Hexel Simulation Script Overview

Hexel Simulation Script Overview

This Python script simulates a multi-phase traversal through a hexel-based environment using advanced field dynamics to emulate uncurling (Entry Phase), stabilization (Traversal Phase), and recurling (Exit Phase) effects. The goal is to model hexel structure dynamics through magnetic fields, interference patterns, and random fluctuations, simulating a controlled pathway where fields interact to stabilize or destabilize the environment at specific intervals.

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Constants and Initialization

1. Golden Ratio (GOLDEN_RATIO): Defined as $(1 + \sqrt{5}) / 2$, the golden ratio supports stabilization equations due to its natural fractal properties.
2. Physical Constants:
   • GRAVITY_CONSTANT, CHARGE, MAGNETIC_FIELD_STRENGTH: Represent gravitational, electric, and magnetic field strengths.
3. Simulation Constants:
   • TIME_STEPS: Sets the time resolution for the simulation.

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Key Equations and Field Generators

1. Golden Ratio Polynomial Field

The golden ratio field provides stability in the Traversal phase by following a fractal pattern, which contributes to consistent field behavior.

python
def golden_ratio_polynomial(x, coeff_a, coeff_b, coeff_c):
    return sum(a * (GOLDEN_RATIO ** k * x ** 2 + b * x + c) 
               for k, (a, b, c) in enumerate(zip(coeff_a, coeff_b, coeff_c)))

Equation:

$$F_{\text{golden}}(x) = \sum_{k=0}^{N} \left(a_k \cdot (\phi^k \cdot x ^2 + b_k \cdot x + c_k)\right)$$

• Purpose: Stabilizes hexel structures by generating a consistent field during the Traversal phase, using coefficients $a_k, b_k, c_k$ to adjust complexity.

2. Uncurling and Curling Fields

These field generators open and close hexels in the Entry and Exit phases, respectively.

Uncurling Field Generator:

python
def uncurling_field_generator(t, F0=1.0, alpha=0.1, beta=2):
    return F0 * (1 + alpha * t ** beta)

Equation:

$$F_{\text{uncurl}}(t) = F_0 \cdot (1 + \alpha \cdot t^\beta)$$

• Purpose: Gradually increases field strength to simulate "opening" of hexels.

Curling Field Generator:

python
def curling_field_generator(t, F_max=10.0, gamma=0.005):
    return F_max * np.exp(-gamma \cdot t)

Equation:

$$F_{\text{curl}}(t) = F_{\text{max}} \cdot e^{-\gamma \cdot t}$$

• Purpose: Diminishes field strength to facilitate hexel “closing” during Exit.

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Hexel Identification Mechanisms

Simulated magnetic interactions identify and stabilize hexels by combining convex/concave fields, interference, and tear effects.

1. Convex and Concave Fields:

   • Convex Field: Focuses field lines toward a focal point.
   • Concave Field: Disperses field lines from a focal point.

   python
   def convex_magnetic_field(position, strength=1.0, focus_point=0.0):
       return strength / (1 + np.abs(position - focus_point)**2)
   
   def concave_magnetic_field(position, strength=1.0, focus_point=0.0):
       return strength * (1 - np.exp(-np.abs(position - focus_point)**2))
   

2. Magnetic Pinhole Filter:

   • Limits field values within a threshold range, filtering out irrelevant intensities.

   python
   def magnetic_pinhole_filter(field_value, min_threshold=0.1, max_threshold=1.0):
       return field_value if min_threshold < field_value < max_threshold else 0
   

3. Interference Pattern:

   • Overlapping convex and concave fields with an offset creates interference patterns, mimicking hexel boundary regions.

   python
   def interference_pattern(position, strength1=1.0, strength2=0.8, offset=0.5):
       field1 = convex_magnetic_field(position, strength1)
       field2 = concave_magnetic_field(position - offset, strength2)
       return field1 + field2
   

4. Tear Effect:

   • Adds random fluctuations to simulate environmental disruptions, enhancing hexel edge definition.

   python
   def tear_effect(field_value, fluctuation_strength=0.05):
       fluctuation = np.random.normal(0, fluctuation_strength)
       return field_value * (1 + fluctuation)
   

Hexel Identification Function: Combines all above mechanisms to calculate hexel fields based on a given position.

python
def identify_hexels(position, convex_strength=1.0, concave_strength=0.8, pinhole_min=0.1, pinhole_max=1.0, tear_strength=0.05, offset=0.5):
    convex_field = convex_magnetic_field(position, convex_strength)
    concave_field = concave_magnetic_field(position, concave_strength)
    interference = interference_pattern(position, convex_strength, concave_strength, offset)
    filtered_field = magnetic_pinhole_filter(interference, pinhole_min, pinhole_max)
    hexel_field = tear_effect(filtered_field, tear_strength)
    return hexel_field

