Finite multi-tape Turing machines and uniform word random access machines compute exactly the same total functions in polynomial time, when both use the length of the canonical binary encoding as input size and the word RAM is required to have a polynomially bounded sufficient word length.

Concretely, a Turing machine must transform the binary encoding of an input word into the binary encoding of its output in polynomially many Turing steps. A word RAM receives the same semantic input as a native list of numbers, physically prefixed by its length so that the input tape is self-delimiting; one program must return the exact output in polynomially many RAM instructions at every sufficiently large word length, and a polynomial in the encoded input length must suffice to represent the length prefix, all input and output entries, and to make the computation correct.

The claim concerns polynomial-time function computation, commonly called $FP$. The usual machine-independence statement for the decision class $P$ is its specialization to functions with Boolean-valued output. No fixed polynomial simulation overhead is claimed, only preservation of the class of polynomial-time computable functions.