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UNNS Simulation Showcase — Guide
What is UNNS?
The Unbounded Nested Number Sequences (UNNS) framework is a mathematical substrate where continuous differential equations are replaced with discrete recursive operations. Time is measured as recursion depth (n ∈ ℕ) rather than a continuous parameter (t ∈ ℝ).
System Architecture
The UNNS Simulation Showcase v1.0 integrates two complementary modules:
- TRE (Temporal Recursion Engine): Simulates forward and reverse UNNS recursion with real-time stability analysis
- KSM (Klein Surface Mapper): Embeds recursion trajectories onto a dynamic non-orientable manifold
- Coupling Architecture: Numeric recursion and geometric deformation as dual projections of the same process
1. TRE — Temporal Recursion Engine
Recursion Equation
Forward: a_{n+1} = α·a_n + β·tanh(a_{n-1}) + δ·n + σ·ε_n
Reverse: a_{n-1} = atanh((a_{n+1} - α·a_n - δ·n) / β)
Where α is damping/persistence, β is coupling coefficient, δ is drift (time-dependent forcing), and σε_n is stochastic perturbation.
Core Features
- Forward Recursion: Expands the temporal cone through iterative operator application
- Reverse Recursion: Requires local invertibility condition |(a_{n+1} - αa_n - δn)/β| < 1
- Stability Analysis: Jacobian magnitude J_n = √((∂F/∂a_n)² + (∂F/∂a_{n-1})²)
- Energy Tracking: E_n = ½a_n² visualizes dissipation vs conservation
- Parameter Space Heatmap: 2D visualization of Jacobian across (α, β) space
Reversibility Domain
The inverse operator F⁻¹ exists only when the transformation is locally invertible. When this condition fails, it represents entropy increase and information loss—the forward recursion becomes irreversible.
TRE Controls
- Damping (α): Controls energy loss per step. α = 1 preserves energy; α < 1 causes decay.
- Coupling (β): Nonlinear feedback strength via tanh. Higher β creates richer dynamics.
- Drift (δ): Linear time-dependent forcing term.
- Noise (σ): Stochastic perturbations; high noise breaks reversibility.
TRE Presets
- Stable Attractor: Strong damping (α = 0.85), converges to fixed point.
- Chaotic Regime: High coupling (β = 1.5), sensitive dependence on initial conditions.
- Periodic Orbit: Minimal damping (α = 1.0), resonant parameters creating cycles.
- Klein Resonance: Parameters tuned for two-step parity-locked oscillation (α = 0.99, opposite-sign initial conditions).
2. KSM — Klein Surface Mapper
Topological Structure
The Klein bottle is a non-orientable manifold with the identification:
(θ, 0) ~ (θ, 1) [A-cycle: periodic]
(0, y) ~ (1, 1-y) [B-cycle: twisted]
This yields w₁ ≠ 0 (first Stiefel-Whitney class nonzero), indicating global non-orientability.
Parametric Embedding
The surface is parametrized by (u,v) ∈ [0,2π]²:
x = (r + a·cos v)·cos u
y = (r + a·cos v)·sin u
z = a·sin v·cos(u/2)
Where r = 2 (major radius), a = 1 (minor radius). The factor cos(u/2) creates the characteristic twist.
Flow Field Dynamics
Particles advect along the surface with velocity u̇ = ωᵤ, where ωᵤ varies with TRE energy, simulating transport along the A-cycle (orientation-preserving direction).
TRE → KSM Coupling
The coupling architecture demonstrates how numeric recursion can drive geometric deformation:
scale = 1 + 0.15·sin(n × 0.05)
hue = f(E_n / E_max)
emissive = g(E_n, depth)
- Scale breathing: Surface expands/contracts with recursion depth
- Color modulation: Shifts from cyan (low energy) to magenta (high energy)
- Emissive intensity: Glows brighter during high-energy states
- Rotation speed: ω = 0.003 + 0.002·E_norm accelerates with energy
Theoretical Correspondences
- Temporal recursion ↔ surface flow: Each depth increment = infinitesimal displacement
- Invertibility ↔ orientability: Non-invertible recursion maps to non-orientable phase space
- Energy decay ↔ metric contraction: Damping compresses surface volume
- Stability cone ↔ recursion cone: Jacobian visualized as causal spread angle
Interactive Features
- Timeline Scrubber: Manual depth navigation for detailed exploration
- Heatmap Toggle: Switch between trajectory view and parameter space analysis
- Global Pause (Spacebar): Synchronized pause across TRE and KSM modules
- Data Export: JSON export for reproducibility and external analysis
- Draggable Legend: Collapsible, movable color guide
Foundational Papers & Technical Documentation
Implementation Summary
Technologies: JavaScript (ES6), Canvas 2D API, WebGL (Three.js r128)
Architecture: Event-driven modular design with TRE→KSM coupling pipeline
Performance: requestAnimationFrame synchronization at ~60 FPS
Data Model: JSON-serializable state for reproducibility
Core Principles
- Recursion depth as time: n ∈ ℕ replaces continuous t ∈ ℝ
- Operator invertibility: F⁻¹ exists only when mapping is bijective and information preserved
- Non-orientability: Global time arrow obstructed by topology when w₁ ≠ 0
- Local vs. Global: Temporal recursion can exist locally while being globally constrained by topology