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.

Exact star discrepancy of the center point

Example

For an integer $d \geq 2$ , let $P = (x)$ be the singleton configuration with $x = (1 / 2, \dots, 1 / 2) \in [0, 1)^{d}$ . Its star discrepancy is $D_{1}^{*} (P) = 1 - 2^{- d}$ . Thus the exact values are $3 / 4$ in dimension two and $7 / 8$ in dimension three.

Facts & Assumptions

Given: An integer $d \geq 2$ and the singleton configuration $P = (1 / 2, \dots, 1 / 2)$ . Compute $D_{1}^{*} (P)$ using the half-open anchored-box convention.

[F1]

Star discrepancy, digital nets, and digital sequences: A point multiset is an ordered list in $[0, 1)^{d}$ , repetitions are counted with multiplicity, and $D_{N}^{*}$ is the supremum over $u \in [0, 1]^{d}$ of the absolute difference between the fraction of listed points in $\prod_{j} [0, u_{j})$ and its volume $\prod_{j} u_{j}$ .

Verification

1.1givenF1

For $u \in [0, 1]^{d}$ , the point $x$ lies in $\prod_{j = 1}^{d} [0, u_{j})$ exactly when $u_{j} > 1 / 2$ for every $j$ , because the box is half-open. The point count is therefore one if all these inequalities hold and zero otherwise.

2.1givenstep 1.1algebra

If some $u_{j} \leq 1 / 2$ , the count is zero, so the discrepancy is $\prod_{j} u_{j} \leq u_{j} \leq 1 / 2$ . Since $d \geq 2$ , $1 - 2^{- d} \geq 3 / 4 > 1 / 2$ , so this case contributes less than $1 - 2^{- d}$ .

2.2givenstep 1.1algebra

If every $u_{j} > 1 / 2$ , the count is one and the discrepancy is $1 - \prod_{j} u_{j}$ . The product exceeds $2^{- d}$ , so this value is less than $1 - 2^{- d}$ .

3.1givenF1step 2.1step 2.2choosealgebra∎

For each integer $k \geq 3$ , the box endpoint $u^{(k)} = (1 / 2 + 1 / k, \dots, 1 / 2 + 1 / k)$ lies in $[0, 1]^{d}$ and has discrepancy $1 - (1 / 2 + 1 / k)^{d} \to 1 - 2^{- d}$ . Thus the supremum is at least $1 - 2^{- d}$ , while steps 2.1–2.2 bound every box error above by this value; hence $D_{1}^{*} (P) = 1 - 2^{- d}$ . Substitution gives $D_{1}^{*} (P) = 3 / 4$ for $d = 2$ and $7 / 8$ for $d = 3$ .

Remarks

The finite calculation is proved only from the definition; it is not an asymptotic-rate claim. For context, Owen reports the planar order $n^{- 1} \log n$ and, for fixed $d \geq 3$ , a classical Roth lower baseline $n^{- 1} (\log n)^{(d - 1) / 2}$ alongside a known construction upper order $n^{- 1} (\log n)^{d - 1}$ . The cited lower bound is not presented as the strongest known lower bound. Dick's version 3 result concerns uniform growing-dimension tractability, and Remark 1.4 keeps fixed dimension separate. These source-reported comparisons do not prove a matching lower bound or resolve the open question.

Depends on

Star discrepancy, digital nets, and digital sequences

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Art B. Owen, Practical Quasi-Monte Carlo, Chapter 15, printed pp. 17–19 and Theorem 15.7 (standard reference, not scraped)
Josef Dick, Optimal Polynomial Tractability Exponents for the Inverse Star Discrepancy, arXiv:2607.23571v3 (standard reference, not scraped)