The Hyperspherical Radius as Cosmic Time — a Geometric Hypothesis for Cosmological Expansion and Quantum Nonlocality

Abstract

In this study, a speculative but falsifiable cosmological framework is formulated in which the observable spatial universe is modeled as a closed three-sphere, with the dimensional hyperspherical radius postulated to increase linearly with cosmic proper time, R(t) = Vₑₓₚt. To avoid the dimensional ambiguity present in conventional scale-factor notation, the normalized FLRW scale factor is defined separately as a(t) = R(t) / R₀ = t / t₀. A coasting background with H(t) = 1/t and q(t) = 0 is implied by this ansatz, yielding a closed-form luminosity-distance relation for radial null geodesics. In angular coordinates, the combination e^(βχ) sin χ, where β = Vₑₓₚ / c, exhibits a vanishing cubic coefficient at β = 1 / √3. This cancellation is identified as a local series property; it is not, by itself, equivalent to an observational demonstration of spatial flatness. Indeed, if R(t) is identified directly with the FLRW curvature radius, the standard relation Ωₖ = −β⁻² creates a decisive consistency condition that must be resolved either by a derived effective metric or by a revised parameter identification. In Version 4, the supplied Phase 8 material is integrated into the preceding six-phase formulation. Three additions are critically developed: a discrete ontic-update model for the future–present–past embedding picture; a finite-kernel line-of-sight phase-diffusion model connected conditionally to cosmic polarization rotation; and an irrotational ADM-shift sector for localized warp-like geometries. The discrete rule Rₙ = nℓ_q, tₙ = nτ_q implies Vₑₓₚ = ℓ_q / τ_q; choosing both Planck length and Planck time therefore fixes Vₑₓₚ = c rather than deriving an arbitrary expansion speed. The proposed phase-variance integral is recalculated using R(χ) = R₀ e^(−βχ), and a common photon phase is distinguished from the helicity-dependent interaction required to convert CMB E-modes into B-modes. The correlator ⟨BEϕ⟩ is identified as a conditional bispectrum null test, whereas anisotropic rotation generally produces nonzero higher-order mode coupling. Finally, a curl-free shift Bᵢ = Dᵢψ on S³ does not by itself establish positive energy, superluminal transport, or chronology protection: these properties depend strictly on the Hamiltonian constraint, the complete stress–energy tensor, and the global causal structure. Exact consequences, conditional calculations, conjectures, and explicit falsification requirements continue to be distinguished throughout the manuscript.