E = mc² from Gravity

Abstract In E = mc² (Einstein 1905) the constant c² descends from spacetime relativity, which rests on the principle of the invariance of the speed of light: the equation is about energy, but its constant is about light. This paper obtains the same number without any reference to light. Dividing the measured fractional rate difference of two clocks at different heights in a gravitational field (Δν/ν) by the mechanically computed change in field energy level between them (ΔΦ) yields a ratio K ≡ ΔΦ ÷ (Δν/ν) = 8.98755 × 10¹⁶ J/kg, constant over six orders of magnitude in height and more than five in field level, from a 22-metre tower to a 20,000-km satellite orbit. K is an energy constant (J/kg) — the energy content of one kilogram of matter — not a speed and not the square of a speed. From this measured K, using only Newtonian mechanics, conservation of energy and the one-mass definition, a conservation cycle gives Δm = ΔE/K, hence E = mK — with no Lorentz transformation, no appeal to the speed of light and no spacetime quantity anywhere. The identity c² = K is a result of this route, not its starting point. On the same foundation two rules (inertia = E/K; dE = v·dp) regenerate seven results of special relativity: γ, momentum, E² = E₀² + p²K, the speed limit, v² = K for rest-energy-free packets, muon lifetime and the full Doppler formula. Dividing the clock-world K by the nuclear-world E/Δm (Rainville et al. 2005) gives the dimensionless number 1 (± 2.5 × 10⁻⁵): two independent worlds declare one exchange rate. The claim is limited and explicit. This is not a new number, not a refutation of relativity and not a unification: it is a foundation for E = mc² that rests on measurement and involves neither the speed of light nor spacetime, and, as its consequence, the coordination of five domains — gravity, electromagnetic oscillation, matter-energy, nuclear binding and motion — under one measured yardstick. The paper's limits — field-generation equations, strong fields — are recorded without concealment in the final section. Keywords: field-energy constant K; gravitational redshift; mass–energy equivalence; conservation cycle; energy account; derivation without light or spacetime; coordination.