plurigrid Graph Grafting Skill
Trit: 0 (ERGODIC - Coordinator)
GF(3) Triad: queryable (-1) ⊗ graftable (0) ⊗ derangeable (+1) = 0
Overview
Combinatorial complex operations replacing GraphQL with pure graph theory:
| Operation | Trit | Description |
|---|---|---|
| Queryable | -1 | Tree-shape decision via bag decomposition |
| Colorable | 0 | GF(3) 3-coloring via sheaf |
| Derangeable | +1 | Permutations with no fixed points |
| Graftable | 0 | Attach rooted tree at vertex |
Mathematical Foundation
Grafting = attaching a rooted tree T at vertex v of graph G:
Graft(T, v, G) → G' where:
- V(G') = V(G) ∪ V(T)
- E(G') = E(G) ∪ E(T) ∪ {(v, root(T))}
- Adhesion = shared labels at attachment point
Quadrant Chart: Colorable × Derangeable
Balanced (GF3=0)
│
Q2 │ Q1 ← OPTIMAL
Identity │ PR#18, Knight Tour
│ SICM Galois
──────────────┼──────────────
Q3 │ Q4
Deadlock │ Phase Trans
│
Fixed Points → Derangement
Usage
using .GraphGrafting
c = GraftComplex(UInt64(1069))
# Build PR tree
root = GraftNode(:pr18, Int8(0), :golden, 0)
alice = GraftNode(:alice, Int8(-1), :baseline, 1)
bob = GraftNode(:bob, Int8(1), :original, 1)
# Graft nodes
graft!(c, root, :none, String[])
graft!(c, alice, :pr18, ["aptos-wallet-mcp"])
graft!(c, bob, :pr18, ["aptos-wallet-mcp"])
# Operations
tree_shape(c) # Queryable
trit_partition(c) # Colorable
derange!(c) # Derangeable
compose(c1, c2, :vertex) # Graftable
# Verify
verify_gf3(c) # → (conserved=true, sum=0)
Neighbors
High Affinity
three-match(-1): Graph coloring verificationderangeable(+1): No fixed pointsbisimulation-game(-1): Attacker/Defender
Example Triad
skills: [graph-grafting, three-match, derangeable]
sum: (0) + (-1) + (+1) = 0 ✓ CONSERVED
References
- Joyal, Combinatorial Species (1981)
- Flajolet & Sedgewick, Analytic Combinatorics (2009)
- Topos Institute, Observational Bridge Types
Scientific Skill Interleaving
This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:
Graph Theory
- networkx [○] via bicomodule
- Graph manipulation and algorithms
Bibliography References
graph-theory: 38 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.