GENESIS Framework R22 Series: Structural Analysis of Chronic Disease Topology — Cusp Bifurcation Classification, Existence Conditions, and Early Warning Signals for Chronic Disease Transitions

The GENESIS framework (Generalized Nonlinear System for Illness Simulation) models chronic disease as a bistable dynamical system with three coupled state variables: stress activation S, regenerative capacity R, and accumulated damage D. Building on the R20. 4 reference architecture and R21. 2 structural analysis (DOI: 10. 5281/zenodo. 18901208), the R22 series completes the structural characterization of GENESIS across four modules. R22. 0 maps the full β×η (damage-sensitization × damage-regen coupling) parameter space, confirming that bistability is maintained across all 625 tested parameter combinations (100% bistable at D=0). No hidden bifurcation exists. Dfunc — the reversibility threshold — varies significantly with β×η: at extreme coupling (β=3, η=1), Dfunc halves from 0. 42 to 0. 20, reducing the therapeutic window for reversible intervention. R22. 1 identifies ecrit ≈ 0. 089 as the critical sensitization threshold: a saddle-node bifurcation point below which chronic disease is structurally impossible regardless of trigger load or damage history. The canonical parameter e=0. 50 lies 5. 6× above this threshold. This result requires a revision of the parameter tier hierarchy: e is reclassified as Tier-0 (Topology Generator), operationally distinct from the Tier-1 topology modulators (r0, k). Three patient phenotype regimes are defined by the combination of ecrit and Dfunc thresholds: monostable-healthy (e ecrit, D ecrit, D > Dfunc). R22. 2 derives the functional form ecrit (D) = Tₑff / (1 + β·D), where Tₑff ≈ 0. 063 is the intrinsic nonlinearity threshold. The form is analytically motivated: since eₑff = e· (1+β·D) and bistability requires eₑff ≥ Tₑff, the critical base sensitization decays hyperbolically with damage. Crucially, ecrit approaches zero asymptotically but never reaches it at finite D, maintaining biological plausibility (residual nonlinearity requirement). Comparison of r0 and e confirms functional analogy (both raise Dfunc when increased) but no formal algebraic symmetry — they operate on different nullclines and are not therapeutically interchangeable. R22. 3 formally classifies GENESIS as a slow-fast Cusp Bifurcation system. The cubic coefficient −eₑff in the S-equation is always negative, establishing topological equivalence to the cusp normal form ẋ = x³ + ax + b. All characteristic cusp signatures are confirmed: S-shaped bifurcation diagram, fold curve in (e, r0) parameter space, hysteresis (Δ ≈ 0. 936 in e-sweep), and critical slowing down (recovery rate correlation with e/ecrit = −0. 99). The hysteresis gap formalizes clinical irreversibility: reducing sensitization to pre-illness levels is insufficient for recovery; the system requires Tier-0 modulation below the lower fold point. Early Warning Signals consistent with critical slowing down theory are identified, yielding the first empirically testable prediction of GENESIS: patients approaching chronification should show decreasing recovery speed from post-exertional events before the chronic transition occurs. GENESIS is formally connected to ecological regime-shift models (Scheffer eutrophic lake, savanna-forest, thermohaline circulation), neurobiological excitability models (FitzHugh-Nagumo class), and energy-based dynamical systems. The R22 series transforms GENESIS from an exploratory simulation framework into a structurally characterized minimal model with analytical thresholds, formal system classification, and testable empirical predictions.