ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

The all-zero loss boundary

Example

Let $n, M \geq 1$ , $B > 0$ , $α \in [0, B)$ , and let every calibration and test loss be zero at every grid point. The rule passes at the first grid point exactly when $α \geq B / (n + 1)$ ; when $α < B / (n + 1)$ it returns the maximum-grid fallback. In either case the selected test loss is zero and its expected risk is zero.

Example references

Finite nested decision families and bounded risk: the bounded finite-grid risk setup.

Finite-grid conformal risk control: the B-corrected pass test and fallback.

Angelopoulos et al., Conformal Risk Control, §1.1

Facts & Assumptions

Given: The finite grid, the all-zero calibration and test losses, $B > 0$ , and $α \in [0, B)$ .

[F1]

The risk family uses a finite deterministic grid and bounded measurable loss rows; expected test risk is the expectation of the selected test loss. Finite nested decision families and bounded risk

[F2]

The algorithm selects the first grid point whose B-corrected calibration average is at most $α$ , and falls back to the maximum point if none passes. Finite-grid conformal risk control

Verification

1.1givenF2algebra

At every grid point, the calibration threshold is $(0 + B) / (n + 1) = B / (n + 1)$ . Hence, if $α \geq B / (n + 1)$ the first point passes, while if $α < B / (n + 1)$ every point fails and the algorithm returns the fallback.

2.1givenF1step 1.1algebra∎

The test loss is zero at every grid point, including the first point and the fallback. Its expectation is therefore $0$ , which is at most $α$ because $α \geq 0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anastasios N. Angelopoulos, Stephen Bates, Adam Fisch, Lihua Lei, and Tal Schuster, Conformal Risk Control, arXiv:2208.02814v4 (standard reference, not scraped)