CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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.

Explicit Euler grows a free oscillator

Physical scope

Pure mathematics; no physical premise.

Statement refuted

For every finite compatible matrix system with $S = 0$ and $j = 0$ , the forward Euler step, defined by $e_{1} = e_{0} + h \dot{e} (t_{0})$ and $b_{1} = b_{0} + h \dot{b} (t_{0})$ for $h > 0$ , preserves the system energy for every initial state.

Statement refuted references

Finite compatible matrix system

Facts & Assumptions

Given: The one-dimensional data $M_{e} = M_{b} = C = [1]$ , $S = 0$ , $D = G = [0]$ , zero forcing, a step size $h > 0$ , and initial scalars $e_{0}, b_{0}$ not both zero.

[F1]

The finite compatible matrix system obeys $M_{e} \dot{e} = C^{T} M_{b} b - S e - j$ and $\dot{b} = - C e$ , with energy $W = \frac{1}{2} (e^{T} M_{e} e + b^{T} M_{b} b)$ (Finite compatible matrix system).

Counterexample

1.1givenF1algebra

The data satisfy the defining matrix assumptions: $M_{e}, M_{b}$ are positive definite, $S$ is positive semidefinite, and $D C = C G = 0$ . With $j = 0$ , [F1] reduces to $\dot{e} = b$ and $\dot{b} = - e$ .

2.1givenstep 1.1algebra

The forward Euler rule in the refuted statement therefore gives $e_{1} = e_{0} + h b_{0}$ and $b_{1} = b_{0} - h e_{0}$ .

3.1givenstep 2.1F1algebra∎

Expanding gives $W (e_{1}, b_{1}) = \frac{1}{2} ((e_{0} + h b_{0})^{2} + (b_{0} - h e_{0})^{2}) = (1 + h^{2}) \frac{1}{2} (e_{0}^{2} + b_{0}^{2}) = (1 + h^{2}) W (e_{0}, b_{0})$ . Because $h > 0$ and the initial pair is nonzero, $W (e_{0}, b_{0}) > 0$ and $1 + h^{2} > 1$ , so the step strictly increases energy. This refutes the universal preservation claim.

Depends on

Finite compatible matrix system