Time Travel CRDT Skill

"Time is of the essence — but the essence is not time." — Kleppmann & Gentle

CRDTs enable time travel: branch, merge, undo, redo without central coordination. GF(3) coloring for causal consistency.

Overview

Time travel in collaborative systems means:

Branching: Diverge from any point in history
Merging: Automatically reconcile divergent branches
Undo/Redo: Navigate the causal graph
Replay: Reconstruct any historical state

This skill connects Diamond Types, Automerge, Eg-walker, and Janus reversible computing.

Core Algorithms

Eg-walker (Gentle & Kleppmann 2025) [ERGODIC: 0]

The Event Graph Walker combines the best of OT and CRDTs:

┌─────────────────────────────────────────────────────────────────────────────┐
│                         EG-WALKER ARCHITECTURE                               │
├─────────────────────────────────────────────────────────────────────────────┤
│                                                                              │
│   Operation Log          Event Graph              Current State              │
│   ┌──────────┐          ┌───────────┐            ┌───────────┐              │
│   │ Insert A │──────────│  A ───┐   │            │           │              │
│   │ Insert B │          │       ▼   │            │  "ABCD"   │              │
│   │ Delete C │──────────│  B ◄── D  │────────────│           │              │
│   │ Insert D │          │       ▲   │            └───────────┘              │
│   └──────────┘          │  C ───┘   │                                       │
│                         └───────────┘                                       │
│                                                                              │
│   Time Complexity:                                                           │
│   - Insert/Delete: O(log n) amortized                                       │
│   - Merge: O(n) worst case, O(1) common case                                │
│   - Memory: O(n) steady state (vs O(n²) for YATA)                           │
│                                                                              │
└─────────────────────────────────────────────────────────────────────────────┘

Key insight: Walk the event graph to compute document state, caching intermediate results.

Diamond Types [PLUS: +1]

bmorphism's eg-walker-reference is a TypeScript reimplementation of Diamond Types:

// Simplified Eg-walker from bmorphism/eg-walker-reference
interface Op {
  id: OpId;           // (agent, seq) tuple
  origin: OpId | null; // causal parent
  content: string;    // inserted content
  deleted: boolean;   // tombstone flag
}

function merge(doc: Doc, remoteOps: Op[]): Doc {
  // Walk the event graph, applying ops in causal order
  for (const op of topoSort(remoteOps)) {
    if (op.deleted) {
      doc = deleteAt(doc, findPosition(doc, op.id));
    } else {
      doc = insertAt(doc, findPosition(doc, op.origin), op.content);
    }
  }
  return doc;
}

Performance (EuroSys 2025):

10x less memory than Automerge
100x faster document loading
Competitive merge performance

Automerge [MINUS: -1]

The original CRDT for JSON-like documents:

import * as Automerge from '@automerge/automerge'

// Create and modify
let doc1 = Automerge.init()
doc1 = Automerge.change(doc1, 'Add title', doc => {
  doc.title = "Time Travel"
})

// Fork (branch)
let doc2 = Automerge.clone(doc1)
doc2 = Automerge.change(doc2, 'Edit on branch', doc => {
  doc.title = "Time Travel CRDTs"
})

// Merge (time travel reconciliation)
doc1 = Automerge.merge(doc1, doc2)

Yjs [MINUS: -1]

Fast CRDT with YATA algorithm:

import * as Y from 'yjs'

const ydoc = new Y.Doc()
const ytext = ydoc.getText('content')

// Observe changes
ytext.observe(event => {
  console.log('Delta:', event.delta)
})

// Time travel via undo manager
const undoManager = new Y.UndoManager(ytext)
undoManager.undo()
undoManager.redo()

Janus Reversible Computing [ERGODIC: 0]

True time travel: run programs backwards.

// Janus: reversible Fibonacci
procedure fib(int n, int x1, int x2)
  if n = 0 then
    x1 += x2
    x1 <=> x2
    n += 1
  else
    n -= 1
    x1 <=> x2
    x1 -= x2
    call fib(n, x1, x2)
  fi x1 = x2

// Run forward: fib(10, 1, 0) → (1, 55)
// Run backward: fib(10, 1, 55) → (1, 0)

Key insight: Every operation has an inverse; no information is lost.

