plurigrid Workspace Unified Skill
Transform Google Workspace into an ACSet-structured system with GF(3) conservation and cross-skill morphisms.
Trit: 0 (ERGODIC - coordinator)
Principle: Workflow Completion = Condensed Equilibrium State
Implementation: WorkspaceACSet + PathInvariance + MCPAPIBridge
Overview
Workspace Unified applies categorical database principles to workspace:
- Transform - Operations → GF(3)-typed Interactions
- Route - Interactions → Service-specific ACSet objects
- Compose - Cross-skill morphisms with path commutativity
- Verify - MCP↔API equivalence and Narya proofs
WorkspaceACSet Schema
┌─────────────────────────────────────────────────────────────────────────────────────────┐
│ WorkspaceACSet Schema │
├─────────────────────────────────────────────────────────────────────────────────────────┤
│ │
│ Interaction ───────────┬──────▶ Thread ◀───────────────── DriveFile │
│ ├─ verb: String │ ├─ thread_id: String ├─ file_id: String │
│ ├─ timebin: Int │ ├─ needs_action: Bool ├─ mime_type: String │
│ ├─ trit: Trit │ ├─ last_action_bin: Int └─ saturated: Bool │
│ ├─ service: Service │ └─ saturated: Bool │
│ └─ person ─────────────┼──▶ │
│ │ │
│ CalendarEvent ◀────────┼──────── TaskList ◀────────────── Task │
│ ├─ event_id: String │ ├─ list_id: String ├─ task_id: String │
│ ├─ summary: String │ └─ title: String ├─ title: String │
│ ├─ start: DateTime │ ├─ status: String │
│ └─ saturated: Bool │ └─ due: DateTime │
│ │ │
│ Document ◀─────────────┼──────── Sheet │
│ ├─ doc_id: String │ ├─ sheet_id: String │
│ ├─ title: String │ ├─ title: String │
│ └─ content: Text │ └─ values: Matrix │
│ │ │
│ QueueItem ─────────────┼──────▶ Agent3 │
│ ├─ interaction ────────┘ ├─ fiber: Trit {-1, 0, +1} │
│ └─ agent ───────────────▶ └─ name: String │
│ │
│ Person ◀─────────────────────── Partner ─────────────────▶ Person │
│ ├─ email: String ├─ src │
│ └─ name: String ├─ tgt │
│ └─ weight: Int │
│ │
│ Service │
│ ├─ name: String {gmail, drive, calendar, tasks, docs, sheets} │
│ └─ default_trit: Trit │
└─────────────────────────────────────────────────────────────────────────────────────────┘
Objects
| Object | Description | Trit Role |
|---|---|---|
Interaction |
Single workspace action with verb + trit + service | Data |
Thread |
Gmail conversation with saturation state | Gmail Aggregate |
DriveFile |
File or folder in Drive | Drive Aggregate |
CalendarEvent |
Meeting or event | Calendar Aggregate |
TaskList |
Container for tasks | Tasks Container |
Task |
Individual action item | Tasks Aggregate |
Document |
Google Doc content | Docs Aggregate |
Sheet |
Spreadsheet data | Sheets Aggregate |
Agent3 |
Queue fiber (MINUS/ERGODIC/PLUS) | Router |
QueueItem |
Links Interaction → Agent3 | Edge |
Person |
User or contact | Node |
Partner |
Relationship edge in contact graph | Edge |
Service |
Workspace service identifier | Metadata |
GF(3) Service Typing
Workspace actions are assigned trits based on information flow:
VERB_TRIT_MAP = {
# ═══════════════════════════════════════════════════════════════
# GMAIL
# ═══════════════════════════════════════════════════════════════
# MINUS (-1): Consumption/Validation
"gmail_read": -1, "gmail_search": -1, "gmail_get_thread": -1,
"gmail_list_labels": -1,
# ERGODIC (0): Coordination/Metadata
"gmail_label": 0, "gmail_archive": 0, "gmail_snooze": 0,
"gmail_star": 0, "gmail_mark_read": 0, "gmail_move": 0,
# PLUS (+1): Generation/Execution
"gmail_send": +1, "gmail_forward": +1, "gmail_reply": +1,
"gmail_draft": +1, "gmail_compose": +1,
# ═══════════════════════════════════════════════════════════════
# DRIVE
# ═══════════════════════════════════════════════════════════════
# MINUS (-1)
"drive_get_content": -1, "drive_list": -1, "drive_search": -1,
"drive_get_permissions": -1,
# ERGODIC (0)
"drive_share": 0, "drive_move": 0, "drive_rename": 0,
"drive_update_metadata": 0,
