# Verification report — AMR-099-0003 (Benjamini's question on points in equilibrium)

Verification date: 2026-10-03 (two independent verification runs, both AI-assisted).

**Verdict.** The answer is yes. Let 1 < s ≤ 2, and let X ⊂ ℝ be locally finite with at least two points, such that for
every x ∈ X the total force Σ_{y≠x} |y − x|^(−s) is finite and the net force Σ_{y≠x} sgn(y − x)|y − x|^(−s) is zero.
Then X is an arithmetic progression. For s = 2 this answers Benjamini's Question 0.1 and Problem 1 of
Georgakopoulos–Kolountzakis for the inverse-square force, under their convention (finite one-sided forces,
increasing labelling). No assumption on the gaps is needed. General strictly decreasing force laws, exponents s > 2
and a principal-value reading of the equilibrium condition are not covered. The note is unrefereed.

## Statement checked
- **Primary source.** I. Benjamini, *Points in equilibrium*, one-page note dated May 2015, Question 0.1. The original
  URL https://www.wisdom.weizmann.ac.il/~itai/equilibrium.pdf is no longer online; the arquivo.pt copy was read.
  - Setting: a locally finite configuration (a_n)_{n∈ℤ} on ℝ and a force F(x, y), for example |x − y|^(−2).
  - Equilibrium: Σ_{i≠n} F(a_i, a_n) sign(a_i − a_n) = 0 for all n.
  - Question: does a_n = αn + β hold for all n?
  - The note does not specify how the series converges, and it does not say that the labelling is increasing.
- **Warwick problem list.** J. Sylvester (ed.), *Open problems from "Random walks on graphs and potential theory"*,
  University of Warwick, 18–22 May 2015. Section 4 (I. Benjamini, by proxy) contains the same question as
  Question 5.
- **Georgakopoulos–Kolountzakis**, *On particles in equilibrium on the real line*, Proc. Amer. Math. Soc. 145 (2017)
  3501–3511, doi:10.1090/proc/13492, arXiv:1604.01649. Both versions were read.
  - Problem 1 asks whether all distances between subsequent particles of an equilibrium must be equal.
  - The arXiv version defines equilibrium by finite one-sided forces and zero net force. The published version puts
    the finiteness assumption on the force law (finite one-sided forces for particles at the integers) and defines
    equilibrium as zero net force; its proofs compare one-sided forces.
  - Results: Theorem 1 (an equilibrium with a gap of maximal or minimal length is an arithmetic progression),
    Corollaries 2–3 (periodic equilibria; finite equilibria on a circle), Theorem 4 (for d^(−2) and uniformly
    discrete configurations, a half-line in equilibrium determines the rest), Corollary 6, Proposition 1 (consecutive
    gap ratios are bounded; sketch), Propositions 2–4, Theorem 7 (non-trivial configurations with all particles but
    one in equilibrium).
  - Both versions describe the aperiodic case as open, even for F(d) = d^(−2), and the problem as open for every
    strictly decreasing F.
- **Corpus record.** ulamai/UnsolvedMath, AMR-099-0003 (status `partially_solved`). It fixes F = |x − y|^(−2) and
  transcribes the question faithfully.

## Readings
| Reading | Answer | Reason |
|---|---|---|
| unordered (absolutely convergent) force sums, increasing labelling, F = d^(−2) (Benjamini; G–K's convention) | yes: every equilibrium is an arithmetic progression | Theorem 1.2 |
| the same for F = d^(−s), 1 < s < 2 | yes | Theorem 1.2 |
| arbitrary bijection ℤ → X as labelling | the literal identity fails trivially (re-enumerate an arithmetic progression); not the intended question | Remark 1.3(b) |
| principal-value sums (infinite one-sided forces, symmetric partial sums cancel) | not covered | Remark 1.3(a) |
| general strictly decreasing F (G–K Problem 1) | not covered; open | Remark 5.4 |
| F = d^(−s), s > 2 | not claimed (a sketch suggests 2 < s ≤ (3+√5)/2) | Remark 5.3 |

