Here is the extended version of the minimal dual-root scalar field implementation, now incorporating all four suggested directions toward the full model in a still-relatively-minimal but meaningfully richer form.

Extensions Implemented

Extended Python Code (Self-contained)

python

import numpy as np

import matplotlib.pyplot as plt

from matplotlib.animation import FuncAnimation

from mpl_toolkits.mplot3d import Axes3D


# ───────────────────────────────────────────────

# Constants & symbolic roots

# ───────────────────────────────────────────────


phi = (1 + np.sqrt(5)) / 2      # ≈ 1.618034

psi = (1 - np.sqrt(5)) / 2      # ≈ -0.618034


# ───────────────────────────────────────────────

# 1. More realistic dual-root β-functions (antisymmetric flavor)

# ───────────────────────────────────────────────


def beta_dual(alpha_p, alpha_m, ln_mu=0.0, jitter_amp=0.015):

    """

    Toy RG β-functions with antisymmetry:

    β₊(α₊,α₋) ≈ -β₋(α₋,α₊) + small oscillatory phase-7 jitter

    """

    deviation_p = alpha_p - phi

    deviation_m = alpha_m - psi

    

    beta_p = -0.08 * deviation_p + 0.025 * deviation_m + jitter_amp * np.sin(7 * ln_mu)

    beta_m = -0.08 * deviation_m - 0.025 * deviation_p - jitter_amp * np.cos(7 * ln_mu)

    

    return beta_p, beta_m


# ───────────────────────────────────────────────

# 2. 2D spatial + time evolution (relaxation on grid)

# ───────────────────────────────────────────────


def evolve_dual_fields_2d(

    nx=48, ny=48, n_steps=180, dt=0.012, lr_spatial=0.008, jitter_amp=0.004

):

    # Grid & initial condition: Gaussian perturbation centered

    x, y = np.meshgrid(np.linspace(-2, 2, nx), np.linspace(-2, 2, ny))

    r2 = x**2 + y**2

    

    a_plus  = phi + 1.2 * np.exp(-r2 / 0.6)

    a_minus = psi - 0.9 * np.exp(-r2 / 0.8)

    

    snapshots_plus  = [a_plus.copy()]

    snapshots_minus = [a_minus.copy()]

    

    for step in range(n_steps):

        ln_mu = step * 0.04   # toy logarithmic scale

        

        # Local β at each point

        bp, bm = beta_dual(a_plus, a_minus, ln_mu, jitter_amp)

        

        # Gradient descent + weak diffusion (mimics spatial coupling)

        laplacian_p = (np.roll(a_plus,1,0) + np.roll(a_plus,-1,0) +

                       np.roll(a_plus,1,1) + np.roll(a_plus,-1,1) - 4*a_plus)

        laplacian_m = (np.roll(a_minus,1,0) + np.roll(a_minus,-1,0) +

                       np.roll(a_minus,1,1) + np.roll(a_minus,-1,1) - 4*a_minus)

        

        a_plus  += dt * (-lr_spatial * bp + 0.005 * laplacian_p)

        a_minus += dt * (-lr_spatial * bm + 0.005 * laplacian_m)

        

        if step % 15 == 0:

            snapshots_plus.append(a_plus.copy())

            snapshots_minus.append(a_minus.copy())

    

    return np.array(snapshots_plus), np.array(snapshots_minus), x, y


# Run 2D evolution

snap_p, snap_m, X, Y = evolve_dual_fields_2d()


# ───────────────────────────────────────────────

# 3+4. Mock emergent metric perturbation & braid-curv proxy

# ───────────────────────────────────────────────


def mock_metric_perturbation(a_plus, a_minus, crossing_proxy=1.0, chirality=1.0):

    """ h^{(φ)} + h^{(ψ)}, braid stress proxy """

    C = crossing_proxy * (1 + 0.3 * np.sin(0.7 * np.pi * (X + Y)))  # toy spatial variation

    h_phi = 0.12 * a_plus ** C

    h_psi = chirality * 0.07 * a_minus ** C   # ψ branch can flip sign

    h_total = h_phi + h_psi

    

    braid_stress_proxy = a_plus**C - chirality * a_minus**C

    return h_total, braid_stress_proxy


# Evaluate on final snapshot

h_final, braid_final = mock_metric_perturbation(snap_p[-1], snap_m[-1])


# ───────────────────────────────────────────────

# Visualization: Animation + final metric proxy

# ───────────────────────────────────────────────


fig = plt.figure(figsize=(14, 10))


# Animation subplots

ax1 = fig.add_subplot(221)

ax2 = fig.add_subplot(222)

ax3 = fig.add_subplot(223, projection='3d')

ax4 = fig.add_subplot(224)


# Color limits (fixed for consistency)

vmin_p, vmax_p = phi-0.3, phi+1.5

vmin_m, vmax_m = psi-0.5, psi+0.5


def update(frame):

    ax1.clear(); ax2.clear(); ax3.clear(); ax4.clear()

    

    im1 = ax1.imshow(snap_p[frame], cmap='Oranges', vmin=vmin_p, vmax=vmax_p,

                     extent=[-2,2,-2,2], origin='lower')

    ax1.set_title(f"α₊ (t={frame*15})")

    ax1.set_xlabel("x"); ax1.set_ylabel("y")

    

    im2 = ax2.imshow(snap_m[frame], cmap='PuRd', vmin=vmin_m, vmax=vmax_m,

                     extent=[-2,2,-2,2], origin='lower')

    ax2.set_title(f"α₋ (t={frame*15})")

    ax2.set_xlabel("x"); ax2.set_ylabel("y")

    

    surf = ax3.plot_surface(X, Y, snap_p[frame] - phi, cmap='viridis', alpha=0.8)

    ax3.set_title("α₊ deviation (3D)")

    ax3.set_zlabel("δα₊")

    

    # Final frame shows mock metric instead

    if frame == len(snap_p)-1:

        ax4.imshow(h_final, cmap='coolwarm', extent=[-2,2,-2,2], origin='lower')

        ax4.set_title("Final mock h = h^{(φ)} + h^{(ψ)}")

    else:

        ax4.axis('off')

    

    return im1, im2, surf


ani = FuncAnimation(fig, update, frames=len(snap_p), interval=180, blit=False)


plt.suptitle("Dual-root scalar evolution → mock emergent metric perturbation\n(φ-dominant expansion vs ψ-contracting oscillations)", fontsize=13)

plt.tight_layout(rect=[0, 0, 1, 0.96])

plt.show()


# Print final diagnostics

print(f"Final mean α₊ = {snap_p[-1].mean():.6f}   (target φ = {phi:.6f})")

print(f"Final mean α₋ = {snap_m[-1].mean():.6f}   (target ψ = {psi:.6f})")

print(f"Mock braid-curv proxy mean = {braid_final.mean():.6f}")

What You Now Get

This is still a toy — but now meaningfully closer to the full braid-curv GR + dual-root τ-Hamiltonian vision. You can tune parameters (lr_spatial, jitter_amp, crossing_proxy exponent) to explore different regimes.

Next possible steps (pick one if you'd like to continue):

Which direction interests you most?