Triangle Metrics Skill

Trit: 0 (ERGODIC - synthesizer/coordinator) Purpose: Unify all triangle inequality skills into a coherent metric space

---

Cross-Referenced Skills

Skill	Guarantee	Integration Point
glass-hopping	≪ order transitivity	TriangleInequality Narya type
world-hopping	Dijkstra pruning	d13 <= d12 + d23 constraint
glass-bead-game	Propagator constraint	world_distance comparisons
epistemic-arbitrage	Knowledge transfer bound	d(A,C) ≤ d(A,B) + d(B,C)
l-space	Navigation metric	:triangle_inequality traversal
open-games	Play/coplay equilibrium	equilibrium ⟺ d(a,c) ≤ d(a,b) + d(b,c)

---

Unified Interface

# Abstract metric interface all skills implement
abstract type TriangleMetric end

struct WorldDistance <: TriangleMetric
    d12::Float64
    d23::Float64
    d13::Float64
end

function triangle_valid(m::WorldDistance)::Bool
    m.d13 ≤ m.d12 + m.d23
end

# Skill-specific implementations
struct GlassHoppingMetric <: TriangleMetric
    h12::Bridge  # W₁ ≪ W₂
    h23::Bridge  # W₂ ≪ W₃
    # Transitivity guarantees h13
end

struct OpenGamesMetric <: TriangleMetric
    play::Strategy    # Forward distance
    coplay::Strategy  # Backward distance
    # Equilibrium ⟺ triangle satisfied
end

---

Mutual Awareness Protocol

When any triangle skill is invoked:

1. Check: Query other loaded triangle skills
2. Validate: Ensure distances are consistent across all
3. Propagate: Share metric updates to siblings
4. Witness: Generate Narya proof if all agree

-- Unified triangle witness
def UnifiedTriangle 
    (glass : GlassHopping.Bridge)
    (world : WorldHopping.Path)
    (game  : OpenGames.Equilibrium)
    : TriangleValidated

---

DuckLake Integration

-- Query triangle-validated interactions
SELECT a.id, a.trit, a.triangle_valid,
       b.id as next_id, b.trit as next_trit,
       c.id as third_id, c.trit as third_trit,
       ABS(c.trit - a.trit) as d13,
       ABS(b.trit - a.trit) + ABS(c.trit - b.trit) as d12_plus_d23
FROM activity_log a
JOIN activity_log b ON b.timestamp > a.timestamp
JOIN activity_log c ON c.timestamp > b.timestamp
WHERE d13 <= d12_plus_d23;  -- Triangle inequality

---

GF(3) Conservation

The unified metric preserves GF(3):

Σ(skill trits) = glass-hopping(0) + world-hopping(0) + 
                 glass-bead-game(0) + epistemic-arbitrage(0) + 
                 l-space(0) + open-games(0) + 
                 triangle-metrics(0) = 0 ✓

---

Usage

# Validate all triangle constraints
just triangle-validate

# Generate unified Narya witness
just triangle-witness

# Query cross-skill distances
just triangle-query

Integration: Load alongside any triangle skill for automatic mutual awareness.

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Graph Theory

• networkx [○] via bicomodule
  • Universal graph hub

Bibliography References

• general: 734 citations in bib.duckdb

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 8. Degeneracy

Concepts: redundancy, fallback, multiple strategies, robustness

GF(3) Balanced Triad

triangle-metrics (+) + SDF.Ch8 (−) + [balancer] (○) = 0

Skill Trit: 1 (PLUS - generation)

Secondary Chapters

• Ch5: Evaluation
• Ch10: Adventure Game Example
• Ch7: Propagators
• Ch4: Pattern Matching

Connection Pattern

Degeneracy provides fallbacks. This skill offers redundant strategies.

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.