einstein_relation
plain-language theorem explainer
The Einstein relation in Recognition Science equates the curvature parameter κ_RS to eight pi times the gravitational constant G_RS in RS units. A physicist deriving Planck's constant from the phi-ladder would cite this identity to confirm consistency with the Einstein field equations. The proof is a term-mode unfolding of the two definitions followed by field simplification that cancels the explicit factors.
Claim. $κ_{RS} = 8π G_{RS}$, where $κ_{RS} := 8 φ^k$ and $G_{RS} := φ^k / π$ for the coherence exponent $k$.
background
The module Planck Constant from RS certifies the RS-native constants ℏ = φ^{-5}, G = φ^5/π and κ = 8φ^5 once the coherence exponent is fixed at 5. G_RS is the explicit definition φ^k / π and κ_RS is the explicit definition 8 φ^k, both noncomputable reals. These follow algebraically from the uniqueness of k=5 proved in CoherenceExponentUniqueness.lean and the forcing chain that sets D=3.
proof idea
The term proof unfolds kappa_RS and G_RS to obtain the expressions 8 * phi ^ coherenceExponent and phi ^ coherenceExponent / Real.pi. It then applies field_simp with the hypothesis Real.pi_ne_zero to cancel the common factors and obtain the identity.
why it matters
This supplies the einstein field to the planckConstantCert bundle that packages the full certification of the RS constants. It closes the structural verification step required by the module after k=5 is pinned, linking the RS-derived G and κ directly to the classical Einstein relation κ = 8πG. The relation is consistent with the eight-tick octave and D=3 forced in the UnifiedForcingChain.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.