Turings h1 is still mysterious. Its almost a reference to several things.
In crypto, its a ref to a function being distinguished from those functions on connected domains that can be expressed algebraically (vs geometrically, say).
In boolean theory, it seems like the truth table based on two bits (and their 4 cases).
In cayley graph quaternion thinking, boolean 4 cases give way to two bits whose names are k and -k. (Or 1 and k, I’m not really sure.)
in automorphic function thinking, we are talking about generating a field (k), one of rational functions.
And when we think of matrices with rational functions on the diagonal, we think of convolutional encoders able to look back a couple of delay lines (and their 4 cases).