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

How statement and proof provenance work

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.

Stability does not imply cell superstability

Statement refuted

Every stable pair potential is cell-superstable.

Facts & Assumptions

Given: An integer $d \geq 1$ and the stability and cell-superstability conventions in the current continuum-interaction definition.

[A1]

The current draft Stability, superstability and regular tails fixes the half-open cubic tiling $C_{k} = ℓ (k + [0, 1)^{d})$ for some $ℓ > 0$ and defines cell superstability by the existence of constants $A_{*} > 0$ , $B_{*} \geq 0$ such that $U_{N} (X) \geq A_{*} \sum_{k} n_{k} (X)^{2} - B_{*} N$ for every finite position list. It defines stability by $U_{N} (X) \geq - B N$ . This counterexample uses these exact interfaces; the supplier is a definition and no proof-status claim is made.

[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

1.1givenA1construct

Set $ϕ (x) = 0$ for every $x \in R^{d}$ . It is even and Borel, vanishes outside every finite range, and has $U_{ϕ} (ω) = 0$ for every finite configuration. Thus it is stable with $B = 0$ under [A1].

2.1givenA1step 1.1construct

Fix any proposed cubical partition with side $s > 0$ and constants $A > 0$ , $B_{s} \geq 0$ for cell superstability. Choose a cell with center $c$ and, for any integer $N \geq 1$ , place the distinct points $x_{j} = c + s (j / (N + 1) - 1 / 2) e_{1}$ , $1 \leq j \leq N$ , in that cell and none elsewhere. All coordinates lie strictly inside the cell, and their first coordinates are distinct. Hence $\sum_{i} N_{C_{i}}^{2} = N^{2}$ while $U_{ϕ} = 0$ .

3.1A1F1step 2.1algebra∎

Apply [F1] to $x = B_{s} / A$ and let $m \leq x < m + 1$ . Set $N = m + 1$ , so $N > x$ . The proposed cell-superstability inequality would require $0 \geq A N^{2} - B_{s} N = A N (N - x) > 0$ , a contradiction. Since every proposed positive $A$ and finite $B_{s}$ fails, the zero potential is stable but not cell-superstable.

Stability does not imply cell superstability

Statement refuted

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

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