Spectral Math Unlocks Bitcoin Private Keys?

[PAPASO]

Code:

# MSE < 1e-7， r = 1~8

import numpy as np
import matplotlib.pyplot as plt

#  (k, x_Q, y_Q)
k_values = np.array([3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80])
x_Q = np.linspace(0.5, 2.0, 12)
y_Q = np.linspace(1.0, 3.0, 12)
xy = x_Q + y_Q

# Hecke  a_p
a_p = np.array([1.0, -1.0, 0.0, 2.0, -2.0, 1.0, -1.0, 3.0, -3.0, 2.0, 0.0, -2.0])

# （r=1~8）
X_parts = []
for i in range(len(a_p)):
    for r in range(1, 9):
        X_parts.append(a_p[i] * (xy ** r))
X = np.column_stack(X_parts)

# 
coeffs, _, _, _ = np.linalg.lstsq(X, k_values, rcond=None)
k_pred = X @ coeffs

# 
mse = np.mean((k_pred - k_values)**2)
max_err = np.max(np.abs(k_pred - k_values))
mean_err = np.mean(np.abs(k_pred - k_values))

# 
print("\nSpectral Scalar k Regression (Enhanced Precision)")
print("="*60)
print(f"MSE:             {mse:.6e}")
print(f"Max Error:       {max_err:.6f}")
print(f"Mean Error:      {mean_err:.6f}")
print(f"k (true):        {k_values.tolist()}")
print(f"k (predicted):   {np.round(k_pred, 6).tolist()}")
print(f"First 6 Coeffs:  {np.round(coeffs[:6], 6).tolist()}")

# 
plt.figure(figsize=(8, 5))
plt.plot(k_values, k_values, 'k--', label='Ideal (k=k)')
plt.plot(k_values, k_pred, 'o-', label='Predicted k')
plt.xlabel("True k")
plt.ylabel("Predicted k")
plt.title("High-Precision Spectral Fit: Hecke a_p → Scalar k (r=1~8)")
plt.grid(True)
plt.legend()
plt.tight_layout()
plt.show()



Spectral Scalar k Regression (Enhanced Precision)
============================================================
MSE:             1.421189e-07
Max Error:       0.000720
Mean Error:      0.000333
k (true):        [3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80]
k (predicted):   [2.999897, 5.00046, 7.99928, 12.000251, 17.000455, 22.999738, 29.999608, 38.00027, 47.00034, 56.999517, 68.000221, 79.999964]
First 6 Coeffs:  [0.268941, -0.527246, 0.449601, -0.18748, 0.048858, -0.007781]
2025-05-10 17:59:36.576 Python[37043:21895956] +[IMKClient subclass]: chose IMKClient_Modern
2025-05-10 17:59:36.577 Python[37043:21895956] +[IMKInputSession subclass]: chose IMKInputSession_Modern

https://zenodo.org/records/15379287

[PAPASO]

# Re-execute full model after reset: multivariable regression using (x_Q, a, b) and a_p
import numpy as np
import matplotlib.pyplot as plt

# Simulated ECC data
n_samples = 12
k_values = np.array([3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80])
x_Q = np.linspace(0.5, 2.0, n_samples)
y_Q = np.linspace(1.0, 3.0, n_samples)
a_curve = np.linspace(1, 3, n_samples)
b_curve = np.linspace(-1, 2, n_samples)
xy = x_Q + y_Q

# Hecke coefficients from ψ(t)
a_p = np.array([1.0, -1.0, 0.0, 2.0, -2.0, 1.0, -1.0, 3.0, -3.0, 2.0, 0.0, -2.0])
P = len(a_p)

# Build multivariable feature matrix
X_parts = []
for i in range(P):
    for r in range(1, 6):
        X_parts.append(a_p * (xy ** r))
        X_parts.append(a_p * (a_curve ** r))
        X_parts.append(a_p * (b_curve ** r))
X = np.column_stack(X_parts)

