Fixed-Point Theorem (v3) Contraction Boundary, Repulsive Pressure, and Dynamical Phase Structure on Pointed dcpos

Abstract We extend the self-referential fixed-point theorem established in v2 from a static existence result to a complete dynamical phase characterization. On a pointed dcpo (L, ≤, ⊥) equipped with sup-norm, v2 established existence and uniqueness of the least fixed point μ* = Φ(μ*, μ*) under Lipschitz contraction constant k < 1. The present work establishes that the existence of μ* is not merely a terminus of iteration but an anchor that determines the full dynamical phase structure of the system. Four results are new. First, we define Contraction Residue CR(w) and Repulsive Pressure P(w) directly from the Lipschitz constant k(w) = (1+α)w, without appeal to Fréchet differentiability. Second, we derive three critical parameter values w = 0.3, w = 1/2, w = 2/3 by inverting the structural conditions of the D-functor. Crucially, w = 2/3 is identified as the regime where joint monotonicity (A1) and single-argument monotonicity (A2) simultaneously fail, collapsing the well-definedness of ψ itself and placing the system entirely outside the scope of Theorem 1. Third, we derive analytically the convergence exponent γ = (1 + ln(1/(1−2w)))·(1−w) from the D-functor’s structural weight partition alone, without appeal to properties of μ*, yielding γ = 1.341 at w = 0.3 as a structural constant. Fourth, we identify the value γ = 1.313 reported in v2 as a finite-grid approximation with 2.1% deviation and provide a complete account of its sources. The unified thesis is: the existence of a unique least fixed point μ* on a pointed dcpo determines, and is inseparable from, the dynamical phase structure of convergence, criticality, and monotonicity collapse that surrounds it. Keywords: self-referential fixed point, dcpo, Lipschitz contraction, repulsive pressure, dynamical phase structure, D-functor, contraction residue, convergence exponent, monotonicity collapse