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.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Mediator confounding changes the natural-effect functional

Example

Let $A$ be randomized with probabilities $1 / 2, 1 / 2$ , independently of $U \sim Bernoulli (1 / 5)$ . Set $M = A x o r U, Y = A + U .$ The mediator and outcome share the unmeasured cause $U$ . The true nested responses are $M (a) = a x o r U$ and $Y (a, m) = a + U$ ; the outcome does not respond to an intervention on $M$ . Both treatment arms have positive probability for each mediator value.

Facts & Assumptions

Given: The definitions of natural effects and the observed-data functional in the mediation formula.

[F1]

$NDE (a, a^{'}) = E [Y (a, M (a^{'})) - Y (a^{'}, M (a^{'}))]$ and $NIE (a, a^{'}) = E [Y (a, M (a)) - Y (a, M (a^{'}))]$ . Controlled and natural mediation effects

[F2]

The mediation functional uses the observed conditional outcome mean $μ_{a} (m) = E [Y ∣ A = a, M = m]$ averaged over the mediator law at the reference treatment; it identifies the nested mean only under the stated assumptions. The natural-effect mediation formula under independent errors

[F3]

The natural-effect formula requires no unmeasured mediator–outcome confounding and the cross-world restriction, as well as positivity. Assumptions for identifying natural effects

Verification

Given: The full-data construction in the Example and treatment levels $a = 1$ , $a^{'} = 0$ .

1.1givenF3algebra

Consistency holds by construction. For each $a$ , $M (a)$ takes both values with positive probability because $0 < P (U = 1) < 1$ ; treatment is randomized and both treatment levels have positive probability. However, $U$ affects both $M (a)$ and $Y (a, m)$ , so the no-unmeasured mediator–outcome-confounding condition in [F3] fails.

1.2F1algebra

The true nested means are $E [Y (1, M (0))] = E [1 + U] = 6 / 5$ , $E [Y (1, M (1))] = 6 / 5$ , and $E [Y (0, M (0))] = E [U] = 1 / 5$ . Thus [F1] gives $NDE (1, 0) = 1$ and $NIE (1, 0) = 0$ .

1.3F2givenalgebra

The observed mediator probabilities are $Q_{0} (1) = 1 / 5$ and $Q_{0} (0) = 4 / 5$ . At $A = 0$ , $M = U$ and $Y = U$ , so $μ_{0} (0) = 0$ , $μ_{0} (1) = 1$ . At $A = 1$ , $M = 1 - U$ and $Y = 1 + U$ , so $μ_{1} (0) = 2$ , $μ_{1} (1) = 1$ . The observed-data functional therefore gives $\tilde{θ} (1, 0) = (4 / 5) 2 + (1 / 5) 1 = 9 / 5$ , while $\tilde{θ} (0, 0) = 1 / 5$ and $\tilde{θ} (1, 1) = 6 / 5$ .

2.1F2step 1.1step 1.2step 1.3algebra∎

Its resulting contrasts are $\overset{}{NDE} (1, 0) = 8 / 5$ and $\overset{}{NIE} (1, 0) = - 3 / 5$ , different from the true values in step 1.2. This computes the formula's observed-data functional but does not apply the identification assertion, whose assumptions fail by step 1.1.

Depends on

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