TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

BPP amplification to inverse-polynomial error

Statement

If $L \in B P P$ and $q$ is a polynomial with positive values on $N$ , then $L$ also has a probabilistic polynomial-time decider whose error on length- $n$ inputs is at most $1 / q (n)$ .

Facts & Assumptions

Given: a language $L \in B P P$ and a positive polynomial $q$ .

[L1]

A BPP machine is correct on every input with probability at least $2 / 3$ (The classes RP, coRP, ZPP, BPP, and PP).

[L2]

Repeating such trials independently and taking the majority answer makes the error probability at most $9 / k$ after $k$ repetitions (Chebyshev bounds the majority error of repeated Bernoulli trials).

Proof

technique · direct

1.1

Let $M$ be a BPP machine for $L$ . On input $x$ , run $M$ independently $k : = ⌈ 9 q (∣ x ∣) ⌉$ times and output the majority answer. Each run uses polynomial time and $k$ is polynomial in $∣ x ∣$ , so the new machine is still probabilistic polynomial-time.

L1givenconstruct

2.1

Fix $x$ . The correctness indicators of the independent runs satisfy the hypotheses of [L2] by [L1], so the probability that the majority answer is wrong is at most $9 / k \leq 1 / q (∣ x ∣)$ .

L1L2step 1.1algebra

3.1

Since $x$ was arbitrary, the amplified machine has inverse-polynomial error on every input length.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Eric Blais, Models of Computation, 14. Randomized Computation (standard reference, not scraped)
Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)