ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Repeated dependent copies do not increase success probability

Example

Let a single response $Y$ be correct with probability $1 / 2$ , and define two candidates by $Y_{1} : = Y_{2} : = Y$ . With a perfect verifier, the probability that at least one candidate is correct is $1 / 2$ . The independent-draw formula at $N = 2$ would give $3 / 4$ , but these duplicated candidates are dependent and do not provide a second opportunity for success.

Example references

The success formula requires independent draws.

Facts & Assumptions

Given: A finite correct/incorrect response law with $\Pr (C (Y) = 1) = 1 / 2$ , and the repeated-copy construction $Y_{1} = Y_{2} = Y$ .

[F1]

Finite truncated response spaces, correctness predicates, and verifiers: correctness is a binary function on the response set.

[F2]

Independent generation success and verifier false-acceptance bounds: $N$ independent draws with one-draw success probability $p$ have success probability $1 - (1 - p)^{N}$ under a perfect verifier.

Verification

Given: The objects and construction in the Example.

1.1givenF1algebra

Since $Y_{1} = Y_{2} = Y$ , the event that at least one candidate is correct equals ${C (Y) = 1}$ , so its probability is $1 / 2$ .

2.1givenF1F2step 1.1algebra∎

The copies are not independent because $\Pr (Y_{1} correct, Y_{2} correct) = 1 / 2 \neq (1 / 2)^{2}$ . Applying [F2] to genuinely independent draws with the same marginal would give $1 - (1 - 1 / 2)^{2} = 3 / 4$ , which differs from the repeated-copy probability in step 1.1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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