= Solution
For positive <masses> and distinct positions, <Newtonian gravitational field> gives
$$
\boxed{m_i\ddot{\mathbf x}_i=-\sum_{j\ne i}\frac{Gm_im_j(\mathbf x_i-\mathbf x_j)}{|\mathbf x_i-\mathbf x_j|^3}}.
$$
Insert $\mathbf x_i=a(t)\mathbf a_i$ with $a>0$. The separation numerator contributes a factor $a$ and the denominator $a^3$, so
$$
m_i\ddot a\,\mathbf a_i=-\frac{\mathbf G_i}{a^2},\qquad \mathbf G_i=\sum_{j\ne i}\frac{Gm_im_j(\mathbf a_i-\mathbf a_j)}{|\mathbf a_i-\mathbf a_j|^3}.
$$
Hence the time-dependent equations are satisfied whenever
$$
\boxed{\mathbf G_i=\Lambda m_i\mathbf a_i\text{ for every }i,\qquad \ddot a=-\frac\Lambda{a^2}}.
$$
The fixed-shape condition is a <Newtonian central configuration>, and the resulting motion is a <homothetic gravitational motion>. It applies away from the collision scale $a=0$; a negative scale would require absolute values in the force scaling rather than the same positive-scale formula.
Multiplying the scalar equation by $\dot a$ gives
$$
\frac{d}{dt}\left(\frac{\dot a^2}{2}-\frac\Lambda a\right)=\dot a\left(\ddot a+\frac\Lambda{a^2}\right)=0.
$$
Thus its <first integral> is $\dot a^2/2-\Lambda/a=k/2$ for some real constant $k$.
In the pair potential
$$
W=-\sum_{j<i}\frac{Gm_im_j}{|\mathbf a_i-\mathbf a_j|},
$$
differentiating a pair with respect to $\mathbf a_i$ produces $Gm_im_j(\mathbf a_i-\mathbf a_j)/|\mathbf a_i-\mathbf a_j|^3$. Every pair incident on $i$ contributes exactly once, so $\nabla_iW=\mathbf G_i$. Moreover $W(\lambda\mathbf a_1,\ldots,\lambda\mathbf a_n)=\lambda^{-1}W$ for $\lambda>0$. Differentiating with respect to $\lambda$ at one proves the relevant <Euler homogeneous function theorem> identity directly:
$$
\sum_i\mathbf a_i\cdot\mathbf G_i=-W.
$$
Dot each fixed-shape equation with $\mathbf a_i$ and sum to obtain
$$
\boxed{\Lambda I=-W,\qquad I=\sum_i m_i|\mathbf a_i|^2}.
$$
For a nontrivial collision-free configuration with at least two positive <masses>, $W<0$ and $I>0$, and therefore $\Lambda>0$. Positivity requires an interacting pair; an isolated one-particle system is the degenerate zero-force exception rather than a positive-$\Lambda$ gravitational configuration. Pairwise cancellation also gives $\sum_i\mathbf G_i=0$, so the nondegenerate configuration has $\sum_i m_i\mathbf a_i=0$: its scaling origin is the <centre of mass>.
Finally, the system's <kinetic energy> is $I\dot a^2/2$ and its <potential energy> is $W/a$. Consequently
$$
\boxed{E_{\rm total}=\frac I2\dot a^2+\frac Wa=I\left(\frac{\dot a^2}{2}-\frac\Lambda a\right)=\frac{kI}{2}}.
$$
This establishes the <energy> relation for every fixed configuration satisfying the algebraic condition, without assuming any particular particle arrangement.
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