## Results in the paper
- **Lemma 2.1.** An s-equilibrium is unbounded in both directions, so it has an increasing enumeration (a_n)_{n∈ℤ}.
- **Lemma 2.2.** Equilibrium of a_n and a_{n+1} gives g_n ≤ r_s g_{n−1}, where r_s solves 2r^(−s) + (1+r)^(−s) = 1;
  by reflection also g_{n−1} ≤ r_s g_n. For s = 2, r_2 = 1.538615…
- **Lemma 2.3.** Subtracting the equilibrium equations at a_n and a_{n+1} (after the index shift j ↦ j+1) gives
  Σ_{j≠n} c(n,j)(g_j − g_n) = 0, where c(n,j) is the divided difference of t ↦ t^(−s) at D = |a_j − a_n| and
  D' = |a_{j+1} − a_{n+1}| with the sign reversed; for s = 2, c(n,j) = (D + D')/(D²D'²). So the gaps form a positive
  harmonic function of a reversible random walk on ℤ with long-range jumps. Remark 2.5: G–K's Theorem 1 is the
  maximum principle for this function.
- **Lemma 2.4.** c(n,j) ≤ s(1+r_s)^(s+1) |a_j − a_n|^(−s−1); for s = 2 also c ≤ (1+r_2)(2+r_2)|a_j − a_n|^(−3) with
  (1+r_2)(2+r_2) = 8.983…; hence π(n) = Σ_j c(n,j) < ∞.
- **Lemmas 3.1–3.3.** Dirichlet principle, Liouville property for positive harmonic functions of recurrent walks, and
  the Doob transform c^h = c·h⊗h, proved for networks that need not be locally finite (only π < ∞ is used).
  Lemma 3.3(c): the transform preserves return probabilities. Lemma 3.3(d): E_c(hφ) = E_{c^h}(φ).
- **Proposition 4.1.** For f_R(n) = φ_R(a_n): E_{c*}(f_R) = O(R^(1−s)) for 1 < s < 2, and O(log R / R) for s = 2. The
  proof uses only the ratio bound and, for s = 2, the point count #{n : |a_n| < 3R} ≤ 1 + 16R²(F⁻_0 + F⁺_0), which
  follows from the finite force at a_0 (any bound with log N(R) = o(R) would do).
- **Theorem 1.2.** By Proposition 4.1 and Lemma 3.1 the transformed walk is recurrent; 1/g is a positive harmonic
  function for it (Lemma 3.3(b)), hence constant (Lemma 3.2). So all gaps are equal.
- **Remark 5.1.** For an equilibrium, the transformed walk is recurrent if and only if the original one is; the
  transform supplies the test functions g_n φ_R(a_n), it does not create recurrence. The informal idea that c may be
  transient when gaps shrink concerns only sequences that are not in equilibrium.

