gworkspace-mcp - Google Workspace MCP Integration with Temporal Consistency

Overview

Integrates Google Workspace services (Gmail, Drive, Calendar, Docs, Sheets, Tasks, Meet) through MCP with:

1. Causal Poset Interaction Time: First-class temporal structure for replay determinism
2. GF(3) Triadic Conservation: Every action classified as PLUS (+1), ERGODIC (0), or MINUS (-1)
3. Cross-Service Atomicity: Two-phase commit for multi-service workflows
4. ANIMA Condensation: Saturation states (Inbox Zero, Task Zero) as fixed points
5. Retry with 1069 Checkpoints: Balanced ternary error recovery

Trit: 0 (ERGODIC) - Coordinates cross-service workflows

Core Formula

InteractionTime ≅ CausalPoset(Events)
GlobalSaturation = (∀s. ServiceSaturated s) ∧ CrossServiceConsistent ∧ TemporalClosure ∧ (Σ trits = 0)
FreeTrace ⊣ CondensedInteractionTime  -- Temporal adjunction

Predicates

Architecture

┌─────────────────────────────────────────────────────────────────────────────────┐
│                    Google Workspace MCP Integration                              │
├─────────────────────────────────────────────────────────────────────────────────┤
│                                                                                  │
│   Services                    Causal Layer                    Condensation       │
│      │                            │                               │              │
│      ▼                            ▼                               ▼              │
│  ┌────────┐    ┌───────────────────────────────────┐    ┌──────────────────┐   │
│  │ Gmail  │───▶│  CausalEvent { id, service,       │───▶│ GlobalSaturation │   │
│  │Calendar│    │    action, trit, seed, timestamp } │    │ CrossConsistent  │   │
│  │ Drive  │    └───────────────────────────────────┘    │ TemporalClosure  │   │
│  │ Docs   │                     │                        └──────────────────┘   │
│  │ Sheets │    ┌───────────────────────────────────┐              │              │
│  │ Tasks  │───▶│  InteractionTime (Causal Poset)  │              ▼              │
│  │ Meet   │    │    reflexive, transitive,         │    ┌──────────────────┐   │
│  └────────┘    │    antisymmetric                  │    │ ANIMA Condensed  │   │
│                └───────────────────────────────────┘    │   InboxZero      │   │
│                                                          │   TaskZero       │   │
│                                                          └──────────────────┘   │
│                                                                                  │
│   Concurrency                 Atomicity                    Error Recovery        │
│      │                            │                               │              │
│      ▼                            ▼                               ▼              │
│  ┌────────────┐    ┌───────────────────────────────┐    ┌──────────────────┐   │
│  │Concurrent  │    │  TwoPhaseCommit               │    │ RetryPolicy      │   │
│  │ActionSet   │    │    phase1_votes → decision    │    │ 1069 checkpoints │   │
│  │ gf3 = 0    │    │    Transaction.gf3_conserved  │    │ [+1,-1,-1,+1...] │   │
│  └────────────┘    └───────────────────────────────┘    └──────────────────┘   │
│                                                                                  │
└─────────────────────────────────────────────────────────────────────────────────┘

GF(3) Trit Assignment for Actions

Triads (GF(3) = 0)

# Core GWorkspace Triads
three-match (-1) ⊗ gworkspace-mcp (0) ⊗ gay-mcp (+1) = 0 ✓  [Core Integration]
temporal-coalgebra (-1) ⊗ gworkspace-mcp (0) ⊗ koopman-generator (+1) = 0 ✓  [Causal Dynamics]
sheaf-cohomology (-1) ⊗ gworkspace-mcp (0) ⊗ oapply-colimit (+1) = 0 ✓  [Cross-Service]
keychain-secure (-1) ⊗ gworkspace-mcp (0) ⊗ gay-mcp (+1) = 0 ✓  [OAuth]
polyglot-spi (-1) ⊗ gworkspace-mcp (0) ⊗ agent-o-rama (+1) = 0 ✓  [Multi-Agent]
shadow-goblin (-1) ⊗ gworkspace-mcp (0) ⊗ pulse-mcp-stream (+1) = 0 ✓  [Event Stream]

MCP Server Configuration

# .ruler/ruler.toml
[mcp_servers.gworkspace]
command = "uvx"
args = ["gworkspace-mcp"]
env = { 
  GOOGLE_CLIENT_ID = "${GOOGLE_CLIENT_ID}",
  GOOGLE_CLIENT_SECRET = "${GOOGLE_CLIENT_SECRET}",
  GF3_SEED = "1069"
}
description = "Google Workspace with causal poset interaction time"

MCP Tools Available

Cross-Service Workflows

Email → Task → Calendar (Balanced Triad)

# Workflow: Process email, create task, schedule calendar
async def email_to_task_calendar(email_id: str, due_date: str):
    # Step 1: Read email (trit = 0)
    email = await mcp.gmail_read(email_id)
    
    # Step 2: Create task from email (trit = +1)
    task = await mcp.tasks_create(
        title=email.subject,
        notes=email.body[:500],
        due=due_date
    )
    
    # Step 3: Create calendar event (trit = +1)
    event = await mcp.calendar_create(
        summary=f"Work on: {email.subject}",
        start=due_date,
        description=f"Task: {task.id}"
    )
    
    # Step 4: Archive email (trit = -1)
    await mcp.gmail_archive(email_id)
    
    # Step 5: Complete placeholder task (trit = -1)
    # GF(3) sum: 0 + 1 + 1 + (-1) + (-1) = 0 ✓
    return {"task": task, "event": event}

