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.

Energy concentration without local equivalence

Statement refuted

Fix $0 < θ < 1$ and $ϵ > 0$ . For every integer $N \geq 1$ , let $P_{N} = Bernoulli (θ)^{\otimes N}$ be the product probability on ${0, 1}^{N}$ , where each coordinate equals $1$ with probability $θ$ , and define $H_{N} (x) = ϵ \sum_{i = 1}^{N} x_{i}$ . The refuted claim is: every sequence of probability laws $A_{N}$ on ${0, 1}^{N}$ for which there are deterministic numbers $e_{N} \to ϵ θ$ with $A_{N} (H_{N} / N = e_{N}) = 1$ has first-coordinate marginals converging in event-supremum total variation to $P_{N}^{(1)} = Bernoulli (θ)$ . Use $∥ P - Q ∥_{T V} = \sup_{A} ∣ P (A) - Q (A) ∣$ . This is a finite-model claim, not an experimental assertion.

Facts & Assumptions

Given: 0<θ<1, ε>0, the Bernoulli product reference, and the configurations above.

Counterexample

Given: $0 < θ < 1$ , $ϵ > 0$ , the independent Bernoulli- $θ$ reference $P_{N}$ , and $H_{N} (x) = ϵ \sum_{i} x_{i}$ .

1.1givenconstructalgebra

Let $x^{(N)}$ have its first $⌊ θ N ⌋$ coordinates equal to one and the rest zero, and set $A_{N} = δ_{x^{(N)}}$ . Then $H_{N} (x^{(N)}) / N = ϵ ⌊ θ N ⌋ / N \to ϵ θ$ , since $0 \leq θ N - ⌊ θ N ⌋ < 1$ . For all sufficiently large $N$ , the first coordinate of $A_{N}$ is one.

2.1step 1.1algebra∎

The event ${1}$ in the first-coordinate space has probability difference $1 - θ$ , so $∥ A_{N}^{(1)} - P_{N}^{(1)} ∥_{T V} \geq 1 - θ > 0$ for every sufficiently large $N$ . In fact equality holds because on a two-point space the event-supremum difference between $δ_{1}$ and Bernoulli- $θ$ is $1 - θ$ . Together with the deterministic energy limit in step 1.1, this refutes the stated implication.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Energy concentration without local equivalence

Statement refuted

Facts & Assumptions

Counterexample

Dependency tree · 0 levels

Sources