ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-27

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.

Numerical local-view plurality decoding

Example

Let $G$ be a $4$ -regular constraint graph over the alphabet $Σ = {a, b, c}$ and put $t : = 1$ , so that $(2 d)^{t} = 8$ ; let $G_{t}$ be its local-view powered graph in the conventions of Plurality decoding of powered local views, and fix a vertex $v$ and a powered labeling $φ$ . Suppose the $8$ length- $t$ lazy patterns read from $v$ have endpoint views that claim the labels $(a, a, a, a, a, b, b, c)$ for $v$ , one pattern each. Then the opinion distribution of $v$ is $p_{v} (a) = 5 / 8$ , $p_{v} (b) = 2 / 8$ , $p_{v} (c) = 1 / 8$ , so the plurality decoding of $φ$ at $v$ is $\hat{φ} (v) = a$ , of frequency $5 / 8$ ; the fixed tie-breaking order is not invoked, because the claimed maximum is attained by the single symbol $a$ .

Facts & Assumptions

Given: a $4$ -regular binary constraint graph $G$ with $t = 1$ (so that the number $(2 d)^{t}$ of length- $t$ patterns read from a vertex is $8$ ), its powered graph $G_{t}$ , a vertex $v$ , a powered labeling $φ$ whose $8$ length- $t$ patterns from $v$ end in views claiming the labels $a, a, a, a, a, b, b, c$ respectively, and a fixed total order on $Σ$ used for tie breaking.

[F1]

For $a^{'} \in Σ$ the number $p_{v} (a^{'})$ is the number of length- $t$ patterns read from $v$ whose endpoint view claims $a^{'}$ for $v$ , divided by the total number $(2 d)^{t}$ of such patterns; the plurality decoding assigns to $v$ the least symbol, in the fixed order, attaining $\max_{a^{'}} p_{v} (a^{'})$ (Plurality decoding of powered local views).

[F2]

The opinion distribution counts length- $t$ patterns with multiplicity, so two distinct patterns ending at the same vertex contribute two claims. The decoding's fixed total order is used only when several symbols attain the maximum (Plurality decoding of powered local views).

Verification

technique · direct

1.1

The eight patterns contribute one claim each, so the claim counts for $v$ are $5$ for $a$ , $2$ for $b$ and $1$ for $c$ ; dividing by the number $8$ of patterns gives $p_{v} (a) = 5 / 8$ , $p_{v} (b) = 2 / 8$ and $p_{v} (c) = 1 / 8$ , which sum to $1$ .

F1algebra

2.1

Since $5 / 8 > 2 / 8 > 1 / 8$ , the maximum of $p_{v}$ is attained only by $a$ , so [F1] gives $\hat{φ} (v) = a$ without any use of the tie-breaking rule; the frequency of the decoded label at $v$ is $p_{v} (a) = 5 / 8$ .

F1step 1.1algebra

3.1

The example illustrates the two conventions that the decoding uses: patterns are counted with multiplicity rather than as distinct centres, so two patterns ending at the same centre contribute their claims twice, and the tie-breaking order matters only when the maximum of $p_{v}$ is attained by several symbols, which does not happen here.

F2step 2.1∎

Depends on

Plurality decoding of powered local views

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

Irit Dinur, The PCP theorem by gap amplification, §6 Equation (4) and Lemma 6.1, printed pp. 19-21. (standard reference, not scraped)
Arora and Barak, Computational Complexity: A Modern Approach, §18.5.1, printed pp. 371-373. (standard reference, not scraped)