🔮 UNNS Axiom → Application Ladder

8 Foundational Axioms Ascending to Advanced Applications

Core Philosophical Insight

Set theory asks: "What can be contained?"
UNNS asks: "What can propagate?"
Together, they frame structure and resonance: skeleton and soul.

8 Axioms • 4 Levels • 2 Axioms per Level

1

Structural Foundation

The bedrock of recurrence mathematics

🔸 Axioms 1-2

  • 1. Recurrence: Every sequence follows a(n+r) = c₁a(n+r-1) + ... + cᵣa(n)
  • 2. Nest Depth: Unique minimal order D for shortest recurrence

🎨 Visualization

Recurrence Flow 1 2 3 D Nest Depth

⚙️ Applications

  • Fibonacci, Lucas, Chebyshev sequence generators
  • Minimal recurrence relation discovery
  • Sequence complexity measurement via depth
2

Algebraic Character

Convergence and algebraic structure

🔸 Axioms 3-4

  • 3. Limit Ratio: Consecutive terms converge to dominant root of characteristic polynomial
  • 4. Coefficient Ring: All coefficients belong to cᵢ ∈ ℤ[α], α a root

🎨 Visualization

φ Coefficient Rings ℤ ℤ[i] ℤ[ω] ℤ[α]

⚙️ Applications

  • Golden ratio φ and algebraic constants
  • Gaussian/Eisenstein integer rings
  • Characteristic polynomial roots
3

Dynamics & Stability

Propagation patterns and stability measures

🔸 Axioms 5-6

  • 5. Propagation: Nests propagate recursively forward unless halted by degeneracy
  • 6. Paradox Index (UPI): Stability constant UPI = D·R/(M+S)

🎨 Visualization

Propagation
SAFE
CAUTION
INSTABILITY

⚙️ Applications

  • Propagation models and branching systems
  • UPI burst detection and stability analysis
  • Degeneracy and periodicity detection
4

Higher Order & Integration

Embedding in lattices and undecidability

🔸 Axioms 7-8

  • 7. Embedding: All nests embed in lattice tower ℤ ⊂ ℤ[i] ⊂ ℤ[ω] ⊂ ...
  • 8. Undecidability: Complex nests contain undecidable branches (Gödel phenomenon)

🎨 Visualization

ℤ ℤ[i] ℤ[ω] ... π e φ Undecidable

⚙️ Applications

  • Gödel incompleteness in nested systems
  • FEEC/DEC for Maxwell equations
  • Cantor expansion ambiguities (0.222... = 1.000...)
  • Quantum field lattice embeddings

Complete Axiomatic Framework

All 8 UNNS axioms are distributed across 4 levels (2 axioms per level).
Each rung adds complexity and explanatory power.
The ladder ensures that applications are grounded in axioms,
making UNNS a discipline with foundations → algebra → dynamics → higher order → applications.