G_RS
G_RS supplies the RS-native gravitational constant as phi to the fifth power over pi. Researchers deriving SM constants from the J-cost equation and coherence uniqueness would cite it to close the Einstein relation and PlanckConstantCert. The definition is a direct substitution once the upstream coherenceExponent is fixed at 5.
claimIn RS-native units the gravitational constant is given by $G = phi^5 / pi$, where $phi$ is the golden ratio and the exponent 5 is the coherence exponent fixed by self-similarity and dimensional constraints at $D=3$.
background
Recognition Science pins constants via the forcing chain T0-T8 and the Recognition Composition Law. The coherenceExponent definition (upstream from CoherenceExponentUniqueness) sets k=5, which yields hbar = phi^{-5} and the present G form. MODULE_DOC states that k=5 follows from both the Fibonacci route (2^D - D = 5) and the integration route (D + 2 = 5) at D=3. Upstream Constants.G gives the general expression lambda_rec^2 c^3 / (pi hbar) that reduces to this algebraic form in RS units with c=1.
proof idea
Direct definition that substitutes the fixed coherenceExponent value 5 into phi^k / Real.pi. It applies the upstream coherenceExponent def from CoherenceExponentUniqueness and the phi constant from the foundation; no tactics are required beyond the expression itself.
why it matters in Recognition Science
This definition supplies the RS form of G that feeds einstein_relation (kappa_RS = 8 pi G_RS) and the PlanckConstantCert structure. It completes the algebraic certification of SM constants once k=5 is established in CoherenceExponentUniqueness. In the framework it realizes the T8 outcome D=3 together with the RCL-derived scaling that produces G = phi^5 / pi.
scope and limits
- Does not derive the numerical SI value of G from CODATA data.
- Does not prove positivity of G_RS (handled by separate theorem G_RS_pos).
- Does not perform unit conversion from RS-native to laboratory units.
- Does not incorporate higher-order quantum or curvature corrections.
Lean usage
example : kappa_RS = 8 * Real.pi * G_RS := by unfold kappa_RS G_RS; field_simp [Real.pi_ne_zero]formal statement (Lean)
39noncomputable def G_RS : ℝ := phi ^ coherenceExponent / Real.pi
proof body
Definition body.
40
41/-- G > 0. -/