Global irreducibility and Galois groups of Chebyshev–Lucas level polynomials

We study the rational levels of the Lucas polynomial \(B_n(X)=∑_j(-1)^j n{2j+1}X^j\). Two incompatible local degree conditions prove that \(Bp^r+1(X)-q\) is irreducible whenever \(v_2(p^r+1-q)=0\) and \(v_p(p^r+1-q)=r\), for an odd prime \(p\). A norm argument supplies a complementary criterion for exceptional even indices that are not perfect powers. We determine explicit specialized symmetric Galois groups, including an infinite congruence family, while keeping their irreducibility hypotheses separate from the group arguments. For every \(n≥5\) the generic group of \(B_n(X)-q\) is the full symmetric group. The proof uses a differential amplitude identity to produce enough individual critical values to rule out polynomial composition; it does not require all critical values to be distinct. Finally, reducible and irreducible rational levels, and in a cubic example three different Galois types, are dense inside the same interval of strict positive interlacing. Thus local arithmetic, generic monodromy and real root geometry control genuinely different aspects of specialization. We further prove irreducibility for a density-one class of even indices by a uniform analytic height bound, construct an \(S₁₁\) realization without the classification of finite simple groups, and show density of full-symmetric rational specializations.