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.

A modification need not be indistinguishable

Statement refuted

A modification of a process must be indistinguishable from it.

Counterexample

Let $U$ be uniform on $[0, 1]$ , let $X_{t} \equiv 0$ , and let $Y_{t} = 1_{{U = t}}$ for $t \in [0, 1]$ .

Given: Lebesgue probability on $[0, 1]$ and its coordinate map $U$ .

1.1

For fixed $t$ , ${X_{t} \neq Y_{t}} = {U = t}$ is measurable and has Lebesgue probability $0$ , so $Y$ is a modification of $X$ under the stated diagonal convention.

given

2.1

For every outcome $u$ , choosing $t = u$ gives $Y_{t} (u) = 1 \neq 0 = X_{t} (u)$ . Thus the equality-for-all-times event is empty, not probability one, and the processes are not indistinguishable.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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 2.3 (standard reference, not scraped)