SCALES. I. Bridging Lensed Clump - Giant Clump Scales in Lensed Galaxies at $z=1-4$
Dense star-forming regions in $z>1$ galaxies, often termed 'clumps', have been studied across two distinct regimes of telescope resolution, each motivating interpretations in tension with the other. In the SCALES series (Star-Forming Clumps As Layered Emergent Structures), we aim to connect star-forming structures across scales, from the $\sim100\,\mathrm{pc}$ clumps in lensed studies to the $\sim\mathrm{kpc}$-scale ''giant'' clumps in unlensed surveys. In this first paper, we present a difference-of-Gaussian (DoG) decomposition that simultaneously models $\sim100-300\,\mathrm{pc}$ (lensed) clumps and $\sim\mathrm{kpc}$-scale component (the extended component, or EC), which would together constitute the giant clumps seen at lower resolution. Here the EC is treated as a scale-separated local background rather than a physical object, absorbing all light that is not a compact clump. Applying this to 23 moderately lensed ($μ\approx 2-9$) galaxies at $z_{spec} = 1-4$ to evaluate clump properties (deferring ECs and giant clumps to the next work), we find their stellar mass to be capped at $\sim 10^{8.5}\,M_{\odot}$. We measure clump stellar mass function slope of $2.19 \pm 0.06$, which is higher than typical, and may reflect a physical truncation at the massive end. We also show that empirical correlations of clump surface densities and clump-to-host mass ratio with redshift are exaggerated by redshift-dependent detection bias, so $\sim100-300\,\mathrm{pc}$ clumps are largely co-similar across our range ($z = 1-4$). Finally, combining clump mass, size and age estimates with literature gas-dispersion measurements at similar scales, we argue that $\sim100\,\mathrm{pc}$ clumps are themselves composed of star-cluster-like sub-components at much smaller scales.