The Multiplicative Spectral Measure Principle: A Conjectural Connection Between Internal Spectral Data and Cosmological Volume

This paper contains one negative result (the 1/3 exponent cannot be derived from known spectral
geometry, Section 2), one structural observation (the additive–multiplicative tension in NCG, Section 3),
and one conjecture (the Multiplicative Spectral Measure Principle, Section 5). These are stated at distinct
confidence levels throughout.

Abstract

A companion paper [1] establishes that the dimensionless spectral invariantL =det(DF)/v24,
constructed from the finite Dirac operator of the Connes–Chamseddine noncommutative geometry
formulation of the Standard Model, is exactly independent of the right-handed neutrino
Majorana mass MR. With framework-derived neutrino Dirac Yukawas (from MR = MG), the
one-loop evaluation gives L−1/3 ≈ 1043.6; the two-loop evaluation gives L−1/3 ≈ 1046.3. The
observed cosmological-to-electroweak scale ratio is approximately 1044.
This paper investigates whether the exponent 1/3 can be derived from known spectral geometry.
The answer is negative: several structural obstructions prevent the standard Connes–
Chamseddine spectral action, 3D-first spectral frameworks, and Weyl asymptotics from producing
the determinant as a dynamical object coupled to spatial volume. The obstructions
share a common root: the spectral action uses additive eigenvalue counting (traces), while L
is a multiplicative eigenvalue measure (determinant).
We propose the Multiplicative Spectral Measure Principle (MSMP): the normalization constant
of the noncommutative integral is determined by the inverse of the internal multiplicative
spectral measure, L−1, so that spatial volume satisfies Vol(Σ) ∝ L−1. The exponent 1/3
then follows from d = 3 observed spatial dimensions: for a round S3 of radius R, R ∝ L−1/3.
MSMP is not derived from the spectral action; it is an additional normalization postulate.
MSMP requires compact spatial topology.
Under MSMP with S3 topology, the proportionality constant is the volume coefficient
c0 = 2π2 of the unit 3-sphere—a geometric quantity determined by the topology, not fitted
to observation. The effective cosmological constant is then Λeff = 3/R2, a consistency relation
(not an independent prediction) that gives Λeff ∼ 6×10−84 GeV2 at one-loop evaluation. The
2.7-decade spread between one-loop and two-loop evaluations of L constitutes the dominant
systematic uncertainty and limits the precision of this consistency check.