ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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.

The harmonic oscillator as a first-order system

Example

The second-order equation

$q^{''} + q = 0$

becomes the first-order system

${(\begin{array}{c} q \\ p \end{array})}^{'} = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}) (\begin{array}{c} q \\ p \end{array}) .$

Its solutions are

$q (t) = q_{0} \cos t + p_{0} \sin t, p (t) = - q_{0} \sin t + p_{0} \cos t .$

Facts & Assumptions

Given: The matrix $A = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ .

[L1]

Autonomous smooth ODEs have unique local solutions (The fundamental theorem for autonomous smooth ODEs).

[L2]

The linear-system example shows how to read a first-order matrix system and its solution operator (A linear system and its fundamental matrix).

Verification

technique · direct

1.1

Setting $p = q^{'}$ turns $q^{''} + q = 0$ into the displayed first-order system, and [given] conversely differentiating the first equation and substituting the second recovers $q^{''} + q = 0$ .

given

2.1

Differentiating the displayed sine-cosine formulas gives [L1, L2, step 1.1] $q^{'} = p$ and $p^{'} = - q$ , so they solve the first-order system with $(q (0), p (0)) = (q_{0}, p_{0})$ . By [L1] the solution is unique, and [L2] identifies the system as the oscillator written in first-order form.

L1L2step 1.1

3.1

Therefore the harmonic oscillator fits the first-order smooth-ODE framework [step 2.1] exactly as claimed.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Nigel Hitchin, Differentiable Manifolds, Appendix §10.2 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.1 (standard reference, not scraped)