## Computations (scripts and outputs in reproducibility/)
- **Lead** (`lead/check_general_s.py`; Parts A–D standard library, Part E numpy; about 12 s).
  - r_s for s ∈ {1.05, 1.25, 1.5, 1.75, 2} to 50 digits.
  - The finite-range identity behind Lemma 2.3 in 60-digit arithmetic for s ∈ {5/4, 3/2, 7/4, 2}: 777 identities,
    relative errors below 2·10^(−59); symmetry and positivity of c.
  - The inequality behind Lemma 2.2 on arbitrary finite configurations: 1006 cases.
  - Lemma 2.4 on adversarial gap patterns; for s = 2 the largest c·D³ found is 3.906.
  - The inequalities of the proof of Proposition 4.1 for s = 5/4 and s = 3/2 on five gap profiles (floating point).
- **Finder** (`finder/`, s = 2).
  - Exact rational checks: the identity of Lemma 2.3 term by term (6240 terms), the comparison bound of Lemma 2.4
    (25,200 pairs), the algebra of Lemma 3.3(b).
  - Numerical illustrations: 900 two-particle equilibrium problems (largest gap ratio 1.4870 < r_2); E*(f_R) for five
    profiles and R up to 300 (E*·R/log R between 2.06 and 6.54).
- **Independent verification run** (`independent/`; AI-assisted, with its own code, not using the finder's scripts).
  - Exact rational checks on configurations with gaps from 10^(−6) to 10^6: the finite-range identity behind Lemma 2.3
    (530 cases), the inequality behind Lemma 2.2 (470 cases), the identities of Lemma 3.3(b) and (d) (40 random
    networks), Lemma 2.4 on adversarial patterns.
  - Every inequality of the proof of Proposition 4.1 (s = 2) on six further profiles (E*·R/log R between 1.29 and 9.96).
  - A numerically solved block of 200 particles in equilibrium between two fixed arithmetic tails: identity (3) holds
    at all 199 interior indices up to 4.9·10^(−8) (after division by π(n); tail truncation) and fails next to the
    fixed particles; the maximum principle and Lemma 2.2 hold wherever their hypotheses apply.
- **Second independent verification run** (`independent_run_2/`; AI-assisted; code written from the text of the paper
  before any packaged script was read).
  - `run2_exact.py` (standard library; exact for s = 2, 100-digit decimals otherwise):
    - r_s for nine values of s in (1, 2];
    - the point count after Definition 1.1 and Lemma 2.1;
    - the finite-range identity behind Lemma 2.3: 1689 identities (s = 2 exact; s = 11/10, 5/4, 3/2, 7/4, 19/10 with
      relative error ≤ 1.4·10^(−99)), together with the symmetry, positivity and closed form of c;
    - the two comparisons in the proof of Lemma 2.2 on arbitrary finite configurations: 1891 cases;
    - Lemma 2.4 on 11,536 pairs. Also D' ≥ D/r_s, and the largest c·D^(s+1) equals the sharp value
      r_s(r_s^s − 1)/(r_s − 1), which is 3.906 for s = 2;
    - Green's identity and Lemma 3.1 on complete-graph networks, exactly;
    - Lemma 3.3(a)–(d) on networks with a harmonic h (with a negative control) and on an infinite birth–death chain
      with a positive harmonic h.
  - `run2_energy.py` (numpy): every inequality of the proof of Proposition 4.1 and its final bound.
    - Cases: s ∈ {1.1, 1.25, 1.5, 1.75, 1.9, 2}, eight gap profiles with consecutive ratios in [1/r_s, r_s], R ≤ 300;
      186 cases.
    - Omitted far pairs are bounded rigorously.
    - No violation; the largest ratio of the energy to the bound of Proposition 4.1 is 0.018.
- All recorded outputs were reproduced from the packaged scripts on 2026-10-03, apart from timing lines. In the second
  run they were reproduced from an extracted copy of the source archive
  (`independent_run_2/rerun_packaged_summary.txt`).
- The numerical runs illustrate the statements; the proofs do not use them.

## Independent verification runs
Both runs were AI-assisted. Each re-derived the proofs line by line and wrote its own code. Neither found a
mathematical error.

### First run
Verdicts (2026-10-03):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED |
| Proofs | CONFIRMED (every step re-derived line by line; no gap found) |
| Computations | CONFIRMED (the finder's scripts reproduce their outputs byte for byte) |
| Answer as posed | CONFIRMED (yes, under G–K's convention: finite one-sided forces, increasing labelling) |
| Novelty | CONFIRMED as far as can be checked (no priority claim) |
| Presentation | CONFIRMED_WITH_FIXES |

All required presentation fixes of the first run were applied:
1. The explanation of the Doob transform was rewritten (Remark 5.1): for an equilibrium, g is harmonic for c, so
   the transformed walk has the same return probabilities and c is recurrent if and only if c* is; the transform
   supplies the test functions, E_c(gφ) = E_{c*}(φ); the transience heuristic concerns only non-equilibrium
   sequences.
2. The conventions are stated inside Theorem 1.2 (finite one-sided forces, increasing enumeration). Remark 1.3
   explains the difference between the arXiv and the published wording of G–K and what is not covered (the
   principal-value reading, general strictly decreasing F).
3. Citations: G–K Theorem 4 (uniqueness of continuation), Theorem 7 (nailed configurations), Proposition 1 (ratio
   bound); two distinct works cite G–K; the Warwick item is Section 4, Question 5.
4. The extension to 1 < s < 2 is written out in full (r_s, the mean-value bound on c, the O(R^(1−s)) energy
   estimate). The range 2 < s ≤ (3+√5)/2 is only a labelled sketch (Remark 5.3) and is not claimed.
5. Optional items: log N(R) = o(R) suffices (Remark 5.2); the Dirichlet principle is cited from Lyons–Peres and
   proved for networks with finite total conductance at each vertex (Lemma 3.1); Duneau–Katz and long-range
   Aubry–Mather theory are discussed as related work.

