The parameter is a function of the input word rather than something carried alongside it. This is what lets one program serve every parameter: a program that must be told the parameter from outside would be a family of programs, one per parameter, and could hide unbounded advice in its literals. The quantifier order says so — the program and the constant come before the instance, the parameter and the word length. A parameter that is a structural property of the encoded object, such as the treewidth of a graph read off the word, is a function of the word all the same, and no computability of that function is required or used.

$Fits$ is the fitting condition, stated as an explicit inequality against $2^w$ rather than through logarithms. It says of each entry $v$ of a word that $c(|x|+v+1)^c \le 2^w$, which at once makes every entry a genuine word and leaves polynomial room in the length, the usual word RAM assumption of a word of at least $c\log|x|$ bits: the memory a polynomial-time algorithm addresses is a polynomial in the length, and a table of size $|x|^2$, such as the adjacency matrix of a graph read off the word, must fit. Entries are not bounded by the length in general: a due date may be any number at all, so quantifying over the entries is what makes a claim about them honest instead of silently assuming they are small.

The bound is $c\,g(k)\,(|x|+1)^c$, elementary rather than asymptotic, with the $+1$ making it meaningful on the empty word. The length enters through a polynomial, the usual form of fixed-parameter tractability, and not linearly: a linear bound would be strictly stronger than the notion the source uses, and would not hold for the algorithms the source cites, such as Lenstra's, whose dependence on the length is polynomial and not linear (nor for an algorithm that first builds a graph from the word, which takes more than linear time in the word). $g$ is an arbitrary function of the parameter: it bounds a fixed program's running time rather than defining it, so no computability requirement on $g$ is needed or intended.

An fpt-reduction from $P$ to $Q$ is a map $f$ on words, computed by one program within $c\,g(k)\,(|x|+1)^c$ instructions, that sends admissible words to admissible words, preserves the answer, and raises the parameter by at most a function of the old parameter. The program is required to run on the admissible words that fit and whose image fits: the image of a word may be exponentially longer than the word (a parameter-sized table), and a machine whose word length is only logarithmic in the input cannot address it, exactly as a decision procedure is only required to run on the words that fit. This is the notion that splits a fixed-parameter tractability proof into a reduction, proved here, and a cited algorithm for the target problem.