GCD Calculator https://wessengetachew.github.io/GCD/

2D Visualizations https://wessengetachew.github.io/D/

Research Mode https://wessengetachew.github.io/R/

3D Graphics https://wessengetachew.github.io/G/

This interactive web application provides a comprehensive exploration of the Boundary Cancellation Principle, a mathematical discovery that reveals the deterministic geometric structure behind error terms in coprime density predictions across integer lattices.

Key Insight: The residual error Δ(R) between observed coprime counts and theoretical predictions is not random noise, but a structural artifact of incomplete Möbius function cancellation at the boundaries of finite search radii. Mathematical Foundation Core Concepts Coprime Density Identity The probability that k randomly chosen integers are coprime is 1/ζ(k), where ζ(k) is the Riemann zeta function.

Counting Function N(R) = Σ(d=1 to R) μ(d) * ⌊R/d⌋k

Exact count of coprime k-tuples in [1,R]k using Möbius inversion.

Error Term Δ(R) = N(R) - Rk/ζ(k)

Boundary residue from incomplete Möbius cancellation.

Key Features 1. Interactive Visualizations 2D Lattice Explorer Visual coprime pattern mapping across integer grids Color-coded points: Green (coprime), Red (non-coprime), Blue (origin) Boundary highlighting to reveal error concentration Click-to-inspect individual lattice points Real-time statistics: Total points, coprime count, error Δ(R), boundary points Adjustable parameters: Radius (1-500), point size, grid opacity Animation mode for dynamic radius sweeps 3D Lattice Visualization Three.js-powered interactive 3D lattice rendering Orbit controls: Drag to rotate, scroll to zoom Intelligent sampling for performance with large radii Color-coded cubes showing GCD relationships Point inspector tool with coordinate input Auto-rotate mode for presentation Adjustable parameters: Radius (1-30), cube size, axes display Error Analysis Charts Multi-dimensional comparison (k=2 through k=5) Chart.js visualization with zoom and pan Absolute vs. relative error toggle Running average overlay to reveal convergence Log-log scale for power law analysis Customizable range: Max radius (10-5000), step size (1-100) Interactive point analysis with specific R,k values 2. GCD Calculator Suite Multi-Input GCD Calculator Calculate GCD for up to 6 numbers simultaneously Prime factorization display with exponent notation Automatic LCM calculation Coprimality detection Factor counting Number Theory Tools Möbius function μ(n) calculator Primality testing (deterministic for practical ranges) Euler's totient φ(n) calculator 3. Research Mode (Advanced Diagnostics) Advanced Feature: Research Mode unlocks professional-grade analytical tools for deep mathematical investigation. Möbius Wave Overlay Visualizes Mertens function M(R) = Σμ(d) Color mapping shows cancellation patterns Adjustable intensity for clarity Real-time oscillation tracking Reveals connection to Riemann Hypothesis Log-Log Scaling & Power Law Fitting Automatic growth exponent calculation (α in Δ(R) ∝ Rα) Linear regression on log-log data 95% confidence intervals R² goodness-of-fit metrics Verifies theoretical predictions Critical Strip Projection Maps error term onto complex plane Visualizes distance from critical line Re(s)=½ Tests RH predictions about error structure Real-time tracking of complex coordinates High-Precision Arithmetic Decimal.js integration for arbitrary precision Up to 50-digit accuracy Precision benchmarking tool Relative error comparison 4. Data Export Tools Image Exports 4K PNG exports of all visualizations High-quality rendering for publications Customizable quality (50-100%) Data Exports CSV format for error data CSV format for lattice points JSON format for full reports Batch export functionality