It has long been understood that non-trivial Conformal Field Theories (CFTs) with vanishing central charge ($c=0$) are logarithmic. So far however, the structure of the identity module -- the (left and right) Virasoro descendants of the identity field -- had not been elucidated beyond the stress-energy tensor $T$ and its logarithmic partner $t$ (the solution of the $cto 0$ catastrophe). In this paper, we determine this structure together with the associated OPE of primary fields up to level $h=bar{h}=2$ for polymers and percolation CFTs. This is done by taking the $cto 0$ limit of $O(n)$ and Potts models and combining recent results from the bootstrap with arguments based on conformal invariance and self-duality. We find that the structure contains a rank-3 Jordan cell involving the field $Tbar{T}$, and is identical for polymers and percolation. It is characterized in part by the common value of a non-chiral logarithmic coupling $a_0=-{25over 48}$.
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