Paper L: Unified Action Principle and Z5 Resonance in Four-Sector Cosmology - Internal Consistency, Slow-Variation Dynamics, and Falsifiable Predictions for Euclid

                                                        Abstract

We present Paper L of the Four-Sector Cosmology (FSC) series, establishing a
unified action principle that integrates the complex scalar field dynamics of Paper F,
the Sector-IV decay mechanism of Paper D, and a novel coupling term between the
field phase θ and the spacetime Ricci scalar Rscalar.
The central contribution is the identification of a two-layer Z5 symmetry: a
geometric Z5 acting on the sector index structure, and a field-theoretic Z5 acting as
a discrete symmetry of the scalar potential V (Φ). These two layers are connected
dynamically through Scoupling = ∫ d4x√−g ξ θRscalar, producing a slow-variation
region in which the phase θ adiabatically tracks the cosmological expansion.
We derive the slow-variation dynamics from first principles, obtaining the adi-
abatic solution φeq(a) = −ξRscalar(a)/(50μv3), which reproduces ∆θ = 0.001 rad
without fine-tuning. A non-trivial internal consistency check is provided by the in-
dependent derivation of δwθ
∣∣a=1 ≈+0.003, confirming the phenomenological result
wa ≈+0.003 of Paper F.
We identify an isomorphism between the cosmological constant problem and the
ξ naturalness problem (Proposition 6.1), suggesting a common dynamical resolution
rooted in Sector-IV decay.
The paper makes a falsifiable prediction:
wFSCa = +0.003 ±0.001,
distinguishable from ΛCDM at the 6.2σ level relative to DESI 2025 (DESI Collabo-
ration, 2025), and testable by Euclid (Laureijs et al., 2011) in the period 2027–2030.
Open problems are identified and assigned to Paper M.
Keywords: Four-Sector Cosmology; complex time; Z5 symmetry; dark energy;
action principle; slow-roll dynamics; cosmological constant; Euclid; DESI