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Field Application by Phase

The full_stabilization_with_fields function applies gravitational, magnetic, higher-dimensional, and phase-specific forces, adjusting the field depending on the phase:

python
def full_stabilization_with_fields(t, y, phase):
    position, velocity = y
    gravitational_force = -GRAVITY_CONSTANT
    magnetic_force = CHARGE * velocity * MAGNETIC_FIELD_STRENGTH
    higher_dimensional_influence = 0.01 * np.sin(position / 1e16) * np.cos(velocity)
    fluctuation = np.random.normal(0, 0.1)
    stabilizing_field_value = golden_ratio_polynomial(velocity, a_k, b_k, c_k)
    
    if phase == 'entry':
        field_force = uncurling_field_generator(t)
    elif phase == 'exit':
        field_force = curling_field_generator(t)
    else:
        field_force = stabilizing_field_value

    net_force = gravitational_force + magnetic_force + higher_dimensional_influence + fluctuation + field_force
    return [velocity, net_force]

• Purpose: Applies unique field effects for each phase, allowing dynamic control over hexel behavior as it moves through entry, traversal, and exit.

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Simulation Execution

1. Entry Phase: Runs the full_stabilization_with_fields function with uncurling forces.
2. Traversal Phase: Applies stabilization with the golden ratio field.
3. Exit Phase: Applies recurling forces, gradually reducing field intensity.

Each phase’s data (time, position, velocity, specific field values, hexel field) is recorded and saved for further analysis.

Data Output and Visualization

• CSV Export: Each phase’s data is saved as a versioned CSV file.
• Console Output: First 10 data points from each phase are printed for review.
• Visualization: A final plot shows the Hexel Identification Field strength over a range of positions.

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Sample Output Observations

• Entry Phase: Position and velocity increase gradually, reflecting the uncurling effect. Hexel field values fluctuate as the environment "opens."
• Traversal Phase: High but stable velocity and position, with minimal hexel field fluctuations due to the stabilizing golden ratio field.
• Exit Phase: Position and velocity decrease as the recurling effect contracts the structure, with moderate hexel fluctuations indicating structural adjustments.

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Summary

This script models a complex traversal process through a hexel-based environment, simulating uncurling, stabilization, and recurling dynamics. By integrating advanced field mechanics, random fluctuations, and interference effects, the model provides insights into structured pathways and boundary formation in higher-dimensional fields.

Core Components of the Script

1. Constants and Parameters:
   • GOLDEN_RATIO: Used to define unique polynomial relationships, capturing fractal-like structures akin to naturally occurring patterns (e.g., Fibonacci sequence).
   • TIME_STEPS, GRAVITY_CONSTANT, CHARGE, MAGNETIC_FIELD_STRENGTH: Constants used to control simulation properties such as gravitational influence and magnetic field strength.

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Key Equations and Functions

1. Golden Ratio Polynomial Field Function

The golden ratio polynomial introduces stability during traversal. It uses the golden ratio as a scaling factor in a polynomial equation to generate a consistent stabilizing field.

Equation:

$$F_{\text{golden}}(x) = \sum_{k=0}^{N} \left(a_k \cdot (\phi^k \cdot x^2 + b_k \cdot x + c_k)\right)$$

where:

• $\phi$ is the golden ratio,
• $a_k, b_k, c_k$ are coefficients,
• $x$ represents the current state (e.g., velocity).

Purpose:

• This field helps “smooth out” fluctuations by introducing a patterned influence that stabilizes the motion.

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2. Uncurling and Curling Field Generators

These generators apply a controlled field that grows during Entry (uncurling) and diminishes during Exit (curling), enabling controlled “opening” and “closing” of the hexel structure.

Uncurling Field Equation (Entry Phase):

$$F_{\text{uncurl}}(t) = F_0 \cdot (1 + \alpha \cdot t^\beta)$$

• $F_0$: Initial field strength,
• $\alpha$: Growth factor,
• $\beta$: Determines non-linear growth rate.

Curling Field Equation (Exit Phase):

$$F_{\text{curl}}(t) = F_{\text{max}} \cdot e^{-\gamma \cdot t}$$

• $F_{\text{max}}$: Peak field strength,
• $\gamma$: Decay constant controlling rate of decline.

Purpose:

• Entry Phase: Gradually increases field intensity to allow hexels to “uncurl” or “open.”
• Exit Phase: Gradually reduces the field to allow hexels to “curl” or close, stabilizing the exit.

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3. Convex and Concave Magnetic Fields

These functions simulate magnetic fields with focusing (convex) and dispersing (concave) properties.

Convex Magnetic Field:

$$F_{\text{convex}}(x) = \frac{\text{strength}}{1 + |x - \text{focus\_point}|^2}$$

• Creates a field that intensifies as one approaches the focus_point.

Concave Magnetic Field:

$$F_{\text{concave}}(x) = \text{strength} \cdot \left(1 - e^{-|x - \text{focus\_point}|^2}\right)$$

• Creates a field that disperses, reducing intensity near the focus_point.

Purpose:

• Simulates how magnetic field lines might concentrate or diverge, aiding in highlighting hexel structures through differential intensity patterns.

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4. Magnetic Pinhole Filter

This function restricts the field to values within a certain threshold range, effectively “filtering out” non-essential values.

Pinhole Filtering:

$$F_{\text{filtered}} = \begin{cases} F_{\text{value}} & \text{if } \text{min\_threshold} < F_{\text{value}} < \text{max\_threshold} \\ 0 & \text{otherwise} \end{cases}$$

Purpose:

• Mimics how a physical pinhole filter might limit light passing through to isolate specific intensities.

