ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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 three-level telescoping estimator and its remaining bias

Example

Take $T = 1$ , $M = 2$ , $L = 2$ , target $P = 0$ , and deterministic level values $P_{ℓ} = 2^{- ℓ}$ . Every correction sample is constant, so sample counts do not change the estimate. The three levels make the telescoping identity and the remaining finest-level bias explicit.

Facts & Assumptions

Given: The deterministic target and level values above, with any positive integer sample counts $N_{0}, N_{1}, N_{2}$ .

[F1]

Geometric MLMC hierarchy and coupled corrections: $Δ_{0} = P_{0}$ and $Δ_{ℓ} = P_{ℓ}^{f} - P_{ℓ - 1}^{c}$ for $ℓ \geq 1$ , with an estimator equal to the sum of level sample means.

[F2]

Expectation, variance, and MSE of a finite MLMC estimator: the estimator mean is $E P_{L}$ and its MSE is its sampling variance plus the squared difference between $E P_{L}$ and the target mean.

Verification

1.1F1givenalgebra

Since each level is deterministic, [F1] gives $Δ_{0} = 1$ , $Δ_{1} = \frac{1}{2} - 1 = - \frac{1}{2}$ , and $Δ_{2} = \frac{1}{4} - \frac{1}{2} = - \frac{1}{4}$ .

2.1F1F2step 1.1algebra∎

Each sample mean equals its constant correction, so ${\hat{μ}}_{2, N} = 1 - \frac{1}{2} - \frac{1}{4} = \frac{1}{4} = E P_{2}$ . Its sampling variance is zero, while [F2] gives MSE $(\frac{1}{4} - 0)^{2} = \frac{1}{16}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Michael B. Giles, Multilevel Monte Carlo Path Simulation, §2 (standard reference, not scraped)