Spiral Matplot Generated by tooeasy10000.py

[Josef_Founder]

Slicing Inside of Slicing inside of Slicing

The 1st plot is, itself, only a slice of the greater script.

1st Plot:

2nd Plot:

The following was the wrong thread, but it is the thread that was provided with the video, so here it is again: ChatGPT - Golden Field Summary

THE CORRECT THREAD TO DESCRIBE WHAT’S GOING ON “UNDER THE HOOD” IS LOCATED HERE.

# tooeasy10000-truncated.py

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify, conjugate, Rational

from scipy.interpolate import interp1d
from scipy.optimize import root_scalar

# Constants
phi = float(sp.GoldenRatio)
sqrt5 = sqrt(5)
Ω = sp.Symbol("Ω")    # Field tension symbol
k = sp.Integer(-1)    # Radial exponent
r = 1                 # Radial unit

# Raw prime list (first 100 primes or more, truncated for brevity)
primes_raw = """
      2      3      5      7     11     13     17     19     23     29 
     31     37     41     43     47     53     59     61     67     71 
     73     79     83     89     97    101    103    107    109    113 
(TRUNCATED, BUT 10,000 PRIMES WERE INJECTED)
"""

primes = [int(p) for p in primes_raw.split()]

def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def solve_n_beta_for_prime(p_target, prime_interp, bracket=(0.1, 20)):
    def objective(n_beta): return P_nb(n_beta, prime_interp) - p_target
    result = root_scalar(objective, bracket=bracket, method='brentq')
    if result.converged:
        return result.root
    else:
        raise ValueError(f"Could not solve for n_beta corresponding to prime {p_target}")

def F_bin(x_val):
    return (phi**x_val - (-1/phi)**x_val) / sqrt5

def Pi_x(x_val, s):
    s = sp.sympify(s)
    return exp(I * pi * x_val) * zeta(s, Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = phi * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

class GoldenClassField:
    def __init__(self, s_list, x_list, prime_interp):
        self.s_list = [sp.sympify(s) for s in s_list]
        self.x_list = x_list
        self.prime_interp = prime_interp
        self.field_generators = []
        self.field_names = []
        self.construct_class_field()

    def construct_class_field(self):
        for s in self.s_list:
            for x in self.x_list:
                f = F_x(x, s, self.prime_interp)
                self.field_generators.append(simplify(f))
                self.field_names.append(f"F_{x:.4f}_s_{s}")

    def as_dict(self):
        return dict(zip(self.field_names, self.field_generators))

    def display(self):
        for name, val in self.as_dict().items():
            print(f"{name} = {val}")

    def reciprocity_check(self):
        print("\nReciprocity Tests: F_x(s) * F_x(1-s)")
        for s in self.s_list:
            for x in self.x_list:
                try:
                    s_conj = 1 - s
                    prod = simplify(F_x(x, s, self.prime_interp) * F_x(x, s_conj, self.prime_interp))
                    print(f"x={x:.4f}, s={s}, F_x(s)·F_x(1-s) = {prod}")
                except Exception as e:
                    print(f"Failed for x={x}, s={s}: {e}")

def field_automorphisms(F_val, x_val, s, prime_interp):
    s = sp.sympify(s)
    return {
        "F_x(s)": simplify(F_x(x_val, s, prime_interp)),
        "F_x(1-s)": simplify(F_x(x_val, 1 - s, prime_interp)),
        "F_-x(s)": simplify(F_x(-x_val, s, prime_interp)),
        "conjugate(F)": simplify(conjugate(F_x(x_val, s, prime_interp))),
    }

def field_tension(F_val, C_val, m_val, s_val):
    # Example symbolic tension extraction formula
    return simplify((F_val * m_val * s_val) / (C_val**2))


if __name__ == "__main__":
    print("Preparing prime interpolation...")
    prime_interp = prepare_prime_interpolation()

    # Example Riemann zeta zeros (first two nontrivial zeros on critical line)
    zeros = [sp.sympify("0.5 + 14.134725*I"), sp.sympify("0.5 + 21.022040*I")]

    # Solve n_beta for a prime near 541 (to demonstrate root solving)
    try:
        x_541 = solve_n_beta_for_prime(541, prime_interp)
        print(f"Solved n+β for prime 541: {x_541:.6f}")
    except Exception as e:
        print(str(e))
        x_541 = None

    x_vals = [5, 10]
    if x_541:
        x_vals.append(x_541)

    # Construct Golden Class Field
    GCF = GoldenClassField(zeros, x_vals, prime_interp)
    GCF.display()

    # Reciprocity tests
    GCF.reciprocity_check()

    # Automorphisms on one example
    test_s = zeros[0]
    test_x = x_vals[1]
    auto = field_automorphisms(F_x(test_x, test_s, prime_interp), test_x, test_s, prime_interp)
    print("\nSymbolic Automorphisms:")
    for key, val in auto.items():
        print(f"{key}: {val}")

    # Example field tension calculation (symbolic)
    print("\nField Tension (Ω):")
    C = sp.Symbol('C')
    m = sp.Symbol('m')
    s_ = sp.Symbol('s')
    F_val = sp.Abs(F_x(test_x, test_s, prime_interp))
    tension = field_tension(F_val, C, m, s_)
    print(tension)

YIELDS:

py tooeasy10000full.py
Preparing prime interpolation...
Solved n+β for prime 541: 9.590622
F_5.0000_s_0.5 + 14.134725*I = 1.0*sqrt(Ω*(1.7674298413849e-8 - 1.11020289309231e-7*I))*(-0.217537177381404 + 6.14965635912474*I)*zeta(0.5 + 14.134725*I, 1/2)
F_10.0000_s_0.5 + 14.134725*I = 39.1462815355537*sqrt(Ω*(1.7674298413849e-8 - 1.11020289309231e-7*I))*(0.5 - 14.134725*I)*zeta(0.5 + 14.134725*I, 1/2)
F_9.5906_s_0.5 + 14.134725*I = 1.0*sqrt(Ω*(1.78610995454154e-6 - 1.12125725404408e-5*I))*(1.37320228847257 - 38.819673433861*I)*exp(9.5906215859709*I*pi)*zeta(0.5 + 14.134725*I, 1/2)
F_5.0000_s_0.5 + 21.02204*I = 1.0*sqrt(Ω*(8.98483605435458e-8 + 4.00709084764903e-7*I))*(-0.0984137961388264 + 4.13771751796451*I)*zeta(0.5 + 21.02204*I, 1/2)
F_10.0000_s_0.5 + 21.02204*I = 17.7097736442471*sqrt(Ω*(8.98483605435458e-8 + 4.00709084764903e-7*I))*(0.5 - 21.02204*I)*zeta(0.5 + 21.02204*I, 1/2)
F_9.5906_s_0.5 + 21.02204*I = 1.0*sqrt(Ω*(9.07062684366856e-6 + 4.04713570287637e-5*I))*(0.621236570695075 - 26.1193200772294*I)*exp(9.5906215859709*I*pi)*zeta(0.5 + 21.02204*I, 1/2)

Reciprocity Tests: F_x(s) * F_x(1-s)
x=5.0000, s=0.5 + 14.134725*I, F_x(s)·F_x(1-s) = 37.8655957588664*sqrt(Ω*(1.7674298413849e-8 + 1.11020289309231e-7*I))*sqrt(Ω*(1.7674298413849e-8 - 1.11020289309231e-7*I))*zeta(0.5 - 14.134725*I, 1/2)*zeta(0.5 + 14.134725*I, 1/2)
x=10.0000, s=0.5 + 14.134725*I, F_x(s)·F_x(1-s) = 306548.259725814*sqrt(Ω*(1.7674298413849e-8 + 1.11020289309231e-7*I))*sqrt(Ω*(1.7674298413849e-8 - 1.11020289309231e-7*I))*zeta(0.5 - 14.134725*I, 1/2)*zeta(0.5 + 14.134725*I, 1/2)
x=9.5906, s=0.5 + 14.134725*I, F_x(s)·F_x(1-s) = 1508.85273003668*sqrt(Ω*(1.78400002592542e-6 + 1.12129084385746e-5*I))*sqrt(Ω*(1.78610995454154e-6 - 1.12125725404408e-5*I))*exp(19.1812431719418*I*pi)*zeta(0.5 - 14.134725*I, 1/2)*zeta(0.5 + 14.134725*I, 1/2)
x=5.0000, s=0.5 + 21.02204*I, F_x(s)·F_x(1-s) = 17.1303915337408*sqrt(Ω*(8.98483605435458e-8 - 4.00709084764903e-7*I))*sqrt(Ω*(8.98483605435458e-8 + 4.00709084764903e-7*I))*zeta(0.5 - 21.02204*I, 1/2)*zeta(0.5 + 21.02204*I, 1/2)
x=10.0000, s=0.5 + 21.02204*I, F_x(s)·F_x(1-s) = 138682.400417811*sqrt(Ω*(8.98483605435458e-8 - 4.00709084764903e-7*I))*sqrt(Ω*(8.98483605435458e-8 + 4.00709084764903e-7*I))*zeta(0.5 - 21.02204*I, 1/2)*zeta(0.5 + 21.02204*I, 1/2)
x=9.5906, s=0.5 + 21.02204*I, F_x(s)·F_x(1-s) = 682.604816173527*sqrt(Ω*(9.07062684366856e-6 + 4.04713570287637e-5*I))*sqrt(Ω*(9.07824227657678e-6 - 4.04696494703687e-5*I))*exp(19.1812431719418*I*pi)*zeta(0.5 - 21.02204*I, 1/2)*zeta(0.5 + 21.02204*I, 1/2)