### Second run
Verdicts (2026-10-03):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED. Benjamini's note (archived copy; the original URL returns 404), the Warwick list (Section 4, Question 5) and both versions of G–K were fetched again and read on rendered pages. The numbering Theorem 1, Theorem 4, Theorem 7, Proposition 1 is correct, and so are the two definitions of equilibrium. |
| Proofs | CONFIRMED line by line: Lemmas 2.1–2.4, Green's identity, Lemmas 3.1–3.3 for networks that are not locally finite, every step and constant of Proposition 4.1, Theorem 1.2. Remark 5.3 is labelled as an unclaimed sketch. The comparison in Lemma 2.4 is valid but not sharp (D' ≥ D/r_s holds); this does not affect the proof. |
| Computations | CONFIRMED with its own code (see Computations), and by re-running every packaged script from an extracted copy of the source archive. |
| Answer as posed | CONFIRMED (yes, under the convention of Remark 1.3) |
| Novelty | No resolution found in a renewed search (see below); no priority claim |
| Presentation | CONFIRMED_WITH_FIXES |

Required fixes of the second run, all applied:
1. The Verification paragraph of the note records both independent verification runs.
2. The record of the literature search in the note ("Scope and priority") and in this report was updated:
   - Semantic Scholar is now reachable;
   - three general web searches were made;
   - the publication lists of the authors of G–K were checked.
3. The wording used for the first run in this report was corrected: both runs were AI-assisted verification runs.
4. The code and outputs of the second run were added as `reproducibility/independent_run_2/`.

## Relation to the literature, novelty and scope
- **Searches (October 2026; repeated by the second verification run).** We searched arXiv, Crossref, OpenAlex,
  zbMATH and Semantic Scholar, and the web (three general searches). Queries covered:
  - particles or points in equilibrium on the line;
  - equally spaced and arithmetic-progression equilibria;
  - inverse-square, Coulomb and Riesz interactions;
  - the Doob transform;
  - long-range Aubry–Mather theory;
  - the authors of G–K.
  Results:
  - OpenAlex and Semantic Scholar list the same two works citing G–K: Bétermin–Petrache 2019, on optimality of
    lattices, and Malikiosis 2018, on formal duality. Neither addresses the question. Semantic Scholar also lists an
    unrelated 2004 book chapter, evidently a misattribution.
  - The publication lists of both authors of G–K contain no follow-up.
  - The web searches found only G–K and copies of it.
  - Nothing reports a resolution of the question or a partial result beyond G–K.
- **Caveats.**
  - MathSciNet and Google Scholar were not searched.
  - Duneau–Katz (1984) was read in its abstract and introduction; it concerns the structural stability of
    one-dimensional lattices arising from finite stable equilibria and does not classify infinite equilibria.
  - Candel–de la Llave (1998) and de la Llave–Valdinoci (2010), on Aubry–Mather theory in statistical mechanics, are
    known to us only bibliographically. Equilibria minimize the formal energy under compact order-preserving
    perturbations, and a statement that such minimizers are ordered with respect to all their translates would imply
    the result. We have not checked whether these works apply to the singular interaction d^(1−s) and to
    configurations that are not uniformly discrete; G–K do not mention such a reduction.

  This negative search is not a proof of priority.
- **Scope.** The note proves the inverse-square case of Benjamini's question, and the d^(−s) cases with 1 < s < 2,
  under finite one-sided forces and increasing labelling. Problem 1 of G–K for general strictly decreasing force laws,
  the exponents s > 2 and the principal-value reading remain open.

## 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/