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.

Multilabel false-negative risk control

Example

Let $K, n \geq 1$ , let $(X, A)$ be a measurable space, give the finite set $2^{[K]}$ its full sigma-algebra, and let $0 \leq λ_{1} < \dots < λ_{M} = 1$ be a deterministic grid in $[0, 1]$ . Fix a measurable score map $f : X \to [0, 1]^{K}$ . Let $(X_{i}, Y_{i})$ , $1 \leq i \leq n + 1$ , be exchangeable random elements of the product measurable space $X \times 2^{[K]}$ , where $Y_{i} \subseteq [K]$ . Define $C_{λ} (x) = {k \in [K] : f_{k} (x) \geq 1 - λ}$ . For nonempty $Y_{i}$ , set $L_{i} (λ) = 1 - ∣ Y_{i} \cap C_{λ} (X_{i}) ∣ / ∣ Y_{i} ∣$ ; for $Y_{i} = \emptyset$ , define $L_{i} (λ) = 0$ . With $B = 1$ and any target $α \in [0, 1)$ , the finite-grid rule selects a threshold whose expected test false-negative loss is at most $α$ .

Example references

Finite nested decision families and bounded risk: exchangeable finite loss rows and endpoint-safe bounded risk.

Finite-grid conformal risk control: the finite-grid calibration rule.

Finite-sample expected-risk control on a finite grid: expected-risk control under loss nesting and endpoint safety.

Angelopoulos et al., Conformal Risk Control, §3.2

Facts & Assumptions

Given: The finite class set, fixed measurable score map, exchangeable example rows, nested threshold sets, explicit empty-label convention, and target $α \in [0, 1)$ .

[F1]

A finite family of bounded loss rows is loss-nested when each row is nonincreasing along the ordered grid, and endpoint-safe when every terminal loss is at most the target. Finite nested decision families and bounded risk

[F3]

For exchangeable, bounded, loss-nested rows with a safe endpoint, the finite-grid rule has expected test loss at most $α$ . Finite-sample expected-risk control on a finite grid

Verification

1.1givenF1construct

For each $k$ and grid point $λ_{j}$ , the event $f_{k} (X_{i}) \geq 1 - λ_{j}$ is measurable; finite intersections, unions, and sums in the displayed loss formula are therefore measurable. The same map is applied to every exchangeable example row, so the resulting loss vectors are exchangeable.

1.2givenalgebra

If $λ \leq λ^{'}$ , then $1 - λ^{'} \leq 1 - λ$ , so $C_{λ} (x) \subseteq C_{λ^{'}} (x)$ . For nonempty $Y_{i}$ , the number of included true labels can only increase, making $L_{i} (λ^{'}) \leq L_{i} (λ)$ ; for empty $Y_{i}$ the loss is constantly zero. The losses lie in $[0, 1]$ .

2.1givenstep 1.2

At $λ_{M} = 1$ , the threshold is zero and every score lies in $[0, 1]$ , so $C_{1} (x) = [K]$ . Hence every nonempty-label loss is zero, and the stated empty-label convention also gives zero. Thus $L_{i} (λ_{M}) = 0 \leq α$ .

3.1givenF3step 1.1step 1.2step 2.1∎

The finite loss rows are exchangeable, bounded by $B = 1$ , nonincreasing, and endpoint-safe by steps 1.1–2.1. Applying [F3] gives $E [L_{n + 1} (\hat{λ})] \leq α$ .

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)