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 documented grouping change and a new-data study

Example

All observations below are synthetic. The descriptive question is how the mean outcome differs between the older and younger age groups. In Study 1, the pre-specified rule assigns ages at most 20 to the younger group and ages above 20 to the older group. The reported contrast is older-group mean minus younger-group mean.

Study 1 record	Age	Outcome
A	18	10
B	19	11
C	21	13
D	22	14

An exploratory reanalysis of the same four rows instead uses a cut point of 18: ages at most 18 are younger, and ages above 18 are older. This changes the grouping rule and therefore the descriptive contrast being computed.

A separate Study 2 collects new observations while addressing the same pre-specified age-group question and using the original cut point of 20.

Study 2 record	Age	Outcome
E	17	9
F	20	10
G	21	12
H	24	14

A transparent account states both cut points, that the second Study 1 analysis was exploratory and chosen after inspecting the first table, and that no rows were excluded. It gives the data and arithmetic below, marks the values as synthetic, and makes no population, causal, or inferential claim. No software is needed for these calculations; a computational report using software would also identify relevant software and environment details. The Study 2 estimate alone does not decide whether the studies' results are consistent: no uncertainty assessment or consistency criterion is supplied here.

Facts & Assumptions

Given: For a nonempty finite group $G$ of records with real outcomes $y_{i}$ , let ${\overset{ˉ}{y}}_{G} = ∣ G ∣^{- 1} \sum_{i \in G} y_{i}$ , where $∣ G ∣$ is the number of records; the mean of an empty group is undefined. Let $Δ$ be the older- group mean minus the younger-group mean, with the age cut points and the two tables as specified in the Example.

[F1]

Computational reproducibility, replicability, and transparent reporting: computational reproducibility keeps the input data, computational steps, methods and code, and analysis conditions the same.

[F2]

Computational reproducibility, replicability, and transparent reporting: replicability compares studies answering the same scientific question using each study's own newly obtained data.

[A1]

The two tables are constructed solely for this example; all four rows in each study are used, group means are unweighted arithmetic means, and no uncertainty model or consistency threshold is assumed.

Verification

1.1givenA1algebra

Under Study 1's pre-specified cut point of 20, the younger outcomes are $10, 11$ and the older outcomes are $13, 14$ . Their means are $21 / 2$ and $27 / 2$ , respectively, so $Δ = 27 / 2 - 21 / 2 = 3$ .

2.1givenA1step 1.1algebra

Under the alternate cut point of 18, the younger mean is $10$ and the older mean is $(11 + 13 + 14) / 3 = 38 / 3$ , so the contrast is $38 / 3 - 10 = 8 / 3$ . The raw rows are unchanged from step 1.1, but the grouping rule and resulting contrast differ.

2.2givenF1step 1.1algebra

Reapplying the Study 1 pre-specified grouping and the same mean calculation to the same rows returns $Δ = 3$ . The table, cut point, formula, row set, and manual arithmetic are unchanged, so this illustrates the same-input computation comparison in [F1]; it concerns reproduction of the calculation, not whether the grouping or conclusion is correct.

3.1givenF2A1algebra∎

Study 2 has its own new rows. Under the same cut point of 20, the younger mean is $(9 + 10) / 2 = 19 / 2$ and the older mean is $(12 + 14) / 2 = 13$ , giving $Δ = 13 - 19 / 2 = 7 / 2$ . It is a replication attempt under [F2], but these point estimates alone cannot determine consistency without uncertainty and a stated comparison criterion.

Depends on

Computational reproducibility, replicability, and transparent reporting

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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

National Academies of Sciences, Engineering, and Medicine, Reproducibility and Replicability in Science (2019), Chapter 3: Understanding Reproducibility and Replicability (standard reference, not scraped)
Greg Wilson, Jennifer Bryan, Karen Cranston, Justin Kitzes, Lex Nederbragt, and Tracy K. Teal, Good enough practices in scientific computing, PLoS Computational Biology 13(6):e1005510 (2017) (standard reference, not scraped)