Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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 two-by-two interaction and error share one residual array

Example

The residual array R=(1−1−11) has zero row and column sums. It can be represented as an interaction γ=R plus projected error e=0, or as interaction γ=0 plus projected error e=R. Both decompositions give the same observed additive residual.

Verification

Given: The 2×2 array R in the Example and the residual-space conclusion of One cell observation confounds unrestricted interaction with residual error.

[F1] For one observation per cell, a double-centered interaction and the additive-model residual both lie in the same residual subspace (One cell observation confounds unrestricted interaction with residual error).

1.1givenF1algebra

Each row and each column of R sums to zero, so R is double-centered and belongs to that residual subspace.

2.1F1step 1.1algebra∎

Both pairs (γ,e)=(R,0) and (γ,e)=(0,R) consist of vectors in the residual subspace and sum to R; hence they yield the identical observed residual with different interaction/error allocations.

Depends on

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