CatSharp Galois Skill

Trit: 0 (ERGODIC - bridge) Color: Yellow (#D8D826)

Overview

Establishes Galois adjunction α ⊣ γ between conceptual spaces:

           α (abstract)
    HERE ─────────────→ ELSEWHERE
      ↑                    │
      │                    │ γ (concretize)
      │    ┌──────────┐    │
      └────│ CatSharp │────┘
           │  Scale   │
           │ (Bridge) │
           └──────────┘
           
    GF(3): (+1) + (0) + (-1) = 0 ✓

• HERE: agent-o-rama Topos (local operations)
• ELSEWHERE: Plurigrid ACT (global cognitive category theory)
• BRIDGE: CatSharp Scale (Mazzola's categorical music theory)

CatSharp Scale Mapping

Pitch classes ℤ₁₂ map to GF(3) trits:

Trit	Pitch Classes	Chord Type	Hue Range
+1 (PLUS)	{0, 4, 8}	Augmented triad	0-60°, 300-360°
0 (ERGODIC)	{3, 6, 9}	Diminished 7th	60-180°
-1 (MINUS)	{2, 5, 7, 10, 11}	Fifths cycle	180-300°

Tritone: The Möbius Axis

The tritone (6 semitones) is the unique self-inverse interval:

6 + 6 = 12 ≡ 0 (mod 12)

This mirrors GF(3) Möbius inversion where μ(3)² = 1.

Galois Connection API

(defn α-abstract
  "Abstraction functor: agent-o-rama → Plurigrid ACT"
  [here-concept]
  (let [trit (or (:trit here-concept)
                 (pitch-class->trit (hue->pitch-class (:H here-concept))))]
    {:type :elsewhere
     :hyperedge (case trit
                  1  :generation
                  0  :verification
                  -1 :transformation)
     :source-trit trit}))

(defn γ-concretize
  "Concretization functor: Plurigrid ACT → agent-o-rama"
  [elsewhere-concept]
  (let [trit (case (:hyperedge elsewhere-concept)
               :generation 1
               :verification 0
               :transformation -1)]
    {:type :here
     :trit trit
     :H (pitch-class->hue (first (trit->pitch-classes trit)))}))

;; Adjunction verification
(defn verify-galois [h e]
  (let [αh (α-abstract h)
        γe (γ-concretize e)]
    (= (= (:hyperedge αh) (:hyperedge e))
       (= (:trit h) (:trit γe)))))

Hyperedge Types

Hyperedge	Trit	HERE Layer	ELSEWHERE Operation
:generation	+1	α.Operadic	ACT.cogen.generate
:verification	0	α.∞-Categorical	ACT.cogen.verify
:transformation	-1	α.Cohomological	ACT.cogen.transform

Color ↔ Pitch Conversion

function hue_to_pitch_class(h::Float64)::Int
    mod(round(Int, h / 30.0), 12)
end

function pitch_class_to_hue(pc::Int)::Float64
    mod(pc, 12) * 30.0 + 15.0
end

function pitch_class_to_trit(pc::Int)::Int
    pc = mod(pc, 12)
    if pc ∈ [0, 4, 8]      # Augmented
        return 1
    elseif pc ∈ [3, 6, 9]  # Diminished
        return 0
    else                    # Fifths
        return -1
    end
end

GF(3) Triads

catsharp-galois (0) ⊗ gay-mcp (-1) ⊗ ordered-locale (+1) = 0 ✓
catsharp-galois (0) ⊗ rubato-composer (-1) ⊗ topos-of-music (+1) = 0 ✓

Commands

# Run genesis with CatSharp bridge
just genesis-catsharp seed=0x42D

# Verify Galois adjunction
just galois-verify here=agent-o-rama elsewhere=plurigrid-act

# Sonify CatSharp scale
just catsharp-play pitch-classes="0 4 7"

Related Skills

• gay-mcp (-1): SplitMix64 color generation
• ordered-locale (+1): Frame structure
• rubato-composer (-1): Mazzola's Rubato system
• topos-of-music (+1): Full Mazzola formalization

References

• Mazzola, G. The Topos of Music (2002)
• Noll, T. "Neo-Riemannian Theory and the PLR Group"
• Heunen & van der Schaaf. "Ordered Locales" (2024)