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.

Digital net sizes are powers of their base

Example

In base $2$ , a digital $(t, m, d)$ -net has exactly $2^{m}$ nodes, so six nodes cannot form one. A six-node rank-one lattice rule is still a valid deterministic quadrature rule, and the Koksma–Hlawka inequality is stated for arbitrary finite nonempty point sets rather than only digital-net sizes.

Facts & Assumptions

Given: Base $2$ , dimension $d = 2$ , and node count $N = 6$ .

[F1]

A $(t, m, d)$ -net in base $b$ has exactly $b^{m}$ nodes. (Star discrepancy, digital nets, and digital sequences)

[F2]

A rank-one lattice rule accepts any integer $N \geq 1$ and a generator $z$ with $\gcd (z_{1}, N) = 1$ , and uses nodes $x_{n, j} = {n z_{j} / N}$ . (Digital point-set and rank-one lattice cube quadrature)

[F3]

Koksma–Hlawka is stated for any finite nonempty point multiset and integrand of finite upper-anchor Hardy–Krause variation. (Koksma–Hlawka integration bound with upper-anchor variation)

[F4]

Since $2 > 1$ , $m \mapsto 2^{m}$ is strictly increasing on the natural exponents, by the monotonicity-in-the-exponent clause of Monotonicity of $x \mapsto x^{n}$ and of $n \mapsto a^{n}$ . The rank-one input with $z_{1} = 1$ and $N = 6$ is admissible: $1$ is a common divisor of $1$ and $6$ , and every common divisor $d$ satisfies $d \leq ∣ 1 ∣ = 1$ by If $d ∣ a$ and $a \neq 0$ then $d \neq 0$ and $∣ d ∣ \leq ∣ a ∣$ ; hence the set of divisors of a nonzero integer is bounded above by $∣ a ∣$ , so $\gcd (1, 6) = 1$ by Common divisor, and the greatest common divisor $\gcd (a, b)$ , with the convention $\gcd (0, 0) : = 0$ .

Verification

technique · direct

1.1F1F4givenalgebra

The positive integer powers of $2$ are $1, 2, 4, 8, \dots$ : by [F4], if $m \leq 2$ then $2^{m} \leq 2^{2} = 4$ , while if $m \geq 3$ then $2^{m} \geq 2^{3} = 8$ . Hence no integer $m \geq 0$ gives $2^{m} = 6$ , so [F1] rules out a base- $2$ digital net with six nodes.

1.2F2F4givenalgebra

With $N = 6$ and $z = (1, 1)$ , [F2] and [F4] generate six nodes; the first coordinate is $n / 6$ for $n = 0, \dots, 5$ , so the nodes are distinct and the lattice-rule inputs are admissible.

2.1F1F3step 1.1step 1.2∎

The star discrepancy and the Koksma–Hlawka statement are defined for this six-node point set even though its size is not a power of $2$ . Thus a finite deterministic quadrature rule exists at this size, while the digital-net label and its elementary-interval guarantee do not apply.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

41 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, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 15 §15.7 and Chapter 16 §16.1 (standard reference, not scraped)
Harald Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, Chapters 4 §4.1 and 5 §5.3 (standard reference, not scraped)