  We prove that if there exists a $C$-subsolution to a constant coefficients
strictly $\Upsilon$-stable general inverse $\sigma_k$ equation, then there
exists a unique solution. As a consequence, this result covers all the
analytical results of the classical strictly $\Upsilon$-stable general inverse
$\sigma_k$ equations, for example, the complex Monge--Amp\`ere equation, the
complex Hessian equation, the J-equation, the deformed Hermitian--Yang--Mills
equation, etc. Hence, we confirm an analytical conjecture by
Collins--Jacob--Yau [arXiv:1508.01934] of the solvability of the deformed
Hermitian--Yang--Mills equation. Their conjecture states that the existence of
a $C$-subsolution to a supercritical phase deformed Hermitian--Yang--Mills
equation gives the solvability.
