The Sentinel Life Equation (SLE) proposes a falsifiable dynamical framework for treating advanced AI systems as long-horizon continuity systems rather than static artifacts. The core claim is that “safety” for evolving AI is a trajectory property: the ability to preserve declared baselines under drift, shocks, operational change and distribution shifts over time. SLE models a continuity state shaped by constructive forces (e.g., integrity, safe adaptability, coherence) and entropic forces (e.g., degradation pressure, misalignment load and erosion), explicitly handling uncertainty and rare shock events. SLE is presented as a candidate model family, not a final law. It includes a validation and falsification posture (calibration checks, sensitivity analysis, ablations and stress-case reasoning) and connects model terms to governance levers – what can be monitored, constrained, strengthened, gated or rolled back without claiming that any single metric “solves alignment”. This paper is the technical foundation for the companion preprints The Cage Paradox (regime map and governed evolution logic) and Sentinel-Grade AI: Continuity Without Cages (technical safety case + evidence posture for audit-grade review in EU AI Act-relevant and mission-critical contexts). Series links (Project Orion): The Sentinel Life Equation (SLE): A Proposed Dynamical Framework for AI Continuity and Alignment – DOI: 10.5281/zenodo.17575603 The Cage Paradox: A Thought Experiment on Stability, Drift and the Evolution of Intelligent Systems – DOI: 10.5281/zenodo.17691117 The Cage Paradox: A Thought Experiment on Stability, Drift and the Evolution of Intelligent Systems – A Non-Technical Introduction to Sentinel-Grade AI – DOI: 10.5281/zenodo.17691383 Sentinel-Grade AI: Continuity Without Cages – DOI: 10.5281/zenodo.18750012 Sentinel-Grade AI: Continuity Without Cages – Non-Technical Companion – DOI: 10.5281/zenodo.18750318 AI-to-AI Diplomacy: Why LLM-Only Negotiation Fails Under Zero Trust and How SLE-Governed Systems Enable Proof-Carrying Compacts – DOI: 10.5281/zenodo.18881155 AI-to-AI Diplomacy: Proof-Carrying Compacts for Zero-Trust AI-to-AI Interoperability – Non-Technical Companion – DOI: 10.5281/zenodo.18881281 Project Hub & updates: Canonical papers and DOIs are archived on Zenodo. For the Project Orion research overview and verification snapshots, see www.auren.one. How it fits in the series: SLE is the mathematical lens: it defines continuity as a time-dependent dynamics problem. The Cage Paradox then uses this lens to map why “frozen” and “free-run” governance fail over long horizons. Continuity Without Cages operationalizes the middle regime – governed evolution – into a safety-case pattern with evidence tiers and rollback semantics. Key points: Treats AI safety as a continuity trajectory, not a snapshot pass/fail result Models continuity using constructive vs entropic forces with uncertainty and shocks Provides a falsification-first posture (ablations, calibration, sensitivity checks) Separates measurable continuity conditions from human-declared ethical baselines Links theory to practical governance levers (monitoring, gating, rollback) Serves as the technical foundation for Cage Paradox and Continuity Without Cages Evidence / verification snippet: This paper is theory-forward and establishes the continuity model and validation posture. Evidence-tiering and audit-grade verification artifacts are addressed and made explicit in Continuity Without Cages via Tier-0 public-safe integrity snapshots and Tier-1 controlled-access verification. Audience line: Researchers and practitioners in AI safety, control theory, cybernetics, governance, mission assurance and institutional due diligence evaluating long-horizon AI risks. Disclaimer line: Independent research preprint. Not a compliance filing, certification claim or regulatory conformity assessment. Not affiliated with any employer or institution.