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For every constant $c>0$, a deterministic multitape Turing machine with more than one tape and time complexity bounded by $T(n)$ has an equivalent machine with the same tape count and bound $T(n)/c+n+O(1)$, allowing a larger finite control and tape alphabet. Blocks of old symbols become single new symbols, and finite control implements blocks of old steps. The linear overhead accounts for initial input packing. If $T(n)/n\to \infty$, the overhead can be absorbed to obtain any fixed constant speed increase for sufficiently large inputs. This does not supply unlimited speed increases for operations on fixed hardware.^[ref]

The linear speedup theorem concerns multitape Turing machines whose finite control and tape alphabet may change. Formally, for $k>1$, if a deterministic $k$-tape Turing machine decides $A$ within $T(n)$ steps on every input of length $n$, then for every constant $c>0$ there is a deterministic $k$-tape Turing machine deciding the same formal language withThe additive constant is independent of $n$; in the usual normalized formulation the bound is $T(n)/c+n+2$, with the integer-rounding convention absorbed in the constant.^[ref]

The construction packs a fixed number of old tape symbols into each new symbol and uses a larger finite control to simulate blocks of old steps. Initial input packing explains the additive linear term. If $T(n)/n\to \infty$, choosing a larger constant $c$ makes the total asymptotic time complexity an arbitrarily small constant multiple of $T(n)$.

It does not promise arbitrary speed increases on fixed hardware. Larger packed symbols and transition tables cease to be single constant-cost hardware operations without limit. The theorem also retains the input-reading overhead. It is a statement about changing the abstract machine description and primitive operations, rather than an unlimited implementation optimization on one processor.