CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

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One finite sequence value does not determine asymptotic growth

Statement refuted

For every positive real sequence $(a_{N})_{N \geq 1}$ , every real $V > 0$ , and every integer $N_{0} \geq 1$ , if $(2 π / N_{0}) \log (a_{N_{0}}) = V$ , then $\lim_{N \to \infty} (2 π / N) \log (a_{N}) = V$ .

Statement references

Hitoshi Murakami and Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, §5

Hitoshi Murakami, An Introduction to the Volume Conjecture, §§1–4

Facts & Assumptions

Given: A positive real sequence $(a_{N})_{N \geq 1}$ , a target value $V > 0$ , and a finite index $N_{0} \geq 1$ .

[F1]

For every positive real $x$ , $\log x$ is defined, and $\log 1 = 0$ .

Counterexample

1.1givenF1constructalgebra

Define $a_{N_{0}} = \exp (V N_{0} / (2 π))$ and $a_{N} = 1$ for every $N \geq 1$ with $N \neq N_{0}$ . Each term is positive, and $(2 π / N_{0}) \log (a_{N_{0}}) = (2 π / N_{0}) (V N_{0} / (2 π)) = V$ .

2.1F1step 1.1algebra∎

For every $N > N_{0}$ , $a_{N} = 1$ , so $(2 π / N) \log (a_{N}) = 0$ . The sequence of normalized logarithms is therefore eventually zero and has limit $0 \neq V$ . This contradicts the asserted conclusion.

Remarks

This is a sequence-level witness about the information in one finite evaluation. It is not asserted to be a colored Jones sequence and does not refute the volume conjecture.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

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

Sources

Hitoshi Murakami and Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, §5 (standard reference, not scraped)
Hitoshi Murakami, An Introduction to the Volume Conjecture, §§1–4 (standard reference, not scraped)