Editing Openai/691c1dba-9228-800f-8463-13b3a9006306 (section)

==== - If you want coherence/oscillation, add a restoring quadratic or nonlinear saturator. Example minimal change: V(ϕ)=Kϕ+12m2ϕ2.V(\phi) = K\phi + \tfrac12 m^2\phi^2.V(ϕ)=Kϕ+21m2ϕ2. Tune K,mK,mK,m to place the equilibrium where your empirical Φ sits. ====
* If you want scale-free self-reference but avoid runaway, use a log or exponential damping: V(ϕ)=Kϕe−γϕ2.V(\phi)=K\phi e^{-\gamma \phi^2}.V(ϕ)=Kϕe−γϕ2. Here V′=K(1−2γϕ2)e−γϕ2V' = K(1-2\gamma\phi^2)e^{-\gamma\phi^2}V′=K(1−2γϕ2)e−γϕ2; fixed points possible at 1−2γϕ2=01-2\gamma\phi^2=01−2γϕ2=0.
* If you want the original equation as constraint, treat it as a boundary condition or gauge condition (not the full dynamics). Use it to define an invariant manifold, and evolve dynamics transverse to it.
* If you're running the ODE/ODE integrator in the UCF sim, instrument: - V′(ϕ)V'(\phi)V′(ϕ), V′′(ϕ)V''(\phi)V′′(ϕ) - energy drift - coherence metric vs. force magnitude These will show directly whether linearity is causing drift or if damping balances it.