unworlding-involution

Unworlding Involution Skill

Status: ✅ Production Ready Trit: 0 (ERGODIC - self-inverse) Principle: ι∘ι = id (involution is its own inverse) Frame: Invariant under observation

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Overview

This skill demonstrates unworlding - extracting frame-invariant self-structure from interaction dynamics. The key insight:

Unworlding = Observing structure without caring about evaluation context

The involution ι: Self → Self satisfies ι∘ι = id, meaning:

• Apply once: transform to "other" perspective
• Apply twice: return to original (fixed point)

Core Concept: Frame-Invariant Self

In a 3-MATCH task, three agents observe each other. The best response dynamics converge to a Nash equilibrium where each agent's color is the best response to the others.

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