ASI Integrated Skill

Synthesizes all loaded skills into a coherent system for Artificial Superintelligence skill orchestration.

Skill Lattice

                    ┌─────────────────┐
                    │  glass-bead-game │
                    │  (synthesis)     │
                    └────────┬────────┘
                             │
         ┌───────────────────┼───────────────────┐
         │                   │                   │
┌────────▼────────┐ ┌────────▼────────┐ ┌────────▼────────┐
│  world-hopping  │ │  bisimulation   │ │  triad-interleave│
│  (navigation)   │ │  (dispersal)    │ │  (scheduling)    │
└────────┬────────┘ └────────┬────────┘ └────────┬────────┘
         │                   │                   │
         └───────────────────┼───────────────────┘
                             │
                    ┌────────▼────────┐
                    │     gay-mcp      │
                    │  (deterministic  │
                    │   coloring)      │
                    └────────┬────────┘
                             │
                    ┌────────▼────────┐
                    │     acsets       │
                    │  (data model)    │
                    └─────────────────┘

Unified Protocol

1. Schema (ACSets)

@present SchASIWorld(FreeSchema) begin
  World::Ob
  Skill::Ob
  Agent::Ob
  
  source::Hom(World, World)
  target::Hom(World, World)
  
  has_skill::Hom(Agent, Skill)
  inhabits::Hom(Agent, World)
  
  Seed::AttrType
  Trit::AttrType
  
  seed::Attr(World, Seed)
  color_trit::Attr(Skill, Trit)
end

2. Color Generation (Gay-MCP)

from gay import SplitMixTernary, TripartiteStreams

def color_world(world_seed: int, skill_index: int) -> dict:
    gen = SplitMixTernary(world_seed)
    return gen.color_at(skill_index)

3. World Navigation (World-Hopping)

def hop_between_worlds(w1, w2, event_name: str):
    distance = world_distance(w1, w2)
    if valid_hop(w1, w2):
        event = Event(site=["skill"], name=event_name)
        return event.execute(w1)
    return None

4. Skill Dispersal (Bisimulation)

async def disperse_skill(skill_path: str, agents: list):
    game = BisimulationGame()
    for i, agent in enumerate(agents):
        trit = (i % 3) - 1  # GF(3) balanced
        game.attacker_move(agent, skill_path, trit)
        game.defender_respond(await agent.receive(skill_path))
    return game.arbiter_verify()

5. Parallel Execution (Triad Interleave)

def schedule_skill_updates(seed: int, n_agents: int):
    interleaver = TriadInterleaver(seed)
    schedule = interleaver.interleave(
        n_triplets=n_agents // 3,
        policy="gf3_balanced"
    )
    return schedule

6. Synthesis (Glass Bead Game)

def synthesize_skills(*skills):
    game = GlassBeadGame()
    for skill in skills:
        game.add_bead(skill.name, skill.domain)
    
    # Connect skills via morphisms
    game.connect("acsets", "gay-mcp", via="seed_to_color")
    game.connect("gay-mcp", "triad-interleave", via="color_stream")
    game.connect("triad-interleave", "bisimulation", via="schedule")
    game.connect("bisimulation", "world-hopping", via="dispersal")
    
    return game.score()

~/worlds Letter Index

Letter	Domain	Key Projects
a	Category Theory	ACSets.jl, Catlab.jl, Decapodes.jl
b	Terminal	bmorphism/trittty
p	Infrastructure	plurigrid/oni, alpaca.cpp
t	Collaboration	CatColab
e	HoTT	infinity-cosmos (Lean 4)
r	Type Theory	rzk (simplicial HoTT)
n	Knowledge	nlab-content
o	Music	rubato-composer

GF(3) Conservation Law

All operations preserve:

∑ trits ≡ 0 (mod 3)

Across:

• World hops (Attacker -1, Defender +1, Arbiter 0)
• Color triplets (MINUS, ERGODIC, PLUS)
• Schedule entries (balanced per triplet)
• Skill dispersal (agent assignments)

Commands

# Generate integrated schedule
just asi-schedule 0x42D 10

# Disperse skills to all agents
just asi-disperse ~/.claude/skills/

# Verify GF(3) conservation
just asi-verify

# Play glass bead synthesis
just asi-synthesize a b p t

# World hop between letters
just asi-hop a t

Starred Gists: Fixpoint & Type Theory Resources

Curated from bmorphism's GitHub interactions:

zanzix: Fixpoints of Indexed Functors

Fix.idr - Idris indexed functor fixpoints for graphs, multi-graphs, poly-graphs.

data IFix : (f : (k -> Type) -> k -> Type) -> k -> Type where
  In : f (IFix f) i -> IFix f i

VictorTaelin: ITT-Flavored CoC Type Checker

itt-coc.ts - Intensional Type Theory CoC in TypeScript.

