ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 raw zero uniform breaks Box–Muller but the midpoint avoids it

Example

A raw grid value $K / 2^{b}$ can equal zero, so the Box–Muller logarithm is undefined on that input. Its midpoint version $(K + 1 / 2) 2^{- b}$ is always strictly positive; an ideal uniform input equals zero only on a null event.

Facts & Assumptions

Given: An integer $K$ uniform on ${0, \dots, 2^{b} - 1}$ and the Box–Muller radius formula $r (u) = \sqrt{- 2 \log u}$ for $u > 0$ .

[F1]

The midpoint value is $(K + 1 / 2) 2^{- b}$ and belongs to $(0, 1)$ . Midpoint bit discretization has bounded coupling and CDF error

[F2]

An ideal bit-stream uniform has CDF $t$ on $[0, 1]$ and hence assigns probability zero to ${0}$ . Disjoint streams of independent fair bits produce independent uniform variables

[F3]

For $0 < u < 1$ , the exact Box–Muller reference radius is $r (u) = \sqrt{- 2 \log u}$ . Box–Muller has a product Gaussian density and Rayleigh second moment

[A1]

The countable fair-bit product supplying [F2] assumes Countable Choice and Dependent Choice. The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

Verification

1.1givenF3algebra

The raw endpoint event is exactly $K = 0$ , which has probability $2^{- b}$ ; the radius expression [F3] has an undefined logarithm at that input.

2.1F1F2F3A1algebra∎

By [F1], the midpoint input is at least $2^{- b - 1} > 0$ and is less than $1$ , so its logarithm is finite. By [F2], the ideal stream hits $0$ with probability zero; a software interface still needs an explicit endpoint convention if it accepts closed-interval inputs.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

49 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

Luc Devroye, Non-Uniform Random Variate Generation, Chapter V §4.4 (standard reference, not scraped)
E. M. Raz, Lecture Notes in Probability, Chapter 5, Exercise 5.2 (standard reference, not scraped)