LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

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.

Distinct Walsh–Hadamard words differ on half the cube

Statement

For $n \geq 1$ and distinct $u, v \in F_{2}^{n}$ , the tables ${WH}_{n} (u)$ and ${WH}_{n} (v)$ differ at exactly $2^{n - 1}$ of their $2^{n}$ coordinates, so their relative distance is one half. For $n = 0$ there are no distinct messages.

Facts & Assumptions

[F1]

${WH}_{n} (u)$ is the truth table of $r \mapsto u \cdot r$ on $F_{2}^{n}$ , indexed by $r$ , and the relative distance is the fraction of disagreeing coordinates. (Walsh–Hadamard encoding and relative Hamming distance)

Proof

Given: Fix $n \geq 1$ and distinct $u, v \in F_{2}^{n}$ .

1.1F1givenconstruct

Put $d = u + v$ . Since $u \neq v$ , $d$ is nonzero. At a mask $r \in F_{2}^{n}$ , the two table values disagree exactly when $(u \cdot r) + (v \cdot r) = d \cdot r = 1$ .

2.1step 1.1algebra

Choose a coordinate $j$ with $d_{j} = 1$ and let $e_{j}$ be its unit vector. The map $r \mapsto r + e_{j}$ is an involution without fixed points, and $d \cdot (r + e_{j}) = d \cdot r + 1$ . Thus it partitions the $2^{n}$ masks into $2^{n - 1}$ pairs, with exactly one mask in each pair satisfying $d \cdot r = 1$ . The coordinate choice exists because $d$ is a nonzero finite binary vector.

3.1F1step 1.1step 2.1algebradischarge-construct∎

By step 2.1 and the disagreement criterion in step 1.1, exactly $2^{n - 1}$ table coordinates differ. Dividing by the $2^{n}$ coordinates gives relative distance $2^{n - 1} / 2^{n} = 1 / 2$ . If $n = 0$ , $F_{2}^{0}$ has only its empty vector, so the statement has no distinct-message pair to check.

Depends on

Walsh–Hadamard encoding and relative Hamming distance

Used by

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

Arora and Barak, Computational Complexity: A Modern Approach, §18.4.1, printed p. 363 (standard reference, not scraped)