This paper provides a rigorous mathematical foundation for resonance as the central structural variable of the Yang Jihoon Dynamics (YJD) research programme. It formalizes resonance as a scale-dependent alignment functional defined on the extended domain [−1,1][-1,1][−1,1], completing the theoretical chain established by three preceding works:(1) Yang Jihoon Dynamics, which introduced coupled phase dynamics and meta-cognitive regulation;(2) Information Asymmetry and Forced Communication, which isolated structural conditions under which resonance becomes inaccessible; and(3) Viability-Structured Field Theory (VSFT), which demonstrated that multi-slack “breathing” architectures are required for indefinite survival across scales. Unlike earlier papers that treated resonance phenomenologically or structurally, the present work defines resonance axiomatically and independently of empirical interpretation. Resonance is constructed as a functional comparing actual adaptive response to an ideal scale-dependent throttle, yielding three regimes: alignment (R > 0), neutral boundary (R = 0), and anti-alignment (R < 0). Fundamental properties—continuity, boundedness, non-binary structure, and scale invariance—are derived purely from real analysis and functional invariance. The central results establish that sustained anti-resonance is structurally unstable and must resolve through either termination or reorganization. This leads to a complete classification of four anti-resonance pathways: voluntary annihilation, involuntary annihilation, voluntary creation, and involuntary creation, determined by two independent binary conditions—outcome (annihilative vs. creative) and agency (meta-cognitively mediated vs. automatic). Building on the coexistence problem posed in Information Asymmetry and Forced Communication, the paper proves a necessary condition for indefinite coexistence between interacting systems: the long-term time-averaged resonance must remain non-negative. Coexistence sustainability is shown to depend on the dominance of creative over annihilative pathways and on the availability of structural reorganization across scales, directly linking to the multi-slack breathing principle formalized in VSFT. All observation scales are proven to be structurally isomorphic, establishing resonance as an intrinsically multi-scale quantity. This result closes the conceptual gap between individual-level dynamics (YJD), inter-agent communication under asymmetry, and the scale-invariant morphology of viable systems identified in VSFT. The theory is entirely non-empirical and domain-independent. States, trajectories, throttles, gates, and meta-cognitive variables are treated as formal mathematical objects without prescribed physical, psychological, or social interpretation. The framework applies universally to any pair of interacting systems capable of adaptive response, regardless of ontological status. By providing a complete axiomatic definition of resonance and a full taxonomy of anti-resonance resolution pathways, this work establishes resonance as the mathematical core of the Yang Jihoon Dynamics programme and supplies the missing structural link between viability theory, information asymmetry, and scale-invariant coexistence.