LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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.

Cell-quadratic lower bound from superstability

Statement

Let $ϕ = ϕ_{s} + ϕ_{r}$ be superstable in the splitting convention of Superstable regular pair interactions and configurational ensemble notation. Suppose $H_{ϕ_{s}} (ω) \geq - B_{s} N (ω)$ and $ϕ_{r} (x) \geq δ > 0$ for $∣ x ∣_{\infty} \leq r_{0}$ , with $ϕ_{r} \geq 0$ elsewhere. For the unit cells $C_{i} = i + [0, 1)^{d}$ , there are constants $A > 0$ and $B \geq 0$ such that every finite configuration satisfies $H_{ϕ} (ω) \geq A \sum_{i \in Z^{d}} N_{C_{i}} (ω)^{2} - B N (ω) .$ Only finitely many summands are nonzero. This is the cell form used in the continuum pressure arguments.

Facts & Assumptions

Given: The potential and splitting above.

[F1]

Superstable regular pair interactions and configurational ensemble notation: superstability supplies a stable part and a nonnegative repulsive part bounded below by $δ$ on a fixed neighborhood of the origin.

Proof

Given: The assumptions and the half-open unit-cell partition.

1.1F1algebra

Choose an integer $k \geq 1$ with $1 / k \leq r_{0}$ . Partition each unit cell $C_{i}$ into $k^{d}$ half-open subcubes of side $1 / k$ , and denote their occupancies by $n_{i, j}$ , $1 \leq j \leq k^{d}$ . Distinct points in one subcube have $∣ x - y ∣_{\infty} < 1 / k \leq r_{0}$ , so their repulsive pair energy is at least $δ$ . Therefore $H_{ϕ_{r}} (ω) \geq \frac{δ}{2} \sum_{i, j} n_{i, j} (n_{i, j} - 1) .$ Writing $\overset{ˉ}{n} = N_{C_{i}} (ω) / k^{d}$ , nonnegativity of $\sum_{j} (n_{i, j} - \overset{ˉ}{n})^{2}$ expands to $\sum_{j} n_{i, j}^{2} \geq N_{C_{i}} (ω)^{2} / k^{d}$ , and $\sum_{i, j} n_{i, j} = N (ω)$ . Thus $H_{ϕ_{r}} (ω) \geq \frac{δ}{2 k^{d}} \sum_{i} N_{C_{i}}^{2} - \frac{δ}{2} N$ .

2.1F1step 1.1algebra∎

The stable part obeys $H_{ϕ_{s}} (ω) \geq - B_{s} N$ . Add this inequality to step 1.1 and take $A = δ / (2 k^{d})$ and $B = B_{s} + δ / 2$ . This proves the cell-quadratic bound. If any term is $+ \infty$ , the inequality holds in the extended order; otherwise all sums are finite.

Depends on

Superstable regular pair interactions and configurational ensemble notation

Cell-quadratic lower bound from superstability

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources