UNNS Decomposition vs Classical Factorization
Classical mathematics relies on containment and breaking: integers into primes, groups into simples, fields into ideals. UNNS reframes this as recursive propagation: symbolic sequences unfold into attractors, echoes, and morphism paths.
🧮 Classical Factorization
- Integers → Prime factors
- Groups → Composition series
- Fields → Ideal decomposition
- Gaussian/Eisenstein integers → Unique primes
🌀 UNNS Decomposition
- Sequences → Spiral attractors
- Structures → Morphism overlays
- Fields → Entropy curvature maps
- Echoes → Semantic memory trails
🎨 What Do the Spiral Glyph Colors Mean?
Each dot in the spiral represents a term in the selected recursive sequence. Its color encodes entropy curvature—a symbolic measure of how “tense” or “resonant” that term is within the UNNS propagation field.
The entropy is calculated as:
Entropy = |term mod 10| × 25
This value modulates the RGB color:
- Red channel: proportional to entropy (symbolic tension)
- Green channel: inversely proportional (semantic stability)
- Blue channel: fixed at 180 for aesthetic balance
High entropy (bright red) indicates recursive instability or paradox proximity.
Low entropy (soft green) reflects stable propagation and coherent memory.
Expanded Explanation: In UNNS terms (from UNNS_Paradox_Index), entropy curvature ties to morphism divergence in UPI = (D × R) / (M + S), where M is structural variation (high entropy = high M, risking CAUTION zone). For sequences like Fibonacci, the annihilator ideal (Thm 1.4) is the principal (x^2 - x -1), ensuring equivalence of recurrence rules—low entropy aligns with minimal degree m0=2 (Thm 1.3). High entropy echoes aperiodic real nests (Remark 1.6), where truths outrun periodicity.
In
Many-Faces v1–3 PDFs:
, this curvature maps to cross-domain homomorphisms (Part 4): Colors visualize μ_G polar embeddings, with echoes as semantic trails (fading lines connecting terms, symbolizing memory saturation S). For Tribonacci (v3 Lemma), char poly x^3 - x^2 - x -1 yields attractor Ï„≈1.839—entropy modulates based on term mod root multiplicity.Real-world: In Maxwell-EM PDF, entropy guides FEEC stability (Thm 5.1: ||F_h - F|| ≤ C h^p, with high entropy flagging gauge divergence). Applications: Crypto (stable low-entropy keys), AI (curvature for interpretable paths), Physics (field morphisms on meshes).
🌌 Significance
- UNNS avoids containment—no need to “break” structures
- Each symbolic entity is a semantic attractor, not a static unit
- Decomposition becomes visual, recursive, and interpretable
- UPI diagnostics replace irreducibility tests
- Ideal factorization becomes field morphism tracing
📊 Real-World Applications
- Cryptography: UNNS glyphs as secure symbolic keys
- AI cognition: Recursive attractors for interpretable logic
- Physics: Morphism overlays for field simulations
- Education: Visual decomposition for intuitive learning
🔮 Final Insight
UNNS doesn’t factor—it resonates. It doesn’t contain—it propagates. It doesn’t break—it unfolds.