A Unifying Differential-Algebraic Framework for Hilbert's Fifth Problem: Constructive Solutions, High-Dimensional Generalizations, and Computable Verification

Key Points

• The aim is to reconstruct and expand Hilbert's Fifth Problem using a differential-algebraic framework.
• Established a unified theory framework called differential algebraic Lie group closure.
• Constructed explicit local coordinates and exponential mapping formulas for local Euclidean topological groups.
• Developed a systematic theory for controlling high-dimensional nonlinear expansions.
• Provided an algorithmic framework and strict verification protocols for computability.
• Proved any local Euclidean topological group can be embedded in a differential closure.
• Created universal recurrence formulas for structural constants in high dimensions.
• Extended the theory to infinite dimensional and non-smooth cases.
• Demonstrated the classical Griess-Montgomery-Zipin theory as a natural corollary.

Abstract