B
   reproduced by `check_note.py` (width 2.5e-65).
8. No internal labels in the note; ABB Prop. 3.6 described as the quenched variant.
9. (second round) Proof of Theorem 3.1: it had said that the 64 functions g_s involve only the six pairs of the final
   formula; in fact 16 of them also contain e^{-4t} terms and two contain e^{-6t} terms, which cancel in f_p (the sums
   over each popcount vanish). The proof now says so and applies the polynomial argument to these pairs as well. The
   same correction for Proposition 5.1 (the pair (3,1) for A) and a remark in Theorem 4.1(b) (the pair (8,3) for C10).
10. (second round) The decimal value of 60206/7137 is 8.43575... (it had been printed as 8.4358...).
11. (second round) Section 6.3: the coverage of the float searches of regular graphs is stated exactly (all connected
   4-regular graphs on 7–10 vertices; 59 of the 60 connected 5-regular graphs on 10 vertices).
12. (second round) Abstract and Section 1.2: for C10 the note now says what Theorem 4.1 proves (f_p(t) > f_p(inf) for
   all large t, every p in (1/2, 1)) instead of "the same conclusion"; a grammatical slip in Section 1.3.
13. (second round) The second round is recorded in the Verification paragraph, the searches paragraph (two rounds,
   eight web searches), Section 7, the package README and here; more vertical space in two rows of Table 1.
None of them changes a theorem.

## Relation to the literature, novelty and scope
- **Searches (20 October 2026, two rounds).** arXiv API (median dynamics / median process; majority dynamics with monotone,
  monotonicity, marginal, conjecture; zero-temperature Glauber with monotonicity; later papers of Amir, Baldasso and
  Beilin); Semantic Scholar (7 works citing ABB, abstracts read); OpenAlex (6 citing works, the same set; first
  request rate-limited, retried); Crossref (is-referenced-by-count 4 for the SPA paper); zbMATH Open (ABB =
  Zbl 1515.60299; the review presents the conjectures as general conjectures with partial results for the complete
  graph and Z); a zbMATH Open search; eight web searches; the OpenAI mathematics release of October 2026 (no entry for this problem).
  - No proof or refutation of OWR Conjecture 1 / ABB Conjecture 3.1 was found.
  - Damron–Sen (PTRF 178 (2020)) introduced the median process on the 3-regular tree (fixation, continuity of the
    fixation probability), not time monotonicity of marginals.
- **What was read.** The OWR abstract in full; ABB arXiv v2 in full (text extraction); the arXiv abstract of
  Damron–Sen; Mossel–Neeman–Tamuz was not read (what the note says is taken from ABB). The Elsevier version of ABB,
  theses and talks were not checked. All four bibliographic records were checked with Crossref on 2026-10-20.
- **Caveats.** The main theorems rest on finite exact computations, re-done by four independent implementations
  and by the exact program of the note; the margin on B is small (9.4e-7) but certified in exact rational
  arithmetic. This negative search is not a proof of priority.
- **Scope.** The note refutes the conjecture as stated (all locally finite graphs). It does not address the
  vertex-transitive case (Z^d, regular trees), connected infinite graphs, or regular graphs of degree >= 4, and it
  makes no conjecture about them.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/