The Pell–Cyclotomic Bridge in K24: An Explicit Trace–Norm Selector and the Arithmetic Rigidity of the 13–14–15 Triangle

We construct an explicit cyclotomic trace–norm selector in the conductor-24 cyclotomic field K24 = Q(ζ24), and apply it to the Pell orbit associated with consecutive Heronian triangles. Using the distinguished selector Δ = c5 c17, we obtain a Gaussian trace invariant whose normalized norm admits the closed form N(β̃n) = 4x² − 3, where x is the Pell coordinate of the orbit. The norm-square condition is shown to hold if and only if n = 1. Consequently, the corresponding geometric configuration is uniquely the classical 13–14–15 triangle. This yields a direct, global, and fully explicit bridge from Euclidean inradius geometry, through a Pell orbit and a cyclotomic trace, to a Gaussian norm-square condition. The proof is constructive, reproducible, and independent of the abstract trace–norm rigidity framework.