$13$ unknowns over quadratic integer rings and Lucas congruences

For every quadratic number field $K$, we prove a uniform $3$-unknown Diophantine definition of integer tuples in ${\mathcal{O}}_K$, allowing finitely many polynomial nonvanishing conditions. This yields an effective $+3$ transfer principle and a $13$-unknown representation of every recursively enumerable integer relation. Consequently, there exists an absolute degree bound $D_0\ge 1$ such that for every quadratic number field $K$, there is no algorithm that, given $$P(Y_1,\dots ,Y_{13})\in \mathbb{Z}[Y_1,\dots ,Y_{13}],\mathrm{d}\mathrm{e}\mathrm{g}P\le D_0,$$ decides whether $P=0$ has a solution in ${\mathcal{O}}_K^{13}$. The arithmetic input is a fourth-order Pell–Lucas congruence. It is a specialization of the norm-one Lucas multiplication formula, which yields exact valuations for the deviation of a Lucas quotient from its linear term, together with deviation criteria for Lucas–Wieferich and Wall–Sun–Sun primes. We also establish local surjectivity and $\mathrm{\ell }$-adic density for second-order correction terms for norm-one Lucas sequences.