BRISM and the NFW Problem: A Technical Note on Resolving Cusp–Core and Universality via Naimark–Stinespring Projection

This technical note presents a structural explanation of the universal Navarro–Frenk–White (NFW) density profile within the BRISM framework (Brane–Resonance–Interface–State Model). BRISM interprets the observable 3D world as the real‑structured image of a phase‑neutral and σ‑additive interface acting on a unitary state in a Naimark–Stinespring dilation space (“bulk”). Under these structural conditions, the Born‑quadratic form emerges as a stability requirement of the interface.Applying the same projection geometry to spherically averaged bulk spectral components yields two generic radial regimes: an inner fold‑caustic producing a ∼ 1/r rise, and an outer geometric dilution with unitary norm conservation producing a ∼ 1/r³ fall‑off. A finite coherence scale of the interface regularizes the unphysical central divergence of the NFW fit and produces an intrinsic core without invoking new particle species or modifying general relativity.The result is a purely structural account of NFW‑type halo profiles: their functional form appears as a geometric consequence of projection, not as a dynamical fine‑tuning. The note outlines observational consequences and provides a basis for quantitative comparison with lensing and cluster‑profile data. profiledensity profilescusp–core problemfold causticNaimark–Stinespring dilationprojection geometryinterface stabilityBorn rule (structural)dark‑matter phenomenologyhalo universalityweak lensingcosmological structure formationspectral stabilitygeometric regularization