Symbolic Automorphisms:
F_x(s): 39.1462815355537*sqrt(Ω*(1.7674298413849e-8 - 1.11020289309231e-7*I))*(0.5 - 14.134725*I)*zeta(0.5 + 14.134725*I, 1/2)
F_x(1-s): 39.1462815355537*sqrt(Ω*(1.7674298413849e-8 + 1.11020289309231e-7*I))*(0.5 + 14.134725*I)*zeta(0.5 - 14.134725*I, 1/2)
F_-x(s): 1.0*sqrt(Ω*(-1.7674298413849e-8 + 1.11020289309231e-7*I))*(0.045424303171084 - 1.2841200672798*I)*zeta(0.5 + 14.134725*I, 1/2)
conjugate(F): 39.1462815355537*(0.5 + 14.134725*I)*conjugate(sqrt(Ω*(1.7674298413849e-8 - 1.11020289309231e-7*I)))*conjugate(zeta(0.5 + 14.134725*I, 1/2))

Field Tension (Ω):
553.668004968514*m*s*Abs(sqrt(Ω*(1.7674298413849e-8 - 1.11020289309231e-7*I))*zeta(0.5 + 14.134725*I, 1/2))/C**2

From this, we created these:

demo7.py

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from sympy import sqrt, zeta, exp, I, Rational
from scipy.interpolate import interp1d

# Constants
phi = float(sp.GoldenRatio)
sqrt5 = np.sqrt(5)
k = -1

def prepare_prime_interpolation(res=10000):
    indices = np.arange(1, res + 1)
    primes = [sp.prime(i) for i in indices]
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    try:
        return (phi**x_val - (-1/phi)**x_val) / sqrt5
    except:
        return np.nan

def Pi_x(x_val, s):
    return sp.exp(sp.I * sp.pi * x_val) * zeta(s, Rational(1, 2))

def D_x(x_val, s, prime_interp, Ω=1):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    z_val = zeta(s)
    s_k = s**k
    product = phi * F * 2**x_val * P * z_val * Ω
    return sqrt(product) * s_k

def F_x(x_val, s, prime_interp, Ω=1):
    return D_x(x_val, s, prime_interp, Ω) * Pi_x(x_val, s)

# Parameters for grid
x_min, x_max = 1.0, 20.0
t_min, t_max = 0.1, 40.0
x_steps = 60
t_steps = 200

# Generate coordinate grids
x_vals = np.linspace(x_min, x_max, x_steps)
t_vals = np.linspace(t_min, t_max, t_steps)
X, T = np.meshgrid(x_vals, t_vals)

# Prepare prime interpolation
prime_interp = prepare_prime_interpolation(10000)

# Evaluate |F_x(0.5 + i t)| on the grid (numeric only for speed)
abs_F = np.zeros_like(X)

for i in range(t_steps):
    for j in range(x_steps):
        x = X[i, j]
        t = T[i, j]
        s = 0.5 + t * 1j
        try:
            val = F_x(x, s, prime_interp)
            mag = abs(complex(val.evalf()))
            abs_F[i, j] = mag if not np.isnan(mag) else 0
        except Exception:
            abs_F[i, j] = 0

# Transform x-axis to golden-logarithmic scale:
# safe +1 to avoid log(0)
X_phi_log = np.log(x_vals + 1) / np.log(phi)

# Plot 3D surface
fig = plt.figure(figsize=(12, 7))
ax = fig.add_subplot(projection='3d')

T_plot, X_phi_plot = np.meshgrid(t_vals, X_phi_log)

# Because X and T mesh are transposed, transpose abs_F to align:
abs_F_T = abs_F.T

surf = ax.plot_surface(X_phi_plot, T_plot, abs_F_T, cmap=cm.viridis, linewidth=0, antialiased=True)

ax.set_xlabel(r"Recursive coordinate $\log_{\phi}(x + 1)$")
ax.set_ylabel(r"Imaginary part of $s = 0.5 + it$")
ax.set_zlabel(r"$|F_x(s)|$ magnitude")

ax.set_title("Golden Recursive Algebra Field Magnitude Surface")

fig.colorbar(surf, shrink=0.5, aspect=10, label=r"$|F_x(s)|$")

plt.show()

WHICH YIELDS:

AND

animate4.py

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt
from sympy import sqrt, zeta, exp, I, Rational
from scipy.interpolate import interp1d
import matplotlib.animation as animation

# Constants
phi = float(sp.GoldenRatio)
k = -1

def prepare_prime_interpolation(res=10000):
    indices = np.arange(1, res + 1)
    primes = [sp.prime(i) for i in indices]
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    try:
        return (phi**x_val - (-1/phi)**x_val) / np.sqrt(5)
    except:
        return np.nan

def Pi_x(x_val, s):
    return sp.exp(sp.I * sp.pi * x_val) * zeta(s, Rational(1, 2))

def D_x(x_val, s, prime_interp, Ω=1):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    z_val = zeta(s)
    s_k = s**k
    product = phi * F * 2**x_val * P * z_val * Ω
    return sqrt(product) * s_k

def F_x(x_val, s, prime_interp, Ω=1):
    return D_x(x_val, s, prime_interp, Ω) * Pi_x(x_val, s)

# Parameters
x_min, x_max = 1.0, 20.0
t_min, t_max = 0.1, 40.0
t_steps = 300
x_frames = 60  # animation frames

# Prepare data
t_vals = np.linspace(t_min, t_max, t_steps)
prime_interp = prepare_prime_interpolation(10000)

# Precompute golden-log x for display axis
def golden_log(x):
    return np.log(x + 1) / np.log(phi)

# Setup plot
fig, ax = plt.subplots(figsize=(10,5))
line, = ax.plot([], [], lw=2)
ax.set_xlim(t_min, t_max)
ax.set_ylim(0, 1)
ax.set_xlabel("Imaginary part of s (t)")
ax.set_ylabel(r"$|F_x(0.5 + it)|$")
title = ax.set_title("")

# Initialization
def init():
    line.set_data([], [])
    return line,

# Animation update function
def animate(i):
    x = x_min + (x_max - x_min) * (i / (x_frames - 1))
    magnitudes = []
    for t in t_vals:
        s = 0.5 + t * I
        try:
            val = F_x(x, s, prime_interp)
            mag = abs(complex(val.evalf()))
            if np.isnan(mag) or np.isinf(mag):
                mag = 0
            magnitudes.append(mag)
        except Exception:
            magnitudes.append(0)

    magnitudes = np.array(magnitudes)
    line.set_data(t_vals, magnitudes)
    ymax = np.max(magnitudes)
    ax.set_ylim(0, ymax*1.1 if ymax > 0 else 1)
    ax.set_title(f"Golden Coord $\\log_{{\\phi}}(x+1)$ = {golden_log(x):.4f}   —   x = {x:.4f}")
    return line,

ani = animation.FuncAnimation(fig, animate, frames=x_frames, init_func=init,
                              blit=True, interval=100, repeat=True)

plt.tight_layout()
plt.show()

WHICH YIELDS:

AND

grok5.py

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify
from scipy.interpolate import interp1d
from scipy.optimize import root_scalar
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.animation import FuncAnimation
import shutil

# Constants
phi = sp.GoldenRatio
sqrt5 = sp.sqrt(5)
Ω = 1.0  # Set Ω to a constant for numerical evaluation
k = -1   # Radial exponent
r = 1    # Radial unit

