The Pell–Cyclotomic Bridge in K24: Norm Compatibility, CM Restrictions, and the Arithmetic Rigidity of the 13–14–15 Triangle

We study the arithmetic of the consecutive triangle family T(a) = (a − 1, a, a + 1), a ≥ 3, and its interaction with the cyclotomic field K₂₄ = ℚ(ζ₂₄). The four Pell-type equations governing this Heronian family are interpreted as real norm identities inside the maximal real subfield K₂₄⁺ = ℚ(√2, √3), while the square and hexagonal CM structures are attached to the CM subfields ℚ(i) and ℚ(√−3). Version 3 corrects the squareclass normalization used in Version 2. The former condition ru = z² is too restrictive and does not contain the fundamental node 13–14–15, since for this triangle r = 4, u = 5, and ru = 20 = 5 · 2². The refined square–CM compatibility condition is therefore ru ∈ 5(ℚ×)², or equivalently, in the integral Pell parametrization, ru = 5z². Under this refined compatibility hypothesis, we prove a conditional rigidity theorem: within the Pell backbone of consecutive Heronian triangles, the only positive nondegenerate solution satisfying ru = 5z² is (a, k, r, u, m) = (14, 24, 4, 5, 9), which corresponds to the triangle 13–14–15. The proof uses the coprimality of r and u, the resulting squareclass splitting into two cases, Ljunggren’s theorem for the equation X² − 3s⁴ = 1, and a reproducible modular certificate excluding the second squareclass case along the Pell orbit. The stronger statement that a full square–CM cyclotomic node structure in K₂₄ should force the refined squareclass condition ru ∈ 5(ℚ×)² is stated separately as a conjectural bridge.