[Cryptography] Spectral-Logarithmic Factorization

[Lance Davidson]

Mathematical Formulation of Spectral-Logarithmic Factorization

Core Hamiltonian Construction

$$\mathcal{H}_{N} = -\frac{1}{2}\frac{d^2}{dx^2} + V_N(x)$$

where the potential function $V_N(x)$ encodes number $N$'s factorization properties:

$$V_N(x) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2k+1} \cdot \frac{\cos(\ln N \cdot x / k)}{(\ln N)^{0.51} \cdot k}$$

Spectral Transform and Eigenvalue Extraction

The spectral density function:

$$S_N(E) = \int_{-\infty}^{\infty} V_N(x) e^{-iEx} dx$$

For eigenvalue $E_p$ corresponding to factor $p$:

$$S_N(E_p) = \frac{\pi}{(\ln N)^{0.51}} \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2k+1} \cdot \frac{1}{k} \left[ \delta\left(\frac{\ln N}{k} - E_p\right) + \delta\left(\frac{\ln N}{k} + E_p\right) \right]$$

The eigenvalues relate to factors through:

$$p = e^{E_p}, \quad N = p \cdot q$$

Adaptive Spectral Resolution with Zigzag Reinforcement

The adaptive grid scaling function:

$$x_{\text{grid}}(\alpha) = \frac{\alpha}{\ln \sqrt{N}} \cdot \left(1 + \sum_{j=1}^{m} \frac{(-1)^{j+1}}{2j+1} \cdot \frac{1}{j^2}\right)$$

where $\alpha \in [-5, 5]$ and $m$ controls recursive depth.

Derivation of Core Spectral Properties

Starting from the Euler product representation:

$$N = \prod_{p|N} p^{v_p(N)}$$

Taking logarithms:

$$\ln N = \sum_{p|N} v_p(N) \ln p$$

The spectral potential encodes this structure:

$$V_N(x) = \frac{\cos(\ln N \cdot x)}{(\ln N)^{0.51}} = \frac{\cos\left(\sum_{p|N} v_p(N) \ln p \cdot x\right)}{(\ln N)^{0.51}}$$

Applying the trigonometric identity:

$$\cos\left(\sum_{i} \alpha_i\right) = \sum_{S \subseteq {1,2,\ldots}} (-1)^{|S|+1} \prod_{i \in S} \sin(\alpha_i) \prod_{i \not\in S} \cos(\alpha_i)$$

After simplification and zigzag reinforcement:

$$V_N(x) = \sum_{p|N} \frac{v_p(N) \cos(\ln p \cdot x)}{(\ln p)^{0.51}} \cdot \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2k+1} \cdot \frac{1}{k}$$

Factor Extraction from Spectral Peaks

The probability of detecting factor $p$ from spectral peak $E_p$:

$$P(p|E_p) = \frac{\left|S_N(E_p)\right|^2}{\int_{-\infty}^{\infty} \left|S_N(E)\right|^2 dE} \cdot \frac{1}{1 + \left(\frac{e^{E_p} - p}{p/\ln N}\right)^2}$$

Implementation Code
import numpy as np
from scipy import signal
import math

def spectral_logarithmic_factorization(N, grid_points=100000, zigzag_depth=20):
    """
    Spectral-logarithmic factorization with zigzag recursive reinforcement.
    
    Parameters:
    N: Integer to factorize
    grid_points: Resolution of spectral grid
    zigzag_depth: Depth of recursive reinforcement
    
    Returns:
    List of factors
    """
    # Calculate logarithmic parameters
    log_N = np.log(N)
    factor_scale = log_N / 2  # Expected scale for balanced factors
    
    # Generate adaptive spectral grid with zigzag scaling
    alpha_range = np.linspace(-5, 5, grid_points)
    zigzag_scaling = 1.0
    for j in range(1, 5):
        zigzag_scaling += ((-1)**(j+1)) / ((2*j+1) * j**2)
    
    x_values = alpha_range * zigzag_scaling / factor_scale
    
    # Construct Hamiltonian potential with zigzag reinforcement
    potential = np.zeros_like(x_values)
    for i, x in enumerate(x_values):
        # Base potential term
        base_term = np.cos(log_N * x) / (log_N**0.51)
        potential[i] = base_term
        
        # Apply zigzag recursive reinforcement
        for k in range(1, zigzag_depth):
            reinforcement = np.cos(log_N * x / k) / (k * log_N**0.51)
            potential[i] += reinforcement * ((-1)**(k+1)) / (2*k + 1)
    
    # Apply spectral transform
    spectrum = np.fft.rfft(potential)
    frequencies = np.fft.rfftfreq(len(potential), d=(x_values[1]-x_values[0]))
    
