Ordinary binary numerals and doubled-bit numerals are polynomially related

Let $\sigma (0)=0$, let $\sigma (n)$ for $n>0$ be the ordinary binary numeral with leading bit $1$, and let $\tau (n)$ be the word obtained by repeating each bit of $\sigma (n)$ twice. Decode $\sigma$ by rejecting the empty word and noncanonical leading zeros, and otherwise reading the binary numeral. Decode $\tau$ by first requiring identical adjacent bit-pairs, replacing each pair by one bit, and then using the decoder for $\sigma$. These total decoders return $\mathtt{m}\mathtt{a}\mathtt{l}$ on every rejected word and recover $n$ on the displayed codewords, so $\sigma$ and $\tau$ are effective encodings. Moreover, $\mathrm{\mid }\tau (n)\mathrm{\mid }=2\mathrm{\mid }\sigma (n)\mathrm{\mid }$ for every $n$, so the two encodings are polynomially related. For example, $\sigma (13)=1101,\tau (13)=11110011.$ Consequently a bound such as $\mathrm{\mid }\tau (n){\mathrm{\mid }}^3$ is also a polynomial bound in $\mathrm{\mid }\sigma (n)\mathrm{\mid }$, namely $8\mathrm{\mid }\sigma (n){\mathrm{\mid }}^3$.