hbar_kappa_octave
plain-language theorem explainer
The product of the reduced Planck constant and Einstein gravitational coupling equals exactly 8 in Recognition Science units. Unification researchers would cite this to establish the forced octave duality between quantum mechanics and gravity. The proof is a one-line wrapper that commutes the factors and invokes the companion theorem establishing the same equality in reverse order.
Claim. $ℏ κ = 8$ in RS-native units, where $ℏ = φ^{-5}$ is the reduced Planck constant and $κ = 8φ^5$ is the Einstein coupling constant.
background
Recognition Science derives all constants from the single J-cost functional, which equals the AM-GM gap of the pair {x, x^{-1}}. The module proves the central quantum-gravity octave duality κ ħ = 8 as the first formal lock between QM and GR. Upstream, Constants defines ħ = φ^{-5} and κ = 8φ^5, while octave is the 8-tick period. The MusicalScale octave supplies the ratio 2 that closes the eight-tick cycle.
proof idea
This is a one-line wrapper proof. It rewrites the product via commutativity of multiplication on reals and then applies the companion theorem kappa_hbar_octave.
why it matters
This theorem supplies the symmetric form of QG-001 inside the module that proves κ ħ = 8. It completes the octave duality forced by the eight-tick cycle (T7) and the phi-fifth powers for ħ and κ. The result feeds the downstream claims on Planck area equaling 1/π and the Fibonacci mass ladder. It closes the first formal link between the J-cost functional and gravitational coupling.
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