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 base rates admit a constant calibrated score

Example

Let $Σ = {σ}$ and $B = {b}$ . In both groups take total mass $N_{t} = 1$ , feature proportion $a_{t σ} = 1$ , positive-outcome probability $p_{σ} = 1 / 2$ , and deterministic assignment $X_{σ b} = 1$ with score $v_{b} = 1 / 2$ . Each group then has base rate $1 / 2$ although the outcome is not determined by the feature, and the constant score is calibrated and balanced for both outcome classes.

Example references

KMR's finite group-conditioned risk-score assignment and calibration: defines the finite group-conditioned law, calibration identity, base rate, and perfect-prediction convention.

Balance for positive and negative outcomes: defines the two outcome-conditioned mean-score balance criteria.

Facts & Assumptions

Given: The one-feature, one-bin model in the example. Verify equal base rates, groupwise calibration, both class-balance criteria, and failure of perfect prediction.

[F1]

KMR's finite group-conditioned risk-score assignment and calibration: calibration is the binwise identity between positive mass and score times assigned mass; the base rate is the positive mass divided by group mass.

[F2]

Balance for positive and negative outcomes: positive and negative balance mean equality of the conditional mean score across groups, separately for the two outcomes.

Verification

1.1givenF1F2algebra

In either group, $ρ_{t} = a_{t σ} p_{σ} = 1 / 2$ . The sole bin has total mass $1$ and positive mass $1 / 2$ , so its calibration identity is $1 / 2 = (1 / 2) \cdot 1$ . Since $S$ is constantly $1 / 2$ , the positive and negative conditional mean scores are both $1 / 2$ in both groups, proving both balances.

2.1givenstep 1.1F1algebra∎

The outcome takes both values with probability $1 / 2$ , while $S = 1 / 2$ always; hence $S \neq Y$ almost surely. This is the equal-base-rate exception without perfect prediction.

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