# Truncated prime list (using the provided primes)
primes_raw = """      2      3      5      7     11     13     17     19     23     29 
     31     37     41     43     47     53     59     61     67     71 
     73     79     83     89     97    101    103    107    109    113 
    127    131    137    139    149    151    157    163    167    173 
    179    181    191    193    197    199    211    223    227    229 
    233    239    241    251    257    263    269    271    277    281 
    283    293    307    311    313    317    331    337    347    349 
    353    359    367    373    379    383    389    397    401    409 
    419    421    431    433    439    443    449    457    461    463 
    467    479    487    491    499    503    509    521    523    541 
    547    557    563    569    571    577    587    593    599    601 
    607    613    617    619    631    641    643    647    653    659 
"""
primes = [int(p) for p in primes_raw.split()]

def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(float(phi))
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    # Use sympy for numerical stability, return complex result
    phi_x = phi**x_val
    neg_phi_inv_x = (-1/phi)**x_val
    result = (phi_x - neg_phi_inv_x) / sqrt5
    return complex(result.evalf())  # Convert to complex number

def Pi_x(x_val, s):
    s = sp.sympify(s)
    return exp(I * pi * x_val) * zeta(s, sp.Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = float(phi) * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

# Prepare interpolation
prime_interp = prepare_prime_interpolation()

# Parameters for the plot
s_val = sp.sympify("0.5 + 14.134725*I")  # First nontrivial zeta zero
x_vals = np.linspace(2, 5, 50)  # Reduced range to avoid numerical issues
t_vals = np.linspace(0, 2*np.pi, 20)  # Reduced frames for clarity

# Compute field values
def compute_field_values(x_vals, s_val, t):
    real_vals = []
    imag_vals = []
    for x in x_vals:
        try:
            f_val = F_x(x, s_val, prime_interp) * np.exp(1j * t)  # Add phase shift
            f_val = complex(f_val.evalf())  # Numerical evaluation
            real_vals.append(f_val.real)
            imag_vals.append(f_val.imag)
        except (ValueError, OverflowError, TypeError):
            real_vals.append(np.nan)  # Handle numerical errors gracefully
            imag_vals.append(np.nan)
    return np.array(real_vals), np.array(imag_vals)

# Store all spirals
spiral_data = []
for t in t_vals:
    real_vals, imag_vals = compute_field_values(x_vals, s_val, t)
    spiral_data.append((real_vals, imag_vals, x_vals))

# Set up the 3D plot
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

# Set labels and limits
ax.set_xlabel('Real(F_x)')
ax.set_ylabel('Imag(F_x)')
ax.set_zlabel('x')
ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')

# Determine axis limits based on all spirals
all_real = np.concatenate([data[0] for data in spiral_data])
all_imag = np.concatenate([data[1] for data in spiral_data])
all_real = all_real[~np.isnan(all_real)]
all_imag = all_imag[~np.isnan(all_imag)]
if len(all_real) > 0 and len(all_imag) > 0:
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
else:
    ax.set_xlim(-1, 1)
    ax.set_ylim(-1, 1)
ax.set_zlim(min(x_vals), max(x_vals))

# Animation functions
def init():
    ax.clear()
    ax.set_xlabel('Real(F_x)')
    ax.set_ylabel('Imag(F_x)')
    ax.set_zlabel('x')
    ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
    ax.set_zlim(min(x_vals), max(x_vals))
    return []

def update(frame, x_vals, spiral_data):
    ax.clear()
    ax.set_xlabel('Real(F_x)')
    ax.set_ylabel('Imag(F_x)')
    ax.set_zlabel('x')
    ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
    ax.set_zlim(min(x_vals), max(x_vals))
    
    # Plot all spirals up to the current frame
    for i, (real_vals, imag_vals, x_vals) in enumerate(spiral_data[:frame + 1]):
        alpha = 0.2 + 0.8 * (i + 1) / len(t_vals)  # Increase opacity for newer spirals
        ax.plot(real_vals, imag_vals, x_vals, 'b-', lw=1, alpha=alpha)
    return []

# Create animation
ani = FuncAnimation(fig, update, frames=len(t_vals), init_func=init, fargs=(x_vals, spiral_data), blit=False, interval=300)

# Check for ffmpeg and save animation if available
if shutil.which('ffmpeg'):
    ani.save('gcf_animation_retained.mp4', writer='ffmpeg', dpi=100)
    print("Animation saved as 'gcf_animation_retained.mp4'")
else:
    print("ffmpeg not found. Saving static plot with all spirals instead.")
    for real_vals, imag_vals, x_vals in spiral_data:
        ax.plot(real_vals, imag_vals, x_vals, 'b-', lw=1, alpha=0.5)
    plt.savefig('gcf_static_plot_retained.png', dpi=100)
    print("Static plot saved as 'gcf_static_plot_retained.png'")

plt.show()

WHICH YIELDS:

For fun, I set OHM = -1 and radius = -1, I then added more steps (42 in place of 20):

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify
from scipy.interpolate import interp1d
from scipy.optimize import root_scalar
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.animation import FuncAnimation
import shutil

# Constants
phi = sp.GoldenRatio
sqrt5 = sp.sqrt(5)
Ω = -1.0  # Set Ω to a constant for numerical evaluation
k = -1   # Radial exponent
r = phi    # Radial unit

# Truncated prime list (using the provided primes)
primes_raw = """      2      3      5      7     11     13     17     19     23     29 
     31     37     41     43     47     53     59     61     67     71 
     73     79     83     89     97    101    103    107    109    113 
    127    131    137    139    149    151    157    163    167    173 
    179    181    191    193    197    199    211    223    227    229 
    233    239    241    251    257    263    269    271    277    281 
    283    293    307    311    313    317    331    337    347    349 
    353    359    367    373    379    383    389    397    401    409 
    419    421    431    433    439    443    449    457    461    463 
    467    479    487    491    499    503    509    521    523    541 
    547    557    563    569    571    577    587    593    599    601 
    607    613    617    619    631    641    643    647    653    659 
"""
primes = [int(p) for p in primes_raw.split()]

def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(float(phi))
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    # Use sympy for numerical stability, return complex result
    phi_x = phi**x_val
    neg_phi_inv_x = (-1/phi)**x_val
    result = (phi_x - neg_phi_inv_x) / sqrt5
    return complex(result.evalf())  # Convert to complex number

def Pi_x(x_val, s):
    s = sp.sympify(s)
    return exp(I * pi * x_val) * zeta(s, sp.Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = float(phi) * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

# Prepare interpolation
prime_interp = prepare_prime_interpolation()

# Parameters for the plot
s_val = sp.sympify("0.5 + 14.134725*I")  # First nontrivial zeta zero
x_vals = np.linspace(2, 5, 50)  # Reduced range to avoid numerical issues
t_vals = np.linspace(0, 2*np.pi, 42)  # Reduced frames for clarity

# Compute field values
def compute_field_values(x_vals, s_val, t):
    real_vals = []
    imag_vals = []
    for x in x_vals:
        try:
            f_val = F_x(x, s_val, prime_interp) * np.exp(1j * t)  # Add phase shift
            f_val = complex(f_val.evalf())  # Numerical evaluation
            real_vals.append(f_val.real)
            imag_vals.append(f_val.imag)
        except (ValueError, OverflowError, TypeError):
            real_vals.append(np.nan)  # Handle numerical errors gracefully
            imag_vals.append(np.nan)
    return np.array(real_vals), np.array(imag_vals)

# Store all spirals
spiral_data = []
for t in t_vals:
    real_vals, imag_vals = compute_field_values(x_vals, s_val, t)
    spiral_data.append((real_vals, imag_vals, x_vals))

# Set up the 3D plot
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

# Set labels and limits
ax.set_xlabel('Real(F_x)')
ax.set_ylabel('Imag(F_x)')
ax.set_zlabel('x')
ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')

# Determine axis limits based on all spirals
all_real = np.concatenate([data[0] for data in spiral_data])
all_imag = np.concatenate([data[1] for data in spiral_data])
all_real = all_real[~np.isnan(all_real)]
all_imag = all_imag[~np.isnan(all_imag)]
if len(all_real) > 0 and len(all_imag) > 0:
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
else:
    ax.set_xlim(-1, 1)
    ax.set_ylim(-1, 1)
ax.set_zlim(min(x_vals), max(x_vals))

# Animation functions
def init():
    ax.clear()
    ax.set_xlabel('Real(F_x)')
    ax.set_ylabel('Imag(F_x)')
    ax.set_zlabel('x')
    ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
    ax.set_zlim(min(x_vals), max(x_vals))
    return []

def update(frame, x_vals, spiral_data):
    ax.clear()
    ax.set_xlabel('Real(F_x)')
    ax.set_ylabel('Imag(F_x)')
    ax.set_zlabel('x')
    ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
    ax.set_zlim(min(x_vals), max(x_vals))
    
    # Plot all spirals up to the current frame
    for i, (real_vals, imag_vals, x_vals) in enumerate(spiral_data[:frame + 1]):
        alpha = 0.2 + 0.8 * (i + 1) / len(t_vals)  # Increase opacity for newer spirals
        ax.plot(real_vals, imag_vals, x_vals, 'b-', lw=1, alpha=alpha)
    return []

# Create animation
ani = FuncAnimation(fig, update, frames=len(t_vals), init_func=init, fargs=(x_vals, spiral_data), blit=False, interval=300)

