ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

Fair-coin cylinder masses and separated blocks

Example

Assume countable choice. Index binary sequences by $0, 1, 2, \dots$ , and let $[a_{0} \dots a_{r - 1}]$ prescribe the first $r$ coordinates. For fair-coin probability $p$ , $p ([0]) = 1 / 2$ , $p ([01]) = 1 / 4$ , $p ([101]) = 1 / 8$ , and $p ([01] \cap σ^{- 3} [10]) = 1 / 16$ .

Facts & Assumptions

[F1]

A cylinder prescribing $k$ distinct coordinates has fair-coin mass $2^{- k}$ . Fair-coin measure on binary sequences.

[F2]

The left shift moves coordinate $j + 1$ into position $j$ . The fair-coin one-sided shift preserves measure and is mixing.

Verification

Given: Assume countable choice. Index binary sequences by $0, 1, 2, \dots$ , and let $[a_{0} \dots a_{r - 1}]$ prescribe the first $r$ coordinates. For fair-coin probability $p$ , $p ([0]) = 1 / 2$ , $p ([01]) = 1 / 4$ , $p ([101]) = 1 / 8$ , and $p ([01] \cap σ^{- 3} [10]) = 1 / 16$ .

1.1

The prefix $[0]$ prescribes the single value $x_{0} = 0$ , so [F1] gives $2^{- 1} = 1 / 2$ . The prefix $[01]$ prescribes $x_{0} = 0, x_{1} = 1$ , so its mass is $2^{- 2} = 1 / 4$ . The prefix $[101]$ prescribes $x_{0} = 1, x_{1} = 0, x_{2} = 1$ , giving $2^{- 3} = 1 / 8$ . Repeated symbols do not reduce the number of distinct prescribed coordinates.

2.1

By [F2], membership in $σ^{- 3} [10]$ prescribes $x_{3} = 1, x_{4} = 0$ . Intersecting with $[01]$ therefore prescribes exactly coordinates $0, 1, 3, 4$ , with respective values $0, 1, 1, 0$ . Coordinate 2 and all later coordinates remain free. The cylinder mass is $2^{- 4} = 1 / 16$ by [F1]. Equivalently this intersection is the disjoint union $[01010] \cup [01110]$ , whose masses sum to $2 (1 / 32) = 1 / 16$ . Countable choice is inherited from the measure construction in [F1], not from the finite coordinate count.

1.1F1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

E–W Examples 2.8–2.9 (standard reference, not scraped)