K_def
Recognition Science defines the dimensionless bridge ratio K to normalize constants on the phi ladder. This lemma records that K equals the positive square root of the golden ratio phi. Researchers deriving RS-native values for G or the fine-structure constant would reference the equality when switching between phi-powers and explicit radicals. The justification is a direct reflexivity step on the noncomputable definition.
claimThe dimensionless bridge ratio satisfies $K = varphi^{1/2}$, where $varphi$ denotes the golden ratio.
background
The module sets the fundamental RS time quantum to one tick. The golden ratio arises as the self-similar fixed point forced by the T5-T6 steps of the unified forcing chain. K is introduced explicitly as the square root of phi to provide a dimensionless scaling factor for later constant expressions such as G = phi^5 / pi.
proof idea
The proof is a one-line wrapper that applies reflexivity to the noncomputable definition of K given in the same module.
why it matters in Recognition Science
The declaration fixes the bridge ratio used in all subsequent constant derivations inside the Recognition framework. It supports the phi-ladder mass formula and the eight-tick octave structure. No downstream theorems are recorded in the used-by graph, leaving open how K will be invoked in curvature or defect-distance calculations.
scope and limits
- Does not prove any inequality or dynamical property of K.
- Does not relate K to the curvature functional defined in LambdaRecDerivation.
- Does not derive K from the Recognition Composition Law or the forcing chain.
- Does not address the alpha inverse band or mass-rung assignments.
formal statement (Lean)
492@[simp] lemma K_def : K = phi ^ (1/2 : ℝ) := rfl
proof body
493