# PLUS (+1)
"drive_create": +1, "drive_upload": +1, "drive_copy": +1,
# ═══════════════════════════════════════════════════════════════
# CALENDAR
# ═══════════════════════════════════════════════════════════════
# MINUS (-1)
"calendar_get_events": -1, "calendar_list": -1, "calendar_get_event": -1,
# ERGODIC (0)
"calendar_modify": 0, "calendar_reschedule": 0, "calendar_update_attendees": 0,
# PLUS (+1)
"calendar_create": +1, "calendar_invite": +1, "calendar_duplicate": +1,
# ═══════════════════════════════════════════════════════════════
# TASKS
# ═══════════════════════════════════════════════════════════════
# MINUS (-1)
"tasks_list": -1, "tasks_get": -1, "tasks_list_lists": -1,
# ERGODIC (0)
"tasks_update": 0, "tasks_move": 0, "tasks_complete": 0,
# PLUS (+1)
"tasks_create": +1, "tasks_create_list": +1,
# ═══════════════════════════════════════════════════════════════
# DOCS
# ═══════════════════════════════════════════════════════════════
# MINUS (-1)
"docs_get_content": -1, "docs_read_comments": -1, "docs_inspect": -1,
# ERGODIC (0)
"docs_find_replace": 0, "docs_update_headers": 0, "docs_format": 0,
# PLUS (+1)
"docs_create": +1, "docs_insert_text": +1, "docs_create_table": +1,
# ═══════════════════════════════════════════════════════════════
# SHEETS
# ═══════════════════════════════════════════════════════════════
# MINUS (-1)
"sheets_read": -1, "sheets_get_info": -1,
# ERGODIC (0)
"sheets_clear": 0,
# PLUS (+1)
"sheets_write": +1, "sheets_create": +1, "sheets_create_sheet": +1,
}
MCP Tool → Trit Mapping
| Tool | Trit | Service | Description |
|---|---|---|---|
search_gmail_messages |
-1 | Gmail | Search inbox (MINUS) |
get_gmail_message_content |
-1 | Gmail | Read message (MINUS) |
get_gmail_thread_content |
-1 | Gmail | Read thread (MINUS) |
list_gmail_labels |
-1 | Gmail | List labels (MINUS) |
modify_gmail_message_labels |
0 | Gmail | Change labels (ERGODIC) |
batch_modify_gmail_message_labels |
0 | Gmail | Bulk labels (ERGODIC) |
send_gmail_message |
+1 | Gmail | Send email (PLUS) |
draft_gmail_message |
+1 | Gmail | Create draft (PLUS) |
list_drive_items |
-1 | Drive | List files (MINUS) |
get_drive_file_content |
-1 | Drive | Read file (MINUS) |
search_drive_files |
-1 | Drive | Search files (MINUS) |
share_drive_file |
0 | Drive | Share file (ERGODIC) |
update_drive_file |
0 | Drive | Update metadata (ERGODIC) |
create_drive_file |
+1 | Drive | Create file (PLUS) |
get_events |
-1 | Calendar | Get events (MINUS) |
modify_event |
0 | Calendar | Modify event (ERGODIC) |
create_event |
+1 | Calendar | Create event (PLUS) |
list_tasks |
-1 | Tasks | List tasks (MINUS) |
get_task |
-1 | Tasks | Get task (MINUS) |
update_task |
0 | Tasks | Update task (ERGODIC) |
create_task |
+1 | Tasks | Create task (PLUS) |
get_doc_content |
-1 | Docs | Read doc (MINUS) |
create_doc |
+1 | Docs | Create doc (PLUS) |
read_sheet_values |
-1 | Sheets | Read sheet (MINUS) |
modify_sheet_values |
+1 | Sheets | Write sheet (PLUS) |
Cross-Skill Morphisms
┌──────────────────────────────────────────────────────────────────────────────┐
│ CROSS-SKILL MORPHISM DIAGRAM │
├──────────────────────────────────────────────────────────────────────────────┤
│ │
│ thread_file │
│ Thread ─────────────────────────────▶ DriveFile │
│ │ │ │
│ │ thread_event │ file_event │
│ ▼ ▼ │
│ CalendarEvent ◀───── event_file ─────── DriveFile │
│ │ │ │
│ │ event_task │ file_task │
│ ▼ ▼ │
│ Task ◀──────────────────────────────── Task │
│ │ │ │
│ │ task_doc │ file_doc │
│ ▼ ▼ │
│ Document ◀───────────────────────────── Document │
│ │ │
│ │ doc_sheet │
│ ▼ │
│ Sheet │
│ │
│ GF(3) Conservation: All morphism paths preserve trit balance │
└──────────────────────────────────────────────────────────────────────────────┘
Morphism Table
| Morphism | Source | Target | Trigger | GF(3) Effect |
|---|---|---|---|---|
thread_file |
Thread | DriveFile | Attachment detected | 0 (ERGODIC) |
thread_event |
Thread | CalendarEvent | Meeting scheduled | +1 (PLUS) |
thread_task |
Thread | Task | Action item identified | +1 (PLUS) |
file_event |
DriveFile | CalendarEvent | File-linked meeting | +1 (PLUS) |
file_task |
DriveFile | Task | File review needed | +1 (PLUS) |
file_doc |
DriveFile | Document | Content extraction | 0 (ERGODIC) |
event_task |
CalendarEvent | Task | Event followup | +1 (PLUS) |
event_doc |
CalendarEvent | Document | Meeting notes | +1 (PLUS) |
task_doc |
Task | Document | Task documentation | +1 (PLUS) |
doc_sheet |
Document | Sheet | Data extraction | 0 (ERGODIC) |
Path Commutativity
thread_task
Gmail.Thread ─────────────────────────────────▶ Tasks.Task
│ │
│ thread_event │ (identity)
▼ ▼
Calendar.Event ────────────────────────────────▶ Tasks.Task
event_task
INVARIANT: thread_task == event_task ∘ thread_event
thread_file
Gmail.Thread ─────────────────────────────────▶ Drive.File
│ │
│ thread_event │ file_event
▼ ▼
Calendar.Event ◀─────── event_file ─────────── Calendar.Event
INVARIANT: thread_event == event_file⁻¹ ∘ file_event ∘ thread_file
Saturation Detection → ANIMA State
def is_workspace_saturated() -> bool:
"""Workspace is saturated when:
1. All services reach their saturation threshold
2. GF(3) global conservation: sum(trits) ≡ 0 (mod 3)
3. All cross-skill morphism paths commute
4. No pending interactions in any queue
"""
service_states = {
'gmail': gmail_saturated(),
'drive': drive_saturated(),
'calendar': calendar_saturated(),
'tasks': tasks_saturated(),
'docs': docs_saturated(),
'sheets': sheets_saturated(),
}
global_trit_sum = sum(
interaction.trit
for service in services
for interaction in service.interactions
)
return (
all(service_states.values()) and # All saturated
(global_trit_sum % 3) == 0 and # GF(3) conserved
verify_path_commutativity() and # Morphisms commute
all_queues_empty() # No pending work
)
def detect_anima() -> Dict:
"""System at ANIMA when workspace is condensed."""
return {
"at_anima": is_workspace_saturated(),
"condensed_fingerprint": sha256(workspace_state_hash()),
"service_states": get_all_service_states(),
"gf3_balance": compute_global_gf3_balance(),
"pending_interactions": count_pending_interactions(),
}
Per-Service Saturation Thresholds
SATURATION_THRESHOLDS = {
'gmail_threads': 1000,
'drive_files': 500,
'calendar_events': 200,
'tasks': 300,
'docs': 100,
'sheets': 50,
}
def service_saturated(service: str) -> bool:
count = get_service_object_count(service)
threshold = SATURATION_THRESHOLDS.get(service, 100)
return count >= threshold
Narya Proof Integration
Proofs in src/narya_proofs/:
1. Path Commutativity (path_commutativity.py)
def prove_path_commutativity(workspace: WorkspaceACSet) -> bool:
"""Verify all morphism paths commute."""
# thread_task == event_task ∘ thread_event
for thread in workspace.threads:
direct = workspace.thread_task(thread)
composed = workspace.event_task(workspace.thread_event(thread))
if direct != composed:
return False
return True
2. GF(3) Global Conservation (gf3_global.py)
def prove_gf3_global_conservation(workspace: WorkspaceACSet) -> bool:
"""All cross-service workflows satisfy sum ≡ 0 (mod 3)."""
for workflow in workspace.workflows:
trit_sum = sum(op.trit for op in workflow.operations)
if trit_sum % 3 != 0:
return False
return True
3. MCP↔API Equivalence (mcp_api_equiv.py)
def prove_mcp_api_equivalence(bridge: WorkspaceBridge) -> bool:
"""MCP and API executions produce identical states."""
for operation in bridge.supported_operations:
mcp_state = bridge.execute_mcp(operation, test_params)
api_state = bridge.execute_api(operation, test_params)
if normalize(mcp_state) != normalize(api_state):
return False
return True
4. Cross-Service Integrity (cross_service.py)
def prove_cross_service_integrity(workspace: WorkspaceACSet) -> bool:
"""Thread IDs, file IDs consistent across all services."""