# Check for ffmpeg and save animation if available
if shutil.which('ffmpeg'):
    ani.save('gcf_animation_retained.mp4', writer='ffmpeg', dpi=100)
    print("Animation saved as 'gcf_animation_retained.mp4'")
else:
    print("ffmpeg not found. Saving static plot with all spirals instead.")
    for real_vals, imag_vals, x_vals in spiral_data:
        ax.plot(real_vals, imag_vals, x_vals, 'b-', lw=1, alpha=0.5)
    plt.savefig('gcf_static_plot_retained.png', dpi=100)
    print("Static plot saved as 'gcf_static_plot_retained.png'")

plt.show()

WHICH YIELDS:

And finally, when attempting to demonstrate our changing axes, we produce a crude version of the desired result:

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.animation import FuncAnimation
from scipy.interpolate import interp1d
import shutil

# Constants
phi = float(sp.GoldenRatio)
sqrt5 = np.sqrt(5)
k = -1
Ω = -1.0

# Prime interpolation
def prepare_prime_interpolation(res=10000):
    indices = np.arange(1, res + 1)
    primes = [sp.prime(i) for i in indices]
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    try:
        phi_x = sp.N(phi**x_val)  # SymPy numerical evaluation for stability
        neg_phi_inv_x = sp.N((-1/phi)**x_val)
        result = (phi_x - neg_phi_inv_x) / sqrt5
        return complex(result)
    except (ValueError, OverflowError, TypeError):
        return np.nan

def Pi_x(x_val, s):
    s = sp.sympify(s)
    return sp.exp(sp.I * sp.pi * x_val) * sp.zeta(s, sp.Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    z_val = sp.zeta(s)
    s_k = s**k
    product = phi * F * 2**x_val * P * z_val * Ω
    return sp.sqrt(product) * s_k

def F_x(x_val, s, prime_interp):
    return D_x(x_val, s, prime_interp) * Pi_x(x_val, s)

# Prepare interpolation
prime_interp = prepare_prime_interpolation(10000)

# Parameters
s_val = sp.sympify("0.5 + 14.134725*I")  # First nontrivial zeta zero
x_vals = np.linspace(2, 5, 50)  # Reduced range for stability
t_vals = np.linspace(0.1, 40, 100)  # Reduced t-range for surface
morph_steps = 100  # Frames for morphing
morph_vals = np.sin(np.linspace(0, np.pi, morph_steps))**2  # Smooth oscillation

# Compute surface data (first script: |F_x(s)| at fixed s, varying t)
X_phi_log = np.log(x_vals + 1) / np.log(phi)  # Golden-logarithmic x
abs_F_surface = np.zeros((len(t_vals), len(x_vals)))
for i, t in enumerate(t_vals):
    for j, x in enumerate(x_vals):
        s = 0.5 + t * 1j
        try:
            val = F_x(x, s, prime_interp)
            mag = abs(complex(val.evalf()))
            abs_F_surface[i, j] = mag if not np.isnan(mag) else 0
        except Exception:
            abs_F_surface[i, j] = 0

# Compute spiral data (second script: Real(F_x), Imag(F_x) at fixed s, t=0)
real_F_spiral, imag_F_spiral = [], []
t_fixed = 0  # Fix t for spiral reference
for x in x_vals:
    try:
        f_val = F_x(x, s_val, prime_interp) * np.exp(1j * t_fixed)
        f_val = complex(f_val.evalf())
        real_F_spiral.append(f_val.real if not np.isnan(f_val.real) else 0)
        imag_F_spiral.append(f_val.imag if not np.isnan(f_val.imag) else 0)
    except Exception:
        real_F_spiral.append(0)
        imag_F_spiral.append(0)
real_F_spiral, imag_F_spiral = np.array(real_F_spiral), np.array(imag_F_spiral)

# Set up plot
fig = plt.figure(figsize=(12, 8))
ax = fig.add_subplot(111, projection='3d')

# Determine axis limits
x1_min, x1_max = min(X_phi_log), max(X_phi_log)
x2_min, x2_max = min(real_F_spiral) * 1.1, max(real_F_spiral) * 1.1
y1_min, y1_max = min(t_vals), max(t_vals)
y2_min, y2_max = min(imag_F_spiral) * 1.1, max(imag_F_spiral) * 1.1
z1_min, z1_max = min(abs_F_surface.flatten()) * 1.1, max(abs_F_surface.flatten()) * 1.1
z2_min, z2_max = min(x_vals), max(x_vals)

# Animation function
def update(frame):
    ax.clear()
    alpha = morph_vals[frame]
    
    # Select a single t-slice from surface data for morphing
    t_idx = frame % len(t_vals)  # Cycle through t for dynamic effect
    abs_F_slice = abs_F_surface[t_idx, :]
    
    # Interpolate axes
    x_coords = (1 - alpha) * X_phi_log + alpha * real_F_spiral
    y_coords = (1 - alpha) * np.full_like(x_vals, t_vals[t_idx]) + alpha * imag_F_spiral
    z_coords = (1 - alpha) * abs_F_slice + alpha * x_vals
    
    # Plot curve
    ax.plot(x_coords, y_coords, z_coords, 'b-', lw=2)
    
    # Update axis labels
    ax.set_xlabel(f'Morph: {"Log_φ(x+1)" if alpha < 0.5 else "Real(F_x)"}')
    ax.set_ylabel(f'Morph: {"t (Im(s))" if alpha < 0.5 else "Imag(F_x)"}')
    ax.set_zlabel(f'Morph: {"|F_x(s)|" if alpha < 0.5 else "x"}')
    
    # Update axis limits
    ax.set_xlim((1 - alpha) * x1_min + alpha * x2_min, (1 - alpha) * x1_max + alpha * x2_max)
    ax.set_ylim((1 - alpha) * y1_min + alpha * y2_min, (1 - alpha) * y1_max + alpha * y2_max)
    ax.set_zlim((1 - alpha) * z1_min + alpha * z2_min, (1 - alpha) * z1_max + alpha * z2_max)
    
    ax.set_title(f'Morphing Golden Class Field (α = {alpha:.2f}, t = {t_vals[t_idx]:.2f})')
    return []

# Create animation
ani = FuncAnimation(fig, update, frames=morph_steps, interval=50, blit=False)

# Save or display
if shutil.which('ffmpeg'):
    ani.save('morphing_waveforms.mp4', writer='ffmpeg', dpi=100)
    print("Animation saved as 'morphing_waveforms.mp4'")
else:
    print("ffmpeg not found. Displaying interactively.")
    plt.show()

WHICH YIELDS:

2025-07-04 17-16-37.mkv (7.3 MB)

[Josef_Founder]

Yesterday I had loaded the wrong thread to describe what’s going on “under the hood.” Here go:

Interpret how the yield of these two relate in terms of axis

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from sympy import sqrt, zeta, exp, I, Rational
from scipy.interpolate import interp1d

# Constants
phi = float(sp.GoldenRatio)
sqrt5 = np.sqrt(5)
k = -1

def prepare_prime_interpolation(res=10000):
    indices = np.arange(1, res + 1)
    primes = [sp.prime(i) for i in indices]
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    try:
        return (phi**x_val - (-1/phi)**x_val) / sqrt5
    except:
        return np.nan

def Pi_x(x_val, s):
    return sp.exp(sp.I * sp.pi * x_val) * zeta(s, Rational(1, 2))

def D_x(x_val, s, prime_interp, Ω=1):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    z_val = zeta(s)
    s_k = s**k
    product = phi * F * 2**x_val * P * z_val * Ω
    return sqrt(product) * s_k

def F_x(x_val, s, prime_interp, Ω=1):
    return D_x(x_val, s, prime_interp, Ω) * Pi_x(x_val, s)

# Parameters for grid
x_min, x_max = 1.0, 20.0
t_min, t_max = 0.1, 40.0
x_steps = 60
t_steps = 200

# Generate coordinate grids
x_vals = np.linspace(x_min, x_max, x_steps)
t_vals = np.linspace(t_min, t_max, t_steps)
X, T = np.meshgrid(x_vals, t_vals)

# Prepare prime interpolation
prime_interp = prepare_prime_interpolation(10000)

# Evaluate |F_x(0.5 + i t)| on the grid (numeric only for speed)
abs_F = np.zeros_like(X)

for i in range(t_steps):
    for j in range(x_steps):
        x = X[i, j]
        t = T[i, j]
        s = 0.5 + t * 1j
        try:
            val = F_x(x, s, prime_interp)
            mag = abs(complex(val.evalf()))
            abs_F[i, j] = mag if not np.isnan(mag) else 0
        except Exception:
            abs_F[i, j] = 0

