G_
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.
claimIn 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 in Recognition Science
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.
scope and limits
- Does not supply conversion factors from native units to SI for G or ℏ.
- Does not include the π factor appearing in the full expression for G.
- Does not derive numerical values or corrections for α or other constants.
- Does not address time-dependent or higher-order dynamical effects.
formal statement (Lean)
161theorem G_ℏ_product : G_rs * ℏ_rs = 1 := by
proof body
Term-mode proof.
162 unfold G_rs ℏ_rs E_coh τ₀
163 simp only [mul_one]
164 -- φ^5 * φ^(-5) = 1
165 rw [zpow_neg, mul_inv_cancel₀]
166 exact pow_ne_zero 5 phi_pos.ne'
167
168/-! ## Fine Structure Constant: α -/
169
170/-- The geometric seed for α: 1/137.
171 This comes from the holographic area count. -/
172noncomputable def α_seed : ℝ := 1 / 137
173
174/-- The gap-45 correction factor.
175 This accounts for the integration gap D²(D+2) = 45 at D = 3. -/