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ASI Elevation Manifest: Categorical Foundations

This document describes the architectural elevation of the plurigrid/asi project to formal, verifiable, and categorically-conceived foundations.

Architecture Overview

┌─────────────────────────────────────────────────────────────────────────────┐
│                    LEVEL 3: VERIFIED FOUNDATIONS (Lean 4)                   │
│  ┌─────────────────────────────────────────────────────────────────────┐   │
│  │  formalization/                                                      │   │
│  │  ├── PolySkill.lean         Skills as polynomial functors (Spivak)  │   │
│  │  ├── TriadOperad.lean       GF(3) triads as operad algebras         │   │
│  │  ├── CoalgebraAgent.lean    Agents as coalgebras (behavioral sem.)  │   │
│  │  ├── SheafSkillGluing.lean  Local→global via Čech cohomology       │   │
│  │  └── EcosystemBridgeType.lean (existing) Federation verification   │   │
│  └─────────────────────────────────────────────────────────────────────┘   │
├─────────────────────────────────────────────────────────────────────────────┤
│                    LEVEL 2: TYPED CONFIGURATION (Dhall/CUE)                 │
│  ┌─────────────────────────────────────────────────────────────────────┐   │
│  │  schema/                                                             │   │
│  │  ├── Skill.dhall            Type-safe skill definitions             │   │
│  │  ├── Triad.dhall            Compile-time GF(3) balance checking     │   │
│  │  └── WiringDiagram.cue      Compositional wiring validation         │   │
│  └─────────────────────────────────────────────────────────────────────┘   │
├─────────────────────────────────────────────────────────────────────────────┤
│                    LEVEL 1: CATEGORICAL RUNTIME (AlgebraicJulia)            │
│  ┌─────────────────────────────────────────────────────────────────────┐   │
│  │  runtime/                                                            │   │
│  │  └── SkillCategory.jl       ACSet-based skill graphs, Poly ops      │   │
│  └─────────────────────────────────────────────────────────────────────┘   │
├─────────────────────────────────────────────────────────────────────────────┤
│                    LEVEL 0: OPERATIONAL ASI (existing)                      │
│  ├── skills/*.md               365+ SKILL.md files                         │
│  ├── src/unworld/bicomodule.py Comonad laws (Python)                       │
│  └── skills.json               Skill registry                               │
└─────────────────────────────────────────────────────────────────────────────┘

Core Mathematical Structures

1. Polynomial Functors (PolySkill.lean)

Skills are polynomial functors p: Set → Set where:

  • Positions = observation types (what the skill can perceive)
  • Directions = action types at each position (what the skill can do)
structure PolySkill where
  poly : Poly
  trit : Trit
  name : String

Key insight: Skill morphisms are dependent lenses, enabling composition that preserves interface contracts.

2. GF(3) Operad Algebras (TriadOperad.lean)

The GF(3) conservation law defines an operad:

  • Operations = n-tuples of trits summing to 0 (mod 3)
  • Composition = substitution preserving balance
structure GF3Operad.Op (n : ℕ) where
  trits : Fin n → GF3
  balanced : (Finset.univ.sum trits) = 0

Standard triads:

  • (+1, 0, -1) = Plus, Ergodic, Minus
  • (+1, +1, +1) = 3 ≡ 0 (mod 3)
  • (-1, -1, -1) = -3 ≡ 0 (mod 3)
  • (0, 0, 0) = Ergodic³

3. Coalgebraic Agents (CoalgebraAgent.lean)

Agents are coalgebras (S, γ: S → F(S)):

  • State = internal agent state
  • Observe = what the agent perceives
  • Transition = how the agent responds
structure Coalgebra (p : Poly) where
  state : Type*
  observe : state → p.positions
  transition : (s : state) → p.directions (observe s) → state

Key insight: Bisimulation captures behavioral equivalence—agents with different implementations but identical observable behavior are bisimilar.

4. Sheaf Gluing (SheafSkillGluing.lean)

Skills form a sheaf over their coverage space:

  • Locality: Sections agreeing everywhere are equal
  • Gluing: Compatible local sections produce global sections
structure SkillSheaf (C : SkillCoverage) (Data : Type*) where
  sections : ∀ U ∈ C.opens, Data
  restrict : ∀ U V, V ⊆ U → sections U → sections V
  locality : ...
  gluing : ...

