ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

Full conformal ranks a candidate in an augmented sample

Example

Take feature space $X = {x_{0}}$ and response space $Y = {1, 2, 3}$ , each with its power-set sigma-algebra, so the product example space is identified with $Y$ by $(x_{0}, y) \mapsto y$ . Let $Z_{1}, Z_{2}, Z_{3}$ be a uniformly random ordering of the pairs $(x_{0}, 1), (x_{0}, 2), (x_{0}, 3)$ , and use the response-coordinate score assignment $A_{i} (z_{1}, z_{2}, z_{3}) = y_{i}$ when $z_{i} = (x_{0}, y_{i})$ . The true candidate's nonrandomized upper-rank p-value is $1$ , $2 / 3$ , or $1 / 3$ , each with probability $1 / 3$ . At $α = 0.4$ , the full conformal set covers with probability $2 / 3$ .

Facts & Assumptions

Given: The six permutations of $((x_{0}, 1), (x_{0}, 2), (x_{0}, 3))$ have equal probability, and the score assignment returns each example's response coordinate.

[F1]

Exchangeability is invariance of the joint law under permutations. Exchangeable random elements

[F2]

The full score assignment is measurable and permutation-equivariant, and the prediction set includes the candidate when its upper-rank p-value exceeds $α$ . Full conformal prediction set

[F3]

The upper-rank p-value is superuniform for exchangeable scores. Conformal rank superuniformity under exchangeability

Verification

1.1givenF1

The six equally likely orderings are carried into one another by coordinate permutations, so the data tuple is exchangeable. Each response-coordinate map is measurable on the finite discrete product, and $A (τ_{π} z) = τ_{π} A (z)$ for every permutation.

1.2givenF2

If the last coordinate is $1$ , $2$ , or $3$ , respectively, the count of scores at least the candidate is $3$ , $2$ , or $1$ . Hence $p_{3}$ equals $1$ , $2 / 3$ , or $1 / 3$ , each with probability $1 / 3$ .

2.1F2F3step 1.2∎

Since $1 > 2 / 5$ and $2 / 3 > 2 / 5$ but $1 / 3 ≯ 2 / 5$ , the candidate is included in four of the six equally likely permutations, giving coverage $4 / 6 = 2 / 3 \geq 3 / 5$ . This is consistent with [F3].

Depends on

Used by

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