sequential_optimization_forces_phi
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.
claimThe 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 in Recognition Science
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.
scope and limits
- Does not prove convergence of finite truncations to φ.
- Does not address fixed points outside the positive reals.
- Does not incorporate non-convex perturbations of J-cost.
- Does not derive numerical approximation rates.
formal statement (Lean)
254theorem sequential_optimization_forces_phi :
255 cfracIteration phi = phi := phi_is_cfrac_fixed_point
proof body
Term-mode proof.
256
257/-- Strong fixed-point form: any positive fixed point of `x ↦ 1 + 1/x` is `φ`.
258
259 (Interpretation) Any sequential optimization with:
260 (1) Self-similarity (same cost at every level)
261 (2) Strict convexity of cost functional
262 (3) Discrete (integer) choices at each level
263
264 converges to φ. This is because:
265 - Self-similarity → scale ratio r
266 - Optimality → r satisfies the Fibonacci recurrence
267 - Convexity → unique positive solution r² = r + 1 → r = φ
268
269 This is exactly the content of `fibonacci_partition_forces_phi`
270 from the Local Cache theorem, applied to continued fractions. -/