= Solution
For the second isotope, consecutive gap differences are approximately $56.4$ MHz, but the smallest gap is about $1.5$ times this value. The higher-$F$ values for the five gaps are therefore $3/2,5/2,7/2,9/2,11/2$. The six levels run from $F=1/2$ to $11/2$. The electronic state is unchanged, so $J=5/2$ as in <2/d>; the endpoints give
$$
\boxed{I_{56}=3,\qquad \frac{|A_{56}|}{h}\simeq56.41\ \mathrm{MHz}.}
$$
For example, fitting $\nu_F$ to $F|A|/h$ with the same equal-weight convention gives $56.408$ MHz. The nearly constant gap differences are compatible with the same leading model of the <magnetic-dipole hyperfine interaction> and the printed rounding.
Within an unchanged electronic state, and neglecting isotope-dependent changes of the electronic <wave function> and <hyperfine anomaly>, $A\propto g_I$. Thus
$$
\boxed{\left|\frac{g_I(^{56}\mathrm{Mn})}{g_I(^{55}\mathrm{Mn})}\right|
\simeq\frac{56.41}{72.42}=0.779,}
$$
or the reciprocal magnitude is about $1.284$. This is a ratio of \b[<nuclear g-factors>], not directly a ratio of <nuclear magnetic moments>: $\mu_I=g_I I\mu_N$, so the moment-magnitude ratio would additionally contain $I_{56}/I_{55}=3/(5/2)$ and is about $0.935$.
The positive transition <frequencies> do not fix either coupling sign. If a signed ratio is required, equal signs must be supplied as an additional assumption or determined by an ordering-sensitive measurement. No absolute <nuclear g-factor> is inferred without a calibrated electronic field coefficient. The second ladder in <2/b> illustrates the inferred half-integer $F$ sequence without turning that sign ambiguity into a falsely measured ordering.
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