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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, ${\epsilon }_{x0}\propto {\gamma }^2{\theta }_b^3$. This follows from the radiation-integral expression ${\epsilon }_{x0}=C_q{\gamma }^2{I}_5/(J_x{I}_2)$: 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, givingThis 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.

If the quoted coupling is $k={\epsilon }_y/{\epsilon }_x$, then ${\epsilon }_y=k{\epsilon }_x$. Combining the horizontal factor of eight with $k$ falling from $0.1$ to $0.02$ givesThis is the usual leading estimate treating the horizontal emittance in part a as unchanged by small coupling.

If one instead accounts for exact redistribution of a fixed natural-emittance sum, ${\epsilon }_x={\epsilon }_0/(1+k)$ and ${\epsilon }_y=k{\epsilon }_0/(1+k)$. Then the horizontal ratio is $(1/8)(1.1/1.02)\simeq 0.1348$ and the vertical ratio is $0.02696$, an improvement by about $37$. These are consistent refinements of the stated approximate factor forty, not a second independent reduction to apply afterward.