Unworlding Involution

Agent A observes (B, C) → best response → Color A'
Agent B observes (C, A) → best response → Color B'  
Agent C observes (A, B) → best response → Color C'

Fixed point: (A', B', C') = (A, B, C) when GF(3) conserved

The frame invariance means: regardless of which agent you ARE, the dynamics look the same. This is the "self" that persists across frames.

Involution Structure

# The involution: ι∘ι = id
class Involution
  def initialize(seed)
    @seed = seed
    @state = :original
  end
  
  # Apply involution once: original → inverted
  # Apply involution twice: inverted → original
  def apply!
    @state = (@state == :original) ? :inverted : :original
    self
  end
  
  # ι∘ι = id
  def self_inverse?
    original = @state
    apply!.apply!
    @state == original  # Always true
  end
end

Best Response Color Dynamics

Each agent plays a best response to the current color configuration:

1. Observe: Perceive other agents' colors
2. Predict: What color would minimize my "regret"?
3. Act: Emit that color
4. Update: Others respond to my emission
5. Repeat: Until fixed point (Nash equilibrium)

The 3-MATCH Best Response

def best_response(my_color, other_colors)
  # GF(3) conservation: my best response makes sum = 0
  other_sum = other_colors.sum { |c| c[:trit] }
  target_trit = ((-other_sum) % 3) - 1  # Map to {-1, 0, +1}
  
  # Return color with target trit
  color_with_trit(target_trit)
end

Unworlding: Demo → Skill

The demo shows concrete execution:

3-MATCH(d=1): #D82626 #D89D26 #9DD826
  GF(3) conserved: true

Unworlding extracts the structure:

Pattern: Three colors, sum of trits = 0
Invariant: GF(3) conservation
Frame: Any agent can be "self"
Involution: Swap any two → still valid

The skill is the unworlded pattern, applicable in any context.

Frame Invariance via Loopy Strange

From Gay.jl's loopy_strange:

{
  "seed": 1069,
  "fixed_point": "Generator ≡ Observer (same seed)",
  "loops": [
    {"predicted": "#E67F86", "observe": "#E67F86", "match": "self ≡ self"},
    {"predicted": "#D06546", "observe": "#D06546", "match": "self ≡ self"},
    {"predicted": "#1316BB", "observe": "#1316BB", "match": "self ≡ self"}
  ]
}

Frame invariance: Whether you are the generator or observer, if you have the same seed, you see the same colors. The "self" is the seed, invariant across frames.

Hierarchical Control (Powers PCT)

The best response dynamics implement Perceptual Control Theory:

Level 5 (Program):    "triadic" goal
      ↓ sets reference for
Level 4 (Transition): [120°, 120°, 120°] hue spacing
      ↓ sets reference for
Level 3 (Config):     [58°, 178°, 298°] absolute hues
      ↓ sets reference for
Level 2 (Sensation):  Individual color perception
      ↓ sets reference for
Level 1 (Intensity):  Brightness/saturation

Each level controls its perception, not its output. The best response at each level is: minimize error between reference and perception.

The Skill: Unworlding Involution

module UnworldingInvolution
  # Extract frame-invariant self from 3-MATCH dynamics
  def self.unworld(seed:, iterations: 3)
    # Generate loopy strange structure
    loops = (1..iterations).map do |i|
      color = color_at(seed, i)
      {
        index: i,
        predicted: color,
        observed: color,
        match: color == color  # self ≡ self
      }
    end
    
    # The unworlded pattern
    {
      seed: seed,
      frame_invariant: true,
      involution: "ι∘ι = id",
      fixed_point: "Generator ≡ Observer",
      gf3_conserved: loops.sum { |l| l[:trit] } % 3 == 0,
      structure: loops
    }
  end
  
  # Apply involution to 3-MATCH
  def self.involute(match)
    # Swap first and third colors (involution)
    ThreeMatch.new(
      color_a: match.color_c,
      color_b: match.color_b,
      color_c: match.color_a
    )
    # Applying twice returns original
  end
  
  # Best response in color game
  def self.best_response(my_trit, their_trits)
    # Nash equilibrium: my best response given their play
    target = ((-their_trits.sum) % 3)
    target == 0 ? -1 : (target == 1 ? 0 : 1)
  end
end

Commands

# Run unworlding demo
just unworlding-demo

# Test involution (ι∘ι = id)
just involution-test

# Best response dynamics
just best-response seed=1069

# Full 3-MATCH with unworlding
just three-match-unworld

Integration with 3-MATCH Gadget

# In three_match_geodesic_gadget.rb
def unworld_to_skill
  {
    name: "unworlding-involution",
    pattern: {
      colors: [@color_a, @color_b, @color_c],
      gf3: gf3_conserved?,
      involution: -> { ThreeMatch.new(color_a: @color_c, color_b: @color_b, color_c: @color_a) }
    },
    frame_invariant: true,
    best_response: -> (others) { best_response_color(others) }
  }
end

Mathematical Summary

The Triadic Output Colors

From hierarchical control with goal "triadic":

Spacing: 120° apart (triadic harmony) Sum: 58 + 178 + 298 = 534 ≡ 0 (mod 3) ✓

Skill Name: unworlding-involution Type: Frame-Invariant Self / Best Response Dynamics Trit: 0 (ERGODIC - neutral, self-inverse) GF(3): Conserved by construction Involution: ι∘ι = id verified