GF(3) Causal Consistency

Color operations by their causal role:

Trit	Color	Operation Type	Role
+1	Red	Insert	Creates content
0	Green	Move/Retain	Preserves structure
-1	Blue	Delete	Removes content (tombstone)

Conservation Law: A well-formed edit sequence maintains:

Σ(trits) ≡ 0 (mod 3)  ⟺  balanced edit

Example:

Insert "hello" (+1)
Insert "world" (+1)
Delete "world" (-1)
Retain structure (0)
───────────────────
Sum: +1 +1 -1 +0 = +1 ≢ 0 (mod 3) → unbalanced!

Fix: Add one more delete or two inserts to balance.

Integration with Unworld

The unworld skill replaces time with derivation chains:

┌─────────────────────────────────────────────────────────────────────────────┐
│                    UNWORLD ↔ CRDT BRIDGE                                     │
├─────────────────────────────────────────────────────────────────────────────┤
│                                                                              │
│   DerivationChain                    CRDT Event Graph                        │
│   ┌─────────────┐                   ┌─────────────────┐                     │
│   │ seed_0      │                   │ (agent_0, 0)    │                     │
│   │    ↓        │                   │      ↓          │                     │
│   │ seed_1 = f(seed_0, trit_0)      │ (agent_0, 1) ◄──│──┐                  │
│   │    ↓        │    ≅              │      ↓          │  │                  │
│   │ seed_2 = f(seed_1, trit_1)      │ (agent_1, 0) ───│──┘                  │
│   │    ↓        │                   │      ↓          │                     │
│   │ ...         │                   │ merge point     │                     │
│   └─────────────┘                   └─────────────────┘                     │
│                                                                              │
│   No time! Only derivation.         Causal order from vector clocks.        │
│                                                                              │
└─────────────────────────────────────────────────────────────────────────────┘

Correspondence:

seed ↔ OpId
trit ↔ operation type (insert/delete/retain)
derivation ↔ causal dependency

bmorphism Interactome

Relevant bmorphism repositories:

Repo	Description	Role
`eg-walker-reference`	TypeScript Eg-walker	Reference implementation
`ewig`	Eternal text editor	Immutable data structures
`Gay.jl`	Splittable PRNG	Deterministic colors for ops
`duck`	DuckDB extensions	Time-travel queries

Cobordism: bmorphism connects Diamond Types (josephg) with Gay.jl coloring for visual debugging of CRDT merge states.

Time Travel Queries (DuckDB)

-- Create temporal versioning table
CREATE TABLE document_history (
    id UUID,
    content TEXT,
    op_type VARCHAR,  -- 'insert', 'delete', 'retain'
    trit INTEGER,     -- GF(3) color
    valid_from TIMESTAMP,
    valid_to TIMESTAMP,
    agent_id VARCHAR,
    seq INTEGER
);

-- Time travel query: document state at specific time
SELECT content
FROM document_history
WHERE valid_from <= '2025-01-01 12:00:00'
  AND (valid_to IS NULL OR valid_to > '2025-01-01 12:00:00')
ORDER BY seq;

-- GF(3) balance check
SELECT 
    SUM(trit) as trit_sum,
    SUM(trit) % 3 as mod3,
    CASE WHEN SUM(trit) % 3 = 0 THEN '✓ Balanced' ELSE '✗ Drift' END as status
FROM document_history
WHERE agent_id = 'agent_0';

Commands

# Diamond Types (Rust)
cargo run --example edit

# Eg-walker reference (TypeScript)
cd eg-walker-reference && npm test

# Automerge
npx automerge-repo-demo

# Janus reversible interpreter
janus run program.jan --reverse

# DuckDB time travel
duckdb -c "SELECT * FROM document_history AS OF '2025-01-01';"

Canonical Triads

# Time Travel Core
automerge (-1) ⊗ eg-walker (0) ⊗ diamond-types (+1) = 0 ✓

# Reversible Computing
janus (-1) ⊗ unworld (0) ⊗ gay-mcp (+1) = 0 ✓

# Temporal Versioning
temporal-coalgebra (-1) ⊗ time-travel-crdt (0) ⊗ koopman-generator (+1) = 0 ✓

# DuckDB Bridge
polyglot-spi (-1) ⊗ time-travel-crdt (0) ⊗ gay-mcp (+1) = 0 ✓

References

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

distributed-systems: 3 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.

time-travel-crdt