The fragmentation mode of high-mass molecular clumps and the properties of the central rotating structures surrounding the most luminous objects have yet to be comprehensively characterised. Using the IRAM NOrthern Extended Millimeter Array (NOEMA) and the IRAM 30-m telescope, the CORE survey has obtained high-resolution observations of 20 well-known highly luminous star-forming regions in the 1.37 mm wavelength regime in both line and dust continuum emission. We present the spectral line setup of the CORE survey and a case study for W3(H2O). At ~0.35 (700 AU at 2 kpc) resolution, the W3(H2O) clump fragments into two cores (West and East), separated by ~2300 AU. Velocity shifts of a few km/s are observed in the dense-gas tracer, CH3CN, across both cores, consistent with rotation and perpendicular to the directions of two bipolar outflows, one emanating from each core. The kinematics of the rotating structure about W3(H2O) W shows signs of differential rotation of material, possibly in a disk-like object. The observed rotational signature around W3(H2O) E may be due to a disk-like object, an unresolved binary (or multiple) system, or a combination of both. We fit the emission of CH3CN (12-11) K = 4-6 and derive a gas temperature map with a median temperature of ~165 K across W3(H2O). We create a Toomre Q map to study the stability of the rotating structures against gravitational instability. The rotating structures appear to be Toomre unstable close to their outer boundaries, with a possibility of further fragmentation in the differentially-rotating core W3(H2O) W. Rapid cooling in the Toomre-unstable regions supports the fragmentation scenario. Combining millimeter dust continuum and spectral line data toward the famous high-mass star-forming region W3(H2O), we identify core fragmentation on large scales, and indications for possible disk fragmentation on smaller spatial scales.
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