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. D_func — the reversibility threshold — varies significantly with β×η: at extreme coupling (β=3, η=1), D_func halves from 0.42 to 0.20, reducing the therapeutic window for reversible intervention. R22.1 identifies e_crit ≈ 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 e_crit and D_func thresholds: monostable-healthy (e < e_crit), bistable-reversible (e > e_crit, D < D_func), and bistable-locked (e > e_crit, D > D_func). R22.2 derives the functional form e_crit(D) = T_eff / (1 + β·D), where T_eff ≈ 0.063 is the intrinsic nonlinearity threshold. The form is analytically motivated: since e_eff = e·(1+β·D) and bistability requires e_eff ≥ T_eff, the critical base sensitization decays hyperbolically with damage. Crucially, e_crit 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 D_func 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 −e_eff 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/e_crit = −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.