Fixed-Point Theorem for Self-Referential Equations in Pointed dcpos with Bounded Second-Argument Dependence

We prove existence and uniqueness of a least solution to the self-referential fixed-point equation μ = Φ(μ, μ) where Φ : L × L → L is a jointly monotone operator on a pointed directed-complete partial order (dcpo) (L, ≤, ⊥). The key condition is bounded second-argument dependence: the auxiliary map ψ(g) = lfp Φ(·, g) is order-contractive with Lipschitz constant k < 1 with respect to the sup-norm on L = [0,1]^O. Under this condition we construct a monotone auxiliary operator ψ : L → L, apply Pataraia's theorem to obtain its least fixed point μ* = lfp ψ, and prove that μ* is the unique least solution of μ = Φ(μ, μ).

The result is motivated by and applied to the Ouroboros model [OuroborosEq2026], a self-consistent fixed-point framework on a finite dcpo in which the retrocausal boundary operator D introduces second-argument dependence bounded by k = (1+α)w with w < 1/3 and α < 1, giving k ≤ 0.435 < 1 for the default parameters (α=0.45, w=0.3).

AI Collaboration Disclosure: This work was developed in collaboration with AI language models: Claude (Anthropic), Grok (xAI), and ChatGPT (OpenAI). Mathematical derivations, proof verification, and manuscript preparation were conducted through iterative dialogue with these systems. The conceptual framework, theoretical claims, and intellectual responsibility remain solely with the author.