kappa_hbar_octave
plain-language theorem explainer
The declaration proves that the Einstein gravitational coupling κ and reduced Planck constant ℏ multiply to exactly 8 in RS-native units. Researchers unifying quantum mechanics with gravity via Recognition Science would reference this as QG-001. Substitution of the explicit φ-power formulas for κ and ℏ reduces the product to 8 via elementary exponent arithmetic.
Claim. In RS-native units the Einstein gravitational coupling κ and the reduced Planck constant ℏ satisfy κ ℏ = 8.
background
The module centers on the relation between gravitational and quantum scales forced by the J-cost functional. Upstream lemmas establish ℏ = φ^{-5} and κ = 8 φ^5, resting on the definition of J-cost as the AM-GM gap J(x) = (x-1)^2 / (2x) for x > 0. This gap enforces the self-similar fixed point φ and the eight-tick octave structure that survives after φ^5 cancellation.
proof idea
The tactic proof first rewrites the goal using kappa_einstein_eq and hbar_eq_phi_inv_fifth. It then applies a calc block that factors the product as 8 · φ^5 · φ^{-5}, invokes Real.rpow_add to combine exponents, reduces the exponent to zero via norm_num, and simplifies to 8.
why it matters
This is the core statement of QG-001, directly feeding hbar_kappa_octave, hbar_eq_eight_div_kappa, kappa_eq_eight_div_hbar, kappa_per_octave_eq_inv_hbar, octave_duality_witness, and qg_octave_cert. It realizes the eight-tick octave by showing that the φ^5 duality between κ and ℏ is exactly offset by the factor 8, closing one link from J-uniqueness to the quantum-gravity scale relation.
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papers checked against this theorem (showing 1 of 1)
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A 38 TeV gauge operator could seed the universe's matter excess
"d_e/e = k_UV g² m_e/(96π² Λ²) = 4.1×10⁻³⁰ cm (k_UV/1.35)(38 TeV/Λ)²"