Here is the next evolutionary step of the toy simulation — now incorporating the three requested extensions in a still relatively minimal (but conceptually richer) form:

1. Mock graviton wave equation sourced by the braid proxy

2. Coupling to deterministic Ω(t) breathing (hourly jumps in the lattice / scalar background)

3. Full quaternion projection + lattice snapshot (simplified but genuine 24-cell style projection using quaternions)

Design Choices Summary

• Keeps 2D spatial grid + time stepping

• Adds deterministic hourly jumps via Ω(t)-style phase (modulo 24, with golden-ratio scaling jump)

• Uses quaternion multiplication for projection (very simplified — maps scalar field value → quaternion → rational part extraction → lattice-like snap)

• Introduces a wave-like equation for mock graviton field h (Klein–Gordon-like with source = braid proxy)

• Visualization now shows: scalar fields + mock graviton ripples + lattice snapshot overlay

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 — golden ratio & model parameters

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

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

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

GRID_SIZE = 64

STEPS_PER_HOUR = 20             # sub-steps per "hour"

TOTAL_HOURS = 8                 # simulate ~8 hours

TOTAL_STEPS = TOTAL_HOURS * STEPS_PER_HOUR

# Lattice jump strength (Ω(t) breathing amplitude)

OMEGA_JUMP_AMPL = 0.08 * phi    # small but visible hourly perturbation

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

# 1. Quaternion projection helper (simplified 24-cell style)

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

def quaternion_project(scalar_value, hour):

    """

    Very simplified: scalar → quaternion q = (w, x, y, z)

    Then extract "rational" part → snap toward lattice-like point

    """

    # Hour-dependent direction (cycling among 4 simple 24-cell-like directions)

    directions = [

        np.array([1, 1, 0, 0]) / np.sqrt(2),

        np.array([1, 0, 1, 0]) / np.sqrt(2),

        np.array([0.5, 0.5, 0.5, 0.5]),

        np.array([1, 0, 0, 1]) / np.sqrt(2)

    ]

    dir_idx = hour % len(directions)

    unit_dir = directions[dir_idx]

    # Scale by golden-powered scalar + small ψ contribution

    scale = scalar_value ** 1.2 + 0.04 * psi * scalar_value

    q = scale * unit_dir

    # "Project": take norm + rationalize by snapping components toward ±0.5, ±1 multiples

    norm = np.linalg.norm(q)

    if norm > 1e-8:

        q = q / norm

    snapped = np.round(q * 2) / 2   # snap to halves (mimics 24-cell / Hurwitz integer lattice)

    return snapped.mean()           # return average component as lattice proxy value

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

# 2. Dual β-functions with Ω(t) hourly breathing

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

def beta_dual_with_breathing(alpha_p, alpha_m, t_global, hour):

    deviation_p = alpha_p - phi

    deviation_m = alpha_m - psi

    # Base RG flow

    bp = -0.07 * deviation_p + 0.022 * deviation_m

    bm = -0.07 * deviation_m - 0.022 * deviation_p

    # Hourly Ω(t) breathing jump (deterministic, periodic)

    phase = 2 * np.pi * (hour % 24) / 24

    jump_p = OMEGA_JUMP_AMPL * np.cos(phase + 0.7) * phi

    jump_m = OMEGA_JUMP_AMPL * np.sin(phase + 1.4) * psi

    bp += jump_p

    bm += jump_m

    return bp, bm

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

# 3. Evolution loop — scalars + mock graviton wave

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

def run_simulation():

    x, y = np.meshgrid(np.linspace(-2.5, 2.5, GRID_SIZE),

                       np.linspace(-2.5, 2.5, GRID_SIZE))

    r2 = x**2 + y**2

    # Initial conditions

    a_plus  = phi + 1.4 * np.exp(-r2 / 0.9)

    a_minus = psi - 1.1 * np.exp(-r2 / 1.2)

    h = np.zeros_like(a_plus)          # graviton field (starts at rest)

    h_velocity = np.zeros_like(h)      # for wave equation

    snapshots = {'plus': [], 'minus': [], 'h': [], 'lattice': []}

    for step in range(TOTAL_STEPS):

        hour = step // STEPS_PER_HOUR

        t_global = step * 0.035

        # RG update with breathing

        bp, bm = beta_dual_with_breathing(a_plus, a_minus, t_global, hour)

