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.

Full and targeted foldovers of a resolution III fraction

Example

Use factors $A, B, C, D, E$ and the regular $2^{5 - 2}$ fraction generated by

$x_{D} = x_{A} x_{B}, x_{E} = x_{C} x_{D} .$

Writing $I$ for the identity word, its defining relations are $I = A B D$ and $I = C D E$ , so $H = {I, A B D, C D E, A B C E}$ and the initial resolution is III. A complete mirror foldover yields the regular $2^{5 - 1}$ fraction $I = A B C E$ with $H_{[5]} = {I, A B C E}$ : each subset $S$ remains paired with $S △ {A, B, C, E}$ , so $A$ remains aliased with $B C E$ while $C$ and $D E$ separate. Reversing only factor $A$ yields the regular $2^{5 - 1}$ fraction $I = C D E$ with $H_{{A}} = {I, C D E}$ : each $S$ remains paired with $S △ {C, D, E}$ , so $A$ separates from $B D$ but $C$ remains aliased with $D E$ . The targeted follow-up therefore leaves resolution III and a specific coefficient ambiguity; this is an exact residual alias list, not a claim that all main effects are clear.

Facts & Assumptions

Given: The five factors and generator equations in the Example, the complete mirror foldover $J = {A, B, C, D, E}$ , the partial foldover $J = {A}$ , and the unrestricted fixed-design linear model with all factorial columns on the augmented runs.

[F1]

Regular two-level fractions, signed defining relations, and resolution: signed words multiply by symmetric difference, and resolution is the shortest nonidentity defining-word length.

[F2]

The defining subgroup gives the exact alias relations: $H$ is generated by the independent signed generators and gives all equal- or opposite-column alias relations.

[F3]

Full and partial foldovers separate aliases by parity: after reversing factor set $J$ , the residual relation subgroup is $H_{J} = {(ϵ, S) \in H : ∣ S \cap J ∣ is even}$ .

[F4]

Aliased factorial columns prevent separate coefficient identification: equal columns in an unrestricted fixed-design model contribute only through the sum of their coefficients, and the two coefficients are not separately identifiable.

Verification

technique · multiply the generators, then apply the exact parity criterion to each follow-up

1.1givenF1F2algebra

The equations give $x_{A} x_{B} x_{D} = + 1$ and $x_{C} x_{D} x_{E} = + 1$ , so the defining words are $A B D$ and $C D E$ . Their product is $A B C E$ because the repeated $D$ cancels in symmetric difference. The two incidence vectors are nonzero and distinct; over $F_{2}$ , two nonzero vectors are dependent only if equal, so these generators are independent. By [F2] these and the identity are exactly the four words in $H$ . Their nonidentity lengths are $3, 3, 4$ , hence the initial resolution is III.

2.1givenF1F2F3step 1.1chooseconstructalgebra

For the complete mirror, [F3] changes the signs of the length-three relations ABD and CDE but preserves ABCE, giving $H_{[5]} = {I, A B C E}$ . The two eight-run blocks are disjoint because ABD has opposite signs on them, and both lie in $Q = {x : x_{A} x_{B} x_{C} x_{E} = + 1}$ ; choosing $x_{A}, x_{B}, x_{C}, x_{D}$ freely determines $x_{E}$ , so $∣ Q ∣ = 16$ and their union is exactly this regular $2^{5 - 1}$ fraction. Its aliases are precisely $S$ with $S △ {A, B, C, E}$ , so $A$ remains paired with $B C E$ , while $C △ D E = C D E$ is absent and that pair separates. The shortest residual word has length $4$ .

2.2givenF1F2F3step 1.1chooseconstructalgebra

For the partial foldover on $J = {A}$ , [F3] changes ABD and ABCE but preserves CDE, giving $H_{J} = {I, C D E}$ . The two eight-run blocks are disjoint because ABD has opposite signs on them, and both lie in $Q^{'} = {x : x_{C} x_{D} x_{E} = + 1}$ ; choosing $x_{A}, x_{B}, x_{C}, x_{D}$ freely determines $x_{E}$ , so their union is all 16 runs in this regular $2^{5 - 1}$ fraction. The complete residual pairs are precisely $S$ and $S △ {C, D, E}$ : this includes $C = D E$ and excludes the original relation $A = B D$ . The shortest residual word has length $3$ .

3.1givenF4step 2.2algebra∎

The retained columns $c_{C} = c_{D E}$ in the partial design contribute as $(β_{C} + β_{D E}) c_{C}$ . By [F4], only this combination enters the fitted vector for this pair, so the partial follow-up does not separately identify $β_{C}$ and $β_{D E}$ without a restriction or further runs.

Sources

Oehlert §§18.2, 18.4, and 18.7 support the generator, resolution, and full-foldover conventions. NIST/SEMATECH §§5.3.3.8.1–5.3.3.8.2 covers mirror and selected-factor follow-ups. All relation products and residual alias sets in this example are computed explicitly.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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, 18.4, and 18.7 (standard reference, not scraped)
NIST/SEMATECH e-Handbook of Statistical Methods, §§5.3.3.8.1–5.3.3.8.2 (standard reference, not scraped)