sequential_optimization_forces_phi
plain-language theorem explainer
The continued fraction recursion fixes the golden ratio as its unique positive fixed point under the map x ↦ 1 + 1/x. Researchers deriving φ from sequential J-cost minimization in Recognition Science cite this when closing the self-similar attractor step. The proof is a one-line wrapper invoking the fixed-point lemma for the iteration.
Claim. The continued-fraction iteration map satisfies $f(φ) = φ$, where $f(x) = 1 + 1/x$ and $φ = (1 + √5)/2$ is the unique positive solution.
background
The module treats continued fractions as sequential J-cost optimizations on positive ratios. J-cost is the functional $J(x) = ½(x + x^{-1}) - 1$, strictly convex on ℝ₊ with unique minimum at x = 1. The self-similar fixed point of the recursion x = 1 + 1/x is forced by the same convexity that appears in T5 of the unified forcing chain. Upstream results supply the cost of recognition events as J-cost and the structure of multiplicative recognizers whose derived cost reproduces the same functional.
proof idea
The proof is a one-line wrapper that applies the lemma phi_is_cfrac_fixed_point establishing the fixed-point property of the continued-fraction map.
why it matters
This declaration supplies the sequential fixed-point core for the Ramanujan bridge, showing that the continued-fraction update lands exactly on the self-similar fixed point forced at T6. It closes the link between J-cost geodesics and the eight-tick octave structure without additional hypotheses. No downstream theorems are recorded yet, leaving open its direct use in mass-ladder or alpha-band derivations.
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