Listing the first few well-formed machine codes in shortlex order

Each tuple $D_i$ has the correct length $6=3+3(3-2)\cdot 1$, satisfies $m=3\ge 3$, $n=1\ge 1$, and $0\le s=0<1$, and records a transition entry with target state in $\{0,1,2\}$, tape symbol $0$, and move bit in $\{0,1\}$. So each $E(D_i)$ is a well-formed machine code.

Any well-formed machine description tuple has arity $3+3(m-2)n\ge 6$, with equality only when $m=3$ and $n=1$; then necessarily $s=0$. Under the fixed encoder, the unary arity header makes every larger-arity code longer than these shortest valid codes. For arity six, shortlex compares the self-delimiting canonical entries in order: target $0$ with move $0$, target $0$ with move $1$, and target $1$ with move $0$ give $E(D_1),E(D_2),E(D_3)$ respectively.

By [L2], decoding $E(D_i)$ recovers exactly the machine described by $D_i$. Reading off the single transition triple of each tuple gives the three concrete machines described in the example statement.

Therefore the shortlex enumeration from [L1] begins by listing these three machines, each exactly once. This shows concretely what the effective enumeration does at its first successful decodings.