Global irreducibility and Galois groups of Chebyshev–Lucas level polynomials

For the Chebyshev–Lucas level polynomials \(B_n(X)-q\), we establish that the generic Galois group over \(\mathbb Q(q)\) (and over \(\mathbb C(q)\)) is the full symmetric group \(S_m\), where \(m=\lfloor(n-1)/2\rfloor\), for every \(n\ge5\). The argument uses the geometry of critical amplitudes and distinguishes generic monodromy from arithmetic specialization. For the critical level \(A_n=B_n-1\), we further prove irreducibility for a density-one set of even indices: among even \(n\le X\), the number of reducible cases is \(O(X^{1/3})\). In particular, sufficiently large even indices that are not perfect odd powers satisfy an explicit irreducibility criterion. We obtain two-prime local criteria, infinitely many specialized groups \(\operatorname{Gal}(A_{2p+2}/\mathbb Q)=S_p\) for primes \(p\equiv3,5\pmod8\), and dense sets of rational levels with full symmetric Galois group. Reducible and irreducible specializations may coexist densely within one real-interlacing interval. These results do not claim universal irreducibility of every Chebyshev–Lucas critical polynomial.