The word RAM of $lax-808846$ reads words of natural numbers; a binary word is handed to it as its list of zeros and ones, one bit an entry, so that the length of the input is the length of the binary word and the machine has no advantage from packing bits into its words. The admissible inputs are the lists of zeros and ones, all of them: a list that is not the encoding of a formula is answered $0$, as it is not in the language. The output is $1$ or $0$; the machine writes it as a single entry.

The bound is proved, not merely asserted, from the program: reading and decoding the word is linear in its length; the implication graph has $2b$ nodes for $b$ the bound on the indices, which is at most the length of the word because an index is written in unary, and at most two edges per clause; each of the at most $2v$ searches costs a number of instructions linear in the size of the graph, and skipping a variable that does not occur costs a constant. The strongly connected components of Aspvall, Plass and Tarjan would bring the second factor down to a constant; that algorithm is not formalized here.

The fitting hypothesis $c\,(|x|+1) \le 2^w$ is the "word is wide enough" condition written as an inequality: every value the program manipulates is a count of nodes, edges, bits or steps, all of them below $c\,(|x|+1)$. The program is quantified before the word length, so it is one algorithm at every word length, not a family.