A Differential-Algebraic Unified Construction Theory for Hilbert's Fourth Problem

This paper establishes a complete unified framework from the first principles of differential algebra,providing a systematic and explicit solution to Hilbert’s Fourth Problem: the construction of all possible line metrics. We construct a novel differential closure Kstraight for line metrics and prove that all analytic metrics satisfying the line axiom can be explicitly represented within this closure. The framework offers the following core contributions: (1) Unification: It incorporates Euclidean,hyperbolic, spherical geometries, as well as the Busemann–Pogorelov construction, Berwald metrics,Douglas metrics, Bryant’s exceptional metrics, etc., into a single theoretical system, rendering them as special cases and corollaries. (2) Constructiveness: A recursive correction algorithm starting from the tangent cone is provided, enabling the computation of explicit metric expressions to arbitrary precision. (3) Completeness: The combinatorial correction terms and branch selection issues in high dimensions are fully addressed, with general formulas and rigorous proofs given. (4) Innovation: New families of line metrics are derived, including mixed-curvature metrics, discrete symmetric metrics,algebraic variety-induced metrics, and exceptional Lie group symmetric metrics. (5) Arithmetic Geometry Connections: The combinatorial coefficients are shown to be intimately related to Bernoulli numbers and multiple zeta values, revealing deep connections with motives and arithmetic quantum theory. (6) Physical Applications: Extensions to Lorentzian signature, renormalization group flows, and quantization are provided, linking to general relativity and quantum field theory.The paper contains a complete chain of derivations from basic definitions to final theorems. All proofs are rigorously conducted within the differential-algebraic framework, accompanied by implementable algorithms and verification methods. The structure is rigorous and logically self-consistent, providing the first systematic explicit constructive solution to Hilbert’s Fourth Problem.