= Solution
The <mean-field approximation> replaces each site's fluctuating surroundings by a self-consistent average. On a <cubic lattice> each spin has <coordination number of a lattice> $z=2D$, even though summing only over positive basis directions counts each bond once. Write $\sigma_n=M+\delta\sigma_n$ and drop products of deviations. Each Ising bond becomes $\sigma_n\sigma_m\simeq M\sigma_n+M\sigma_m-M^2$, so the approximate <statistical Hamiltonian> is
$$
H_{\rm MF}=-zJM\sum_n\sigma_n+\frac{NzJ}{2}M^2.
$$
The constant corrects counting each bond twice. The resulting independent-spin <partition function> is $Z_{\rm MF}=e^{-\beta_TNzJM^2/2}[2\cosh(\beta_TzJM)]^N$, where $\beta_T=1/(k_BT)$. The one-spin probabilities give
$$
\langle\sigma\rangle=\frac{e^{\beta_TzJM}-e^{-\beta_TzJM}}{e^{\beta_TzJM}+e^{-\beta_TzJM}}.
$$
Equating this average with the assumed $M$ gives
$$
\boxed{M=\tanh\!\left(\frac{2DJM}{k_BT}\right),\qquad T_c=\frac{2DJ}{k_B}.}
$$
Here $J>0$ is assumed; a ferromagnetic <order parameter> is not the appropriate choice for an antiferromagnetic coupling. Put $A=T_c/T$. For $A\leq1$, $\tanh(AM)<AM\leq M$ at any $M>0$, so only $M=0$ is possible. For $A>1$, the function $\tanh(AM)-M$ has positive slope at zero, is <strictly concave> on $M>0$ and is negative at $M=1$. It therefore has exactly one positive zero and a spin-reversed negative zero. Expanding at the bifurcation gives
$$
0=(A-1)M-\frac{A^3M^3}{3}+O(M^5),\qquad M^2=\frac{3(A-1)}{A^3}+O((A-1)^2)\sim3\frac{T_c-T}{T_c}.
$$
The transition is continuous, with <order-parameter critical exponent> $\beta=1/2$. Stability can also be checked from the <Bragg-Williams free energy of the Ising model> per <Ising spin>:
$$
f_{\rm BW}(M)=-\frac{zJ}{2}M^2+k_BT\left\{\frac{1+M}{2}\log\frac{1+M}{2}+\frac{1-M}{2}\log\frac{1-M}{2}\right\}.
$$
Its <derivative> is $-zJM+k_BT\operatorname{arctanh}M$, giving the same <self-consistency equation>. Near zero,
$$
f_{\rm BW}(M)=-k_BT\log2+\frac{k_BT-zJ}{2}M^2+\frac{k_BT}{12}M^4+O(M^6).
$$
The quartic coefficient is positive at $T_c$, the stable minima approach zero continuously, and the minimized <free energy> has no first-derivative jump. Thus \b[<mean-field approximation> predicts a second-order transition at $T_c=2DJ/k_B$]. This is the <mean-field Ising critical bifurcation>.
For short-range interactions, large dimension gives many neighbours and suppresses the relative fluctuations of their average. More quantitatively, <Gaussian field theory> fluctuations averaged over a <correlation volume> have <variance> of order $\xi^{2-D}$, while the squared ordinary mean-field <order parameter> is of order $|t|\sim\xi^{-2}$. Their ratio is proportional to $\xi^{4-D}$. The <Ginzburg criterion> therefore makes neglect of critical fluctuations asymptotically consistent above four dimensions, inconsistent below four, and marginal at four, where logarithms occur. Large dimension is a route to validity; sufficiently long-range or infinite-range interactions can also produce mean-field behaviour. In low-dimensional short-range systems even the existence and location of a transition can differ from this approximation.
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