Spherical Light: Wavelength as Sphere Size, Redshift as Dilution, and the Universal Rule m × R = n × ħc

Every wave in nature is spherical, yet textbooks define wavelength as a distance along a line. This paper takes the spherical nature of waves seriously and follows it to its consequences. Wavelength becomes a sphere size (R = ħc/E), frequency becomes energy per unit area, and the speed of light becomes a sphere expansion rate. The proton is identified as a sphere that stopped expanding, with an internal standing wave whose mode count is (4/3) π — the volume of a unit sphere in D = 3. A single conservation law — OE + W = 1 — connects five scales: quarks (fm), the proton wall (sub-fm), multi-nucleon nuclei (fm), electron shells (pm), and cosmology (Gpc). The same transmission fraction p = e^ (−3/4) that determines proton confinement also yields the Hubble constant H₀ = 70. 05 km/s/Mpc and the matter fraction Ωₘ = 0. 2784, with zero fitted parameters across 41 orders of magnitude. The "equal budget per shell" rule (Area × ρ = constant inside the proton; Capacity × OE = constant in electron shells) operates on both sides of the proton wall, with the wall as the mirror point. A full electron shell absorbs 3/4 of the nuclear OE — the same fraction as the proton wall. Key predictions: • Proton mode count: mₚ × Rₚ / ħc = (4/3) π (0. 007% match) • Electron mode count: mₑ × a₀ / ħc = 1/α (exact) • Shell budget: Cap × OE = 6 (Zα) ² per shell (algebraically exact) • Most stable nucleus: A (Fe) = (mₚ/mₑ) × (4/3) π × α = 56. 13 (0. 2% match) • Stability balance: ΔA + 6 (Zα) ² = 1 at iron (0. 4% from unity) • Relativistic transition: OE exceeds 1 at Z ≈ 79 (matches Dirac equation threshold) Nine results derived from first principles with zero fitted parameters. Seven non-trivial numerical predictions match observation. Five open problems identified. Part of the Clausius-Mossotti / Speed Gap Framework (Singh 2026a–2026v). All calculations reproducible via GitHub: https: //github. com/singhmandy25-gif/speed-gap-framework/tree/main/cm-spherical-light