phi
The definition supplies the golden ratio φ as the explicit real (1 + sqrt(5))/2. Workers on the Recognition Science forcing chain cite this value for all self-similar scaling calculations. The declaration is a bare assignment with no computational steps or lemmas.
claimThe golden ratio is defined by the equation $φ = (1 + √5)/2$.
background
The Constants module introduces RS-native constants with the fundamental time quantum τ₀ = 1 tick. Phi is the golden ratio, which the framework forces as the unique positive real satisfying the self-similarity equation x = 1 + 1/x. No upstream results are required for this definition.
proof idea
The proof body is empty because the declaration is a direct definition. The right-hand side is the standard closed-form expression for the golden ratio.
why it matters in Recognition Science
This constant is the self-similar fixed point at T6 in the unified forcing chain. It determines the eight-tick octave period and enters the mass formula as the base of the phi-ladder. The value also fixes G = phi^5 / pi and hbar = phi^{-5} in RS units.
scope and limits
- Does not prove irrationality or other number-theoretic properties of phi.
- Does not derive phi from the Recognition Composition Law.
- Does not specify units or scaling relative to τ₀.
formal statement (Lean)
17noncomputable def phi : ℝ := (1 + Real.sqrt 5) / 2
proof body
Definition body.
18