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 spectralgeometry, 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 distinctconfidence 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 geometryformulation of the Standard Model, is exactly independent of the right-handed neutrinoMajorana mass MR. With framework-derived neutrino Dirac Yukawas (from MR = MG), theone-loop evaluation gives L−1/3 ≈ 1043.6; the two-loop evaluation gives L−1/3 ≈ 1046.3. Theobserved 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 producingthe determinant as a dynamical object coupled to spatial volume. The obstructionsshare a common root: the spectral action uses additive eigenvalue counting (traces), while Lis a multiplicative eigenvalue measure (determinant).We propose the Multiplicative Spectral Measure Principle (MSMP): the normalization constantof the noncommutative integral is determined by the inverse of the internal multiplicativespectral measure, L−1, so that spatial volume satisfies Vol(Σ) ∝ L−1. The exponent 1/3then 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 coefficientc0 = 2π2 of the unit 3-sphere—a geometric quantity determined by the topology, not fittedto 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. The2.7-decade spread between one-loop and two-loop evaluations of L constitutes the dominantsystematic uncertainty and limits the precision of this consistency check.