Galois Separation of Perron–Frobenius Frequencies for Arnoux–Rauzy Substitutions

We study the one-dimensional Arnoux–Rauzy substitution familyσ_{a,b}: 1 ↦ 1^b 2, 2 ↦ 1^a 3, 3 ↦ 1 with b ≥ a ≥ 1, and thePerron–Frobenius frequency vector f(a,b) associated with itsincidence matrix. We show that the field ℚ(λ_{a,b}) generatedby the dominant root of x³ − bx² − ax − 1 does not classify thefamily: in the range N ≤ 30 there are 40 pairs with coincidentfield discriminant, grouped into 21 classes, of which 33 haveisomorphic cubic fields. We develop a Galois method that comparesfrequency vectors through their Galois conjugates. The methodyields an unconditional proof of separation for 16 of the 40pairs: the 7 pairs with distinct cubic fields and the 9 pairswith isomorphic cubic fields but distinct real subfields. For theremaining 24 pairs we provide bounded exhaustive computationalevidence that the frequency vectors are separated (with explicitbound computed per pair). Under the expected identification ofthe canonical trace with the Perron–Frobenius frequency pairing,this would imply trace separation; the unconditional statementover GL(3,Z) remains open.