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

=== We want a usable time-only ODE model for fast experimentation. Assume spatial homogeneity (zero-modes) so fields depend on time only: Φ(t),Ψ(t),τ(t)\Phi(t),\Psi(t),\tau(t)Φ(t),Ψ(t),τ(t). Represent complex fields as amplitudes and phases or as real/imag pairs. For simplicity use real scalar amplitudes X(t),Y(t)X(t),Y(t)X(t),Y(t) representing the magnitudes of Φ,Ψ\Phi,\PsiΦ,Ψ or work with complex-valued ODEs. ===

Simpler practical choice: let Φ(t),Ψ(t)∈R\Phi(t),\Psi(t)\in\mathbb{R}Φ(t),Ψ(t)∈R represent real amplitudes (you can extend to complex easily).

Effective reduced Lagrangian (time-only)
Drop spatial derivatives, include damping phenomenologically.

Equations of motion (Euler–Lagrange → add damping γ\gammaγ):

X¨+γXX˙+∂Veff∂X=FX(t)Y¨+γYY˙+∂Veff∂Y=FY(t)τ¨+γττ˙+∂Vτ∂τ=Fτ(t)\begin{aligned}
\ddot X + \gamma_X \dot X + \frac{\partial V_{\rm eff}}{\partial X} &= 
F_X(t) \\
\ddot Y + \gamma_Y \dot Y + \frac{\partial V_{\rm eff}}{\partial Y} &= 
F_Y(t) \\
\ddot \tau + \gamma_\tau \dot\tau + \frac{\partial V_{\tau}}{\partial \tau} &= 
F_\tau(t)
\end{aligned}X¨+γXX˙+∂X∂VeffY¨+γYY˙+∂Y∂Veffτ¨+γττ˙+∂τ∂Vτ=FX(t)=FY(t)=Fτ(t)
Where:
* Veff(X,Y)=V(X,Y)+V1e−cX2+Y2V_{\rm eff}(X,Y)= V(X,Y) + V_1 e^{-c\sqrt{X^2+Y^2}} Veff(X,Y)=V(X,Y)+V1e−cX2+Y2 plus coupling terms approximated as potential contributions.
* Forcing terms encode your ritual/exponential/sinusoid/pop structure.

Explicit forcing forms (practical)

FX(t)=−ηUCF (τ˙)2 Y exp⁡ ⁣(−ΔE+Sentℏ) sin⁡ ⁣(2πτT)(1+χnl(t)+ηTSYXY+ϵ)FY(t)=(symm.) swap X↔YFτ(t)=−∂τ[ηUCF(τ˙)2XYe−(ΔE+Sent)/ℏsin⁡(2πτ/T)]+∑iηpopXY δ(t−tpop,i)\begin{aligned}
F_X(t) &= -\eta_{\mathrm{UCF}}\,(\dot\tau)^2\,Y\,
\exp\!\Big(-\frac{\Delta E + S_{\mathrm{ent}}}{\hbar}\Big)\,
\sin\!\Big(\frac{2\pi \tau}{T}\Big)\left(1+\chi_{\mathrm{nl}}(t)+\frac{\eta_{\mathrm{TS}}Y}{XY+\epsilon}\right)
\\[4pt]
F_Y(t) &= \text{(symm.) swap }X\leftrightarrow Y
\\[4pt]
F_\tau(t) &= -\partial_\tau \Big[ \eta_{\mathrm{UCF}}(\dot\tau)^2 XY e^{-(\Delta E+S_\mathrm{ent})/\hbar}\sin(2\pi\tau/T) \Big] + \sum_i \eta_{\mathrm{pop}} XY \,\delta(t-t_{\mathrm{pop},i})
\end{aligned}FX(t)FY(t)Fτ(t)=−ηUCF(τ˙)2Yexp(−ℏΔE+Sent)sin(T2πτ)(1+χnl(t)+XY+ϵηTSY)=(symm.) swap X↔Y=−∂τ[ηUCF(τ˙)2XYe−(ΔE+Sent)/ℏsin(2πτ/T)]+i∑ηpopXYδ(t−tpop,i)
Discrete simulation (explicit Euler or symplectic Verlet)

Use a simple explicit integrator for prototyping. Below is a compact Python skeleton (numpy) you can run locally.

