Absolutely summing Toeplitz operators on weighted Fock spaces

We study absolutely $r$-summing Toeplitz operators $T_f:F_{\phi }^p\to F_{\phi }^q$ with complex symbols. The weight has a uniformly positive and bounded real Hessian. The main difficulty is that local distance to holomorphic functions does not detect an entire part of the symbol, while the Toeplitz operator does. Using the complete classification of absolutely summing diagonal maps, we characterize absolutely summing Fock–Carleson embeddings. An IDA decomposition, a Rademacher argument, and local scalar recovery then detect the missing analytic part of a complex symbol. We obtain equivalent criteria in terms of the local $L^q$ mean, the complex ball average, and the Berezin transform for every $1\le p,q,r<\mathrm{\infty }$. At $q=1$ the criterion is $L^{s_p}$, where $s_p=2p\mathrm{/}(3p-2)$ for $p\le 2$ and $s_p=1$ for $p>2$. We also obtain a sectorial-symbol criterion without an IDA assumption and prove sharpness by separated models.