cfracIteration
plain-language theorem explainer
The continued fraction iteration is the map sending a positive real x to 1 plus its reciprocal. Researchers analyzing golden-ratio attractors and sequential J-cost minimization cite this definition when deriving fixed-point properties in Recognition Science. It is introduced as a direct algebraic assignment with no lemmas or tactics.
Claim. The continued fraction iteration is the function given by the rule $xmapsto 1 + 1/x$ for real $x$.
background
The module situates continued fractions inside Recognition Science as sequential J-cost optimizations. The J-cost functional is $J(x) = frac12(x + x^{-1}) - 1$, which is strictly convex on the positive reals and attains its global minimum at $x=1$. The golden ratio $phi$ is the unique positive solution to the fixed-point equation $x = 1 + 1/x$, making it the attractor of the recursion. Upstream structures on phi forcing and ledger factorization supply the algebraic setting in which this iteration appears as the ground-state geodesic.
proof idea
The declaration is a direct definition that implements the algebraic expression $1 + 1/x$. No lemmas are invoked and no tactics are applied; the body is simply the right-hand side of the intended map.
why it matters
This definition supplies the primitive recursion used by the downstream theorems phi_is_cfrac_fixed_point, cfrac_ground_state_is_phi, phi_worst_approximable_core and sequential_optimization_forces_phi. It realizes the T6 self-similar fixed point of the forcing chain and the J-cost interpretation of Ramanujan-type continued fractions. The construction closes the link between classical continued-fraction identities and the Recognition Composition Law.
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