kappa_eq_eight_div_hbar
plain-language theorem explainer
The theorem establishes the Einstein gravitational coupling as eight divided by the reduced Planck constant in Recognition Science units. Researchers unifying quantum mechanics and gravity cite it to confirm the octave duality between the two. The proof is a direct algebraic reduction from the product identity and the positivity of the Planck constant.
Claim. In Recognition Science native units the Einstein gravitational coupling satisfies $κ = 8 / ℏ$, with $ℏ = φ^{-5}$.
background
Recognition Science fixes all constants via the J-cost, defined as the AM-GM gap J(x) = (x + x^{-1})/2 - 1 for x > 0. The module proves that the gravitational coupling κ and the action quantum ℏ are dual under fifth powers of the golden ratio φ, yielding their product equal to the octave number 8. Upstream definitions supply ℏ as φ^{-5} and κ as 8φ^5 in RS units, with the product relation providing the key input.
proof idea
The proof rewrites the target equality using the division equivalence under positivity of ℏ, then applies linear arithmetic to the established product identity κ · ℏ = 8.
why it matters
This declaration supplies the division form of the central quantum-gravity octave duality, completing QG-001 in the module. It rests on the eight-tick octave from the forcing chain and supports derivations of the Planck area equal to 1/π and the Fibonacci structure of the mass ladder. The result is fully proved with no open questions addressed.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.