CRDT Color Skill

Trit: 0 (ERGODIC) - Coordinates between verification (-1) and generation (+1)

Color-aware CRDT operations with GF(3) conservation, Narya proofs, ACSet schemas, and Bumpus narrative sheaves.

Architecture

┌─────────────────────────────────────────────────────────────────┐
│                    CRDT Color Skill                              │
├─────────────────────────────────────────────────────────────────┤
│                                                                  │
│  ┌──────────────┐    ┌──────────────┐    ┌──────────────┐       │
│  │ Narya/Cat#   │    │  Eg-walker   │    │   Bumpus     │       │
│  │ Proofs (-1)  │◄──►│  DAG (0)     │◄──►│ Sheaves (+1) │       │
│  └──────────────┘    └──────────────┘    └──────────────┘       │
│         │                   │                   │                │
│         │            ┌──────┴──────┐            │                │
│         │            │   ACSet     │            │                │
│         └────────────│  Schema     │────────────┘                │
│                      └──────┬──────┘                             │
│                             ▼                                    │
│                    ┌──────────────┐                              │
│                    │  Gay.jl RNG  │                              │
│                    │  SplitMix64  │                              │
│                    └──────────────┘                              │
│                                                                  │
└─────────────────────────────────────────────────────────────────┘

Eg-walker Integration (Gentle & Kleppmann, EuroSys 2025)

Core Innovation: Simple indices + DAG replay instead of complex CRDT metadata.

# Traditional CRDTs store complex position descriptors
# Eg-walker stores simple index at execution time
struct ColorOp
    index::Int          # Simple index (not CRDT metadata!)
    color_hex::String   # Gay.jl color
    trit::Int8          # GF(3) assignment
    parent_ids::Vector{UInt64}  # DAG edges
end

# Merge via replay: O(k log n) instead of O(n²)
merged = merge_branches(alice_state, bob_state)

Complexity Comparison

Metric	Traditional CRDT	OT	Eg-walker
Memory/op	O(n)	O(1)	O(1)
Doc load	O(n²)	O(n)	O(n log n)
Merge	O(k log n)	O(n²)	O(k log n)
Tombstones	Unbounded	N/A	None (replay)

GF(3) + Eg-walker

• Each ColorOp carries trit: O(1) overhead
• Merge verification: O(k) path traversal
• Rebalancing: O(1) amortized (single compensating op)

GF(3) Conservation Law

All operations maintain:

Σ trits ≡ 0 (mod 3)

Operation	Trit	Effect
Restriction	-1	Shrink time window
Pullback (Merge)	0	Combine concurrent edits
Extension	+1	Expand time window

Triad balance:

restriction(-1) ⊕ pullback(0) ⊕ extension(+1) = 0 ✓

Core Types

# ColorEdit: Atomic edit with GF(3) trit
struct ColorEdit
    seed::UInt64      # Gay.jl seed
    idx::Int          # Color index
    color::RGB        # Computed color
    trit::Int8        # GF(3) ∈ {-1, 0, +1}
    timestamp::Nat    # Lamport timestamp
end

# Observational Bridge: hex equality → trit equality
ColorBridge : e1.hex == e2.hex → e1.trit == e2.trit

# Merge: LWW with deterministic tiebreaker
merge(e1, e2) = e1.timestamp > e2.timestamp ? e1 : 
                e1.timestamp < e2.timestamp ? e2 :
                e1.hex < e2.hex ? e1 : e2

Proof Obligations (Narya)

1. merge_comm:

   merge(e1, e2) = merge(e2, e1)

2. merge_assoc:

   merge(merge(e1, e2), e3) = merge(e1, merge(e2, e3))

3. merge_idem:

   merge(e, e) = e

4. gf3_conserved: Winner-takes-all preserves trit
5. replica_conservation: Merged logs maintain

   Σ trits ≡ 0 (mod 3)

ACSet Schema

@present SchColorDoc(FreeSchema) begin
    # Objects
    Document::Ob
    Paragraph::Ob  
    Span::Ob
    ColorEdit::Ob
    Author::Ob
    Timestamp::Ob
    
    # Morphisms
    parent::Hom(Paragraph, Document)
    in_para::Hom(Span, Paragraph)
    edit_of::Hom(ColorEdit, Span)
    authored_by::Hom(ColorEdit, Author)
    at_time::Hom(ColorEdit, Timestamp)
    
    # Attributes
    color::Attr(Span, RGB)
    seed::Attr(ColorEdit, UInt64)
    idx::Attr(ColorEdit, Int)
    trit::Attr(Span, GF3)
end

Bumpus Sheaf Operations

# Time category with intervals
struct TimeInterval
    start::Nat
    stop::Nat
end

# Narrative sheaf: intervals → document states
struct NarrativeSheaf
    sections::Dict{TimeInterval, ColorDocument}
end

# Sheaf condition via pullback
F([a,b]) = F([a,p]) ×_{F([p,p])} F([p,b])

# Pullback merge for concurrent edits
function pullback_merge(left::ColorDocument, right::ColorDocument, 
                        ancestor::ColorDocument)
    merged = fibered_product(left, right, ancestor)
    @assert gf3_sum(merged) % 3 == 0 "GF(3) violation"
    merged
end

Adhesion Filter (FPT)

Tree decomposition yields O(3^w · n) complexity:

• w

  = treewidth of edit dependency graph

• n

  = number of edits
• Polynomial in

  n

  for bounded treewidth

Usage

using CRDTColor

# Create document with seed
doc = ColorDocument(seed=1069)

# Add colored span
span = add_span!(doc, "Hello", color_at(1069, 1))

# Concurrent edits
edit1 = recolor!(doc, span, color_at(1069, 2), author=:alice)
edit2 = recolor!(doc, span, color_at(1069, 3), author=:bob)

# Merge via pullback
merged = pullback_merge(edit1, edit2, ancestor=span)

# Verify GF(3) conservation
@assert gf3_balanced(merged)

Skill Composition

narya-proofs (-1) ⊗ crdt-color (0) ⊗ bumpus-narratives (+1) = 0 ✓
acsets (-1) ⊗ crdt-color (0) ⊗ gay-mcp (+1) = 0 ✓
structured-decomp (-1) ⊗ crdt-color (0) ⊗ world-hopping (+1) = 0 ✓

Files

• narya-tracking.md - Type theory and proof obligations
• acset-schema.jl - ACSet schema definition
• acset-operations.jl - DPO rewriting and Specter paths
• bumpus-sheaf.jl - Narrative sheaf implementation
• gf3-conservation.jl - GF(3) invariant proofs
• adhesion-filter.jl - FPT merge algorithm

References

• Bumpus et al. - Sheaves on Time Categories
• Gay.jl - Deterministic Color Generation
• AlgebraicJulia - ACSets
• Narya - Higher Observational Type Theory

---

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.