phi_pow_pos
Powers of the golden ratio remain positive for every integer exponent. Results establishing stability of the φ-ladder cite this fact to confirm that every rung qualifies as a stable position. The proof is a one-line wrapper that applies the general lemma on positivity of positive bases under integer exponentiation.
claimLet φ denote the golden ratio. Then φ^n > 0 for every integer n.
background
The φ-Emergence module derives the golden ratio from J-cost minimization, where J(x) = (x + x^{-1})/2 - 1 is strictly convex with unique minimum at x = 1. The golden ratio appears as the unique positive self-similar fixed point forced by the Recognition Science chain. Upstream structures on ledger factorization and spectral emergence supply the discrete φ-tiers that label nuclear densities and gauge content.
proof idea
This is a one-line wrapper that applies the general fact that a positive real base raised to any integer power remains positive, instantiated at the golden ratio which satisfies the positivity hypothesis from its definition as the positive fixed point.
why it matters in Recognition Science
The result is invoked by phi_ladder_stable to show that every position φ^n is stable. This anchors the hypothesis that stable positions coincide exactly with the φ-ladder, supporting the eight-tick octave and the emergence of D = 3 in the forcing chain.
scope and limits
- Does not prove that φ exceeds 1.
- Does not extend to non-integer exponents.
- Does not apply to the negative conjugate root.
- Does not establish closure or ratio properties of the ladder.
Lean usage
constructor · exact phi_pow_pos n · exact ⟨n, rfl⟩formal statement (Lean)
86theorem phi_pow_pos (n : ℤ) : φ^n > 0 := by
proof body
Term-mode proof.
87 exact zpow_pos phi_pos n
88
89/-- Adjacent rungs of the ladder have ratio φ -/