The parameter is read off the input word, and the program is fixed before it: the quantifier order puts the program and the constant before the instance, the parameter and the word length. This is the uniformity of the definition. A definition that let the program be chosen after $k$ would describe a family of programs, one per parameter, which could hide unbounded advice in its literals.

$Fits$ 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 $c(|x|+v+1) \le 2^w$, which makes every entry a word and leaves room for the constant multiple of the length that bounds the addresses and the step count. Entries are not bounded by the length in general: a parameter, a weight or a deadline may be any number at all, so the claim quantifies over the entries instead of assuming they are small.

A reduction's output has to fit as well. A machine at word length $w$ reduces what it writes modulo $2^w$, so a word whose image does not fit is one the program cannot emit; the admissible set therefore constrains the image too. Like every fitting condition here this restricts the inputs rather than appearing as a hypothesis: as a hypothesis it would be empty, since no word length accommodates every encoding of a fixed instance at once.

The bound is $c * g k * (x.length + 1)$, elementary rather than asymptotic, with the $+ 1$ making it meaningful on the empty word. Its dependence on the length is linear, where the usual definition of FPT allows $g(k)\,|x|^{O(1)}$, so both notions defined here are the stricter ones: a problem that is fixed-parameter tractable in this sense is so in the usual sense, and a reduction that meets this bound is an fpt-reduction in the usual sense. A membership and a hardness proved against these definitions therefore imply their standard forms. Nothing weaker is defined because nothing weaker is needed — the dynamic program of Theorem 3 and the reduction of Theorem 1 both run within a linear bound.

$g$ 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. Some presentations of FPT require $g$ to be computable; the definition here does not, and the $g$ that Theorem 3 supplies is a closed-form expression in the parameter, so that theorem meets the definitions that require it as well.

Only the timed notions are defined. Plain computability is the special case in which the bound is unconstrained, and is not what any statement of this submission needs.