The radius of the 2nd orbit of Li2+ is x. The expected radius of the 3rd orbit of Be3+ is:

🧠 The Dual-Factor Scale Bohr radius ($r$) is a tension between distance ($n^2$) and nuclear pull ($Z$). We must compare two different ions at two different levels.

🗺️ The Ratio Mapping

1. Radius Law: $r \propto n^2/Z$.
2. Setup State 1 (Li²⁺): $n=2, Z=3 \implies r_1 = K \cdot (4/3) = x$.
3. Setup State 2 (Be³⁺): $n=3, Z=4 \implies r_2 = K \cdot (9/4)$.
4. Solving for x: $K = 3x/4$. $r_2 = (3x/4) \times (9/4) = \mathbf{27x/16}$.

⚡ The "Proportionality Grid" Ratio $= \frac{(n_{new}/n_{old})^2}{(Z_{new}/Z_{old})} = \frac{(3/2)^2}{(4/3)} = \frac{9/4}{4/3} = \frac{27}{16}$.

$\boxed{\text{Answer: 27x/16}}$