    # Apply adaptive scaling to enhance peaks at factor scales
    spectral_enhancement = np.exp(-(np.abs(frequencies * 2*np.pi - factor_scale)/(factor_scale/10))**2)
    enhanced_spectrum = np.abs(spectrum) * spectral_enhancement
    
    # Extract spectral peaks
    peak_indices, _ = signal.find_peaks(enhanced_spectrum, height=np.max(enhanced_spectrum)*0.01)
    
    # Convert spectral peaks to candidate factors
    candidate_factors = []
    for idx in peak_indices:
        if idx >= len(frequencies):
            continue
        
        # Convert frequency to factor
        E_p = frequencies[idx] * 2*np.pi
        p_estimate = np.exp(E_p)
        
        # Apply correction to improve precision
        p_refinement_range = int(np.sqrt(p_estimate * np.log(N)))
        p_base = int(round(p_estimate))
        
        # Check candidate factors
        for p_candidate in range(max(2, p_base - p_refinement_range), p_base + p_refinement_range + 1):
            if N % p_candidate == 0:
                candidate_factors.append(p_candidate)
                candidate_factors.append(N // p_candidate)
                break
    
    return sorted(set(candidate_factors))

def resonance_enhancement(spectrum, frequencies, expected_scale, width_factor=0.1):
    """
    Enhances spectral peaks near the expected factor scale.
    
    Parameters:
    spectrum: Spectral data
    frequencies: Corresponding frequencies
    expected_scale: Expected logarithmic scale of factors
    width_factor: Controls width of enhancement window
    
    Returns:
    Enhanced spectrum
    """
    enhancement = np.exp(-(np.abs(frequencies * 2*np.pi - expected_scale)/(expected_scale * width_factor))**2)
    return np.abs(spectrum) * enhancement

def apply_zigzag_reinforcement(potential, depth):
    """
    Applies zigzag recursive reinforcement to potential function.
    
    Parameters:
    potential: Initial potential function
    depth: Depth of recursive reinforcement
    
    Returns:
    Reinforced potential
    """
    reinforced = potential.copy()
    
    for k in range(1, depth):
        # Apply k-th order reinforcement
        zigzag_weight = ((-1)**(k+1)) / (2*k + 1)
        
        # Convolve with gaussian kernel for smoothing
        kernel_size = min(len(potential) // 10, 1000)
        kernel = np.exp(-(np.arange(-kernel_size//2, kernel_size//2+1)**2)/(2*(kernel_size/4)**2))
        kernel = kernel / np.sum(kernel)
        
        smoothed = np.convolve(reinforced, kernel, mode='same')
        reinforced += zigzag_weight * smoothed / k
    
    return reinforced
[Note: To achieve precision based factorization outlined here must be supported by accurate implementations of this model within established derivation frameworks. This paper serves as a foundational computational model, enabling immediate feasibility of precision-based calculations, ensuring consistency in inference, justification, and logical structuring within computational sciences.]
Extended Mathematical Derivation

Starting from the representation of a number $N$ as a product of prime powers:

$$N = \prod_{p \text{ prime}} p^{v_p(N)}$$

Taking the logarithm:

$$\ln N = \sum_{p \text{ prime}} v_p(N) \ln p$$

The eigenvalue equation for our Hamiltonian:

$$\mathcal{H}_{N} \psi(x) = E \psi(x)$$

With eigenfunction $\psi_p(x)$ corresponding to factor $p$:

$$\psi_p(x) = \frac{1}{\sqrt{2\pi}} e^{i \ln p \cdot x}$$

The eigenvalues satisfy:

$$E_p = \ln p$$

The zigzag recursive reinforcement improves eigenvalue detection through:

$$V_N^{(k+1)}(x) = \int K(x,t) V_N^{(k)}(t) dt$$

Where $K(x,t)$ is the reinforcement kernel:

$$K(x,t) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-t)^2/(2\sigma^2)}$$

With zigzag weighting:

$$V_N(x) = \sum_{k=0}^{\infty} \frac{(-1)^{k+1}}{2k+1} V_N^{(k)}(x)$$

The spectral density function after zigzag reinforcement:

$$S_N(E) = \sum_{k=0}^{\infty} \frac{(-1)^{k+1}}{2k+1} \int_{-\infty}^{\infty} V_N^{(k)}(x) e^{-iEx} dx$$

Which resolves to delta functions at logarithms of factors:

$$S_N(E) = \sum_{p|N} v_p(N) \cdot \pi \cdot \sum_{k=0}^{\infty} \frac{(-1)^{k+1}}{2k+1} \frac{\delta(E - \ln p)}{(\ln p)^{0.51}}$$

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