Key insight: Local skill correctness (each skill works in isolation) implies global system correctness (composed skills work together).

Configuration Layer

Dhall: Type-Safe Skill Definitions

let Triad = {
  s1 : Skill,
  s2 : Skill,
  s3 : Skill
}

let assertBalanced : Triad → Triad = λ(t : Triad) →
  assert : isBalanced t === True
  t

Dhall provides:

  • Totality: No infinite loops, guaranteed termination
  • Compile-time balance checking: Unbalanced triads are rejected before runtime
  • Hermetic evaluation: No side effects, reproducible configs

CUE: Wiring Diagram Validation

#WiringDiagram: {
  skills: [Name=string]: #Skill & {name: Name}
  wires: [...#Wire]
  
  // Constraint: GF(3) conservation
  _tritSum: list.Sum([for s in skills {s.trit}])
  _balanced: mod(_tritSum, 3) == 0
}

CUE provides:

  • Structural constraints: Wire types must match
  • GF(3) validation: Diagrams flagged as balanced or not
  • Compositionality: Triads compose into larger diagrams

Runtime Layer

AlgebraicJulia: SkillCategory.jl

# Skills as ACSet objects
@present SchSkill(FreeSchema) begin
    Skill::Ob
    Bridge::Ob
    source::Hom(Bridge, Skill)
    target::Hom(Bridge, Skill)
    skill_trit::Attr(Skill, Trit)
end

# Polynomial composition
function compose_skills(s1::PolySkill, s2::PolySkill)
    PolySkill(
        "($(s1.name) ◃ $(s2.name))",
        s1.trit + s2.trit,
        compose_poly(s1.poly, s2.poly)
    )
end

AlgebraicJulia provides:

  • ACSet-based skill graphs: Efficient categorical data structures
  • Wiring diagram execution: Catlab's compositional semantics
  • Runtime GF(3) checking: Balance verified at composition time

Modelica Integration

Modelica's acausal semantics fit naturally:

Modelica Concept Categorical Structure
Connector Lens (effort × flow, effort)
Equation Morphism in Poly
Acausal composition Bimodule over polynomial
Causality assignment Right action computed by solver

The modelica skill (trit 0, ERGODIC) bridges to:

  • langevin-dynamics (+1, PLUS): Stochastic generation
  • fokker-planck (-1, MINUS): Probabilistic verification

Key Theorems

Theorem Status File
Skill composition is associative sorry PolySkill.lean
GF(3) conserved under compose Proving PolySkill.lean
Triad composition produces ergodic Proving TriadOperad.lean
Cocycle condition on 26 worlds sorry SheafSkillGluing.lean
Bisimilar agents have same behavior sorry CoalgebraAgent.lean
Local correctness → global correctness sorry SheafSkillGluing.lean

Usage

Create a balanced triad (Dhall)

let t = Triad.mkTriad 
  (Skill.plusSkill "generator" "Generates things" "core")
  (Skill.ergodicSkill "coordinator" "Coordinates" "core")
  (Skill.minusSkill "verifier" "Verifies" "core")
  "my-triad"
  "A balanced triad"

Validate wiring (CUE)

cue eval schema/WiringDiagram.cue

Run categorical operations (Julia)

using .SkillCategory
g = unworld_federation()
@assert isbalanced(g)

References

  1. Spivak, D. "Polynomial Functors and Wiring Diagrams"
  2. Shapiro & Spivak "Dynamic Categories, Machines, and Polynomial Functors"
  3. Rutten, J. "Universal Coalgebra: A Theory of Systems"
  4. Mac Lane & Moerdijk "Sheaves in Geometry and Logic"
  5. Powers, W. "Behavior: The Control of Perception" (PCT)

Next Steps

  1. Complete Lean proofs using Aristotle MCP
  2. Generate SKILL.md files from Dhall schemas
  3. Integrate CUE validation into skill registration pipeline
  4. Connect Julia runtime to Python bicomodule layer
  5. Prove 26-world cocycle formally