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.

A small first stage magnifies reduced-form changes

Example

Consider two hypothetical pairs of observed mean contrasts. In each, the first-stage contrast is $Δ_{A} = 0.02$ ; the reduced-form contrasts are $Δ_{Y} = 0.06$ and $Δ_{Y} = 0.07$ . The ratio formula gives $3$ and $3.5$ , respectively. This is arithmetic illustrating the ratio's sensitivity to a small denominator; it makes no claim about a sampling distribution, estimator variance, or causal validity.

Facts & Assumptions

Given: $Δ_{A} = 0.02$ and, in turn, $Δ_{Y} = 0.06$ or $0.07$ . Compute the two Wald ratios and their difference.

[F1]

The Wald ratio is $Δ_{Y} / Δ_{A}$ when $Δ_{A} \neq 0$ . (Binary-instrument Wald ratio)

Verification

1.1givenF1algebra

Since $0.02 \neq 0$ , the two ratios are defined by [F1] and equal $0.06 / 0.02 = 3$ and $0.07 / 0.02 = 3.5$ .

2.1givenstep 1.1algebra∎

Their difference is $0.5$ , exactly the reduced-form change $0.01$ divided by the common first stage $0.02$ . The calculation is a property of the ratio only and implies no sampling or causal conclusion.

Depends on

Binary-instrument Wald ratio

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hernán and Robins, Causal Inference: What If, Chapter 16, §§16.2 and 16.5, printed pp. 212–222 (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Unmeasured Confounding (Instrumental Variables), instrument strength and LATE sections (standard reference, not scraped)