UPDATE: I built this out, it may help with better prime production on-the-fly, and can be modified to suit need -
# Extended primes list (up to 10,000)
def generate_primes(n):
sieve = [True] * (n + 1)
sieve[0] = sieve[1] = False
for i in range(2, int(np.sqrt(n)) + 1):
if sieve[i]:
for j in range(i * i, n + 1, i):
sieve[j] = False
return [i for i in range(n + 1) if sieve[i]]
PRIMES = generate_primes(104729)[:10000] # First 10,000 primes, up to ~104,729
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.
# ✅ 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.
See also:
