Cored Galactic Halos Exclude a Dominant Cold Component: A Quantitative No-Go for Hybrid Dark-Sector Models

A broad class of proposed alternatives to cold dark matter (CDM) replaces CDM at galactic scales with a mechanism producing cored, isothermal-like halos — thereby inheriting the well-known advantage of cored profiles over NFW on rotation-curve data — while retaining a cold, collisionless residual at cosmological scales to supply the pre-recombination potential wells that the CMB acoustic-peak structure demands. We quantify, for the first time explicitly, the internal consistency limit of this hybrid construction. Fitting a two-component halo (cuspy NFW carrying a mass fraction f, cored isothermal carrying 1-f) to 171 SPARC galaxies, with the NFW concentration fixed by the concentration-mass relation of Dutton & Maccio (2014), the median reduced chi-squared degrades monotonically from 0.50 at f=0 to the NFW-equivalent value already at f~0.4, and to 2.15 at f=1. The rotation-curve data therefore tolerate a cuspy halo fraction of at most f < 0.2-0.3. By contrast, a full Boltzmann-code (CAMB) scan shows the CMB third-to-second peak ratio to be a steep function of the cold density that excludes any appreciable reduction of the residual below the full Planck cold-dark-matter density: the residual must equal essentially the entire present-day dark sector (Omega_cdm = 0.264 +/- 0.004), placing the required galactic cuspy fraction at f~1.0, at which the median fit quality (chi-squared = 2.15) is far past NFW level. There is no overlap between the f-range the CMB demands and the f-range the rotation curves tolerate. The result is a dilemma — internal inconsistency or redundancy — that applies generically to hybrid models combining cored galactic phenomenology with a cold cosmological residual, independently of the specific core-producing mechanism, provided that mechanism operates only after recombination.