UNNS Arithmetic as Modular Computation
🧮 UNNS Arithmetic as Modular Computation
Exploring Nested Number Sequences through Modular Domain Mathematics
📐 Theoretical Foundation
UNNS arithmetic operates within modular domains ℤₙ, where each nest N defines a computational space with specific mathematical properties. The core formula creates structured relationships between modulus values and their corresponding nests.
f(M, N) = (M × N) + (M ÷ N) + (M - N) + (M + N)
Modular Interpretation:
• M ∈ ℤ (modulus/position parameter)
• N ∈ ℤ⁺ (nest/domain parameter)
• f(M, N) ∈ ℚ (rational output space)
Integer-Preserving Condition:
f(M, N) ∈ ℤ ⟺ N | M (N divides M)
When M = kN: f(kN, N) = k(N + 1)²
Each nest N creates a modular computational domain where UNNS arithmetic operates. Values within the domain follow specific mathematical relationships.
Operations: (+, -, ×, ÷) mod n
UNNS enables mapping between different modular domains, creating cross-nest validation opportunities through mathematical relationships.
φ(x) = f(x, n) mod n
Integer-preserving properties emerge at specific modulus values, creating predictable validation points within the modular domain.
Density: 1/N per domain
Nested domains create recursive computational structures where values from one nest can validate in lower-order nests.
Chain: ℤₙ → ℤₙ₋₁ → ... → ℤ₁
🎮 Interactive Modular Computation
🔬 Computational Properties
1. Domain Closure:
For nest N, UNNS values modulo N create closed computational domains
2. Integer Density Theorem:
In domain ℤₙ, exactly 1/N of UNNS values are integers
Proof: Integers occur when N | M, giving density 1/N
3. Cross-Domain Homomorphism:
UNNS operations preserve structure across modular domains
φ(f(a,n) ⊕ f(b,n)) = φ(f(a,n)) ⊕ φ(f(b,n))
4. Recursive Validation:
Values in ℤₙ can validate in ℤₙ₋₁, ℤₙ₋₂, ..., ℤ₁
Creating multi-level validation chains
A comprehensive interactive exploration of UNNS arithmetic as modular computation.
Core Features:
- Theoretical Foundation: Mathematical framework explaining UNNS within modular domains ℤₙ
- Interactive Modular Computation: Live generation of UNNS sequences with configurable parameters
- Multiple Display Modes: Grid view, sequence view, and detailed analysis view
- Modular Operations: Demonstrations of addition, multiplication, composition, and inversion in UNNS context
Key Mathematical Concepts Covered:
- Domain Structure: Each nest N creates a modular computational domain ℤₙ
- Integer-Preserving Properties: Visual identification of when f(M,N) ∈ ℤ occurs (when N|M)
- Cross-Domain Mapping: Validation between different modular spaces
- Recursive Structure: How values cascade between nested domains
Interactive Elements:
- Modular Visualization: Grid showing integer-preserving values vs. standard UNNS values
- Cross-Nest Analysis: Mapping validation between multiple modular domains
- Operation Demonstrations: Live examples of modular arithmetic with UNNS values
- Performance Metrics: Real-time computation statistics and theoretical property verification
Educational Value:
- Mathematical Rigor: Proper notation and formal definitions for modular arithmetic
- Visual Learning: Color-coded cells and interactive elements for pattern recognition
- Theoretical Proofs: Embedded mathematical properties and verification examples
- Practical Applications: Shows how UNNS arithmetic operates in computational contexts
The tool effectively bridges abstract mathematical theory with interactive exploration, making modular UNNS computation accessible while maintaining mathematical accuracy.