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 verification
• derangeable (+1): No fixed points
• bisimulation-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.