ExampleConstruction: 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.

A finite two-stage village strategy comparison

Example

There are four groups, each with three units. The first stage assigns exactly two groups to $ψ$ and two to $ϕ$ . In each group, strategy $ϕ$ treats one unit uniformly and strategy $ψ$ treats two units uniformly. Let every unit's potential outcome be the total number treated in its own group: $Y_{i j} (z_{i}) = \sum_{ℓ = 1}^{3} z_{i ℓ} .$ The model allows within-group interference because a unit's outcome changes with its peers' assignments.

Verification

Given: The design and estimands in Two-stage randomized design and strategy-specific effects, with $N = 4$ , $n_{i} = 3$ , $C = 2$ , $K_{i}^{ϕ} = 1$ , and $K_{i}^{ψ} = 2$ .

[F1] Two-stage randomized design and strategy-specific effects: under strategy $s$ , each group has exactly $K_{i}^{s}$ treated units; the target means average the observed potential outcomes over the within-group design and the fixed groups.

[F2] Two-stage strategy-specific effect estimators are design-unbiased: the four source strategy-contrast estimators use the within-group treatment-specific means, the across-group strategy means, and their stated differences.

1.1givenF1algebra

In every assignment under strategy $s$ , the outcome of every unit in that group equals the fixed treated count $K_{i}^{s}$ , independently of which particular units are treated. Thus all conditional and unconditional group means under $s$ equal $K_{i}^{s}$ .

2.1step 1.1F1algebra

Consequently $\overset{ˉ}{Y} (0; ϕ) = \overset{ˉ}{Y} (1; ϕ) = \overset{ˉ}{Y} (ϕ) = 1$ and $\overset{ˉ}{Y} (0; ψ) = \overset{ˉ}{Y} (1; ψ) = \overset{ˉ}{Y} (ψ) = 2$ . Substitution into the definitions gives $D E (ψ) = 0$ , $I E (ϕ, ψ) = - 1$ , $T E (ϕ, ψ) = - 1$ , and $O E (ϕ, ψ) = - 1$ .

3.1step 2.1F1F2algebra∎

In each realized group, every observed outcome equals its strategy's fixed count. The within-group mean in either treatment arm and the full group mean therefore equal that same count. Averaging over the groups assigned each strategy yields hatted means $1$ under $ϕ$ and $2$ under $ψ$ , so all four observed contrasts equal the targets from step 2.1 for every assignment.

Depends on

Used by

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Sources

Michael G. Hudgens and M. Elizabeth Halloran, Toward Causal Inference With Interference, §§2.2–4.1 (standard reference, not scraped)