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The instantaneous kinematic viscosity law is $\bar{\nu }(r,\Sigma )$, with no explicit constitutive memory. Expand its transported quantity at the possibly evolving background:The viscous transport response exponent satisfies ${\partial }_{\Sigma }(\bar{\nu }\Sigma ){|}_0=q{\bar{\nu }}_0$. Subtract the background equation and retain only linear terms to obtainThis remains a linear equation with space- and time-dependent background coefficients; neither a steady background nor constant $q$ is required for this step. The sign of the response coefficient is the negative-diffusion criterion for viscous disk instability.

The negative-diffusion criterion for viscous disk instability is ${\partial }_{\Sigma }(\bar{\nu }\Sigma )<0$. A surface density enhancement then transports angular momentum less effectively and grows. Freezing the Keplerian viscous diffusion equation locally gives growth proportional to the square of radial wavenumber, within the thin-disc transport range; wavelengths must still be sufficiently long compared with the disk scale height for that closure. The formal short-wave limit is a backward heat equation.

Within an instantaneous vertically averaged transport closure, the local Keplerian viscous diffusion equation has effective diffusion coefficient $3{\partial }_{\Sigma }(\bar{\nu }\Sigma )$. A positive coefficient smooths short radial Fourier modes in the allowed thin-disc wavelength range; a negative coefficient gives the negative-diffusion criterion for viscous disk instability. A zero coefficient is linearly degenerate and does not establish strict decay. This local response criterion alone does not settle thermal, self-gravitating or boundary-driven stability.