# Transform x-axis to golden-logarithmic scale:
# safe +1 to avoid log(0)
X_phi_log = np.log(x_vals + 1) / np.log(phi)

# Plot 3D surface
fig = plt.figure(figsize=(12, 7))
ax = fig.add_subplot(projection='3d')

T_plot, X_phi_plot = np.meshgrid(t_vals, X_phi_log)

# Because X and T mesh are transposed, transpose abs_F to align:
abs_F_T = abs_F.T

surf = ax.plot_surface(X_phi_plot, T_plot, abs_F_T, cmap=cm.viridis, linewidth=0, antialiased=True)

ax.set_xlabel(r"Recursive coordinate $\log_{\phi}(x + 1)$")
ax.set_ylabel(r"Imaginary part of $s = 0.5 + it$")
ax.set_zlabel(r"$|F_x(s)|$ magnitude")

ax.set_title("Golden Recursive Algebra Field Magnitude Surface")

fig.colorbar(surf, shrink=0.5, aspect=10, label=r"$|F_x(s)|$")

plt.show()

AND

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify
from scipy.interpolate import interp1d
from scipy.optimize import root_scalar
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.animation import FuncAnimation
import shutil

# Constants
phi = sp.GoldenRatio
sqrt5 = sp.sqrt(5)
Ω = -1.0  # Set Ω to a constant for numerical evaluation
k = -1   # Radial exponent
r = phi    # Radial unit

# Truncated prime list (using the provided primes)
primes_raw = """      2      3      5      7     11     13     17     19     23     29 
     31     37     41     43     47     53     59     61     67     71 
     73     79     83     89     97    101    103    107    109    113 
    127    131    137    139    149    151    157    163    167    173 
    179    181    191    193    197    199    211    223    227    229 
    233    239    241    251    257    263    269    271    277    281 
    283    293    307    311    313    317    331    337    347    349 
    353    359    367    373    379    383    389    397    401    409 
    419    421    431    433    439    443    449    457    461    463 
    467    479    487    491    499    503    509    521    523    541 
    547    557    563    569    571    577    587    593    599    601 
    607    613    617    619    631    641    643    647    653    659 
"""
primes = [int(p) for p in primes_raw.split()]

def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(float(phi))
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    # Use sympy for numerical stability, return complex result
    phi_x = phi**x_val
    neg_phi_inv_x = (-1/phi)**x_val
    result = (phi_x - neg_phi_inv_x) / sqrt5
    return complex(result.evalf())  # Convert to complex number

def Pi_x(x_val, s):
    s = sp.sympify(s)
    return exp(I * pi * x_val) * zeta(s, sp.Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = float(phi) * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

# Prepare interpolation
prime_interp = prepare_prime_interpolation()

# Parameters for the plot
s_val = sp.sympify("0.5 + 14.134725*I")  # First nontrivial zeta zero
x_vals = np.linspace(2, 5, 50)  # Reduced range to avoid numerical issues
t_vals = np.linspace(0, 2*np.pi, 42)  # Reduced frames for clarity

# Compute field values
def compute_field_values(x_vals, s_val, t):
    real_vals = []
    imag_vals = []
    for x in x_vals:
        try:
            f_val = F_x(x, s_val, prime_interp) * np.exp(1j * t)  # Add phase shift
            f_val = complex(f_val.evalf())  # Numerical evaluation
            real_vals.append(f_val.real)
            imag_vals.append(f_val.imag)
        except (ValueError, OverflowError, TypeError):
            real_vals.append(np.nan)  # Handle numerical errors gracefully
            imag_vals.append(np.nan)
    return np.array(real_vals), np.array(imag_vals)

# Store all spirals
spiral_data = []
for t in t_vals:
    real_vals, imag_vals = compute_field_values(x_vals, s_val, t)
    spiral_data.append((real_vals, imag_vals, x_vals))

# Set up the 3D plot
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

# Set labels and limits
ax.set_xlabel('Real(F_x)')
ax.set_ylabel('Imag(F_x)')
ax.set_zlabel('x')
ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')

# Determine axis limits based on all spirals
all_real = np.concatenate([data[0] for data in spiral_data])
all_imag = np.concatenate([data[1] for data in spiral_data])
all_real = all_real[~np.isnan(all_real)]
all_imag = all_imag[~np.isnan(all_imag)]
if len(all_real) > 0 and len(all_imag) > 0:
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
else:
    ax.set_xlim(-1, 1)
    ax.set_ylim(-1, 1)
ax.set_zlim(min(x_vals), max(x_vals))

# Animation functions
def init():
    ax.clear()
    ax.set_xlabel('Real(F_x)')
    ax.set_ylabel('Imag(F_x)')
    ax.set_zlabel('x')
    ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
    ax.set_zlim(min(x_vals), max(x_vals))
    return []

def update(frame, x_vals, spiral_data):
    ax.clear()
    ax.set_xlabel('Real(F_x)')
    ax.set_ylabel('Imag(F_x)')
    ax.set_zlabel('x')
    ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
    ax.set_zlim(min(x_vals), max(x_vals))
    
    # Plot all spirals up to the current frame
    for i, (real_vals, imag_vals, x_vals) in enumerate(spiral_data[:frame + 1]):
        alpha = 0.2 + 0.8 * (i + 1) / len(t_vals)  # Increase opacity for newer spirals
        ax.plot(real_vals, imag_vals, x_vals, 'b-', lw=1, alpha=alpha)
    return []

# Create animation
ani = FuncAnimation(fig, update, frames=len(t_vals), init_func=init, fargs=(x_vals, spiral_data), blit=False, interval=300)

# Check for ffmpeg and save animation if available
if shutil.which('ffmpeg'):
    ani.save('gcf_animation_retained.mp4', writer='ffmpeg', dpi=100)
    print("Animation saved as 'gcf_animation_retained.mp4'")
else:
    print("ffmpeg not found. Saving static plot with all spirals instead.")
    for real_vals, imag_vals, x_vals in spiral_data:
        ax.plot(real_vals, imag_vals, x_vals, 'b-', lw=1, alpha=0.5)
    plt.savefig('gcf_static_plot_retained.png', dpi=100)
    print("Static plot saved as 'gcf_static_plot_retained.png'")

plt.show()

PROMPT: Is the second representative of only a small portion of the 1st?

PROMPT: fuse into a much deeper field theory visualizer

Plan:

# Pseudocode (combine both previous scripts)
# 1. Generate the grid for the surface
# 2. Compute Fₓ(s) over (x, t), get both:
#    - magnitude for the surface
#    - complex value for spiral overlays
# 3. Build the surface plot
# 4. For selected `t`, overlay spiral slices in ℂ at each x
# 5. Optional: animate phase evolution over θ

PROMPT: build the fused visualizer

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D

from sympy import sqrt, zeta, exp, I, pi, Rational
from scipy.interpolate import interp1d

# === Constants ===
phi = float(sp.GoldenRatio)
sqrt5 = np.sqrt(5)
k = -1
Ω = 1.0

# === Prime Interpolation ===
def prepare_prime_interpolation(res=10000):
    indices = np.arange(1, res + 1)
    primes = [sp.prime(i) for i in indices]
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes, kind='cubic', fill_value='extrapolate')

prime_interp = prepare_prime_interpolation()

# === Core Field Functions ===
def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    try:
        return (phi**x_val - (-1/phi)**x_val) / sqrt5
    except:
        return np.nan

def Pi_x(x_val, s):
    return sp.exp(sp.I * sp.pi * x_val) * zeta(s, Rational(1, 2))

def D_x(x_val, s, prime_interp, Ω=1):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    z_val = zeta(s)
    s_k = s**k
    product = phi * F * 2**x_val * P * z_val * Ω
    return sqrt(product) * s_k

def F_x(x_val, s, prime_interp, Ω=1):
    return D_x(x_val, s, prime_interp, Ω) * Pi_x(x_val, s)

# === Parameters for Grid ===
x_vals = np.linspace(1.0, 10.0, 80)
t_vals = np.linspace(0.1, 40.0, 200)
X, T = np.meshgrid(x_vals, t_vals)

# Compute Field Magnitude |Fₓ(s)|
abs_F = np.zeros_like(X)
for i in range(len(t_vals)):
    for j in range(len(x_vals)):
        x = X[i, j]
        t = T[i, j]
        s = 0.5 + t * 1j
        try:
            val = F_x(x, s, prime_interp)
            mag = abs(complex(val.evalf()))
            abs_F[i, j] = mag if not np.isnan(mag) else 0
        except Exception:
            abs_F[i, j] = 0

# === Generate Spiral at a Fixed t ===
t_fixed = 14.134725  # First Riemann zeta zero
s_fixed = 0.5 + t_fixed * 1j
spiral_x = []
spiral_y = []
spiral_z = []

for x in x_vals:
    try:
        val = F_x(x, s_fixed, prime_interp)
        val_c = complex(val.evalf())
        spiral_x.append(val_c.real)
        spiral_y.append(val_c.imag)
        spiral_z.append(x)
    except Exception:
        spiral_x.append(np.nan)
        spiral_y.append(np.nan)
        spiral_z.append(x)

