DN-Orthogonal Freedom in the Canonical Seesaw: Flavor Invariants and Physical Non-Equivalence of F-Classes

We study basis-independent structures in the Type-I seesaw mechanism for light Majorana neutrinos, assuming the canonical scenario with three heavy right-handed (sterile) neutrinos. Let mν denote the 3×3 mass matrix of light neutrinos, obtained at tree level from heavy Majorana singlets with a diagonal mass matrix DN=diag(M1,M2,M3) and a Dirac matrix mD. We show that all right actions F on the seesaw matrix that leave mν unchanged form the group G=DN1/2O(3,C)DN−1/2. While oscillation data determine the PMNS matrix UPMNS and the mass-squared splittings, they do not fix the F-class within G. We classify basis-invariant quantities into those that are class-blind (e.g., det η) and class-sensitive (e.g., Trη, Trη2, an alignment measure, and CP-odd traces relevant to leptogenesis), where η denotes the non-unitarity matrix of the light sector. We provide explicit formulas and both high-scale and GeV-scale benchmark examples that illustrate these invariant fingerprints and their scaling with DN. This converts the degeneracy at fixed mν into measurable, basis-invariant fingerprints.