ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-09-07

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.

Sequential repetition of a constant-gap protocol

Example

For a protocol with completeness $3 / 4$ and soundness $1 / 4$ , run $k$ independent copies and accept a strict majority.

Verification

Given: the repeated protocol with odd $k$ .

1.1given

Its honest and cheating parameters satisfy the hypotheses $c > 1 / 2 > s$ of Sequential repetition amplifies completeness and soundness gaps.

2.1step 1.1choose∎

Thus each error probability is exponentially small in $k$ ; choosing an odd $k$ with that bound below $10^{- 6}$ gives completeness at least $1 - 10^{- 6}$ and soundness at most $10^{- 6}$ .

Depends on

Sequential repetition amplifies completeness and soundness gaps

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

Arora and Barak, §8.2 (standard reference, not scraped)