The Spectral Scale Ratio : Seesaw Invariance, Structural Obstructions, and an Exploratory Remark

This paper contains one proven result (the seesaw invariance of det(DF), Theorem 1), one structuralanalysis (five independent obstructions to deriving the 1/3 exponent), and one exploratory remark (theproximity of L−1/3 to the cosmological scale ratio). These are stated at distinct confidence levelsthroughout. Abstract The determinant of the finite Dirac operator DF in the Connes–Chamseddine noncommutativegeometry formulation of the Standard Model is exactly independent of the right-handedneutrino Majorana mass MR, by the seesaw block-determinant identity. The full multiplicativespectral content of the Standard Model is therefore determined entirely by Dirac Yukawacouplings. We compute the dimensionless spectral invariant L = det(DF)/v24 using full colormultiplicity and Standard Model renormalization group evolution. The one-loop evaluationgives L−1/3 ≈ 1043.6; the two-loop evaluation gives L−1/3 ≈ 1046.3. The observed ratio betweenthe cosmological radius and the electroweak length scale is approximately 1044, whichfalls within this range. The 2.7-decade spread between loop orders reflects the extreme sensitivityof a 48-power Yukawa product to perturbative corrections and constitutes the dominantsystematic uncertainty. We identify five independent structural obstructions to deriving the exponent1/3 from known spectral geometry. The obstruction is precise: the spectral action usesadditive eigenvalue counting (traces), while L is a multiplicative eigenvalue measure (determinant).The observation is that the spectral invariant falls in the vicinity of the cosmologicalscale ratio with no fitted parameters; it is not a precise numerical match, and its significancecannot currently be assessed. A companion paper [6] proposes a conjectural principle thatwould bridge this gap.