NFW Profiles as Fold Caustics: Fixed Core Regularization and the Case of IC2574

The Navarro–Frenk–White (NFW) profile occupies a central role in the description of dark-matter halos. Despite its remarkable empirical and numerical success, the physical origin of its universal inner scaling \[\rho(r)\propto r^{-1}\] remains a subject of ongoing discussion, particularly in connection with the longstanding cusp–core problem. This paper explores the possibility that the inner NFW cusp may admit a geometric interpretation. Motivated by the universal structure of fold caustics in singularity theory, we investigate whether the characteristic inner NFW scaling can be understood as the radial manifestation of a projected fold singularity rather than solely as a consequence of collisionless halo dynamics. A central observation motivates the present work. Ideal fold caustics are singular only in idealized mathematical limits in which finite physical scales are removed. Across wave optics, gravitational lensing, collisionless phase-space dynamics, and other physical systems, the observed realization of a caustic is generically regularized by finite intrinsic scales. The observed finite structure is not the exception. The singularity is. This suggests a different perspective on the cusp–core problem. If the NFW cusp represents the ideal mathematical form of an underlying fold-like structure, then the observed finite core may not require a separate explanation. Instead, the core may represent the physically expected realization of that structure, while the perfectly unregularized cusp corresponds to an idealized limiting case. To explore this possibility, we consider a simple fixed-width regularization of the NFW cusp, \(\rho(r)=\frac{\rho_s}{(r/r_s+\epsilon)(1+r/r_s)^2},\) with a representative dimensionless regularization scale of order \(\epsilon\sim0.1\). As a proof-of-concept example, the dwarf galaxy IC2574 is used to test whether such a fixed regularization can reproduce an observed core-dominated system while preserving the characteristic outer behavior of the NFW profile. The purpose of the present work is not to challenge the cosmological success of \(\Lambda\)CDM, modify gravity, or introduce a new dark-matter model. Rather, it investigates whether the universal NFW cusp may admit a complementary geometric interpretation as a fold-caustic structure and whether the observed galactic cores may naturally arise as the physical regularization of that structure.