---
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)
