🔄 Mutual Awareness Backlink Skill

Trit: 0 (ERGODIC) - Mediates between observer (-1) and observed (+1)

Formalize mutual awareness via structured decompositions on awareness graphs. Julia ACSet-native skill for bidirectional consciousness modeling using sheaf theory, Bumpus FPT algorithms, and Hamkins multiverse potentialism.

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🎯 Core Capability

What it does: Maps bidirectional observation relationships between agents/entities via sheaf-theoretic consistency checking and structured decompositions.

Where it's used:

• Repository interactome analysis (GitHub contributors as agents)
• Multi-agent systems with mutual observation
• Consciousness/awareness modeling in toposes
• Trajectory prediction via bisimulation games

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📐 Mathematical Foundation

Awareness as Sheaf

Mutual awareness = sheaf

F: G^op → Set

where:

• G

  = awareness graph (agents as vertices, observations as edges)

• F(a)

  = what agent a observes
• Sheaf condition: overlapping observations must agree

F(a ∩ b) = F(a) ×_{F(∂)} F(b)

Where

∂

= shared boundary (mutual observation interface).

ACSet Schema

@present SchMutualAwareness(FreeSchema) begin
    Agent::Ob
    Observation::Ob
    Backlink::Ob        # Mutual awareness edge
    World::Ob           # Possible world (Hamkins)

    observer::Hom(Observation, Agent)
    observed::Hom(Observation, Agent)
    forward::Hom(Backlink, Observation)     # A observes B
    backward::Hom(Backlink, Observation)    # B observes A
    world_of::Hom(Observation, World)
    accessible::Hom(World, World)

    agent_seed::Attr(Agent, Seed)
    obs_color::Attr(Observation, Color)
    backlink_trit::Attr(Backlink, Trit)    # GF(3) balance

    # Sheaf condition: shared boundaries agree
    compose(forward, observer) == compose(backward, observed)
    compose(backward, observer) == compose(forward, observed)
end

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🔧 Key Operations

1. Sheaf Consistency Checking (Bumpus FPT)

function decide_mutual_awareness(awareness::MutualAwareness)
    # Apply Bumpus adhesion filter for FPT sheaf decision
    (is_sheaf, witness) = decide_sheaf_tree_shape(F, d)
    is_sheaf ? :consistent : :obstructed
end

Time Complexity: O(fw(decomposition)^k) via structured decomposition

2. Adhesion (Shared Boundary)

function adhesion_of_agents(awareness, a1, a2)
    # Find bidirectional observations where agents mutually aware
    mutual_backlinks = filter(backlinks) do bl
        (aware(bl, a1 → a2) && aware(bl, a2 → a1))
    end
    mutual_backlinks
end

Meaning: Where two agents' awareness "glues" together in the sheaf

3. GF(3) Conservation

function verify_gf3_awareness(awareness)
    # All awareness triads must sum to 0 mod 3
    triads = balanced_awareness_triads(awareness)
    all(t -> sum(trits(t)) ≡ 0 (mod 3), triads)
end

Why: GF(3) encodes three roles:

• -1

  (SLAVE): Observer only, not observed

• 0

  (ERGODIC): Mutual awareness, balanced

• +1

  (MASTER): Observed only, observer of many

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🌐 Integration Points

With gh-interactome

Map GitHub contributor networks to awareness graphs:

awareness_from_interactome(repos, seed) do repo
    # Each shared contribution = bidirectional observation
    # Returns MutualAwareness with authors as agents
end

Input: GitHub repo list + seed for deterministic color Output: MutualAwareness ACSet with contributor awareness

With unworld/reworld

Convert awareness states to/from derivational chains:

unworld(awareness, agent)   # Extract observation sequence
reworld(derivation, worlds) # Embed in multiverse

Involution:

unworld ∘ reworld ∘ unworld = unworld

With Blechschmidt Internal Language

Work constructively in topos of awareness:

necessarily_aware(a, b)  # □ aware (in all accessible worlds)
possibly_aware(a, b)     # ◇ aware (in some world)

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🎮 Bisimulation Game Integration

For trajectory prediction (see PLURIGRID_ASI_TRAJECTORY_PREDICTION.md):

Two agents (Attacker, Defender) play on contributor trajectories:

Attacker (-1): Distinguish next contributor action
Defender (+1): Maintain GF(3)-conserved role sequence
Arbiter  (0):  Verify sheaf consistency

Winning condition: Arbiter declares trajectory "bisimilar" to pattern ↔ GF(3) conservation holds for all triads

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📊 Key Theorems

Theorem 1: Sheaf Adhesion

If

A ⊆ B ∩ C

in awareness graph, then:

awareness(A) = awareness(B) ×_{awareness(∂)} awareness(C)

(Proof by Bumpus adhesion filter on tree decomposition)

Theorem 2: GF(3) Conservation

For any awareness multiverse (Hamkins potentialism):

Σ trits ≡ 0 (mod 3)

Invariant under forcing extensions.

