= Solution
For fixed, magnetically linear geometry, <self-inductance> relates a coil's own <magnetic flux linkage> to its <electric current>, $\Lambda_i=L_iI_i$. <mutual inductance> relates the linkage in one coil to the current in the other, $\Lambda_{21}=MI_1$. These coefficients enter the induced <electromotive forces> through $\mathcal E_i=-d\Lambda_i/dt$; reversing a chosen winding orientation reverses the sign assigned to $M$.
In the ideal confined-field <toroid>, a circular <magnetostatic Ampère law> contour encloses $N_1I$, so
$$
\boxed{H_\phi(r)=\frac{N_1I}{2\pi r}\quad(a<r<b).}
$$
Write $w=b-a$ for the height of the square cross-section. The <inductance of a rectangular-section toroid> follows by integrating the <magnetic flux density> over that cross-section; it cannot be replaced by its value at an arbitrary mean <radius>:
$$
\Phi=w\int_a^b\mu_0\mu_rH_\phi(r)\,dr
=\frac{\mu_0\mu_rN_1Iw}{2\pi}\ln\frac ba.
$$
Consequently, for winding normals chosen consistently,
$$
\boxed{L_i=\frac{\mu_0\mu_rN_i^2(b-a)}{2\pi}\ln\frac ba,\qquad
M_0=\frac{\mu_0\mu_rN_1N_2(b-a)}{2\pi}\ln\frac ba=\sqrt{L_1L_2}.}
$$
The equality means complete shared <magnetic flux linkage>; with leakage one instead has $|M|<\sqrt{L_1L_2}$.
For the crack, use the <radial-path approximation for a cracked toroid>, with an air <length> $d$ on each path and core <length> $2\pi r-d$. Continuity of the normal <magnetic flux density> at a crack face gives $H_g=\mu_rH_c$. Hence
$$
N_1I=\frac{B(r)}{\mu_0\mu_r}(2\pi r-d)+\frac{B(r)}{\mu_0}d,
\qquad
B(r)\simeq\frac{\mu_0\mu_rN_1I}{2\pi r+(\mu_r-1)d}.
$$
Integrating the radial paths yields
$$
M_d\simeq\frac{\mu_0\mu_rN_1N_2w}{2\pi}
\ln\frac{b+q}{a+q},\qquad q=\frac{(\mu_r-1)d}{2\pi},
\qquad
\boxed{\frac{M_d-M_0}{M_0}\simeq
\frac{\ln[(b+q)/(a+q)]}{\ln(b/a)}-1=-0.529.}
$$
Here $q=1.5915\,\mathrm{cm}$ and $M_d/M_0=0.47079$: \b[about a 53% reduction]. The small ratio $d/a$ does not justify expanding in $\mu_rd/a$, which is about ten. The crack's low <magnetic permeability> gives appreciable <magnetic reluctance>. This is an estimate within a confined-flux model; actual redistribution near the slit is outside its accuracy. Narrow-gap circuit models assume small gap width relative to transverse dimensions and approximately directed flux.<Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher/Magnetic Circuits>[§9.7, Example 9.7.1 and the paragraph after it: air-gap circuit approximation and need for a more general field solution]{ref}
For the intact ring, ideal poloidal winding currents and the no-leakage boundary condition admit $\mathbf H=N_1I\widehat{\boldsymbol\phi}/(2\pi r)$ with zero radial and vertical components. It is curl-free inside the current-free core, divergence-free for uniform <magnetic permeability>, tangent to the core boundary, and has the required <circulation>; <uniqueness of the Neumann problem> removes an additional homogeneous scalar-potential field. This explains zero non-azimuthal components in the stipulated intact-ring idealization. A literal slit breaks that rotational <symmetry>, so \b[zero transverse components everywhere is not exact for the cracked ring]: <magnetic fringing> and flux redistribution occur. In fact, extending the approximate $B(r)\propto1/(r+q)$ unchanged around a current-free core would give $[\nabla\times\mathbf H]_z\ne0$. Thus that radial-path calculation and a claim of exact purely azimuthal field in the cracked geometry cannot both be exact. The numerical answer above retains the intended circuit approximation while making its limitation explicit.
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