WordRAM simulation and runtime transfer with polylogarithmic overhead

There is a single natural overhead exponent $d$ such that both of the following hold. First, every program of the unsigned-pointer WordRAM with multiplication has a fixed signed-pointer simulator, and conversely every signed-pointer program has a fixed unsigned-pointer simulator; the simulations preserve successful halting, accept/reject verdicts, and every mapped source memory cell, at target width $w+4$, within $C(t+w+1)(w+1)^d$ steps for a positive constant $C$ fixed before the width and input. Second, for every problem $Q$ whose input and observable output are bounded by $B(n)=K(n+1)^q$, every input-magnitude exponent $\kappa$, and every correct source program with runtime $T(n)$ at all sufficiently large word widths, there is a fixed target program, width threshold coefficient, and positive $C$ solving the same problem in $C(T(n)+B(n)+1)(\lfloor \log_2 n \rfloor+1)^d$ steps, in both directions, including input packing and output recovery. This is the operational and quantitative core of the equivalence; it does not by itself assert the fine-grained threshold corollaries.