G_
plain-language theorem explainer
The product of the RS gravitational constant and reduced Planck constant equals unity in native units, confirming that gravitational and quantum scales cancel to a dimensionless quantity. Researchers deriving constants from the forcing chain and composition law would cite this to verify scaling consistency. The proof reduces the product via unfolding definitions to φ^5 times φ^{-5} and applies algebraic cancellation using power and inverse rules.
Claim. In Recognition Science native units the gravitational constant $G$ and reduced Planck constant $ℏ$ satisfy $G ℏ = 1$.
background
Recognition Science derives constants from the composition law and J-cost function J(x) = ½(x + x^{-1}) - 1, with φ the self-similar fixed point and the eight-tick period fixing τ₀. G_rs is defined as φ^5, the curvature extremum scaled by fundamental mass M₀ = 1/φ^5. ℏ_rs is constructed as E_coh · τ₀, yielding φ^{-5} in these units. The module shows constants as algebraic ratios of RS-native quantities rather than free parameters, with c = 1 fixing the causal bound.
proof idea
Term-mode proof unfolds G_rs, ℏ_rs, E_coh and τ₀ to expose φ^5 · φ^{-5}. It applies simp to reduce the product, rewrites with zpow_neg, then invokes mul_inv_cancel₀ together with phi_pos.ne' to obtain exact cancellation to 1.
why it matters
This theorem closes the scaling relation for G and ℏ inside the constant-derivation chain, confirming the RS version of ℏG/c³ is dimensionless when c = 1. It aligns with the T5 J-uniqueness and T6 φ fixed-point steps of the unified forcing chain, supplying an algebraic identity rather than an empirical input. The result supports later derivations such as the geometric seed for α^{-1} ≈ 137.
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