LemmaStatement: Literature-sourcedProof: 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.

Constant private-coin error reduction by repetition

Statement

If a private-coin simultaneous-message Equality protocol has worst-case error at most 1/3 and each sender sends at most c bits, then for any fixed odd t independent repetitions followed by majority have error at most exp(−t/48) and each sender sends at most t(c+1) bits under a fixed-length encoding.

Facts & Assumptions

Given: A classical private-coin SMP Equality protocol whose error on every promised pair is at most $1 / 3$ , whose individual messages have length at most $c$ bits, and an odd integer $t \geq 1$ . Run $t$ independent copies and output the majority decision.

[F1]

A Chernoff bound for sums of independent Bernoulli trials: For independent Bernoulli $(p)$ variables $X_{1}, \dots, X_{k}$ , $S = \sum_{i} X_{i}$ , $μ = p k$ , and $0 < α < 1$ , $\Pr (S \leq (1 - α) μ) \leq \exp (- α^{2} μ / 2)$ .

Proof

technique · direct

1.1givenF1algebra

Fix a promised input. Let $X_{i}$ indicate that copy $i$ is correct. The independent private random tapes make these Bernoulli variables independent with common success probability $p \geq 2 / 3$ . Thus $μ = p t \geq 2 t / 3$ . Since $t$ is odd, majority is wrong only if $S < t / 2$ , an event contained in $S \leq (3 / 4) μ$ because $(3 / 4) μ \geq t / 2$ .

2.1step 1.1F1algebra∎

Apply [F1] with $α = 1 / 4$ : the error is at most $\exp (- μ / 32) \leq \exp (- t / 48)$ . Each original message is a binary string of length at most $c$ , so has fewer than $2^{c + 1}$ possibilities. A tuple of $t$ such messages has at most $2^{t (c + 1)}$ possibilities and can be injectively encoded by a fixed-length string of at most $t (c + 1)$ bits. Thus repetition preserves private independence, yields the stated error, and charges at most $t (c + 1)$ bits to each sender.

Depends on

Used by

Private-coin SMP equality requires Ω(√n) bits Theorem

Dependency tree · two levels

12 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

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)
Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)