= Solution
For a real steady amplitude, integrate the conserved-field equation twice. Periodicity forces the integration slope to vanish, and the zero mean fixes the remaining constant. Hence
$$
\boxed{B=\kappa(\langle A^2\rangle-A^2),\qquad\kappa=\mu/\sigma.}
$$
Substituting this into the steady <amplitude equation> gives the nonlocal <ordinary differential equation>
$$
A''+(1-\kappa\langle A^2\rangle)A+(\kappa-1)A^3=0.
$$
For $A=R\operatorname{sech}(cX)$, use the <hyperbolic secant> identity $A''=c^2A-2c^2A^3/R^2$. Vanishing of the two independent coefficients gives
$$
\boxed{\kappa\langle A^2\rangle=1+c^2,\qquad \kappa=1+\frac{2c^2}{R^2}.}
$$
Choose $c>0$ and $R>0$; an overall negative sign is another solution. Thus $\kappa>1$. The exact average of the proposed profile over the finite interval is
$$
\langle A^2\rangle=\frac{2R^2}{cL}\tanh\frac{cL}{2}.
$$
For $cL\gg1$, the <hyperbolic tangent> differs from one by $O(e^{-cL})$, so $cL\langle A^2\rangle\simeq2R^2$. Eliminating $R^2=2c^2/(\kappa-1)$ gives the requested leading relation
$$
\boxed{L(c^2+1)=\frac{4c\kappa}{\kappa-1}.}
$$
Equivalently $c+c^{-1}=K$, with $K=4\kappa/[L(\kappa-1)]$. Positive real roots exist precisely when $K\geq2$; they are $(K\pm\sqrt{K^2-4})/2$ and are reciprocal. Each candidate must additionally satisfy $cL\gg1$. For $L>2$, the leading necessary range is $1<\kappa\leq L/(L-2)$, with equality at the coalescence $c=1$. Away from this leading-order fold, the inequality is strict.
The strict finite-window condition can also be seen by retaining the average exactly. Its consistency relation is $c+c^{-1}=K\tanh(cL/2)$. Since $\tanh(cL/2)<1$, it implies $K>2$, and hence
$$
\boxed{1<\kappa<\frac L{L-2}\quad\text{provided }L>2.}
$$
The strict endpoint distinction is exponentially small in the large-$cL$ approximation; the leading polynomial alone permits its double root. When $0<L\leq2$, this algebra imposes no finite upper bound on $\kappa>1$. The printed interval therefore needs $L>2$: for example $L=1$, $\kappa=1.1$ gives a large positive root $c\simeq43.98$ of the leading equation and $cL\gg1$, despite the printed upper endpoint being negative.
Finally, the single <localized pulse of a conserved-field convection model> is an approximate periodic solution. Its values agree at the interval ends, but its derivatives have opposite signs there; their magnitude is exponentially small, $O(cR e^{-cL/2})$. Periodic-tail corrections are needed for an exact smooth <periodic boundary condition>. The range derived from the truncated profile is a consistency condition, not a proof that every permitted parameter gives an exact stable periodic pulse.
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