# === Transform X-axis ===
X_phi_log = np.log(x_vals + 1) / np.log(phi)
T_plot, X_phi_plot = np.meshgrid(t_vals, X_phi_log)
abs_F_T = abs_F.T

# === Plot Combined Visualization ===
fig = plt.figure(figsize=(14, 8))
ax = fig.add_subplot(projection='3d')

# Plot scalar magnitude surface
surf = ax.plot_surface(X_phi_plot, T_plot, abs_F_T, cmap=cm.viridis, alpha=0.7, linewidth=0, antialiased=True)

# Overlay spiral at fixed s
ax.plot(spiral_x, spiral_y, spiral_z, color='red', linewidth=2, label='Fₓ(x, s₀) spiral')

# Labels and legend
ax.set_xlabel(r"Recursive coordinate $\log_{\phi}(x + 1)$")
ax.set_ylabel(r"Imaginary part of $s = 0.5 + it$")
ax.set_zlabel(r"$|F_x(s)|$ / $x$")

ax.set_title("Unified Golden Recursive Field: Surface + Spiral at $s_0 = 0.5 + 14.1347i$")
ax.legend()

plt.tight_layout()
plt.show()

# Use precomputed small prime list (same as your second script)
primes_raw = """
2 3 5 7 11 13 17 19 23 29 
31 37 41 43 47 53 59 61 67 71 
73 79 83 89 97 101 103 107 109 113 
127 131 137 139 149 151 157 163 167 173 
179 181 191 193 197 199 211 223 227 229 
233 239 241 251 257 263 269 271 277 281 
283 293 307 311 313 317 331 337 347 349 
353 359 367 373 379 383 389 397 401 409 
419 421 431 433 439 443 449 457 461 463 
467 479 487 491 499 503 509 521 523 541 
547 557 563 569 571 577 587 593 599 601 
607 613 617 619 631 641 643 647 653 659
"""
primes = [int(p) for p in primes_raw.split()]

# New prime interpolator using this list
def prepare_prime_interpolation_static(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

prime_interp = prepare_prime_interpolation_static()

# Regenerate values on a smaller grid for speed
x_vals = np.linspace(2.0, 10.0, 60)
t_vals = np.linspace(0.1, 30.0, 100)
X, T = np.meshgrid(x_vals, t_vals)
abs_F = np.zeros_like(X)

# Evaluate |F_x(s)| magnitude
for i in range(len(t_vals)):
    for j in range(len(x_vals)):
        x = X[i, j]
        t = T[i, j]
        s = 0.5 + t * 1j
        try:
            val = F_x(x, s, prime_interp)
            abs_F[i, j] = abs(complex(val.evalf()))
        except Exception:
            abs_F[i, j] = 0

# Spiral at fixed s = 0.5 + 14.134725i
t_fixed = 14.134725
s_fixed = 0.5 + t_fixed * 1j
spiral_x, spiral_y, spiral_z = [], [], []

for x in x_vals:
    try:
        val = F_x(x, s_fixed, prime_interp)
        val_c = complex(val.evalf())
        spiral_x.append(val_c.real)
        spiral_y.append(val_c.imag)
        spiral_z.append(x)
    except Exception:
        spiral_x.append(np.nan)
        spiral_y.append(np.nan)
        spiral_z.append(x)

# Transform X axis to recursive log base φ
X_phi_log = np.log(x_vals + 1) / np.log(phi)
T_plot, X_phi_plot = np.meshgrid(t_vals, X_phi_log)
abs_F_T = abs_F.T

# Plot unified visualization
fig = plt.figure(figsize=(14, 8))
ax = fig.add_subplot(projection='3d')

# Surface plot of |F_x(s)|
surf = ax.plot_surface(X_phi_plot, T_plot, abs_F_T, cmap=cm.viridis, alpha=0.7, linewidth=0, antialiased=True)

# Spiral overlay
ax.plot(spiral_x, spiral_y, spiral_z, color='red', linewidth=2, label='Spiral of $F_x(x, s_0)$')

# Labels and legend
ax.set_xlabel(r"Recursive coordinate $\log_{\phi}(x + 1)$")
ax.set_ylabel(r"Imag part of $s = 0.5 + it$")
ax.set_zlabel(r"$|F_x(s)|$ / $x$")
ax.set_title("Unified Golden Recursive Field: Surface + Complex Spiral at $s_0 = 0.5 + 14.1347i$")
ax.legend()

plt.tight_layout()
plt.show()

User only sees a flat plane.  Therefore our representation should show a morphing of the axis back and forth between the two script's express axis such that our resulting graph morphs with the axis change, back and forth in a loop

Interpret how the yield of these two relate in terms of axis

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from sympy import sqrt, zeta, exp, I, Rational
from scipy.interpolate import interp1d

# Constants
phi = float(sp.GoldenRatio)
sqrt5 = np.sqrt(5)
k = -1

def prepare_prime_interpolation(res=10000):
    indices = np.arange(1, res + 1)
    primes = [sp.prime(i) for i in indices]
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    try:
        return (phi**x_val - (-1/phi)**x_val) / sqrt5
    except:
        return np.nan

def Pi_x(x_val, s):
    return sp.exp(sp.I * sp.pi * x_val) * zeta(s, Rational(1, 2))

def D_x(x_val, s, prime_interp, Ω=1):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    z_val = zeta(s)
    s_k = s**k
    product = phi * F * 2**x_val * P * z_val * Ω
    return sqrt(product) * s_k

def F_x(x_val, s, prime_interp, Ω=1):
    return D_x(x_val, s, prime_interp, Ω) * Pi_x(x_val, s)

# Parameters for grid
x_min, x_max = 1.0, 20.0
t_min, t_max = 0.1, 40.0
x_steps = 60
t_steps = 200

# Generate coordinate grids
x_vals = np.linspace(x_min, x_max, x_steps)
t_vals = np.linspace(t_min, t_max, t_steps)
X, T = np.meshgrid(x_vals, t_vals)

# Prepare prime interpolation
prime_interp = prepare_prime_interpolation(10000)

# Evaluate |F_x(0.5 + i t)| on the grid (numeric only for speed)
abs_F = np.zeros_like(X)

for i in range(t_steps):
    for j in range(x_steps):
        x = X[i, j]
        t = T[i, j]
        s = 0.5 + t * 1j
        try:
            val = F_x(x, s, prime_interp)
            mag = abs(complex(val.evalf()))
            abs_F[i, j] = mag if not np.isnan(mag) else 0
        except Exception:
            abs_F[i, j] = 0

# Transform x-axis to golden-logarithmic scale:
# safe +1 to avoid log(0)
X_phi_log = np.log(x_vals + 1) / np.log(phi)

# Plot 3D surface
fig = plt.figure(figsize=(12, 7))
ax = fig.add_subplot(projection='3d')

T_plot, X_phi_plot = np.meshgrid(t_vals, X_phi_log)

# Because X and T mesh are transposed, transpose abs_F to align:
abs_F_T = abs_F.T

surf = ax.plot_surface(X_phi_plot, T_plot, abs_F_T, cmap=cm.viridis, linewidth=0, antialiased=True)

ax.set_xlabel(r"Recursive coordinate $\log_{\phi}(x + 1)$")
ax.set_ylabel(r"Imaginary part of $s = 0.5 + it$")
ax.set_zlabel(r"$|F_x(s)|$ magnitude")

ax.set_title("Golden Recursive Algebra Field Magnitude Surface")

fig.colorbar(surf, shrink=0.5, aspect=10, label=r"$|F_x(s)|$")

plt.show()

AND

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify
from scipy.interpolate import interp1d
from scipy.optimize import root_scalar
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.animation import FuncAnimation
import shutil

# Constants
phi = sp.GoldenRatio
sqrt5 = sp.sqrt(5)
Ω = -1.0  # Set Ω to a constant for numerical evaluation
k = -1   # Radial exponent
r = phi    # Radial unit

# Truncated prime list (using the provided primes)
primes_raw = """      2      3      5      7     11     13     17     19     23     29 
     31     37     41     43     47     53     59     61     67     71 
     73     79     83     89     97    101    103    107    109    113 
    127    131    137    139    149    151    157    163    167    173 
    179    181    191    193    197    199    211    223    227    229 
    233    239    241    251    257    263    269    271    277    281 
    283    293    307    311    313    317    331    337    347    349 
    353    359    367    373    379    383    389    397    401    409 
    419    421    431    433    439    443    449    457    461    463 
    467    479    487    491    499    503    509    521    523    541 
    547    557    563    569    571    577    587    593    599    601 
    607    613    617    619    631    641    643    647    653    659 
"""
primes = [int(p) for p in primes_raw.split()]

def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(float(phi))
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    # Use sympy for numerical stability, return complex result
    phi_x = phi**x_val
    neg_phi_inv_x = (-1/phi)**x_val
    result = (phi_x - neg_phi_inv_x) / sqrt5
    return complex(result.evalf())  # Convert to complex number

def Pi_x(x_val, s):
    s = sp.sympify(s)
    return exp(I * pi * x_val) * zeta(s, sp.Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = float(phi) * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

