Triadic Skill Orchestrator

Orchestrates skills in GF(3)-balanced triplets with deterministic trit assignment.

Core Workflow

1. Trit Assignment — Assign skills to MINUS(-1)/ERGODIC(0)/PLUS(+1) based on seed
2. GF(3) Conservation — Verify Σ trits ≡ 0 (mod 3)
3. Parallel Dispatch — Fan out to 3 skills simultaneously
4. Role Mapping — VALIDATOR/COORDINATOR/GENERATOR per trit
5. Color Integration — gay-mcp provides deterministic coloring

Trit Assignment Algorithm

(defn assign-trits [skill-names seed]
  (let [groups (partition-all 3 skill-names)]
    (mapcat (fn [[s0 s1 s2]]
              (let [t0 (mod (sha256-int (str seed "::0::" s0)) 3)
                    t1 (mod (sha256-int (str seed "::1::" s1)) 3)
                    t2 (mod (- 0 t0 t1) 3)]  ; Force conservation
                [{:skill s0 :trit (- t0 1)}
                 {:skill s1 :trit (- t1 1)}
                 {:skill s2 :trit (- t2 1)}]))
            groups)))

GF(3) Conservation Check

(defn verify-gf3 [assignments]
  (let [sum (reduce + (map :trit assignments))]
    {:conserved (zero? (mod sum 3))
     :sum sum
     :mod3 (mod sum 3)}))

Role Assignments

Trit	Role	Function	Color Range
-1	VALIDATOR	Verify, constrain, check	Cold (180-300°)
0	COORDINATOR	Mediate, synthesize, balance	Neutral (60-180°)
+1	GENERATOR	Create, execute, produce	Warm (0-60°, 300-360°)

Denotation

This skill coordinates triadic skill applications to ensure no trit imbalance exists, dispatching skills in GF(3)-balanced triplets with deterministic seed propagation.

Effect: SkillSet → (Task × Seed) → [Result₋₁, Result₀, Result₊₁]
Invariant: ∀ dispatch: Σ(trit) ≡ 0 (mod 3)
Fixed Point: When skill outputs stabilize across reruns with different seeds

Invariant Set

Invariant	Definition	Verification
Conservation	Σ(trit) ≡ 0 (mod 3) after every dispatch	Sum check on each triplet
TritBalance	Each skill receives exactly -1, 0, or +1	Assignment validation
SeedDeterminism	Same seed → same trit assignment	Replay test
RoleInjection	Each role (VALIDATOR/COORDINATOR/GENERATOR) appears exactly once	Role counting

Narya Compatibility

Field	Definition
before	Skill states before dispatch
after	Skill states after dispatch
delta	Trit assignments made
birth	Initial skill set with no assignments
impact	1 if any skill changed equivalence class

Condensation Policy

Trigger: When a skill triplet produces identical outputs for 3 consecutive dispatches with different seeds.

Action: Mark triplet as saturated, collapse to canonical representative.

Parallel Dispatch

(defn dispatch-skill [skill-assignment task]
  (let [{:keys [skill trit]} skill-assignment
        role (case trit
               -1 "VALIDATOR"
               0  "COORDINATOR"
               1  "GENERATOR")]
    {:skill skill :trit trit :role role :task task}))

Integration with gay-mcp

Each skill gets a deterministic color from its trit:

from gay import SplitMixTernary

gen = SplitMixTernary(seed=0x42D)
for skill in skills:
    color = gen.color_at(skill.index)
    trit = color['trit']  # Maps to role
    hex_color = color['hex']  # Visual identifier

Justfile Recipes

# Orchestrate 3 skills for a task
triadic-orchestrate TASK SEED="0x42D":
  bb scripts/triadic_skill_orchestrator.bb run "{{TASK}}" {{SEED}} \
    finder-color-walk google-workspace gay-mcp

# Verify GF(3) conservation for skill triplet
triadic-verify *SKILLS:
  bb scripts/triadic_skill_orchestrator.bb verify {{SKILLS}}

# List available skills
triadic-list:
  bb scripts/triadic_skill_orchestrator.bb list

# Run orchestrator with custom skills
triadic-custom TASK SEED *SKILLS:
  bb scripts/triadic_skill_orchestrator.bb run "{{TASK}}" {{SEED}} {{SKILLS}}

Example Output

╔═══════════════════════════════════════════════════════════════╗
║     TRIADIC SKILL ORCHESTRATOR                                ║
╚═══════════════════════════════════════════════════════════════╝

Task: Sync files with Drive
Seed: 0x42D
Skills: 3

Assignments:
  finder-color-walk: +1 (PLUS)
  google-workspace: -1 (MINUS)
  gay-mcp: 0 (ERGODIC)

GF(3) Conservation: ✓ OK (sum=0, mod3=0)

Dispatching...
[GENERATOR] finder-color-walk (trit=+1): Sync files with Drive
[VALIDATOR] google-workspace (trit=-1): Sync files with Drive
[COORDINATOR] gay-mcp (trit=0): Sync files with Drive

Valid Triads

Skill 1 (MINUS)	Skill 2 (ERGODIC)	Skill 3 (PLUS)
sheaf-cohomology	ordered-locale	gay-mcp
bisimulation-game	google-workspace	triad-interleave
say-narration	parallel-fanout	finder-color-walk

Reference Script

See scripts/triadic_skill_orchestrator.bb for full implementation.

---

Skill Name: triadic-skill-orchestrator
Type: Skill Composition / Parallel Dispatch
Trit: 0 (ERGODIC - coordinator)
GF(3): Enforced via third-trit correction

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.

Forward Reference

• unified-reafference (B3 Poset triadic structure)