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 two-factor interaction depends on response scale

Example

In a two-factor full factorial, let the four positive responses be $y_{- -} = 1$ , $y_{+ -} = 2$ , $y_{- +} = 3$ , and $y_{+ +} = 6$ . The coded $A B$ effect is $1$ on the raw response scale and $0$ on the logarithmic scale. The same design therefore shows interaction on one response scale and no interaction on another.

Facts & Assumptions

[F1]

Two-level full factorial designs and coded effects: for a nonempty index subset $S$ , its factorial effect is the mean response among runs with $x_{S} = + 1$ minus the mean among runs with $x_{S} = - 1$ .

Verification

Given: The four positive responses in the Example, with factor levels coded $- 1, + 1$ .

1.1givenF1algebra

Here $x_{{1, 2}} = x_{1} x_{2}$ is $+ 1$ on runs $- -$ and $+ +$ , and $- 1$ on runs $+ -$ and $- +$ . By [F1], the raw-scale $A B$ effect is $\frac{1 + 6}{2} - \frac{2 + 3}{2} = 1$ .

1.2givenF1algebra

The log-transformed responses remain indexed by these same four runs, so [F1] uses the same two sign groups. The log-scale $A B$ effect is $\frac{\log 1 + \log 6}{2} - \frac{\log 2 + \log 3}{2} = 0$ , using $\log 1 = 0$ and $\log 6 = \log 2 + \log 3$ .

2.1step 1.1step 1.2algebra∎

The raw-scale interaction effect is nonzero while the log-scale interaction effect is zero, so this table's interaction depends on the declared response scale. The run set alone specifies no response additivity or physical mechanism.

Depends on

Two-level full factorial designs and coded effects

Used by

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