[proofplan] The statement is the Fontaine-Mazur Conjecture, so a complete proof cannot be supplied: the conjecture remains open in general. The appropriate verification is therefore a status argument, not a derivation from the hypotheses on $\rho$. We record precisely what the conjecture asserts, isolate the known obstruction to writing a proof, and explain why no proof can conclude the claimed geometric realization in full generality. [/proofplan] [step:Identify the assertion as a conjectural prediction rather than a proved theorem] The displayed representation \begin{align*} \rho: G_{\mathbb Q} \to GL_n(\overline{\mathbb Q}_\ell) \end{align*} is assumed continuous, irreducible, geometric, and unramified outside finitely many primes. The conclusion asserted in the statement is precisely the Fontaine-Mazur prediction: such a geometric $\ell$-adic Galois representation should come from the $\ell$-adic cohomology of an algebraic variety over $\mathbb Q$, up to Tate twist, and in the two-dimensional odd case should correspond to a cuspidal modular eigenform up to the stated twists. Because this assertion is a conjecture and is open in general, there is no known argument that derives the stated geometric origin from the listed hypotheses for arbitrary $n$ and arbitrary such $\rho$. [/step] [step:Explain why the hypotheses cannot be consumed in a complete general proof] A proof of the stated conclusion would need to construct, from the representation $\rho$, an algebraic variety $X$ over $\mathbb Q$, a cohomological degree $i \in \mathbb Z_{\geq 0}$, and a Tate twist integer $m \in \mathbb Z$ such that $\rho$ appears as a subquotient of the $G_{\mathbb Q}$-representation \begin{align*} H^i_{\acute{e}t}(X_{\overline{\mathbb Q}}, \overline{\mathbb Q}_\ell)(m). \end{align*} No such construction is known in the stated generality. Thus the assumptions of continuity, irreducibility, geometricity, and finite ramification cannot presently be assembled into a complete proof of the asserted cohomological realization. [/step] [step:Record the valid mathematical status of the theorem entry] The correct mathematical status is therefore that this entry states an open conjecture, not a theorem with a complete proof. Special cases are known through modularity and potential modularity theorems, especially in dimension two under additional hypotheses, but those results do not prove the full statement as written. Consequently, the proof field must stop at this status declaration unless the theorem statement is weakened to a known special case. [/step]