Sub-kiloparsec Test of the Kennicutt-Schmidt Relation in a Strongly Lensed Dusty Star-Forming Galaxy at z~2.78
[abridged] Star formation is a key process driving galaxy evolution, but understanding how it proceeds within the interstellar medium (ISM) requires resolving cold molecular gas and dust on sub-kpc scales. At z~2-3, near the peak of cosmic star formation, such resolution can generally only be achieved through strong gravitational lensing. We present an ALMA study of SMMJ0658, a 20x-magnified main-sequence star-forming galaxy at z=2.7768 +/- 0.0002, lensed by the Bullet Cluster (z=0.296) into three images. We use 0.2" and 0.6" ALMA Band-3 observations to test the Kennicutt-Schmidt (KS) relation at sub-kpc scales at cosmic noon. Using a JWST-based strong-lensing model of the Bullet Cluster, we reconstruct source-plane CO(3-2) and rest-frame 803um dust-continuum maps down to ~200pc scales. We derive spatially resolved molecular gas and star formation rate surface densities, depletion times, and the scale dependence of the CO-to-dust flux ratio over 200pc-3.2kpc apertures. The 0.2" ALMA data reveal massive, dense star-forming clumps, with Sigma_mol ~1.3-2.3 x 10^3 M_sun pc^-2, Sigma_SFR ~0.5-2.3 M_sun yr^-1 kpc^-2, and tau_dep ~0.82-2.56 Gyr. CO-dust spatial decorrelation produces a strong scale dependence in tau_dep. We find a KS-relation breakdown scale of 0.8 +/- 0.1 kpc, consistent with individual regions tracing distinct stages of the star-formation cycle. The ALMA and JWST data also reveal evidence for a lensed galaxy pair at z~2.78, although whether SMMJ0658 is undergoing an early-stage interaction remains uncertain. Our results show that the apparent universality of the KS relation breaks down once individual star-forming regions are spatially resolved, in line with observations from the local universe to z~1. We extend this picture to z~2.78, providing new insight into the resolved properties of the clumpy cold ISM and the star-formation cycle at cosmic noon.