phi_pow_pos
plain-language theorem explainer
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.
Claim. Let 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
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.
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