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

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A negative mode defeats a symmetric-potential bound

Statement refuted

Use the Euclidean norm on $R^{N}$ . For $v : R \to R^{N}$ , define $∥ v ∥_{2}^{2} : = \int_{R} ∣ v ∣^{2} d x$ and $∥ v ∥_{H^{1}}^{2} : = \int_{R} (∣ v ∣^{2} + ∣ v_{x} ∣^{2}) d x$ .

For every integer $N \geq 1$ , every smooth real symmetric $N \times N$ matrix potential $V (x)$ on $R$ , and every smooth solution $U : R_{t} \times R_{x} \to R^{N}$ such that, for every nonempty compact $K \subset R$ , integer $k \geq 0$ , and integer $m \geq 0$ , $\sup_{t \in K, x \in R} (1 + ∣ x ∣)^{m} ∣ \partial_{x}^{k} U (t, x) ∣ < \infty,$ the solution of $U_{t t} - U_{x x} + V (x) U = 0$ has a finite constant $C (V, N)$ , depending only on $V$ and $N$ , such that $∥ U (t, \cdot) ∥_{H^{1} (R)} + ∥ U_{t} (t, \cdot) ∥_{2} \leq C (V, N) (∥ U (0, \cdot) ∥_{H^{1} (R)} + ∥ U_{t} (0, \cdot) ∥_{2}) (t \geq 0) .$

Statement references

Positive matrix-potential wave system energy

Facts & Assumptions

Given: The false assertion above and the abstract positive-potential estimate for scope comparison.

[F1]

Positive matrix-potential wave system energy: the energy estimate assumes the matrix potential is positive semidefinite; it makes no claim for an indefinite potential.

[A1]

It suffices to test the false universal assertion at $N = 1$ with a real smooth scalar potential and one solution.

Counterexample

1.1givenA1algebra

Let $\cosh x : = (e^{x} + e^{- x}) / 2$ , $w (x) : = 1 / \cosh x$ , and $V (x) : = - 2 w (x)^{2}$ . These are smooth real functions. Put $τ (x) : = (e^{x} - e^{- x}) / (e^{x} + e^{- x})$ ; direct differentiation gives $w^{'} = - w τ$ , $τ^{'} = 1 - τ^{2} = w^{2}$ , and $w^{2} = 1 - τ^{2}$ . Therefore $w^{''} = w (2 τ^{2} - 1) = w - 2 w^{3}$ . Inductively each derivative $w^{(k)}$ is $w$ times a polynomial $P_{k} (τ)$ ; since $∣ τ ∣ \leq 1$ and $\cosh x \geq \frac{1}{2} e^{∣ x ∣}$ , there is a finite constant $C_{k}$ with $∣ w^{(k)} (x) ∣ \leq C_{k} e^{- ∣ x ∣}$ for all $x$ . In particular, the integrals $\int_{R} ∣ w ∣^{2} d x$ and $\int_{R} (∣ w ∣^{2} + ∣ w^{'} ∣^{2}) d x$ are finite, so the stated $L^{2}$ and $H^{1}$ norms are finite. Consequently the differential expression $A : = - \partial_{x}^{2} + V (x)$ satisfies $A w = - w$ .

2.1givenF1step 1.1algebra∎

Define $U (t, x) = e^{t} w (x)$ . Then $U_{t t} = e^{t} w$ and $- U_{x x} + V U = e^{t} (- w^{''} - 2 w^{3}) = - e^{t} w$ by step 1.1. For every compact $K \subset R$ , $e^{t}$ is bounded on $K$ ; for every integer $m \geq 0$ , $(1 + ∣ x ∣)^{m} e^{- ∣ x ∣}$ is bounded on $R$ . The derivative bounds from step 1.1 therefore give the exact local-uniform decay condition in the refuted Statement for every spatial derivative of $U$ . Thus $U$ is an admissible solution of that assertion. Therefore $U_{t t} - U_{x x} + V U = 0$ , while $∥ U (t) ∥_{H^{1}} + ∥ U_{t} (t) ∥_{2} = e^{t} (∥ w ∥_{H^{1}} + ∥ w ∥_{2}) ⟶ \infty .$ The initial norm is finite and nonzero, so no time-uniform constant $C (V, N)$ as asserted exists. The example refutes only the claim that symmetry alone suffices; its potential violates the positive-semidefinite hypothesis of the supplied theorem and is not asserted to be an Einstein–Maxwell master potential.

Remarks

The positive-potential assumption cannot be deleted from a general wave-energy claim. Establishing the sign and symmetry of an actual coupled gravitational/electromagnetic master system is a separate argument.

Depends on

Positive matrix-potential wave system energy