CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge 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.

Cross-group contamination violates partial interference

Statement refuted

An outcome that changes when only a different group's treatment assignment changes can still be represented as a function of the focal unit's own-group assignment.

Facts & Assumptions

Given: Two one-unit groups with binary assignments $Z_{11}, Z_{21}$ and all four assignments allowed.

[F1]

Finite-population potential outcomes under partial interference: partial interference requires $Y_{i j} (z) = Y_{i j} (z^{'})$ whenever the assignments within group $i$ agree.

Counterexample

Let the first group's outcome schedule be $Y_{11} (z_{11}, z_{21}) = z_{21}$ , so it depends only on the other group's assignment. All outcomes are real and the assignment space is finite.

1.1givenF1construct

The global assignments $(z_{11}, z_{21}) = (0, 0)$ and $(0, 1)$ have the same first-group restriction $z_{11} = 0$ , but their first-group outcomes are respectively $0$ and $1$ .

2.1step 1.1F1algebra∎

The inequality of those outcomes violates the invariance in [F1]. Therefore no function of the first group's assignment alone can represent this schedule, and partial interference fails.

Depends on

Finite-population potential outcomes under partial interference

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Michael G. Hudgens and M. Elizabeth Halloran, Toward Causal Inference With Interference, §2.1 (standard reference, not scraped)