Establishing a Rigorous Connection between Soft Numbers/Möbius Geometry and Kinetic-to-Fluid Derivations

Develop a rigorous mathematical connection between the Soft Numbers and Möbius-strip geometric framework and the derivation of fluid equations from Newtonian hard-sphere particle systems via Boltzmann’s kinetic theory (as in Deng–Hani–Ma), clarifying how the Soft Numbers formalism captures the local-to-global emergence of macroscopic laws.

Background

Deng, Hani, and Ma provide a rigorous derivation of macroscopic fluid equations from microscopic dynamics via Boltzmann’s kinetic theory, addressing the passage from local to global behavior. The present paper proposes a complementary geometric perspective using Soft Numbers and the Möbius-strip, whose topology mirrors local-to-global phenomena.

The authors explicitly state that formalizing a rigorous bridge between these two approaches—kinetic-to-fluid derivations and Soft Numbers/Möbius geometry—remains open, suggesting a potentially unifying framework for Hilbert’s sixth problem.

References

The topological properties of the Möbius-strip-locally two-sided yet globally one-sided — provide a natural geometric model for the kind of emergent macroscopic behavior that Hilbert's sixth problem seeks to axiomatize. Developing a rigorous connection between these two approaches remains an open and promising direction for future research.

— Hilbert's Sixth Problem and Soft Logic  (2603.29969 - Klein et al., 31 Mar 2026) in Section 6, Discussion

From a physical viewpoint, the Navier–Stokes are more appropriate in the presence of boundaries or shocks; however, it is not clear whether they arise in this case from a pure scaling.

— Some Considerations on the Fluid-Dynamical Limit of Particle Systems  (2609.18550 - Pulvirenti et al., 16 Sep 2026) in Section 5, “Concluding remarks”