Abstract Background: The NOAH6 rotational regulatory model describes biological organisms as hierarchically organized nonlinear dynamical systems capable of existing in multiple stable attractor states. Previous versions (v27, v28) established the mathematical framework and introduced an immune checkpoint gating axis (G) with a testable temporal signature (R4 → G → R5). Objective: This third version extends the model by formalizing the concept of critical transition thresholds between attractor states. We introduce the parameter κ (kappa) – effective immune system strength – which determines whether the system resides in a homeostatic or tumor-stabilized attractor. The model explains heterogeneous responses to PD-1 blockade, spontaneous regressions, and post-response recurrence within a unified deterministic framework. Methods: Building on the existing nonlinear controlled dynamical system: Ẋ = (A-D) X + g eₖ + f (X) we derive the reduced dynamics of the tumor compartment (R5) and define κ as a composite function of biological terrain (R4), checkpoint gating (G), and immune effector efficiency (α). Bifurcation analysis identifies the critical threshold κ₂ₑ₈ₓ at which the system transitions between attractors. For medical operability, we additionally define a κ-index: a practical, biomarker-based approximation of κ designed for prospective validation. Results: The model yields three distinct perturbation regimes: subthreshold (tumor attractor stable), near-threshold optimal (transition to homeostasis), and suprathreshold (diminishing returns with increased risk of compensatory stabilization). The existence of a narrow transition zone around κ₂ₑ₈ₓ explains why small perturbations can produce dramatic clinical responses, while identical interventions in different baseline states yield minimal effects. Rotational regulation functions as a mechanism to maintain κ near – but not beyond – the critical threshold while reducing adaptive compensation. Conclusion: The extended NOAH6-v3 framework provides a mathematically closed explanation for observed immune-oncological phenomena that remain paradoxical in linear models. The model generates precise, falsifiable predictions regarding the relationship between systemic regulatory state and PD-1 blockade outcomes. Model Interpretation Note: All temporal sequences and control signals described in this manuscript represent abstract constructs within a dynamical systems framework and do not constitute medical treatment recommendations. The model is theoretical and not intended for clinical use.
Zakir Causevic (Mon,) studied this question.