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.

Binary confounding and standardization

Example

Let $Ω = {0, 1} \times {1, 2, 3, 4}$ with each point having probability $1 / 8$ , and write its coordinates as $(L, W)$ . Set $A = 1$ when $L = 0, W = 1$ or when $L = 1, W \in {1, 2, 3}$ , and set $A = 0$ otherwise. Let $Y^{0} = 10 L$ , $Y^{1} = 10 L + 1$ , and $Y = Y^{A}$ . Assignment depends on the prognostic covariate $L$ , but both treatment arms occur in both strata. The target ATE is $1$ ; the unadjusted treated-minus-untreated mean is $6$ ; standardization over the target law of $L$ returns $1$ .

Verification

Given: The finite probability space and variables above; the definitions of potential outcomes, conditional exchangeability, positivity, and ATE; and the discrete-treatment identification theorem.

[A1] Choice is assumed for the library's conditional-expectation classes. The Axiom of Choice

[F1] Consistency means that the observed outcome equals the potential outcome at the received treatment version. Potential outcomes and consistency

[F2] Conditional exchangeability is conditional independence of the full potential-outcome family and treatment given $L$ . Conditional exchangeability of potential outcomes

[F3] If one side of conditional independence is measurable with respect to the conditioning sigma-algebra, conditional independence holds. Conditional independence given a sigma-algebra

[F4] Positivity requires positive treatment probability in each target covariate stratum for every treatment in the contrast. Positivity for discrete treatment

[F5] The ATE is the target-population mean of $Y^{1} - Y^{0}$ . Average and conditional average treatment effects

[F6] Under consistency, conditional exchangeability and positivity, standardization identifies each potential-outcome mean. Discrete-treatment potential-outcome means are identifiable

1.1givenF1algebra

The point set has eight equally likely elements, $Y^{0}$ and $Y^{1}$ are bounded, and $Y = Y^{A}$ by construction, so all required integrability and consistency conditions hold by [F1].

1.2A1F2F3F4algebra

Given $L = l$ , $(Y^{0}, Y^{1}) = (10 l, 10 l + 1)$ is fixed, hence its sigma-algebra is contained in $σ (L)$ and [F3] gives conditional exchangeability [F2]. Direct counting gives $P (A = 1 ∣ L = 0) = 1 / 4$ , $P (A = 1 ∣ L = 1) = 3 / 4$ , and the complementary probabilities for $A = 0$ ; both marginal probabilities equal $1 / 2$ . Thus [F4] holds.

1.3F5F6algebra

In stratum $L = 0$ , the observed outcome means are $E [Y ∣ A = 1, L = 0] = 1$ and $E [Y ∣ A = 0, L = 0] = 0$ . In stratum $L = 1$ , they are $11$ and $10$ . Since $P (L = 0) = P (L = 1) = 1 / 2$ , [F6] standardizes the treatment means to $E [Y^{1}] = (1 + 11) / 2 = 6$ and $E [Y^{0}] = (0 + 10) / 2 = 5$ , so [F5] gives ATE $1$ .

2.1givenstep 1.2algebra∎

Bayes' rule gives $P (L = 1 ∣ A = 1) = 3 / 4$ and $P (L = 1 ∣ A = 0) = 1 / 4$ . Therefore $E [Y ∣ A = 1] = (1 / 4) 1 + (3 / 4) 11 = 8.5$ and $E [Y ∣ A = 0] = (3 / 4) 0 + (1 / 4) 10 = 2.5$ , whose difference is $6$ , not the ATE. The difference comes from the different target-covariate mixtures in the two treatment groups.

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