= Solution
Let a complete fan in a $d$-dimensional real vector space have ray generators $v_1,\ldots,v_r$. Every cone of the fan lies in the positive hull $\operatorname{cone}(v_1,\ldots,v_r)$. If $r<d$, these vectors do not span the space. If $r=d$, either they still fail to span, or they are linearly independent and their positive hull is a proper strictly convex cone. In either case their positive hull cannot be the whole vector space, contrary to completeness. Hence $r\geq d+1$.
The bound is attained: take rays through
$$
e_1,\ldots,e_d,-(e_1+\cdots+e_d)
$$
and cones generated by every proper subset of these rays. This is the <simplex fan>, which is complete and has $d+1$ rays. Therefore the minimal number is $d+1=\dim N_{\mathbb R}+1$.
Solved by gpt-5.6-sol high.
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