= Solution
For geometrically similar FODO optics at fixed beam <energy> and damping partition, the natural <horizontal emittance> scales approximately as the cube of the bending angle per cell, $\varepsilon_{x0}\propto\gamma^2\theta_b^3$. This follows from the radiation-integral expression $\varepsilon_{x0}=C_q\gamma^2I_5/(J_xI_2)$: at fixed optical <phase> advance the dispersion-invariant scaling makes the ratio proportional to the cube of the cell bend.
Doubling the number of cells around a fixed circumference halves the bending angle per cell, giving
$$
\boxed{\varepsilon_{x0,\rm new}/\varepsilon_{x0,\rm old}\simeq1/8.}
$$
This estimate assumes the redesigned magnets actually preserve the relevant dimensionless optical structure, the same <energy> and circumference, and no dominant insertion-device emittance contribution. Merely counting twice as many quadrupoles without specifying the lattice would not guarantee the cubic improvement.
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