The Abebe Tensility Framework — The Geometry of Failure —

A candidate cross-domain framework for identifying and monitoring structural instability across heterogeneous systems. At its core is the Abebe Tensility Index (Φ_z), the dimensionless ratio of destabilizing pressure (τ) to cohesive integrity (V_k). Key Contributions • Canonical Ratio: Formalizes Φ_z = τ / V_k and identifies a recurrent transition regime near Φ_z ≈ 2.0, associated with the progression from organized structural coherence toward fragmentation (parity exhaustion). • Cross-Domain Calibration: Provides illustrative empirical alignment across mechanical systems (ductile fracture in structural alloys), biological systems (oxygen extraction limits in human physiology), and macro-financial systems (debt-to-liquidity dynamics during systemic stress). • Dynamic Extension: Introduces Tensility Velocity (Φ̇_z) and Tensility Acceleration (Φ̈_z) for trajectory-based monitoring, distinguishing static stress from active progression toward failure. • Regime Architecture: Defines a four-stage progression — Stable (Φ_z < 1.6), Transition (1.6 ≤ Φ_z < 1.8), Falling Manifold (1.8 ≤ Φ_z < 2.0), and Critical Regime (Φ_z ≈ 2.0). The mappings are operational and illustrative (first-order representations of structural similarity) rather than strictly equivalent. This work supplies the foundational formulation for the analysis of structural stability and the monitoring of systemic failure dynamics. Series Position Core Paper #4 (Operational Framework & Geometry of Failure) Companion Papers (Version 2) 1. The Abebe Unified Field of Integrity (vision) 2. The Abebe Parity Identity (mathematics) 3. The Abebe Parity Identity II (physical embedding) 5. Five Millennia of Structural Tension – Empirical Study I 6. THE Abebe SENSOR PROTOCOL (2026–2027)