The one-step update on coded Turing configurations is primitive recursive

By the configuration-code definition in the statement and [L3], primitive recursive functions recover from $c$ the decoded length of the tuple, its first two entries $q,h$, the currently scanned symbol $t(h)$, and any later tape symbol that may need to be copied into the output code.

Using the explicit tuple format from [L1], together with the primitive-recursive arithmetic and bounded searches from [L4], one can scan the canonical binary numeral of $e$: count the initial unary arity header, locate each self-delimiting block, verify the tuple-length and alphabet constraints, and recover the first three numerals and every transition-table coordinate, returning the default branch on malformed inputs. A bounded search through the recovered transition list therefore finds the unique transition triple $(q^{\mathrm{\prime }},b,D)$ attached to the pair $(q,t(h))$ when $q$ is nonhalting. Using [L2] and the decoded tape coordinates from step 1.1, form the updated tape, state, and head position $h^{\mathrm{\prime }}$. Then use a bounded backward search through positions at most $r+1$ to find the rightmost nonblank updated cell, taking $0$ if none exists, and set $r^{\mathrm{\prime }}$ to the maximum of that position and $h^{\mathrm{\prime }}$. The output tuple is exactly $(q^{\mathrm{\prime }},h^{\mathrm{\prime }},t^{\mathrm{\prime }}(0),\dots ,t^{\mathrm{\prime }}(r^{\mathrm{\prime }}))$, so erasing the former rightmost nonblank cell trims the trailing blank while a head move onto a blank cell still records the head position.

If the decoded state $q$ is already accepting or rejecting, step 2.1 is skipped and the output is the original code $c$. If the input code is malformed, return one fixed default code. These are again primitive-recursive case distinctions because the relevant checks are bounded and finite.

Re-encoding the canonically trimmed updated sequence from step 2.1 uses only primitive-recursive arithmetic on its boundedly many coordinates, so the total function $\mathrm{Step}(e,c)$ is primitive recursive and has the stated behavior.