phi5_is_both_quantum_and_gravitational
plain-language theorem explainer
The theorem establishes that φ^5 equals both 1/ℏ in the quantum sector and G π in the gravitational sector. Researchers in quantum gravity unification cite it to demonstrate that the fundamental couplings share a common origin in the recognition parameter φ without separate tuning. The proof is a term-mode split that rewrites the quantum half via the established identity ℏ = φ^{-5} and pairs it with the pre-proved gravitational equality.
Claim. In RS-native units, $1/ℏ = φ^5$ and $G π = φ^5$.
background
In the Recognition Science framework the reduced Planck constant satisfies ℏ = φ^{-5} by the identity proved in Constants (C-004.1). The gravitational constant is defined as G = λ_rec² c³ / (π ℏ) and reduces in native units (c = 1) to the relation G π = φ^5. The module QuantumGravityOctaveDuality places these scalings inside the octave duality κ ℏ = 8, where κ = 8 φ^5 is the Einstein coupling derived from the J-cost functional equation.
proof idea
The proof is a term-mode construction that splits the conjunction into two conjuncts. The left conjunct is discharged by rewriting with the lemma hbar_eq_phi_inv_fifth followed by power simplification. The right conjunct is supplied directly by the pre-proved equality G_pi_eq_phi5.
why it matters
This declaration supplies the explicit φ^5 identification that underpins the zero-free-parameter claim in the quantum-gravity octave duality. It directly supports the module's central result κ ℏ = 8 by exhibiting the dual scalings of the quantum and gravitational sectors. The double appearance of φ^5 formalizes that both arise from the same J-cost fixed point (T5) without independent constants.
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