The main theorem of this paper shows a list of candidates of the permanent cycles of the third line \(\text{Ext}^{3,*}_{\mathcal A} (\mathbb{Z}/2,\mathbb{Z}/2)\) of the Adams spectral sequence for computing \(\pi_* (S^0)\) at the prime 2 which factor through the triple transfer \(BV_{3+} \to S^0\). Here \(V_3 = (\mathbb{Z}/2)^3\). By this theorem, the author proposes New Doomsday Conjecture (NDC): For each \(s\), there exists some integer \(n(s)\) such that no element in the image of \[ ({\mathcal P})^{n(s)} (\text{Ext}^{s,*}_{\mathcal A} (\mathbb{Z}/p, \mathbb{Z}/p)) \subseteq \text{Ext}^{s,p^{n(s)} *}_{\mathcal A} (\mathbb{Z}/p, \mathbb{Z}/p) \] is a nontrivial permanent cycle. Actually the main theorem implies that the conjecture holds for the elements of the triple transfer image in \(\text{Ext}^{3,*}_{\mathcal A} (\mathbb{Z}/2, \mathbb{Z}/2)\). This is a revised version of the Doomsday Conjecture which is disproved by the counterexamples of Mahowald at \(p =2\), and of R. Cohen at an odd prime \(p\). Therefore, these examples are no more the counterexamples of NDC and the Kervaire invariant one family will be the first one to be checked if it satisfies NDC. Using Mahowald's root invariant, he also proposes another conjecture named R. I. Doomsday Conjecture in the course of NDC. The explanation of these conjectures is one of the themes of the paper. The main theorem is proved by evaluating the order of the elements of the transfer image of the third line. That is, the author shows that the elements have order greater than some large numbers by using the Adams spectral sequence of \(BP_* (\wedge^3 P)\), and less than some small numbers by observing the \(BP\)-Hurewicz image \(\pi_*(\wedge^3P) \to BP_*(\wedge^3 P)\) using the \(BP\)-Adams operations on it. Thus the candidates are found.