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.

Störmer–Verlet has bounded oscillatory energy error on the harmonic oscillator

Example

For $H (q, p) = \frac{1}{2} (q^{2} + p^{2})$ , use the real-parameter kick and drift shears of One-step leapfrog map as force and drift shears and define the signed kick-drift-kick step $S_{h} : = K_{h / 2} \circ D_{h} \circ K_{h / 2}$ for real $h$ ; for $h > 0$ this is the published one-step map, and for $h < 0$ it is its same-form signed-step extension. If $0 < ∣ h ∣ < 2$ , this step preserves the positive quadratic form ${\tilde{H}}_{h} (q, p) = \frac{1}{2} ((1 - h^{2} / 4) q^{2} + p^{2}) .$ Consequently its original-energy error is bounded uniformly over all step counts by $∣ H (q_{n}, p_{n}) - H (q_{0}, p_{0}) ∣ \leq \frac{h^{2}}{8} (\frac{2 {\tilde{H}}_{h} (q_{0}, p_{0})}{1 - h^{2} / 4} + q_{0}^{2}) .$

Facts & Assumptions

Given: The published kick-drift-kick map, $M = 1$ , $V (q) = q^{2} / 2$ , and $0 < ∣ h ∣ < 2$ .

[F1]

The force kicks and drifts are defined for real parameters, with formulas $K_{a} (q, p) = (q, p - a q)$ and $D_{b} (q, p) = (q + b p, p)$ ; their signed composition is $S_{h} = K_{h / 2} \circ D_{h} \circ K_{h / 2}$ . For $h > 0$ it is the one-step map in One-step leapfrog map as force and drift shears.

Verification

1.1F1givenalgebra

The first kick gives $p_{1 / 2} = p - h q / 2$ , the drift gives $q_{1} = q + h p_{1 / 2} = (1 - h^{2} / 2) q + h p$ , and the last kick gives $p_{1} = p_{1 / 2} - h q_{1} / 2 = - h (1 - h^{2} / 4) q + (1 - h^{2} / 2) p$ . Thus $S_{h} = (\begin{array}{cc} a & h \\ - h s & a \end{array}), a = 1 - h^{2} / 2, s = 1 - h^{2} / 4.$

2.1step 1.1algebra

Direct multiplication gives $S_{h}^{T} (\begin{array}{cc} s & 0 \\ 0 & 1 \end{array}) S_{h} = (\begin{array}{cc} s (a^{2} + h^{2} s) & 0 \\ 0 & a^{2} + h^{2} s \end{array}) = (\begin{array}{cc} s & 0 \\ 0 & 1 \end{array}),$ because $a^{2} + h^{2} s = 1$ . Hence ${\tilde{H}}_{h}$ is exactly invariant. Since $s > 0$ , invariance implies $q_{n}^{2} \leq 2 {\tilde{H}}_{h} (q_{0}, p_{0}) / s$ for every $n$ .

3.1step 2.1algebra∎

The identity ${\tilde{H}}_{h} = H - (h^{2} / 8) q^{2}$ gives $H (q_{n}, p_{n}) - H (q_{0}, p_{0}) = \frac{h^{2}}{8} (q_{n}^{2} - q_{0}^{2}) .$ Apply the bound from step 2.1 and the triangle inequality to obtain the stated uniform error estimate.

Depends on

One-step leapfrog map as force and drift shears

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Ernst Hairer, Geometric Numerical Integration, TU München Lecture 3: Backward error analysis (standard reference, not scraped)
Nikolaj Nordkvist and Poul G. Hjorth, Classical Mechanics and Symplectic Integration (standard reference, not scraped)