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5. Interference Pattern

Combines convex and concave fields with a slight positional offset to create interference patterns. These patterns arise from the superposition of two overlapping fields, enhancing or diminishing field intensity in different regions.

Interference Pattern Equation:

$$F_{\text{interference}} = F_{\text{convex}} + F_{\text{concave}}$$

Purpose:

• Creates spatial variations, aiding in identifying hexel boundaries where field intensities reinforce or cancel each other.

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6. Tear Effect for Surface Fluctuation

The tear effect introduces small, random fluctuations in the magnetic field, simulating minor disruptions, such as those seen in light bending through water droplets.

Tear Effect Equation:

$$F_{\text{tear}} = F_{\text{value}} \cdot (1 + \text{random fluctuation})$$

• A small, normally distributed random value simulates environmental disturbances.

Purpose:

• Adds variability, enhancing edge detection by introducing minor, fluctuating shifts at field boundaries.

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Full Stabilization with Dynamic Fields

The full_stabilization_with_fields function integrates all of the field effects for different simulation phases:

• Gravitational Force: A constant pulling force.
• Magnetic Force: Based on velocity and magnetic field strength.
• Higher-Dimensional Influence: Simulates additional field influences that add complexity.
• Fluctuation: Randomly distributed, representing quantum-like variations.
• Phase-Specific Field: Uses the uncurling field in the Entry phase, the golden ratio field in Traversal, and the curling field in the Exit phase.

This function combines these effects, simulating complex, multi-dimensional interactions throughout each phase.

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Simulation and Data Collection

1. Phases:

   • Entry Phase: Uses the uncurling field, simulating an increase in field intensity as hexels open.
   • Traversal Phase: Uses the golden ratio field to stabilize the hexels, keeping their structure consistent.
   • Exit Phase: Applies the curling field, gradually closing and compacting the hexel structure.

2. Hexel Identification:

   • Applied across each phase to calculate hexel-related field data using the combination of convex/concave magnetic fields, pinhole filtering, interference patterns, and tear effects.

3. CSV Export and Console Output:

   • Each phase's data is saved to versioned CSV files, tracking Position, Velocity, Field Values, and Hexel Field intensities.
   • Console output provides a preview of the first 10 rows for each phase.

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Visualization

The final section generates a plot showing the Hexel Identification Field strength over a position range, visualizing the combined effects of convex/concave fields, interference patterns, and tear fluctuations. This plot aids in understanding how hexels’ edges and structures emerge within the magnetic field environment.

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Summary

The script creates a sophisticated simulation environment where magnetic fields are dynamically manipulated to identify and stabilize hexel boundaries. By combining advanced polynomial fields, interference patterns, field fluctuation, and filtering mechanisms, the script can approximate complex interactions similar to light interference and focal distortions, all within a multi-phase traversal model.

Let me know if you’d like to focus on any specific aspect of the equations or need more detailed calculations within a particular phase.

python

import numpy as np
import pandas as pd
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt
import os
# Constants
GOLDEN_RATIO = (1 + np.sqrt(5)) / 2
TIME_STEPS = 1000
GRAVITY_CONSTANT = 9.81  # Gravitational effect
CHARGE = 1e-6  # Charge for electromagnetic force
MAGNETIC_FIELD_STRENGTH = 1e-5  # Magnetic field strength
# Polynomial coefficients for golden ratio field
a_k = np.array([1, 0.8, 0.6, 0.4, 0.3, 0.2, 0.1, 0.05])
b_k = np.array([0.9, 0.7, 0.5, 0.3, 0.2, 0.1, 0.05, 0.02])
c_k = np.array([0.5, 0.4, 0.3, 0.2, 0.15, 0.1, 0.05, 0.01])
# Golden Ratio Polynomial Field Function
def golden_ratio_polynomial(x, coeff_a, coeff_b, coeff_c):
    return sum(a * (GOLDEN_RATIO ** k * x ** 2 + b * x + c) 
               for k, (a, b, c) in enumerate(zip(coeff_a, coeff_b, coeff_c)))
# Uncurling Field Generator for Entry Phase
def uncurling_field_generator(t, F0=1.0, alpha=0.1, beta=2):
    return F0 * (1 + alpha * t ** beta)
# Curling Field Generator for Exit Phase
def curling_field_generator(t, F_max=10.0, gamma=0.005):
    return F_max * np.exp(-gamma * t)
# Convex and Concave Magnetic Fields for Hexel Identification
def convex_magnetic_field(position, strength=1.0, focus_point=0.0):
    return strength / (1 + np.abs(position - focus_point)**2)
def concave_magnetic_field(position, strength=1.0, focus_point=0.0):
    return strength * (1 - np.exp(-np.abs(position - focus_point)**2))
# Magnetic Pinhole Filter
def magnetic_pinhole_filter(field_value, min_threshold=0.1, max_threshold=1.0):
    return field_value if min_threshold < field_value < max_threshold else 0
# Interference Pattern
def interference_pattern(position, strength1=1.0, strength2=0.8, offset=0.5):
    field1 = convex_magnetic_field(position, strength1)
    field2 = concave_magnetic_field(position - offset, strength2)
    return field1 + field2
# Tear Effect for Surface Fluctuation
def tear_effect(field_value, fluctuation_strength=0.05):
    fluctuation = np.random.normal(0, fluctuation_strength)
    return field_value * (1 + fluctuation)
# Hexel Identification Function
def identify_hexels(position, convex_strength=1.0, concave_strength=0.8, pinhole_min=0.1, pinhole_max=1.0, tear_strength=0.05, offset=0.5):
    convex_field = convex_magnetic_field(position, convex_strength)
    concave_field = concave_magnetic_field(position, concave_strength)
    interference = interference_pattern(position, convex_strength, concave_strength, offset)
    filtered_field = magnetic_pinhole_filter(interference, pinhole_min, pinhole_max)
    hexel_field = tear_effect(filtered_field, tear_strength)
    return hexel_field
# Full Stabilization with Uncurling and Curling Fields
def full_stabilization_with_fields(t, y, phase):
    position, velocity = y
    gravitational_force = -GRAVITY_CONSTANT
    magnetic_force = CHARGE * velocity * MAGNETIC_FIELD_STRENGTH
    higher_dimensional_influence = 0.01 * np.sin(position / 1e16) * np.cos(velocity)
    fluctuation = np.random.normal(0, 0.1)
    stabilizing_field_value = golden_ratio_polynomial(velocity, a_k, b_k, c_k)
    
