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 structural
analysis (five independent obstructions to deriving the 1/3 exponent), and one exploratory remark (the
proximity of L−1/3 to the cosmological scale ratio). These are stated at distinct confidence levels
throughout.

 
Abstract

The determinant of the finite Dirac operator DF in the Connes–Chamseddine noncommutative
geometry formulation of the Standard Model is exactly independent of the right-handed
neutrino Majorana mass MR, by the seesaw block-determinant identity. The full multiplicative
spectral content of the Standard Model is therefore determined entirely by Dirac Yukawa
couplings. We compute the dimensionless spectral invariant L = det(DF)/v24 using full color
multiplicity and Standard Model renormalization group evolution. The one-loop evaluation
gives L−1/3 ≈ 1043.6; the two-loop evaluation gives L−1/3 ≈ 1046.3. The observed ratio between
the cosmological radius and the electroweak length scale is approximately 1044, which
falls within this range. The 2.7-decade spread between loop orders reflects the extreme sensitivity
of a 48-power Yukawa product to perturbative corrections and constitutes the dominant
systematic uncertainty. We identify five independent structural obstructions to deriving the exponent
1/3 from known spectral geometry. The obstruction is precise: the spectral action uses
additive eigenvalue counting (traces), while L is a multiplicative eigenvalue measure (determinant).
The observation is that the spectral invariant falls in the vicinity of the cosmological
scale ratio with no fitted parameters; it is not a precise numerical match, and its significance
cannot currently be assessed. A companion paper [6] proposes a conjectural principle that
would bridge this gap.