This study focuses on addressing the following phases:

I. Stability phase: explore how σ changes for ρ ∈ [0.1, 0.999] and k ∈ [−100, 1].

II. Traffic intensity phase: examine system behaviour when ρ = 1 and k ∈ [−100, 1].

III. Chaotic phase: investigate the system response for ρ > 1 (e.g., 2, 3, 4) and k ∈ [-100, −1].

In other words, we are looking at Eq. (7) from a different end of the spectrum, by choosing k =

−1, −2, −3, −4, …

The current breakthrough uses computational tools to visualize these relationships, ensuring an accurate representation of transitions between phases for the unexplored phenomenal negative values of the number of sets of phases, namely, k. In mathematical terms, the current exposition marks a new phenomenal chapter of rethinking the totality of Pointwise Stationary Fluid Approximation (PSFFA) theory to shift the longstanding unresearched topic of PSFFA into a higher level of thinking between such unparallel scientific disciplines, within chaos, quantum mechanics, random walks, Brownian motion, and much more. Having seen the results of this research, it is anticipated that further investigations into more unexplored cases will expand on Ismail’s Contemporary PSFFA Theory and merge it with numerous multi-interdisciplinary spectrums of human knowledge.

import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D from scipy.optimize import fsolve import os # Function to compute sigma numerically def sigma_equation(sigma, k, rho):

return ((k * rho + sigma - 1) / (k * rho))**k - sigma def compute_sigma(k, rho):

if k == 1:

return 1.0 initial_guess = 0.5 if rho < 1 else 1.5 sol, = fsolve(sigma_equation, initial_guess, args=(k, rho)) return sol # Prepare output directory os.makedirs("3d_phase_plots", exist_ok=True) # k values k_values = np.arange(1, 101) # --- Stability Phase --- rho_stability = np.linspace(0.01, 0.99, 50) K, RHO = np.meshgrid(k_values, rho_stability) SIGMA = np.vectorize(compute_sigma)(K, RHO) fig = plt.figure(figsize=(12, 8)) ax = fig.add_subplot(111, projection='3d') ax.plot_surface(K, RHO, SIGMA, cmap='viridis') ax.set_title('Stability Phase (ρ in (0,1))') ax.set_xlabel('k') ax.set_ylabel('ρ') ax.set_zlabel('σ') plt.tight_layout() plt.savefig("3d_phase_plots/stability_phase_3d.png") plt.show() # --- Traffic Intensity Phase --- rho_traffic = np.array([1.0]) K, RHO = np.meshgrid(k_values, rho_traffic) SIGMA = np.vectorize(compute_sigma)(K, RHO) fig = plt.figure(figsize=(12, 6)) ax = fig.add_subplot(111) ax.plot(k_values, SIGMA.flatten(), color='red') ax.set_title('Traffic Intensity Phase (ρ = 1)') ax.set_xlabel('k') ax.set_ylabel('σ') plt.tight_layout() plt.savefig("3d_phase_plots/traffic_intensity_phase_2d.png") plt.show() # --- Chaotic Phase --- rho_chaotic = np.linspace(1.1, 3.0, 50) K, RHO = np.meshgrid(k_values, rho_chaotic) SIGMA = np.vectorize(compute_sigma)(K, RHO) fig = plt.figure(figsize=(12, 8)) ax = fig.add_subplot(111, projection='3d') ax.plot_surface(K, RHO, SIGMA, cmap='plasma') ax.set_title('Chaotic Phase (ρ > 1)') ax.set_xlabel('k') ax.set_ylabel('ρ') ax.set_zlabel('σ') plt.tight_layout() plt.savefig("3d_phase_plots/chaotic_phase_3d.png") plt.show() print("✅ All 3D phase plots generated in '3d_phase_plots/' folder and displayed.") The pseudocode is structured as follows:

The pseudocode provided in the flowchart serves as a visual guide to implementing the iterative computational framework for analyzing σ in dynamic systems with negative k. It encapsulates the algorithm's step-by-step logic for exploring stability, traffic intensity, and chaotic phases. Below is a detailed explanation of the key components:

I. Initialization The algorithm begins by defining the equation:

kρ+σ−1

σ = ( ) .

The initial value of σ is set to 0.50 as a starting point for the iterative approximation.

II. Iterative process The algorithm repeatedly calculates a new value of σ based on the equation above.

Convergence is determined by checking if the absolute difference between the new σ and the previous σ falls below a specified tolerance (10−6).

III. Error handling Division by zero or undefined operations are managed by assigning Not a Number (NaN) to σ, ensuring the algorithm remains robust and does not crash.

IV. Phases: stability phase:

− Iterates over a range of values from 0.1 to 0.999 and k from −100 to −1.

− Generates a 3D plot to visualize σ as a function of k and ρ.

Traffic intensity phase:

− Fixes = 1 and varies k, producing a 2D plot of σ versus k.

Chaotic phase:

− Explores > 1 with varying k, generating 3D visualizations of chaotic behaviour.

Significance: the pseudocode is instrumental in systematically exploring the behavior of σ under negative k, providing the foundation for the visualizations and insights presented in the results section. It emphasizes robustness and adaptability, accurately representing system dynamics across phases.

I. Critical transitions: the stability phase is highly sensitive to small changes in ρ, whereas the chaotic phase exhibits larger, unpredictable variations.

II. Algorithm robustness: the iterative approach converges reliably under most conditions, with exceptions managed through error handling.

The current study has some limitations, including the inability to visualize the unexplored dynamics at k=∞. In other words, what are the mathematical and physical interpretations at sufficiently large values of k? This would be a next-generation vision of delving into a different universe within the reach of infinity, if it exists as a number rather than a mathematical concept [8–10], to showcase our inability as mathematicians to count beyond a certain limit at the far edge of the counting spectrum. It is an informed guess based on future research exploring ways to solve this complex open problem from mathematical and computational perspectives.