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.

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A two-test BH family distinguishes FDR from FWER

Example

Take two hypotheses and $q = 1 / 2$ . Under the specified law, $H_{1}$ is true with $P_{1} \sim Unif [0, 1]$ , while $H_{2}$ is false with $P_{2} = 0$ almost surely. The p-values are independent. The BH procedure has $FDR = 1 / 4$ and $FWER = 1 / 2$ , so its FDR bound is attained while FDR and FWER differ.

Facts & Assumptions

Given: $m = 2$ , $q = 1 / 2$ , $I_{0} = {1}$ , $P_{1} \sim Unif [0, 1]$ , and $P_{2} = 0$ almost surely under the fixed law.

[F1]

False discovery proportion, FDR, and FWER: if $V$ counts false rejections and $R$ counts all rejections, then $FDP = V / (R \lor 1)$ , $FDR = E [FDP]$ , and $FWER = P (V \geq 1)$ .

[F2]

Benjamini–Hochberg step-up procedure: BH rejects ranks through the largest $k$ with $p_{(k)} \leq k q / m$ .

[F3]

Independent random elements: random elements on one probability space are independent when their generated sigma-algebras are independent.

[F4]

Valid p-values: a p-value statistic takes values in $[0, 1]$ and is valid under a null law when $P (p \leq u) \leq u$ for every $u \in [0, 1]$ .

[F5]

Benjamini–Hochberg FDR under independence: under mutual independence and exactly uniform true-null p-values, BH has FDR $(m_{0} / m) q$ .

[A1]

In this example, $P_{2}$ is a constant random element and hence is independent of $P_{1}$ ; $P_{1}$ is exactly uniform and therefore valid under $H_{1}$ .

[A2]

For the global-null comparison in step 3.1, all hypotheses in the finite family are true, so $I_{0} = [m]$ .

Verification

1.1givenA1F2F3F4algebra

For Borel sets $A, B$ , $P (P_{1} \in A, P_{2} \in B) = P (P_{1} \in A) 1 {0 \in B} = P (P_{1} \in A) P (P_{2} \in B)$ , so [F3] confirms independence; also $P (P_{1} \leq u) = u$ for $u \in [0, 1]$ , so [F4] confirms true-null validity. The ordered values are $0, P_{1}$ : the first BH threshold is $q / 2 = 1 / 4$ and always passes, while the second is $q = 1 / 2$ and passes exactly when $P_{1} \leq 1 / 2$ . Thus both hypotheses are rejected when $P_{1} \leq 1 / 2$ , and only the false null $H_{2}$ is rejected otherwise.

2.1givenA1F1F5step 1.1algebra

If $P_{1} \leq 1 / 2$ , then $V = 1$ , $R = 2$ , and FDP $= 1 / 2$ ; if $P_{1} > 1 / 2$ , then $V = 0$ , $R = 1$ , and FDP $= 0$ . Since $P (P_{1} \leq 1 / 2) = 1 / 2$ , [F1] gives $FDR = 1 / 4$ and $FWER = 1 / 2$ . Here $(m_{0} / m) q = (1 / 2) (1 / 2) = 1 / 4$ , which attains the guarantee [F5].

3.1A2F1algebra∎

Under [A2], $V = R$ pointwise; when $R > 0$ , both $V / (R \lor 1)$ and $1 {V \geq 1}$ equal $1$ , and when $R = 0$ both equal $0$ . Thus the FDP is $1 {V \geq 1}$ pointwise, and evaluating its simple-function integral gives $E [1 {V \geq 1}] = 1 \cdot P (V \geq 1) + 0 \cdot P ((V \geq 1)^{c}) = P (V \geq 1)$ . By [F1], this gives $FDR = FWER$ under the global null.

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