hbar_eq_eight_div_kappa
plain-language theorem explainer
The theorem states that the reduced Planck constant equals eight divided by the Einstein gravitational coupling in Recognition Science units. Unification researchers would cite it as the explicit form of the octave duality between quantum action and gravitational strength. The proof is a one-line algebraic rearrangement that invokes the positivity of kappa and the already-proved product identity kappa times hbar equals eight.
Claim. $ℏ = 8 / κ$ where $ℏ$ is the reduced Planck constant and $κ$ is the Einstein gravitational coupling, both in RS-native units with $c=1$.
background
The Quantum-Gravity Octave Duality module centers on the identity $κ · ℏ = 8$, forced by the single J-cost functional. J-cost is the arithmetic-geometric mean gap: for $x > 0$, Jcost $x = (x-1)^2/(2x)$, which is exactly AM($x$, $x^{-1}$) minus GM($x$, $x^{-1}$) and yields J ≥ 0 with equality only at $x=1$. Upstream constants fix $ℏ = φ^{-5}$ and $κ = 8φ^5$ via the forcing chain T5–T8 and the Recognition Composition Law, so their product is exactly the octave factor 8.
proof idea
Term-mode proof: rewrite the target via eq_div_iff using the positivity lemma kappa_einstein_pos, then discharge the resulting product equation by linarith on the upstream lemma kappa_hbar_octave.
why it matters
This declaration supplies the explicit division form of QG-001, the quantum-gravity octave duality that locks quantum action to gravitational coupling by the eight-tick cycle (T7). It instantiates the central theorem of the module and supports downstream relations such as G · ℏ = 1/π and the Planck area equaling 1/π. The result is fully proved with zero sorry and closes the duality statement inside the J-cost framework.
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