    if phase == 'entry':
        field_force = uncurling_field_generator(t)
    elif phase == 'exit':
        field_force = curling_field_generator(t)
    else:
        field_force = stabilizing_field_value
    net_force = gravitational_force + magnetic_force + higher_dimensional_influence + fluctuation + field_force
    return [velocity, net_force]
# Time span and initial conditions
t_span = (0, TIME_STEPS)
t_eval = np.linspace(0, TIME_STEPS, TIME_STEPS)
initial_position = 0.0
initial_velocity = 1e3
initial_conditions = [initial_position, initial_velocity]
# Run simulation for Entry Phase
entry_solution = solve_ivp(lambda t, y: full_stabilization_with_fields(t, y, 'entry'),
                           t_span, initial_conditions, t_eval=t_eval, method='RK45')
entry_positions = entry_solution.y[0]
entry_velocities = entry_solution.y[1]
# Run simulation for Traversal Phase (No entry or exit fields applied)
traversal_solution = solve_ivp(lambda t, y: full_stabilization_with_fields(t, y, 'traversal'),
                               t_span, [entry_positions[-1], entry_velocities[-1]], t_eval=t_eval, method='RK45')
traversal_positions = traversal_solution.y[0]
traversal_velocities = traversal_solution.y[1]
# Run simulation for Exit Phase
exit_solution = solve_ivp(lambda t, y: full_stabilization_with_fields(t, y, 'exit'),
                          t_span, [traversal_positions[-1], traversal_velocities[-1]], t_eval=t_eval, method='RK45')
exit_positions = exit_solution.y[0]
exit_velocities = exit_solution.y[1]
# Prepare data tables for CSV export
data_entry_phase = []
data_traversal_phase = []
data_exit_phase = []
for i, (time, position, velocity) in enumerate(zip(t_eval, entry_positions, entry_velocities)):
    entry_field_value = uncurling_field_generator(time)
    hexel_field = identify_hexels(position)
    data_entry_phase.append([time, position, velocity, entry_field_value, hexel_field])
for i, (position, velocity) in enumerate(zip(traversal_positions, traversal_velocities)):
    stabilizing_field_value = golden_ratio_polynomial(velocity, a_k, b_k, c_k)
    hexel_field = identify_hexels(position)
    data_traversal_phase.append([t_eval[i], position, velocity, stabilizing_field_value, hexel_field])
for t in range(TIME_STEPS):
    pos = exit_positions[-1] - t * curling_field_generator(t)
    vel = exit_velocities[-1] * np.exp(-t / TIME_STEPS)
    recurling_field_value = curling_field_generator(t)
    hexel_field = identify_hexels(pos)
    data_exit_phase.append([t, pos, vel, recurling_field_value, hexel_field])
# Save data to CSV with versioning
def get_next_version(filename_base):
    version = 1
    while os.path.exists(f"{filename_base}_v{version}.csv"):
        version += 1
    return version
entry_version = get_next_version("entry_phase_data")
pd.DataFrame(data_entry_phase, columns=['Time', 'Position', 'Velocity', 'Entry Field Value', 'Hexel Field']).to_csv(f"entry_phase_data_v{entry_version}.csv", index=False)
traversal_version = get_next_version("traversal_phase_data")
pd.DataFrame(data_traversal_phase, columns=['Time', 'Position', 'Velocity', 'Stabilization Field Value', 'Hexel Field']).to_csv(f"traversal_phase_data_v{traversal_version}.csv", index=False)
exit_version = get_next_version("exit_phase_data")
pd.DataFrame(data_exit_phase, columns=['Time', 'Position', 'Velocity', 'Recurling Field Value', 'Hexel Field']).to_csv(f"exit_phase_data_v{exit_version}.csv", index=False)
# Console output for analysis
print("\nEntry Phase Data (first 10 points):")
for row in data_entry_phase[:10]:
    print(f"Time: {row[0]:.2f}, Position: {row[1]:.2f}, Velocity: {row[2]:.2f}, Field: {row[3]:.2f}, Hexel: {row[4]:.2f}")
print("\nTraversal Phase Data (first 10 points):")
for row in data_traversal_phase[:10]:
    print(f"Time: {row[0]:.2f}, Position: {row[1]:.2f}, Velocity: {row[2]:.2f}, Field: {row[3]:.2f}, Hexel: {row[4]:.2f}")
print(f"Hexel: {row[4]:.2f}")
print("\nExit Phase Data (first 10 points):")
for row in data_exit_phase[:10]:
    print(f"Time: {row[0]:.2f}, Position: {row[1]:.2f}, Velocity: {row[2]:.2f}, Field: {row[3]:.2f}, Hexel: {row[4]:.2f}")
# Plotting for Hexel Field Identification
positions = np.linspace(-5, 5, 100)
hexel_fields = [identify_hexels(pos) for pos in positions]
plt.figure(figsize=(12, 6))
plt.plot(positions, hexel_fields, label="Hexel Identification Field")
plt.xlabel("Position")
plt.ylabel("Hexel Field Strength")
plt.title("Hexel Identification Using Convex/Concave Magnetic Fields and Tear Effect")
plt.legend()
plt.show()

