A binary language $A$ belongs to $\mathrm{NP}$ if there are a verifier language $V\in\mathrm{P}$ and a polynomial $p\in\mathbb{N}[X]$ such that $x\in A$ precisely when some binary certificate $y$ with $|y|\le p(|x|)$ satisfies $\langle x,y\rangle\in V$. The verifier is a single deterministic polynomial-time decider on the explicit pair encoding, and its running time is polynomial in the combined input and certificate lengths. The certificate bound depends only on the original input length. This is the standard certificate definition of NP.