Two-Phase Commit for Atomicity

async def atomic_workflow(operations: List[Operation]):
    """Execute operations atomically across services."""
    transaction = Transaction(
        id=next_transaction_id(),
        operations=operations,
        services_involved=list(set(op.service for op in operations))
    )
    
    # Phase 1: Prepare
    votes = {}
    for service in transaction.services_involved:
        votes[service] = await prepare_service(service, transaction)
    
    # Phase 2: Commit or Abort
    if all(votes.values()):
        for op in operations:
            await commit_operation(op)
        return TransactionState.Committed
    else:
        for op in operations:
            await rollback_operation(op)
        return TransactionState.Aborted

Causal Closure

# When an action triggers dependent actions
DEPENDENCY_GRAPH = {
    ("gmail", "read"): [("tasks", "create")],      # Reading email may create task
    ("tasks", "create"): [("calendar", "create")], # Task may need calendar slot
    ("calendar", "create"): [("meet", "schedule")],# Event may need meeting
}

async def causal_closure(trigger: CausalEvent) -> List[CausalEvent]:
    """Compute all events causally triggered by an action."""
    deps = DEPENDENCY_GRAPH.get((trigger.service, trigger.action), [])
    result = []
    
    for target_service, target_action in deps:
        new_event = CausalEvent(
            id=trigger.id + len(result) + 1,
            service=target_service,
            action=target_action,
            trit=action_trit(target_action),
            seed=trigger.seed,
            timestamp=trigger.timestamp + 1
        )
        result.append(new_event)
        result.extend(await causal_closure(new_event))
    
    return result

Retry with 1069 Checkpoints

# Balanced ternary checkpoint pattern from mathpix-gem
CHECKPOINT_1069 = [+1, -1, -1, +1, +1, +1, +1]  # Sums to +3 ≡ 0 mod 3

async def retry_with_checkpoints(operation, max_attempts=7):
    for attempt, trit in enumerate(CHECKPOINT_1069[:max_attempts]):
        try:
            result = await operation()
            if result.confidence >= trit_to_confidence(trit):
                return result
        except APIError as e:
            if attempt == max_attempts - 1:
                raise
            # Adjust strategy based on trit
            if trit == +1:
                await enhance_request()
            elif trit == -1:
                await try_alternative()
            else:
                await validate_partial()
            await asyncio.sleep(2 ** attempt)

Saturation States

Inbox Zero

def check_inbox_zero(gmail_state: ServiceState) -> bool:
    """Check if Gmail has reached Inbox Zero saturation."""
    return (
        gmail_state.pending_ops == [] and
        all(e.action in ["read", "archive"] for e in gmail_state.committed_ops)
    )

Task Zero

def check_task_zero(tasks_state: ServiceState) -> bool:
    """Check if Tasks has reached Task Zero saturation."""
    return (
        tasks_state.pending_ops == [] and
        all(e.action == "complete" for e in tasks_state.committed_ops)
    )

Global Saturation

def check_global_saturation(state: GlobalState) -> bool:
    """Check if entire workspace has reached ANIMA condensation."""
    return (
        all(s.pending_ops == [] for s in state.services) and
        cross_service_consistent(state) and
        temporal_closure(state) and
        sum(e.trit for s in state.services for e in s.committed_ops) % 3 == 0
    )

Formal Specification

See narya_formal_proofs/SkillAdmissibility.nry for:

• PART 14: InteractionTime as causal poset
• PART 15: ConcurrentActionSet with path invariance
• PART 16: GlobalSaturation with temporal closure
• PART 17: FreeTrace ⊣ CondensedInteractionTime adjunction
• PART 18: Causal closure operator
• PART 19: Transaction with two-phase commit
• PART 20: RetryPolicy with 1069 checkpoints
• PART 21: GWorkspaceService enumeration and theorems

BDD Feature Tests

Feature: Google Workspace MCP Integration

  @causal-poset
  Scenario: Interaction time maintains causal ordering
    Given a sequence of Gmail operations
    When I construct the InteractionTime poset
    Then CausallyPrecedes should be transitive
    And concurrent operations should be independent

  @gf3-conservation
  Scenario: Workflow maintains GF(3) conservation
    Given an email-to-task-calendar workflow
    When I sum all action trits
    Then the total should be 0 mod 3

  @atomicity
  Scenario: Cross-service operations are atomic
    Given a transaction across Gmail and Calendar
    When one service fails in phase 1
    Then all operations should rollback
    And no partial state should persist

  @saturation
  Scenario: Inbox Zero triggers global saturation check
    Given all emails have been processed
    When I check GlobalSaturation
    Then InboxZero predicate should hold
    And TemporalClosure should be satisfied

Neighbor Awareness

Commands

just gworkspace-auth        # OAuth authentication flow
just gworkspace-inbox-zero  # Process inbox to zero
just gworkspace-task-zero   # Complete all tasks
just gworkspace-saturate    # Achieve global saturation
just gworkspace-workflow    # Run cross-service workflow
just gworkspace-test        # Run BDD feature tests

References

• SkillAdmissibility.nry - Formal specification
• mathpix-ocr skill - 1069 checkpoint pattern
• mcp-builder skill - MCP server patterns
• temporal-coalgebra skill - State observation

Status: ✅ L4 Admissible (Typed, Documented, Compositional, Predicates + Neighbors) Trit: 0 (ERGODIC) Date: 2025-12-25

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.