In the definition on $KSUM$ problem we are given $n$ input integers and we have to decide if $K$ of them sum to $0$.
$KSUM$ is $FPT$ if there is a $O(f(K)poly(n))$ algorithm for it. However Downey and Fellows in their book give definition as
These two definitions do not seem to be entirely consistent.
We know $W[P]=FPT\implies W=FPT$.
We also know this puts $KSUM$ is $FPT$.
Does it mean $KSUM$ is $FPT$ at a fixed $K$ and independent of length of integer inputs or can it also imply some asymptotic $FPT$ result where $K$ depends non-trivially on number of input integers and length of input integers?
I looked for this information in Grohe's or Downey's book and I am unable to find this information and could someone weight in on this subtlety? That is we might have an algorithm for $KSUM$ that is $FPT$ if input integers have only $O(polylog(n))$ bits each. However if input bitlength is polynomial in $n$ or exponential in $K$ then the $FPT$ algorithm complexity might not be of shape $O(f(K)\cdot poly(n))$. This could still be $FPT$ under Downey's definition. Would such an algorithm still be considered $FPT$ for $KSUM$ as well? It should be but from the working definitions of $FPT$ for $KSUM$ in literature this is unclear.
Is it consistent $W[P]=FPT$ while $KSUM$ is not $FPT$ at bounded or fixed $K$?