Closure-channel identifiability and two-channel recovery in monatomic kinetic normal shocks

Residual agreement in a kinetic or moment equation does not automatically identify every higher-order closure variable entering a nonequilibrium shock. We formulate this issue as an observability problem for the fourth-order closure content of monatomic normal shocks and follow it through a hierarchy of collision models and diagnostics. The kinematic part of the result is independent of the collision operator: the one-dimensional heat-flux budget observes the projected fourth-order channel $S=R_{xx}^{\text{cl}}+\mathrm{\Delta }\mathrm{/}3$, not the tensorial R26-level moment $R_{xx}^{\text{cl}}$ separately from the scalar fourth-order excess $\mathrm{\Delta }$. The observation map therefore has a one-dimensional null space, so a heat-flux residual can be small while the split between tensorial anisotropy and isotropic tail intensity remains wrong. A DVM-consistent scalar-excess budget supplies the missing channel and gives the two-channel reconstruction $R_{xx}^{\text{cl}}=S-\mathrm{\Delta }\mathrm{/}3$ without direct $R_{xx}^{\text{cl}}$ data. Across BGK shocks at Mach 2–5, this reduces the active-zone $R_{xx}^{\text{cl}}$ error from about $63$–$64\mathrm{\%}$ to $2.4$–$4.1\mathrm{\%}$. Sparse scalar-excess interpolation is used only as an information-reduction test: a representative 24-probe operating point gives $R_{xx}^{\text{cl}}$ errors below $4.5\mathrm{\%}$, and below $4.7\mathrm{\%}$ with $1\mathrm{\%}$ probe noise. Collision-model diagnostics then separate the invariant observation channel from the model-dependent source law. Shakhov changes the heat-flux relaxation to the correct Prandtl number but is neutral in the even $\mathrm{\mid }\mathit{c}{\mathrm{\mid }}^4$ scalar-excess source; a direct discrete Shakhov channel check recovers $S$, $\mathrm{\Delta }$ and $R_{xx}^{\text{cl}}$ with errors $6.4\times {10}^{-4}$, $2.1\times {10}^{-7}$ and $1.0\times {10}^{-3}$, respectively.