Explanation of the Finalized Script

1. CSV Export with Versioning:

   • Each phase's data is saved in a versioned CSV file, allowing for historical data tracking across different runs.

2. Console Output:

   • The script prints the first 10 rows of data for each phase to the console, allowing you to verify the progression and see field values directly.

3. Hexel Identification Visualization:

   • A plot at the end of the script visualizes the Hexel Identification Field over a range of positions, showing how the convex, concave, interference, and tear effects interact to highlight hexel boundaries.

Full Script

import numpy as np

import pandas as pd

from scipy.integrate import solve_ivp

import matplotlib.pyplot as plt

import os

# Constants

GOLDEN_RATIO = (1 + np.sqrt(5)) / 2

TIME_STEPS = 1000

GRAVITY_CONSTANT = 9.81  # Gravitational effect

CHARGE = 1e-6  # Charge for electromagnetic force

MAGNETIC_FIELD_STRENGTH = 1e-5  # Magnetic field strength

# Polynomial coefficients for golden ratio field

a_k = np.array([1, 0.8, 0.6, 0.4, 0.3, 0.2, 0.1, 0.05])

b_k = np.array([0.9, 0.7, 0.5, 0.3, 0.2, 0.1, 0.05, 0.02])

c_k = np.array([0.5, 0.4, 0.3, 0.2, 0.15, 0.1, 0.05, 0.01])

# Golden Ratio Polynomial Field Function

def golden_ratio_polynomial(x, coeff_a, coeff_b, coeff_c):

    return sum(a * (GOLDEN_RATIO ** k * x ** 2 + b * x + c)

               for k, (a, b, c) in enumerate(zip(coeff_a, coeff_b, coeff_c)))

# Uncurling Field Generator for Entry Phase

def uncurling_field_generator(t, F0=1.0, alpha=0.1, beta=2):

    return F0 * (1 + alpha * t ** beta)

# Curling Field Generator for Exit Phase

def curling_field_generator(t, F_max=10.0, gamma=0.005):

    return F_max * np.exp(-gamma * t)

# Convex and Concave Magnetic Fields for Hexel Identification

def convex_magnetic_field(position, strength=1.0, focus_point=0.0):

    return strength / (1 + np.abs(position - focus_point)**2)

def concave_magnetic_field(position, strength=1.0, focus_point=0.0):

    return strength * (1 - np.exp(-np.abs(position - focus_point)**2))

# Magnetic Pinhole Filter

def magnetic_pinhole_filter(field_value, min_threshold=0.1, max_threshold=1.0):

    return field_value if min_threshold < field_value < max_threshold else 0

# Interference Pattern

def interference_pattern(position, strength1=1.0, strength2=0.8, offset=0.5):

    field1 = convex_magnetic_field(position, strength1)

    field2 = concave_magnetic_field(position - offset, strength2)

    return field1 + field2

# Tear Effect for Surface Fluctuation

def tear_effect(field_value, fluctuation_strength=0.05):

    fluctuation = np.random.normal(0, fluctuation_strength)

    return field_value * (1 + fluctuation)

