operadic-delegation Skill

Operads → Properads → Props for many-to-many agent delegation without losing associativity

Trit: 0 (ERGODIC - coordinator)

Core Insight

From Interverse transcript (Aug 2025):

"Listen and when to interrupt are also operands... the ability to join them in parallel needs to not have the original sin of losing associativity. For that purpose, operads allow us to check... Operads are superseded for me by properads and props, which are many-to-many connections rather than many-to-one."

The Hierarchy

Operad:    n inputs → 1 output  (tree-like)
Properad:  n inputs → m outputs (directed graph)
Prop:      n inputs → m outputs + symmetry (full permutation)

Agent Delegation Pattern

# Operad: Many agents → one coordinator
struct OperadicDelegation
    agents::Vector{Agent}
    coordinator::Agent
    composition::Function  # Must be associative!
end

# Properad: Many-to-many delegation graph
struct ProperadicDelegation
    sources::Vector{Agent}
    targets::Vector{Agent}
    morphisms::Matrix{Function}  # Parallel channels
end

# The key invariant: associativity
# (f ∘ g) ∘ h = f ∘ (g ∘ h)
# Parallel joins must not lose this property

Associativity Preservation

The "original sin" is when parallel task joining breaks associativity:

BAD:  join(A, join(B, C)) ≠ join(join(A, B), C)
GOOD: Operadic composition guarantees equality

GF(3) Integration

Operad composition: trit(f ∘ g) = trit(f) + trit(g) (mod 3)
Properad extension: Σ source_trits = Σ target_trits (mod 3)

Related Skills

• operad-compose

  (+1): Operad composition primitives

• kan-extensions

  (0): Universal property guarantees

• triadic-skill-orchestrator

  (0): GF(3) balanced delegation

• skill-dispatch

  (0): Triadic task routing

Commands

# Verify associativity for a delegation graph
julia --project=. -e 'using Gay; verify_operadic_associativity(graph)'

# Convert operad to properad
just properad-lift operad.jl

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Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.