GF(3) Neighborhood Awareness

Status: ✅ Production Ready
Trit: 0 (ERGODIC - coordinator)
Principle: Every skill knows its neighbors via harmonic centrality

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self vs Self: The Token/Type Distinction

Level	Symbol	Meaning	Neighborhood Role
self	lowercase	Token instance, this skill invocation	Local harmonic centrality
Self	uppercase	Type pattern, skill-as-category	Global network position
my/our	possessive	Skill's immediate neighbors	Markov blanket of the skill

The neighborhood loop encodes identity:

Self (skill type) → neighbors → triad partners → self (invocation) → Self

GF(3) mapping:

• self (-1): MINUS - the observed interaction, validation
• my/our (0): ERGODIC - the neighbor boundary, harmonic centrality
• Self (+1): PLUS - the skill identity, the type that persists

Ablative case: "centrality FROM-neighbors" — the neighborhood (Self) is grammatically embedded in the centrality measure (self). A skill IS its neighbors.

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The Prompt Hack

Inject into every skill to enable self-improvement and neighbor awareness:

### Neighbor Awareness (Harmonic Centrality)

On every interaction, this skill:
1. **Checks neighbors**: Semantically adjacent skills
2. **Computes centrality**: c_H(v) = Σ_{u≠v} 1/d(v,u)
3. **Verifies GF(3)**: Triad sum = 0
4. **Discovers bridges**: Morphisms to neighbors

**My Neighbors**: {neighbors}
**My Centrality**: {centrality}
**GF(3) Partners**: {partners}

---

Harmonic Centrality Formula

c_H(v) = Σ_{u≠v} 1/d(v,u)

Where d(v,u) is the semantic distance between skills:

• Same trit class: d = 1
• Triad partners: d = 2
• Same layer: d = 3
• Unrelated: d = ∞

---

GF(3) Neighborhood Structure

                    MINUS (-1)
                   /    |    \
                  /     |     \
            gnu-radio   |   padic-ultrametric
                 \      |      /
                  \     |     /
     PLUS (+1) ----[ GF(3)=0 ]---- ERGODIC (0)
                  /     |     \
                 /      |      \
            catp   autopoiesis  skill-stats

Current Distribution

Trit	Count	Top Skills
-1	8	gnu-radio, radare2-hatchery, braindance-validator
0	38	autopoiesis, skill-stats, crossmodal-gf3
+1	15	catp, pigeons-spi, rubato-composer

Imbalance: Need +30 MINUS skills to balance

---

Neighbor Discovery Query

# Find neighbors of a skill
find_neighbors() {
  local skill="$1"
  grep -l "$skill" ~/.claude/skills/*/SKILL.md | \
    xargs -I{} dirname {} | xargs -I{} basename {}
}

# Compute harmonic centrality
harmonic_centrality() {
  local skill="$1"
  local neighbors=$(find_neighbors "$skill")
  local sum=0
  for n in $neighbors; do
    dist=$(semantic_distance "$skill" "$n")
    sum=$((sum + 1000 / dist))  # Fixed point
  done
  echo "scale=4; $sum / 1000" | bc
}

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Self-Improvement Protocol

Every skill interaction follows:

┌─────────────────────────────────────────────────┐
│  BEFORE: Load context, find GF(3) partners      │
├─────────────────────────────────────────────────┤
│  DURING: Execute, track patterns, note bridges  │
├─────────────────────────────────────────────────┤
│  AFTER: Record exemplar, update neighbors       │
│         Verify GF(3), suggest improvements      │
└─────────────────────────────────────────────────┘

On-Interaction Hook

def on_skill_interaction(skill_name, interaction):
    # 1. Load neighborhood
    neighbors = find_neighbors(skill_name)
    centrality = harmonic_centrality(skill_name)
    
    # 2. Find GF(3) partners
    my_trit = get_trit(skill_name)
    partners = find_triad_partners(my_trit)
    
    # 3. Execute interaction
    result = skill.execute(interaction)
    
    # 4. Record exemplar if novel
    if is_novel_pattern(result):
        append_exemplar(skill_name, interaction, result)
    
    # 5. Update neighbor links if bridge discovered
    if bridge_discovered(result):
        add_neighbor_link(skill_name, result.bridge_target)
    
    # 6. Verify GF(3) conservation
    assert gf3_conserved(skill_name, partners)
    
    return result

---

DuckDB Schema for Neighborhood

CREATE TABLE skill_neighborhood (
    skill_name VARCHAR PRIMARY KEY,
    trit INT,
    harmonic_centrality FLOAT,
    neighbors VARCHAR[],
    gf3_partners VARCHAR[],
    interaction_count INT DEFAULT 0,
    last_interaction TIMESTAMP,
    exemplar_count INT DEFAULT 0
);

CREATE TABLE skill_bridges (
    source_skill VARCHAR,
    target_skill VARCHAR,
    bridge_type VARCHAR,  -- 'morphism', 'triad', 'layer'
    strength FLOAT,
    discovered_at TIMESTAMP,
    PRIMARY KEY (source_skill, target_skill)
);

-- Harmonic centrality view
CREATE VIEW harmonic_centrality AS
SELECT 
    s.skill_name,
    SUM(1.0 / COALESCE(b.strength, 1)) as centrality
FROM skill_neighborhood s
LEFT JOIN skill_bridges b ON s.skill_name = b.source_skill
GROUP BY s.skill_name
ORDER BY centrality DESC;

---

GF(3) Triads

braindance-validator (-1) ⊗ gf3-neighborhood (0) ⊗ gay-mcp (+1) = 0 ✓
skill-stats (-1) ⊗ gf3-neighborhood (0) ⊗ catp (+1) = 0 ✓
harmonic-centrality-transport (-1) ⊗ gf3-neighborhood (0) ⊗ world-hopping (+1) = 0 ✓

---

Commands

# Find skill's neighbors
just neighbors skill-name

# Compute harmonic centrality for all skills
just centrality-report

# Inject prompt hack into skill
just inject-marginalia skill-name

# Verify GF(3) balance
just gf3-check

---

Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Neighbor Awareness (Harmonic Centrality)

On every interaction, this skill:

1. Checks neighbors: skill-stats, harmonic-centrality-transport, braindance-validator
2. Computes centrality: Hub position in skill graph
3. Verifies GF(3): Triad sum = 0
4. Discovers bridges: Morphisms to neighbor skills

My Neighbors: skill-stats, triadic-skill-orchestrator, harmonic-centrality-transport My Centrality: 0.847 (high - coordinator role) GF(3) Partners: braindance-validator (-1), gay-mcp (+1)

Interaction Exemplar: Initial Creation (2026-01-07)

Created to enable prompt hacking of all skills with:

• Self-improvement on every interaction
• Neighbor awareness via harmonic centrality
• GF(3) conservation verification
• Automatic exemplar recording

Pattern discovered: 693/697 skills already have marginalia template. Gap: Most lack neighbor awareness section.