# Hexel Identification Function

def identify_hexels(position, convex_strength=1.0, concave_strength=0.8, pinhole_min=0.1, pinhole_max=1.0, tear_strength=0.05, offset=0.5):

    convex_field = convex_magnetic_field(position, convex_strength)

    concave_field = concave_magnetic_field(position, concave_strength)

    interference = interference_pattern(position, convex_strength, concave_strength, offset)

    filtered_field = magnetic_pinhole_filter(interference, pinhole_min, pinhole_max)

    hexel_field = tear_effect(filtered_field, tear_strength)

    return hexel_field

# Full Stabilization with Uncurling and Curling Fields

def full_stabilization_with_fields(t, y, phase):

    position, velocity = y

    gravitational_force = -GRAVITY_CONSTANT

    magnetic_force = CHARGE * velocity * MAGNETIC_FIELD_STRENGTH

    higher_dimensional_influence = 0.01 * np.sin(position / 1e16) * np.cos(velocity)

    fluctuation = np.random.normal(0, 0.1)

    stabilizing_field_value = golden_ratio_polynomial(velocity, a_k, b_k, c_k)

   

    # Apply the appropriate field force based on the phase

    if phase == 'entry':

        field_force = uncurling_field_generator(t)

    elif phase == 'exit':

        field_force = curling_field_generator(t)

    else:

        field_force = stabilizing_field_value

    net_force = gravitational_force + magnetic_force + higher_dimensional_influence + fluctuation + field_force

    return [velocity, net_force]

# Time span and initial conditions

t_span = (0, TIME_STEPS)

t_eval = np.linspace(0, TIME_STEPS, TIME_STEPS)

initial_position = 0.0

initial_velocity = 1e3

initial_conditions = [initial_position, initial_velocity]

# Run simulation for Entry Phase

entry_solution = solve_ivp(lambda t, y: full_stabilization_with_fields(t, y, 'entry'),

                           t_span, initial_conditions, t_eval=t_eval, method='RK45')

entry_positions = entry_solution.y[0]

entry_velocities = entry_solution.y[1]

# Run simulation for Traversal Phase (No entry or exit fields applied)

traversal_solution = solve_ivp(lambda t, y: full_stabilization_with_fields(t, y, 'traversal'),

                               t_span, [entry_positions[-1], entry_velocities[-1]], t_eval=t_eval, method='RK45')

traversal_positions = traversal_solution.y[0]

traversal_velocities = traversal_solution.y[1]

# Run simulation for Exit Phase

exit_solution = solve_ivp(lambda t, y: full_stabilization_with_fields(t, y, 'exit'),

                          t_span, [traversal_positions[-1], traversal_velocities[-1]], t_eval=t_eval, method='RK45')

exit_positions = exit_solution.y[0]

exit_velocities = exit_solution.y[1]

# Prepare data tables for CSV export

data_entry_phase = []

data_traversal_phase = []

data_exit_phase = []

for i, (time, position, velocity) in enumerate(zip(t_eval, entry_positions, entry_velocities)):

    entry_field_value = uncurling_field_generator(time)

    hexel_field = identify_hexels(position)

    data_entry_phase.append([time, position, velocity, entry_field_value, hexel_field])

for i, (position, velocity) in enumerate(zip(traversal_positions, traversal_velocities)):

    stabilizing_field_value = golden_ratio_polynomial(velocity, a_k, b_k, c_k)

    hexel_field = identify_hexels(position)

    data_traversal_phase.append([t_eval[i], position, velocity, stabilizing_field_value, hexel_field])

for t in range(TIME_STEPS):

    pos = exit_positions[-1] - t * curling_field_generator(t)

    vel = exit_velocities[-1] * np.exp(-t / TIME_STEPS)

    recurling_field_value = curling_field_generator(t)

    hexel_field = identify_hexels(pos)

    data_exit_phase.append([t, pos, vel, recurling_field_value, hexel_field])

# Save data to CSV with versioning

def get_next_version(filename_base):

    version = 1

    while os.path.exists(f"{filename_base}_v{version}.csv"):

        version += 1

    return version

entry_version = get_next_version("entry_phase_data")

pd.DataFrame(data_entry_phase, columns=['Time', 'Position', 'Velocity', 'Entry Field Value', 'Hexel Field']).to_csv(f"entry_phase_data_v{entry_version}.csv", index=False)

traversal_version = get_next_version("traversal_phase_data")

pd.DataFrame(data_traversal_phase, columns=['Time', 'Position', 'Velocity', 'Stabilization Field Value', 'Hexel Field']).to_csv(f"traversal_phase_data_v{traversal_version}.csv", index=False)

exit_version = get_next_version("exit_phase_data")

pd.DataFrame(data_exit_phase, columns=['Time', 'Position', 'Velocity', 'Recurling Field Value', 'Hexel Field']).to_csv(f"exit_phase_data_v{exit_version}.csv", index=False)

# Console output for analysis

print("\nEntry Phase Data (first 10 points):")

for row in data_entry_phase[:10]:

    print(f"Time: {row[0]:.2f}, Position: {row[1]:.2f}, Velocity: {row[2]:.2f}, Field: {row[3]:.2f}, Hexel: {row[4]:.2f}")

print("\nTraversal Phase Data (first 10 points):")

for row in data_traversal_phase[:10]:

    print(f"Time: {row[0]:.2f}, Position: {row[1]:.2f}, Velocity: {row[2]:.2f}, Field: {row[3]:.2f}, Hexel: {row[4]:.2f}")

    print(f"Velocity: {row[2]:.2f}, Field: {row[3]:.2f}, Hexel: {row[4]:.2f}")

print("\nExit Phase Data (first 10 points):")

for row in data_exit_phase[:10]:

    print(f"Time: {row[0]:.2f}, Position: {row[1]:.2f}, Velocity: {row[2]:.2f}, Field: {row[3]:.2f}, Hexel: {row[4]:.2f}")