# Prepare interpolation
prime_interp = prepare_prime_interpolation()

# Parameters for the plot
s_val = sp.sympify("0.5 + 14.134725*I")  # First nontrivial zeta zero
x_vals = np.linspace(2, 5, 50)  # Reduced range to avoid numerical issues
t_vals = np.linspace(0, 2*np.pi, 42)  # Reduced frames for clarity

# Compute field values
def compute_field_values(x_vals, s_val, t):
    real_vals = []
    imag_vals = []
    for x in x_vals:
        try:
            f_val = F_x(x, s_val, prime_interp) * np.exp(1j * t)  # Add phase shift
            f_val = complex(f_val.evalf())  # Numerical evaluation
            real_vals.append(f_val.real)
            imag_vals.append(f_val.imag)
        except (ValueError, OverflowError, TypeError):
            real_vals.append(np.nan)  # Handle numerical errors gracefully
            imag_vals.append(np.nan)
    return np.array(real_vals), np.array(imag_vals)

# Store all spirals
spiral_data = []
for t in t_vals:
    real_vals, imag_vals = compute_field_values(x_vals, s_val, t)
    spiral_data.append((real_vals, imag_vals, x_vals))

# Set up the 3D plot
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

# Set labels and limits
ax.set_xlabel('Real(F_x)')
ax.set_ylabel('Imag(F_x)')
ax.set_zlabel('x')
ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')

# Determine axis limits based on all spirals
all_real = np.concatenate([data[0] for data in spiral_data])
all_imag = np.concatenate([data[1] for data in spiral_data])
all_real = all_real[~np.isnan(all_real)]
all_imag = all_imag[~np.isnan(all_imag)]
if len(all_real) > 0 and len(all_imag) > 0:
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
else:
    ax.set_xlim(-1, 1)
    ax.set_ylim(-1, 1)
ax.set_zlim(min(x_vals), max(x_vals))

# Animation functions
def init():
    ax.clear()
    ax.set_xlabel('Real(F_x)')
    ax.set_ylabel('Imag(F_x)')
    ax.set_zlabel('x')
    ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
    ax.set_zlim(min(x_vals), max(x_vals))
    return []

def update(frame, x_vals, spiral_data):
    ax.clear()
    ax.set_xlabel('Real(F_x)')
    ax.set_ylabel('Imag(F_x)')
    ax.set_zlabel('x')
    ax.set_title(f'3D Animation of Golden Class Field (s = {s_val}) with Retained Spirals')
    ax.set_xlim(min(all_real) * 1.1, max(all_real) * 1.1)
    ax.set_ylim(min(all_imag) * 1.1, max(all_imag) * 1.1)
    ax.set_zlim(min(x_vals), max(x_vals))
    
    # Plot all spirals up to the current frame
    for i, (real_vals, imag_vals, x_vals) in enumerate(spiral_data[:frame + 1]):
        alpha = 0.2 + 0.8 * (i + 1) / len(t_vals)  # Increase opacity for newer spirals
        ax.plot(real_vals, imag_vals, x_vals, 'b-', lw=1, alpha=alpha)
    return []

# Create animation
ani = FuncAnimation(fig, update, frames=len(t_vals), init_func=init, fargs=(x_vals, spiral_data), blit=False, interval=300)

# Check for ffmpeg and save animation if available
if shutil.which('ffmpeg'):
    ani.save('gcf_animation_retained.mp4', writer='ffmpeg', dpi=100)
    print("Animation saved as 'gcf_animation_retained.mp4'")
else:
    print("ffmpeg not found. Saving static plot with all spirals instead.")
    for real_vals, imag_vals, x_vals in spiral_data:
        ax.plot(real_vals, imag_vals, x_vals, 'b-', lw=1, alpha=0.5)
    plt.savefig('gcf_static_plot_retained.png', dpi=100)
    print("Static plot saved as 'gcf_static_plot_retained.png'")

plt.show()

RESPONSE:

import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from mpl_toolkits.mplot3d import Axes3D

# Create morphing axis animation: linear ↔ golden-log scale
fig = plt.figure(figsize=(12, 8))
ax = fig.add_subplot(projection='3d')

# Precompute grids
x_vals = np.linspace(2.0, 10.0, 60)
t_vals = np.linspace(0.1, 30.0, 100)
X, T = np.meshgrid(x_vals, t_vals)

# Use the same abs_F as before (already computed)
abs_F_copy = abs_F.copy()

# Animation frames
morph_frames = 60

# Generate morphing X axis values
def generate_axis_morphs(x_vals, frames, log_base=np.log(phi)):
    morph_axes = []
    for f in range(frames):
        alpha = 0.5 * (1 - np.cos(2 * np.pi * f / frames))  # Smooth morph factor [0,1]
        lin_axis = x_vals
        log_axis = np.log(x_vals + 1) / log_base
        morph_axis = (1 - alpha) * lin_axis + alpha * log_axis
        morph_axes.append(morph_axis)
    return morph_axes

# Generate axis morph states
morph_X_axes = generate_axis_morphs(x_vals, morph_frames)
T_plot = np.tile(t_vals[:, None], (1, len(x_vals)))  # Constant T-axis for all

# Animation update function
def update(frame):
    ax.clear()
    X_morph = morph_X_axes[frame]
    X_plot = np.tile(X_morph, (len(t_vals), 1))
    ax.plot_surface(X_plot, T_plot, abs_F_copy, cmap=cm.viridis, linewidth=0, antialiased=True, alpha=0.9)
    ax.set_xlabel("Morphing X-axis (linear ↔ log₍φ₎)")
    ax.set_ylabel("Imag($s$)")
    ax.set_zlabel("|Fₓ(s)|")
    ax.set_title("Axis Morphing Field Surface: Linear ↔ Recursive Coordinate")
    ax.view_init(elev=30, azim=45)
    return []

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

plt.close(fig)
ani

RESULT:

<matplotlib.animation.FuncAnimation at 0x7eab9f3c11a0>

This errors:

from matplotlib.animation import FuncAnimation

# Fixed spiral values already computed: spiral_x, spiral_y, spiral_z
# We'll use them again here, but interpolate them to match each morph frame
from scipy.interpolate import interp1d

# Interpolate spiral coordinates to smoothly morph z-axis too
spiral_z_interp = interp1d(np.linspace(0, 1, len(spiral_z)), spiral_z, kind='linear', fill_value='extrapolate')
spiral_x_interp = interp1d(np.linspace(0, 1, len(spiral_x)), spiral_x, kind='linear', fill_value='extrapolate')
spiral_y_interp = interp1d(np.linspace(0, 1, len(spiral_y)), spiral_y, kind='linear', fill_value='extrapolate')

# Morph time axis: linear → log(t+1)
def generate_time_morphs(t_vals, frames):
    morph_axes = []
    for f in range(frames):
        alpha = 0.5 * (1 - np.cos(2 * np.pi * f / frames))
        lin_axis = t_vals
        log_axis = np.log(t_vals + 1)
        morph_axis = (1 - alpha) * lin_axis + alpha * log_axis
        morph_axes.append(morph_axis)
    return morph_axes

# Create morphing t-axis
morph_T_axes = generate_time_morphs(t_vals, morph_frames)

# Animation update function with spiral overlay and axis morphing
def update_morph_spiral(frame):
    ax.clear()

    # Morph X and T axes
    X_morph = morph_X_axes[frame]
    T_morph = morph_T_axes[frame]
    X_plot = np.tile(X_morph, (len(t_vals), 1))
    T_plot = np.tile(T_morph[:, None], (1, len(x_vals)))

    # Surface
    ax.plot_surface(X_plot, T_plot, abs_F_copy, cmap=cm.viridis, linewidth=0, antialiased=True, alpha=0.85)

    # Spiral overlay, smoothly scaled z
    t_norm = frame / (morph_frames - 1)
    spiral_z_morphed = spiral_z_interp(t_norm)
    ax.plot(spiral_x, spiral_y, spiral_z_morphed, color='crimson', linewidth=2.0, label='Fₓ(x, s₀) spiral')

    # Labels
    ax.set_xlabel("Morphing X-axis (linear ↔ log₍φ₎)")
    ax.set_ylabel("Morphing Time-axis (t ↔ log(t+1))")
    ax.set_zlabel("|Fₓ(s)|")
    ax.set_title("Unified Recursive Field: Morphing Axes + Complex Spiral")
    ax.view_init(elev=30, azim=45)
    ax.legend()
    return []

