= Solution
Set $N=\int_VP^2\,dV$, $D=\int_{\mathbb R^3}|\nabla P|^2\,dV$, $M=\int_{\mathbb R^3}|\mathbf B|^2\,dV$, and $q=\max_V|Q|$. The confined flow satisfies the <impermeability condition> $\mathbf u\cdot\mathbf n=0$, so $Q=0$ at $r=a$. Multiplying the <radial magnetic induction scalar> equation by $P$ and integrating, the <advection> term vanishes by the <divergence theorem>:
$$
\int_VP\,\mathbf u\cdot\nabla P\,dV=\frac12\int_{\partial V}P^2\mathbf u\cdot\mathbf n\,dS=0.
$$
The same <integration by parts>, using $\nabla\cdot\mathbf B=0$, gives
$$
\int_VP\,\mathbf B\cdot\nabla Q\,dV
=\int_{\partial V}PQ\,\mathbf B\cdot\mathbf n\,dS-\int_VQ\,\mathbf B\cdot\nabla P\,dV
=-\int_VQ\,\mathbf B\cdot\nabla P\,dV.
$$
For diffusion, <Green's first identity> in the sphere gives $\int_VP\Delta P=-\int_V|\nabla P|^2+\int_{\partial V}P\partial_rP$. In the exterior, $\Delta P=0$ and the normal on its inner boundary is $-\hat{\mathbf r}$, so
$$
\int_{\mathbb R^3\setminus V}|\nabla P|^2\,dV=-\int_{\partial V}P\partial_rP\,dS.
$$
The infinity term vanishes for the isolated decaying field. The <insulating boundary condition for the radial magnetic scalar> permits matching the two traces, and therefore the exterior contribution is essential: it turns the diffusive term into $-\eta D$. Altogether,
$$
\frac12N'=-\int_VQ\,\mathbf B\cdot\nabla P\,dV-\eta D
\leq q\int_{\mathbb R^3}|\mathbf B\cdot\nabla P|\,dV-\eta D.
$$
Applying the pointwise <Cauchy-Schwarz inequality> and then its integral version,
$$
\int_{\mathbb R^3}|\mathbf B\cdot\nabla P|
\leq\int_{\mathbb R^3}|\mathbf B||\nabla P|
\leq\sqrt{MD},
\qquad
\frac12N'\leq q\sqrt{MD}-\eta D.
$$
Thus a nonzero field with stationary or growing $N$ must satisfy the <radial-flow dynamo energy bound>:
$$
\boxed{(\max_V|Q|)^2\geq\eta^2
\frac{\displaystyle\int_{\mathbb R^3}|\nabla P|^2\,dV}
{\displaystyle\int_{\mathbb R^3}|\mathbf B|^2\,dV}.}
$$
The temporal qualification matters. For general time-dependent <dynamo action>, nondecay does not make $N'\geq0$ at every instant; the boxed bound applies at nondecreasing instants, to a marginal or growing mode, or in an appropriate long-time sense. A uniform strict violation, $q\sqrt{M/D}\leq\eta-\varepsilon$ with $\varepsilon>0$, gives $N'\leq-2\varepsilon D$. The whole-space <Sobolev inequality> and <Holder inequality> give $N\leq |V|^{2/3}\|P\|_6^2\leq C_aD$, forcing exponential decay of $N$. Alternatively, for a bounded statistically steady field, time averaging the energy balance and applying <Cauchy-Schwarz> once more gives $q_*^2\geq\eta^2\langle D\rangle/\langle M\rangle$, where $q_*=\sup_tq(t)$. Sustained <dynamo action> cannot have a uniform negative dissipation gap.
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