# Least squares regression
coeffs, _, _, _ = np.linalg.lstsq(X, k_values, rcond=None)
k_pred = X @ coeffs

# Error metrics
mse = np.mean((k_pred - k_values) ** 2)
max_err = np.max(np.abs(k_pred - k_values))
mean_err = np.mean(np.abs(k_pred - k_values))

# Output
print("\nMultivariable Spectral Regression (k from ψ(t), x_Q, a, b)")
print("=" * 65)
print(f"MSE:             {mse:.6e}")
print(f"Max Error:       {max_err:.6f}")
print(f"Mean Error:      {mean_err:.6f}")
print(f"k (true):        {k_values.tolist()}")
print(f"k (predicted):   {np.round(k_pred, 6).tolist()}")
print(f"First 6 Coeffs:  {np.round(coeffs[:6], 6).tolist()}")

# Visualization
plt.figure(figsize=(8, 5))
plt.plot(k_values, k_values, 'k--', label='Ideal')
plt.plot(k_values, k_pred, 'o-', label='Predicted')
plt.xlabel("True k")
plt.ylabel("Predicted k")
plt.title("Spectral Fit: Multivariable Hecke Model (k ≈ ψ(x,a,b))")
plt.grid(True)
plt.legend()
plt.tight_layout()
plt.show()


Multivariable Spectral Regression (k from ψ(t), x_Q, a, b)
=================================================================
MSE:             1.142697e-25
Max Error:       0.000000
Mean Error:      0.000000
k (true):        [3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80]
k (predicted):   [3.0, 5.0, 8.0, 12.0, 17.0, 23.0, 30.0, 38.0, 47.0, 57.0, 68.0, 80.0]
First 6 Coeffs:  [0.000527, 0.00011, 0.003505, 0.006097, 0.00205, -0.004563]
2025-05-10 19:10:55.795 Python[37714:21939219] +[IMKClient subclass]: chose IMKClient_Modern
2025-05-10 19:10:55.795 Python[37714:21939219] +[IMKInputSession subclass]: chose IMKInputSession_Modern

[PAPASO]

#  MSE < 1e-7

import numpy as np
import matplotlib.pyplot as plt

# Step 1:  (n=1000)
n_samples = 1000
k_values = np.linspace(1e3, 1e6, n_samples)  # 更大 k 量级
x_Q = np.linspace(0.5, 2.0, n_samples)
y_Q = np.linspace(1.0, 3.0, n_samples)
a_curve = np.linspace(1, 3, n_samples)
b_curve = np.linspace(-1, 2, n_samples)
xy = x_Q + y_Q

# Hecke
a_p = np.array([1.0, -1.0, 0.0, 2.0, -2.0, 1.0, -1.0, 3.0, -3.0, 2.0, 0.0, -2.0])
P = len(a_p)

# Step 2:  X（r = 1~5）
X_parts = []
for i in range(P):
    for r in range(1, 6):
        X_parts.append(a_p * (xy ** r))
        X_parts.append(a_p * (a_curve ** r))
        X_parts.append(a_p * (b_curve ** r))
X = np.column_stack(X_parts)

# Step 3:
coeffs, _, _, _ = np.linalg.lstsq(X, k_values, rcond=None)
k_pred = X @ coeffs

# Step 4:
mse = np.mean((k_pred - k_values) ** 2)
max_err = np.max(np.abs(k_pred - k_values))
mean_err = np.mean(np.abs(k_pred - k_values))

#
print("\nMultivariable Spectral Regression (n = 1000)")
print("=" * 65)
print(f"MSE:             {mse:.6e}")
print(f"Max Error:       {max_err:.6f}")
print(f"Mean Error:      {mean_err:.6f}")
print(f"First 5 k (true):     {np.round(k_values[:5], 3).tolist()}")
print(f"First 5 k (predicted):{np.round(k_pred[:5], 3).tolist()}")

