Physical Constants as Geometric Snapshots in an Expanding 3-Sphere Universe

Abstract The foundational question addressed in this paper is whether the distinction between derivable and irreducible physical constants is a question of geometry. Working within a closed three-sphere (S³) of cosmic radius R, a characteristic curvature acceleration and expansion rate are derived from the manifold's topology: a* = c²/πR and H = πc/R. The framework has one degree of freedom, R, and three geometric postulates: G = c²/πR, H = πc/R, and R = ct. Given these, the seven SI base units reduce to two dimensional primitives (Lp, Tp) plus the vacuum impedance Z₀ as the one remaining irreducible electromagnetic input. Four experimentally independent routes return R = 4.286×10²⁶ m within the 22 ppm uncertainty of G. The ratio G/H = c/π² is confirmed at 0.6% with current data. The predicted secular drift Ġ/G = −H exceeds the lunar laser ranging bound by three orders of magnitude, with resolution depending on whether atomic frequency standards co-scale with R. The global Hubble constant is predicted as H = 67.80 km/s/Mpc at zero free parameters, and the Hubble tension is recast as a local density readout. The ratios are eternal; the snapshots are ours. Version history This is v2. Changes from v1 (Geometric Origin of Physical Constants in an Expanding 3-Sphere Universe, Zenodo, 25 May 2026): Title and framing. The title is revised to Physical Constants as Geometric Snapshots in an Expanding 3-Sphere Universe, which more accurately reflects the paper's scope: G and H are derived from S³ geometry; h and e follow as algebraic consequences; Z₀ and α remain irreducible primitives. The abstract is completely rewritten to name the three postulates explicitly, introduce the epistemic taxonomy, acknowledge the formulaic dependence of the h and e routes through Lp, state the LLR tension quantitatively, and confirm the G/H null test against existing data. New: geometric derivation of [kg] = L². A new subsection establishes that the unit assignment [kg] = L² is not a convention but a constraint — the unique choice consistent with three independent requirements: E = mc², dimensional balance of the Einstein field equations under G = c²/πR, and the identification of mass as a solid-angle region R²Ω on the S³ manifold. New: quantitative G/H null test. The v1 claim "agreement to current measurement precision" is replaced with an explicit numerical comparison: G/H|obs = 3.057×10⁷ m/s vs. c/π² = 3.038×10⁷ m/s, a residual of 0.6% within the 1.3% uncertainty of the Planck 2018 H determination, which dominates the error budget by a factor of 27 over G. New: circularity acknowledged explicitly. A dedicated subsection addresses the circularity question for the h and e routes: both formulas share Lp = √(ℏG/c³) with the G route. The distinction between formula dependence and experimental independence is made precise, and the consistency is characterised as non-trivial rather than guaranteed. New: quantitative Hubble tension analysis. The local expansion discussion is extended to a full quantitative treatment: H_local = f·H_global is derived, f = 1.077 is computed from the SH0ES value, the implied local overdensity δ = 7.7% and local radius R_local = 3.979×10²⁶ m are stated, and a G-bias correction is derived showing that standard virial, hydrostatic, and peculiar-velocity mass estimates overestimate δ by factor f. A summary table gives H_local as a function of f bracketed by κ = 1 + π⁻². The γ mechanism is explicitly distinguished from standard GR gravitational time dilation (which would require |Φ|/c² ~ 0.07, far exceeding local potentials of ~10⁻⁶). New: BBN and CMB early-universe discussion. A new paragraph addresses the implications of G ∝ R⁻¹ for big bang nucleosynthesis and CMB acoustic peaks, noting that standard constraints on Ġ/G from these epochs are derived within ΛCDM and are not directly applicable to a framework in which both G and particle masses co-scale with R. A dedicated analysis of dimensionless nuclear and photon–baryon transport ratios under WT scaling is identified as necessary future work. New: Bekenstein framework comparison. A new section places WT within the standard covariant framework for varying constants. The key structural distinction is stated: in WT, the variation of G arises from the evolution of the background geometric parameter R(t), not from a dynamical scalar field with matter coupling. Consequently there is no fifth force of the Bekenstein type and the weak equivalence principle is preserved. The LLR bound and the co-scaling resolution are treated in full. Conclusion expanded. The conclusion now lists five falsifiable predictions (up from four in v1), adding the Hubble tension as a density prediction with the κ upper bracket H_κ = 74.67 km/s/Mpc. The closing paragraph threads GR and Conformal Cyclic Cosmology explicitly, with mass-as-geometry and continuous conformal rescaling as the two connecting claims.