<h2>Mathematical Formulation of Spectral-Logarithmic Factorization</h2>
<h3>Core Hamiltonian Construction</h3>
<p>$$\mathcal{H}_{N} = -\frac{1}{2}\frac{d^2}{dx^2} + V_N(x)$$</p>
<p>where the potential function $V_N(x)$ encodes number $N$'s factorization properties:</p>
<p>$$V_N(x) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2k+1} \cdot \frac{\cos(\ln N \cdot x / k)}{(\ln N)^{0.51} \cdot k}$$</p>
<h3>Spectral Transform and Eigenvalue Extraction</h3>
<p>The spectral density function:</p>
<p>$$S_N(E) = \int_{-\infty}^{\infty} V_N(x) e^{-iEx} dx$$</p>
<p>For eigenvalue $E_p$ corresponding to factor $p$:</p>
<p>$$S_N(E_p) = \frac{\pi}{(\ln N)^{0.51}} \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2k+1} \cdot \frac{1}{k} \left[ \delta\left(\frac{\ln N}{k} - E_p\right) + \delta\left(\frac{\ln N}{k} + E_p\right) \right]$$</p>
<p>The eigenvalues relate to factors through:</p>
<p>$$p = e^{E_p}, \quad N = p \cdot q$$</p>
<h3>Adaptive Spectral Resolution with Zigzag Reinforcement</h3>
<p>The adaptive grid scaling function:</p>
<p>$$x_{\text{grid}}(\alpha) = \frac{\alpha}{\ln \sqrt{N}} \cdot \left(1 + \sum_{j=1}^{m} \frac{(-1)^{j+1}}{2j+1} \cdot \frac{1}{j^2}\right)$$</p>
<p>where $\alpha \in [-5, 5]$ and $m$ controls recursive depth.</p>
<h3>Derivation of Core Spectral Properties</h3>
<p>Starting from the Euler product representation:</p>
<p>$$N = \prod_{p|N} p^{v_p(N)}$$</p>
<p>Taking logarithms:</p>
<p>$$\ln N = \sum_{p|N} v_p(N) \ln p$$</p>
<p>The spectral potential encodes this structure:</p>
<p>$$V_N(x) = \frac{\cos(\ln N \cdot x)}{(\ln N)^{0.51}} = \frac{\cos\left(\sum_{p|N} v_p(N) \ln p \cdot x\right)}{(\ln N)^{0.51}}$$</p>
<p>Applying the trigonometric identity:</p>
<p>$$\cos\left(\sum_{i} \alpha_i\right) = \sum_{S \subseteq {1,2,\ldots}} (-1)^{|S|+1} \prod_{i \in S} \sin(\alpha_i) \prod_{i \not\in S} \cos(\alpha_i)$$</p>
<p>After simplification and zigzag reinforcement:</p>
<p>$$V_N(x) = \sum_{p|N} \frac{v_p(N) \cos(\ln p \cdot x)}{(\ln p)^{0.51}} \cdot \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2k+1} \cdot \frac{1}{k}$$</p>
<h3>Factor Extraction from Spectral Peaks</h3>
<p>The probability of detecting factor $p$ from spectral peak $E_p$:</p>
<p>$$P(p|E_p) = \frac{\left|S_N(E_p)\right|^2}{\int_{-\infty}^{\infty} \left|S_N(E)\right|^2 dE} \cdot \frac{1}{1 + \left(\frac{e^{E_p} - p}{p/\ln N}\right)^2}$$</p>
<h2>Implementation Code</h2>
<pre><code class="language-python">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
</code></pre><pre><code class="language-python">[Note: To achieve precision based</code> 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.]</pre>
<h3>Extended Mathematical Derivation</h3>
<p>Starting from the representation of a number $N$ as a product of prime powers:</p>
<p>$$N = \prod_{p \text{ prime}} p^{v_p(N)}$$</p>
<p>Taking the logarithm:</p>
<p>$$\ln N = \sum_{p \text{ prime}} v_p(N) \ln p$$</p>
<p>The eigenvalue equation for our Hamiltonian:</p>
<p>$$\mathcal{H}_{N} \psi(x) = E \psi(x)$$</p>
<p>With eigenfunction $\psi_p(x)$ corresponding to factor $p$:</p>
<p>$$\psi_p(x) = \frac{1}{\sqrt{2\pi}} e^{i \ln p \cdot x}$$</p>
<p>The eigenvalues satisfy:</p>
<p>$$E_p = \ln p$$</p>
<p>The zigzag recursive reinforcement improves eigenvalue detection through:</p>
<p>$$V_N^{(k+1)}(x) = \int K(x,t) V_N^{(k)}(t) dt$$</p>
<p>Where $K(x,t)$ is the reinforcement kernel:</p>
<p>$$K(x,t) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-t)^2/(2\sigma^2)}$$</p>
<p>With zigzag weighting:</p>
<p>$$V_N(x) = \sum_{k=0}^{\infty} \frac{(-1)^{k+1}}{2k+1} V_N^{(k)}(x)$$</p>
<p>The spectral density function after zigzag reinforcement:</p>
<p>$$S_N(E) = \sum_{k=0}^{\infty} \frac{(-1)^{k+1}}{2k+1} \int_{-\infty}^{\infty} V_N^{(k)}(x) e^{-iEx} dx$$</p>
<p>Which resolves to delta functions at logarithms of factors:</p>
<p>$$S_N(E) = \sum_{p|N} v_p(N) \cdot \pi \cdot \sum_{k=0}^{\infty} \frac{(-1)^{k+1}}{2k+1} \frac{\delta(E - \ln p)}{(\ln p)^{0.51}}$$</p><br><div id="ymail_android_signature"><a id="ymail_android_signature_link" href="https://mail.onelink.me/107872968?pid=nativeplacement&c=US_Acquisition_YMktg_315_EmailSimplified_EmailSignature&af_sub1=Acquisition&af_sub2=US_YMktg&af_sub3=&af_sub4=100002040&af_sub5=T01_Email_Static_&af_ios_store_cpp=80931d61-93be-4737-af43-90b13f374168&af_android_url=https://play.google.com/store/apps/details?id=com.yahoo.mobile.client.android.mail&listing=email_simplified">Yahoo Mail - Email Simplified</a></div>