The Ouroboros Equation: A Self-Referential Fixed-Point Framework on Directed-Complete Partial Orders

Abstract 

We define the Ouroboros Equation — a self-referential fixed-point equation on a directed-complete partial order (dcpo) in which the least fixed point μF serves simultaneously as its own future boundary condition: μF = F(μF, μF). The composite operator F = R ∘ ((D ⊗ G) ⊕ H) is built from monotone components; the central innovation is the D-functor, D(f, μF)(t) = (1−2w)·f(t) + w·f(t−1) + w·μF(t+1), w = 0.3. All parameters are derived from first-principles structural constraints: α ∈ (0,1) from the monotonicity condition on R; w = 0.3 as the extremal value of the time-symmetry constraint set. We prove existence and uniqueness of μF (Theorem 1), establish that μF is a bidirectional interpretive closure — invariant under a complete forward/backward cycle (Theorem 2) — and confirm convergence numerically: λ(α) = 2.775·α^1.341 (R² > 0.999), scale-invariant across grid sizes S ∈ {5,...,50}. The framework is self-contained and admits extension to any domain where a self-referential fixed-point structure with bounded second-argument dependence can be defined. 

Keywords: self-referential fixed point, dcpo, directed-complete partial order, Ouroboros Equation, D-functor, bidirectional interpretive closure, self-consistent boundary condition, monotone operator