kappa_per_octave_eq_inv_hbar
plain-language theorem explainer
The theorem establishes that the Einstein gravitational coupling κ divided by the octave factor 8 equals the reciprocal of the reduced Planck constant ℏ. Researchers working on quantum-gravity unification via the J-cost functional would cite it to confirm the locking of gravitational and quantum scales. The proof is a short algebraic reduction that rewrites the division equality and applies linear arithmetic to the product identity κ ℏ = 8.
Claim. $k / 8 = 1 / h$ where $k$ is the Einstein gravitational coupling and $h$ the reduced Planck constant in Recognition Science units.
background
The Quantum Gravity Octave Duality module proves the central identity κ · ℏ = 8 as QG-001. This follows from the definitions κ = 8 φ^5 and ℏ = φ^{-5} in RS-native units, where φ is the self-similar fixed point of the J-cost functional J(x) = (x + x^{-1})/2 - 1. The module documentation states that each of the eight ticks contributes φ^5 curvature per unit energy.
proof idea
The term proof rewrites the division equality into an equivalent cross-multiplication statement via the division lemma, using the facts that 8 is nonzero and hbar is positive. It then closes the goal by linear arithmetic on the sibling product identity κ ℏ = 8.
why it matters
This isolates the per-octave form of the QG-001 duality and feeds into the QG Octave Certificate that packages all duality results. It directly instantiates the eight-tick octave (T7) from the forcing chain, showing how J-uniqueness forces the gravitational coupling and quantum of action to be reciprocal up to the octave factor.
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