naturality-factor

--- name: naturality-factor description: 'Naturality Factor Skill' version: 1.0.0 --- # Naturality Factor Skill > *"The naturality condition ensures transformations commute with structure."* Measures how well transformations preserve conservation laws across musical and categorical structures. ## Overview **Trit**: 0 (ERGODIC - Coordinator) **Location**: `lib/conserved_quantity.rb`, `lib/rubato_bridge.rb` **Dependencies**: GF(3), Z/12Z chromatic, Rubato morphisms ## Core Concept In category theory, a **natural transformation** η: F → G satisfies: ``` η_A F(A) ───→ G(A) │ │ F(f) G(f) ↓ ↓ F(B) ───→ G(B) η_B ``` The **naturality factor** ν ∈ [0,1] measures how well this square commutes: - ν = 1.0 → perfectly natural (conservation preserved) - ν = 0.0 → maximally unnatural (conservation violated) ## Mazzola's Insight From *Topos of Music*: "Conservation" in music IS naturality of functors. Transposition preserves intervals because the naturality square closes. ## Classes ### NaturalityFactor ```ruby nf = ConservedQuantity::NaturalityFactor.new( conservation: ConservedQuantity::Laws::CHROMATIC, source_functor: ->(x) { x }, target_functor: ->(x) { x } ) result = nf.compute( eta: ->(x) { x + 7 }, # Transposition morphism: ->(x) { x }, # Identity morphism object_a: 60, # C4 object_b: 64 # E4 ) # => { factor: 1.0, defect: 0, natural?: true } ``` ### Chromatic Naturality ```ruby # Does transposition preserve intervals? result = ConservedQuantity::NaturalityFactor.chromatic_naturality( interval: 7, # Perfect fifth notes: [0, 4, 7] # C major triad ) # => { factor: 1.0, natural?: true, original_intervals: [4, 3] } ``` ### Triadic Naturality (GF(3)) ```ruby # Does doubling preserve trit balance? result = ConservedQuantity::NaturalityFactor.triadic_naturality( transform: ->(x) { x * 2 }, objects: [0, 1, 2, 3, 4, 5], charge_fn: ->(x) { x % 3 } ) # => { factor: 1.0, defect: 0, natural?: true } ``` ### Yoneda Conservation Objects determined by ALL their relationships (Yoneda lemma): ```ruby yoneda = ConservedQuantity::YonedaConservation.new( conservation: ConservedQuantity::Laws::TRIADIC, objects: [0, 1, 2, 3, 4, 5, 6, 7, 8] ) yoneda.yoneda_charge(3) # Sum of all relationships yoneda.yoneda_balanced?(0, 3, 6) # Check if balanced ``` ## Rubato Integration ### NaturalMorphism Rubato morphisms with naturality tracking: ```ruby t7 = RubatoBridge::Morphisms.transposition(7) transposed_score = t7.apply(score) # Check naturality result = t7.compute_naturality(notes) puts t7.naturality_factor # => 1.0 ``` ### Standard Morphisms | Morphism | Naturality | Preserves | |----------|------------|-----------| | `transposition(n)` | 1.0 | Intervals | | `inversion(axis:)` | 1.0 | Interval magnitudes | | `retrograde` | 1.0 | Pitch content | | `augmentation(f)` | varies | Depends on f | ## GF(3) Triads Naturality factor connects to skill triads: ``` three-match (-1) ⊗ naturality-factor (0) ⊗ gay-mcp (+1) = 0 ✓ sheaf-cohomology (-1) ⊗ naturality-factor (0) ⊗ rubato-composer (+1) = 0 ✓ persistent-homology (-1) ⊗ naturality-factor (0) ⊗ topos-generate (+1) = 0 ✓ ``` ## Commands ```bash # Run conservation demo with naturality ruby lib/conserved_quantity.rb # Run Rubato bridge with naturality demo ruby lib/rubato_bridge.rb # Check naturality of a transposition just naturality-check T7 "0,4,7" ``` ## Mathematical Foundation ### Defect Calculation For transformation η with conservation law C: ``` defect = C.combine(charge(left_path), -charge(right_path)) factor = defect == identity ? 1.0 : 1.0 / (1.0 + |defect|) ``` ### Conservation Laws Supported | Law | Modulus | Use Case | |-----|---------|----------| | GF(3) | 3 | Trit balance | | Chromatic | 12 | Pitch classes | | Diatonic | 7 | Scale degrees | | Parity | 2 | XOR operations | | Integer | ∞ | Unbounded | ## See Also - [conserved_quantity.rb](file:///Users/bob/ies/music-topos/lib/conserved_quantity.rb) - [rubato_bridge.rb](file:///Users/bob/ies/music-topos/lib/rubato_bridge.rb) - [ctp-yoneda skill](.agents/skills/ctp-yoneda/SKILL.md) - [rubato-composer skill](.agents/skills/rubato-composer/SKILL.md)