You totally misunderstood what the 42 pages are. They are not an implementation of LC, but an implementation of LISP in LC.
> Lisp gives you a way to talk about lambda calculus, in a way that executes.
LC gives you that too; it allows you to encode lambda terms as bitstrings, which can be represented as lambda terms themselves.
> But lambda calculus doesn't explain what x is
In the binary encoding, x is a de-Bruijn index: a natural number indicating the nesting level of the binding lambda.
> Lisp closes the circle
So does Binary Lambda Calculus.