Lib

# HyJAX Thread Relational Skill

**Date**: 2025-12-22  
**Status**: IMPLEMENTED + CLI  
**Purpose**: Apply relational thinking (ACSets) to Amp thread analysis using HyJAX

## Current Database State

| Metric | Value |
|--------|-------|
| Threads | 23 |
| Messages | 2,290 |
| Concepts | 26 |
| Relations | 36 |
| Entropy | 4.41 bits (93.7% efficient) |
| GF(3) | RED=14, YELLOW=5, BLUE=7 |

## CLI Tool

```bash
uv run python query_thread_lake.py concepts    # Top concepts with GF(3)
uv run python query_thread_lake.py hubs        # Hub scores
uv run python query_thread_lake.py paths skill # 2-hop paths from concept
uv run python query_thread_lake.py cluster ACSet # Threads by concept
uv run python query_thread_lake.py sexpr       # Colored S-expression
```

---

## Architecture

```
┌─────────────────────────────────────────────────────────────────┐
│                    HYJAX THREAD RELATIONAL SYSTEM               │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  ┌──────────────┐    ┌──────────────┐    ┌──────────────┐      │
│  │   Threads    │───▶│  ThreadACSet │───▶│  ColoredSExpr │      │
│  │  (Amp Data)  │    │   (Schema)   │    │   (Output)    │      │
│  └──────────────┘    └──────────────┘    └──────────────┘      │
│         │                   │                   │               │
│         ▼                   ▼                   ▼               │
│  ┌──────────────┐    ┌──────────────┐    ┌──────────────┐      │
│  │   Messages   │───▶│  Morphisms   │───▶│  DuckDB      │      │
│  │   Concepts   │    │  (Relations) │    │  (Persist)   │      │
│  │   Files      │    │              │    │              │      │
│  └──────────────┘    └──────────────┘    └──────────────┘      │
│                                                                 │
├─────────────────────────────────────────────────────────────────┤
│  LAYERS:                                                        │
│  1. ThreadACSet - Category-theoretic schema (Objects+Morphisms) │
│  2. ColoredSExpr - Semantic tree with color annotations         │
│  3. Entropy Interleaving - Information-theoretic ordering       │
│  4. Bidirectional Learning - Read ↔ Write coupling              │
│  5. Network Perspectives - Interperspectival analysis           │
│  6. DuckDB Bridge - SQL persistence and queries                 │
└─────────────────────────────────────────────────────────────────┘
```

---

## Files Created

| File | Purpose |
|------|---------|
| [thread_relational_hyjax.hy](./thread_relational_hyjax.hy) | Main HyJAX analyzer |
| [thread_duckdb_bridge.hy](./thread_duckdb_bridge.hy) | DuckDB persistence bridge |
| [hy_skill_loader.hy](./hy_skill_loader.hy) | Original HyJAX + PyZX integration |

---

## ACSet Schema

```
@present SchThread(FreeSchema) begin
  Thread::Ob
  Message::Ob
  File::Ob
  Concept::Ob
  
  thread_msg::Hom(Message, Thread)    # Message belongs to Thread
  mentions::Hom(Message, File)         # Message mentions File
  discusses::Hom(Message, Concept)     # Message discusses Concept
  related::Hom(Concept, Concept)       # Concept relates to Concept
  
  Text::AttrType
  Timestamp::AttrType
  Entropy::AttrType
  
  content::Attr(Message, Text)
  time::Attr(Message, Timestamp)
  info_gain::Attr(Message, Entropy)
end
```

---

## DuckDB Tables

```sql
-- Objects (Thread, Message, File, Concept)
CREATE TABLE thread_acset_objects (
  object_id VARCHAR PRIMARY KEY,
  object_type VARCHAR,
  data JSON,
  created_at TIMESTAMP
);

-- Morphisms (relations between objects)
CREATE TABLE thread_acset_morphisms (
  morphism_id VARCHAR PRIMARY KEY,
  morphism_type VARCHAR,
  source_id VARCHAR,
  target_id VARCHAR,
  data JSON,
  created_at TIMESTAMP
);

-- Entropy-maximized sequences
CREATE TABLE thread_entropy_sequences (
  sequence_id VARCHAR PRIMARY KEY,
  message_ids JSON,
  final_entropy DOUBLE,
  message_count INTEGER,
  strategy VARCHAR,
  created_at TIMESTAMP
);

-- Colored S-expressions
CREATE TABLE thread_colored_sexprs (
  sexpr_id VARCHAR PRIMARY KEY,
  acset_id VARCHAR,
  colored_tree JSON,
  root_color VARCHAR,
  depth INTEGER,
  created_at TIMESTAMP
);
```

