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

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

# 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))

# 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):

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:

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

# 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
• rubato_bridge.rb
• ctp-yoneda skill
• rubato-composer skill