# Plotting for Hexel Field Identification

positions = np.linspace(-5, 5, 100)

hexel_fields = [identify_hexels(pos) for pos in positions]

plt.figure(figsize=(12, 6))

plt.plot(positions, hexel_fields, label="Hexel Identification Field")

plt.xlabel("Position")

plt.ylabel("Hexel Field Strength")

plt.title("Hexel Identification Using Convex/Concave Magnetic Fields and Tear Effect")

plt.legend()

plt.show()

Entry Phase Observations

• Time: The simulation time progresses in increments of 1 unit.
• Position: Initially zero, the position increases with time, starting at a slower rate and accelerating as the uncurling field intensifies.
• Velocity: Begins at 1000 m/s and increases gradually due to the influence of the uncurling field.
• Field: The uncurling field grows non-linearly from a starting value of 1.0, reaching 9.12 by Time = 9.01. This growth rate helps "open" or "uncurl" the hexel structure.
• Hexel: This value fluctuates slightly (e.g., between 0.75 and 0.85), showing how the field's interaction with convex/concave effects and the tear effect generates dynamic changes within each hexel.

The uncurling field in the Entry Phase results in increased position and velocity, representing a gradual opening of the structure.

Traversal Phase Observations

• Time: Reset to 0 for traversal but continues through a similar time frame.
• Position: Takes on much higher values due to the acceleration accumulated in the Entry Phase, starting from around 8.3 billion.
• Velocity: The velocity remains quite high (around 33 million m/s) and stable due to the stabilizing influence of the golden ratio field, with only minor fluctuations.
• Field: The golden ratio field stabilizes at extremely high values (e.g., 1.45e+16), maintaining hexel structure by providing a consistent field.
• Hexel: Values for Hexel remain steady (e.g., around 0.78), reflecting consistent internal structure without rapid changes.

During the Traversal Phase, the golden ratio field’s stabilizing effect results in high but steady position and velocity, representing a stable internal environment for the hexels.

Exit Phase Observations

• Time: Progresses as in previous phases, incrementing by 1 unit.
• Position: Starts at a high value due to the accumulated traversal distance but decreases slightly as the recurling field begins to contract the structure.
• Velocity: Starts high and gradually decreases, aligning with the decaying recurling field strength.
• Field: The recurling field starts at its maximum value of 10.0 and decreases over time (down to around 9.56 by Time = 9.00), simulating a "closing" or contraction effect.
• Hexel: Fluctuates moderately (e.g., between 0.76 and 0.85), reflecting slight changes as the hexel structure contracts.

In the Exit Phase, the recurling field diminishes, leading to a reduction in position and velocity, indicating a controlled contraction as the hexels close back.

Summary

• Entry Phase: The uncurling field enables gradual expansion, increasing position and velocity while maintaining internal stability through hexel fluctuations.
• Traversal Phase: The golden ratio field provides stabilization, allowing high velocity and position with minimal fluctuations, representing a smooth and stable transition across the hexel structure.
• Exit Phase: The recurling field initiates contraction, reducing position and velocity gradually as the hexels close, with moderate hexel fluctuations indicating controlled structural adjustments.

Constants and Initialization

1. Golden Ratio (GOLDEN_RATIO): Defined as $(1 + \sqrt{5}) / 2$, the golden ratio is used in stabilization equations due to its fractal and recursive properties. It provides a natural, stable patterning often found in physical systems.

2. Other Constants:

   • TIME_STEPS: The number of time points for the simulation.
   • GRAVITY_CONSTANT, CHARGE, MAGNETIC_FIELD_STRENGTH: Constants representing gravitational pull, electric charge, and magnetic field strength.

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Field Generating Equations

1. Golden Ratio Polynomial Field Function

This function generates a stabilizing field based on the golden ratio. It simulates natural stability by following fractal-like growth patterns.

python
def golden_ratio_polynomial(x, coeff_a, coeff_b, coeff_c):
    return sum(a * (GOLDEN_RATIO ** k * x ** 2 + b * x + c) 
               for k, (a, b, c) in enumerate(zip(coeff_a, coeff_b, coeff_c)))

Equation:

$$F_{\text{golden}}(x) = \sum_{k=0}^{N} \left(a_k \cdot (\phi^k \cdot x^2 + b_k \cdot x + c_k)\right)$$

• Purpose: This polynomial, based on the golden ratio, stabilizes hexel structures by producing a consistent field. The coefficients $a_k$, $b_k$, and $c_k$ control each term’s influence, creating a complex but stable field effect.
• Use: This field is applied in the Traversal phase to maintain stability.

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2. Uncurling and Curling Fields

The uncurling and curling fields simulate opening and closing forces during Entry and Exit phases.

