Generic Hodge and special Hodge--Witt polygons

We prove that the Hodge polygon of the generic fiber of a proper smooth $p$-adic formal scheme over ${\mathcal{O}}_C$ lies on or above the generalized Hodge–Witt polygon of its special fiber. The argument combines prismatic cohomology with Ekedahl’s diagonal theory of coherent Raynaud complexes. For each integral cutoff and each finite iteration length, a finite iterated Nygaard construction produces a perfect $A_{\mathrm{i}\mathrm{n}\mathrm{f}}$-complex. Its individual-degree lengths along the generic divisor branches are weighted Hodge sums. A maximal-minor specialization argument bounds their sum by the corresponding crystalline length. On the special fiber, an exact derived modification of coherent Raynaud complexes has length growth governed by weighted Hodge–Witt numbers, with bounded error. Passing to the limit in these numerical lengths gives the comparison, allowing negative Hodge–Witt numbers and crystalline torsion. No projectivity, algebraization, or descent to a discretely valued field is assumed.