plurigrid 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:
- Check: Query other loaded triangle skills
- Validate: Ensure distances are consistent across all
- Propagate: Share metric updates to siblings
- 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
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.