The amplification tree has polynomially many leaves

The ternary majority tree used for error amplification has depth $d(n) = \lceil \log_2 (n+1) \rceil + 3$ on inputs of length $n$, hence $3^{d(n)}$ leaves. This theorem records that the leaf count is polynomially bounded:

The bound follows from $3 \le 2^2$ together with $2^{\lceil \log_2 m \rceil} \le 2m$, which give $3^{\lceil \log_2 (n+1)\rceil} \le \big(2^{\lceil \log_2 (n+1)\rceil}\big)^2 \le 4 (n+1)^2$; multiplying by the constant factor $3^3 = 27$ yields the claim. Since the amplified verifier makes one call to the base verifier per leaf, this is precisely the fact that error amplification only costs a polynomial blow-up in running time.