Turing machines admit an effective enumeration

Enumerate all binary words in shortlex order and run the total decoder from [L2] on each word. Whenever the decoder reports malformed input, output nothing; whenever it returns a coded machine, output that machine.

By [L1], every coded machine $M$ has some binary code $<mi\; mathvariant="normal">⌜</mi>M<mi\; mathvariant="normal">⌝</mi>$, and shortlex enumeration eventually reaches that code. At that stage the decoder returns $M$, so every coded machine appears somewhere in the output stream.

By injectivity in [L2], two different binary words cannot decode to the same coded machine. Therefore the procedure from step 1.1 lists each coded deterministic one-tape Turing machine exactly once and gives an effective enumeration.