This paper provides a complete, unified, and constructive solution to Hilbert’s fourth problem based on differential algebra. We construct a specialized differential closure Kstraight in which all analytic Finsler metrics whose geodesics are straight lines (projectively flat Finsler metrics) can be explicitly represented. By establishing a recursive correction algorithm, a complete theory of combinatorial coefficients, and a rigorous obstruction theory,we develop a systematic construction method. This framework unifies all classical theories of straight metrics (Euclidean, hyperbolic, spherical, Busemann-Pogorelov, Berwald, Douglas, Bryant, etc.) and embeds them as special cases. Using this constructive capability, we systematically discover and rigorously prove the existence of several new families of metrics, including metrics of mixed curvature, metrics with discrete symmetries, metrics induced by algebraic varieties, and metrics with exceptional Lie group symmetries. Finally, we implement a complete computational system (DACS/SMCL) that provides tools from theory to computational verification, with rigorous interval arithmetic validation. The core innovations of this paper are: 1) the construction of a specialized differential closure Kstraight for projectively flat Finsler metrics, transforming an infinite-dimensional geometric problem into a tractable algebraic one; 2) the development of a systematic algorithm combining recursive correction, obstruction analysis, and branch selection with complete proofs;3) the unification of all classical theories while providing effective tools for systematically discovering and rigorously proving the existence of new families of metrics.
shifa liu (Wed,) studied this question.