ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 zero-particle prefix does not prove a zero event probability

Example

Take $Ω = {0, 1}$ with uniform law, a single level $A_{1} = {0}$ , and $N = 2$ SIR particles. The event probability is positive, but the all-zero first event-potential branch has positive probability and returns the zero estimate.

Facts & Assumptions

Given: $P$ is uniform on ${0, 1}$ , $A_{1} = {0}$ , and the full-resampling SIR splitting algorithm is used with $L = 1, N = 2$ .

[F1]

Nested rare-event levels and conditional laws: A one-level target has probability $P (A_{1})$ .

[F2]

Full-resampling fixed-level SIR splitting with invariant mutations: If any incremental event-weight average is zero, the algorithm returns status zero-particle-prefix and estimate zero.

[F3]

Fixed-level full-resampling SIR splitting is unbiased: The estimator is unbiased in expectation under the theorem's positive-prefix model.

Verification

technique · direct

1.1F1givenalgebra

$P (A_{1}) = 1 / 2 > 0$ . The two initial particles are IID from $P$ , so the probability that both equal $1$ and therefore both have zero $A_{1}$ -weight is $(1 / 2)^{2} = 1 / 4$ .

2.1F2F3step 1.1given∎

On this branch resampling the initial population still selects only particles equal to $1$ , and the identity transition $M_{1}$ leaves them there. Thus the first average event weight, from $G_{1} = 1_{A_{1}}$ , is zero, so [F2] returns estimate zero despite $P (A_{1}) > 0$ . This does not contradict [F3]: unbiasedness is an expectation over all branches and does not imply that each realization is positive.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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