ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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.

The alias classes in a $2^{3 - 1}$ fraction

Example

Take $k = 3$ with factors $A, B, C$ and the positive generator equation $x_{A} x_{B} x_{C} = + 1$ , conventionally written $I = A B C$ . The four runs, in $(A, B, C)$ order, are $(- 1, - 1, + 1)$ , $(- 1, + 1, - 1)$ , $(+ 1, - 1, - 1)$ , and $(+ 1, + 1, + 1)$ . The complete equal-column alias classes are

${I, A B C}, {A, B C}, {B, A C}, {C, A B} .$

In the unrestricted model with all eight factorial terms, each pair contributes through its coefficient sum. Separating a main effect from its aliased interaction therefore requires additional runs or an explicit sparsity or hierarchy assumption.

Facts & Assumptions

Given: $k = 3$ , the generator equation $x_{A} x_{B} x_{C} = + 1$ , the four runs displayed in the Example, and the unrestricted fixed-design model with columns for all subsets of ${A, B, C}$ .

[F1]

Regular two-level fractions, signed defining relations, and resolution: a regular fraction is specified by independent generator equations; $H$ is generated by their signed words, and equal-column aliases are columns that agree on every run.

[F2]

The defining subgroup gives the exact alias relations: $A$ and $B$ have equal columns exactly when $(+ 1, A △ B) \in H$ .

[F3]

Aliased factorial columns prevent separate coefficient identification: if two distinct columns satisfy $c_{A} = σ c_{B}$ , their contribution depends only on $σ β_{A} + β_{B}$ , so the coefficients are not separately identifiable in the unrestricted model.

Verification

technique · enumerate the fraction and multiply its one defining word by every effect word

1.1givenF1constructchoose

Choose $x_{A}, x_{B} \in {- 1, + 1}$ freely and set $x_{C} = x_{A} x_{B}$ . This gives exactly the four runs listed in the Example, one for each of the four choices of $(x_{A}, x_{B})$ , and every run satisfies $x_{A} x_{B} x_{C} = + 1$ .

2.1F1F2step 1.1algebra

Here $H = {I, A B C}$ . By [F2], effects have equal columns exactly when their symmetric difference is either $\emptyset$ or ${A, B, C}$ . Pairing each subset with its complement gives precisely ${I, A B C}$ , ${A, B C}$ , ${B, A C}$ , and ${C, A B}$ ; these four pairs exhaust the eight subsets.

3.1givenF3step 2.1algebra∎

In each listed pair the columns are equal, so [F3] shows that the corresponding terms enter the fitted vector through $β_{I} + β_{A B C}$ , $β_{A} + β_{B C}$ , $β_{B} + β_{A C}$ , or $β_{C} + β_{A B}$ , respectively. The design by itself does not identify the two coefficients within any pair separately.

Sources

Oehlert §18.2 and NIST/SEMATECH §§5.3.3.4.1–5.3.3.4.3 give the standard $I = A B C$ half-fraction and its alias notation. The four-row enumeration and coefficient ambiguity are verified locally.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Gary W. Oehlert, A First Course in Design and Analysis of Experiments, Chapter 18, §18.2 (standard reference, not scraped)
NIST/SEMATECH e-Handbook of Statistical Methods, §§5.3.3.4.1–5.3.3.4.3 (standard reference, not scraped)

The alias classes in a 23−123−1 fraction

Example

Facts & Assumptions

Verification

Sources

Depends on

Used by

Dependency tree · two levels

Sources

The alias classes in a $2^{3 - 1}$ fraction