= Solution
For hydrogen, the leading characteristic scales are
$$
\boxed{E_{\rm gross}\sim\alpha^2m_ec^2,\qquad
E_{\rm fine}\sim\alpha^4m_ec^2,\qquad
E_{\rm hf}\sim\alpha^4\frac{m_e^2}{m_p}c^2.}
$$
Order-one <spin> and nuclear magnetic factors, and state-dependent factors, are suppressed here. <Gross atomic structure> is Coulomb binding. <Fine structure> includes relativistic kinetic and spin-orbit effects, with <Thomas precession> affecting the spin-orbit coefficient. <Hyperfine structure> couples electronic motion and <spin> to the <nuclear magnetic moment>, whose inverse nuclear <mass> makes it much weaker. The contact <spin> interaction can remain in an s state where orbital spin-orbit coupling vanishes.
For <positronium>, the <reduced mass> is $m_e/2$. Its gross levels and separation scale are
$$
\boxed{E_n=-\frac{\alpha^2m_ec^2}{4n^2}\simeq-\frac{6.80\,\mathrm{eV}}{n^2},
\qquad a_{\rm Ps}=2a_0.}
$$
Thus the mean ground-state <electron>–<Positron> separation is twice the infinite-proton-mass hydrogen value: both means have the same factor $3/2$ multiplying their respective Bohr radii. Retaining <proton> recoil makes the ratio $2m_p/(m_p+m_e)$, very slightly below two. Separation is not the distance of either <positronium> constituent from its <centre of mass>.
Interpret the <spin> operators in units of $\hbar$, as required by the printed Zeeman term, so $A$ has units of <energy>. Two <spin> halves give \b[four eigenstates]. At zero field, total <spin> $F=0,1$ and projection $m_F$ label the singlet and triplet. At high field relative to the hyperfine scale, the individual projections $m_{s,e},m_{s,p}=\pm1/2$ become approximately good <quantum numbers>. Total projection remains exact at every field, but $F^2$ does not for the two mixed states.
Choose the quantization axis along $\mathbf B$. The aligned-spin states $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$ both have $E=A/4$ because their Zeeman shifts cancel. In the other product-state sector,
$$
H_0=\begin{pmatrix}-A/4+2\mu_BB&A/2\\A/2&-A/4-2\mu_BB\end{pmatrix},
\qquad
\boxed{E_\pm=-\frac A4\pm\frac12\sqrt{A^2+16\mu_B^2B^2}.}
$$
For physical <positronium> $A>0$. At $B=0$ there is a triplet at $A/4$ and a singlet at $-3A/4$. At large positive field the mixed states approach $|\uparrow\downarrow\rangle$ and $|\downarrow\uparrow\rangle$, with energies $-A/4+2\mu_BB$ and $-A/4-2\mu_BB$ respectively, while the aligned pair remains degenerate at $A/4$.
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2014/paper-1-aop-positronium-levels.png]
{title=Four positronium spin states: an exactly degenerate aligned pair and two mixed branches}
The dimensionless mixing parameter is $x=4\mu_BB/A$. A crossover estimate is $x\sim1$, rather than a sharp transition. The scale $A\sim\alpha^4m_ec^2$ already gives a field of a few tesla. Using the physical hyperfine interval $A/h\simeq203.39\,\mathrm{GHz}$ gives
$$
\boxed{B_\times\sim\frac{A}{4\mu_B}\simeq3.63\,\mathrm T.}
$$
The numerical hyperfine interval is external physical input, not a number supplied by the symbolic <Hamiltonian>.<New Precision Measurement of Hyperfine Splitting of Positronium by A. Ishida and others>[arXiv:1310.6923, abstract, measured interval approximately 203.39 GHz]{ref} The high-field limit here still presumes that orbital distortion is negligible compared with Coulomb binding.
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