Pro-modularity at p = 2: odd absolutely irreducible 2-adic representations occur in the completed Hecke algebra (auxiliary tame level allowed)

Theorem 1.1: for every continuous, odd, absolutely irreducible $r:G_{\mathbb Q}\to\mathrm{GL}_2(E)$, $E/\mathbb Q_2$ finite, unramified outside finitely many primes, there are an odd $N$ divisible by the odd ramified primes and a continuous $\lambda:\mathbb T_2(N)\to\mathcal O_E$ with $\lambda(T_\ell)=\mathrm{tr}\,r(\mathrm{Frob}_\ell)$ and $\lambda(\ell S_\ell)=\det r(\mathrm{Frob}_\ell)$ for $\ell\nmid2N$. Scalar and reducible residual representations are allowed; no de Rham hypothesis. It is an occurrence result and does not assert classical modularity; the tame level may contain auxiliary primes, unlike Emerton's formulation.

Theorem 1.1: for every continuous, odd, absolutely irreducible $r:G_{\mathbb Q}\to\mathrm{GL}_2(E)$, $E/\mathbb Q_2$ finite, unramified outside finitely many primes, there are an odd $N$ divisible by the odd ramified primes and a continuous $\lambda:\mathbb T_2(N)\to\mathcal O_E$ with $\lambda(T_\ell)=\mathrm{tr}\,r(\mathrm{Frob}_\ell)$ and $\lambda(\ell S_\ell)=\det r(\mathrm{Frob}_\ell)$ for $\ell\nmid2N$. Scalar and reducible residual representations are allowed; no de Rham hypothesis. It is an occurrence result and does not assert classical modularity; the tame level may contain auxiliary primes, unlike Emerton's formulation.