K3 Surfaces and the Riemann-Roch Program C: Skeletal Filtration Construction of Derived Riemann-Roch Theorem for Twisted Sheaves and Its Applications on K3 Stacks

This paper establishes a noncommutative twisted version of the Grothendieck
Riemann-Roch (GRR) theorem on rank-2 polarized K3 stacks, reconstructing the
 Euler characteristic of divisor classes as alternating sums of Ext groups through
 skeletal filtration constructions in derived categories. Based on the 23 cohomologi
cal jump trajectories observed on Picard lattice U(2)-type K3 surfaces in previous
 work, this paper proves that these phenomena correspond to nontrivial extensions
 of canonical skeletal filtrations of twisted sheaf complexes. The core contribution
 lies in providing the explicit formula χ(Oα
 X(D)) = ∑2
 n=0(−1)n dimExtn
 Db(X,α)(En,Fn),
 where {En} is a sequence of rigid sheaves determined by the quasi-wall struc
ture. Theoretical results are numerically verified on quartic surface stacks using
 Macaulay2, covering effective divisor classes in the parameter range |a|,|b| ≤ 15,
 consistent with predictions.