#
plt.figure(figsize=(8, 5))
plt.plot(k_values[:100], k_values[:100], 'k--', label='Ideal')
plt.plot(k_values[:100], k_pred[:100], 'o-', label='Predicted')
plt.xlabel("True k")
plt.ylabel("Predicted k")
plt.title("Spectral Regression: Hecke a_p → Scalar k (n = 1000)")
plt.grid(True)
plt.legend()
plt.tight_layout()
plt.show()


Multivariable Spectral Regression (n = 1000)
=================================================================
MSE:             3.990343e-16
Max Error:       0.000000
Mean Error:      0.000000
First 5 k (true):     [1000.0, 2000.0, 3000.0, 4000.0, 5000.0]
First 5 k (predicted):[1000.0, 2000.0, 3000.0, 4000.0, 5000.0]
2025-05-10 19:29:44.285 Python[37892:21950858] +[IMKClient subclass]: chose IMKClient_Modern
2025-05-10 19:29:44.285 Python[37892:21950858] +[IMKInputSession subclass]: chose IMKInputSession_Modern

[PAPASO]

# n = 10000

import numpy as np
import matplotlib.pyplot as plt

# Step 1:
n_samples = 10000
k_values = np.linspace(1e3, 1e6, n_samples)
x_Q = np.linspace(0.5, 2.0, n_samples)
y_Q = np.linspace(1.0, 3.0, n_samples)
a_curve = np.linspace(1, 3, n_samples)
b_curve = np.linspace(-1, 2, n_samples)
xy = x_Q + y_Q

# Hecke
a_p = np.array([1.0, -1.0, 0.0, 2.0, -2.0, 1.0, -1.0, 3.0, -3.0, 2.0, 0.0, -2.0])
P = len(a_p)

# Step 2:
X_parts = []
for i in range(P):
    for r in range(1, 6):
        X_parts.append(a_p * (xy ** r))
        X_parts.append(a_p * (a_curve ** r))
        X_parts.append(a_p * (b_curve ** r))
X = np.column_stack(X_parts)

# Step 3:
coeffs, _, _, _ = np.linalg.lstsq(X, k_values, rcond=None)
k_pred = X @ coeffs

# Step 4:
mse = np.mean((k_pred - k_values) ** 2)
max_err = np.max(np.abs(k_pred - k_values))
mean_err = np.mean(np.abs(k_pred - k_values))

#
print("\nMultivariable Spectral Regression (n = 10000)")
print("=" * 70)
print(f"MSE:             {mse:.6e}")
print(f"Max Error:       {max_err:.6f}")
print(f"Mean Error:      {mean_err:.6f}")
print(f"First 5 k (true):     {np.round(k_values[:5], 3).tolist()}")
print(f"First 5 k (predicted):{np.round(k_pred[:5], 3).tolist()}")

#
plt.figure(figsize=(8, 5))
plt.plot(k_values[:100], k_values[:100], 'k--', label='Ideal')
plt.plot(k_values[:100], k_pred[:100], 'o-', label='Predicted')
plt.xlabel("True k")
plt.ylabel("Predicted k")
plt.title("Multivariable Spectral Fit: Hecke a_p → Scalar k (n = 10000)")
plt.grid(True)
plt.legend()
plt.tight_layout()
plt.show()


Multivariable Spectral Regression (n = 10000)
======================================================================
MSE:             3.320705e-16
Max Error:       0.000000
Mean Error:      0.000000
First 5 k (true):     [1000.0, 1099.91, 1199.82, 1299.73, 1399.64]
First 5 k (predicted):[1000.0, 1099.91, 1199.82, 1299.73, 1399.64]
2025-05-10 19:40:21.196 Python[38006:21957039] +[IMKClient subclass]: chose IMKClient_Modern
2025-05-10 19:40:21.196 Python[38006:21957039] +[IMKInputSession subclass]: chose IMKInputSession_Modern

[BADecker]

The point is that a whole system of encrypted wallets should be developed. This would mean that not only would people be able to trade P2P with Bitcoin, they would be able to trade P2P with wallets... something like double encryption squared.

A new wallet system could take all the info in the OP posts into account, and make the system stronger than ever.