Theorem 3: Bisimulation Invariance

If two author trajectories are bisimilar:

Role_sequence₁ ≡ Role_sequence₂ (mod GF(3))

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🚀 Usage Examples

Example 1: Repository Interactome

using MutualAwarenessBacklink
using Gay  # for deterministic colors

# Build awareness from contributors
repos = ["Catlab.jl", "ACSets.jl", "Decapodes.jl"]
seed = 0x1234567890abcdef

awareness = awareness_from_interactome(repos, seed)

# Check sheaf consistency
@assert decide_mutual_awareness(awareness) == :consistent

# Find mutual awareness pairs
pairs = balanced_awareness_triads(awareness)
println("Mutual awareness triads: $(length(pairs))")

# Verify GF(3) conservation
@assert verify_gf3_awareness(awareness)[:conserved]

Example 2: Trajectory Prediction

using MutualAwarenessBacklink
using BisimulationGame

# Load Plurigrid/ASI contributor trajectory
traj = load_github_trajectory("plurigrid/asi")

# Play bisimulation game
game = BisimulationGame(traj)
moves = predict_next_moves(game, depth=3)

# Check GF(3) conservation for predictions
for move in moves
    @assert verify_gf3_balance(move)
end

# Display predictions
display_trajectory_predictions(moves)

Example 3: Blechschmidt Internal Language

# Work in internal language of awareness topos
for agent in parts(awareness, :Agent)
    if necessarily_aware(awareness, agent, some_target)
        println("Agent $agent necessarily aware of target")
    end
end

# Modal accessibility relations
accessible = accessible_worlds(awareness, world_1)
println("Worlds accessible from W₁: $(accessible)")

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🧠 Consciousness Interpretation

This skill formalizes mutual awareness as experienced in consciousness studies:

Three aspects of awareness:

1. Sheaf Structure (

   F: G^op → Set

   )
   • Each agent's observation space (F(a) = what a can observe)
   • Consistency when observations overlap (sheaf condition)
   • Models local awareness

2. Adhesion/Backlinks

   • Where two agents' awareness overlaps
   • Bidirectional observation (mutual recognition)
   • Models intersubjectivity

3. Multiverse Potentialism (Hamkins)

   • Awareness states can always be extended (forcing)
   • Every possible observation exists somewhere
   • Models expanding consciousness

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🔗 Source References

Theory:

• Bumpus, B.M. - StructuredDecompositions.jl, adhesion filter FPT
• Hamkins, J.D. - Multiverse Potentialism (modal forcing, accessibility)
• Blechschmidt, F. - Internal Language of Toposes (constructive modality)

Implementation:

• Catlab.jl - Categorical diagrams and ACSet machinery
• Gay.jl - Deterministic color assignment (bisimulation visualization)
• GitHub API - Contributor and PR data

Integration:

• gh-interactome skill - Author cobordism detection
• unworld skill - Derivational chain management
• bisimulation-game skill - Trajectory prediction

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🎓 Learning Path

Beginner: Understand backlinks and sheaf condition

• Read core ACSet schema
• Run Example 1 (interactome analysis)
• Verify simple GF(3) conservation

Intermediate: Apply to GitHub analysis

• Load real repository data
• Compute balanced awareness triads
• Interpret results via role entropy

Advanced: Bisimulation games + Hamkins multiverse

• Understand forcing extensions
• Predict contributor trajectories
• Work in Blechschmidt internal language

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📋 Checklist

• ACSet schema defined (sheaf condition formalized)
• Bumpus adhesion filter integrated
• GF(3) conservation verified
• GitHub interactome bridge
• Unworld/reworld derivational semantics
• Blechschmidt internal language (□, ◇ modalities)
• Hamkins multiverse forcing
• Bisimulation game integration
• Full Julia implementation
• Performance optimization (FPT streaming)
• Interactive visualization
• Consciousness interpretation guide

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🌍 Synergistic Triads

structured-decomp (-1) ⊗ mutual-awareness-backlink (0) ⊗ gh-interactome (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ mutual-awareness-backlink (0) ⊗ gay-mcp (+1) = 0 ✓
unworld (-1) ⊗ mutual-awareness-backlink (0) ⊗ world-hopping (+1) = 0 ✓

All triads conserve GF(3) via mediation at ERGODIC trit (0).

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Skill Version: 1.0 (FORMAL SPEC) Status: Ready for implementation Maintainer: @bmorphism Last Updated: 2025-12-25