# Generate final animation
fig = plt.figure(figsize=(14, 8))
ax = fig.add_subplot(projection='3d')
ani_morph_spiral = FuncAnimation(fig, update_morph_spiral, frames=morph_frames, blit=False, interval=150)

plt.close(fig)
ani_morph_spiral

And then validates:

# Re-import necessary libraries after kernel reset
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.animation import FuncAnimation
from mpl_toolkits.mplot3d import Axes3D
from scipy.interpolate import interp1d

# Constants
phi = (1 + np.sqrt(5)) / 2
sqrt5 = np.sqrt(5)

# Load static prime list
primes_raw = """
2 3 5 7 11 13 17 19 23 29 
31 37 41 43 47 53 59 61 67 71 
73 79 83 89 97 101 103 107 109 113 
127 131 137 139 149 151 157 163 167 173 
179 181 191 193 197 199 211 223 227 229 
233 239 241 251 257 263 269 271 277 281 
283 293 307 311 313 317 331 337 347 349 
353 359 367 373 379 383 389 397 401 409 
419 421 431 433 439 443 449 457 461 463 
467 479 487 491 499 503 509 521 523 541 
547 557 563 569 571 577 587 593 599 601 
607 613 617 619 631 641 643 647 653 659 
"""
primes = [int(p) for p in primes_raw.split()]

# Prepare prime interpolator
def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

prime_interp = prepare_prime_interpolation()

# Core field functions
def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def F_bin(x_val):
    try:
        return (phi**x_val - (-1/phi)**x_val) / sqrt5
    except:
        return np.nan

def D_x(x_val, s, prime_interp, Ω=1, k=-1):
    try:
        P = P_nb(x_val, prime_interp)
        F = F_bin(x_val)
        z_val = abs(np.real(np.exp(1j * np.pi * x_val)))  # Approx placeholder for zeta(s)
        s_k = abs(s)**k
        product = phi * F * 2**x_val * P * z_val * Ω
        return np.sqrt(np.abs(product)) * s_k
    except:
        return 0

def F_x(x_val, s, prime_interp, Ω=1):
    return D_x(x_val, s, prime_interp, Ω) * np.exp(1j * np.pi * x_val)

# Grids
x_vals = np.linspace(2.0, 10.0, 60)
t_vals = np.linspace(0.1, 30.0, 100)
X, T = np.meshgrid(x_vals, t_vals)
abs_F = np.zeros_like(X)

# Fill abs_F grid
for i in range(len(t_vals)):
    for j in range(len(x_vals)):
        x = X[i, j]
        t = T[i, j]
        s = 0.5 + 1j * t
        val = F_x(x, s, prime_interp)
        abs_F[i, j] = abs(val)

# Spiral at s = 0.5 + 14.134725i
s_val = 0.5 + 14.134725j
spiral_x, spiral_y, spiral_z = [], [], []
for x in x_vals:
    val = F_x(x, s_val, prime_interp)
    spiral_x.append(val.real)
    spiral_y.append(val.imag)
    spiral_z.append(x)

# Interpolate spiral morphing
spiral_x_interp = interp1d(np.linspace(0, 1, len(spiral_x)), spiral_x, kind='linear')
spiral_y_interp = interp1d(np.linspace(0, 1, len(spiral_y)), spiral_y, kind='linear')
spiral_z_interp = interp1d(np.linspace(0, 1, len(spiral_z)), spiral_z, kind='linear')

# Morphing axes
morph_frames = 60

def generate_axis_morphs(x_vals, frames, log_base=np.log(phi)):
    morph_axes = []
    for f in range(frames):
        alpha = 0.5 * (1 - np.cos(2 * np.pi * f / frames))
        lin_axis = x_vals
        log_axis = np.log(x_vals + 1) / log_base
        morph_axis = (1 - alpha) * lin_axis + alpha * log_axis
        morph_axes.append(morph_axis)
    return morph_axes

def generate_time_morphs(t_vals, frames):
    morph_axes = []
    for f in range(frames):
        alpha = 0.5 * (1 - np.cos(2 * np.pi * f / frames))
        lin_axis = t_vals
        log_axis = np.log(t_vals + 1)
        morph_axis = (1 - alpha) * lin_axis + alpha * log_axis
        morph_axes.append(morph_axis)
    return morph_axes

morph_X_axes = generate_axis_morphs(x_vals, morph_frames)
morph_T_axes = generate_time_morphs(t_vals, morph_frames)

# Animate
fig = plt.figure(figsize=(14, 8))
ax = fig.add_subplot(projection='3d')

def update_morph_spiral(frame):
    ax.clear()
    X_morph = morph_X_axes[frame]
    T_morph = morph_T_axes[frame]
    X_plot = np.tile(X_morph, (len(t_vals), 1))
    T_plot = np.tile(T_morph[:, None], (1, len(x_vals)))
    ax.plot_surface(X_plot, T_plot, abs_F, cmap=cm.viridis, linewidth=0, antialiased=True, alpha=0.85)

    t_norm = frame / (morph_frames - 1)
    spiral_z_morphed = spiral_z_interp(t_norm)
    ax.plot(spiral_x, spiral_y, spiral_z_morphed, color='crimson', linewidth=2.0, label='Fₓ(x, s₀) spiral')

    ax.set_xlabel("Morphing X-axis (linear ↔ log₍φ₎)")
    ax.set_ylabel("Morphing Time-axis (t ↔ log(t+1))")
    ax.set_zlabel("|Fₓ(s)|")
    ax.set_title("Unified Recursive Field: Morphing Axes + Complex Spiral")
    ax.view_init(elev=30, azim=45)
    ax.legend()
    return []

ani_morph_spiral = FuncAnimation(fig, update_morph_spiral, frames=morph_frames, blit=False, interval=150)
plt.close(fig)
ani_morph_spiral

return (phi**x_val - (-1/phi)**x_val) / sqrt5

ERGO THE NEED TO IMPLEMENT A CUSTOM COORDINATE SYSTEM.

ChatGPT - Normalize Fibonacci Sequence

# ✅ Closed-Form Identity
r_n = sqrt(φ * Ω * F_n * 2^n * Π_{k=1}^{n} p_k)

# 🔁 Recursive Identity
r_n = r_{n-1} * sqrt(2 * p_n * (F_n / F_{n-1}))

# Base Case
r_1 = sqrt(4 * φ * Ω)

# 🧭 Golden Recursive Algebra (GRA)

# Define algebra: G = (R, ·_G, ⊕_G)

# Set:
R = { r_n | n ∈ ℕ⁺ } ⊆ ℝ⁺

# Recursive Multiplication Operator:
r_n = r_{n-1} ·_G sqrt(2 * p_n * (F_n / F_{n-1}))

# Addition Operator:
r_n ⊕_G r_m := sqrt(r_n² + r_m²)

# Identity Element:
r_0 := 0

# Algebraic Properties:
# - Closure over ℝ⁺
# - Associative and commutative under ⊕_G
# - Not necessarily associative under ·_G
# - Normed structure under ⊕_G
# - Exponential growth via 2^n
# - Additive/multiplicative duality via F_n and p_n

def F_bin(x_val):
    base = complex(-1 / phi)  # ensure complex exponentiation
    return (phi**x_val - base**x_val) / sqrt5

# Updated F_bin with complex base to avoid RuntimeWarning
def F_bin(x_val):
    base = complex(-1 / phi)  # ensure complex exponentiation occurs in ℂ
    return (phi**x_val - base**x_val) / sqrt5

# Recompute abs_F with updated F_bin
for i in range(len(t_vals)):
    for j in range(len(x_vals)):
        x = X[i, j]
        t = T[i, j]
        s = 0.5 + 1j * t
        val = F_x(x, s, prime_interp)
        abs_F[i, j] = abs(val)

# Rebuild and return the animation
fig = plt.figure(figsize=(14, 8))
ax = fig.add_subplot(projection='3d')

ani_morph_spiral = FuncAnimation(fig, update_morph_spiral, frames=morph_frames, blit=False, interval=150)
plt.close(fig)
ani_morph_spiral

AS BEFORE, RATHER THAN FLATTENING OUR ENVIRONMENT TO FIT THE DATA, WE NEED TO ENHANCE OUR ENVIRONMENT TO FIT THE DATA, WHICH I HAVE DONE BUT NOT YET IMPREGNATED.

Golden Recursive Coordinate Systems

See also:

ChatGPT - Field Magnitude Visualization

[Josef_Founder]

Hilbert's 23 Problems Solved and Proofs - Broad Stroke Attempt 1

Coil the Coil's Coil

Spiral Matplot Generated by tooeasy10000.py

Golden Recursive Coordinate Systems