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.

Fallback when the calibration correction exceeds the target

Example

Let $n, M \geq 1$ , $B > 0$ , and $0 \leq α < B / (n + 1)$ . Consider any loss-nested finite-grid family with calibration losses in $[0, B]$ and endpoint losses $L_{i} (λ_{M}) \leq α$ almost surely. The B-corrected calibration threshold is too large to pass at any grid point, so the rule falls back to $λ_{M}$ . The endpoint condition then bounds the realized test loss, and the expected test risk is at most $α$ .

Example references

Finite nested decision families and bounded risk: nonnegative bounded losses and the safe maximum-grid endpoint.

Finite-grid conformal risk control: the pass threshold includes the B correction and otherwise returns the maximum grid point.

Angelopoulos et al., Conformal Risk Control, §1.1

Facts & Assumptions

Given: A finite grid, $n \geq 1$ , $B > 0$ , $0 \leq α < B / (n + 1)$ , calibration losses in $[0, B]$ , and endpoint test loss at most $α$ almost surely.

[F1]

Losses are nonnegative and endpoint-safe, with $L_{n + 1} (λ_{M}) \leq α$ almost surely. Finite nested decision families and bounded risk

[F2]

At grid point $λ_{j}$ , the algorithm tests $(\sum_{i = 1}^{n} L_{i} (λ_{j}) + B) / (n + 1)$ and uses $λ_{M}$ when no point passes. Finite-grid conformal risk control

[F3]

Expectation preserves almost-sure inequalities for integrable real random variables. Linearity, monotonicity, and the modulus bound for expectation

Verification

1.1givenF2algebra

Since every calibration loss is nonnegative, for each $j$ the algorithm's threshold satisfies $(\sum_{i = 1}^{n} L_{i} (λ_{j}) + B) / (n + 1) \geq B / (n + 1) > α$ . No point passes, so the rule returns $λ_{M}$ .

2.1givenF1F3step 1.1∎

The endpoint condition gives $0 \leq L_{n + 1} (λ_{M}) \leq α$ almost surely. The loss is bounded and integrable, so [F3] implies $E [L_{n + 1} (λ_{M})] \leq α$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)