Polyphasic Cybernetics and Phase-Plane Geometry

This appendix establishes a clean mathematical bridge between classical dynamical-systems neuroscience(phase portraits, nullclines, separatrices, bifurcations, excitability, bistability, resonant oscillations) and thePolyphasic / RHEA‑UCM control architecture (Ψ, Φ, Δ, trust, entropy, glyph triggers, reseal events). The goalis not to claim that neurons “are” the same thing as a symbolic cyber-defense engine, but to show that thestability grammar of neurodynamics provides an accepted, observation-rooted template for constructingand validating Polyphasic regulation. We proceed by (i) formalizing the phase-plane geometry used tointerpret excitability, hysteresis, and Hopf/Saddle-Node transitions; (ii) mapping those structures ontoPolyphasic state variables and gating logic; and (iii) deriving a compact “Mathematical Core” for a Polyphasiccontroller expressed as a piecewise-smooth dynamical system with event-driven resealing.