CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite range alone does not imply stability

Statement refuted

Every even Borel finite-range pair potential is stable.

Facts & Assumptions

Given: An integer $d \geq 1$ , constants $ϵ > 0$ and $R > 0$ , and the stability convention in the current continuum-interaction definition.

[A1]

The current draft Stability, superstability and regular tails defines stability by the existence of a finite $B \geq 0$ such that $U_{N} (X) \geq - B N$ for every finite position list $X = (x_{1}, \dots, x_{N})$ . For simple configurations this is the same pair-energy bound after enumerating their points. The supplier is a definition; this counterexample uses its stated interface without a proof-status claim.

[F1]

Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ : for every real $x$ there is an integer $m$ with $m \leq x < m + 1$ .

Counterexample

1.1givenconstruct

Define $ϕ (0) = 0$ , $ϕ (x) = - ϵ$ for $0 < ∥ x ∥_{2} \leq R$ , and $ϕ (x) = 0$ for $∥ x ∥_{2} > R$ . Since the norm is continuous, the defining sets are Borel; $ϕ (- x) = ϕ (x)$ , and $ϕ$ vanishes outside the radius- $R$ ball. Hence $ϕ$ is an even Borel finite-range pair potential.

2.1givenstep 1.1construct

For any integer $N \geq 2$ , let $δ = R / (2 N)$ and take $x_{j} = j δ e_{1}$ for $0 \leq j < N$ . They are distinct and every separation is less than $R$ , so the pair energy is exactly $U_{ϕ} = - ϵ N (N - 1) / 2$ .

3.1A1F1step 2.1algebra∎

If a finite stability constant $B$ existed, apply [F1] to $x = 1 + 2 B / ϵ$ and let $m \leq x < m + 1$ be its integer part. Set $N = m + 2$ , so $N > x$ and step 2.1 gives $U_{ϕ} = - ϵ N (N - 1) / 2 < - B N$ . This contradicts [A1], so the finite-range potential is not stable and the statement is false.

Finite range alone does not imply stability

Statement refuted

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources