G_pos
Positivity of the RS-native gravitational constant follows from its definition as φ^5 where φ is the golden ratio. Researchers deriving constants from the Recognition Science foundation cite this when confirming signs for G in the constant set. The proof is a one-line term-mode reduction that unfolds the definition and applies the power positivity lemma.
claim$G_{rs} > 0$ where $G_{rs} := φ^5$ and $φ = (1 + √5)/2$ is the golden ratio.
background
The ConstantDerivations module derives physical constants from the RS foundation using the composition law and J-cost function. G_rs is defined as φ^5 in RS-native units and emerges as the curvature extremum in recognition geometry, with the derivation involving the holographic bound and ledger capacity at mass scale M_0 = 1/φ^5. This builds on upstream results such as the G definition in Constants.GravitationalConstant (G_rs = φ^5 / π) and positivity lemmas for c and ħ.
proof idea
The term-mode proof unfolds G_rs to expose φ^5 and applies the lemma pow_pos to phi_pos, yielding the strict inequality directly.
why it matters in Recognition Science
This result supports parent theorems including lambda_rec_pos and the Physical structure in Bridge.DataCore, which invoke G_pos under physical assumptions. It fills the G > 0 step in the constant derivation chain from the RS foundation (T5 J-uniqueness through T8 D = 3), ensuring positivity on the phi-ladder scaling for G = φ^5 / π in native units.
scope and limits
- Does not relate G_rs to the CODATA numerical value of G.
- Does not derive the expression for G from the Recognition Composition Law.
- Does not include the π factor present in other G definitions.
- Does not address unit conversions or higher-order corrections.
Lean usage
example : G_rs > 0 := G_posformal statement (Lean)
152theorem G_pos : G_rs > 0 := by
proof body
Term-mode proof.
153 unfold G_rs
154 exact pow_pos phi_pos 5
155
156/-- G is algebraic in φ. -/