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

Abstract

This article formulates a speculative but falsifiable cosmological framework in which the observable spatial universe is modeled as a closed three-sphere and the dimensional hyperspherical radius is postulated to increase linearly with cosmic proper time, R(t) = V_exp · t. 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. The ansatz implies a coasting background, H(t) = 1/t and q(t) = 0, and yields a closed-form luminosity-distance relation for radial null geodesics. In angular coordinates, the combination e^(βχ) sin(χ), where β = V_exp / c, has a vanishing cubic coefficient at β = 1/√3. This cancellation is 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 Ω_k = −β⁻² creates a decisive consistency condition that must be resolved by a derived effective metric or by a revised parameter identification. Version 6 retains the discrete-update, phase-diffusion, irrotational-shift, and disformal constructions and subjects the two-metric sector to the multimessenger constraint from GW170817/GRB 170817A. For a homogeneous scalar ϕ = ϕ₀ (t₀ / t)³, the choice D(ϕ) = D₀ (ϕ / ϕ₀)^(−8/3) produces a constant effective lapse N² = 1 − γ² when D₀ = γ² c² t₀² / (9 ϕ₀²). Universal matter coupling to the disformal metric gives τ = N t, β_eff = β / N, and Ω̃_k = −N² / β². If tensor modes instead propagate on the ontic metric while photons follow the matter metric, then c_g / c = 1/N. The GW170817 propagation bound forces N to differ from unity by no more than approximately 7 × 10⁻¹⁶; at β = 1/√3 this returns Ω̃_k ≈ −3 and excludes split-cone curvature screening. The only retained route is a common-cone physical-frame completion whose quadratic tensor action satisfies ℱ_T = 𝒢_T > 0, or equivalently α_T = 0. A minimally coupled Einstein–Hilbert action written in the physical metric realizes this equality, but it must still generate the assumed background and pass scalar-sector, stability, amplitude-propagation, and cosmological tests. The profile D(ϕ) ∝ ϕ^(−8/3) and the Yoneda lemma do not establish luminal propagation by themselves. Exact N = 0 remains noninvertible, and a time-dependent lapse N ∝ t^(−δ) ceases to be coasting in matter-frame proper time. The disformal sector is therefore sharpened into a falsifiable branch analysis rather than presented as a completed solution of the curvature crisis.