NP-hardness of approximating Uniform Sparsest Cut within every constant factor

Theorem 1.1: for every fixed $C>1$ there is a polynomial-time reduction from 3SAT producing uniform-demand instances with a multiplicative gap $C$, so approximating Uniform Sparsest Cut within any constant factor is NP-hard. The constants and exponents of the reduction depend on $C$. It does not give a superconstant (for instance $\sqrt{\log N}$-type) hardness factor, does not address Balanced Separator directly, and says nothing about the Small-Set Expansion hypothesis itself.

Theorem 1.1: for every fixed $C>1$ there is a polynomial-time reduction from 3SAT producing uniform-demand instances with a multiplicative gap $C$, so approximating Uniform Sparsest Cut within any constant factor is NP-hard. The constants and exponents of the reduction depend on $C$. It does not give a superconstant (for instance $\sqrt{\log N}$-type) hardness factor, does not address Balanced Separator directly, and says nothing about the Small-Set Expansion hypothesis itself.