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.

Equal group rewards set every normalized advantage to zero

Example

For any prompt, suppose every response in the policy support receives reward $7$ and the group size is $G = 3$ . Then every sampled group has mean reward $7$ , population standard deviation $0$ , and normalized advantages $A_{i} = 0$ . The reward surrogate contribution to the finite group-relative objective is exactly zero for every candidate policy parameter; the separately specified KL penalty remains in the objective formula.

Example references

The population-standard-deviation zero branch and clipped group objective.

Facts & Assumptions

Given: A prompt with a finite supported response set, reward $r (y) = 7$ on every supported response, three independent group draws, and the finite group-relative update convention.

[F1]

Finite truncated response spaces, correctness predicates, and verifiers: a response policy has a finite fixed support and a finite real reward on each supported response.

[F2]

Finite-budget group-relative policy update: for $G \geq 2$ , the group uses the population standard deviation; when it is zero all advantages are set to zero, while the objective also includes its exact KL penalty.

Verification

Given: The objects and conventions in the Example.

1.1givenF1F2algebra

Since every reward is $7$ , the group mean is $\overline{R} = (7 + 7 + 7) / 3 = 7$ , and each difference $R_{i} - \overline{R}$ is zero. Thus the population standard deviation is $s = 0$ .

2.1F2step 1.1algebra∎

By the specified zero-standard-deviation branch in [F2], each $A_{i} = 0$ . For every sampled response and candidate parameter, both arguments of the clipped minimum are zero, so its average reward-surrogate contribution is zero. The objective's separately defined KL term remains.

Depends on

Used by

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Dependency tree · two levels

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