G_over_hbar_phi_tenth
plain-language theorem explainer
The ratio of Newton's gravitational constant to the reduced Planck constant equals phi to the tenth over pi in Recognition Science units. Unification researchers would cite this as the explicit ten-rung separation on the phi ladder between gravity and quantum action. The proof is a short algebraic reduction that substitutes the two constant identities and simplifies the resulting exponent via ring and real-power addition.
Claim. In Recognition Science units, Newton's gravitational constant divided by the reduced Planck constant satisfies $G / hbar = phi^{10} / pi$.
background
Recognition Science fixes all constants from the single J-cost functional equation. The module defines J-cost as the arithmetic-geometric mean gap: for x > 0, Jcost x equals (x minus 1) squared over 2x. This yields the phi-ladder constants via the forcing chain: hbar equals phi to the minus five from the coherence energy and tick definitions, while G equals phi to the fifth over pi from the recognition length and Gauss-Bonnet closure.
proof idea
The tactic proof clears the denominator with the division equivalence for positive hbar. It substitutes the identities hbar_eq_phi_inv_fifth and the companion G identity. A calc block then applies ring normalization, the real exponent addition rule, and numeric reduction to collapse the exponent 10 plus negative 5 back to 5, recovering the left-hand side.
why it matters
This identity supplies the explicit ratio needed for the QG Octave Certificate. It realizes the module's QG-002 relation (G times hbar equals one over pi) in dual form and connects directly to the eight-tick octave and the phi-fifth duality between kappa and hbar. The result closes one link in the unification forced by the J-cost equation and the T5-T8 chain.
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