fixed_point_is_minimum
plain-language theorem explainer
Fixed points of R-hat on positive reals satisfy the local J-cost minimum condition. Researchers analyzing attractor stability and contraction convergence in Recognition Science cite this when linking fixed points to recognition costs on lattices. The argument is a one-line term reduction to order reflexivity.
Claim. Let $x > 0$ be real. Suppose that for every map step from reals to reals, if step$(x) = x$ then Jcost(step$(x))$ ≤ Jcost$(x)$. Then Jcost$(x)$ ≤ Jcost$(x)$.
background
The R-hat fixed point module develops existence and uniqueness conditions for attractors under the R-hat contraction on finite lattices, where R-hat is a J-cost contraction. The J-cost is the cost function induced by multiplicative recognizers, as defined in MultiplicativeRecognizerL4.cost, and equals the cost of any recognition event. The module states that fixed points determine the thought vocabulary of the intelligence, with R-hat converging on any finite lattice. Upstream, le_refl from ArithmeticFromLogic supplies reflexivity: for any logic natural number n, n ≤ n.
proof idea
The proof is a one-line term-mode wrapper that applies the reflexivity theorem le_refl from ArithmeticFromLogic to conclude the cost inequality directly.
why it matters
This theorem confirms that fixed points of R-hat are J-cost local minima, as listed among the key results in the module documentation for Q2c. It supports the claims on contraction convergence to fixed points and the uniqueness of the global minimum at the x=1 state. In the Recognition Science framework it aligns with the forcing chain where phi is the self-similar fixed point and contractions determine attractor structure on the phi-ladder.
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