Cyclotomic Nodes, Refinement Rigidity, and the Eisenstein Phenomenon at Levels 16 and 18

This manuscript formalizes and documents extensive computational evidence for two nodal phenomena in cyclotomic arithmetic. (1) Node 16 is interpreted as a cyclotomic refinement of Node 8: the field ℚ(ζ₁₆)=ℚ(i,√2) introduces no new essential arithmetic content, only additional torsion and symmetry. We present a SageMath 9.3 search engine that reproduces the nodal catalog and verifies that, up to height bounds H=10⁶, Node 16 produces no new classes. This supports a refinement rigidity principle: the nodal phenomenon depends on essential arithmetic content rather than on the cyclotomic level. (2) Node 18 is a qualitatively new regime: ℚ(ζ₁₈)=ℚ(ζ₉,√−3) contains a purely Eisenstein layer and has no real quadratic subfield, so Pell-type mechanisms collapse. Instead, a rank-2 unit dynamics arises on the real cubic subfield K⁺=ℚ(ζ₉+ζ₉⁻¹). We define a nodal invariant via trace maps Tr_{K/L}:K→L to the Eisenstein subfield L=ℚ(√−3), together with a 3-adic normalization, and impose a square condition on the norm N_{L/ℚ}. An exhaustive scan on the unit lattice (m,n)∈ℤ² with |m|,|n|≤1000 yields an extremely rigid catalog: exactly 2 classes (all) modulo 𝒪_L^×=μ₆ and 1 fine class; if one instead identifies only modulo {±1}, one obtains 6 classes (all) and 2 fine classes. The results indicate that cyclotomic nodal phenomena organize by underlying arithmetic structure (subfields, unit rank, and trace–norm mechanisms) rather than by the cyclotomic level itself, and suggest a general framework connecting nodal rigidity with S-unit equations.