Harder-Narasimhan filtration of torsion F-gauges

We establish a Harder-Narasimhan theory for torsion coherent $F$-gauges over a perfect field $k$ of positive characteristic and completely classify its stable objects. Our construction uses Ekedahl’s derived equivalence with coherent complexes over the Raynaud ring, where the theory refines his half-integral type filtration on diagonal dominoes. At each finite rational slope $d\mathrm{/}q$ in lowest terms, there is a unique stable diagonal domino up to Breuil-Kisin twist, constructed explicitly from the corresponding Christoffel word. The semistable category of this slope is equivalent to a product of $q$ copies of the zero-slope category. The classification of stable objects corrects Ekedahl’s classification of weakly simple diagonal dominoes of type $\frac{1}{2}$. The theory upgrades to a locally finite Bridgeland stability condition on ${\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}}_{\mathrm{t}\mathrm{o}\mathrm{r}}(k^{\mathrm{S}\mathrm{y}\mathrm{n}})$.