CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 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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Continuous-time frequencies alias under fixed discrete sampling

Statement refuted

For a fixed sampling interval $h > 0$ , if a real continuous-time signal has the form $y_{ω} (t) = \cos (ω t)$ for an unknown frequency $ω \geq 0$ , then its entire noiseless sample sequence $(y_{ω} (k h))_{k \in N_{0}}$ uniquely determines $ω$ .

Statement references

Fixed sample interval and discrete observations

The common period $2 π$ of sine and cosine

Facts & Assumptions

Given: A fixed sampling interval $h > 0$ , the two frequencies $ω_{0} = 0$ and $ω_{1} = 2 π / h$ , and the sampled output convention at times $k h$ .

[F1]

Discrete linear input-state-output experiments: sample $k$ is observed at time $k h$ for a fixed interval $h > 0$ .

[F2]

The zero sets of sine and cosine and the least positive common period 2 pi: cosine has period $2 π$ , so $\cos (2 π k) = \cos 0 = 1$ for every integer $k$ .

Counterexample

1.1givenF1F2algebra

Since $h > 0$ , $ω_{1} = 2 π / h > 0 = ω_{0}$ , so the frequencies are distinct. By [F1], the $k$ th samples occur at $t_{k} = k h$ . The two continuous outputs are $\cos (ω_{0} t_{k}) = \cos 0 = 1$ and $\cos (ω_{1} t_{k}) = \cos (2 π k) = 1$ by [F2]. Thus their entire noiseless sample sequences agree.

2.1givenstep 1.1algebra∎

The distinct admissible frequencies in step 1.1 have identical complete noiseless data. Therefore the data do not uniquely determine the frequency, which refutes the stated claim.

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