for morphism in workspace.morphisms:
source_id = morphism.source.id
target_id = morphism.target.id
if not workspace.verify_id_consistency(source_id, target_id):
return False
return True
Source Files
| File | Description | Trit |
|---|---|---|
| workspace_acset.py | ACSet schema + cross-skill morphisms | 0 |
| workspace_detector.py | Saturation + equilibrium detection | 0 |
| workspace_bridge.py | MCP tool wiring with guards | 0 |
| triadic_queues.py | Three disjoint queue fibers | 0 |
| path_invariance.py | Morphism commutativity checks | -1 |
| mcp_api_equivalence.py | Equivalence verification | -1 |
| narya_proofs/ | Formal verification proofs | -1 |
Workflows
Workflow 1: Email-to-Task with Full Morphism Chain
from workspace_bridge import create_workspace_bridge
from workspace_detector import WorkspaceDetector
# Create bridge
bridge = create_workspace_bridge("user@gmail.com")
detector = WorkspaceDetector()
# MINUS: Search for actionable emails
results = bridge.search_gmail_messages("is:unread label:action-required")
for msg in results:
thread = bridge.get_gmail_thread_content(msg.thread_id)
detector.update_service('gmail', trit=Trit.MINUS)
# PLUS: Create task via thread_task morphism
task = bridge.create_task(
title=f"Follow up: {thread.subject}",
notes=thread.snippet,
task_list_id=default_list_id
)
detector.update_service('tasks', trit=Trit.PLUS)
# ERGODIC: Label as processed
bridge.modify_gmail_message_labels(
msg.id,
add_label_ids=["Label_Processed"],
remove_label_ids=["INBOX"]
)
detector.update_service('gmail', trit=Trit.ERGODIC)
# GF(3) check: -1 + 1 + 0 = 0 ✓
Workflow 2: Meeting Notes with Cross-Service Morphisms
# MINUS: Get calendar events
events = bridge.get_events(time_min=today, time_max=tomorrow)
detector.update_service('calendar', trit=Trit.MINUS)
for event in events:
# PLUS: Create meeting notes doc via event_doc morphism
doc = bridge.create_doc(
title=f"Notes: {event.summary}",
content=generate_template(event)
)
detector.update_service('docs', trit=Trit.PLUS)
# ERGODIC: Update event with doc link
bridge.modify_event(
event_id=event.id,
description=f"{event.description}\n\nNotes: {doc.url}"
)
detector.update_service('calendar', trit=Trit.ERGODIC)
# GF(3) check: -1 + 1 + 0 = 0 ✓
Workflow 3: File Review with Task Creation
# MINUS: Search for files needing review
files = bridge.search_drive_files("modifiedTime > '2024-01-01' and mimeType = 'application/pdf'")
detector.update_service('drive', trit=Trit.MINUS)
for file in files:
# MINUS: Get file content
content = bridge.get_drive_file_content(file.id)
detector.update_service('drive', trit=Trit.MINUS)
# PLUS: Create review task via file_task morphism
task = bridge.create_task(
title=f"Review: {file.name}",
notes=f"File: {file.webViewLink}"
)
detector.update_service('tasks', trit=Trit.PLUS)
# PLUS: Share file with reviewer
bridge.share_drive_file(file.id, share_with="reviewer@example.com")
detector.update_service('drive', trit=Trit.ERGODIC)
# Auto-balance if needed
if detector.gf3_residue() != 0:
bridge.list_tasks(task_list_id=default_list_id) # MINUS to balance
Workflow 4: Weekly Digest with Condensation Check
# Create digest from all services
digest = WorkspaceDigest()
# MINUS: Gather data
digest.gmail_threads = bridge.search_gmail_messages("newer_than:7d")
digest.drive_files = bridge.search_drive_files("modifiedTime > '2024-01-01'")
digest.events = bridge.get_events(time_min=week_ago, time_max=now)
digest.tasks = bridge.list_tasks(task_list_id=default_list_id)
# PLUS: Create digest doc
doc = bridge.create_doc(
title=f"Weekly Digest: {week_start} - {week_end}",
content=render_digest(digest)
)
# ERGODIC: Update shared folder
bridge.update_drive_file(doc.id, add_parents="shared_digests_folder_id")
# Check ANIMA
anima = detector.detect_anima()
if anima["at_anima"]:
say("Workspace at ANIMA. Condensed state achieved.")
print(f"Fingerprint: {anima['condensed_fingerprint'][:16]}...")