---

## Colored S-Expression Output

```lisp
(acset-root-gold 
  (threads-red 
    (thread-scarlet "T-hyjax-001" "HyJAX Exploration")
    (thread-scarlet "T-acsets-002" "ACSets Relational"))
  (concepts-green
    (concept-emerald "HyJAX" 2)
    (concept-emerald "entropy" 5)
    (concept-emerald "ACSet" 3))
  (files-blue
    (file-azure "music-topos/lib/hy_skill_loader.hy"))
  (relations-purple
    (relation-violet "HyJAX" "adjacent" "ColoredShape")
    (relation-violet "entropy" "adjacent" "ACSet")))
```

---

## Key Patterns

### 1. Entropy-Maximized Interleaving
From AGENT.md Layer 5 - arrange messages to maximize information gain:

```hy
(defn entropy-maximized-interleave [messages]
  ;; Find ordering that maximizes Shannon entropy at each step
  ;; Returns {:sequence ordered-msgs :final-entropy H :message-count N}
  ...)
```

### 2. Bidirectional Learning
From worlding_skill_omniglot_hyjax.hy - read ↔ write coupling:

```hy
(defclass ThreadBidirectionalLearner []
  (defn encode-thread [self acset] ...)   ; READ: ACSet → Latent
  (defn decode-thread [self latent] ...)  ; WRITE: Latent → Template
  (defn bidirectional-loss [self acset] ...) ; Coupling loss
)
```

### 3. Network Perspectives
From AGENT.md Layer 7 - interperspectival analysis:

```hy
(defclass ThreadNetworkPerspective []
  (defn analyze-concept-flow [self acset] ...)
  (defn find-concept-hubs [self acset] ...)
  (defn consensus-view [self] ...)
)
```

---

## Usage

```hy
;; Create analyzer
(setv analyzer (ThreadRelationalAnalyzer))

;; Ingest threads
(analyzer.ingest-thread "T-001" "Title" [{:content "..." :timestamp 1.0}])

;; Extract concepts
(analyzer.extract-concepts ["HyJAX" "ACSet" "entropy"])

;; Run analysis
(setv result (analyzer.analyze))

;; Get Colored S-expression
(setv sexpr (acset-to-colored-sexpr analyzer.acset))

;; Persist to DuckDB
(setv bridge (DuckDBBridge "threads.db"))
(bridge.save-acset analyzer.acset "my-analysis")
```

---

## Connection to Prior Work

| Source | Pattern | Integration |
|--------|---------|-------------|
| hy_skill_loader.hy | `defoperad`, `defmonadic`, HyJAX | Compositional rules |
| worlding_skill_omniglot_hyjax.hy | ColoredShape, bidirectional learning | Tensor semantics |
| music-topos/AGENT.md | 7-layer architecture, entropy interleaving | System design |
| ducklake.sql | amp_threads, GF(3) colors | Persistent storage |
| ACSets skill | C-Set schema, pullback queries | Relational formalism |

---

## Relational Thinking Patterns

1. **Objects as Types**: Thread, Message, File, Concept
2. **Morphisms as Relations**: thread_msg, mentions, discusses, related
3. **Pullback Queries**: "Get all files mentioned in thread T-001"
4. **Pushout Composition**: Merge threads by shared concepts
5. **Entropy Ordering**: Maximize information gain in message sequence
6. **Colored Output**: Semantic color names on S-expression tree

---

## Future Extensions

- [ ] Connect to `find_thread` MCP tool for live thread ingestion
- [ ] Add JAX gradients for concept similarity (HyJAX transforms)
- [ ] Integrate with Gay.jl GF(3) coloring for balanced ternary
- [ ] Build visualization with PyZX-style diagrams
- [ ] Add meta-learning for skill discovery (from Omniglot patterns)