Constraints on the Pop III Sky Surface Brightness from High-Redshift Caustic Transients in MACS0416

Population III (Pop III) stars are hypothetical zero-metallicity stellar structures formed from primordial hydrogen and helium. They are theorized to span a wide mass range, extending to several 100 solar masses, and may have played important roles in nucleosynthesis and reionization. Their expected fluxes are far below JWST NIRCam detection limits, making direct observation unlikely. However, extreme magnification near the caustics of massive foreground galaxy clusters may enable their detection. We search for overlooked high-redshift (7 <= z <= 17) caustic transits in the lensing cluster MACS J0416.1-2403. Using three observations spanning 126 days, we create difference images to identify potential candidates. Critical curves for sources at 7 <= z <= 17, derived from a strong-lensing model, guided visual inspection of three difference-images. No additional transits were found. The longer caustics in our model sweep a larger source-plane area, increasing the probability of detecting an event. Combining this increased statistical sensitivity with deeper imaging, we establish a fainter limit for the unresolved stellar population at z >= 7. From this null result, we constrain the 100 solar mass 2 micron Pop III sky surface brightness to >= 32.8 +/- 0.6 mag arcsec-2. Modeling the non-detection as a Poisson process gives a posterior mean caustic-transit rate of 0.29 cluster-1 yr-1 and a 95 percent upper credible limit of lambda 95 = 0.86 cluster-1 yr-1. The inferred rate remains consistent with the adopted fiducial Pop III caustic-transit model and provides an empirical benchmark for future multi-epoch monitoring campaigns. This search exploits the fact that individual stars projected close to source-plane caustics can be briefly magnified far beyond their unlensed fluxes. These constraints provide a direct test of the abundance of luminous Pop III stars at early cosmic times.