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.

Positivity failure leaves the target unsupported

Example

Let $Ω = {0, 1}$ with each point having probability $1 / 2$ , and set $L = A = ω$ . Consider two full-data laws on this same observed space. In both, $Y^{0} = 0$ and $Y = Y^{A}$ . In law $Q_{1}$ , set $Y^{1} = 1$ everywhere; in law $Q_{2}$ , set $Y^{1} = L$ . Both laws give the same observed distribution: $(L, A, Y) = (0, 0, 0)$ or $(1, 1, 1)$ with probability $1 / 2$ each. Both satisfy consistency and conditional exchangeability given $L$ , but positivity fails. Their ATEs are respectively $1$ and $1 / 2$ , so the observed law and conditional exchangeability alone do not identify the ATE.

Verification

Given: The two full-data laws described above; the definitions of potential outcomes, conditional exchangeability, positivity, ATE and identifiability; and the theorem's positivity hypothesis.

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

[F1] Consistency requires $Y = Y^{A}$ almost surely. Potential outcomes and consistency

[F2] Conditional exchangeability is conditional independence of potential outcomes and treatment given $L$ . Conditional exchangeability of potential outcomes

[F3] Conditional independence holds when one of its sigma-algebras is contained in the conditioning sigma-algebra. Conditional independence given a sigma-algebra

[F4] Positivity for a contrast requires positive treatment probability in each treatment arm almost surely on the target covariate law. Positivity for discrete treatment

[F5] The ATE is $E [Y^{1} - Y^{0}]$ . Average and conditional average treatment effects

[F6] Identifiability means the estimand is constant across parameter values with the same observed law. Identifiability of an estimand

[F7] The displayed standardization theorem assumes positivity for each treatment value in the contrast. Discrete-treatment potential-outcome means are identifiable

1.1givenF1algebra

In both laws, when $L = 0$ one has $A = 0, Y = Y^{0} = 0$ , and when $L = 1$ one has $A = 1, Y = Y^{1} = 1$ . Thus the observed law is identical in $Q_{1}, Q_{2}$ , and consistency [F1] holds in each.

1.2A1F2F3F4F7algebra

In each law the potential-outcome vector is $σ (L)$ -measurable and $A = L$ is $σ (L)$ -measurable. By [F3], conditional exchangeability [F2] holds. However, $P (A = 1 ∣ L = 0) = 0$ and $P (A = 0 ∣ L = 1) = 0$ , so [F4] fails despite $P (A = 0) = P (A = 1) = 1 / 2$ . Hence the hypotheses of [F7] are not met.

2.1F5F6step 1.1algebra∎

Under $Q_{1}$ , $E [Y^{1} - Y^{0}] = 1$ . Under $Q_{2}$ , $E [Y^{1} - Y^{0}] = E [L] = 1 / 2$ . The same observed law therefore corresponds to two target values, so [F6] says the ATE is not identifiable under consistency and conditional exchangeability without positivity.

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