= Solution
For an infinite <cardinal>, the <Gimel function> is $\gimel(\kappa)=\kappa^{\operatorname{cf}(\kappa)}$. The <Gimel hypothesis> asserts, for every <singular cardinal> $\kappa$,
$$
\boxed{\gimel(\kappa)=\max\{2^{\operatorname{cf}(\kappa)},\kappa^+\}.}
$$
These are the unavoidable lower bounds supplied by monotonicity of exponentiation and <König theorem for cardinal numbers>. The hypothesis imposes the least allowed value at singular <cardinals>; it does not constrain the continuum <function> on regular <cardinals> to their successors. Thus it is weaker than <Generalized continuum hypothesis>. In the case $2^{\operatorname{cf}(\kappa)}<\kappa$, it says $\kappa^{\operatorname{cf}(\kappa)}=\kappa^+$, the usual <singular cardinals hypothesis> case.
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