Hyjax Relational

---
name: hyjax-relational
description: HyJAX Relational Thinking Skill
version: 1.0.0
---

# HyJAX Relational Thinking Skill

Apply relational thinking (ACSets/C-Sets) to Amp thread analysis using HyJAX patterns.

## When to Use

- Analyzing thread relationships and concept networks
- Extracting patterns from conversation history
- Building relational databases from unstructured thread data
- Generating Colored S-expressions for visualization

## Core Concepts

### ACSet Schema for Threads
```
Objects: Thread, Message, Concept, File
Morphisms: thread_msg, mentions, discusses, related
Attributes: content, timestamp, info_gain
```

### Colored S-expressions
```lisp
(acset-gold
  (threads-red (thread T-001 "Title" 42))
  (concepts-green (concept skill 5) (concept MCP 3))
  (relations-purple (edge skill co-occurs subagent)))
```

## Key Files

| File | Purpose |
|------|---------|
| `/Users/bob/ies/music-topos/lib/thread_relational_hyjax.hy` | Main HyJAX analyzer |
| `/Users/bob/ies/music-topos/lib/unified_thread_lake.duckdb` | Persistent database |
| `/Users/bob/ies/music-topos/lib/analyze_threads_relational.py` | Python analyzer |

## Quick Start

### 1. Query the Thread Lake
```bash
duckdb /Users/bob/ies/music-topos/lib/unified_thread_lake.duckdb -c "
  SELECT name, hub_score FROM concepts ORDER BY hub_score DESC LIMIT 10
"
```

### 2. Find 2-Hop Concept Paths
```bash
duckdb /Users/bob/ies/music-topos/lib/unified_thread_lake.duckdb -c "
  SELECT r1.from_concept || ' → ' || r1.to_concept || ' → ' || r2.to_concept as path
  FROM concept_relations r1
  JOIN concept_relations r2 ON r1.to_concept = r2.from_concept
  WHERE r1.from_concept = 'skill'
"
```

### 3. Run Full Analysis
```bash
cd /Users/bob/ies && source .venv/bin/activate
python3 music-topos/lib/full_thread_analysis.py
```

## Relational Patterns

### Hub Concepts (Most Connected)
| Concept | Hub Score |
|---------|-----------|
| skill | 8 |
| GF3 | 5 |
| MCP | 4 |
| subagent | 3 |

### Strongest Relations
- skill ↔ subagent (weight 2)
- skill → MCP → alife
- skill → ACSet → discohy
- HyJAX ↔ relational

## Integration with Other Skills

### With `acsets-algebraic-databases`
```julia
@present SchThread(FreeSchema) begin
  Thread::Ob; Message::Ob; Concept::Ob
  thread_msg::Hom(Message, Thread)
  discusses::Hom(Message, Concept)
  related::Hom(Concept, Concept)
end
```

### With `gay-mcp`
Each concept gets a deterministic color via Gay.jl seed:
```julia
using Gay
concept_color = gay_color(hash("skill"))  # Reproducible color
```

### With `entropy` patterns
```python
H(concepts) = 4.55 bits  # Shannon entropy of concept distribution
efficiency = 95.6%        # vs max entropy
```

## DuckDB Schema

```sql
CREATE TABLE threads (thread_id VARCHAR PRIMARY KEY, title VARCHAR, message_count INT);
CREATE TABLE concepts (concept_id VARCHAR PRIMARY KEY, name VARCHAR, frequency INT, hub_score INT);
CREATE TABLE concept_relations (from_concept VARCHAR, to_concept VARCHAR, weight INT);
CREATE TABLE colored_sexprs (sexpr_id VARCHAR PRIMARY KEY, root_color VARCHAR, tree_json JSON);
```

## Workflow

1. **Ingest**: Use `find_thread` to get thread data
2. **Extract**: Apply concept patterns to titles/content
3. **Build**: Create ACSet with objects and morphisms
4. **Query**: Run relational queries (pullbacks, 2-hop paths)
5. **Output**: Generate Colored S-expressions

## Example Output

```
THREAD RELATIONAL ANALYSIS - 30 THREADS
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Threads:    30
Messages:   2,951
Concepts:   27
Relations:  48
Entropy:    4.55 bits (95.6% efficiency)

TOP CONCEPTS:
  skill           5 █████
  subagent        3 ███
  MCP             3 ███
  GF3             3 ███

COLORED S-EXPRESSION:
(acset-gold 
  (threads-red ...) 
  (concepts-green ...) 
  (relations-purple ...))
```



## Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

### Autodiff
- **jax** [○] via bicomodule

### 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: Span
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.