Searching for Inflationary Physics with the CMB Trispectrum: 3. Constraints from Planck

Is there new physics hidden in the four-point function of the cosmic microwave background (CMB)? We conduct a detailed analysis of the Planck PR4 temperature and polarization trispectrum for $\ell\in[2,2048]$. Using the theoretical and computational tools developed in Paper 1 and Paper 2, we search for 33 template amplitudes, encoding a variety of effects from inflationary self-interactions to particle exchange. We find no evidence for primordial non-Gaussianity and set stringent constraints on both phenomenological amplitudes and couplings in the inflationary Lagrangian. Due to the use of optimal estimators and polarization data, our constraints are highly competitive. For example, we find $\sigma(g_{\rm NL}^{\rm loc})=4.8\times 10^4$ and $\tau_{\rm NL}^{\rm loc}<1500$ (95\% CL), a factor of two improvement on Effective Field Theory amplitudes, and a $43\sigma$ detection of gravitational lensing. Many templates are analyzed for the first time, such as direction-dependent trispectra and the collapsed limit of the `cosmological collider', across a range of masses and spins. We perform a variety of validation tests; whilst our results are stable, the most relevant systematics are found to be lensing bias, residual foregrounds, and mismatch between simulations and data. The techniques discussed in this series can be extended to future datasets, allowing the primordial Universe to be probed at even higher sensitivity.