CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

One-dimensional marginals alone do not specify a joint law

Statement refuted

The one-dimensional marginals of a process determine all its finite-dimensional laws.

Counterexample

Let $Z$ be a fair bit. The pair $(Z, Z)$ and a pair $(Z, W)$ of independent fair bits have the same one-dimensional marginals.

Given: Fair bits $Z$ and $W$ , with $Z, W$ independent in the second pair.

1.1

Each displayed coordinate has Bernoulli $(1 / 2)$ law in both pairs.

given

2.1

But $P (Z = Z) = 1$ for the first pair, whereas $P (Z = W) = 1 / 2$ for the independent pair. Their two-dimensional laws differ, refuting the statement.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Biskup, MATH 275D notes, Section 2.1 (standard reference, not scraped)