The Causal Hourglass: A Geometric Framework for Node Formation in Relational Causality

A geometric framework for modeling constraint accumulation, maneuverability loss, and regime transition in complex systems. The Causal Hourglass introduces a substrate-independent framework for modeling how constraint accumulation shapes the evolution of complex systems. The framework focuses on how the volume of accessible system trajectories contracts under layered constraints, eventually reaching a point at which meaningful intervention becomes impossible. The model represents this process geometrically as an hourglass. Systems begin in a regime of high maneuverability where many trajectories remain available. As constraints accumulate and feedback amplification increases, the accessible trajectory volume contracts. The system eventually reaches a structural threshold termed Earliest Domain Closure (EDC) — the earliest point at which intervention feasibility falls below the level required to redirect system dynamics. Beyond this point, regime transition becomes effectively unavoidable without large-scale structural reconfiguration. Rather than treating determinism and probability as ontologically distinct regimes, the framework models them as context-dependent phases within causal processes. Systems far from closure exhibit wide probabilistic trajectory spaces, while systems approaching closure become increasingly constrained and locally predictable. The framework formalizes this contraction using a normalized flexibility variable Sc(t)S_c(t)Sc(t), representing accessible trajectory volume, together with nonlinear amplification dynamics and threshold-based intervention modeling. The resulting structure captures how gradual constraint accumulation combined with feedback mechanisms can produce abrupt regime transitions without requiring large external shocks. Two complementary documents are provided: • Introductory Framework Paper — presents the geometric intuition of the hourglass model, introduces the concept of trajectory accessibility, and provides minimal dynamical examples and cross-domain interpretations. • Technical Version — develops the formal mathematical structure, including contextual entropy dynamics, relevance kernels for historical influence weighting, and environmental coupling mechanisms that govern closure behavior. The framework is intentionally cross-domain and can be applied to financial systems, infrastructure networks, ecological regimes, institutional dynamics, and other constraint-driven systems. A computational toy model demonstrates the core dynamics and produces empirically testable signatures such as flexibility contraction, nonlinear amplification near mid-range constraint, loss of intervention efficacy near closure, and constrained divergence after regime transition. By shifting analytical focus from isolated crisis events to the geometry of maneuverability loss, the Causal Hourglass provides a unified framework for studying constraint-driven systemic transitions across domains.