        # Very weak diffusion

        lap_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)

        lap_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)

        da_p = -0.012 * bp + 0.004 * lap_p

        da_m = -0.012 * bm + 0.004 * lap_m

        a_plus  += da_p

        a_minus += da_m

        # Quaternion-projected lattice snapshot (once per hour)

        if step % STEPS_PER_HOUR == 0:

            lat_p = np.vectorize(lambda v: quaternion_project(v, hour))(a_plus)

            snapshots['lattice'].append(lat_p)

        else:

            snapshots['lattice'].append(snapshots['lattice'][-1] if snapshots['lattice'] else lat_p)

        # Mock graviton wave equation: □h = source = braid proxy

        C_proxy = 1.0 + 0.4 * np.sin(0.9 * (x + y + t_global))

        braid_source = (a_plus ** C_proxy - 0.6 * (a_minus ** C_proxy))

        lap_h = (np.roll(h,1,0) + np.roll(h,-1,0) +

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

        # Klein-Gordon like: ∂²h/∂t² = c² Δh - m² h + source

        # (here discretized very crudely — for visualization only)

        accel = 0.18 * lap_h - 0.03 * h + 0.07 * braid_source

        h_velocity += accel * 0.8

        h += h_velocity * 0.8

        h *= 0.995  # artificial damping

        # Save every 5 sub-steps

        if step % 5 == 0:

            snapshots['plus'].append(a_plus.copy())

            snapshots['minus'].append(a_minus.copy())

            snapshots['h'].append(h.copy())

    return snapshots, x, y

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

# Run & Animate

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

snapshots, X, Y = run_simulation()

frames = len(snapshots['plus'])

fig, axes = plt.subplots(2, 2, figsize=(13, 10))

def update(f):

    for ax in axes.flat:

        ax.clear()

    axes[0,0].imshow(snapshots['plus'][f], cmap='Oranges', vmin=phi-0.4, vmax=phi+1.6,

                     extent=[-2.5,2.5,-2.5,2.5], origin='lower')

    axes[0,0].set_title(f"α₊  (hour ≈ {f//(STEPS_PER_HOUR//5)})")

    axes[0,1].imshow(snapshots['minus'][f], cmap='PuRd', vmin=psi-0.6, vmax=psi+0.4,

                     extent=[-2.5,2.5,-2.5,2.5], origin='lower')

    axes[0,1].set_title(f"α₋")

    im_h = axes[1,0].imshow(snapshots['h'][f], cmap='coolwarm', vmin=-0.4, vmax=0.4,

                            extent=[-2.5,2.5,-2.5,2.5], origin='lower')

    axes[1,0].set_title("Mock graviton h (sourced by braid proxy)")

    lat = snapshots['lattice'][f]

    axes[1,1].imshow(lat, cmap='viridis', vmin=-0.7, vmax=0.7,

                     extent=[-2.5,2.5,-2.5,2.5], origin='lower')

    axes[1,1].set_title("Quaternion-projected lattice snapshot")

    for ax in axes.flat:

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

ani = FuncAnimation(fig, update, frames=frames, interval=120, blit=False)

plt.suptitle("Dual scalars + Ω(t) breathing + mock graviton + quaternion lattice projection", fontsize=13)

plt.tight_layout()

plt.show()

What this version demonstrates

• Ω(t) breathing — deterministic, periodic jumps every ~20 sub-steps (hourly), using golden-ratio modulated phase

• Quaternion projection — simplified but genuine: scalar → quaternion → norm + snap → lattice proxy field (updated hourly)

• Mock graviton — wave-like propagation sourced by braid proxy (α₊^C − ε α₋^C term)

• All fields live on the same 2D grid and interact indirectly through the source term

Limitations & Next Natural Steps

• Wave equation is extremely crude (no proper finite-difference scheme, no boundary conditions)

• Quaternion projection is toy-level (only 4 directions, simple snapping)

• No full τ-Hamiltonian or Painlevé structure yet

• No proper 24-cell coordinates table (could be added from literature values)

Possible continuations:

• Replace mock wave with 1+1D or 2D proper wave equation solver

• Use real tabulated 24-cell vertices (there are only 24 → can hard-code)

• Add phase-7 specific jump enhancement

• Export animation frames or save lattice time-series for later analysis

Let me know which aspect you'd like to deepen next.