We develop a rigorous, substrate-independent mathematical framework for addiction based on the extractive oscillator with sensor degradation (EOS) dynamical systems class (Kriger, 2017; doi:10.5281/zenodo.18529185). The binge–remission–tension–relapse cycle, progressive erosion of self-assessment capacity, sunk-cost entrenchment, and social isolation are derived as necessary structural consequences of a coupled two-agent system satisfying five axiomatic conditions (K1–K5). The framework is extended in three directions: (1) a developmental vulnerability model formalizing the adolescent endorphin decline and self-regulation maturation gap; (2) a threat analysis of neural interface addiction as the zero-path-length limiting case, including a balanced assessment of both increased addictive potential and elimination of injection-related harms, toxic metabolites, and recurring procurement costs; (3) a formal theory of intervention in which each therapeutic strategy is derived as a structural attack on a specific theorem, feedback loop, or phase transition. The article establishes formal correspondence with the unified dynamical systems theory (DST) of psychiatric disorders (Kriger, 2026; doi:10.5281/zenodo.18556979). The DST framework provides five capabilities that EOS alone does not: prediction of the initial healthy-to-addicted transition via critical slowing down (measurable early warning signals before the first binge), quantification of recovery difficulty via Kramers escape rates, comorbidity modeling on a shared attractor landscape (interaction of addiction with depression, PTSD, anxiety), operationalization of abstract variables to clinical measurement instruments (EMA, PHQ-9, actigraphy, NLP), and formalization of the "fear of relapse" as an autocatalytic destabilizing loop that erodes recovery resilience. The empirical basis for substrate independence is strengthened by analysis of six behavioral addiction types (gambling, gaming, social media, pornography, workaholism, food addiction), demonstrating that identical dynamical structures emerge across substrates sharing no common chemistry. Cross-addiction is explained as transition between isomorphic potential wells. Five appendices provide: (A) full proof of limit cycle existence via Poincaré–Bendixson/Filippov theory with explicit trapping region construction; (B) full proof of the phase transition in self-assessment with identified order parameter, finite-time collapse demonstration, and Azuma–Hoeffding concentration bound; (C) numerical worked example calibrated to opioid relapse data (Hser et al., 2001), producing ~85-day cycle period, ~2.2-year sensor degradation, and quantified sunk-cost threshold; (D) substantive engagement with five competing models (Redish RL, Koob–Le Moal allostasis, Robinson–Berridge incentive salience, Borsboom network theory, Gutkin et al. dynamical systems), identifying what EOS uniquely explains; (E) complete DST mathematics for substance use disorders reproduced from the parent framework, including the Kriger system general definition with boundary cases addressing falsifiability. Eight falsifiable predictions are presented, including joint DST–EOS predictions. The framework applies with equal force to heroin, social media, gambling, and future neural implant technologies. Keywords: addiction; dynamical systems; extractive oscillator; sensor degradation; neural interfaces; reward hijacking; substrate independence; intervention theory; relaxation oscillator; sunk-cost trap; adolescent vulnerability; attractor landscape; anosognosia; critical slowing down; Kramers escape rate; comorbidity; DSM-5-TR; phase transition; Poincaré–Bendixson; falsifiable predictions Related works: Kriger, B. (2017). Extractive Oscillators with Sensor Degradation: A Dynamical Systems Class and Its Manifestation in Quasi-Narcissistic Relational Dynamics. Zenodo. https://doi.org/10.5281/zenodo.18529185 (isPartOf) Kriger, B. (2026). Formalization of Mental Disintegration Phenomena Through Dynamical Systems Theory: With Applications to DSM-5-TR Diagnostic Categories. Zenodo. https://doi.org/10.5281/zenodo.18556979 (isSupplementedBy) Kriger, B. (2023). Positive Alternatives to Drugs. Altaspera Publishing, Canada. (references)