Ideal linear Penning mode-frequency invariant

Let $q\mathrm{\ne }0$ be a signed charge in coulombs, $M>0$ a rest-mass energy in joules, $c>0$ the speed of light in metres per second, and $\mathbf{B}\in {\mathbb{R}}^3\setminus \{0\}$ a constant magnetic-field vector in teslas. Set $m:=M\mathrm{/}{c}^2>0$. Let $K\in {\mathbb{R}}^{3\times 3}$ be a real symmetric matrix with $\mathrm{tr}K=0$, with units volts per square metre, and define the quadratic scalar potential $\mathrm{\Phi }(\mathbf{r}):=\frac{1}{2}{\mathbf{r}}^{T}K\mathbf{r}$ for $\mathbf{r}\in {\mathbb{R}}^3$ in metres. Thus ${\mathrm{\partial }}_i{\mathrm{\partial }}_j\mathrm{\Phi }=K_{ij}$ and ${\sum }_{i=1}^3{\mathrm{\partial }}_i^2\mathrm{\Phi }=\mathrm{tr}K=0$. Put $\alpha :=q{c}^2\mathrm{/}M=q\mathrm{/}m$, $D:=-\alpha K$, and define $C:{\mathbb{R}}^3\to {\mathbb{R}}^3$ by $C\mathbf{w}:=\alpha \mathbf{w}\times \mathbf{B}$. The ideal linear model is the real linear map $A:{\mathbb{R}}^3\times {\mathbb{R}}^3\to {\mathbb{R}}^3\times {\mathbb{R}}^3$, $A(\mathbf{r},\mathbf{v}):=(\mathbf{v},D\mathbf{r}+C\mathbf{v})$, equivalently $A=\left(\begin{array}{cc}0& I_3\\ D& C\end{array}\right)$. Equivalently, its equations are $\dot{\mathbf{r}}=\mathbf{v}$ and $\dot{\mathbf{v}}=-(q{c}^2\mathrm{/}M)K\mathbf{r}+(q{c}^2\mathrm{/}M)\mathbf{v}\times \mathbf{B}$. The entries of $\mathbf{r},\mathbf{v}$ are their real numerical coordinates in the fixed SI units. A state solution on an open time interval $I\subseteq \mathbb{R}$ is a $C^1$ map $y=(\mathbf{r},\mathbf{v}):I\to {\mathbb{R}}^3\times {\mathbb{R}}^3$ satisfying the displayed equations for every $t\in I$. Here $\mathbf{v}$ has units metres per second, $D$ has units inverse seconds squared, and $C$ has units inverse seconds.

Define a linear mode to be a complex eigenvector of the complexified state matrix $A$; its eigenvalue $+i\omega$ or $-i\omega$ gives the time factor $e^{\pm i\omega t}$, with angular frequency $\omega >0$ in radians per second. Assume the complexification of $A$ is diagonalizable with eigenvalues $+i{\omega }_1,-i{\omega }_1,+i{\omega }_2,-i{\omega }_2,+i{\omega }_3,-i{\omega }_3$, where each ${\omega }_j$ is counted with its eigenvalue multiplicity. Define the free cyclotron angular frequency ${\omega }_c:=\mathrm{\mid }q\mathrm{\mid }{c}^2\mathrm{\mid }\mathbf{B}\mathrm{\mid }\mathrm{/}M$. Then

${\omega }_c^2={\omega }_1^2+{\omega }_2^2+{\omega }_3^2\mathrm{.}$

Consequently, if ${\nu }_c:={\omega }_c\mathrm{/}(2\pi )$ and ${\nu }_j:={\omega }_j\mathrm{/}(2\pi )$ are the corresponding ordinary frequencies in hertz, then ${\nu }_c^2={\nu }_1^2+{\nu }_2^2+{\nu }_3^2$. The claim is conditional on the stated uniform-field, harmonic-quadrupole, and complete stable-mode hypotheses; it is an identity for this linear model.

This is a mathematical deduction about the displayed first-order matrix. It does not prove that an actual trap realizes the model or spectral hypothesis. The statement excludes higher spatial multipoles, magnetic-field inhomogeneity, relativistic shifts, radiation or cavity shifts, and nonlinear motion. The Brown–Gabrielse derivation treats the broader quadratic model with field misalignment and ellipticity; the trace proof above supplies the local argument under the stated matrix hypotheses.

The current draft Definition QFT example units, metric, gauge, and pairing conventions has been inspected and its mass-energy interface matches this use. This conditional matrix deduction does not certify the supplier or any physical trap.

Define $R:\mathbb{R}\times (0,\mathrm{\infty })\to \mathbb{R}$ by $R(a,c):=1+a\mathrm{/}c$. Here $c$ is only the positive denominator variable in this lemma, not the speed of light. For real $\delta a,\delta c$ and nonnegative bounds ${ϵ}_a,{ϵ}_c$ satisfying $\mathrm{\mid }\delta a\mathrm{\mid }\le {ϵ}_a$ and $\mathrm{\mid }\delta c\mathrm{\mid }\le {ϵ}_c<c$, both denominators are positive and

$\mid R(a+\delta a,c+\delta c)-R(a,c)\mid \le \frac{c{ϵ}_a+\mathrm{\mid }a\mathrm{\mid }{ϵ}_c}{c(c-{ϵ}_c)}\mathrm{.}$

For the first-order statement, write $x=(a,c)$, $h=(u,v)$, and $\mathrm{\parallel }h{\mathrm{\parallel }}_2:=\sqrt{u^2+v^2}$. Suppose the segment $\{x+th:0\le t\le 1\}$ lies in the closed rectangle $Q=[A_{-},A_{+}]\times [c_{-},c_{+}]$, where $A_{-}\le A_{+}$ and $0<c_{-}\le c_{+}$. Define $A_{\ast }:=\max (\mathrm{\mid }{A}_{-}\mathrm{\mid },\mathrm{\mid }{A}_{+}\mathrm{\mid })$, $K_Q:=(A_{\ast }+c_{+}\mathrm{/}2)\mathrm{/}{c}_{-}^3$, and the real-linear map $L_x(u,v):=u\mathrm{/}c-av\mathrm{/}{c}^2$. The total derivative of $R$ at $x$ means a real-linear map $L$ such that ${\lim }_{h\to 0}\mathrm{\mid }R(x+h)-R(x)-L(h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }h{\mathrm{\parallel }}_2=0$. Then $L_x$ is the total derivative, and

$\mid R(x+h)-R(x)-L_x(h)\mid \le K_Q\mathrm{\parallel }h{\mathrm{\parallel }}_2^2\mathrm{.}$

The two local bounds are deterministic consequences of positive denominators and algebra. The proof uses no statistical uncertainty model, independence assumption, Taylor theorem, chain rule, compactness theorem, or probability supplier. Penning-trap frequency ratios motivate the application; the exact finite-error and quadratic-remainder bounds are proved here and are not attributed to an external source.