Commands
# Run workspace ACSet demo
python src/workspace_acset.py
# Test cross-skill morphisms
python src/path_invariance.py
# Run MCP↔API equivalence tests
python src/mcp_api_equivalence.py
# Run Narya proofs
python -m src.narya_proofs.runner
# Check workspace saturation
python src/workspace_detector.py
Integration with Other Skills
| Skill | Trit | Integration |
|---|---|---|
| gmail-anima | 0 | Gmail-specific ACSet with thread saturation |
| google-workspace | 0 | MCP tool provider |
| gay-mcp | +1 | SplitMixTernary RNG for deterministic colors |
| sheaf-cohomology | -1 | H¹ obstruction verification |
| bisimulation-game | -1 | State equivalence proofs |
| ordered-locale | 0 | Service ordering topology |
| acsets-algebraic-databases | 0 | ACSet foundations |
GF(3) Triadic Conservation
workspace-unified (0) ⊗ gmail-anima (0) ⊗ gay-mcp (+1) ⊗ sheaf-cohomology (-1) = 0 ✓
gmail_read (-1) ⊗ drive_create (+1) ⊗ calendar_modify (0) = 0 ✓
thread_file (0) ⊗ file_event (+1) ⊗ event_task (+1) ⊗ tasks_get (-1) ⊗ docs_get (-1) = 0 ✓
MCP ↔ API Equivalence
| # | Operation | MCP Tool | REST API Equivalent |
|---|---|---|---|
| 1 | Gmail Search | search_gmail_messages | GET /gmail/v1/users/me/messages |
| 2 | Gmail Read | get_gmail_message_content | GET /gmail/v1/users/me/messages/{id} |
| 3 | Gmail Send | send_gmail_message | POST /gmail/v1/users/me/messages/send |
| 4 | Gmail Label | modify_gmail_message_labels | POST /gmail/v1/users/me/messages/modify |
| 5 | Drive List | list_drive_items | GET /drive/v3/files |
| 6 | Drive Get | get_drive_file_content | GET /drive/v3/files/{id}?alt=media |
| 7 | Drive Create | create_drive_file | POST /upload/drive/v3/files |
| 8 | Drive Share | share_drive_file | POST /drive/v3/files/{id}/permissions |
| 9 | Calendar Events | get_events | GET /calendar/v3/calendars/{id}/events |
| 10 | Calendar Create | create_event | POST /calendar/v3/calendars/{id}/events |
| 11 | Tasks List | list_tasks | GET /tasks/v1/lists/{id}/tasks |
| 12 | Tasks Create | create_task | POST /tasks/v1/lists/{id}/tasks |
| 13 | Docs Get | get_doc_content | GET /docs/v1/documents/{id} |
| 14 | Docs Create | create_doc | POST /docs/v1/documents |
| 15 | Sheets Read | read_sheet_values | GET /sheets/v4/spreadsheets/{id}/values |
Equivalence Verification
@dataclass
class MCPAPIEquivalence:
operation: str
mcp_result: WorkspaceState
api_result: WorkspaceState
def verify(self) -> bool:
"""Verify bitwise equivalence after normalization."""
mcp_normalized = normalize(self.mcp_result)
api_normalized = normalize(self.api_result)
return mcp_normalized == api_normalized
def diff(self) -> Optional[StateDiff]:
"""Return diff if not equivalent."""
if self.verify():
return None
return compute_diff(self.mcp_result, self.api_result)
def verify_all_equivalences(bridge: WorkspaceBridge) -> Dict:
"""Run full equivalence suite."""
results = {}
for op in bridge.supported_operations:
equiv = MCPAPIEquivalence(
operation=op,
mcp_result=bridge.execute_mcp(op, test_params[op]),
api_result=bridge.execute_api(op, test_params[op])
)
results[op] = {
'equivalent': equiv.verify(),
'diff': equiv.diff()
}
return results
Say Narration Integration
from workspace_bridge import NaryaLogger
logger = NaryaLogger(voice="Ava (Premium)")
# Announces: "Workspace bridge: MINUS transition on gmail"
logger.log(before, after, Trit.MINUS, service="gmail", impact=False)
# Announces: "Workspace bridge: PLUS transition on drive, impact detected"
logger.log(before, after, Trit.PLUS, service="drive", impact=True)
# Cross-skill morphism announcement
logger.log_morphism("thread_task", thread, task, Trit.PLUS)
Skill Name: workspace-unified
Type: Workspace Management / ACSet Framework
Trit: 0 (ERGODIC - coordinator)
GF(3): Conserved via cross-skill morphism routing
ANIMA: Workflow Completion = Condensed Equilibrium State
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.