G_algebraic_in_
plain-language theorem explainer
The RS-native gravitational constant equals the fifth power of the golden ratio. Researchers deriving physical constants from the Recognition Science foundation cite this to confirm G is algebraically fixed by phi rather than a free parameter. The proof is a term-mode witness that supplies the exponent 5 and reduces directly via the definition of G_rs.
Claim. There exists an integer $n$ such that the RS-native gravitational constant $G_{rs}$ satisfies $G_{rs} = phi^n$, where $phi$ is the golden ratio.
background
The Constant Derivations module obtains c, hbar, G, and alpha as ratios of RS quantities rather than independent inputs. G_rs is introduced as the curvature extremum in recognition geometry, with the fundamental mass scale fixed at 1/phi^5 so that G_rs equals phi_val raised to the fifth power. This step follows the forcing chain from the Composition Law through J-uniqueness and the eight-tick period that fixes D=3.
proof idea
The proof is a term-mode construction that supplies the witness 5 for the existential quantifier and applies simplification to the definition of G_rs to confirm the equality.
why it matters
This result shows G is algebraic in phi, completing one link in the module's derivation chain from the RS foundation to physical constants. It aligns with the explicit statement that constants are ratios of RS-native quantities, all algebraic in phi, after the eight-tick octave and D=3 are fixed. No immediate downstream theorems are listed, but the declaration supports the broader claim that the framework eliminates free parameters for G.
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