Recursion, Collapse, Recombination: A Family of Non-Binary Dynamical Systems

We present Recursion-Collapse-Recombination (RCR), a family of five dynamical systems in which three coupled operators — self-amplifying Recursion (F), restoring-force Collapse (C), and dispersion-scaled Recombination (G) — produce Active Metastability (AM): persistent structured minority dispersion with continuous dynamical activity under physical stability constraints. Six minimal axioms (A1–A6) define the system class. Nine formal theorems (Theorems 0–8 plus Corollary 2) establish the framework: Theorem 0 (Classification) proves A1–A5 (with A4 in its multi-attractor form, k ≥ 2) are necessary and sufficient for AM within the RCR operator class; Theorem 1 (Stochastic Recurrence) proves all trajectories return to a compact set almost surely; Theorem 2 (Persistent Metastability) proves dispersion cannot collapse to zero, with a computable floor δ = ηE|ε|/ (2ζ), where ζ is the Cₗocal collapse coefficient of Definition 2 (distinct from the F-recursion parameter α) ; Theorem 3 (Existence) proves the AM regime is inhabited; Theorem 4 (Necessity) proves endogenous responsive attractors are necessary, not merely sufficient; Theorem 5 (Endogenous Stochastic Closure) proves the noise amplitude Σ (R) is endogenously bounded and coupled to operator tension. Theorem 6 (Reduction to Classical Dynamics) proves RCR strictly contains Hegselmann-Krause, Deffuant, Langevin, and mean-field consensus as limiting cases. Theorem 7 (h (r) Operator Balance Boundary) characterizes the AM regime boundary ∂ΛAM via the closed-form operator balance function h (r) and derives the exact parameter thresholds χ* = 0. 0891, χₕard = 0. 1714, χₐnn = 0. 2400. Theorem 8 (Coarse Density in Ω′) proves ΛAM² is 0. 06-dense in the restricted parameter space Ω′ = Ω \ E: every ball of radius ε > 0. 06 contains an AM point, establishing AM genericity at that resolution. Lemma 1 (Operator Irreducibility) proves RCR cannot be reduced to any single-state system. A Model 5 extension with state-dependent noise establishes strict containment ΛAM² ⊆ ΛAM⁵, with 76 parameter regions accessible only via endogenous noise. Empirically: 30-seed validation (composite 2. 256 ± 0. 061), 500-point Latin hypercube genericity sweep (AM rate 16. 2%, eight structured components at conservative sweep scale — a sampling artifact that dissolves at 5× density; see §15. 3), global sensitivity analysis (Sobol 10 corrected: S₁ (χ) = 0. 29, ST (χ) = 0. 83, χ the only parameter with CI excluding zero; σ/μ near-negligible S₁ but large total effects ST = 0. 73/0. 62), and out-of-sample political polarization prediction (calibrated adversarial-baseline error 0. 031 vs 0. 094 for next-best model; raw bimodality index error: 0. 117 vs 0. 430 for standard social models — see §14. 3) jointly support the framework. Claim scope is explicit: theorems establish existence of AM, not inevitability. Most dynamical systems models of collective behavior converge — populations settle into consensus, species reach equilibrium, opinions homogenize. The harder problem is explaining persistence: why do minorities survive, why do dissenting configurations endure in the face of forces that should eliminate them? This paper introduces a dynamical framework — Recursion, Collapse, Recombination (RCR) — in which a third outcome is possible: Active Metastability, a regime of persistent structured dispersion coupled with continuous dynamical activity, stable in the physical sense but never converging. We prove this regime exists, characterize the parameter space it occupies (16. 2% of the admissible range, eight structured components), and prove that endogenously responsive attractors are necessary — not merely sufficient — for the phenomenon to occur. A five-model family is presented, with Model 5 demonstrating that state-dependent noise strictly enlarges the accessible regime. The framework recovers Hegselmann-Krause 6, Deffuant 7, Langevin, and mean-field consensus as limiting cases, placing RCR as a strict generalization of existing opinion dynamics models 6, 7. Out-of-sample prediction of US political polarization (1994–2024) provides empirical grounding. Model 5 closes the autopoietic loop: the system generates the uncertainty it then metabolizes.