cfrac_ground_state_is_phi
plain-language theorem explainer
The theorem establishes that the continued fraction iteration map fixes the golden ratio φ, confirming it as the attractor of sequential J-cost minimization. Researchers in number theory studying Ramanujan-type expansions or in Recognition Science modeling self-similar geodesics would cite this result. The proof is a direct one-line application of the already-established fixed-point property for φ under the iteration.
Claim. The continued fraction iteration map satisfies $f(φ) = φ$, where $φ = (1 + √5)/2$ is the golden ratio and $f$ encodes the self-similar recursion arising from minimal J-cost at each nesting level.
background
J-cost is defined by $J(x) = ½(x + x^{-1}) - 1$, which is strictly convex on the positive reals with a unique minimum at $x=1$. The golden ratio φ satisfies the fixed-point equation $x = 1 + 1/x$, making it the unique positive attractor of the recursion that arises when each level of a continued fraction chooses the ratio minimizing local J-cost subject to the next level's value. The module treats a continued fraction as a sequential optimization path on the choice manifold, with the infinite self-similar case converging to φ because J is convex and the ground-state geodesic passes through the fixed point.
proof idea
One-line wrapper that applies the fixed-point theorem for the continued fraction iteration at φ.
why it matters
This result closes the Ramanujan bridge by showing why nested continued fractions evaluate to algebraic expressions in φ: they compute the ground state of J-cost minimization. It directly instantiates the self-similar fixed point forced at T6 of the unified forcing chain and supplies the mathematical reason that Ramanujan's deepest continued fractions always involve φ. No downstream theorems are listed, so the declaration serves as a terminal link between the classical continued-fraction literature and the Recognition Science cost geometry.
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