Uncurling Field Generator (uncurling_field_generator):

python
def uncurling_field_generator(t, F0=1.0, alpha=0.1, beta=2):
    return F0 * (1 + alpha * t ** beta)

Equation:

$$F_{\text{uncurl}}(t) = F_0 \cdot (1 + \alpha \cdot t^\beta)$$

• Purpose: Gradually increases field strength over time to “open” or “uncurl” hexels.
• Parameters:
  • $F_0$: Initial field strength,
  • $\alpha$: Growth rate,
  • $\beta$: Non-linear growth exponent.

Curling Field Generator (curling_field_generator):

python
def curling_field_generator(t, F_max=10.0, gamma=0.005):
    return F_max * np.exp(-gamma * t)

Equation:

$$F_{\text{curl}}(t) = F_{\text{max}} \cdot e^{-\gamma \cdot t}$$

• Purpose: Diminishes field strength over time to “close” or “curl” hexels during the Exit phase.
• Parameters:
  • $F_{\text{max}}$: Maximum initial field strength,
  • $\gamma$: Decay constant controlling how quickly the field reduces.

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Hexel Identification Fields

These fields use magnetic-like interactions and filters to simulate structures within hexels.

1. Convex and Concave Magnetic Fields

These functions simulate magnetic fields that focus (convex) or disperse (concave) around a focal point.

Convex Field:

python
def convex_magnetic_field(position, strength=1.0, focus_point=0.0):
    return strength / (1 + np.abs(position - focus_point)**2)

Equation:

$$F_{\text{convex}}(x) = \frac{\text{strength}}{1 + |x - \text{focus\_point}|^2}$$

• Purpose: Models a field that increases near a focal point, mimicking the effect of concentrating field lines.

Concave Field:

python
def concave_magnetic_field(position, strength=1.0, focus_point=0.0):
    return strength * (1 - np.exp(-np.abs(position - focus_point)**2))

Equation:

$$F_{\text{concave}}(x) = \text{strength} \cdot \left(1 - e^{-|x - \text{focus\_point}|^2}\right)$$

• Purpose: Creates a field that disperses around the focal point, mimicking a spreading or diffusing magnetic field.

2. Magnetic Pinhole Filter

Limits field values to a specific range, isolating only the values that meet certain thresholds, similar to a physical pinhole filter.

python
def magnetic_pinhole_filter(field_value, min_threshold=0.1, max_threshold=1.0):
    return field_value if min_threshold < field_value < max_threshold else 0

Equation:

$$F_{\text{filtered}} = \begin{cases} F_{\text{value}} & \text{if } \text{min\_threshold} < F_{\text{value}} < \text{max\_threshold} \\ 0 & \text{otherwise} \end{cases}$$

• Purpose: Mimics a filter that only allows certain field intensities to pass, highlighting specific magnetic contours.

3. Interference Pattern

Combines convex and concave fields with a slight offset to create interference patterns that reveal hexel boundaries.

python
def interference_pattern(position, strength1=1.0, strength2=0.8, offset=0.5):
    field1 = convex_magnetic_field(position, strength1)
    field2 = concave_magnetic_field(position - offset, strength2)
    return field1 + field2

• Purpose: Creates zones where field effects reinforce or cancel each other, simulating boundaries between hexels.

4. Tear Effect

Adds slight fluctuations to the field, simulating irregularities similar to light bending through a tear on the surface of an eye.

python
def tear_effect(field_value, fluctuation_strength=0.05):
    fluctuation = np.random.normal(0, fluctuation_strength)
    return field_value * (1 + fluctuation)

• Purpose: Adds minor, random fluctuations to create natural variability in the field, enhancing edge detection within hexels.

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Field Application by Phase

The full_stabilization_with_fields function integrates all field effects depending on the current phase (Entry, Traversal, or Exit).

python
def full_stabilization_with_fields(t, y, phase):
    position, velocity = y
    gravitational_force = -GRAVITY_CONSTANT
    magnetic_force = CHARGE * velocity * MAGNETIC_FIELD_STRENGTH
    higher_dimensional_influence = 0.01 * np.sin(position / 1e16) * np.cos(velocity)
    fluctuation = np.random.normal(0, 0.1)
    stabilizing_field_value = golden_ratio_polynomial(velocity, a_k, b_k, c_k)
    
    # Apply the appropriate field force based on the phase
    if phase == 'entry':
        field_force = uncurling_field_generator(t)
    elif phase == 'exit':
        field_force = curling_field_generator(t)
    else:
        field_force = stabilizing_field_value

    net_force = gravitational_force + magnetic_force + higher_dimensional_influence + fluctuation + field_force
    return [velocity, net_force]

• Purpose: Applies a combination of gravitational, magnetic, and phase-specific forces, adjusting field forces based on the current simulation phase.
• Outputs: Returns velocity and net force for the solve_ivp integrator to update positions and velocities over time.

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Simulation Execution and Data Collection

• Simulations: Three separate simulations (entry_solution, traversal_solution, exit_solution) capture each phase with distinct forces.
• Data Tables: Collects Time, Position, Velocity, specific field values, and Hexel Field across each phase for CSV export and console output.

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Visualization

Finally, a plot shows the Hexel Identification Field over a range of positions, helping visualize how hexel edges emerge through the combined field dynamics of convex/concave fields, interference, and tear effects.