Practice studio
Compose Lab
Adjust multi-ary slots and arities, watch the composite land in the right P(n), then run the compositionality checklist against a system you care about.
Live composition lab
Root operation α has 2 slots. Each child contributes its arity. Composite lives in P(3).
α ∘ (β, γ) ∈ P(3) · arities (2+1)
β
arity P(2)
γ
arity P(1)
Unit check: set any child arity to represent a pass-through only when that child is the unit in P(1) and you intend identity substitution — changing structure without the unit law breaks compositionality.
Tree view
Operad composition tree
Nested form: α ∘ (β, γ) lives in P(3+1)=P(4). Tree shape encodes substitution.
Checklist
Ports complete?
Every external dependency appears as a port.
Common failures
| Mode | Symptom | Repair |
|---|---|---|
| Doc-only hierarchy | Nesting exists in slides, not in interfaces | Make subassemblies real ported boxes |
| Algebra drift | Teams interpret the same wire differently | Publish α and a golden composition test |
| Port inflation | Everything is public; no encapsulation | Hide internal wires; minimize outer boundary |
| Fake compositionality | Marketing claims without preservation law | Demand the equation or demote the claim |
Prefer the guided path? Open Lesson 6