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

Abstract

This note is a versioned corrigendum to The Hyperspherical Radius as Cosmic Time (Version 6). It records three changes that the reader of Version 6 must apply. First, Version 6 omitted the substantial prior literature on linearly expanding cosmologies; the relevant antecedents and the existing observational constraints on that class are supplied here, together with an explicit statement of where the present closed construction differs from them. Second, the defining ansatz R(t) = Vₑₓₚ t with β ≡ Vₑₓₚ/c = 1/√3 is a constant-expansion-rate model, and such models are already conclusively disfavoured by published joint analyses of Type Ia supernovae with baryon acoustic oscillations and with the acoustic scale; independent fits by the present author against current DESI distances reproduce that conclusion at higher significance. Third, and new to this note, the acoustic scale is computed within the model rather than imported from ΛCDM. Because the model's own sound horizon grows logarithmically, χₛ = ln(t*/tᵢ) at β = 1/√3, and because multi-traversal fixes |sin χ*| = 0.437 rather than allowing hyperbolic growth, the predicted acoustic multipole is ℓ_A = π|sin χ*|/χₛ. Reproducing ℓ_A ≃ 220 then requires the coasting description to switch on at tᵢ = 0.9938 t*, that is, within the final 0.6% of the time before recombination; a single e-fold of coasting history places the acoustic scale at ℓ_A ≃ 1.4, below the observable multipole range entirely. This failure is specific to the closed geometry: in an open coasting model the corresponding factor is sinh χ* = 545 rather than |sin χ*| = 0.437, a difference of 1246 in angular distance, which is why concordance results reported for open linear coasting do not transfer here. It is further noted that exact coasting has a constant comoving Hubble radius, so no mode ever crosses it, and no cosmological event horizon exists; the stochastic sector of Version 6 therefore cannot generate a primordial spectrum, and any amplitude derived from a Gibbons–Hawking temperature is withdrawn. The value β = 1/√3 is retired. The geometric and ontological construction, which requires only a global time function and a monotonic R(t), is retained with Ṙ no longer assumed constant.