Torsion of rational elliptic curves over ${\mathbb{Z}}_p$-extensions of quadratic fields for $p\ge 5$, with a slope analysis for $p=3$

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve, let $K$ be a quadratic field, and let $L\mathrm{/}K$ be a ${\mathbb{Z}}_p$-extension. We revisit Avcı’s theorem on the equality $E(L{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}=E(K{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}$ for $p>5$ and clarify a Galois-over-$\mathbb{Q}$ point needed in its proof: the auxiliary descent lemmas are finite Galois-over-$\mathbb{Q}$ statements, while finite layers of a general mixed ${\mathbb{Z}}_p$-extension of an imaginary quadratic field need not be Galois over $\mathbb{Q}$. We repair this by replacing finite layers by finite Galois envelopes built from the cyclotomic and anti-cyclotomic directions. We then prove the equality $E(L{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}=E(K{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}$ for every $p\ge 5$, including the formerly exceptional prime $p=5$, by excluding the remaining cyclic $25$-torsion obstruction via quadratic twists and the theorem of Chou-Daniels-Krijan-Najman over ${\mathbb{Q}}_{\mathrm{\infty },5}$. Finally, for $p=3$ and imaginary quadratic $K\mathrm{\ne }\mathbb{Q}(\sqrt{-3})$, we give a slope stratification: the odd part of $E(L{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}$ comes from $L\cap K_{\text{cyc}}$, while the $2$-primary part comes from $L\cap K(E[2])$. In particular, the two residue classes with both slope coordinates $3$-adic units have no torsion growth, the anti-cyclotomic residue class can contribute only $2$-primary growth, and all new odd torsion is forced into the cyclotomic residue class. The possible orders of odd cyclic subgroups are explicitly bounded, and $13$-torsion is excluded in the cases covered by the stratification.