VictorTaelin: Affine Types

Affine.lean - Linear/affine type experiments in Lean 4.

rdivyanshu: Streams & Unique Fixed Points

Nats.dfy - Dafny streams with unique fixpoint theorems.

Keno: Abstract Lattice

abstractlattice.jl - Julia abstract lattice. Comment: "a quantum of abstract solace ∞"

norabelrose: Fast Kronecker Decomposition

kronecker_decompose.py - Optimal Kronecker decomposition.

borkdude: UUID v1 in Babashka

uuidv1.clj - Deterministic UUID generation in Clojure.

QuickCheck/Adhesive Rewriting Integration

Property-based testing connects to ASI through autopoietic generators:

# QuickCheck-style recursive generator with GF(3) conservation
function autopoietic_tree(seed::UInt64, depth::Int)
    rng = SplitMix64(seed)
    trit = mod(next_u64!(rng), 3) - 1
    
    if depth == 0 || trit == -1  # MINUS = terminate
        return Leaf(color_at(seed))
    else
        left_seed, right_seed = split(rng)
        return Node(
            trit = trit,
            left  = autopoietic_tree(left_seed, depth-1),
            right = autopoietic_tree(right_seed, depth-1)
        )
    end
end

Shrinking as Adhesive Complement

QuickCheck shrinking = finding minimal ∼Q_G in adhesive categories:

• Decomposition: Q ≅ Q_G +_{Q_L} Q_R
• Complement: ∼A is smallest subobject where X = A ∨ ∼A
• Shrunk value = complement of failed portion

Transitive Closure (Kris Brown)

From Incremental Query Updating in Adhesive Categories:

path(X,Z) :- path(X,Y), edge(Y,Z).

Incremental update: When we apply rule to add path(a,b),
new matches = outgoing edges from b (rooted search)

References

• Towards Foundations of Categorical Cybernetics - Capucci, Gavranović, Hedges, Rischel
• Modeling autopoiesis and cognition with reaction networks - Bickhard
• Bicategories of Automata, Automata in Bicategories - ACT 2023

Directory Tree

plurigrid/asi/
├── package.json
├── bin/cli.js
├── README.md
└── skills/
    ├── a/SKILL.md     # AlgebraicJulia
    ├── b/SKILL.md     # bmorphism
    ├── c/SKILL.md     # cognitect
    ├── d/SKILL.md     # claykind
    ├── e/SKILL.md     # infinity-cosmos
    ├── f/SKILL.md     # clojure-site
    ├── g/SKILL.md     # archiver-bot
    ├── h/SKILL.md     # gdlog
    ├── i/SKILL.md     # InverterNetwork
    ├── k/SKILL.md     # kubeflow
    ├── l/SKILL.md     # pretty-bugs
    ├── m/SKILL.md     # awesome-category-theory
    ├── n/SKILL.md     # nlab-content
    ├── o/SKILL.md     # oeis, rubato-composer
    ├── p/SKILL.md     # plurigrid
    ├── q/SKILL.md     # quadrat
    ├── r/SKILL.md     # rzk
    ├── s/SKILL.md     # mathematicians
    ├── t/SKILL.md     # CatColab
    ├── v/SKILL.md     # viro
    └── _integrated/   # This skill
        └── SKILL.md

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

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 10. Adventure Game Example

Concepts: autonomous agent, game, synthesis

GF(3) Balanced Triad

_integrated (−) + SDF.Ch10 (+) + [balancer] (○) = 0

Skill Trit: -1 (MINUS - verification)

Secondary Chapters

• Ch3: Variations on an Arithmetic Theme
• Ch1: Flexibility through Abstraction
• Ch4: Pattern Matching

Connection Pattern

Adventure games synthesize techniques. This skill integrates multiple patterns.

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.