Anti-Fourier heat flux does not certify the fourth-order closure state of a rarefied cavity

Cold-to-hot heat transfer in rarefied cavities is usually treated as a signature of Fourier-law failure. Here it is used to ask whether a correct anti-Fourier heat-flux field certifies the flux-side fourth-order closure state. In a two-dimensional monatomic flow, the heat-flux hierarchy observes the divergence of the composite R26-level tensor $A_{ij}=R_{ij}^{\text{cl}}+\mathrm{\Delta }{\delta }_{ij}\mathrm{/}3$, not the tensorial fourth-order anisotropy $R_{ij}^{\text{cl}}$ and scalar fourth-order excess $\mathrm{\Delta }$ separately. Unlike the one-dimensional shock problem, the null space is not a single algebraic direction: it is the function space of divergence-free symmetric tensor fields, including an exactly invisible out-of-plane channel $A_{zz}$. DSMC data for argon lid-driven cavities show that the size of the anti-Fourier region is strongly regime dependent: it is suppressed when the lid speed is increased from $100$ to $200\mathrm{m}{\mathrm{s}}^{-1}$, but enlarged when the Knudsen number is increased from $0.05$ to $0.10$. In all cases, the anti-Fourier channel is primarily tensorial, while scalar-excess effects remain a smaller local modulation. Hidden Airy and out-of-plane states, scaled relative to the measured RMS composite tensor, change $R^{\text{cl}}$ and $\mathrm{\Delta }$ by order-one amounts while leaving the in-plane heat-flux observable below the seed-to-seed statistical resolution, or exactly unchanged for the $A_{zz}$ mode. These shifted states satisfy necessary scalar Cauchy and contracted fourth-order Gram-positivity checks. Thus anti-Fourier heat-flux agreement is a physical validation target, but it is not a certificate of full R26-level closure recovery.