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.

A linear system and its fundamental matrix

Example

Consider the linear planar system

$x^{'} (t) = A x (t), A = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) .$

Its solution through $(x_{1}^{0}, x_{2}^{0})$ is $x (t) = (e^{t} x_{1}^{0}, e^{- t} x_{2}^{0})$ , and a fundamental matrix is

$Φ (t) = (\begin{array}{cc} e^{t} & 0 \\ 0 & e^{- t} \end{array}) .$

Facts & Assumptions

Given: The constant matrix $A = diag (1, - 1)$ .

[F1]

The variational equation along a solution is a linear matrix ODE (The variational equation along an ODE solution).

[L1]

Linear matrix ODEs have unique solutions on compact intervals (Linear matrix ODEs have unique global solutions on a fixed interval).

[L2]

A fundamental matrix is invertible at every time (A fundamental matrix is invertible).

Verification

technique · direct

1.1

Differentiating the displayed formula gives [given] $x^{'} (t) = (e^{t} x_{1}^{0}, - e^{- t} x_{2}^{0}) = A x (t)$ , and $x (0) = (x_{1}^{0}, x_{2}^{0})$ . So it is a solution.

given

2.1

The matrix $Φ$ satisfies $Φ^{'} (t) = A Φ (t)$ and $Φ (0) = I_{2}$ , so by [F1, L1, L2, step 1.1] [F1] and [L1] it is the fundamental matrix of this system. Its determinant is $e^{t} e^{- t} = 1$ , which is consistent with [L2].

F1L1L2step 1.1

3.1

Therefore the linear system has the stated solution operator and [step 2.1] fundamental matrix.

step 2.1∎

Depends on

Used by

The harmonic oscillator as a first-order system Example

Dependency tree · two levels

13 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 §10.3 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.4 (standard reference, not scraped)