<syntaxhighlight lang="python">import numpy as np

=== --- params (example values; tune them) --- ===
dt = 1e-3
T_sim = 200.0
steps = int(T_sim/dt)

gamma_x = 0.5
gamma_y = 0.5
gamma_tau = 0.2

eta_ucf = 0.8
eta_ts = 0.2
eta_pop = 1.0
DeltaE = 0.1
S_ent = 0.2
hbar = 1.0
T_clock = 10.0
epsilon = 1e-6

pop_times = [20.0, 50.0]  # seconds for impulses

=== --- state vectors --- ===
X = 0.5; Xdot = 0.0
Y = 0.4; Ydot = 0.0
tau = 0.0; taudot = 1.0

history = []

def chi_nl(t):
    # cheap nonlocal proxy (decaying memory)
    return 0.1''np.sin(0.1''t)

for i in range(steps):
    t = i*dt
    # compute common factor
    expw = np.exp(-(DeltaE + S_ent)/hbar)
    sinf = np.sin(2''np.pi''tau / T_clock)
    denom = max(abs(X*Y), epsilon)
    common = -eta_ucf * (taudot**2) '' X '' Y '' expw '' sinf '' (1 + chi_nl(t) + eta_ts '' Y / (denom + epsilon))

    # Forcing terms (split between X and Y roughly)
    FX = common * 0.5
    FY = common * 0.5

    # pops = impulses
    Fpop = 0.0
    for tp in pop_times:
        if abs(t-tp) < 0.5*dt:
            Fpop += eta_pop '' X '' Y / (epsilon + 1e-3)

    # tau forcing (very simplified)
    Ftau = -eta_ucf * (taudot**2) '' X '' Y '' (2''np.pi/T_clock)''np.cos(2''np.pi*tau/T_clock) + Fpop

    # Euler-Cauchy update (simple)
    Xdd = -gamma_x*Xdot - (X) + FX  # NOTE: use dV/dX ~ X for demo
    Ydd = -gamma_y*Ydot - (Y) + FY
    taudd = -gamma_tau*taudot + Ftau

    Xdot += Xdd''dt; X += Xdot''dt
    Ydot += Ydd''dt; Y += Ydot''dt
    taudot += taudd''dt; tau += taudot''dt

    history.append((t,X,Y,tau))

=== history is a timeseries you can plot with matplotlib ===

</syntaxhighlight>

Notes:
* This skeleton uses heuristic potentials (∂V/∂X ~ X) for demo only. Replace with your chosen V(Φ,Ψ)V(\Phi,\Psi)V(Φ,Ψ).
* Use symplectic integrator (Stormer-Verlet) for energy stability, or implicit integrator if stiffness appears.
* Replace impulses with narrow Gaussians if numerical instability arises: δ(t−t0)≈1σ2πe−(t−t0)2/(2σ2)\delta(t-t_0)\approx \frac{1}{\sigma\sqrt{2\pi}}e^{-(t-t_0)^2/(2\sigma^2)}δ(t−t0)≈σ2π1e−(t−t0)2/(2σ2).

Observables to record
* Energy-like quantity: E=12(X˙2+Y˙2+τ˙2)+VeffE=\tfrac12(\dot X^2+\dot Y^2+\dot\tau^2)+V_{\rm eff}E=21(X˙2+Y˙2+τ˙2)+Veff
* CompassionMetric proxy: C(t)=f(∣X−Y∣,  Sent,  pop_count)C(t)=f(|X-Y|,\;S_{\text{ent}},\;\text{pop\_count})C(t)=f(∣X−Y∣,Sent,pop_count)
* Memory ingot generation: hash(log-lines) same as Carousel approach